ÓÅÑÁÖÅÉÌ ÊÁÑÁÌÐÏÃÉÁÓ
ΣΕΡΑΦΕΙΜ ΚΑΡΑΜΠΟΓΙΑΣ
ΣΗΜΑΤΑ
ÓÇÌÁÔÁ
ÊÁÉ
ΚΑΙ
ÓÕÓÔÇÌÁÔÁ
ΣΥΣΤΗΜΑΤΑ
õåéó(t) õåî(t)
Ät
A B
R
õåéó(t) i(t) õåî(t)
C
T0 t T0 t
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ΣΕΡΑΦΕΙΜ ΚΑΡΑΜΠΟΓΙΑΣ
Επίκουρος Καθηγητής
Σήματα και Συστήματα
Σήματα και Συστήματα
Συγγραφή
Σεραφείμ Καραμπογιάς
Κριτικός αναγνώστης
Στέφανος Κόλλιας
ISBN: 978-960-603-327-8
Copyright © ΣΕΑΒ, 2015
Το παρόν έργο αδειοδοτείται υπό τους όρους της άδειας Creative Commons Αναφορά Δημιουργού - Μη Εμπορική
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ΣΥΝΔΕΣΜΟΣ ΕΛΛΗΝΙΚΩΝ ΑΚΑΔΗΜΑΪΚΩΝ ΒΙΒΛΙΟΘΗΚΩΝ
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Ηρώων Πολυτεχνείου 9, 15780 Ζωγράφου
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ÐÑÏËÏÃÏÓ
To bibl o autì apeujÔnetai se es pou parakolouje te Tm mata Plhroforik
kai Thlepikoinwni¸n kai apotele katastlagma th empeir a apì th didaskal a tou
maj mato S mata kai Sust mata gia arket qrìnia.
Sth shmerin epoq th Koinwn a th Plhrofor a , pou qarakthr zetai apì th
sÔgklish kai enopo hsh diaforetik¸n mèqri t¸ra episthmonik¸n perioq¸n, to ped o
twn shmtwn kai twn susthmtwn apotele plèon èna enia o sÔnolo basik¸n kai
jemeliwd¸n gn¸sewn gia èna eurÔ fsma perioq¸n pou sqet zontai me ton èna l-
lo trìpo me thn paragwg , thn epexergas a, thn apoj keush kai th metdosh th
plhrofor a .
Basikì skopì tou bibl ou e nai na eisaggei ton anagn¸sth sti basikè teqnikè
anlush kai melèth twn shmtwn kai susthmtwn me enia o trìpo kai na tou pro-
sfèrei ta katllhla majhmatik ergale a, me ta opo a mpore na qeiriste ta s mata
kai ta sust mata. àqei katablhje prospjeia na dojoÔn oi jewrhtikè ènnoie me
aplì trìpo kai na sundejoÔn me ant stoiqe ènnoie th fusik . H epilog twn
paradeigmtwn ègine me gn¸mona th qrhsimìtht tou ¸ste na bohjoÔn ton anagn¸sth
na emped¸sei th jewr a kai na katano sei ti teqnikè prosèggish kai antimet¸pish
twn problhmtwn.
Oi basikè jematikè enìthte tou Maj mato S mata kai Sust mata anaptÔs-
sontai se Ept keflaia.
Episkìphsh tou Bibl ou
Sto Keflaio 1 d netai mia genik eikìna tou ti e nai s ma kai parousizontai oi
idiìthte twn shmtwn. Ep sh or zontai merik stoiqei¸dh s mata, ta opo a pa zoun
idia tero rìlo sth jewr a twn shmtwn.
Sto Keflaio 2 d netai o orismì tou sust mato , parousizontai oi kathgor e
susthmtwn, oi trìpoi sÔndes tou kai merikè basikè idiìthtè tou . To keflaio
autì odhge ton anagn¸sth sthn katanìhsh jemeliwd¸n sqèsewn, ìpw h sqèsh pou
sundèei to s ma eisìdou kai to s ma exìdou enì sust mato , kai basik¸n ennoi¸n,
ii Prìlogo
ìpw oi ènnoie th grammikìthta , th eustjeia kai th aitiìthta .
Sto Keflaio 3 perigrfetai h mèjodo anlush enì analogikoÔ s mato kat
Fourier, dhlad w upèrjesh stoiqeiwd¸n hmitonoeid¸n shmtwn kai eisgontai oi
ènnoie th seir Fourier kai tou metasqhmatismoÔ Fourier. Ep sh anafèrontai oi
idiìthte tou metasqhmatismoÔ Fourier
Sto Keflaio 4 parousizontai basikè efarmogè tou metasqhmatismoÔ Fourier
suneqoÔ qrìnou. Ep sh eisgetai h ènnoia tou f ltrou.
Sto Keflaio 5 perigrfetai h mèjodo anlush enì s mato diakritoÔ qrìnou
kat Fourier, dhlad , w upèrjesh stoiqeiwd¸n hmitonoeid¸n shmtwn diakritoÔ qrìnou
kai eisgontai oi ènnoie th seir Fourier, tou metasqhmatismoÔ Fourier diakri-
toÔ qrìnou, tou diakritoÔ metasqhmatismoÔ Fourier kai tou taqèo metasqhmatismoÔ
Fourier. Ep sh anafèrontai oi idiìthte twn metasqhmatism¸n Fourier. Tèlo parousi-
zontai basikè efarmogè tou metasqhmatismoÔ Fourier diakritoÔ qrìnou.
Sto Keflaio 6 or zetai o metasqhmatismì Laplace, anafèrontai oi idiìthtè tou
kai parousizetai h sten sqèsh tou me to metasqhmatismì Fourier suneqoÔ qrìnou.
Ep sh parousizetai o monìpleuro metasqhmatismì Laplace. Tèlo , parousizetai
h qr sh tou metasqhmatismoÔ Laplace sthn anlush analogik¸n susthmtwn kai th
melèth th eustjeia kai th aitiìtht tou .
Sto Keflaio 7 or zetai o metasqhmatismì z, anafèrontai oi idiìthtè tou kai
parousizetai h sten sqèsh tou me to metasqhmatismì Fourier diakritoÔ qrìnou.
Ep sh parousizetai o monìpleuro metasqhmatismì z. Tèlo , parousizetai h
qr sh tou metasqhmatismoÔ z sthn anlush susthmtwn diakritoÔ qrìnou kai th
melèth th eustjeia kai th aitiìtht tou .
To bibl o oloklhr¸netai me tr a parart mata. Sto pr¸to Parrthma parousi-
zontai basik stoiqe a gia tou migadikoÔ arijmoÔ . Sto deÔtero Parrthma parou-
sizetai o trìpo anptuxh mia rht sunrthsh se apl klsmata. Ta stoiqe a
twn dÔo aut¸n pararthmtwn e nai dh gnwst apì to LÔkeio. Tèlo sto tr to
Parrthma parat jentai qr simoi majhmatiko tÔpoi.
Se kje keflaio uprqoun probl mata. Met apì th melèth kje kefala ou,
sa sunistoÔme na prospaje te na ta lÔnete. Ta probl mata aut sa d noun th
dunatìthta na parakolouje te thn prìodì sa kai exasfal zoun ìti èqete afomoi¸sei
to ulikì tou ant stoiqou kefala ou.
Ja prèpei ed¸ na shmei¸soume ìti to bibl o autì èqei kajar eisagwgikì qara-
kt ra. Sto tèlo tou uprqei bibliograf a sthn opo a mpore te na anatrèxete gia
plèon exeidikeumène gn¸sei .
Ja jela na euqarist sw tou spoudastè eke nou , proptuqiakoÔ kai meta-
ptuqiakoÔ , pou me prosoq melèthsan mèrh tou bibl ou kai me ti parathr sei tou
bo jhsan sth belt ws tou.
An kai katabl jhke h mègisth dunat prospjeia na mhn uprqoun parorma-
Prìlogo iii
ta, sthn per ptwsh pou diapistwje ìti uprqoun, ja epishma nontai sth dieÔjunsh
http://sig-sys-book.di.uoa.gr/ tou diktÔou. Sth dieÔjunsh aut , h opo a euelpistoÔme
ìti ja apotelèsei mia anoikt gramm epikoinwn a , ja parousizoume nèe ask sei ,
¸ste na èqete perissìtere dunatìthte exskhsh .
Serafe m Karampogi
Ep kouro Kajhght
EjnikoÔ kai KapodistriakoÔ
Panepisthm ou Ajhn¸n
ÐÅÑÉÅ×ÏÌÅÍÁ
PROLOGOS vi
1 EISAGWGH STA SHMATA 1
1.1 Taxinìmhsh Shmtwn 2
1.1.1 S mata suneqoÔ qrìnou analogik s mata, 2
1.1.2 S mata diakritoÔ qrìnou, 2
1.1.3 Yhfiak s mata, 2
1.2 Idiìthte Analogik¸n Shmtwn 3
1.2.1 Periodik kai mh periodik s mata, 3
1.2.2 Aitita kai mh aitiat s mata, 4
1.2.3 S mata peperasmèna kai s mata peperasmènh kai peirh dirkeia , 5
1.2.4 rtia kai peritt s mata, 5
1.2.5 Energeiak s mata - S mata isqÔo , 6
1.2.6 Aitiokratik kai Tuqa a - Stoqastik s mata, 7
1.3 Metatropè S mato w pro to Qrìno 8
1.3.1 Anklash, 8
1.3.2 Allag kl maka qrìnou, 8
1.3.3 Qronik metatìpish, 9
1.4 Stoiqei¸dh S mata 10
1.4.1 Migadikì ekjetikì s ma suneqoÔ qrìnou, 10
1.4.2 Migadikì ekjetikì s ma diakritoÔ qrìnou, 14
1.4.3 Idiìthte twn ekjetik¸n shmtwn, 15
1.4.4 H sunrthsh monadia ou b mato suneqoÔ qrìnou, 18
1.4.5 H kroustik sunrthsh suneqoÔ qrìnou Sunrthsh dèlta, 19
1.4.6 O orjog¸nio palmì , 24
1.4.7 O trigwnikì palmì , 24
1.4.8 H sunrthsh kl sh , 25
1.4.9 H sunrthsh pros mou, 25
1.4.10 Monadia a bhmatik akolouj a - Monadia o b ma diakritoÔ qrìnou, 26
Perieqìmena v
1.4.11 To monadia o de gma - Kroustik akolouj a, 26
Probl mata 28
2 EISAGWGH STA SUSTHMATA 31
2.1 Orismì Sust mato - Kathgor e Susthmtwn 32
2.2 Sundèsei Susthmtwn 35
2.3 Idiìthte Susthmtwn 37
2.3.1 Grammikìthta, 37
2.3.2 Aitiìthta, 37
2.3.3 Antistrèyima kai mh antistrèyima sust mata, 38
2.3.4 Sust mata statik kai dunamik, 39
2.3.5 Qronik anallo wta sust mata, 39
2.3.6 Eustjeia, 40
2.4 Sqèsh metaxÔ Eisìdou - Exìdou Sust mato 41
2.4.1 Grammik qronik anallo wta sust mata suneqoÔ qrìnou. -
To olokl rwma th sunèlixh , 41
2.4.2 Idiìthte th sunèlixh , 45
2.4.3 Grafikì prosdiorismì th sunèlixh , 47
2.4.4 Grammik qronik anallo wta sust mata diakritoÔ qrìnou. -
To jroisma th sunèlixh , 49
2.5 Apìkrish Grammik¸n Susthmtwn se Ekjetikè Eisìdou 54
2.5.1 Suneq per ptwsh, 54
2.5.2 Diakrit per ptwsh, 56
Probl mata 60
3 ANAPTUGMA - METASQHMATISMOS FOURIER
ANALOGIKWN SHMATWN 63
3.1 H Idèa tou Q¸rou twn Shmtwn 63
3.1.1 To sÔnolo twn orjogwn wn ekjetik¸n periodik¸n shmtwn, 67
3.1.2 To sÔnolo twn orjogwn wn trigwnometrik¸n periodik¸n shmtwn, 68
3.2 Anptugma Fourier - Seir Fourier 69
3.2.1 Ekjetik seir Fourier, 69
3.2.2 Trigwnometrik seir Fourier, 70
3.2.3 Seirè Fourier periodik¸n shmtwn, 72
3.2.4 Ìparxh seir Fourier, 72
3.2.5 Tautìthta tou Parseval, 77
3.2.6 Fainìmeno Gibbs, 81
vi Perieqìmena
3.3 Metasqhmatismì Fourier 82
3.3.1 Ìparxh tou metasqhmatismoÔ Fourier, 91
3.3.2 Idiìthte tou metasqhmatismoÔ Fourier, 93
3.3.3 Metasqhmatismì Fourier periodik¸n shmtwn, 109
3.4 Enèrgeia kai IsqÔ 110
3.4.1 Energeiak s mata, 111
3.4.2 S mata isqÔo , 113
Probl mata 115
4 EFARMOGES TOU METASQHMATISMOU FOURIER 119
4.1 Apìkrish Suqnìthta Sust mato 119
4.1.1 H apìkrish suqnìthta gia sust mata ta opo a perigrfontai
apì diaforikè exis¸sei me stajeroÔ suntelestè , 120
4.2 Upologismì tou Antistrìfou MetasqhmatismoÔ Fourier 122
4.3 Diagrmmata Bode 125
4.4 Idanikì F ltro Basik Z¸nh - Katwperatì F ltro 128
Probl mata 133
5 SEIRA - METASQHMATISMOS FOURIER
DIAKRITOU QRONOU 137
5.1 Parstash Periodik¸n Shmtwn sto Ped o tou Qrìnou
Seir Fourier DiakritoÔ Qrìnou 138
5.1.1 Prosdiorismì th seir Fourier diakritoÔ qrìnou, 138
5.2 Metasqhmatismì Fourier DiakritoÔ Qrìnou 143
5.3 Idiìthte tou FM DiakritoÔ Qrìnou 151
5.3.1 Apodektish sto qrìno, 152
5.3.2 Parembol , 154
5.3.3 jroisma, 156
5.3.4 Idiìthta th diamìrfwsh , 157
5.3.5 O MF diakritoÔ qrìnou gia periodik s mata, 160
5.4 S mata apì deigmatolhy a sto Ped o Suqnot twn 163
5.4.1 Je¸rhma deigmatolhy a , 167
5.4.2 Oi Suntelestè Fourier w de gmata se m a per odo tou MF, 168
5.5 Diakritì Metasqhmatismì Fourier 170
5.5.1 Kuklik anklash akolouj a , 173
5.5.2 Kuklik ol sjhsh akolouj a , 174
5.5.3 Kuklik sunèlixh akolouji¸n, 175
Perieqìmena vii
5.5.4 Idiìthte tou diakritoÔ metasqhmatismoÔ Fourier, 176
5.5.5 H grammik sunèlixh me th bo jeia tou diakritoÔ MF, 177
5.5.6 O diakritì metasqhmatismì Fourier se morf pinkwn, 180
5.5.7 TaqÔ metasqhmatismì Fourier, 181
5.6 Efarmogè tou MF DiakritoÔ Qrìnou 186
5.6.1 H apìkrish suqnìthta gia sust mata ta opo a qarakthr zontai
apì grammikè exis¸sei diafor¸n me stajeroÔ suntelestè , 187
Probl mata 195
6 METASQHMATISMOS LAPLACE 199
6.1 Orismo 200
6.1.1 Metasqhmatismì Laplace stoiqeiwd¸n shmtwn, 202
6.1.2 Idiìthte th perioq sÔgklish -Ìparxh metasqhmatismoÔ Laplace, 206
6.1.3 Idiìthte tou metasqhmatismoÔ Laplace, 209
6.2 Ant strofo Metasqhmatismì Laplace 211
6.2.1 Upologismì tou ant strofou MetasqhmatismoÔ Laplace, 213
6.3 O monìpleuro Metasqhmatismì Laplace 216
6.4 Efarmogè twn Metasqhmatism¸n Laplace 219
6.4.1 Ep lush grammik diaforik ex swsh me th bo jeia MML, 220
6.4.2 H qr sh tou ML sthn anlush GQA susthmtwn, 221
6.4.3 Parathr sei gia thn perioq sÔgklish tou ML, 226
Probl mata 231
7 METASQHMATISMOS Z 233
7.1 Orismo 234
7.1.1 Metasqhmatismì z stoiqeiwd¸n akolouji¸n, 236
7.1.2 Idiìthte th perioq sÔgklish -Ìparxh metasqhmatismoÔ z, 241
7.2 Idiìthte tou MetasqhmatismoÔ z 245
7.3 O Monìpleuro Metasqhmatismì z 250
7.4 O Ant strofo Metasqhmatismì z 256
7.4.1 Upologismì tou antistrìfou Mz gia rhtè sunart sei , 257
7.4.2 Upologismì me anptuxh se apl klsmata, 257
7.4.2 Upologismì me anptuxh se dunamoseir, 261
7.5 Efarmogè twn Metasqhmatism¸n z 262
7.5.1 Sust mata ta opo a qarakthr zontai apì grammikè exis¸sei diafor¸n
me stajeroÔ suntelestè , 263
7.5.1 Melèth GQA sust mato me th bo jeia Mz, 270
viii Perieqìmena
Probl mata 274
PARARTHMA A: MERIKA BASIKA STOIQEIA
GIA TOUS MIGADIKOUS ARIJMOUS 279
A.1 Parstash migadikoÔ arijmoÔ sto migadikì ep pedo 279
A.2 Suzug migadikì arijmì - Idiìthte 280
Probl mata 282
PARARTHMA B: ANAPTUXH RHTHS SUNARTHSHS
SE APLA KLASMATA 283
B.1 O bajmì tou N (x) e nai mikrìtero tou bajmoÔ tou D (x) 283
B.1.1 R ze diakekrimène kai pragmatikè , 284
B.1.2 R ze pollaplè kai pragmatikè , 285
B.1.3 Migadikè r ze , 286
B.2 O bajmì tou N (x) e nai megalÔtero so tou bajmoÔ tou D (x) 286
PARARTHMA G: QRHSIMOI MAJHMATIKOI TUPOI 287
G.1 Trigwnometr a 287
G.2 Aìrista oloklhr¸mata 288
G.2.1 Rht¸n algebrik¸n sunart sewn, 288
G.2.2 Trigwnometrik¸n sunart sewn, 289
G.2.3 Ekjetik¸n sunart sewn, 289
G.2.4 Orismèna oloklhr¸mata, 289
G.3 Gewmetrikè seirè 290
EURETHRIO 291
P naka suntomeÔsewn-akrwnÔmia
dB Desibel
A/D analogoyhfiakì metatropèa analog to digital converter
IDFT ant strofo diakritì metasqhmatismì Foureir inverse disctete Fourier transform
bajuperatì f ltro
LPF f ltro basik z¸nh lowpass filter
f ltro dièleush qamhl¸n suqnot twn
GS grammikì sÔsthma linear system
GQAS grammikì qronik anallo wto SÔsthma linear time invariant system
DFT diakritì metasqhmatismì Fourier discrete Fourier transform
2-D disdistata s mata two-dimensional signals
FEFE eustjeia fragmènh eisìdou bounded input bounded output
fragmènh exìdou
BPF zwnoperatì f ltro
zwnodiabatì f ltro bandpass filter
f ltro dièleush z¸nh suqnot twn
BSF zwnofraktikì f ltro bandstop filter
f ltro apokop z¸nh suqnot twn
MF metasqhmatismì Fourier Fourier transform
DTFT metasqhmatismì Fourier diakritoÔ qrìnou discrete time Fourier transform
ML metasqhmatismì Laplace Laplace transform
Mz metasqhmatismì z z Transform
1-D monadistata s mata One-dimensional signal
MML monìpleuro metasqhmatismì Laplace unilateral Laplace transform
MMz monìpleuro metasqhmatismì z unilateral z Transform
PS ped o sÔgklish Region of convergence
ROC ped o sÔgklish Region of convergence
DTFS seir Fourier diakritoÔ qrìnou discrete-time Fourier series
QAS sÔsthma qronik anallo wto Time invariant system
MISO sust mata me pollè eisìdou -m a exìdou multi-input, single-output
MIMO sust mata me pollè eisìdou -pollè exìdou multi-input, multi-output
SISO sust mata m a eisìdou-m a exìdou single-input, single-output
HPF uyiperatì f ltro highpass filter
f ltro dièleush uyhl¸n suqnot twn
D/A yhfioanalogikì metatropèa digital to analog converter
P naka basik¸n sumbìlwn
j tetragwnik r za tou -1 squere root of -1
jzj mètro migadikoÔ arijmoÔ z magnitude of complex quantity z
argfzg fsh (ìrisma) tou z phase angle of z
<efzg pragmatikì mèro tou z imaginary part of z
=mfzg fantastikì mèro tou z imaginary part of z
z? suzug migadikì arimjì tou z complex conjugate of z
! (kuklik ) suqnìthta gia s mata suneqoÔ qrìnou frequency for continuous-time signal
(kuklik ) suqnìthta gia s mata diakritoÔ qrìnou frequency for discrete-time signal
u(t) bhmatik sunrthsh step function
Æ(t) sunrthsh dèlta (dirac)-kroustik sunrthsh unit impulse
(t) orjog¸nio palmì rectangular pulse
(t) trigwnikì palmì triangular pulse
r(t) sunrthsh kl sh ramb function
sgn(t) sunrthsh pros mou signum function
? sumbol zei to olokl rwma th sunèlixh detotes convolution operation
N kuklik sunèlixh N -shme wn N -point circular convolousion
h(t) kroustik sunrthsh impulse response
H (s) sunrthsh metafor transfer function
H (!) apìkrish suqnìthta frequency response
sin (t) sunrthsh deigmatolhy a sampling function
X (!) ()
metasqhmatismì Fourier tou x t ()
Fourier transform of x t
X( ) ()
metasqhmatismì Fourier th x n ()
Fourier transform of x n
X (k) ()
diakritì metasqhmatismì Fourier th x n ()
disrete Fourier transform of x n
X (s) ()
metasqhmatismì Laplace tou x t ()
Laplace transform of x t
X (s) monìpleuro metasqhmatismì Laplace tou x t() ()
unilateral Laplace transform of x t
X (z ) ()
metasqhmatismì z th x n ()
z transform of x n
X (z) monìpleuro metasqhmatismì z th x n () ()
unilateral z transform of x n
F! upodeiknÔei zeÔgo metasqhmatism¸n Fourier Fourier transform pair
L
! upodeiknÔei zeÔgo metasqhmatism¸n Laplace Laplace transform pair
L! upodeiknÔei monìpleuro metasqhmatismì Laplace unilateral Laplace transform pair
Z
! upodeiknÔei zeÔgo metasqhmatism¸n z z transform pair
Z! upodeiknÔei monìpleuro metasqhmatismì z unilateral z transform pair
ÊÅÖÁËÁÉÏ 1
ÅÉÓÁÃÙÃÇ ÓÔÁ ÓÇÌÁÔÁ
Skopì tou kefala ou autoÔ e nai na d¸sei m a genik eikìna tou ti e nai s ma kai
na katatxei ta difora s mata se kathgor e anloga me ti basikè tou idiìthte .
Epiprìsjeta ja oristoÔn antiproswpeutik s mata, ta opo a èqoun idia terh shmas a
sth jewr a shmtwn.
To keflaio autì, pragmateÔetai ta krit ria bsei twn opo wn or zontai ta s ma-
ta kai taxinome ta s mata. Ep sh , anafèrei ti basikè idiìthte pou parousizoun
ta s mata kai ti metatropè s mato w pro to qrìno. Sth sunèqeia or zontai
stoiqei¸dh s mata, ta opo a pa zoun ènan idia tero rìlo sth jewr a shmtwn, w
ergale a gia th melèth poluplokìterwn shmtwn. Shmei¸netai ìti sto Parrthma A
uprqoun merik basik stoiqe a gia tou migadikoÔ arijmoÔ .
Eisagwg
W s ma or zetai èna fusikì mègejo to opo o metablletai se sqèsh me to qrìno to
q¸ro me opoiad pote llh anexrthth metablht metablhtè . Gia pardeigma, to s -
ma omil a antistoiqe sti metabolè th akoustik p esh se sqèsh me to qrìno kai
proèrqetai apì ti kin sei twn fwnhtik¸n qord¸n. To s ma eikìna antistoiqe sti
metabolè th fwteinìthta se sqèsh me ti dÔo qwrikè metablhtè . lla parade g-
mata shmtwn e nai ta seismik s mata, ta iatrik s mata (ìpw to kardiogrfhma),
o et sio de kth tim¸n katanalwt , o de kth tou posostoÔ anerg a an m na k.l.p.
Apì majhmatik poyh, èna s ma ekfrzetai w sunrthsh mia perissìterwn
anexrthtwn metablht¸n.
t ! x(t)
H anexrthth metablht t e nai sun jw o qrìno , h opo a mpore na èqei kai llh
()
fusik shmas a. Me x t sumbol zetai h tim tou s mato th qronik t.
Anloga me to pl jo twn anexart twn metablht¸n ta s mata qarakthr zontai
w monodistata s mata (1-D), disdistata (2-D), poludistata s mata.
2 Eisagwg sta S mata Keflaio 1
1.1 TAXINOMHSH SHMATWN
Anloga me ton tÔpo th anexrthth th exarthmènh metablht th sunrthsh
ta s mata katatssontai sti paraktw kathgor e :
1.1.1 S mata suneqoÔ qrìnou analogik s mata
S mata suneqoÔ qrìnou analogik s mata e nai ta s mata twn opo wn h anexrthth
metablht metablletai se èna suneqè disthma tim¸n. Sta monodistata s mata
to ped o orismoÔ tou s mato e nai disthma th euje a twn pragmatik¸n arijm¸n.
Sto Sq ma 1.1 èqei sqediaste èna analogikì s ma. Epeid h anexrthth metablht t
sun jw e nai o qrìno , ta s mata aut onomzontai s mata suneqoÔ qrìnou s mata
suneqoÔ metablht .
x(t)
t
Sq ma 1.1 Grafik anaparstash enì analogikoÔ s mato .
1.1.2 S mata diakritoÔ qrìnou
()
àna analogikì s ma xa t e nai dunatì na anaparaqje apì ti timè twn deigmtwn
()
tou x n pou lambnontai an qronik diast mata TÆ . Ta qronik diast mata TÆ
kajor zontai anloga me to e do tou analogikoÔ s mato kai thn epijumht pistìthta
anaparagwg .
x(n) xa (nTÆ ); n = 0; 1; 2; ::: (1.1.1)
H diadikas a onomzetai deigmatolhy a kai to qronikì disthma metaxÔ diadoqik¸n
deigmtwn per odo deigmatolhy a TÆ . Me th diadikas a th deigmatolhy a enì
analogikoÔ s mato prokÔptei èna s ma diakritoÔ qrìnou.
S mata diakritoÔ qrìnou e nai ta s mata twn opo wn to ped o orismoÔ e nai kpoio
diakritì sÔnolo p.q. to sÔnolo twn akera wn arijm¸n en¸ h exarthmènh metablht
e nai dunatìn na lambnei opoiad pote tim . To s ma sto Sq ma 1.2a e nai èna s ma
diakritoÔ qrìnou.
Sti efarmogè sti opo e e nai anagka o na metad doume na apojhkeÔoume èna
analogikì s ma metad doume apojhkeÔoume ta de gmat tou.
Enìthta 1.2 Idiìthte Analogik¸n Shmtwn 3
x(n) x(n)
n n
(á) ( â)
Sq ma 1.2 Grafik anaparstash (a) enì s mato diakritoÔ qrìnou kai (b) enì yhfiakoÔ
s mato .
1.1.3 Yhfiak s mata
H anaparstash twn analogik¸n deigmatolhpthmènwn tim¸n x n ; n () =0 1 2
; ; ; ::: me
èna peperasmèno sÔnolo epitrepìmenwn tim¸n lègetai kbntish. Me th diadikas a th
()
kbntish oi timè x n enì diakritoÔ s mato stroggulopoioÔntai sthn plhsièsterh
epitrepìmenh tim kai ètsi prokÔptei èna yhfiakì s ma.
Yhfiak s mata e nai ta s mata sta opo a tìso h anexrthth metablht , ìso kai
h exarthmènh metablht mporoÔn na lambnoun mìno diakritè timè . Sto Sq ma 1.2b
fa netai èna yhfiakì s ma.
AfoÔ deigmatolhpthje kai kbantiste h èxodo m a analogik phg plhro-
for a , dhmiourge tai m a akolouj a apì kbantismène timè . Kje kbantismènh st-
jmh kwdikopoie tai se m a duadik akolouj a m kou , ìpou N =2
e nai o arijmì
twn epitrepìmenwn tim¸n.
àna sÔsthma yhfiak metdosh katagraf qou metatrèpei èna akoustikì
s ma se m a seir apì arijmoÔ bit tou opo ou metad dei katagrfei p.q. se optikì
d sko. H metatrop enì analogikoÔ s mato se m a duadik akolouj a onomzetai
palmokwdik diamìrfwsh.
1.2 IDIOTHTES ANALOGIKWN SHMATWN
Sthn enìthta aut parousizontai merikè basikè idiìthte pou èqoun ta analogik
s mata. Anloge idiìthte èqoun kai ta s mata diakritoÔ qrìnou.
1.2.1 Periodik kai Mh Periodik S mata
()
àna analogikì s ma x t lègetai periodikì, ìtan uprqei èna jetikì arijmì T gia
()= ( + )
ton opo o isqÔei x t x t T gia kje tim tou t. Sto Sq ma 1.3 èqei sqediaste èna
periodikì s ma. O stajerì arijmì T lègetai per odo . H elaq sth dunat per odo
e nai gnwst w jemeli¸dh per odo kai sumbol zetai me T0 . Sthn prxh pollè forè
anaferìmaste apl¸ sthn per odo kai ennooÔme th jemeli¸dh.
4 Eisagwg sta S mata Keflaio 1
x(t)
-2T -T 0 T 2T t
Sq ma 1.3 Periodikì s ma suneqoÔ qrìnou.
Pardeigma periodikoÔ s mato e nai to hmitonoeidè s ma
x(t) = A sin(!t + ) (1.2.1)
me per odo T =2 =!. To ! e nai gnwstì w kuklik suqnìthta kai e nai ! = 2f ,
ìpou f h suqnìthta tou hm tonou.
àna llo periodikì s ma e nai to migadikì s ma
y(t) = Aej!t (1.2.2)
me thn dia per odo T =2
=!. An kai ta migadik s mata den èqoun fusik upìstash,
e nai “elkustik” apì poyh majhmatikoÔ formalismoÔ, epeid aplousteÔoun thn
lgebra twn prxewn. Gia pardeigma, pollaplasiasmì dÔo ekjetik¸n shmtwn
antistoiqe apl¸ sthn prìsjesh twn ekjet¸n tou . Prgmati, an y1 t Aej!1 t () =
kai y2 t()= j! t
Ae , tìte y1 t y2 t A e
2 2 (
( ) ( ) =
j !1 + !2 ) t . Sto Parrthma A uprqei m a
sÔntomh parous ash twn migadik¸n arijm¸n kai twn idiot twn tou .
1.2.2 Aitiat kai Mh Aitiat S mata
àna s ma x(t) lègetai aitiatì, en e nai mhdenikì gia arnhtikè timè tou qrìnou t,
dhlad
x(t) = 0 gia t<0 (1.2.3)
Sthn ant jeth per ptwsh, to s ma lègetai mh aitiatì. Sto Sq ma 1.4 eikon zontai èna
aitiatì kai èna mh aitiatì s ma.
x(t) x(t)
0 t 0 t
(á) (â)
Sq ma 1.4 Pardeigma (a) aitiatoÔ s mato kai (b) mh aitiatoÔ s mato .
Enìthta 1.2 Idiìthte Analogik¸n Shmtwn 5
1.2.3 S mata Peperasmèna kai S mata Peperasmènh kai peirh Dirkeia
()
àna s ma x t lègetai peperasmèno an x t < j ( )j 1
, gia kje tim tou qrìnou t. àna
()
s ma x t lègetai s ma peperasmènh dirkeia an
x(t) = 0; t T1
0; t T2 (1.2.4)
( )
ìpou T1 kai T2 , T1 < T2 , e nai peperasmènoi arijmo . An T1 g nei so me to me on
peiro kai T2 g nei so me to peiro, tìte to s ma èqei peirh dirkeia.
1.2.4 rtia kai Peritt s mata
àna s ma x(t) lègetai rtio parousizei rtia summetr a an
x( t) = x(t); 1 < t < +1 (1.2.5)
Ant jeta, lègetai perittì parousizei peritt summetr a an
x( t) = x(t); 1 < t < +1 (1.2.6)
To s ma sto Sq ma 1.5a parousizei rtia summetr a kai to s ma sto Sq ma 1.5b
x(t) x(t)
0 t
0 t
(á) (â)
Sq ma 1.5 S mata suneqoÔ qrìnou ta opo a parousizoun (a) rtia summetr a kai (b)
peritt summetr a.
peritt summetr a. Kje s ma migadikì pragmatikì mpore na ekfrasje w jroi-
()
sma enì rtiou (even), xe t , kai enì perittoÔ (odd) s mato , xo t , ()
x(t) = xe (t) + xo (t) (1.2.7)
ìpou
xe (t) =
1 [x(t) + x ( t)℄ xo (t) =
1 [x(t) x ( t)℄
2 2 (1.2.8)
me ? dhl¸netai o suzug migadikì arijmì .
6 Eisagwg sta S mata Keflaio 1
Pardeigma 1.2.1
D netai to s ma
x(t) = 3t; t<0
t; t0 (1.2.9)
Na ekfrsete to s ma w jroisma enì rtiou kai enì perittoÔ s mato .
LÔsh To rtio s ma xe (t) e nai
1
1
xe (t) = [x(t) + x( t)℄ = 1 [t + 3t℄; t 0 = 2t;2t; tt <
2 [ 3 t t℄ ; t < 0 0
2 2 0 (1.2.10)
en¸ to perittì s ma xo (t) e nai
1
1
xo (t) = [x(t) x( t)℄ = 21 [[t 3t3+t℄;t℄; tt < 0 =) xo (t) = t
2 2 0 (1.2.11)
Sto Sq ma 1.6 eikon zontai to s ma x(t), to xe (t) kai to xo (t). Apì ti grafikè
parastsei twn shmtwn aut¸n fa netai ìti
x(t) = xe (t) + xo (t) (1.2.12)
x(t) xe(t) xï(t)
-t
-3t t -2t 2t
0 t
0 t 0 t
(á) (â) (ã)
Sq ma 1.6 Grafikè parastsei gia ta s mata (a) x(t), (b) xe (t) kai (g) xo (t).
1.2.5 Energeiak S mata - S mata IsqÔo
Gia kje analogikì s ma x(t), h enèrgeia tou s mato Ex, d netai apì th sqèsh
Z T
Ex = Tlim
!1
jx(t)j2 dt (1.2.13)
T
j ( )j
ìpou x t e nai to mètro tou s mato . Gia kje s ma diakritoÔ qrìnou x(n), h
E
enèrgei tou x , d netai apì th sqèsh
1
X
Ex = jx(n)j2 (1.2.14)
n= 1
Enìthta 1.2 Idiìthte Analogik¸n Shmtwn 7
j ( )j
ìpou x n e nai to mètro tou s mato kai TÆ e nai h per odo deigmatolhy a .
àna s ma qarakthr zetai w energeiakì s ma an
0 < Ex < 1 (1.2.15)
H mèsh isqÔ Px, tou analogikoÔ s mato x(t), d netai apì th sqèsh
Px = Tlim 1 Z T jx(t)j2 dt
!1 2T T
(1.2.16)
An to s ma e nai periodikì, tìte h mèsh isqÔ tou Px d netai apì th sqèsh
Px = T1
Z
jx(t)j2 dt (1.2.17)
0 <T0 >
H mèsh isqÔ Px, tou s mato diakritoÔ qrìnou x(n), d netai apì th sqèsh
Px = Nlim 1 N
X
jx(n)j2
!1 2N + 1
(1.2.18)
n= N
An to s ma e nai periodikì, tìte h mèsh isqÔ tou Px d netai apì th sqèsh
0 1
Px = N1
NX
jx(n)j2 (1.2.19)
0 n=0
àna s ma qarakthr zetai w s ma isqÔo an
0 < Px < 1 (1.2.20)
Shmei¸netai ìti gia pragmatik s mata jx(t)j2 = x2 (t) (blèpe Parrthma A).
1.2.6 Aitiokratik kai Tuqa a - Stoqastik S mata
ätan oi timè pou pa rnei èna s ma se kje qronik stigm or zontai qwr abebaiìth-
ta, to s ma qarakthr zetai w aitiokratikì s ma nomoteleiakì s ma. àna tètoio s -
ma, gia pardeigma, e nai to sunhm tono (Sq ma 1.7a). Sthn prxh, ìmw , sunantme
poll s mata, ìpw o jermikì jìrubo , sta opo a h tim se opoiad pote qronik
stigm den mpore na prokajoriste me bebaiìthta prin emfanistoÔn. Ta s mata aut
onomzontai tuqa a stoqastik s mata (Sq ma 1.7b). Gia na epexergastoÔme tètoiou
e dou s mata anagkastik katafeÔgoume sth jewr a Pijanot twn kai Statistik .
Sto bibl o autì ja perioristoÔme mìno sta aitiokratik s mata.
8 Eisagwg sta S mata Keflaio 1
x(t)=A cos(2 ð f0 t + ö)
x(t)
A
Acos(ö)
0 t 0
T0 t
-A
(á) (â)
Sq ma 1.7 Pardeigma (a) nomoteleiakoÔ s mato kai (b) stoqastikoÔ s mato .
1.3 METATROPES SHMATOS WS PROS TON QRONO
Pollè forè sthn prxh parousizontai s mata ta opo a sqet zontai metaxÔ tou me
allag th anexrthth metablht , dhlad tou qrìnou. Sth sunèqeia anafèrontai
oi basikè metatropè s mato w pro to qrìno.
1.3.1 Anklash
àna s ma y(t) apotele thn anklash tou s mato x(t) w pro t = 0 an
y(t) = x( t) (1.3.1)
H metatrop th anklash èqei w apotèlesma thn enallag metaxÔ “pareljìnto ”
()
kai “mèllonto ” enì s mato . An to s ma x t e nai h èxodo enì magnhtof¸nou,
( )
tìte to s ma x t e nai h èxodo tou id ou magnhtof¸nou, ìtan autì peristrèfetai
ant jeta. Sto Sq ma 1.8 èqei sqediaste èna s ma suneqoÔ qrìnou kai h anklas
tou w pro t =0
.
x(t) x(- t)
0 t 0 t
(á) (â)
Sq ma 1.8 (a) àna s ma suneqoÔ qrìnou kai (b) h anklas tou w pro t = 0.
1.3.2 Allag Kl maka Qrìnou
To s ma x1 (t) apotele m a qronik sustol tou s mato x(t), an
x1 (t) = x(at) me a > 1 (1.3.2)
Enìthta 1.3 Metatropè S mato w pro to Qrìno 9
To s ma x2 (t) apotele m a qronik diastol tou s mato x(t), an
x2 (t) = x(at) me 0 < a < 1 (1.3.3)
()
An to s ma x t e nai h èxodo enì magnhtof¸nou, tìte to s ma x t e nai h (2 )
èxodo tou id ou magnhtof¸nou, ìtan autì peristrèfetai me diplsia taqÔthta kai
( 2)
x t= e nai h èxodo , ìtan autì peristrèfetai me upodiplsia taqÔthta. Sto Sq ma
1.9 èqoun sqediaste h qronik sustol kai diastol enì s mato .
x(t) x(2 t) x(t/2)
-t 0 0 t0 t t0 0 t0 t -2 t0 0 2 t0 t
2 2
(á) (â) (ã)
Sq ma 1.9 (a) S ma, (b) h qronik sustol tou kai (g) h qronik diastol tou.
1.3.3 Qronik Metatìpish
àna s ma y(t) e nai m a qronik metatopismènh kat t0 morf tou s mato x(t) an
y(t) = x(t t0 ) (1.3.4)
()
Sto Sq ma 1.10 èqei sqediaste èna s ma x t kai h qronik metatopismènh morf tou.
H qronik metatìpish e nai m a polÔ sunhjismènh metabol sthn prxh. Se peript¸-
sei metdosh enì s mato èqoume qronikè kajuster sei , oi opo e exart¸ntai apì
ti idiìthte tou mèsou metdosh . Gia pardeigma, se èna thlepikoinwniakì sÔsthma
to s ma pou lambnei o dèkth e nai qronik kajusterhmèno se sqèsh me autì pou
ekpèmpetai apì ton pompì.
x(t) x(t-t0)
0 t 0 t0 t
(á) (â)
Sq ma 1.10 (a) To s ma x(t) kai (b) h qronik metatopismènh morf tou.
Pardeigma 1.3.1
D netai to s ma 8
< 2t + 2; 1 t < 0
x(t) = 2 t; 0 t < 2 (1.3.5)
:
0; alli¸
10 Eisagwg sta S mata Keflaio 1
Na sqedisete to s ma y(t) = x( t).
LÔsh To s ma x(t) eikon zetai sto Sq ma 1.11a. To s ma y(t) apotele thn anklash
tou s mato x(t). Ja prosdior soume th sunrthsh tou s mato y(t)
8
< 2( t) + 2; 1 t < 0
y(t) = x( t) = 2 ( t); 0 t < 2
:
8
0; alli¸
< 2t + 2; 1 t > 0
= : 2 + t; 0 t > 2
8
0; alli¸
< t + 2; 2t<0
= : 2 2t; 0 t < 1 (1.3.6)
0; alli¸
H grafik parstash tou s mato y (t) = x( t) d netai sto Sq ma 1.11b.
x(t) y(t)=x(-t)
2 2
2 t+2 2-t t+2 2-2t
-1 0 2 t -2 0 1 t
(á) (â)
Sq ma 1.11 H grafik parstash (a) tou s mato x(t) kai (b) tou s mato y(t) sto
Pardeigma 1.3.1
1.4 STOIQEIWDH SHMATA
H anlush enì s mato se aploÔstera s mata, twn opo wn h sumperifor e nai e te
gnwst e te eukolìtero na melethje , apotele basik mejodolog a sthn epexergas a
s mato . Sth sunèqeia ja or soume ènan arijmì stoiqeiwd¸n shmtwn pou pa zoun
ènan idia tero rìlo sth jewr a shmtwn, w ergale a gia th melèth poluplokìterwn
shmtwn.
1.4.1 Migadikì ekjetikì s ma suneqoÔ qrìnou
To migadikì ekjetikì s ma suneqoÔ qrìnou or zetai apì th sqèsh
x(t) = est (1.4.1)
ìpou s = + j! ètsi x(t) = et ej!t kai èqei ti akìlouje idiìthte :
Enìthta 1.4 Stoiqei¸dh S mata 11
1. E nai antistrèyimo ( est ) 1 = 1 e st .
2.
d ( est ) = sest .
E nai diafor simo dt
M a shmantik kathgor a ekjetik¸n shmtwn suneqoÔ qrìnou prokÔptei an to
s e nai pragmatikì arijmì , s ( = )
opìte to x t ()=
et onomzetai pragmatikì
ekjetikì s ma, kai parousizei asumptwtik sumperifor anloga me ti timè tou
(blèpe Sq ma 1.12).
x(t) x(t) x(t)
c c c
0 t 0 t 0 t
(á) (â) (ã)
Sq ma 1.12 To pragmatikì ekjetikì s ma (a) gia < 0, (b) gia > 0 kai gia = 0.
M a llh shmantik kathgor a ekjetik¸n shmtwn suneqoÔ qrìnou prokÔptei an
to s e nai fantastikì arijmì (s =
j!0 ), dhlad , x t ()=
ej!0 t . To s ma x t ej!0 t ()=
e nai periodikì me per odo T , ìtan
!0 = 0, tìte x(t) = 1, to opo o mpore na jewrhje periodikì gia kje T .
!0 =
6 0, tìte h jemeli¸dh per odo T0 , dhlad h mikrìterh tim tou T , e nai
T0 = 2=j!0 j. Prgmati, apì ton orismì isqÔei:
ej!0 t = ej!0 (t+T ) ) ej!0 t = ej!0 t ej!0 T ) ej!0 T = 1 )
os(!0 T ) + j sin(!0 T ) = 1 ) !0T = 2k ) T0 = 2=j!0 j
ìpou qrhsimopoi jhke h sqèsh tou Euler ej = os + j sin .
To gnwstì sunhmitonoeidè s ma x(t) = A os(!0 t + ') (blèpe Sq ma 1.13) e nai,
ep sh , periodikì me jemeli¸dh analogik per odo T0 ,
jemeli¸dh analogik kuklik
suqnìthta !0 kai jemeli¸dh analogik suqnìthta f0 ìpou f0 = 1=T0 kai !0 = 2f0 .
To sunhmitonoeidè s ma sqet zetai mesa me to migadikì ekjetikì s ma. Prg-
mati, an qrhsimopoi soume th sqèsh tou Euler, mporoÔme na ekfrsoume to migadikì
ekjetikì s ma me th bo jeia hmitonoeid¸n shmtwn th id a jemeli¸dou periìdou
apì th sqèsh
ej (!0 t+') = os( + ) + sin( + )
!0 t ' j !0 t ' (1.4.2)
MporoÔme, profan¸ , na gryoume
os(!0t + ') = <e[ej(!0 t+') ℄ kai sin(!0t + ') = =m[ej(!0 t+') ℄ (1.4.3)
12 Eisagwg sta S mata Keflaio 1
ìpou <e[℄ sumbol zei to pragmatikì kai =m[℄ to fantastikì mèro migadikoÔ arij-
moÔ.
x(t)=A cos(2 ð f0 t + ö)
T0 2ð
A ù0
Acos(ö)
T0 t
Sq ma 1.13 To sunhmitonoeidè
-A
s ma suneqoÔ qrìnou.
H sqèsh tou Euler antistrèfetai kai ètsi mporoÔme na ekfrsoume to hm tono
to sunhm tono me th bo jeia ekjetik¸n migadik¸n ìrwn
os(!0t) = e
j!0 t +e j!0 t
sin(!0t) = e
j!0 t e j!0 t
2 kai
2j (1.4.4)
To migadikì ekjetikì s ma ej!0 t , ìpw to (sun)hmitonoeidè s ma !0 t , !0 t , os( ) sin( )
e nai gnwst w s mata mia suqnìthta s mata apl suqnìthta . äpw ja doÔme, ta
s mata aut qrhsimopoioÔntai gia na perigryoun ta qarakthristik poll¸n fusik¸n
diadikasi¸n.
Sto Sq ma 1.14 d nontai tr a parade gmata sunhmitonoeid¸n shmtwn me diafore-
tik kuklik suqnìthta kai per odo. ParathroÔme ìti, ìtan h kuklik suqnìthta
auxnei, !1 < !2 < !3 , h jemeli¸dh per odo elatt¸netai, T1 > T2 > T3 , kai
auxnei o rujmì twn talant¸sewn tou s mato , dhlad auxnei o rujmì metabol
tou s mato . àna s ma qamhl suqnìthta metablletai me argì rujmì se ant jesh me
èna s ma uyhl suqnìthta pou metablletai me gr goro rujmì.
H genik per ptwsh migadikoÔ ekjetikoÔ s mato suneqoÔ qrìnou e nai
x(t) = est; ìpou = j jej ; kai s = + j!0 (1.4.5)
ètsi
x(t) = j j ej e(+j!0 )t = j j et ej (!0 t+) (1.4.6)
Me th bo jeia th sqèsh tou Euler èqoume:
x(t) = j j et os(!0t + ) + j j j et sin(!0t + )
= j j et os(!0t + ) + j j j et os(!0t + =2) (1.4.7)
Enìthta 1.4 Stoiqei¸dh S mata 13
x1(t)= cos(ù1 t )
T1 t
x2(t)= cos(ù2 t )
T2 t
x3(t)= cos(ù3 t )
Sq ma 1.14 H sumperifor tou
T3 t sunhmitìnou gia diaforetikè ku-
klikè suqnìthte !1 < !2 < !3 .
Gia =0to pragmatikì kai to fantastikì mèro (tm ma) e nai (sun)hmitonoeid
s mata (Sq ma 1.15a).
0
Gia > ta ant stoiqa (sun)hmitonoeid s mata pollaplasizontai me ènan
( )
auxanìmeno ekjetikì pargonta et (Sq ma 1.15b).
0
Gia < ta ant stoiqa (sun)hmitonoeid s mata pollaplasizontai me ènan
( )
ekjetikì pargonta et pou fj nei (Sq ma 1.15g). Ta s mata aut e nai gnw-
st w fj nonta hmitonoeid s mata kai emfan zontai sti fj nouse armonikè
mhqanikè hlektrikè talant¸sei ìpw ja doÔme.
Sto Sq ma 1.15 oi diakekommène grammè antistoiqoÔn sti sunart sei j j et
kai apoteloÔn thn peribllousa th kampÔlh talntwsh .
ℜe x(t) = c cos(ù0 t+è) ℜe x(t) = c eót cos(ù0 t+è) ℜe x(t) = c eót cos(ù0 t+è)
c eót c eót
c
t t t
c eót c eót
(á) ( â) (ã)
Sq ma 1.15 Grafik anaparstash tou pragmatikoÔ mèrou tou migadikoÔ ekjetikoÔ s ma-
to (a) gia = 0, (b) gia > 0 kai (g) gia < 0.
14 Eisagwg sta S mata Keflaio 1
1.4.2 Migadikì ekjetikì s ma diakritoÔ qrìnou
To migadikì ekjetikì s ma diakritoÔ qrìnou or zetai apì th sqèsh
x(n) = n (1.4.8)
ìpou kai e nai genik migadiko arijmo (enallaktik or zetai apì th x (n) = e n
ìpou =e ).
x(n) x(n)
(a) n (â) n
x(n) x(n)
n n
(ã) (ä)
Sq ma 1.16 To pragmatikì ekjetikì s ma diakritoÔ qrìnou (a) gia > 1, (b) gia 0 < < 1,
(g) gia < 1 kai (d) gia 1 < <0.
1. An kai e nai pragmatiko arijmo , èqoume ti grafikè parastsei tou Sq -
mato 1.16
2. An e nai fantastikì arijmì j 0 kai( = ) =
Aej , tìte x n Aej 0 n . ( )=
To s ma autì sundèetai me to (sun)hmitonoeidè s ma diakritoÔ qrìnou x n ( )=
A os( + )
0 n me th bo jeia th sqèsh tou Euler. Prgmati,
A
A os( 0 n + ) = 2 ej ej 0n + A2 e j e j 0 n
ìpou 0 e nai h jemeli¸dh yhfiak kuklik suqnìthta.
3. H genik per ptwsh =j j ej kai =j j ej 0 d nei
x(n) = j jj jej( 0 n+)
= j jj jn os( 0 n + ) + j j jj jn sin( 0 n + ) (1.4.9)
Sto Sq ma 1.17 eikon zontai to pragmatikì mèro tou migadikoÔ ekjetikoÔ s mato
diakritoÔ qrìnou, gia ti peript¸sei ìpou a < kai a > . jj 1 jj 1
An to n den èqei diastsei , tìte h yhfiak kuklik suqnìthta 0 kai h gwn a
èqoun diastsei gwn a (rad).
Enìthta 1.4 Stoiqei¸dh S mata 15
ℜe x(t) ℜe x(t)
c an c an
n n
c an c an
(á) (â)
Sq ma 1.17 To pragmatikì mèro tou migadikoÔ ekjetikoÔ s mato diakritoÔ qrìnou (a) gia
j j < 1, kai (b) gia j j > 1.
1.4.3 Idiìthte twn ekjetik¸n shmtwn
1. H pr¸th idiìthta afor thn periodikìthta tou migadikoÔ ekjetikoÔ s mato
diakritoÔ qrìnou, w pro th suqnìthta. Ta migadik ekjetik s mata suneqoÔ
qrìnou, ej!1 t kai ej!2 t , an !1 6=
!2 e nai diaforetik s mata. To migadikì
ekjetikì s ma diakritoÔ qrìnou, me kuklik suqnìthta 0 +2
e nai to dio me
to ant stoiqo th kuklik suqnìthta 0 . Prgmati,
ej ( 0 +2)n = ej 2n ej 0n = ej 0n
àtsi, to ekjetikì s ma diakritoÔ qrìnou me kuklik suqnìthta 0 e nai to dio
me ta ekjetik s mata pou èqoun kuklikè suqnìthte 0 ; 0 ; ::: +2 +4
kai gia to lìgo autì to migadikì ekjetikì s ma diakritoÔ qrìnou qreizetai na
perigrafe sto disthma kuklik¸n suqnot twn 0< 0 2
0 < .
Sto ekjetikì s ma suneqoÔ qrìnou parathroÔme ìti ìso auxnei h w tìso
auxnei kai o rujmì twn talant¸sewn. Sta ekjetik s mata diakritoÔ qrìnou,
ìso to 0 auxnei apì 0 mèqri thn tim , èqoume s mata me rujmì talntwsh
pou ep sh auxnetai (Sq ma 1.18). An to 0 auxnetai apì thn tim mè-
qri thn tim 2
, èqoume t¸ra me wsh tou rujmoÔ talntwsh . àtsi, s mata
diakritoÔ qrìnou, ta opo a parousizoun mikroÔ rujmoÔ metabol (qamhlè
suqnìthte ), apoteloÔntai apì suqnìthte pou br skontai sth perioq tou 0 kai
se kje rtio pollaplsio tou . Ant jeta, s mata diakritoÔ qrìnou, ta opo a
parousizoun meglou rujmoÔ metabol (uyhlè suqnìthte ), apoteloÔntai
apì suqnìthte sthn perioq tou kai se kje perittì pollaplsio tou .
2. H deÔterh idiìthta afor thn periodikìthta tou migadikoÔ ekjetikoÔ s mato
diakritoÔ qrìnou, w pro th metablht n. An to ej 0 n e nai periodikì me
0
per odo N > , prèpei
ej 0 (n+N ) = ej 0n =) ej 0N = 1 =) os( 0 N ) + j sin( 0 N ) = 1
16 Eisagwg sta S mata Keflaio 1
x(n)=cos(0n)=cos(2ðn) x(n)=cos (ð n = cos (158ð n(
8 (
(á) (â)
x(n)=cos (ð n = cos (74ð n( x(n)=cos (ð n = cos (32ð n(
2 (
4 (
(ã) ( ä)
x(n)=cos(ðn)
(å)
Sq ma 1.18 Hmitonoeid s mata diakritoÔ qrìnou gia suqnìthte (a) 0 kai 2p, (b) p/8 kai
15p/8, (g) p/4 kai 7p/4, (d) p/2 kai 3p/2 kai (e) p.
dhlad , to 0N prèpei na e nai pollaplsio tou 2p ètsi
0
0 N = 2k =) 2 = Nk (1.4.10)
ParathroÔme ìti to migadikì ekjetikì s ma diakritoÔ qrìnou den e nai genik
2
periodikì, e nai periodikì an 0 = e nai rhtì arijmì .
àqoume, loipìn, gia ta migadik ekjetik s mata diakritoÔ qrìnou
An x(n) e nai periodikì me jemeli¸dh per odo N , h jemeli¸dh suqnìthta e nai
2=N .
Gia na e nai periodikì prèpei 0=2 = k=N . An oi k kai N e nai pr¸toi metaxÔ
tou , tìte h jemeli¸dh per odo e nai N.
H jemeli¸dh per odo mpore na grafe N = k(2= 0 ).
Ta migadik ekjetik s mata suneqoÔ qrìnou pou èqoun kuklikè suqnìthte
pollaplsie th jemeli¸dou !0 =T0 (armonikè ), ejk(2=T0 )t , e nai di-
=2
Enìthta 1.4 Stoiqei¸dh S mata 17
aforetik, dhlad
2 2
ejk T t 6= ejm T t k 6= m
Den isqÔei ìmw to dio gia ta diakritoÔ qrìnou, ìpou lìgw th ej ( 0 +2)n =
ej 0 n ta s mata fk n ejk(2=N )n ; k
( )= = 0 1
; ; ::: e nai ta dia gia timè
tou k pou diafèroun pollaplsio tou N . Prgmati,
fk+N n ej (k+N ) N n ej 2n ejk N n fk n
2 2
( )= = = ()
ParathroÔme ìti uprqoun mìno N diaforetik migadik ekjetik s mata diakri-
toÔ qrìnou twn opo wn oi suqnìthte e nai pollaplsia th jemeli¸dou . Ta s mata
aut or zoun to sÔnolo A =f ( ) () ()
f0 n ; f1 n ; f2 n ; :::; fN 1 n . An fk n den( )g ()
an kei sto A, tìte e nai dio me èna apì aut, dhlad fN n ( )= ( )
f0 n ; f 1 n ( )=
()
fN 1 n kai oÔtw kajex .
()
àstw x n h akolouj a diakritoÔ qrìnou, h opo a proèrqetai apì th deigmatolh-
y a tou ekjetikoÔ s mato ej!0 t se shme a ta opo a isapèqoun kat qronik diast -
mata sa me TÆ
x n ej!0 nTÆ ej (!0 TÆ )n
( )= =
An 0 e nai h yhfiak kuklik suqnìthta tìte x n ( ) = ej 0 n . Sugkr nonta ti dÔo ek-
()
frsei tou x n èqoume th sqèsh metaxÔ analogik kai yhfiak kuklik suqnìth-
ta
0 = !0 TÆ (1.4.11)
To analogikì s ma x t ( ) = os( )
!0 t e nai gia kje tim th !0 periodikì. To s -
ma diakritoÔ qrìnou x n ( ) = os( )
0 t e nai periodikì mìno ìtan 0 = k=N 2 =
2 = ()
!0 TÆ = k=N . ParathroÔme ìti to s ma diakritoÔ qrìnou x n , to opo o prokÔptei
apì to periodikì analogikì s ma x t( ) = os( )
!0 t , e nai periodikì an o lìgo th pe-
riìdou deigmatolhy a TÆ pro thn per odo T0 tou analogikoÔ s mato e nai rhtì
arijmì , dhlad
TÆ
T0
= Nk (1.4.12)
Prgmati, an deigmatolhpt soume to periodikì analogikì s ma x t t( ) = os(2 )
me periìdo TÆ = 1 12
= , prokÔptei to s ma diakritoÔ qrìnou x n ( ) = os(2 12)
n= ,
(blèpe Sq ma 1.19a), to opo o e nai periodikì, afoÔ ikanopoie tai h sunj kh (1.4.12).
ParathroÔme ìti h per odo tou analogikoÔ s mato , grammoskiasmènh perioq , sum-
p ptei me thn per odo tou s mato diakritoÔ qrìnou. Se anloga sumpersmata
katal goume ìtan h per odo deigmatolhy a e nai TÆ = 4 31
= (blèpe Sq ma 1.19b).
Sthn per ptwsh aut h per odo tou s mato diakritoÔ qrìnou e nai sh me tèsseri
periìdou tou analogikoÔ s mato . Ant jeta, an h per odo e nai TÆ = , to = 1 12
() ()
s ma diakritoÔ qrìnou x n , se ant jesh me to x t , den e nai periodikì (blèpe Sq ma
1.19g).
18 Eisagwg sta S mata Keflaio 1
••• •••
x(n)=cos (2ð n(
12
(á)
••• •••
x(n)=cos (8ð n(
12 31 31
(â)
••• •••
x(n)=cos ( 12
1 n
(
(ã)
Sq ma 1.19 Hmitonoeid s mata diakritoÔ qrìnou, (a) kai (b) periodik (g) mh periodikì.
1.4.4 H sunrthsh monadia ou b mato suneqoÔ qrìnou
M a eidik morf s mato e nai h sunrthsh monadia ou b mato suneqoÔ qrìnou, h
opo a or zetai w
u(t) = 0; t<0
1; t>0 (1.4.13)
kai èqei th morf tou Sq mato 1.20a.
u(t) uÄ(t)
1 1
0 t 0 Ä t
(á) (â)
Sq ma 1.20 (a) H sunrthsh monadia ou b mato (suneqoÔ qrìnou) kai (b) h suneq
prosèggish th sunrthsh monadia ou b mato .
()
H sunrthsh u t e nai asuneq kai den or zetai sto t = 0. àna llo trìpo
()
na doÔme th sunrthsh u t e nai w ìrio th
8
< 0; t < 0
u (t) = 1 t; 0 < t <
:
(1.4.14)
1; t
Enìthta 1.4 Stoiqei¸dh S mata 19
H sunrthsh u (t) fa netai sto Sq ma 1.20b. ParathroÔme ìti u (t) = lim!0 u(t).
1.4.5 H kroustik sunrthsh suneqoÔ qrìnou sunrthsh dèlta
Idia tero endiafèron parousizei h pargwgo th sunrthsh u (t)
8
du (t) < 0; t < 0
Æ (t) =: 1
dt ; 0 < t < (1.4.15)
0; t >
h opo a den or zetai sta shme a asunèqeia 0 kai kai fa netai sto Sq ma 1.21a.
äÄ(t) ä(t)
1 1
Ä
0 Ä t 0 t
(á) (â)
Sq ma 1.21 (a) H pargwgo th sunrthsh u(t) kai (b) h sunrthsh dèlta Æ(t).
()
ParathroÔme ìti to embadì th Æ t e nai so me th monda gia kje tim th
()
kai ìti h sunrthsh Æ t e nai sh me to mhdèn èxw apì to disthma t . ätan 0
!0 h qronik dirkeia tou palmoÔ elatt¸netai, kai auxnetai to plto tou, en¸
to embadì paramènei stajerì kai so me th monda.
Sto ìrio !0 to eÔro tou palmoÔ te nei sto mhdèn kai to plto te nei sto
peiro. Or zoume th sunrthsh Æ t w ()
Æ(t) = lim Æ (t) (1.4.16)
!0
H Æ(t) onomzetai sunrthsh dèlta
sunrthsh dirac kroustik sunrthsh.
ParathroÔme ìti h kroustik sunrthsh e nai sh me thn pargwgo th sunrthsh
monadia ou b mato .
du(t)
Æ(t) = (1.4.17)
dt
àna genikìtero orismì th Æ(t) e nai
Æ(t) = 0; t 6= 0 (1.4.18)
kai Z 1
x(t) Æ(t t0 ) dt = x(t0 ) (1.4.19)
1
20 Eisagwg sta S mata Keflaio 1
ìpou x(t) e nai suneq sunrthsh sto t0 . H (1.4.19) anafèretai kai w idiìthta
ol sjhsh th kroustik sunrthsh . Apì thn (1.4.19) parathroÔme ìti, an x t , ( )=1
Z 1
Æ(t) dt = 1 (1.4.20)
1
()
H sunrthsh Æ t grafik paristnetai ìpw sto Sq ma 1.21b. To mètro tou dia-
nÔsmato , to opo o qrhsimopoioÔme gia na apod¸soume th sunrthsh dèlta, epilègetai
¸ste na e nai so me to embadì th , dhlad so me 1.
M a basik idiìthta th sunrthsh dèlta e nai
Æ(t) = Æ( t)
M a llh qr simh idiìthta th Æ(t) e nai h idiìthta th allag kl maka qrìnou
1
Æ( t) = Æ(t); >0 (1.4.21)
Pardeigma 1.4.1
Na exetsete an ta paraktw s mata e nai periodik ìqi. An to s ma e nai periodikì,
na upologiste h jemeli¸dh suqnìtht tou.
1. x(t) = 3 os(5t + =4)
2. x(t) = 2ej(t 1)
x(t) = 1
P (t 2n)2
3. n= 1 e
4. x(t) = ( os(2t))u(t)
LÔsh
1. Exetzoume an uprqei jetikì arijmì T gia ton opo o x(t + T ) = x(t) gia kje
tim t
tou qrìnou . àtsi èqoume
x(t + T ) = x(t) =) 3 os(5t + 5T + =4) = 3 os(5t + =4) (1.4.22)
Gnwr zoume ìmw ìti an os = os , tìte ' = 2k, kai epomènw prokÔptei
ìti:
(5t + 5T + =4) + (5t + =4) = 2k =) T = (2k)=5 2t =10 (1.4.23)
(5t + 5T + =4) (5t + =4) = 2k =) T = (2k)=5 (1.4.24)
Apì thn (1.4.23) den prokÔptei stajer tim gia thn per odo. Apì thn (1.4.24)
parathroÔme ìti to s ma e nai periodikì me jemeli¸dh per odo T0 = 2=5 kai
jemeli¸dh suqnìthta f0 = 5=2.
Enìthta 1.4 Stoiqei¸dh S mata 21
2. Me ìmoio trìpo èqoume
x(t + T ) = x(t) ) 2ej(t+T 1) = 2ej(t 1) ) ejT =1
os(T ) + j sin(T ) = 1 ) T = 2k ) T = 2k
ParathroÔme ìti to s ma e nai periodikì me jemeli¸dh per odo T0 = 2 kai
jemeli¸dh suqnìthta f0 = 1=2.
3. H qronik metatopismènh kat T morf tou s mato
1
X
e (t 2n)
2
x(t) = (1.4.25)
n= 1
x(t + T ) = 1 e (t+T 2n) jètonta T = 2k èqoume x(t + 2k) =
P 2
e nai
P1 n = 1
n= 1 e
[(t 2(n k)℄2 me allag metablht n k = m to s ma apokt th
morf
X1
e (t 2m)
2
x(t + 2k) = (1.4.26)
m= 1
Sugkr nonta ti (1.4.25) kai (1.4.26) parathroÔme ìti x(t + 2k ) = x(t), ra to
s ma e nai periodikì me per odo T = 2k . H jemeli¸dh per odo tou s mato
e nai T0 = 2 kai h jemeli¸dh suqnìthta f0 = 1=2.
4. ParathroÔme ìti to s ma
x(t) = ( os(2t))u(t) = 0os(2
;
t); t>0
alli¸
(1.4.27)
den e nai periodikì.
Pardeigma 1.4.2
An x(t) e nai to s ma pou d netai sto Pardeigma 1.3.1 (1.3.5), na upologistoÔn ta
1. y1 (t) = x(t)u(t)
2. y2 (t) = x(t)Æ (t 1)
x(t) y1(t) y2(t)
2 2 2
2 t+2 2- t 2- t 1
-1 0 1 2 t -1 0 1 2 t -1 0 1 2 t
Sq ma 1.22 Grafikè parastsei gia ta s mata x(t), y1 (t) kai y2 (t).
LÔsh
1. Me th bo jeia th sqèsh orismoÔ th sunrthsh u(t) parathroÔme
ìti to s ma
y1 (t) e nai to aitiatì tm ma tou s mato
2
x(t), dhlad , y1 (t) = 0; t; 0t<2 .
alli¸
22 Eisagwg sta S mata Keflaio 1
2. To s ma y2 (t) e nai h sunrthsh dèlta qronik metatopismènh kat 1 me plto
x(1) = 1, dhlad , y2 (t) = x(1)Æ(t 1).
Sto Sq ma 1.22 eikon zontai ta s mata x(t), y1 (t) kai y2 (t).
Pardeigma 1.4.3
Na anaptuqje èna tuqa o analogikì s ma se jroisma apì olisj sei th kroustik
sunrthsh .
x(t) x(t)
x(t)
t -Ä 0 Ä 2Ä kÄ t
(a1)
(a2)
x(-2Ä)äÄ(t+2Ä)Ä
x(-2Ä)
-2Ä 0
(â)
t
x(-Ä)äÄ(t+Ä)Ä
x(Ä)
-Ä 0
(ã) t
x(0)äÄ(t)Ä
x(0)
0 Ä t
(ä)
Sq ma 1.23 Anptugma s mato suneqoÔ qrìnou se olisj sei th kroustik sunrthsh .
LÔsh àstw to tuqa o analogikì s ma x(t) tou Sq mato 1:23a1. JewroÔme to kli-
makwt morf s ma x^(t), tou Sq mato 1:23a2, to opo o prosegg zei to s ma x(t).
Upenjum zoume ìti h Æ(t t0) e nai èna palmì me arq th qronik stigm t0 , me
dirkeia kai plto so me èna. O palmì , tou Sq mato 1.23b, me arq th qronik
stigm t = 2 kai Ôyo so me thn tim tou s mato thn dia qronik stigm , x( 2)
ekfrzetai apì thn
x( 2)Æ(t + 2) (1.4.28)
Me anlogo trìpo ekfrzontai kai oi lloi palmo , oi opo oi prosdior zontai apì to
Enìthta 1.4 Stoiqei¸dh S mata 23
s ma x^(t) (blèpe Sq ma 1.23g kai d). àtsi to s ma x^(t) ekfrzetai apì thn ex swsh
1
X
x^(t) = x(k)Æ (t k) (1.4.29)
k= 1
An ! 0, to s ma x^(t) ! x(t), ètsi
1
X
x(t) = lim
!0
x(k)Æ (t k) (1.4.30)
k= 1
ätan ! 0, to k g netai h suneq metablht , to parapnw jroisma grfetai
w olokl rwma kai epeid Æ(t) = lim!0 Æ(t), to s ma x(t) d netai apì thn ex swsh
Z 1 Z 1
x(t) = x( )Æ(t ) d = x( )Æ( t) d (1.4.31)
1 1
To pardeigma autì anadeiknÔei mia fusik proèktash th (1.4.19) kai ja ma fane
qr simh sto epìmeno keflaio.
Pardeigma 1.4.4
D netai to s ma tou Sq mato 1.24a. Na sqedisete ta s mata x+ (t) = x(t)u(t) kai
x (t) = x( t)u(t). Parathr +
ste ìti ta x (t) kai x (t) e nai aitiat s mata kai to
x(t) mpore na ekfraste w
mh aitiatì s ma jroisma twn dÔo aut¸n shmtwn me th
bo jeia th sqèsh
x(t) = x+ (t) + x ( t) (1.4.32)
x(t) x(-t)
1 1
-2 -1 0 1 t -1 0 1 2 t
(a) ( â)
x(t) u(t) x(-t)u(t)
1 1
-2 -1 0 1 t -1 0 1 2 t
(ã) (ä)
Sq ma 1.24 Ta s mata tou Parade gmato 1.4.4.
LÔsh Sto Sq ma 1.24a eikon zetai to s ma x(t), kai sto Sq ma 1.24b h anklas tou
x( t). To aitiatì tm ma tou s mato x(t), x+ (t) = x(t)u(t) eikon zetai sto Sq ma
1.24g kai to aitiatì tm ma tou s mato x( t), x (t) = x( t)u(t) eikon zetai sto
Sq ma 1.24d.
24 Eisagwg sta S mata Keflaio 1
ParathroÔme apì ta diagrmmata ìti to s ma x(t) e nai so me to jroisma tou s mato
x+ (t) kai th anklash x ( t) tou s mato x (t), dhlad , e nai
x(t) = x+ (t) + x ( t)
1.4.6 O orjog¸nio palmì
O orjog¸nio palmì monadia ou pltou me monadia a qronik dirkeia sumbol zetai
w ( )
t kai d netai apì to majhmatikì tÔpo
tj < 12
(t) = 10;; jalli¸ (1.4.33)
O orjog¸nio palmì ekfrzetai w diafor dÔo katllhla olisjhmènwn bhmatik¸n
sunart sewn. Prgmati,
(t) = u u t
1 t+
1
2 2 (1.4.34)
Sto Sq ma 1.25 uprqei h grafik parstash tou A(t=T1 ), dhlad , enì orjog¸niou
palmoÔ qronik dirkeia T1 kai pltou A.
A Ð ( Tt1 (
A
Sq ma 1.25 O orjog¸nio palmì qronik
T1 0 T1 t
2 2 dirkeia T1 kai pltou A.
1.4.7 O trigwnikì palmì
O trigwnikì palmì monadia ou pltou sumbol zetai w (t) kai d netai apì to
majhmatikì tÔpo
8
< t + 1; 1t<0
(t) = : t + 1; 0t<1 (1.4.35)
0; alli¸
Sto Sq ma 1.26 uprqei h grafik parstash trigwnikoÔ palmoÔ.
Enìthta 1.4 Stoiqei¸dh S mata 25
Ë(t)
1
-1 0 1 t Sq ma 1.26 O trigwnikì palmì (t).
1.4.8 H sunrthsh kl sh
H sunrthsh kl sh sumbol zetai w r(t) kai d netai apì to majhmatikì tÔpo
r(t) = t; t 0
0; t < 0 (1.4.36)
H sunrthsh kl sh ekfrzetai kai
r(t) = tu(t) (1.4.37)
Sto Sq ma 1.27 uprqei h grafik parstash th sunrthsh kl sh .
r(t)
0 t Sq ma 1.27 H sunrthsh kl sh r(t).
1.4.9 H sunrthsh pros mou
H sunrthsh pros mou sumbol zetai w sgn(t) kai d netai apì to majhmatikì tÔpo
sgn(t) = 1;1; tt >< 00 (1.4.38)
Sto Sq ma 1.28 uprqei h grafik parstash th sunrthsh pros mou.
sgn(t)
1
t
-1 Sq ma 1.28 H sunrthsh pros mou sgn(t).
26 Eisagwg sta S mata Keflaio 1
1.4.10 Monadia a bhmatik akolouj a - Monadia o b ma diakritoÔ qrìnou
H monadia a bhmatik akolouj a to monadia o b ma diakritoÔ qrìnou lambnetai apì
th sunrthsh monadia ou b mato , an antikatast soume to t me to n kai upolog soume
autì mìno gia akèraie timè tou qrìnou. àtsi, èqoume
u(n) = 0; n<0
1; n0 (1.4.39)
Sto Sq ma 1.29 èqoume th grafik parstash tou monadia ou b mato diakritoÔ
qrìnou.
u(n)
-4 -2 0 2 4 n
Sq ma 1.29 H monadia a bhmatik akolouj a.
1.4.11 To monadia o de gma - Kroustik akolouj a
To monadia o de gma h kroustik akolouj a or zetai me th sqèsh
Æ(n) = 1; n=0
0; alli¸
(1.4.40)
Sto Sq ma 1.30 èqoume th grafik parstash th kroustik akolouj a . H monadi-
a a bhmatik akolouj a sundèetai me th kroustik akolouj a me th sqèsh
1
X
u(n) = Æ (n k ) (1.4.41)
k=0
en¸ h kroustik akolouj a sundèetai me th monadia a bhmatik akolouj a me th sqèsh
Æ(n) = u(n) u(n 1) (1.4.42)
ä(n)
-4 -2 0 2 4 n
Sq ma 1.30 H kroustik akolouj a.
Pardeigma 1.4.5
Na anaptuqje to s ma diakritoÔ qrìnou x(n), tou Sq mato 1.31a, se jroisma apì
olisj sei monadia ou de gmato .
Enìthta 1.4 Stoiqei¸dh S mata 27
x(n)
-2 2 4 n (a)
-4 0
x (-2)ä(n+2)
n (â)
-4 -2 0 2 4
x (-1)ä(n+1)
n (ã)
-4 -2 0 2 4
x(0) ä(n)
(ä)
-4 -2 0 2 4 n
x(1) ä(n-1)
n (å)
-4 -2 0 2 4
x(2) ä(n-2)
n (óô)
-4 -2 0 2 4
Sq ma 1.31 Anptugma s mato diakritoÔ qrìnou se olisj sei monadia ou de gmato .
LÔsh Upenjum zoume ìti Æ(n n0 ) e nai h akolouj a th opo a ìla ta stoiqe a e nai
mhdenik, ektì apì to stoiqe o gian = n0 , to opo o e nai so me èna. H akolouj a tou
Sq mato 1.31b, th opo a ìla ta stoiqe a e nai sa me mhdèn, ektì apì to stoiqe o th
qronik stigm n= 2, to opo o e nai so me x( 2), ekfrzetai apì th
x( 2)Æ(n + 2)
Me anlogo trìpo ekfrzontai kai oi akolouj e sta Sq mata 1.31g èw st. àtsi to
s ma diakritoÔ qrìnou x(n) tou Sq mato 1.31a, mpore na ekfraste w
1
X
x(n) = x(k)Æ(n k) (1.4.43)
k= 1
SÔnoyh Kefala ou
Sto keflaio autì dìjhke o orismì th ènnoia “s ma" kai h majhmatik th
èkfrash. Katatxame ta s mata se trei kathgor e , ta analogik s mata, ta s mata
diakritoÔ qrìnou kai ta yhfiak s mata. Perigryame ti basikè idiìthte pou èqoun
ta analogik kai yhfiak s mata, pou apoteloÔn to antike meno autoÔ tou bibl ou,
kai anafèrame ti metabolè pou uf statai èna s ma w pro to qrìno.
28 Eisagwg sta S mata Keflaio 1
Ep sh , sto keflaio autì perigryame to migadikì ekjetikì s ma kai to hmito-
noeidè s ma. Anafèrame dÔo shmantikè sunart sei , th sunrthsh monadia ou b -
mato kai th sunrthsh dèlta. Tèlo , anafèrame th sunrthsh tou orjog¸niou pal-
moÔ, thn trigwnik sunrthsh, th sunrthsh kl sh kai th sunrthsh pros mou.
1.5 PROBLHMATA
1.1 Poi apì ta s mata e nai periodik;
1. x1 (t) = sin(10t)
2. x2 (t) = sin(20t)
3. x3 (t) = sin(31t)
4. x4 (t) = x1 (t) + x2 (t)
5. x5 (t) = x1 (t) + x3 (t)
1.2 D netai to s ma, 8
< 2t + 2; 1 t < 0
x(t) = 2 t; 0 t < 2
:
0; alli¸
Na sqedisete ta s mata
1. y1 (t) = x(t + 1)
2. y2 (t) = x(2t)
3. y3 (t) = x(t=2)
4. y4 (t) = x(1 t)
5. y5 (t) = x(2t 1)
1.3 Na sqediasjoÔn ta s mata
1. x1 (t) = (2t + 6)
2. x2 (t) = (2t 1)
3. x3 (t) = r( 0; 5t + 2)
4. x4 (t) = sin (2(t 3))
1.4 D netai to s ma 8
< t; 0t<1
x(t) = t + 2; 1t<2
:
0; alli¸
Na sqediasjoÔn ta s mata
Enìthta 1.5 Probl mata 29
y1 (t) = xo (t)u(t)
1.
2. y2 (t) = xe (t)u(t)
ìpou xo (t) e nai to perittì mèro tou s mato x(t) kai xe (t) e nai to rtio mèro
tou.
1.5 Na exetsete an ta paraktw s mata e nai periodik ìqi. An to s ma e nai
periodikì, na upologiste h jemeli¸dh suqnìtht tou.
1. x1 (t) = 2 os(3t + =4)
2. x2 (t) = ej (t 1)
3. x3 (n) = os(8n=7 + 2)
4. x4 (n) = ej (n=8 )
5. x5 (n) = sin2 (t =6)
6. x6 (n) = os(n2 =8)
7. x7 (n) = os(n=4) os(n=4)
8. x8 (n) = 2 os(n=4) + sin(n=8) 2 os(n=2 =6)
P
9. x9 (t) = 1 n= 1 e
(t 3n)2
1.6 Na exetaste an ta s mata e nai energeiak s mata s mata isqÔo kai na
upologiste h enèrgei tou h isqÔ tou .
1. x1 (t) = e t u(t); >0
2. x2 (t) = A os(!0 t + )
3. x3 (t) = Aej (!0 t+)
1.7 Na sqediasjoÔn ta s mata.
1. x(t) = 2Æ (t) + 3Æ (t 1) + 5Æ(t 2) = 1
2. y(t) = u(t + 2) u(t 1)
1.8 D netai to s ma x(t) tou Sq mato 1.32.
x(t) g(t) y(t) z(t)
1 1
1 1
-1 0 1 t -4 -3 -2 -1 0 t -2 -1 0 1 t -4 -3 -2 -1 0 t
Sq ma 1.32 Ta s mata tou Probl mato 1.8.
30 Eisagwg sta S mata Keflaio 1
1. Na ekfrsete to s ma x(t) me th bo jeia th bhmatik sunrthsh u(t).
2. Na ekfrsete ta s mata g (t), y (t) kai z (t) tou Sq mato 1.32 me th bo jeia
()
tou s mato x t .
1.9 Qrhsimopoi¸nta ti metatropè s mato w pro to qrìno, na g nei h grafik
parstash se sunrthsh me to qrìno tou s mato x t t ìpou( ) = (2 3)
t ( )
e nai o trigwnikì palmì
1.10 D netai to s ma x(t) tou Sq mato 1.33.
x(t)
2Volts
1
-1 0 1 2 3 t (sec) Sq ma 1.33 To s ma tou Probl mato 1.10.
1. Na ekfraste to s ma x(t) me th bo jeia th sunrthsh kl sh kai tou
orjog¸niou palmoÔ.
2. Na breje h enèrgeia tou s mato .
1.11 Na breje h pargwgo twn shmtwn
1. tou orjog¸niou palmoÔ (t)
2. tou trigwnikoÔ palmoÔ (t)
3. th suntrhsh pros mou sgn(t).
kai na g nei h grafik parstash twn ant stoiqwn parag¸gwn.
Bibliograf a
1.1 S. Jeodwr dh , K. Mpermper dh , L. Kof dh , “Eisagwg sth Jewr a Shmtwn
kai Susthmtwn”, Tupwj tw - Gi¸rgo Dardanì , Aj na 2003.
1.2 A. Mrgarh , “S mata kai Sust mata SuneqoÔ kai DiakritoÔ Qrìnou ”, Ekdì-
sei Tziìla 2012.
1.3 S. Haykin, B. Veen, “Signal and Systems”, John & Wiley Sons, Inc. 2003
1.4 A. V. Oppenheim, R. S. Willsky, I. T. Young, “Signal and Systems”, Prentice -
Hall Inc., N. Y., 1983.
1.5. R. E. Siemer, W. H. Tranter, D. R. Fannin, “Signals & Systems Continuous and
Discrete”, Prentice Hall, 1998.
ÊÅÖÁËÁÉÏ 2
ÅÉÓÁÃÙÃÇ ÓÔÁ ÓÕÓÔÇÌÁÔÁ
Skopì tou kefala ou autoÔ e nai na d¸sei m a genik eikìna tou ti e nai sÔsthma,
na katatxei ta sust mata anloga me ton arijmì kai to e do twn epitrepìmenwn
eisìdwn kai exìdwn kai na perigryei ti basikè idiìthtè tou . Sto keflaio autì ja
perigrafe h mèjodo prosdiorismoÔ th exìdou enì sust mato , ìtan gnwr zoume thn
e sodì tou, kaj¸ kai thn èxodì tou ìtan h e sodì tou diege retai apì th sunrthsh
dèlta. Sth sunèqeia ja de xoume ìti se m a eidik kathgor a susthmtwn, ìtan h
e sodo e nai to migadikì ekjetikì s ma kuklik suqnìthta !0 , tìte kai h ant stoiqh
èxodo e nai, ep sh , èna migadikì ekjetikì s ma me thn dia kuklik suqnìthta, to
plto kai h fsh tou opo ou èqoun uposte m a allag pou prokale to sÔsthma.
Tèlo , ja efarmìsoume ta parapnw se apl hlektrik kai mhqanik sust mata.
Eisagwg
Sto prohgoÔmeno keflaio asqolhj kame me basikoÔ orismoÔ kai ènnoie pou
aforoÔn sta s mata. Sto keflaio autì ja asqolhjoÔme me ta sust mata. To
perieqìmeno th ènnoia tou sust mato e nai genikì. Suqn qrhsimopoioÔme th lè-
xh “sÔsthma” gia na anaferjoÔme se èna “sÔnolo dom¸n” kai “leitourgi¸n”. Eme
ìmw ja estisoume to endiafèron ma se m a eidik shmas a th ènnoia tou susth-
mto , aut n pou èqei mesh sqèsh me ta s mata. Sugkekrimèna, sth Jewr a Susth-
mtwn sÔsthma e nai h ontìthta eke nh pou epexergzetai, metabllei, katagrfei,
metad dei s mata. Gia pardeigma, èna sÔsthma yhfiak katagraf qou meta-
trèpei èna akoustikì s ma se mia seir apì arijmoÔ (bits) tou opo ou katagrfei,
p.q., se optikì d sko. Ant jeta, to CD player e nai èna sÔsthma to opo o diabzei
tou arijmoÔ , oi opo oi e nai apojhkeumènoi ston optikì d sko, kai anapargei to
hqhtikì s ma to opo o mporoÔme na akoÔsoume. àna sÔsthma epikoinwn a metafèrei
plhrofor a, p.q. to s ma fwn , apì èna shme o tou q¸rou, pou lègetai phg , se èna
llo shme o, pou e nai o proorismì qr sh th .
32 Eisagwg sta Sust mata Keflaio 2
2.1 ORISMOS SUSTHMATOS - KATHGORIES SUSTHMATWN
W sÔsthma or zoume, thn ontìthta eke nh h opo a epenerg¸nta se èna s ma x(t) èqei
w apotèlesma èna llo s ma y(t).
Apì majhmatik poyh, èna sÔsthma mpore na
jewrhje w èna metasqhmatismì S ()
pou metasqhmat zei èna s ma, x t , se èna llo
s ma y t ( ) = Sf ( )g
x t . H drsh enì sust mato perigrfetai sqhmatik sto Sq ma
()
2.1. To arqikì s ma x t , to opo o diege rei to sÔsthma, lègetai s ma eisìdou apl
e sodo tou sust mato , en¸ to apotèlesma sth diadikas a diègersh , dhlad to s ma
()
y t lègetai s ma exìdou apl èxodo tou sust mato .
Åßóïäïò Óýóôçìá ¸îïäïò Sq ma 2.1 Sqhmatik perigraf
x(t) S y(t) sust mato .
O parapnw orismì e nai polÔ genikì kai mpore na perigryei poll fusik
sust mata, ìpw : hlektrik kukl¸mata (p.q. radiìfwno), mhqanik sust mata (p.q.
autok nhto, èna rompotikì braq ona), èna epikoinwniakì kanli, ènan hlektronikì
upologist kai poll lla.
Anloga me ton arijmì kai to e do twn epitrepìmenwn eisìdwn kai exìdwn, ta
sust mata diakr nontai se:
1. Sust mata m a eisìdou - m a exìdou SISO (Single-Input, Single-Output). Ta
pio apl sust mata m a eisìdou - m a exìdou e nai o bajmwtì pollaplasi-
ast y t()= ()
ax t kai to sÔsthma kajustèrhsh y t x t t0 . ()= ( )
2. Sust mata me pollè eisìdou kai m a èxodo pou e nai gnwst w sust mata
MISO (Multi-Input, Single-Output). àna tètoio sÔsthma e nai o ajroist dÔo
perissìterwn shmtwn y(t) = x1(t) + x2 ( )t kai o pollaplasiast y t ()=
( ) ( )
x1 t x2 t .
3. Sust mata me pollè eisìdou kai pollè exìdou , gnwst w sust mata MIMO
(Multi-Input, Multi -Output).
Anloga me th fÔsh twn epitrepìmenwn eisìdwn kai exìdwn, ta sust mata diakr -
nontai w ex
1. Sust mata suneqoÔ qrìnou analogik sust mata, ìtan ta s mata eisìdou kai
ta s mata exìdou e nai analogik s mata. ätan ta s mata eisìdou kai exìdou
e nai s mata diakritoÔ qrìnou, tìte ta sust mata qarakthr zontai w sust -
mata diakritoÔ qrìnou.
2. Aitiokratik sust mata,ìtan ta s mata eisìdou kai exìdou e nai aitiokratik
s mata. ätan ta s mata eisìdou kai exìdou e nai stoqastik s mata, ta sust -
mata qarakthr zontai w stoqastik sust mata.
Enìthta 2.1 Orismì Sust mato - Kathgor e Susthmtwn 33
àna pardeigma analogikoÔ kai sugqrìnw stoqastikoÔ sust mato e nai to gnw-
stì kÔklwma anìrjwsh kai exomlunsh th enallassìmenh tsh tou Sq mato
2.2, sto opo o to s ma eisìdou e nai h efarmozìmenh tsh eis t kai s ma exìdou h ()
anaptussìmenh tsh sta kra th ant stash R, ex t . ()
D
õåéó(t) R C õåî(t)
Sq ma 2.2 Pardeigma analogikoÔ
sust mato .
àna sÔsthma epikoinwn a e nai èna stoqastikì sÔsthma efìson h e sodì tou kai
h èxodì tou e nai stoqastik s mata.
Uprqoun, ep sh , sust mata ta opo a metasqhmat zoun analogikè eisìdou se
diakritè exìdou kai antijètw . Tètoia sust mata e nai gnwst w ubridik sust -
mata. O analogoyhfiakì metatropèa (A/D Analog to Digital converter), o opo o
metatrèpei èna analogikì s ma se yhfiakì kai o yhfioanalogikì metatropèa (D/A),
e nai ubridik sust mata.
Sth sunèqeia ja prosdior soume thn ex swsh h opo a perigrfei th sqèsh metaxÔ
tou s mato eisìdou kai tou s mato exìdou enì hlektrikoÔ kai enì mhqanikoÔ
sust mato .
Se èna hlektrikì kÔklwma gnwr zoume ìti h tsh R t sta kra mia wmik ()
ant stash R, pou diarrèetai apì reÔma èntash i t , e nai ()
R (t) = R i(t) (2.1.1)
H tsh L (t) sta kra phn ou, autepagwg L, pou diarrèetai apì reÔma èntash
()
i t e nai
di(t)
L (t) = L (2.1.2)
dt
kai h èntash tou reÔmato fìrtish enì puknwt , qwrhtikìthta C e nai
d C (t)
i(t) = C (2.1.3)
dt
ìpou C (t) e nai h tsh sta kra tou puknwt .
Pardeigma 2.1.1
Na diatupwje h sqèsh th tsh eisìdou eis (t) kai th tsh exìdou (t)
ex
gia to
kÔklwma tou Sq mato 2.3.
LÔsh Efarmìzonta to deÔtero kanìna tou Kirchhoff sto brìqo tou kukl¸mato
èqoume
Ri(t) + (t) =
ex eis (t) (2.1.4)
34 Eisagwg sta Sust mata Keflaio 2
R
i(t)
õåéó(t) C õåî(t)
Sq ma 2.3 To kÔklwma tou
Parade gmato 2.1.1.
Lambnonta upìyh thn (2.1.3), h (2.1.4) grfetai
d (t)
RC
dt
ex
+ (t) = ex eis (t) (2.1.5)
H (2.1.5) e nai m a diaforik ex swsh me stajeroÔ suntelestè , h opo a perigrfei th
sqèsh metaxÔ eisìdou kai exìdou tou sust mato . H ex swsh èqei th genik morf
dy(t)
dt
+ y(t) = x(t) (2.1.6)
H txh tou sust mato prosdior zetai apì th megalÔterh pargwgo th exìdou y(t),
h opo a emfan zetai sth diaforik ex swsh. àtsi, h (2.1.6) perigrfei èna sÔsthma
pr¸th txh .
Se èna mhqanikì sÔsthma gnwr zoume ìti isqÔei o jemeli¸dh nìmo th mhqanik
X X d2 x
Fk = ma Fk = m 2 (2.1.7)
k k
dt
ìpou Fk e nai oi dunmei oi opo e askoÔntai sth mza m, x h jèsh th kai a h
epitquns th .
O nìmo tou Hook, o opo o d nei to mètro th dÔnamh Fel pou aske tai apì èna
elat rio, w sunrthsh th metabol tou m kou tou kat x, e nai
F el = kx (2.1.8)
ìpou k e nai h stajer tou elathr ou.
Ep sh , gnwr zoume ìti h dÔnamh h opo a antidr sthn k nhsh enì s¸mato (dÔ-
namh apìsbesh ) e nai anlogh th taqÔtht tou, , kai d netai apì th sqèsh
dx
F ap = b F ap = b
dt
(2.1.9)
ìpou b e nai h stajer apìsbesh tou sust mato .
Pardeigma 2.1.2
Na diatupwje h sqèsh metaxÔ efarmozìmenh dÔnamh F (t) kai metatìpish x(t) gia
th mza m tou Sq mato 2.4.
Enìthta 2.2 Sundèsei Susthmtwn 35
x(t) k
F(t) Fåë(t)
m
Sq ma 2.4 To mhqanikì sÔsth-
Fáð(t)
ma gia to Pardeigma 2.1.2.
LÔsh Efarmìzonta to jemeli¸dh nìmo th mhqanik gia th mza m èqoume
F F ap Fel
= ma (2.1.10)
Qrhsimopoi¸nta ti (2.1.8) kai (2.1.9) pa rnoume
dx d2 x d2 x b dx k 1
F b kx = m 2 2 + + x= F (2.1.11)
dt dt dt m dt m m
h opo a e nai mia diaforik ex swsh me stajeroÔ suntelestè kai perigrfei to gram-
mikì talantwt me apìsbesh. Sth diaforik ex swsh (2.1.11) perièqetai h deÔterh
pargwgo th exìdou me apotèlesma, to sÔsthma, me e sodo th dÔnamh F kai èxodo
thn apomkrunsh x na e nai deÔterh txh .
2.2 SUNDESEIS SUSTHMATWN
Se pollè peript¸sei , h anlush enì polÔplokou sust mato dieukolÔnetai shma-
ntik an doÔme to sÔsthma w apotèlesma diasÔndesh ligìtero polÔplokwn susth-
mtwn.
Oi pio basikè sundèsei metaxÔ susthmtwn e nai h seiriak , h parllhlh, h
meikt kai h sÔndesh me anatrofodìthsh andrash (Sq ma 2.5).
H sqhmatik anaparstash dÔo susthmtwn ta opo a èqoun sundeje seiriak
fa netai sto Sq ma 2.5a. ParathroÔme ìti, ìtan dÔo sust mata 1 kai 2 sundèontai S S
S S
seiriak, me to sÔsthma 1 na prohge tai tou 2 , h èxodo tou 1 e nai e sodo tou S
S2 . M a shmantik diadikas a, h opo a sqet zetai me th seiriak sÔndesh, e nai h
antistrof sust mato , gia thn opo a ja mil soume sthn Enìthta 2.3.3.
H sqhmatik anaparstash dÔo susthmtwn, ta opo a èqoun sundeje parllh-
la, fa netai sto Sq ma 2.5b. ParathroÔme ìti h dia e sodo trofodote kai ta
dÔo sust mata. Aut leitourgoÔn tautìqrona kai oi dÔo epimèrou èxodoi ajro -
zontai kai pargoun thn èxodo th parllhlh sÔndesh twn dÔo susthmtwn. H
ulopo hsh enì sust mato , to opo o perigrfetai apì th sqèsh eisìdou-exìdou:
( )= ( )+ ( )
y t a x t x t t0 , g netai me thn parllhlh sÔndesh enì bajmwtoÔ sust -
mato kai enì sust mato pou prokale kajustèrhsh kat t0 .
Sth meikt sÔndesh, Sq ma 2.5g, èqoume ta sust mata 1 kai 2 , ta opo a èqoun S S
S
sundeje parllhla, kai to sÔsthma 3 , to opo o èqei sundeje seiriak.
36 Eisagwg sta Sust mata Keflaio 2
Åßóïäïò Óýóôçìá Óýóôçìá ¸îïäïò
x(t) S1 S2 y(t)
(á)
Óýóôçìá
S1
Åßóïäïò ¸îïäïò
x(t) y(t)
Óýóôçìá
S2
(â)
Óýóôçìá
S1
Åßóïäïò Óýóôçìá ¸îïäïò
x(t) S3 y(t)
Óýóôçìá
S2
(ã)
Åßóïäïò Óýóôçìá ¸îïäïò
x(t) S1 y(t)
Óýóôçìá
S2
(ä)
Sq ma 2.5 (a) Seiriak sÔndesh dÔo susthmtwn, (b) parllhlh sÔndesh dÔo susthmtwn
(g) meikt sÔndesh susthmtwn kai (d) sÔndesh me anatrofodìthsh.
To sÔsthma autìmath plo ghsh enì oq mato dèqetai w e sodo m a troqi
thn opo a jèloume na diagryei to ìqhma. To sÔsthma parakolouje thn troqi
pou èqei to ìqhma, sugkr nei th jèsh tou oq mato me thn epijumht jèsh, dhlad
prosdior zei to sflma kai, ìtan exwteriko pargonte prokaloÔn parèklish apì
thn prokajorismènh troqi, proba nei sti anagka e rujm sei , ¸ste to ìqhma na
akolouje thn prodiagegrammènh troqi. To parapnw sÔsthma mpore na parasta-
S S
je sqhmatik me ta sust mata 1 kai 2 ta opo a èqoun sundeje me anatrofodìthsh.
S S
Sto Sq ma 2.5d fa netai h sÔndesh twn 1 kai 2 me anatrofodìthsh. ParathroÔme
ìti anatrofodotoÔme thn èxodo tou sust mato S S
1 sto sÔsthma 2 kai, afoÔ thn
epexergastoÔme me ton elegkt 2 , S
thn sugkr noume me to s ma anafor x t kai ()
qrhsimopoioÔme to apotèlesma th sÔgkrish gia na odhg soume to sÔsthma 1 . Me S
S ()
lla lìgia, to sÔsthma 2 tropopoie thn èxodo y t me trìpo ¸ste na mpore na
sugkrije me to epijumhtì s ma eisìdou.
Enìthta 2.3 Idiìthte Susthmtwn 37
2.3 IDIOTHTES SUSTHMATWN
Sthn enìthta aut parousizontai merikè basikè idiìthte pou èqoun ta sust ma-
ta. Prin anafèroume ti idiìthte twn susthmtwn e nai skìpimo na perigryoume
m a basik ènnoia, h opo a pollè forè parale petai sta egqeir dia. Ja lème ìti
èna sÔsthma br sketai se katstash hrem a th qronik stigm t0 , en autì den èqei
uposte diègersh apì llo s ma gia kje qronik stigm t < t0 . Apì fusik poyh,
èna sÔsthma pou e nai se katstash hrem a se dedomènh qronik stigm t0 , shma nei
ìti den e qe apojhkeumènh enèrgeia th qronik stigm t t0 . =
2.3.1 Grammikìthta
àna sÔsthma pou e nai se arqik hrem a ja lègetai grammikì sÔsthma (GS), an, kai
() ()
mìno an, dojèntwn dÔo shmtwn x1 t kai x2 t isqÔei:
Sfa x1(t) + b x2(t)g = a Sfx1(t)g + b Sfx2(t)g (2.3.1)
ìpou a kai b stajerè . Dhlad h apìkrish tou sust mato se m a e sodo, pou e nai o
grammikì sunduasmì dÔo shmtwn, isoÔtai me ton ant stoiqo grammikì sunduasmì
twn apokr sewn tou sust mato sto kajèna apì ta s mata aut. Sto Sq ma 2.6
perigrfetai sqhmatik h idiìthta th grammikìthta dÔo susthmtwn.
Åßóïäïé Åßóïäïé
x1(t) a Ãñáììéêü a
x1(t)
¸îïäïò Óýóôçìá ¸îïäïò
Ãñáììéêü y(t) y(t)
Óýóôçìá
x2(t) Ãñáììéêü
x2(t) b Óýóôçìá b
Sq ma 2.6 Sqhmatik perigraf th grammikìthta enì sust mato .
H parapnw idiìthta genikeÔetai gia opoiod pote grammikì sunduasmì pepera-
smènou arijmoÔ shmtwn eisìdou. Gen keush th (2.3.1) odhge sthn akìloujh sqèsh:
S fPk ak xk (t)g = Pk ak yk (t) (2.3.2)
ìpou yk (t) e nai èxodo tou sust mato , ìtan h e sodo e nai to xk (t).
2.3.2 Aitiìthta
( )
àna sÔsthma e nai aitiatì, an h èxodì tou th qronik stigm t0 , y t0 , exarttai apì
ti timè tou s mato eisìdou, x t , gia t ()
t0 . Dhlad , gia kje s ma eisìdou x t , ()
()
h ant stoiqh èxodo y t exarttai mìno apì thn paroÔsa kai prohgoÔmene timè
38 Eisagwg sta Sust mata Keflaio 2
th eisìdou. Me lla lìgia, èna sÔsthma e nai aitiatì, an oi metabolè sthn èxodo
(apotèlesma) tou sust mato potè den prohgoÔntai twn metabol¸n pou epiteloÔntai
sthn e sodo tou sust mato (ait a).
Ta sust mata ta opo a perigrfontai apì ti exis¸sei
y(t) = a x(t); y(t) = b x(t 1) kai y(t) =
1 Z t
x( )d (2.3.3)
C 1
e nai aitiat, en¸ to sÔsthma diakritoÔ qrìnou upologismoÔ mèsh tim pou peri-
grfetai apì thn ex swsh
y(n) =
1 X M
2M + 1 k= M x(n k) (2.3.4)
e nai mh aitiatì sÔsthma.
2.3.3 Antistrèyima kai mh antistrèyima sust mata
àna sÔsthma lègetai antistrèyimo, an h gn¸sh th exìdou kajist efiktì ton upolo-
gismì tou s mato eisìdou. H diadikas a antistrof enì sust mato sun stataiS
ston prosdiorismì enì sust mato , to opo o sundeìmeno se seir me to sÔsthma , S
S
parèqei sthn èxodì tou to s ma eisìdou tou sust mato . H antistrof parousize-
tai se pollè efarmogè sti opo e e nai epijumht h afa resh th ep drash enì
sust mato pnw se èna s ma. Se èna epikoinwniakì sÔsthma, to opo o èqei stìqo thn
ankthsh tou metadidìmenou s mato apì to lambanìmeno s ma, o dèkth apotele
èna antistrofèa tou kanalioÔ, pou katapolem ta difora fainìmena diataraq¸n. Ta
sust mata ta opo a perigrfontai apì ti sqèsei
n
X
y(t) = x(t) kai y(n) = x(k) (2.3.5)
k= 1
e nai antistrèyima kai èqoun w ant strofa ta sust mata me sqèsei eisìdou-exìdou
1
y(t) = x(t) kai y(n) = x(n) x(n 1) (2.3.6)
ant stoiqa. Se ant jesh, to sÔsthma
y(t) = x2 (t) (2.3.7)
den e nai antistrèyimo, giat kje tim th exìdou mpore na proèrqetai apì dÔo di-
aforetikè timè th eisìdou.
Enìthta 2.3 Idiìthte Susthmtwn 39
2.3.4 Sust mata Statik kai Dunamik
àna sÔsthma kale tai statikì sÔsthma qwr mn mh, en gia kje s ma eisìdou h
ant stoiqh èxodo , gia kje qronik stigm , exarttai mìno apì thn tim th eisìdou
thn dia qronik stigm (Sq ma 27a). H wmik ant stash e nai èna pardeigma sust -
mato qwr mn mh, afoÔ h tsh sta kra th R t (èxodo ) kje qronik stigm ()
()
exarttai apì thn èntash tou reÔmato i t (e sodo ) apì thn opo a diarrèetai thn
dia qronik stigm .
R t R i t ()= () (2.3.8)
x(t) y(t)
Óôáôéêü
óýóôçìá
0 t0 t 0 t0 t
(a)
x(t) y(t)
Äõíáìéêü
óýóôçìá
0 t0 t 0 t0 t
(â)
Sq ma 2.7 H e sodo kai h èxodo (a) enì statikoÔ sust mato kai (b) enì dunamikoÔ
sust mato .
En èna sÔsthma den e nai statikì, kale tai dunamikì sÔsthma me mn mh (Sq ma
2.7b). O puknwt , an jewrhje w sÔsthma me èxodo thn tsh sta kra tou, C t , ()
()
kai e sodo to reÔma pou to fort zei i t , e nai èna sÔsthma me mn mh, afoÔ h tsh
kje qronik stigm e nai apotèlesma tou ìlou istorikoÔ th sunrthsh i t ()
C (t) =
1 Z t
i( )d (2.3.9)
C 1
2.3.5 Qronik Anallo wta Sust mata
àna sÔsthma lègetai qronik anallo wto (QA) (ametblhto) an, kai mìno an, qronikè
olisj sei tou s mato eisìdou metafrzontai se ant stoiqe qronikè olisj sei
()
sthn èxodo. Me lla lìgia, an y t e nai h èxodo se èna s ma eisìdou x t , tìte ()
( ) ( )
gia e sodo x t t0 pargetai h èxodo y t t0 . Dhlad , to s ma exìdou paramènei
to dio, anexrthta apì to poia qronik stigm diege roume thn e sodo. To mìno pou
uf statai e nai h ant stoiqh qronik metatìpish. Sto Sq ma 2.8 d netai èna pardeigma
shmtwn eisìdou-exìdou enì qronik anallo wtou sust mato .
40 Eisagwg sta Sust mata Keflaio 2
x(t) y(t)
×ñïíéêÜ
áíáëëïßùôï
0 t1 t óýóôçìá
0 t1 t
x(t-t0) y(t-t0)
×ñïíéêÜ
áíáëëïßùôï
0 t1+t0 t óýóôçìá 0 t1+t0 t
(a) (â)
Sq ma 2.8 (a) H e sodo kai (b) h èxodo enì sust mato qronik anallo wtou.
2.3.6 Eustjeia
àna sÔsthma lègetai ìti e nai FEFE eustajè (Eustjeia Fragmènh Eisìdou Frag-
mènh Exìdou) (Bounded Input Bounded Output (BIBO) stable), an kai mìnon an gia
kje fragmènh e sodo h èxodì tou paramènei fragmènh. Me lla lìgia, èna sÔsthma
lègetai FEFE - eustajè , an gia kje jetikì arijmì M1 < gia ton opo o isqÔei 1
jx(t)j M1 (2.3.10)
uprqei jetikì arijmì M2 < 1 gia ton opo o isqÔei
jy(t)j M2 (2.3.11)
ParathroÔme ìti h apa ths ma gia eustjeia enì sust mato taut zetai me thn
apa thsh ta s mata eisìdou kai exìdou na paramènoun peperasmèna se plto (Sq -
ma 2.9).
ÖñáãìÝíç ÖñáãìÝíç
åßóïäïò ÅõóôáèÝò Ýîïäïò
x(t) M1 óýóôçìá y(t) M1
(a)
ÖñáãìÝíç
åßóïäïò Ìç Ìç öñáãìÝíç
åõóôáèÝò Ýîïäïò
x(t) M1 óýóôçìá
(â)
Sq ma 2.9 (a) SÔsthma eustajè kai (b) sÔsthma mh eustajè , h èxodo te nei sto peiro.
Enìthta 2.4 Sqèsh MetaxÔ Eisìdou-Exìdou Sust mato 41
Pardeigma 2.3.1
To sÔsthma to opo o perigrfetai apì thn paraktw sqèsh eisìdou x(t) exìdou y(t)
d
y(t) = S fx(t)g = x(t) (2.3.12)
dt
anafèretai w diaforist . Na exetsete, an to sÔsthma e nai grammikì, qronik anal-
lo wto, aitiatì kai antistrèyimo.
LÔsh An to s ma x1 (t) e nai h e sodo tou diaforist , tìte h èxodo tou e nai h
pargwgo x_ 1 (t) tou s mato eisìdou. Omo w , an x2 (t) e nai to s ma eisìdou, h èxodo
e nai h pargwgo x _ 2 (t). An h e sodo tou sust mato e nai o grammikì sunduasmì
ax1 (t) + bx2 (t), tìte h èxodo e nai
d
dt 1
[ax (t) + bx2(t)℄ = a dtd x1(t) + b dtd x2 (t) (2.3.13)
ParathroÔme ìti to sÔsthma e nai grammikì. O diaforist e nai qronik anallo wto
sÔsthma, afoÔ
d d
x(t t0 ) = x(t) (2.3.14)
dt dt t=t t0
O diaforist e nai aitiatì sÔsthma afoÔ h èxodì tou exarttai mìno apì thn paroÔsa
tim th eisìdou tou. O diaforist den antistrèfetai, giat dÔo s mata ta opo a
diafèroun kat m a stajer èqoun thn dia pargwgo.
2.4 SQESH METAXU EISODOU - EXODOU SUSTHMATOS
Sthn enìthta aut ja diatup¸soume m a basik sqèsh th jewr a susthmtwn. Me
th bo jeia th sqèsh aut ja mporoÔme na prosdior zoume thn èxodo y t enì ()
()
grammikoÔ sust mato , an gnwr zoume a) thn e sodo x t tou sust mato kai b) thn
apìkrish tou sust mato , ìtan autì diege retai apì th sunrthsh Æ t . ()
2.4.1 Grammik qronik anallo wta sust mata suneqoÔ qrìnou. -
To olokl rwma th sunèlixh
Apì to Pardeigma 1.4.3 gnwr zoume ìti kje s ma suneqoÔ qrìnou mpore na proseg-
giste , ìpw sto Sq ma 2.9a, apì èna s ma th morf
1
X
x^(t) = x(k)Æ (t k) (2.4.1)
k= 1
^ ()
àstw h0 t h èxodo tou upì melèth grammikoÔ sust mato , ìtan h e sodo e nai o
()
palmì Æ t . Lìgw th grammikìthta , ìtan h e sodo tou sust mato e nai o palmì
42 Eisagwg sta Sust mata Keflaio 2
x(t)
x(t)
0Ä kÄ t (a)
x(0)ä(t)Ä x(0)h0(t)Ä
x(0)
Ãñáììéêü
óýóôçìá
0Ä t (â) 0 t
x(Ä)ä(t-Ä)Ä x(Ä)hÄ(t)Ä
x(Ä)
Ãñáììéêü
óýóôçìá
0Ä t (ã) 0Ä t
x(kÄ)ä(t-kÄ)Ä x(kÄ)hkÄ(t)Ä
Ãñáììéêü
x(kÄ) óýóôçìá
0 kÄ t (ä) 0 kÄ t
x(t) y(t)
Ãñáììéêü
óýóôçìá
0 t (å)
0 t
x(t) y(t)
Ãñáììéêü
óýóôçìá
0 t (óô) 0 t
Sq ma 2.10 H grafik ermhne a th apìkrish enì grammikoÔ qronik metaballìmenou
sust mato , ìpw aut ekfrzetai apì thn Ex swsh (2.4.6).
x(0)Æ (t)4, h èxodo tou e nai h x h0 t (0)^ ( )
(blèpe Sq ma 2.10b). Genik, an hk ^ (t)
e nai h apìkrish tou grammikoÔ sust mato sthn e sodo Æ t k , dhlad , ( )
h^ k (t) = S fÆ (t k)g (2.4.2)
( ) (
tìte gia e sodo x k Æ t k )
, h èxodo ja e nai x k ( )h^ k (t) (blèpe Sq ma
2.10d).
An efarmoste sthn e sodo tou sust mato to s ma x ^(t), tìte h èxodì tou ja
e nai ( )
1
X
y^(t) = S fx^(t)g = S x(k)Æ (t k) (2.4.3)
k= 1
Gia thn pleionìthta twn shmtwn kai twn susthmtwn pou sunantme sthn prxh, h
Enìthta 2.4 Sqèsh MetaxÔ Eisìdou-Exìdou Sust mato 43
grammikìthta isqÔei kai gia peirou ìrou . àtsi, h (2.4.3) g netai
1
X
y^(t) = x(k)S fÆ (t k)g (2.4.4)
k= 1
ètsi apì thn (2.4.4) kai me th bo jeia th (2.4.2), h èxodo tou sust mato d netai apì
1
x(k)h^ k (t)
X
y^(t) = (2.4.5)
k= 1
^( )
Sto Sq ma 2.10e èqei sqediaste h èxodo y t tou sust mato , ìtan h e sodì tou
^( )
e nai to s ma x t . H èxodo tou sust mato , lìgw th idiìthta th grammikìthta ,
e nai sh me to jroisma twn epimèrou exìdwn tou sust mato , pou eikon zontai sta
Sq mata 2.10b-d.
ParathroÔme ìti gia na prosdior soume thn èxodo enì grammikoÔ sust mato gia
opoiod pote s ma eisìdou me th bo jeia th sqèsh (2.4.5), qreiazìmaste thn h^ k (t)
gia kje tim k.
tou
^(t) ! x(t), dhlad , x(t) = lim!1 P1k= 1 x(k)-
An ! 0, Æ (t) ! Æ (t) kai x
Æ (t k). àtsi, an h e sodo tou sust mato e nai to s ma x(t), h èxodo tou
sust mato ja e nai
1
x(k)h^ k (t)
X
y(t) = lim y^(t) = lim (2.4.6)
!1 !1 k= 1
()
Sto Sq ma 2.10st èqei sqediaste h e sodo x t , pou e nai to ìrio th klimakwt
^( ) ()
morf sunrthsh x t , kai h èxodo y t , h opo a e nai to ìrio th y t . ^( )
()
àstw h t h èxodo tou sust mato pou pargetai apì thn e sodo Æ t , ( )
dhlad ,
h t ( ) = Sf (
Æt )g (2.4.7)
An h qronikdirkeia
twn palm¸n mikra nei kai te nei sto mhdèn, , to !0
k g netai h suneq metablht k ( ! )
, to jroisma sto dexiì mèlo th (2.4.6)
grfetai w olokl rwma kai h èxodo tou sust mato d netai apì th sqèsh
Z 1
y(t) = x( )h ( )d (2.4.8)
1
An to grammikì sÔsthma e nai kai qronik anallo wto, tìte h apìkrish tou sust -
() ( )
mato h t , ìtan autì diege retai apì th Æ t , e nai dia me thn h t h0 t all () ()
qronik metatopismènh kat , dhlad h t ()= ( )
h t . àtsi, h èxodo tou sust -
mato d netai apì th sqèsh
Z 1
y(t) = x( )h(t )d (2.4.9)
1
44 Eisagwg sta Sust mata Keflaio 2
H (2.4.9) e nai gnwst kai w olokl rwma th sunèlixh , kai sumbol zetai w
y(t) = h(t) ? x(t) (2.4.10)
ParathroÔme ìti se èna GQA sÔsthma arke h gn¸sh mia mìno sunrthsh , th
h(t), gia na perigrafe pl rw h sqèsh metaxÔ tou s mato eisìdou x(t) kai tou s mato
exìdou y (t) tou sust mato me th bo jeia tou oloklhr¸mato th sunèlixh . H prxh h
opo a sunduzei dÔo s mata x(t) kai h(t) gia to sqhmatismì tou s mato y (t) kale tai
sunèlixh.
()
H sunrthsh h t , h opo a e nai h èxodo tou sust mato , ìtan autì diege retai
apì th sunrthsh Æ t ()
ht Æt ( ) = S f ( )g (2.4.11)
kale tai kroustik apìkrish tou sust mato .
H (2.4.9) grfetai kai l go diaforetik. Allzonta ti metablhtè e nai polÔ
eÔkolo na doÔme ìti mporoÔme na gryoume
Z 1
y(t) = h( )x(t )d (2.4.12)
1
Pardeigma 2.4.1
Na de xete ìti èna GQA sÔsthma e nai FEFE eustajè , an h kroustik apìkris tou
e nai apìluta oloklhr¸simh, dhlad , an
Z 1
jh(t)j dt < +1 (2.4.13)
1
LÔsh JewroÔme ìti h e sodo enì GQA sust mato e nai fragmènh, dhlad ,
jx( )j M < 1 (2.4.14)
ìpou M m a jetik stajer. H èxodo tou sust mato d netai apì to olokl rwma th
sunèlixh
Z 1
y(t) = x( )h(t )d (2.4.15)
1
apì thn opo a sunepgetai ìti
Z 1 Z 1
jy(t)j = x( )h(t )d jx( )j jh(t )j d
1 1
Z 1
M jh(t )j d (2.4.16)
1
Met thn allag metablht èqoume
Z 1
jy(t)j M jh( )j d (2.4.17)
1
Enìthta 2.4 Sqèsh MetaxÔ Eisìdou-Exìdou Sust mato 45
kai lìgw th (2.4.13) èpetai ìti h èxodo tou sust mato e nai ep sh fragmènh, opìte
to sÔsthma e nai FEFE eustajè . Mpore na apodeiqje ìti h sunj kh aut e nai kai
anagka a.
Pardeigma 2.4.2
Na upologiste h kroustik apìkrish tou sust mato mèsh tim
y(t) =
1 Z t
x( )d (2.4.18)
T t T
LÔsh H kroustik apìkrish tou sust mato mèsh tim , h(t), e nai sh me thn èxodo
tou sust mato an autì diegerje apì th sunrthsh Æ(t), dhlad ,
h(t) =
1 Z t Æ( )d = 1 Z t du( )
T t T T t T
= 1 [u(t) u(t T
T )℄
!
= T1 t T
2 (2.4.19)
T
ìpou lbame upìyh ìti Æ( )d = du( ) kai (t) e nai o orjog¸nio palmì .
2.4.2 Idiìthte th Sunèlixh
H sunèlixh èqei ti akìlouje idiìthte :
Antimetajetik idiìthta
h1 (t) ? h2 (t) = h2 (t) ? h1 (t) (2.4.20)
H apìdeixh th parapnw idiìthta aporrèei apì ton orismì th sunèlixh . Prgmati,
allzonta th metablht t èqoume=
Z 1
h1 (t) ? h2 (t) = h1 ( )h2 (t ) d
Z
1
t =' 1
= h1 (t )h2 ( ) d
Z
1
1
= h2 ( )h1 (t ) d
1
= h2 (t) ? h1 (t) (2.4.21)
H fusik shmas a th idiìthta aut fa netai sto Sq ma 2.11, apì to opo o parath-
roÔme ìti, an dÔo sust mata e nai sundedemèna se seir mporoÔme na enallxoume th
seir sÔndes tou .
46 Eisagwg sta Sust mata Keflaio 2
x(t) h1(t) h2(t) y(t) x(t) h2(t) h1(t) y(t)
Sq ma 2.11 H fusik shmas a th antimetajetik idiìthta th sunèlixh .
Prosetairistik idiìthta
h2 (t) ? [h1 (t) ? x(t)℄ = [h2 (t) ? h1 (t)℄ x(t) (2.4.22)
H apìdeixh th idiìthta akolouje thn dia pore a me thn prohgoÔmenh. H fusik
shmas a th prosetairistik idiìthta fa netai sto Sq ma 2.12. ParathroÔme ìti,
ìtan dÔo sust mata sundèontai se seir, mporoÔn na antikatastajoÔn me èna tr -
to sÔsthma, to opo o èqei kroustik apìkrish sh me th sunèlixh twn kroustik¸n
apokr sewn twn dÔo susthmtwn pou èqoun sundeje se seir.
x(t) h1(t) h2(t) y(t) x(t) h1(t) * h2(t) y(t)
Sq ma 2.12 H fusik shmas a th prosetairistik idiìthta th sunèlixh .
Epimeristik idiìthta
[h1 (t) + h2 (t)℄ ? x(t) = h1 (t) ? x(t) + h2 (t) ? x(t) (2.4.23)
H apìdeixh aporrèei mesa apì tou orismoÔ . H fusik shmas a th epimeristik
idiìthta fa netai sto Sq ma 2.13. Lìgw th epimeristik idiìthta , an dÔo sust -
mata èqoun sundeje parllhla, tìte mporoÔn na antikatastajoÔn apì èna tr to
sÔsthma, tou opo ou h kroustik apìkrish e nai sh me to jroisma twn kroustik¸n
apokr se¸n tou .
h1(t)
x(t) y(t) x(t) h1(t) + h2(t) y(t)
h2(t)
Sq ma 2.13 H fusik shmas a th epimeristik idiìthta th sunèlixh .
Tautotik idiìthta
h(t) ? Æ(t) = h(t) (2.4.24)
H tautotik idiìthta e nai apìrroia tou orismoÔ th kroustik apìkrish sust ma-
to .
Enìthta 2.4 Sqèsh MetaxÔ Eisìdou-Exìdou Sust mato 47
2.4.3 Grafikì prosdiorismì th sunèlixh
Gia na upolog soume thn èxodo enì GQA sust mato me th bo jeia tou oloklhr¸-
mato th sunèlixh
Z 1
y(t) = x( )h(t )d (2.4.25)
1
gia kje qronik stigm t akoloujoÔme ta b mata:
1. B ma: Anklash. Antistrèfoume thn kroustik apìkrish, dhlad prosdior -
( )
zoume thn h .
2. B ma: Qronik Metatìpish. Metatop zoume thn h( ) kat t kai ètsi pros-
dior zoume thn h(t ).
3. B ma: Pollaplasiasmì . Prosdior soume to ginìmeno x( ) h(t ).
4. B ma: Olokl rwsh Embadomètrhsh. Oloklhr¸noume to ginìmeno autì ( upo-
log soume to embadì tou s mato tou dhmiourge tai apì th grafik parstash
tou ginomènou kai tou xona tou qrìnou ). To apotèlesma ja e nai so me thn
()
èxodo tou sust mato y t thn ant stoiqh qronik stigm t.
5. B ma: Epanlhyh. Ta b mata aut epanalambnontai gia ti difore timè tou
qrìnou.
Efarmìzoume ta parapnw sto pardeigma pou akolouje .
Pardeigma 2.4.3
Na upologiste , h èxodo enì grammikoÔ qronik anallo wtou sust mato pou èqei
kroustik apìkrish
h(t) = 1 t; 0t1
0; alli¸
(2.4.26)
an h e sodo tou e nai to s ma:
x(t) = 1; 0 t 2
0; alli¸ (2.4.27)
LÔsh To GQA sÔsthma kai h e sodì tou perigrfontai sto Sq ma 2.14a. Sto Sq ma
2.14b d netai h kroustik apìkrish tou sust mato .
Sto Sq ma 2.14g e nai h katoptrik morf th kroustik apìkrish tou sust mato ,
h( ). Sto Sq ma 2.14d h katoptrik morf th kroustik apìkrish èqei metatopis-
te kat t < 0, h(t ). ParathroÔme ìti to ginìmeno h(t ) x( ) e nai so me mhdèn
gia kje tim tou qrìnou t mikrìterh tou mhdenì . àtsi, h èxodo tou sust mato e nai
y(t) = 0 gia t<0 (2.4.28)
48 Eisagwg sta Sust mata Keflaio 2
x(t) y(t)=x(t)* h(t)
1
ÃXA
óýóôçìá (a)
0 2 t 0 t
h(ô)
1 1-ô, 0≤ô<1
h(ô)=
0, áëëßùò
(â)
0 1 ô
h(-ô)
1 1+ô, -1<ô≤0
h(-ô)=
0, áëëßùò
(ã)
-1 0 ô
x(ô) 1-t+ô, t-1<ô≤t
h(t-ô) 1
h(t-ô)=
t<0 0, áëëßùò
(ä)
t-1 t 0 2 ô
x(ô)
1
1- t
t<0<1
h(t-ô) (å)
t-1 0 t 2 ô
x(ô)
1
1<t<2 h(t-ô)
( óô )
0 t-1 t 2 ô
x(ô) h(t-ô)
1
2<t<3 3- t
(æ)
0 t-1 2 t ô
x(ô) h(t-ô)
1
3<t (ç)
0 2 t-1 t ô
Sq ma 2.14 O grafikì prosdiorismì th exìdou enì GQA sust mato me th bo jeia tou
oloklhr¸mato th sunèlixh .
Sto Sq ma 2.14e h katoptrik morf th kroustik apìkrish èqei metatopiste kat
0 t < 1. Qrhsimopoi¸nta ti (2.4.26) kai (2.4.27) h èxodo tou sust mato d netai
apì th sqèsh
Z 1 Z t 2
y(t) = x( ) h(t ) d = 1 (1 t + ) d = t t2 gia 0 t < 1 (2.4.29)
1 0
kai e nai sh me to embadì tou grammoskiasmènou trapez ou sto Sq ma 2.14e.
Sto Sq ma 2.14st h katoptrik morf th kroustik apìkrish èqei metatopiste kat
Enìthta 2.4 Sqèsh MetaxÔ Eisìdou-Exìdou Sust mato 49
1 t < 2. Qrhsimopoi¸nta ti (2.4.26) kai (2.4.27) br skoume ìti h èxodo tou
sust mato e nai sh me
= 21 ;
Z t
y(t) = 1 (1 t + ) d gia 1t<2 (2.4.30)
t 1
kai e nai sh me to embadì tou grammoskiasmènou trig¸nou sto Sq ma 2.14st.
Sto Sq ma 2.14z h katoptrik morf th kroustik apìkrish èqei metatopiste kat
2 t < 3. H èxodo tou sust mato d netai t¸ra apì th sqèsh
2
= 21 (3 t)2 ;
Z
y(t) = 1 (1 t + ) d gia 2t<3 (2.4.31)
t 1
h opo a e nai sh me to embadì tou grammoskiasmènou trig¸nou tou Sq mato 2.14z.
Tèlo , ìpw parathroÔme sto Sq ma 2.14h, to ginìmeno h(t )x( ) e nai so me mhdèn
gia kje tim tou qrìnou t megalÔterh sh apì 3. àtsi, h èxodo tou sust mato e nai
y(t) = 0; gia 3t (2.4.32)
H èxodo , loipìn, tou sust mato e nai:
8
>
> t t2 =2; 0t1
y(t) =
<
1=2; 1t2
>
> (3 t)2 =2; 2t3 (2.4.33)
:
0; alli¸
Sto Sq ma 2.15 èqoume sqedisei thn e sodo kai thn èxodo tou GQA sust mato .
x(t) y(t)
1 1
2
ÃXA
óýóôçìá
0 1 2 3 t 0 1 2 3 t
(a) (â)
Sq ma 2.15 (a) H e sodo kai (b) h èxodo tou GQA sust mato tou Parade gmato 2.4.3.
2.4.4 Grammik qronik anallo wta sust mata diakritoÔ qrìnou. -
To jroisma th sunèlixh
Sto Pardeigma 1.4.5 de xame ìti kje s ma diakritoÔ qrìnou mpore na analuje se
jroisma apì olisj sei monadia ou de gmato
1
X
x(n) = x(k)Æ(n k) (2.4.34)
k= 1
50 Eisagwg sta Sust mata Keflaio 2
H èxodo enì grammikoÔ sust mato d netai apì th
( 1 ) 1
X X
y(n) = S fx(n)g = S x(k)Æ(n k) = x(k)S fÆ(n k)g (2.4.35)
k= 1 k= 1
ìpou qrhsimopoi jhke h idiìthta th grammikìthta tou sust mato . An gia kje
()
akèraio k , hk n e nai h èxodo tou sust mato pou pargetai apì thn e sodo Æ n k , ( )
dhlad hk n ( ) = Sf ( )g
Æ n k , tìte h oikogèneia twn shmtwn apìkrish
fhk (n)g ; 1<k<1 (2.4.36)
metafèrei ìlh thn plhrofor a pou qreiazìmaste gia na kajor soume thn èxodo pou
pargetai apì èna s ma peperasmènh èktash me th bo jeia th
1
X
y(n) = x(k)hk (n) (2.4.37)
k= 1
Diafwt zoume thn teleuta a sqèsh me to akìloujo pardeigma.
Pardeigma 2.4.4
JewroÔme to grammikì sÔsthma S to opo o trofodote tai me to s ma x(n), pou peri-
grfetai sto Sq ma 2.16.
x(n)
Óýóôçìá
x(n) S y(n)
-1
-2 0 1 2 3 n
Sq ma 2.16 Grammikì sÔsthma diakritoÔ qrìnou kai h e sodo tou.
Oi apokr sei h 1 (n); h0 (n) kai h1 (n), gia eisìdou Æ(n +
tou grammikoÔ sust mato
1); Æ(n); kai Æ(n 1) ant stoiqa, èqoun sqediaste sto Sq ma 2.17. P1 H apìkrish tou
grammikoÔ sust mato , ìpw aut ekfrzetai apì th sqèsh y (n) = k= 1 x(k )hk (n)
prosdior zetai grafik sto Sq ma 2.18.
h-1(n) h0(n) h1(n)
2 1 -1
-3 -2 -1 0 1 3 n -3 -2 -1 0 2 3 n -3 -2 1 2 3 n
Sq ma 2.17 Oi apokr sei tou grammikoÔ sust mato gia eisìdou Æ(n +1), Æ(n) kai Æ(n 1).
ätan to sÔsthma e nai kai qronik anallo wto, tìte
hk (n) = h0 (n k) (2.4.38)
Enìthta 2.4 Sqèsh MetaxÔ Eisìdou-Exìdou Sust mato 51
dhlad , ìpw h Æ n ( ) ()
k e nai mia qronik ol sjhsh th Æ n ètsi kai h apìkrish
() ()
hk n e nai mia qronik ol sjhsh th apìkrish h0 n , thn opo a gia eukol a thn
sumbol zoume me h n , h n ( ) ( ( ) ( ) = S f ( )g)
h0 n Æ n . àtsi, h èxodo tou sust mato
d netai apì th sqèsh
1
X 1
X
y(n) = x(k)h(n k) y(n) = h(k)x(n k) (2.4.39)
k= 1 k= 1
H (2.4.39) e nai gnwst w jroisma th sunèlixh .
x(-1) ä(n+1) x(-1) h-1(n)
Ãñáììéêü
-1 óýóôçìá -1 1
-2 0 1 2 3 n -2 2 3 n
x(0) ä(n) x(0) h0(n)
Ãñáììéêü
óýóôçìá 1
-2 -1 0 1 2 3 n -2 -1 2 3 n
x(1) ä(n-1) x(1) h1(n)
Ãñáììéêü
óýóôçìá -1
-2 -1 0 1 2 3 n -2 1 2 3 n
+1
+1
x(n)= Ók=-1 x(k) ä(n-k) y(n)=Ók=-1 x(k) hk(n)
Ãñáììéêü
-1 óýóôçìá -1 1
-2 0 1 2 3 n -2 2 3 n
Sq ma 2.18 H grafik ermhne a th apìkrish enì grammikoÔ sust mato , ìpw aut
ekfrzetai apì to jroisma th sunèlixh .
H sumperifor enì grammikoÔ qronik anallo wtou sust mato qarakthr zetai
()
apì to s ma h n , to opo o kale tai apìkrish monadia ou de gmato kroustik apìkri-
sh. Me lla lìgia se èna GQA sÔsthma arke h gn¸sh m a kai mìno sunrthsh th
()
kroustik apìkrish h n gia na perigrafe pl rw h sqèsh eisìdou x n kai exìdou ()
()
y n tou sust mato apì to jroisma th sunèlixh .
()
H prxh h opo a sunduzei dÔo s mata x n kai h n gia to sqhmatismì tou ()
()
s mato y n , kale tai sunèlixh kai sumbol zetai w
y(n) = h(n) ? x(n) (2.4.40)
52 Eisagwg sta Sust mata Keflaio 2
H sqèsh eisìdou - exìdou aitiat¸n GQA susthmtwn diakritoÔ qrìnou e nai
n
X
y(n) = x(k)h(n k) (2.4.41)
k= 1
( )=0
afoÔ h n ; n< . 0
An h e sodo e nai aitiatì s ma, h èxodo d netai se kje qronik stigm n apì to
peperasmèno jroisma
n
X
y(n) = x(k)h(n k); 0n<1 (2.4.42)
k=0
Pardeigma 2.4.5
Na upologiste h èxodo enì grammikoÔ qronik anallo wtou sust mato diakritoÔ
qrìnou pou èqei kroustik apìkrish
h(n) =
n; 0n6
0; alli¸
(2.4.43)
ìtan h e sodì tou e nai to s ma
x(n) = 1; 0 n 4
0; alli¸ (2.4.44)
LÔsh To GQA sÔsthma kai h e sodo tou perigrfontai sto Sq ma 2.19a. Sto Sq ma
2.19b d netai h kroustik apìkrish tou sust mato . Sto Sq ma 2.19g e nai h katoptrik
morf th kroustik h( k). Sto Sq ma 2.19d h katoptrik
apìkrish tou sust mato
morf th kroustik apìkrish n < 0, h(n k).
èqei metatopiste kat
1) Apì to Sq ma 2.19d parathroÔme ìti gia n < 0 to ginìmeno x(k )h(n k) e nai so
me mhdèn gia kje tim tou n mikrìterh tou mhdenì . àtsi, h èxodo tou sust mato
e nai:
y(n) = 0 (2.4.45)
2) Apì to Sq ma 2.19e parathroÔme ìti gia 0 n 4 to ginìmeno x(k)h(n k) e nai:
k) = 0; ; 0alli¸
kn
n k
x(k)h(n (2.4.46)
ètsi, h èxodo tou sust mato e nai
n n
X n k=r X
y(n) = n k = r
k=0 r=0
n+1
= 11 (2.4.47)
Enìthta 2.4 Sqèsh MetaxÔ Eisìdou-Exìdou Sust mato 53
x(n) y(n)=x(n) * y(n)
Ã×Á
óýóôçìá (a)
-2 -1 0 1 2 3 4 5 6 n n
h(k)
-8 -6 -4 -2 0 1 2 3 4 5 6 7 8 k (â)
h(-k)
(ã)
-8 -6 -4 -2 -1 0 1 2 3 4 5 6 7 8 k
h(n-k) x(k) n<0
(ä)
n-6 n 0 4 k
h(n-k) 0<n<4
(å)
n-6 0 n 4 k
h(n-k) 4<n<6
( óô )
n-6 0 4 n k
h(n-k) 6 < n < 10
(æ)
0 n-6 4 n
k
x(k) h(n-k) 10 < n
(ç)
0 4 n-6 n k
Sq ma 2.19 O grafikì prosdiorismì th exìdou enì GQA diakritoÔ sust mato me th
bo jeia tou ajro smato th sunèlixh . Sta sq mata e, st kai z den èqei sqediasje to s ma
eisìdou. H skiasmènh perioq prosdior zei thn perioq sthn opo a x(k) 6= 0.
3) Apì to Sq ma 2.19st parathroÔme ìti gia 4 < n 6 h èxodo tou sust mato e nai:
4
X 4
X 4
y(n) = n k = n 1 k = an
k=0 k=0 1
n 4 n+1
= 1 (2.4.48)
54 Eisagwg sta Sust mata Keflaio 2
4) Apì to Sq ma 2.19z parathroÔme ìti gia 6 < n 10 h èxodo tou sust mato e nai:
4 10Xn 10Xn
y(n) =
X
n k k n+6=r
= 6 r = 6 1 r = 6 10 n
k=n 6 r=0 r=0 1
16 n 7
= 1 (2.4.49)
5) Apì to Sq ma 2.19h parathroÔme ìti gia 10 < n h èxodo tou sust mato e nai:
y(n) = 0 (2.4.50)
2.5 APOKRISH GRAMMIKWN SUSTHMATWN SE EKJETIKES EISODOUS
2.5.1 Suneq per ptwsh
Gnwr zoume ìti h e sodo x t kai hRèxodo
1 x
() y(t) enì GQA sust mato sundèontai me to
olokl rwma th sunèlixh y t 1 ( )= ( )h(t )d . An h e sodo tou sust mato
e nai to ekjetikì migadikì s ma
x(t) = Aest (2.5.1)
ìpou s migadikì arijmì , tìte h èxodo e nai
Z 1 Z 1 Z 1
y(t) = h(t )x( )d = h( )x(t )d = h( )Aest e s d
Z
1
1 1
1
= h( )e s d Aest (2.5.2)
1
Telik,
y(t) = H (s)Aest (2.5.3)
ìpou Z 1
H (s) = h( )e s d (2.5.4)
1
()
To H s e nai èna migadikì arijmì gia thn tim s tou ekjetikoÔ s mato . En
apeleujer¸soume to s kai to af soume na metablletai, tìte to H s e nai m a mi- ()
gadik sunrthsh th migadik metablht s kai or zei, ìpw ja doÔme sto Keflaio
6, to Metasqhmatismì Laplace th kroustik apìkrish tou sust mato h t kai ()
kale tai sunrthsh metafor tou sust mato .
An s =
j!0 , dhlad , h e sodo tou sust mato e nai to s ma
x(t) = Aej!0 t (2.5.5)
Enìthta 2.5 Apìkrish Grammik¸n Susthmtwn se Ekjetikè Eisìdou 55
h èxodo tou sust mato y(t) e nai
y(t) = H (!0 )Aej!0 t (2.5.6)
ìpou Z 1
H (!) = h(t)e j!t dt (2.5.7)
1
()
To H ! e nai èna migadikì arijmì gia thn tim ! tou ekjetikoÔ s mato . En h !
()
metablletai, tìte to H ! e nai m a migadik sunrthsh th pragmatik metablht
()
!. H sunrthsh H ! exarttai ep sh kai apì thn kroustik apìkrish tou sust -
() ()
mato h t . äpw ja doÔme sto Keflaio 3, h H ! or zei to Metasqhmatismì Fourier
th kroustik apìkrish tou sust mato kai onomzetai apìkrish suqnìthta tou
sust mato . H apìkrish suqnìthta tou sust mato grfetai w migadik sunrthsh
me th morf :
H! H ! ej arg H (!)
( ) = j ( )j (2.5.8)
j ( )j
ìpou H ! onomzetai apìkrish pltou kai arg ( )
H ! apìkrish fsh tou sust -
mato . àtsi, h èxodo tou sust mato , an to s ma eisìdou e nai x t Aej (!0 t+ ) ,
()=
e nai:
y (t) = jH (!0 )jej arg H (!0 ) Aej(!0 t+ )
= jH (!0 )jAej(!0 t+ +arg H (!0 )) (2.5.9)
ParathroÔme ìti:
1. An h e sodo enì GQA sust mato e nai to migadikì ekjetikì s ma me kuklik
suqnìthta !0 , h èxodì tou e nai ep sh migadikì ekjetikì s ma me thn dia
kuklik suqnìthta !0 .
2. To plto th exìdou e nai so me to ginìmeno tou pltou th eisìdou ep to
mètro th apìkrish suqnìthta tou sust mato upologismènou sthn kuklik
j ( )j
suqnìthta !0 , dhlad , A H !0 .
3. H fsh th exìdou tou sust mato e nai metatopismènh w pro th fsh th
eisìdou kai prosdior zetai w jroisma th fsh tou s mato eisìdou kai th
fsh th apìkrish suqnìthta prosdiorismènh sthn kuklik suqnìthta !0
tou s mato eisìdou.
Me lla lìgia, èna grammikì sÔsthma metabllei to mètro kai th fsh tou s ma-
to eisìdou, all ìqi thn kuklik suqnìtht tou. H diat rhsh th kuklik suqnìth-
ta apotele m a basik idiìthta twn grammik¸n susthmtwn. Anakefalai¸nonta ,
sumpera noume ìti, an sthn e sodo enì grammikoÔ sust mato efarmoste s ma to
opo o apotele tai apì jroisma shmtwn apl¸n suqnot twn, tìte kai h èxodì tou
ja apotele tai apì upèrjesh twn diwn shmtwn me diaforetikì plto kai metatopi-
smènwn kat fsh.
56 Eisagwg sta Sust mata Keflaio 2
An h e sodo tou sust mato e nai
x(t) = A os(!0t + ) (2.5.10)
me ìmoio trìpo br skoume ìti h èxodo tou sust mato e nai
y(t) = jH (!0 )jA os(!0t + + arg H (!0 )) (2.5.11)
gia thn opo a isqÔoun anloge parathr sei , ìpw kai sthn per ptwsh ìpou h e sodo
tou sust mato e nai to migadikì ekjetikì s ma.
Apì ta parapnw anadeiknÔetai h fusik shmas a th apìkrish suqnìthtwn kai
dikaiologe tai to ìnom th w apìkrish suqnìthta tou sust mato .
Lìgw th basik idiìthta th diat rhsh th suqnìthta , pou èqoun ta GQA
sust mata, ìtan diege rontai apì migadik ekjetik s mata, ìpw ja doÔme sto epìme-
no keflaio, e nai epijumhtì sthn prospjei ma na broÔme thn apìkrish enì GQA
sust mato se tuqa o s ma na prosdior soume trìpou anptuxh enì tuqa ou s ma-
to se jroisma ekjetik¸n migadik¸n shmtwn. àtsi, ekmetalleuìmenoi thn idiìthta
th grammikìthta prokÔptei ìti h èxodo tou sust mato ja e nai sh me to jroisma
twn ekjetik¸n aut¸n migadik¸n shmtwn me ti die suqnìthte , twn opo wn to pl-
to kai h fsh èqoun uposte thn allgh pou prokale to sÔsthma se kje ekjetikì
migadikì s ma, anloga me th suqnìtht tou, kai h opo a prosdior zetai apì to mètro
kai th fsh th apìkrish suqnìthta tou sust mato gia th suqnìthta aut . Me ton
trìpo autì ja èqoume th dunatìthta na epexergastoÔme s mata polÔplokh morf
me th bo jeia twn aploustèrwn aut¸n ekjetik¸n shmtwn.
2.5.2 Diakrit per ptwsh
() ()
Gnwr zoume ìti h e sodo x n kai h èxodo y n enì diakritoÔ GQA sust mato
P1
sundèontai me to jroisma th sunèlixh y n ()=
k= 1 h k x n k . An h ( ) ( )
e sodo tou sust mato e nai to ekjetikì migadikì s ma apl suqnìthta
x(n) = z n (2.5.12)
ìpou z = rej migadik metablht , tìte h èxodo e nai
1 " 1 #
X X
y(n) = h(k) zn k = h(k) z k zn
k= 1 k= 1
telik
y(n) = H (z ) z n (2.5.13)
Enìthta 2.5 Apìkrish Grammik¸n Susthmtwn se Ekjetikè Eisìdou 57
ìpou
1
X
H (z ) = h(k) z k (2.5.14)
k= 1
ParathroÔme ìti h èxodo lambnetai apì to dio ekjetikì s ma eisìdou dia-
() ()
bajmismèno me H z . H sunrthsh H z , ìpw ja doÔme sto Keflaio 7, e nai o
Metasqhmatismì z th kroustik apìkrish kai kale tai sunrthsh metafor tou
sust mato .
An jèsoume z =
ej , dhlad , h e sodo tou sust mato e nai to s ma
x(n) = ej n (2.5.15)
h èxodo tou sust mato y(n) d netai apì th
y(n) = H ( )ej n (2.5.16)
ìpou
1
X
H( )= h(k) e j k (2.5.17)
k= 1
H sunrthsh H ( )
exarttai apì thn kuklik yhfiak suqnìthta kai apì th sunrth-
sh pou perigrfei thn kroustik apìkrish tou sust mato . äpw ja doÔme sto Ke-
flaio 5, h H ( )
apotele to Metasqhmatismì Fourier diakritoÔ qrìnou th kroustik
apìkrish tou sust mato kai onomzetai apìkrish suqnìthta tou sust mato .
H diat rhsh th suqnìthta apotele , ep sh , basik idiìthta twn grammik¸n
susthmtwn diakritoÔ qrìnou.
Pardeigma 2.5.1
Me th bo jeia th (2.5.6), na upologiste h apìkrish suqnìthta tou kukl¸mato tou
Sq mato 2.20, tou opo ou h e sodo e nai h tsh in (t) kai èxodo h tsh o (t).
R
i(t)
õin(t) C õo(t)
Sq ma 2.20 To kÔklwma tou
Parade gmato 2.5.1.
LÔsh Efarmìzonta to deÔtero kanìna tou Kirchhoff sto brìqo tou kukl¸mato
èqoume
Ri(t) + o (t) = in (t) (2.5.18)
Lambnonta upìyh thn (2.1.3) h (2.5.18) grfetai
d (t)
RC o + o (t) = in (t) (2.5.19)
dt
58 Eisagwg sta Sust mata Keflaio 2
An h e sodo tou sust mato e nai
in (t) = ej!t (2.5.20)
tìte, sÔmfwna me th (2.5.6), h èxodo tou kukl¸mato e nai
o(t) = H (!)ej!t (2.5.21)
Kai epeid
d o (t)
dt
= H (!)j!ej!t (2.5.22)
h (2.5.19) d nei
RCH (!)j!ej!t + H (!)ej!t = ej!t (2.5.23)
apì thn opo a upolog zetai h apìkrish suqnìthta tou sust mato
H (!) =
1
1 + jRC! (2.5.24)
Sto pardeigma pou akolouje anadeiknÔetai h fusik shmas a th apìkrish suqnìth-
ta enì sust mato .
Pardeigma 2.5.2
H apìkrish suqnìthta enì sust mato e nai
H (!) =
2
p
2+j 3! (2.5.25)
Na upologiste h èxodo tou sust mato , an h e sodo e nai to s ma
x(t) = 4ej2t (2.5.26)
kai na g noun oi grafikè parastsei tou pragmatikoÔ mèrou tou s mato eisìdou
kai tou s mato exìdou tou sust mato se sunrthsh me to qrìno.
LÔsh Gnwr zoume ìti, an h e sodo GQA sust mato e nai èna migadikì ekjetikì s ma
x(t) = Aej!0 t , h èxodo tou sust mato d netai apì th sqèsh
y(t) = jH (!0 )jAej(!0 t+ +arg H (!0 )) (2.5.27)
H kuklik suqnìthta tou s mato eisìdou x(t) = Aej!0 t e nai !0 = 2. H apìkrish
suqnìthta tou sust mato gia !0 = 2 e nai
1 (1
2p p
H (2) = = j 3)
2+j 32 4 (2.5.28)
h opo a se polik morf grfetai
1
H (2) = e j 3
2 (2.5.29)
Enìthta 2.5 Apìkrish Grammik¸n Susthmtwn se Ekjetikè Eisìdou 59
àtsi, h èxodo tou sust mato e nai
1
y(t) = e j 3 4ej2t = 2e j(2t 3 )
2 (2.5.30)
Gia to pragmatikì mèro th eisìdou kai th exìdou èqoume ant stoiqa
<e [x(t)℄ = 4 os(2t) (2.5.31)
<e [y(t)℄ = 2 os 2t 3 (2.5.32)
ParathroÔme ìti h èxodo èqei thn dia suqnìthta me thn e sodo, plto so me to
1/2 tou pltou th eisìdou kai diafor fsh me thn e sodo sh me =3, dhlad ,
parousizei qronik kajustèrhsh kat t = T ( =2) = =6. Sto Sq ma 2.21 èqei
sqediaste to pragmatikì mèro tou s mato eisìdou kai exìdou tou sust mato , sto
opo o parathroÔme ti allagè pou epibllei to sÔsthma sto plto kai th fsh tou
s mato eisìdou.
Re[ x(t)] Re[ y(t)]
Ät= ð
6
4 4
2
0
ð 2ð t H(ù) 0
ð 2ð t
-2
-4 -4
(a) (â)
Sq ma 2.21 To pragmatikì mèro (a) tou s mato eisìdou kai (b) tou s mato exìdou gia to
sÔsthma tou Parade gmato 2.5.2.
SÔnoyh Kefala ou
\
Sto keflaio autì dìjhke h ermhne a th ènnoia sÔsthma" kai h majhmatik th
èkfrash. Anafèrame tou trìpou sÔndesh susthmtwn kai perigryame ti basikè
idiìthte twn susthmtwn.
Parathr same ìti gia na perigrafe pl rw h sqèsh eisìdou-exìdou enì GQA
sust mato , me th bo jeia tou oloklhr¸mato th sunèlixh ( tou ajro smato th
sunèlixh ), arke h gn¸sh th kroustik apìkrish tou sust mato .
E dame ìti, an h e sodo enì GQA sust mato e nai èna s ma apl suqnìthta ,
tìte kai h ant stoiqh èxodo e nai s ma th dia suqnìthta , to opo o èqei uposte al-
lag , pou epibllei to sÔsthma sto plto kai th fsh tou kai h opo a prosdior zetai
apì th sunrthsh apìkrish suqnìthta tou sust mato .
Tèlo , rjame se m a pr¸th gnwrim a me thn ènnoia tou MetasqhmatismoÔ Fouri-
er, tou MetasqhmatismoÔ Laplace kai tou MetasqhmatismoÔ z , oi opo oi ja ma a-
pasqol soun se epìmena keflaia.
60 Eisagwg sta Sust mata Keflaio 2
2.6 PROBLHMATA
2.1 Na breje h sqèsh th tsh eisìdou in (t) kai th tsh exìdou o (t) gia to
kÔklwma tou Sq mato 2.22.
L
õin(t) i(t)
R õo(t)
Sq ma 2.22 To kÔklwma tou Probl -
mato 2.1.
2.2 To sÔsthma to opo o perigrfetai apì thn paraktw sqèsh eisìdou x (t), exìdou
y(t)
y(t) = jx(t)j (2.6.1)
anafèretai w sÔsthma pl rou anìrjwsh . Na exetsete an to sÔsthma e nai
grammikì, qronik anallo wto kai aitiatì.
2.3 Na de xete ìti to sÔsthma, sto opo o h sqèsh eisìdou x(t) kai exìdou y(t) e nai
y(t) = x2 (t) (2.6.2)
e nai mh grammikì sÔsthma.
2.4 H èxodo enì grammikoÔ qronik anallo wtou sust mato (Sq ma 2.23b) e nai o
trigwnikì palmì tou Sq mato 2.23g, ìtan h e sodo e nai o orjog¸nio palmì
tou Sq mato 2.23a. Na breje h èxodo tou sust mato ìtan h e sodo e nai to
()
s ma x1 t tou Sq mato 2.24a to s ma x2 t tou Sq mato 2.24b. ()
x(t) y(t)
1 1
ÃXA
óýóôçìá
0 1 2 3 t 0 1 2 3 t
(a) (â) (ã)
Sq ma 2.23 H e sodo kai h èxodo tou GQA sust mato sto Prìblhma 2.4
x1(t) x2(t)
1 2
1
0 1 2 3 4 t
-1 0 1 2 3 t
(a) (â)
Sq ma 2.24 Oi e sodoi tou GQA sust mato sto Prìblhma 2.4
Enìthta 2.6 Probl mata 61
2.5 To sÔsthma to opo o perigrfetai apì thn paraktw sqèsh eisìdou x (t), exìdou
y(t) Z t
y(t) = S fx(t)g = x( )d (2.6.3)
1
anafèretai w oloklhrwt . Na exetsete an to sÔsthma e nai grammikì, qronik
anallo wto, aitiatì kai na prosdioriste h kroustik apìkris tou.
2.6 To sÔsthma to opo o perigrfetai apì thn paraktw sqèsh eisìdou x (t), exìdou
y(t)
y(t) = S fx(t)g = x(t) os(!0 t) (2.6.4)
anafèretai w diamorfwt . Na exetsete an to sÔsthma e nai grammikì, qronik
anallo wto kai aitiatì kai na prosdioriste h kroustik tou apìkrish.
2.7 H kroustik apìkrish tou kukl¸mato RC se seir sto Sq ma 2.3 e nai
1
h(t) = e u(t)
t
(2.6.5)
ìpou = RC h stajer qrìnou. An = 1se , na upologiste h èxodo tou
sust mato , ìtan h e sodì tou e nai to s ma
x(t) = u(t) u(t 2) (2.6.6)
2.8 H kroustik apìkrish enì GQA sust mato e nai
h(t) = u(t) u(t 2) (2.6.7)
Me th bo jeia tou oloklhr¸mato th sunèlixh na upolog sete thn èxodo tou
sust mato , ìtan h e sodo e nai
x(t) = u(t 1) u(t 2) (2.6.8)
2.9 H kroustik apìkrish enì GQA sust mato e nai
h(t) = u(t 1) u(t 3) (2.6.9)
Me th bo jeia tou oloklhr¸mato th sunèlixh na upolog sete thn èxodo tou
sust mato , ìtan h e sodì tou e nai to s ma
x(t) = sin(t); 0 t 2
0; alli¸
(2.6.10)
62 Eisagwg sta Sust mata Keflaio 2
2.10 An ta stoiqe a tou kukl¸mato , pou perigrfetai sto Sq ma 2.20, e nai R =
103 kai C 6 na upologiste h èxodì tou, ìtan h e sodo tou e nai to
= 10 F
s ma
p
x(t) = 2 sin 1000t + 3 (2.6.11)
2.11 H kroustik apìkrish enì GQA sust mato e nai
h(n) = (0; 9)n u(n) (2.6.12)
Me th bo jeia tou ajro smato th sunèlixh na upolog sete thn èxodo tou
sust mato , ìtan h e sodo e nai to diakritì s ma
x(n) = u(n) u(n 4) (2.6.13)
2.12 H kroustik apìkrish enì GQA sust mato e nai
h(n) = [ 3; 2; 1℄ (2.6.14)
"
ìpou to bèlo pro ta pnw dhl¸nei to de gma gia n =0
. Me th bo jeia tou
ajro smato th sunèlixh na upolog sete thn èxodo tou sust mato , ìtan h
e sodo e nai to s ma
xn ( )=[1 2 3 4℄
; ; ; (2.6.15)
"
2.13 Na de xete ìti to sÔsthma to opo o perigrfetai apì th diaforik ex swsh
dy(t)
dt
+ 10y(t) + 5 = x(t) (2.6.16)
den e nai grammikì.
Bibliograf a
2.1 S. Jeodwr dh , K. Mpermper dh , L. Kof dh , “Eisagwg sth Jewr a Shmtwn
kai Susthmtwn”, Tupwj tw - Gi¸rgo Dardanì , Aj na 2003.
2.2 A. Mrgarh , “S mata kai Sust mata SuneqoÔ kai DiakritoÔ Qrìnou ”, Ekdì-
sei Tziìla 2012.
2.3 S. Haykin, B. Veen, “Signal and Systems”, John & Wiley Sons, Inc. 2003
2.4 A. V. Oppenheim, R. S. Willsky, I. T. Young, “Signal and Systems”, Prentice -
Hall Inc., N. Y., 1983.
2.5. R. E. Siemer, W. H. Tranter, D. R. Fannin, “Signals & Systems Continuous and
Discrete”, Prentice Hall, 1998.
ÊÅÖÁËÁÉÏ 3
ÁÍÁÐÔÕÃÌÁ - ÌÅÔÁÓ×ÇÌÁÔÉÓÌÏÓ FOURIER
ÁÍÁËÏÃÉÊÙÍ ÓÇÌÁÔÙÍ
Sthn prxh, pollè forè qreizetai na prosdior soume thn èxodo enì sust -
mato , ìtan autì diege retai apì èna s ma. Sto prohgoÔmeno keflaio, e dame ìti
h èxodo enì GQA sust mato perièqei ti die suqnìthte me to s ma eisìdou, me
diaforetikì, ìmw , mètro kai fsh. Sto keflaio autì ja eisaggoume kai ja melet -
soume majhmatik ergale a, ta opo a ma epitrèpoun na analÔoume èna sÔnjeto s ma
se s mata apl¸n suqnot twn. M a tètoia prosèggish ma dieukolÔnei ¸ste na up-
olog soume thn èxodo enì sust mato , to opo o diege retai apì èna sÔnjeto s ma, me
th bo jeia twn apokr sewn tou sust mato sti epimèrou sunist¸se sugkekrimènwn
suqnot twn. Sth sunèqeia ja efarmìsoume ti mejìdou autè ¸ste na analÔsoume
èna arijmì shmtwn, ta opo a sunantme suqn sthn prxh.
Eisagwg
Sto prohgoÔmeno keflaio, e dame ìti an h e sodo enì GQA sust mato e nai to
migadikì ekjetikì s ma to hmitonoeidè s ma, tìte h èxodì tou e nai s ma th dia
suqnìthta me diaforetikì plto kai fsh. àtsi, parathr same ìti e nai polÔ eÔko-
lo na prosdior soume thn èxodo tou GQA sust mato , ìtan to s ma pou efarmìzetai
sthn e sodì tou analuje se s mata sugkekrimènh suqnìthta .
Sto keflaio autì ja anaptÔxoume kai ja melet soume trìpou anlush enì
s mato se s mata sugkekrimènh suqnìthta . àtsi, an to s ma autì diege rei èna
sÔsthma, ekmetalleuìmenoi thn idiìthta th grammikìthta , ja prosdior zoume thn
èxodo tou sust mato w jroisma shmtwn pou èqoun ti die suqnìthte me autè
pou perièqei to s ma eisìdou, twn opo wn to plto kai h fsh èqei uposte allag
pou prokale to sÔsthma.
3.1 H IDEA TOU QWROU TWN SHMATWN
H gènnhsh kai oi r ze th jewr a aut ofe lontai sto Gllo Majhmatikì Jean Bap-
tiste Joseph Fourier (1768-1830), o opo o eis gage thn anlush mia sunrthsh se
sunart sei sugkekrimènwn suqnot twn gia na melet sei fainìmena didosh th jer-
64 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
mìthta . H anlush mia sÔnjeth posìthta se aploÔstere sunist¸se , me skopì
h melèth enì probl mato na g netai eukolìterh, e nai m a genikìterh mejodolog -
a. Gia pardeigma, sth grammik lgebra, èna dinusma analÔetai se mia bsh pou
perigrfei to q¸ro. Me aform thn parat rhsh aut a doÔme giat kai ìla ta s -
[ ℄
mata pou or zontai se èna disthma a; b mporoÔn na perigrafoÔn kai w dianÔsmata.
Prgmati, o grammikì sunduasmì dÔo h perissotèrwn s matwn d nei èna nèo s ma
sto dio disthma. Ep sh , o pollaplasiasmì enì s mato me mia stajer posìth-
ta d nei èna nèo s ma sto dio disthma. Exetzonta ta s mata w dianÔsmata se
èna ant stoiqo q¸ro ma ano getai o drìmo na analÔsoume èna s ma se jroisma
aploustèrwn shmtwn, sto dianusmatikì q¸ro twn shmtwn sto disthma a; b . E - [ ℄
nai skìpimo ed¸ na jumhjoÔme merikè basikè ènnoie apì th grammik lgebra kai
sth sunèqeia na or soume ant stoiqe ènnoie sta s mata.
Gia pardeigma, èna dinusma sto q¸ro twn tri¸n diastsewn paristnetai me th
bo jeia twn probol¸n tou sta monadia a dianÔsmata tou q¸rou, ta opo a apoteloÔn
th “bsh tou q¸rou. àtsi, to dinusma a mpore na ekfraste w
a = a1 e1 + a2 e2 + a3 e3 (3.1.1)
ìpou e1 , e2 kai e2 e nai ta monadia a dianÔsmata sti trei basikè dieujÔnsei tou
q¸rou (Sq ma 3.1). To dinusma a mpore isodÔnama na parastaje me thn trida
(
suntetagmènwn a1 ; a2 ; a3 )
z
e3 a
e2 y
e1
x Sq ma 3.1 Anlush dianÔsmato .
Sto q¸ro twn n diastsewn, o opo o èqei bsh e1 ; e2 ; ::: ; en , kje dinusma
paristnetai w
n
X
a= ai ei (3.1.2)
i=1
To dinusma a paristnetai, isodÔnama, apì ti suntetagmène (a1 ; a2 ; :::; an ).
H distash tou q¸rou n, e nai o arijmì twn monadia wn dianusmtwn ta opo a
e nai anagka a kai ikan gia na ekfrsoun kje dinusma tou q¸rou. Ta dianÔsma-
ta a1 ; a2 ; :::; an e nai anexrthta, an kanèna apì aut den mpore na ekfraste w
grammikì sunduasmì twn llwn.
Enìthta 3.1 H Idèa tou Q¸rou twn Shmtwn 65
Pn
To dÔo dianusmtwn n diastsewn a
eswterikì ginìmeno = Pni=1 aiei kai b =
b e
i=1 i i , sumbol zetai me h i
a; b kai or zetai apì th sqèsh
n
X
ha; bi = aT b = ai bi (3.1.3)
i=1
ìpou aT e nai to anstrofo dinusma tou a. To eswterikì ginìmeno èqei ti akìlouje
idiìthte
E nai jetik orismèno ha; ai > 0 ìtan a 6= 0 kai h0; 0i = 0
Antimetajetik ha; bi = (hb; ai)?
Epimeristik h(a + b); i = ha; i + hb; i
Pollaplasiasmì me stajer h a; bi = ha; bi
upenjum zetai ìti me “?” dhl¸netai o suzug migadikì . DÔo dianÔsmata e nai or-
h
jog¸nia an, a; b i=0.
To mètro (norm) m ko enì dianÔsmato a sumbol zetai me k a k kai e nai o mh
arnhtikì arijmì pou or zetai apì th sqèsh
k a k= ha; ai 12 = Pn
i=1 ai
2 12 (3.1.4)
dhlad , to mètro enì dianÔsmato e nai so me thn tetragwnik r za tou eswterikoÔ
ginomènou tou dianÔsmato me ton eautì tou.
( )
àna sÔnolo dianusmtwn a1 ; a2 ; :::; an kale tai orjokanonikì, an aut e nai an
dÔo orjog¸nia kai ìla èqoun mètro so me th monda, dhlad ,
hak ; am i = Æ(k m) = 10;; kk ==6 m
m (3.1.5)
(
Gia m a orjokanonik bsh dianusmtwn oi suntetagmène a1 ; a2 ; :::; an enì di- )
anÔsmato a e nai oi probolè tou a se kje èna apì ta dianÔsmata bsh kai pros-
dior zontai apì th sqèsh
ai = ha; ei i i = 1; 2; :::; n (3.1.6)
àtsi, to a grfetai
n
X
a= ha; ei i ei (3.1.7)
i=1
66 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
Sth sunèqeia ja or soume ti ant stoiqe ènnoie gia ta s mata. To eswterikì
ginìmeno () ()
dÔo shmtwn x t kai y t , ta opo a or zontai sto disthma a; b , sum- [ ℄
h ( ) ( )i
bol zetai me x t ; y t kai or zetai w
Z b
hx(t); y(t)i = x(t)y? (t)dt (3.1.8)
a
()
ìpou y ? t e nai to suzugè s ma tou y(t). E nai eÔkolo na doÔme ìti o orismì autì
plhro ti akìlouje idiìthte
E nai jetik orismèno hx(t); x(t)i 0 kai hx(t); x(t)i = 0 , x(t) = 0
Antimetajetik hx(t); y(t)i = hy(t); x(t)i?
Epimeristik hx(t) + y(t); z(t)i = hx(t); z(t)i + hy(t); z(t)i
Pollaplasiasmì me stajer h x(t); y (t)i = hx(t); y (t)i
DÔo s mata e nai orjog¸nia an, hx(t); y (t)i = 0.
àna migadikì q¸ro efodiasmèno me eswterikì ginìmeno kale tai kai (migadikì )
Eukle dio q¸ro . Se ènan Eukle dio q¸ro to eswterikì ginìmeno or zei tautìqrona
kai to mètro (norm) enì s mato w thn tetragwnik r za tou eswterikoÔ ginomènou
tou s mato me ton eautì tou, dhlad ,
qR
k x(t) k2 = hx(t); x(t)i 12 = b
a jx(t)j2 dt (3.1.9)
jj ( ) k
Profan¸ , o x t 2 e nai pnta èna mh arnhtikì pragmatikì arijmì .
JewroÔme èna sÔnolo orjokanonik¸n shmtwn n t ,n f ( )g = 1 2
; ; ::: P
sto disth-
[ ℄ h ()
ma a; b , gia ta opo a isqÔei k t ; m t Æ k m kai èstw ìti h seir 1
( )i = ( )
n=1 xn n (t)
()
sugkl nei se èna s ma x t sto disthma a; b , dhlad , [ ℄
1
X
x(t) = xn n (t) (3.1.10)
n=1
Tìte oi suntelestè xn ikanopoioÔn th sqèsh
Z b
xn = hx(t); n (t)i = x(t); ?
n (t)dt (3.1.11)
a
H apìdeixh th (3.1.11) e nai profan , arke na pollaplasisoume kai ta dÔo mèlh
th (3.1.10) diadoqik me ti n t , n ( ) =1 2
; ; ::: kai na oloklhr¸soume. ParathroÔme
ìti sto grammikì Eukle dio q¸ro, pou dhmiourgoÔn ta s mata n t , n ; ; ::: () =1 2
kai ta ìri tou , h (3.1.10) e nai to anptugma tou s mato x t w pro ta s mata ()
Enìthta 3.1 H Idèa tou Q¸rou twn Shmtwn 67
()
n t kai xn den e nai t pota llo apì ti probolè tou x t se kje èna apì ta ()
orjokanonik s mata n t . ()
An sthn (3.1.8) jèsoume w ìria th olokl rwsh 1
, o Eukle dio q¸ro pou
( )
prokÔptei e nai gnwstì kai w q¸ro L2 R , dhlad ,
L2 (R) = fx(t); t 2 ( 1; +1) :k x(t) k2 < +1g
To mètro kk
2 e nai gnwstì kai w L2 -mètro. Profan¸ , sto q¸ro autì an koun
ìla ta s mata peperasmènh enèrgeia .
Sti epìmene paragrfou ja asqolhjoÔme me sugkekrimèna orjokanonik s ma-
ta kai anaptÔgmata th morf (3.1.10). M a austhr majhmatik melèth tou L2 R ( )
e nai pèra apì ta pla sia tou parìnto egqeirid ou. O endiaferìmeno anagn¸sth
parapèmpetai sta sqetik bibl a th proteinìmenh bibliograf a .
3.1.1 To sÔnolo twn orjog¸niwn analogik¸n ekjetik¸n periodik¸n shmtwn
äpw èqoume de sthn Pargrafo 1.4.1, to ekjetikì s ma ej!0 t e nai periodikì me
jemeli¸dh per odo T =2
=!0 . Ta ekjetik s mata, pou èqoun kuklik suqnìth-
(
ta pollaplsia th !0 ejk!0 t ; me k = 0 1 2 )
; ; ; ::: , e nai ep sh periodik me
jemeli¸dei periìdou Tk =2
=k!0 , ant stoiqa. ParathroÔme, ìti h per odo T =
2=!0 e nai koin per odo gia ìla ta s mata ejk!0t ; me k ; ; ; :::, afoÔ = 0 1 2
T = 2! = k k!
2 = k T
k
0 0
Ta ekjetik s mata, ejk!0 t ; me k = 0; 1; 2; :::, se opoiod pote peperasmèno
qronik disthma [t0 ; t0 + T ℄, dirkeia T = 2=!0 , kaloÔntai armonik susqetizìmena
ekjetik s mata, epeid oi jemeli¸dei kuklikè suqnìthtè tou e nai akèraia pol-
laplsia th kuklik suqnìthta !0 kai sqhmat zoun èna orjog¸nio sÔnolo, dhlad ,
e nai an dÔo orjog¸nia. Prgmati, to eswterikì ginìmeno twn ekjetik¸n s matwn
ejk!0t kai ejm!0 t , e nai
Z T Z T
hejk!0t; ejm!0 t i = ejk!0 t e jm!0 t dt = ej (k m)!0 t dt
0 0
An k 6= m, tìte
hejk!0t; ejm!0 t i = j (k 1m)! ej(k m)!0 t = j (k 1m)! ej(k m) 2T T e0 = 0
T h i
0 0 0
En¸, an k = m, e nai
Z T Z T
hejk!0t ; ejm!0 ti = ej (k m)!0 t dt = 1dt = T (3.1.12)
0 0
68 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
Telik,
hejk!0t ; ejm!0 ti = T Æ(k m) (3.1.13)
H apìdeixh ègine gia to disthma [0; T ℄. Me ìmoio trìpo g netai gia opoiod pote
disthma me m ko T .
3.1.2 To sÔnolo twn orjog¸niwn analogik¸n trigwnometrik¸n periodik¸n
shmtwn
äpw èqoume de sthn Pargrafo 1.4.1, ta s mata, k!0 t kai os( ) sin(
k!0 t me k ) =
0 1 2
; ; ; ::: e nai periodik me jemeli¸dei periìdou Tk =2
=k!0 , ant stoiqa.
=2
ParathroÔme, ìti h per odo T =!0 e nai koin per odo gia ìla ta s mata.
Ta s mata, os( ) sin( ) 1
k!0 t kai 1
k!0 t < k < , se opoiod pote peperasmèno
[ + ℄
qronikì disthma t0 ; t0 =2
T dirkeia T =!0 , kaloÔntai armonik susqetizì-
mena kai sqhmat zoun èna orjog¸nio sÔnolo. Prgmati, to eswterikì ginìmeno twn
sin( ) sin( )
trigwnometrik¸n s matwn k!0 t kai m!0 t e nai
Z T
hsin(k!0t); sin(m!0t)i = sin(k!0t)sin(m!0 t)dt
0Z
= 12 1 Z T os[(k + m)!0t℄dt
T
os[(k m)!0 t℄dt
2 0
0
ParathroÔme ìti, an k 6=m ta oloklhr¸mata sto deÔtero mèlo th ex swsh e nai
sa me mhdèn, dedomènou ìti h olokl rwsh g netai se m a per odo. Ant jeta, gia
= 6= 0
k m to pr¸to olokl rwma e nai so me T , en¸ to deÔtero e nai so me mhdèn.
àtsi, to eswterikì ginìmeno twn sin( )
k!0 t kai m!0 t , e nai sin( )
hsin(k!0 t); sin(m!0t)i = T2 Æ(k m) (3.1.14)
Me ìmoio trìpo apodeiknÔontai kai oi sqèsei
h os(k!0 t); os(m!0t)i = T2 Æ(k m) (3.1.15)
hsin(k!0t); os(m!0 t)i = 0; gia kje k kai m (3.1.16)
H apìdeixh ègine gia to disthma [0; T ℄ kai me ìmoio trìpo g netai gia opoiod pote
disthma me m ko T .
Enìthta 3.2 Anptugma Fourier - Seir Fourier 69
3.2 ANAPTUGMA FOURIER - SEIRA FOURIER
3.2.1 Ekjetik seir Fourier
Sthn Enìthta 3.1.1, e dame ìti ta ekjetik stoiqei¸dh s mata, ejk!0 t , k ; ; ; :::, = 0 1 2
pou or zontai se opoiod pote peperasmèno qronik disthma dirkeia t0 ; t0 T, [ + ℄
ìpou !0 =2=T kai t0 pragmatikì arijmì , sqhmat zoun èna orjog¸nio sÔnolo.
()
àstw t¸ra èna s ma x t sto disthma t0 ; t0 [ + ℄
T , kai a upojèsoume ìti e nai
dunatìn na anaptuqje se jroisma ekjetik¸n stoiqeiwd¸n shmtwn,
1
X
x(t) = ak ejk!0 t (3.2.1)
k= 1
H ( 3.2.1) apotele thn ekjetik seir Fourier to anptugma Fourier tou s mato
()
x t . O upologismì twn suntelest¸n ak g netai an pollaplasisoume kai ta dÔo
mèlh th ( 3.2.1) me e jn!0 t
1
X
x(t)e jn!0 t = ak ejk!0 t e jn!0 t (3.2.2)
k= 1
kai oloklhr¸soume apì t0 èw t0 + T
Z t0 +T 1
X Z t0 +T
x(t)e jn!0 t dt = ak ejk!0t e jn!0 t dt
t0 k= 1 t0
1
X
= ak hejk!0 t ; ejn!0 t i (3.2.3)
k= 1
Lìgw th (3.1.13) ìloi oi ìroi tou ajro smato sto deÔtero mèlo th (3.2.3) e nai
soi me to mhdèn, ektì apì ton ìro k =
n, o opo o e nai so me T . Apì thn (3.2.3),
èqoume loipìn ìti
Z t0 +T Z t0 +T
x(t)e jn!0 t dt = T an , an =
1 x(t)e jn!0 t dt (3.2.4)
t0 T t0
àtsi, an uprqei to anptugma Fourier tou s mato x(t) qarakthr zetai apì to zeÔgo
twn exis¸sewn
+1
X
x(t) = ak ejk!0 t ; t 2 [t0 ; t0 + T ℄ Ex swsh sÔnjesh (3.2.5)
k= 1
70 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
ak =
1 Z t0 +T x(t)e jk!0 t dt Ex swsh anlush (3.2.6)
T t0
Oi migadiko suntelestè ak kaloÔntai suntelestè Fourier fasmatikè grammè tou
()
x t kai or zoun to fsma tou s mato . H stajer a0 e nai h suneq h stajer
sunist¸sa tou fsmato . Kje ak antistoiqe sthn probol tou s mato x t pnw ()
sthn k sth orjog¸nia sunist¸sa ejk!0 t , dhl¸nei to fasmatikì perieqìmeno tou x t ()
sth suqnìthta k!0 kai onomzetai k sth armonik sunist¸sa. Prèpei na toniste ìti to
[ + ℄
anptugma Fourier isqÔei mìno sto disthma t0 ; t0 T kai to eÔro T kajor zei th
basik suqnìthta.
En parathr soume thn ekjetik seir Fourier (3.2.5), diapist¸noume ìti sto
jroisma uprqoun kai arnhtikè timè tou k , oi opo e , bèbaia, antistoiqoÔn se arnh-
tikè suqnìthte gia ti opo e den uprqei fusik ènnoia. Oi arnhtikè suqnìthte
upeisèrqontai sto jroisma epeid anaptÔssoume to s ma, pou e nai m a pragmatik
sunrthsh, me th bo jeia migadik¸n sunart sewn, ejk!0 t . Ja epanèljoume sto shme o
autì argìtera.
3.2.2 Trigwnometrik seir Fourier
Sthn Enìthta 3.1.2, e dame ìti ta periodik trigwnometrik s mata, k!0 t kai os( )
sin( ) 1
k!0 t , 1
< k < , pou or zontai se opoiod pote peperasmèno qronikì disth-
[ + ℄
ma dirkeia t0 ; t0 T , ìpou !0 =2
=T kai t0 pragmatikì arijmì , sqhmat zoun èna
() [
orjog¸nio sÔnolo. àstw t¸ra ìti to s ma x t , sto disthma t0 ; t0 T , anaptÔs- + ℄
setai se jroisma trigwnometrik¸n shmtwn pou to kajèna apì aut èqei jemeli¸dh
kuklik suqnìthta k!0 , dhlad ,
1
X 1
X
x(t) = a0 + bk os(k!0t) + k sin(k!0t) (3.2.7)
k=1 k=1
Ja upolog soume tou suntelestè th trigwnometrik seir Fourier (3.2.7), a0 ,
()
b1 ; b2 ; :::; kai 1 ; 2 ; :::, gia opoiod pote s ma x t gia to opo o uprqei èna tètoio anp-
tugma. Gia na prosdior soume th sqèsh me thn opo a upolog zetai o a0 oloklhr¸noume
+
thn (3.2.7) apì t0 èw t0 T kai parathroÔme ìti ìla ta oloklhr¸mata sto deÔtero
mèlo , efìson h olokl rwsh g netai se m a per odo, e nai sa me to mhdèn, ektì apì
to pr¸to to opo o e nai so me T . àtsi, o suntelest a0 d netai apì thn
Z t0 +T
a0 =
1 x(t)dt (3.2.8)
T t0
kai e nai so me th mèsh tim tou s mato .
Enìthta 3.2 Anptugma Fourier - Seir Fourier 71
O upologismì twn suntelest¸n, bk , g netai an pollaplasizoume kai ta dÔo mèlh
th (3.2.7) me os( )
n!0 t kai oloklhr¸soume apì t0 èw t0 T , w ex +
Z t0 +T Z t0 +T
x(t) os(n!0t)dt = a0 os(n!0t)
t0 t0
1
X Z t0 +T
+ bk os(k!0t) os(n!0t)dt
k=1 t0
1
X Z t0 +T
+ bk sin(k!0 t) os(n!0t)dt
k=1 t0
T
= 2 bn (3.2.9)
ìpou allxame th seir olokl rwsh kai jroish . To pr¸to olokl rwma sto
deÔtero mèlo th (3.2.9) e nai so me mhdèn. Epiplèon, lìgw th (3.1.16) ìla ta
oloklhr¸mata sto deÔtero jroisma th (3.2.9) e nai sa me mhdèn, en¸ lìgw th
(3.1.15) apì ta oloklhr¸mata sto pr¸to jroisma, mìno to olokl rwma gia k n =
2
e nai so me T= , en¸ ìla ta lla e nai sa me mhdèn. àtsi, oi suntelestè bn d nontai
apì thn
Z t0 +T
bn =
2 x(t) os(n!0 t)dt; n = 1; 2; ::: (3.2.10)
T t0
Me ìmoio trìpo br sketai ìti oi suntelestè n prosdior zontai apì thn
Z t0 +T
n=
2 x(t)sin(n!0 t)dt; n = 1; 2; ::: (3.2.11)
T t0
Apì ta parapnw prokÔptei ìti h trigwnometrik anaparstash Fourier twn peri-
odik¸n shmtwn qarakthr zetai apì ti exis¸sei sÔnjesh kai anlush
1
X 1
X
x(t) = a0 + bk os(k!0 t) + k sin(k!0 t); t 2 [t0 ; t0 + T ℄ (3.2.12)
k=1 k=1
Z t0 +T
a0
1
= T x(t)dt
t0
Z t0 +T
bk = 2 x(t) os(k!0 t)dt (3.2.13)
T t0
Z t0 +T
k = T2 x(t)sin(k!0 t)dt
t0
72 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
Me th bo jeia gnwst trigwnometrik tautìthta èqoume
bk os(k!0 t) + k sin(k!0t) = Ak os(k!0t + k )
ìpou q
Ak = b2k + 2k kai k = tan 1 k
bk
àtsi, to trigwnometrikì anptugma Fourier (3.2.12) mpore na grafe kai w
1
X
x(t) = a0 + Ak os(k!0t + k ) (3.2.14)
k=1
()
Apì thn (3.2.14) parathroÔme ìti to s ma x t èqei analuje se èna jroisma sunhm -
tonwn, kje èna apì ta opo a èqei diaforetikì plto kai fsh. Ep sh , a shmeiwje
ìti ed¸ den upeisèrqontai arnhtikè suqnìthte . H suneisfor kje suqnìthta sto -
jroisma prosdior zetai apì thn tim tou suntelest Ak , tou ant stoiqou sunhm tonou.
An oi suntelestè twn ìrwn me qamhlè suqnìthte e nai sqetik megalÔteroi apì tou
suntelestè twn ìrwn me uyhlè suqnìthte , tìte h taqÔthta metabol tou s mato
w pro to qrìno e nai mikr kai to s ma qarakthr zetai w s ma qamhl¸n suqnot twn.
Ant jeta, an oi suntelestè twn ìrwn me qamhlè suqnìthte e nai sqetik mikrìteroi
apì tou suntelestè twn ìrwn me uyhlè suqnìthte , tìte h taqÔthta metabol tou
s mato w pro to qrìno e nai meglh kai to s ma qarakthr zetai w s ma uyhl¸n
suqnot twn. Fusik, oi ènnoie qamhl¸n uyhl¸n suqnot twn e nai ènnoie sqetikè
kai exart¸ntai apì thn kje efarmog .
3.2.3 Seirè Fourier periodik¸n shmtwn
Mèqri t¸ra or same to anptugma se seir Fourier enì s mato se èna disthma
[t0; t0 + T ℄. àxw apì to disthma autì h seir Fourier den sugkl nei kat' angkh sto
s ma x(t). A doÔme ìmw ti g netai, en to s ma e nai periodikì me per odo T , dhlad ,
x(t) = x(t + T ). To anptugma
P
Fourier tou x(t) se èna qronikì disthma eÔrou T , so
me m a per odo, e nai x(t) = jk!0 t me ! = 2=T . Epeid ejk!0 t = ejk!0 (t+T ) , h
k ak e 0
seir Fourier e nai periodik me per odo sh me thn per odo tou s mato , ra sugkl nei
sto x(t) se ìlo to disthma 1 < k < 1. To dio isqÔei kai gia thn trigwnometrik
seir Fourier. Shmei¸noume ìti, ìtan to s ma e nai periodikì, h olokl rwsh sti
exis¸sei anlush mpore na g nei se èna auja reto disthma eÔrou T .
3.2.4 Ìparxh seir Fourier
To er¸thma pou t¸ra t jetai e nai en kai ktw apì poie pro pojèsei èna s ma
mpore na anaptuqje se seir Fourier. Mèqri t¸ra apl¸ upojèsame ìti to anp-
tugma autì uprqei. ApodeiknÔetai ìti, an plhroÔntai orismène sunj ke uprqei to
anptugma enì s mato se seir Fourier.
Enìthta 3.2 Anptugma Fourier - Seir Fourier 73
Ikan Sunj kh 1. H sunrthsh (s ma) x(t) na e nai apìluta oloklhr¸simh sto
disthma eÔrou T , dhlad ,
Z t0 +T
jx(t)jdt < +1 (3.2.15)
t0
H sunj kh aut exasfal zei ìti kje suntelest ak e nai peperasmèno . Prgmati,
gia kje suntelest ak e nai
Z t0 +T
ja j 1 jx(t)e jk!0 t jdt 1 Z t0 +T jx(t)jdt
k
T t0 T t0
kai lìgw th (3.2.15), èqoume ak < j j 1. àna s ma to opo o paraba nei th sunj kh
aut e nai to s ma tou Sq mato 3.2a
x(t) = ;
1 0<t1
t
x(t) x(t) x(t)
1
1
0
2
1
4
1 t
0 1 t 0 2 4 6 8 t
(a) (â) (ã)
Sq ma 3.2 S mata to opo a den ikanopoioÔn ti sunj ke Dirichlet.
Ikan Sunj kh 2. H sunrthsh (s ma), x t , se kje peperasmèno qronikì ()
disthma prèpei na e nai suneq perièqei peperasmèno arijmì asuneqei¸n, kajem -
a apì ti opo e e nai peperasmènou Ôyou . àna s ma sto disthma ; to opo o [0 8℄
ikanopoie thn pr¸th sunj kh, en¸ paraba nei th deÔterh sunj kh e nai to s ma x t ()
8
1; 0 t < 4
>
>
1=2; 4 t < 6
>
<
x(t) = 1=4; 6 t < 7
>
> >
: .. ..
. .
Sto Sq ma 3.2b blèpoume to grfhma tou s mato x t . ParathroÔme ìti to embadì ()
()
ktw apì th x t se qronikì disthma T =8
e nai mikrìtero apì 8 me apotèlesma h
pr¸th sunj kh ikanopoie tai.
Ikan Sunj kh 3. H sunrthsh (s ma), x t na e nai fragmènh kÔmansh , ()
dhlad , na uprqei peperasmèno arijmì meg stwn kai elaq stwn sto disthma. àna
74 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
s ma to opo o ikanopoie thn pr¸th sunj kh kai paraba nei thn tr th e nai to s ma
tou Sq mato 3.2g
x(t) = sin
2 ; 0 < t 1
t
Oi sunj ke , autè e nai gnwstè w sunj ke Dirichlet. Sunart sei pou plhroÔn
ti sunj ke 2 kai 3 qarakthr zontai w tmhmatik omalè . En èna s ma plhro ti
sunj ke Dirichlet, tìte uprqei o metasqhmatismì Fourier tou.
Me to pardeigma pou akolouje ja prospaj soume na anade xoume th fusik
shmas a th ekjetik seir Fourier kai na prosdior soume th sqèsh th me th tri-
gwnometrik seir Fourier.
Pardeigma 3.2.1
JewroÔme to periodikì s ma x(t) me jemeli¸dh kuklik suqnìthta 2, to opo o d netai
apì thn
+5
X
x(t) = ak ejk!0 t (3.2.16)
k= 5
a0 = 1; a1 =a 1= 1 ; a2 = a 2 = 0; a3 =a 3= 1 ; a4 = a 4 = 0 kai
ìpou
2 6
a5 = a 5 = 101 .
Me antikatstash twn suntelest¸n sthn (3.2.16) èqoume
x(t) = 1 +
1 1
ej2t + e j2t + ej6t + e j6t +
1 ej10t + e j10t
2 6 10 (3.2.17)
kai me th bo jeia th tautìthta tou Euler to s ma grfetai
x(t) = 1 + os(2t) + 13 os(6t) + 15 os(10t) (3.2.18)
H (3.2.18) apotele to trigwnometrikì anptugma tou x(t). H kataskeu tou s mato
x(t), apì armonik sunhmitonoeid s mata, fa netai sto Sq ma 3.3.
Ja doÔme ìti ta sumpersmata tou Parade gmato 3.2.1 genikeÔontai gia kje pra-
()
gmatikì s ma. An to x t e nai pragmatikì, tìte x? t x t epomènw ()= ()
+1
X +1
X +1
X +1
X
a?k e jk!0 t = ak ejk!0 t ) a? k ejk!0t = ak ejk!0 t
k= 1 k= 1 k= 1 k= 1
dhlad , prokÔptei ìti
a? k = ak a?k = a k (3.2.19)
An oi suntelestè Fourier ak e nai pragmatiko arijmo , ja e nai ak =a k.
Enìthta 3.2 Anptugma Fourier - Seir Fourier 75
x0(t)=1
1
0 t
x0(t)+x1(t)
x1(t)=cos(2ðt ) 2
1 1
0 t 0 t
-1
x0(t)+x1(t)+x2(t)
2
x2(t)= 13 cos(6ðt )
1
0 t 0 t
x0(t)+x1(t)+x2(t)+x3(t)
2
x3(t)= 15 cos(10ð t)
1
0 t 0 t
Sq ma 3.3 Kataskeu tou s mato x(t) apì armonik susqetizìmena sunhm tona.
An diaspsoume to jroisma sto deÔtero mèlo th (3.2.5), èqoume
X1 +1
X
x(t) = ak ejn!0 t + a0 + ak ejk!0t
k= 1 k=1
+1
X +1
X
= a ke jk!0 t + a0 + ak ejk!0t )
k=1 k=1
+1 h
X i
= a0 + a ke jk!0 t + ak ejk!0t
k=1
+1 h
X i
= a0 + ak ejk!0 t + a?k e jk!0 t
k=1
kai me th bo jeia th ( 3.2.19) kai epeid <e[z℄ = (z + z? )=2 èqoume
1
X h i
x(t) = a0 + 2<e ak ejk!0 t (3.2.20)
k=1
76 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
An ekfrsoume to ak se polikè suntetagmène ak = jak jejk , èqoume
+1
X h i
x(t) = a0 + 2<e jak jej(k!0t+k ) (3.2.21)
k=1
+1
X
x(t) = a0 + Ak os(k!0t + k ) (3.2.22)
k=1
ìpou Ak = 2j j
ak . Parathr ste ìti gia th dhmiourg a tou sunhmitìnou suqnìthta k!0
summetèqoun h ant stoiqh jetik kai arnhtik suqnìthta tou ekjetikoÔ anaptÔgmato .
Apì ti (3.2.6), (3.2.10) kai (3.2.11) me th bo jeia th tautìthta tou Euler èqoume
2 =
ak bk j k . àtsi, an antikatast soume to ak sthn (3.2.20), èqoume
1
X
x(t) = a0 + <e [(bk j k )( os(k!0t) + j sin(k!0 t))℄
k=1
1
X
= a0 + [bk os(k!0 t) + k sin(k!0t)℄ (3.2.23)
k=1
Ta trigwnometrik anaptÔgmata Fourier, pou perigrfoun oi (3.2.22) kai (3.2.23), gi-
a pragmatik s mata e nai isodÔnama me thn ekjetik seir Fourier kai perièqoun
mìno jetikè suqnìthte . H anptuxh, ìmw , se migadik s mata e nai pio eÔkolh apì
majhmatik poyh kai me aut n kur w ergazìmaste. Ston P naka 3.1 uprqoun oi
exis¸sei sÔnjesh th ekjetik kai th trigwnometrik seir Fourier enì peri-
odikoÔ s mato kai oi sqèsei metaxÔ twn ant stoiqwn suntelest¸n.
PINAKAS 3.1 SEIRES FOURIER
ak = T1 tt00 +T x(t)e jk!0 t dt
P+1 R
x(t) = k= 1 ak ejk!0 t
x(t) = a0 + +k=1
P 1
Ak os(k!0 t + k ) Ak = 2jak j
P1
x(t) = a0 + k=1 [bk os(k!0 t) + k sin(k!0 t)℄ 2ak = bk j k
Pardeigma 3.2.2
Na upologiste h mèsh isqÔ kje ìrou th ekjetik seir Fourier (3.2.5).
LÔsh Apì thn (1.2.16) h mèsh isqÔ s mato e nai
Px = Tlim 1 Z T
jx(t)j2 dt
!1 2T
(3.2.24)
T
An h olokl rwsh g nei se qronikì disthma [t0 ; t0 + T ℄, èqoume
Px = T1
Z
jx(t)j2 dt (3.2.25)
<T >
Enìthta 3.2 Anptugma Fourier - Seir Fourier 77
O k sto
ìro tou migadikoÔ anaptÔgmato Fourier, ejk!0 t prosfèrei sto jroisma mèsh
isqÔ
Px = T1
Z
ak ejk!0 t a?k e jk!0 t dt = jak j2 (3.2.26)
<T >
3.2.5 Tautìthta tou Parseval
H olik mèsh isqÔ enì periodikoÔ s mato e nai sh me to jroisma twn isqÔwn
ìlwn twn ìrwn th ekjetik seir Fourier, dhlad
1 Z 1
X
Px = T
jx(t)j2 dt = jak j2 (3.2.27)
<T > k= 1
H sqèsh aut onomzetai tautìthta tou Parseval kai ekfrzei th dunatìthta upolo-
gismoÔ th isqÔo sto ped o qrìnou kai sto ped o suqnot twn.
Apìdeixh
H (3.2.25) me th bo jeia th ex swsh anlush d nei diadoqik
1
Px = T1
Z X
x(t) a?k e jk!0 t dt
<T > k= 1
1
X 1 Z
= T
a?k x(t)e jk!0 t dt
k= 1 <T >
1
X 1
X
= a?k ak = jak j2 (3.2.28)
k= 1 k= 1
An to s ma x(t) e nai pragmatikì lìgw th (3.2.19), èqoume
1
Px = T1
Z X
jx(t)j2 dt = ja0 j2 + 2 jak j2 (3.2.29)
<T > k=1
Pardeigma 3.2.3
Na upologiste h mèsh isqÔ kje ìrou th trigwnometrik seir Fourier (3.2.14).
LÔsh O k sto
ìro th os(k!0t + k ) kai prosfèrei isqÔ
(3.2.14) e nai o Ak
Px = T1
Z
A2k os2 (k!0 t + k ) dt
<T >
2Z 1 + os2(k!0t + k ) dt
= ATk 2
<T >
A2k A2k
Z Z
= 2T dt +
2T os2(k!0t + k ) dt
<T > <T >
(3.2.30)
78 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
epeid to pr¸to olokl rwma e nai so me T kai to deÔtero olokl rwma e nai so me
mhdèn, èqoume
2
Pk = A2k (3.2.31)
H mèsh isqÔ tou s mato lìgw th (3.2.31) ja e nai
1 A2k
Px = T1
Z X
jx(t)j2 dt = ja0 j2 + 2 2 (3.2.32)
<T > k=1
pou prokÔptei kai apì thn (3.2.29) gia Ak = 2jak j. An ekfrsoume to ak sthn (3.2.29)
se kartesianè suntetagmène 2ak = bk j k , èqoume
1 Z
1X 1
Px = x(t)j2 dt = ja0 j2 + 2 2
T <T >
j 4 k=1 bk + k (3.2.33)
Apì ta Parade gmata 3.2.2 kai 3.2.3 èqoume ti akìlouje parathr sei
1. Gia pragmatik s mata, epeid a?k a k èqoume ak=a k . àtsi, h isqÔ thj j=j j
2
P =j j
k armonik , k ak , sthn ekjetik seir Fourier (3.2.5) e nai sh me thn
sth
isqÔ th ksth armonik , k a k 2 .
P =j j
2. Gia pragmatik s mata isqÔei ìti Ak = 2j j
ak , dhlad ta plth tou trigw-
nometrikoÔ anaptÔgmato e nai sa me to diplsio twn ant stoiqwn suntelest¸n
tou ekjetikoÔ anaptÔgmato gia ti jetikè timè tou k . Epomènw , h isqÔ th
ksth armonik sthn trigwnometrik seir Fourier (3.2.22), k A2k = , e nai P = 2
sh me to jroisma th isqÔo th k sth armonik kai th ksth armonik sthn
ekjetik seir. H Ôparxh arnhtik suqnìthta gia pragmatik s mata e nai
apìrroia th anaparstash tou s mato me th bo jeia migadik¸n shmtwn kai
èqei w apotèlesma na moirzei ex sou thn isqÔ metaxÔ jetik kai arnhtik
armonik . Sthn pragmatikìthta to arnhtikì mèro tou fsmato den ma parè-
qei kami plhrofor a, prgma pou epibebai¸netai apì ti exis¸sei (3.2.22) kai
(3.2.23). Fusik, autì den isqÔei gia migadik s mata.
Pardeigma 3.2.4
Na upologistoÔn oi suntelestè th ekjetik seir Fourier tou s mato x(t) =
os(!0t).
LÔsh Me th bo jeia th sqèsh tou Euler to s ma grfetai
1 1
x(t) = ej!0 t + e j!0 t
2 2 (3.2.34)
H sÔgkrish th teleuta a ex swsh me thn ex swsh sÔnjesh d nei
1
a1 = a 1 = ; ak = 0; k = 0; 2; 3; :::
2 (3.2.35)
Enìthta 3.2 Anptugma Fourier - Seir Fourier 79
sto Sq ma 3.4 èqoun sqediaste oi suntelestè th ekjetik seir Fourier.
1
2
1
2
an
-2 -1 0 1 2 3 n
Sq ma 3.4 Oi suntelestè Fourier tou s mato x(t) = os(!0 t).
Pardeigma 3.2.5
Na upologistoÔn oi suntelestè th ekjetik seir Fourier tou s mato x(t) =
sin(!0t).
LÔsh Me th bo jeia th sqèsh tou Euler to s ma grfetai
x(t) =
1 j!0t 1 e j!0 t
2j e 2j (3.2.36)
H sÔgkrish th teleuta a ex swsh me thn ex swsh sÔnjesh d nei
a1 =
1 1
2j ; a 1 = 2j ; ak = 0; k = 0; 2; 3; ::: (3.2.37)
sto Sq ma 3.5 èqei sqediaste to mètro kai h fsh twn suntelest¸n th ekjetik seir
Fourier.
1 1 an
2 2 ð arg an
2
1
-2 -1 0 1 2 3 n -2 -1 0 2 3 n
ð
2
Sq ma 3.5 To mètro kai h fsh twn suntelest¸n Fourier tou s mato x(t) = sin(!0 t).
Pardeigma 3.2.6
Na upologistoÔn oi suntelestè th ekjetik seir kai th trigwnometrik seir
Fourier gia to periodikì orjog¸nio s ma
x(t) = 1; jtj < T1
0; T1 < jtj < T0=2 (3.2.38)
LÔsh Sto Sq ma 3.6 èqei sqediaste to periodikì orjog¸nio s ma. ParathroÔme ìti
to s ma èqei jemeli¸dh per odo T0 kai kuklik suqnìthta !0 = 2=T0. Oi suntelestè
Fourier upolog zontai me th bo jeia th ex swsh anlush , ìpou h olokl rwsh g netai
apì T0=2 èw T0 =2.
àtsi, gia k = 0 èqoume
a0 =
1 Z T1
dt =
2T1 (3.2.39)
T0 T1 T0
80 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
x(t)
1
T0 T0 T1 0 T1 T0 T0 2T0 t
2 2
Sq ma 3.6 To periodikì orjog¸nio s ma tou Parade gmato 3.2.6.
en¸ gia k 6= 0 èqoume
a0
1
= T0
Z T1
e jk!0 t dt
T1
= 1 e jk!0 t T1
jk!0 T0 T1
= 1 ejk!0 T1 e jk!0 T1
jk!0 T0
= 2 sin( k!0 T1)
k! T 0 0
(3.2.40)
kai epeid !0T0 = 2, èqoume
ak =
sin(k!0T1) (3.2.41)
k
Gia thn per ptwsh, ìpou T0 = 4T1, to x(t) e nai summetrikì orjog¸nio s ma, dedomènou
ìti isqÔei ìti !0 T1 = =2 oi suntelestè Fourier d nontai apì th sqèsh
ak
k
= sin
k 2
; k 6= 0 kai a0 =
1
2 (3.2.42)
a1 = a 1 = 1 1 1
àtsi,
; a3 = a 3 = 3 ; a5 = a 5 = 5 ; : : : kai ak = 0, an k rtio .
Sto Sq ma 3.7 èqoun sqediaste oi suntelestè Fourier gia to periodikì orjog¸nio s ma,
an to T1 e nai stajerì, kai gia diaforetikè timè sthn per odo T0 .
Me th bo jeia twn (3.2.13) upolog zoume tou suntelestè th trigwnometrik seir
Fourier
a0 = 2TT01 ;
bk = 2 sin(kk!0 T1 )
; gia k = 1; 3; ::: kai bk = 0; gia k = 2; 4; :::
k = 0; gia k = 1; 2; 3; ::: (3.2.43)
Enìthta 3.2 Anptugma Fourier - Seir Fourier 81
sin ( kð
2 ( Ô0 = 4Ô1
ak kð
-2 0 2 k
sin ( kð
4 ( Ô0 = 8Ô1
ak kð
-4 0 4 k
sin ( kð
8 ( Ô0 = 16Ô1
ak kð
-8 0 8 k
Sq ma 3.7 Oi suntelestè Fourier gia to periodikì orjog¸nio s ma gia stajerì T1 kai
diaforetikè timè sthn per odo T0 .
3.2.6 Fainìmeno Gibbs
A doÔme t¸ra ti sumba nei an prospaj soume na prosegg soume to periodikì s ma
()
x t apì to peperasmèno jroisma
N
X
xN (t) = ak ejk!0 t (3.2.44)
k= N
2
sto opo o qrhsimopoioÔntai h suneq kai mìno N armonikè sunist¸se tou fs-
mato . To sflma prosèggish e nai eN t ()= ()
x t xN t . Sto Sq ma 3.8 èqoume ()
sqedisei thn prosèggish tou periodikoÔ orjog¸niou s mato apì thn (3.2.44) gia
difore timè th paramètrou N .
()
H prosèggish enì s mato , x t , to opo o parousizei asunèqeie peperasmènou
Ôyou , apì èna jroisma me s mata sugkekrimènwn suqnot twn th morf ejk!0 t ,
ta opo a e nai suneqe sunart sei , dhmiourge sto shme o asunèqeia tou s mato
()
x t talant¸sei . Epiplèon, sta shme a asunèqeia to grfhma tou s mato xN t ()
dièrqetai apì to mèso th asunèqeia pou parousizei to s ma x t sto shme o autì, ()
dhlad mpore na apodeiqje
1 [x(t ) + x(t+ )℄
xN (t) =
2 (3.2.45)
ìpou x(t ) kai x(t+ ) e nai ta ìria tou s mato x(t) apì ta arister kai dexi, an-
t stoiqa, sto shme o asunèqeia .
82 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
xN(t) N=1
t
xN(t) N=3
t
xN(t) N=7
t
xN(t) N=19
t
Sq ma 3.8 H prosèggish tou periodikoÔ orjog¸niou s mato apì to merikì jroisma (3.2.44)
gia difore timè th paramètrou N.
To plto twn talant¸sewn e nai anexrthto tou pl jou twn suqnot twn pou
suneisfèroun sthn prosèggish tou s mato ()
x t apì th (3.2.44). äso to N auxne-
tai tìso perissìtere suqnìthte suneisfèroun sthn prosèggish tou s mato . ätan
N !1 , tìte ìle oi armonikè suqnìthte lambnoun mèro kai to s ma x t anak- ()
ttai pl rw . Ant jeta, an to N e nai peperasmèno, uprqoun suqnìthte pou de
lambnontai upìyh sto jroisma. Autì èqei w apotèlesma na parathroÔntai talan-
t¸sei sto shme o asunèqeia .
Se ant jesh me to plto twn talant¸sewn pou paramènei anallo wto ìso to N
auxnetai, to eÔro th perioq , sthn opo a entop zontai oi talant¸sei , te nei sto
mhdèn. To fainìmeno autì e nai gnwstì w fainìmeno Gibbs.
3.3 METASQHMATISMOS FOURIER
Sthn prohgoÔmenh enìthta, e dame pw èna periodikì s ma mpore na anaptuqje sto
disthma ( 1 1)
; se m a seir Fourier, dhlad , na parastaje w èna grammikì
sunduasmì ape rwn armonik¸n ekjetik¸n shmtwn. Sthn enìthta aut , ja doÔme
ìti ta apotelèsmata aut mporoÔn na epektajoÔn kai se mh periodik s mata, sto
disthma ( 1 1)
; . Epishma noume, gia llh m a for, ìti gia mh periodik s mata
to anptugma se seir Fourier e nai dunatì se peperasmènou eÔrou diast mata.
Enìthta 3.3 Metasqhmatismì Fourier 83
Sto Pardeigma 3.2.6 e dame ìti oi suntelestè th seir Fourier tou periodikoÔ
orjog¸niou s mato e nai
ak =
2sin(k!0T1 ) (3.3.1)
k!0 T0
ìpou T0 e nai h per odo kai !0 =2
=T0 h kuklik suqnìthta. Upenjum zetai ìti oi
suntelestè Fourier ak fasmatikè grammè tou s mato prosdior zoun th suneis-
for kje suqnìthta sto anptugma Fourier tou s mato ìpw ep sh ìti apoteloÔn
to fsma tou s mato , to opo o gia to lìgo autì qarakthr zetai w grammikì fsma.
H ex swsh (3.3.1) mpore na apokt sei th morf
ak =
2T1 sin(k!0T1 ) = 2T1 sin(x) (3.3.2)
T0 k!0 T1 T0 x
ìpou x =
k!0 T1 .
sin(x)
H sunrthsh x apotele thn peribllousa tou fsmato , dhlad , oi fas-
matikè grammè , oi opo e br skontai sti suqnìthte k!0 , e nai fragmène apì th
sunrthsh aut , ìpw fa netai sto Sq ma 3.9.
Apì to Sq ma 3.9 èqoume ti akìlouje parathr sei gia to fsma:
ak 2T1
a0 T0
ÐåñéâÜëëïõóá
2ð 2T1 sin(ùT1 )
Äù T0 T0 ùT1
0 ù
ð 2ð ð
ù T1 ù0 T0
ù T1
Sq ma 3.9 Oi suntelestè Fourier kai h peribllous tou gia to periodikì orjog¸nio kÔma.
1. H suneq sunist¸sa tou fsmato e nai a0 = 2TT01 .
2. H jemeli¸dh suqnìthta e nai !0 = 2T0 .
3. H apìstash metaxÔ twn fasmatik¸n gramm¸n e nai ! = 2T0 .
4. O pr¸to mhdenismì th peribllousa tou fsmato g netai ìtan
sin(k!0 T1) = 0 ) k!0T1 = ) k = 2TT0
1
(an 2TT01 den e nai akèraio arijmì , tìte den uprqei fasmatik gramm sth
suqnìthta aut ).
84 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
5. H suqnìthta tou pr¸tou mhdenismoÔ e nai !=
T1 .
A upojèsoume t¸ra ìti h per odo T0 auxnetai en¸ diathroÔme stajerì to T1 .
To apotèlesma th aÔxhsh aut sth morf tou orjog¸niou kÔmato fa netai sta
Sq mata 3.10a1 ; a2 ; a3 , sta opo a èqei sqediaste to periodikì kÔma gia T0 T1 , =4
T0 =8 T1 kai T0 = 16
T1 . ParathroÔme ìti krat¸nta thn tim tou T1 stajer ,
diathroÔme stajer th qronik dirkeia twn orjog¸niwn palm¸n kai auxnonta thn
tim th periìdou T0 auxnoume thn orizìntia apìstash twn orjog¸niwn palm¸n
pou apoteloÔn to orjog¸nio kÔma. Sta Sq mata 3.10 1 ; 2 ; 3 , èqoume sqedisei ta
ant stoiqa fsmata, gia ta opo a parathroÔme ìti kaj¸ auxnetai h per odo tou
orjog¸niou kÔmato
ak 1
2
Ô0 = 4Ô1
2T1
Ô0 = 4Ô1
t ð
0 ð ù
T1 T1
( a1) ( â1)
ak 1
4
2T1
Ô0 = 8Ô1
t ð
0 ð ù
Ô0 = 8Ô1 T1 T1
( a2) ( â2)
ak 1
2T1 8
Ô0 = 16Ô1
Ô0 = 16Ô1
t ð
0 ð ù
T1 T1
( a3) ( â3)
Sq ma 3.10 To periodikì orjog¸nio kÔma kai oi fasmatikè grammè tou gia stajer tim
T1 kai gia diaforetikè timè periìdou T0.
1. To plto twn fasmatik¸n gramm¸n elatt¸netai.
Enìthta 3.3 Metasqhmatismì Fourier 85
2. H apìstash metaxÔ twn fasmatik¸n gramm¸n elatt¸netai
3. To pl jo twn fasmatik¸n gramm¸n, pou perièqontai ston kentrikì lobì, auxne-
tai.
4. H suqnìthta tou pr¸tou mhdenismoÔ den metablletai
5. H peribllousa tou fsmato ,
sin(x) , diathre stajer morf .
x
ätan T0 !1 , to arqikì periodikì kÔma metatrèpetai sto mh periodikì s ma
tou orjog¸niou palmoÔ. Ep sh , ìtan T0 !1
, èqoume th dhmiourg a enì ape rou
pl jou fasmatik¸n gramm¸n me plto to opo o te nei sto mhdèn, kai h metaxÔ tou
apìstash te nei ep sh sto mhdèn.
H pore a thn opo a perigryame den e nai katllhlh gia na petÔqoume to fsma
enì aploÔ palmoÔ. Sth sunèqeia ja doÔme pw sto stìqo autì mporoÔme na ftsoume
w m a oriak per ptwsh twn ìswn dh gnwr zoume.
Gia to periodikì orjog¸nio kÔma tou Parade gmato 3.2.6 to ginìmeno th periìdou
T0 ep to suntelest ak grfetai
T0 ak =
2sin(k!0 T1) = 2sin(!T1 ) (3.3.3)
k!0 ! !=k!0
Sto Sq ma 3.11 apod dontai oi grafikè parastsei tou ginomènou T0 ak se sunrthsh
me thn kuklik suqnìthta ! gia stajer tim tou T1 kai gia diaforetikè timè tou
T0 .
ätan to T0 auxnetai, to pl jo twn suntelest¸n th seir Fourier g netai ìlo
kai megalÔtero, en¸ ta ant stoiqa ginìmena paramènoun stajer. To disthma metaxÔ
twn deigmtwn T0 ak g netai ìlo kai mikrìtero, dhlad ta de gmata plhsizoun ìlo
kai perissìtero metaxÔ tou kai mporoÔme na poÔme ìti telik, ìtan T0 !1
, to
sÔnolo twn ginomènwn T0 ak plhsizei thn peribllousa. Kai to ant stoiqo fsma
g netai suneqè . Aut e nai h basik idèa tou metasqhmatismoÔ Fourier.
2 sin(!T1 ) , h opo a e nai h peribllousa twn T a apotele
H suneq sunrthsh ! 0 k
to metasqhmatismì Fourier tou orjog¸niou palmoÔ. Oi suntelestè tou anaptÔgmato
Fourier tou orjog¸niou kÔmato e nai isapèqonta de gmata th peribllousa , dhlad ,
tou metasqhmatismoÔ Fourier. H de apìstas tou prosdior zetai apì thn per odo
mèsw th sqèsh
! = 2T (3.3.4)
0
Sth sunèqeia ja genikeÔsoume ta parapnw sumpersmata gia kje mh periodikì
()
s ma. àstw èna mh periodikì s ma x t peperasmènh dirkeia , dhlad x t ()=0
jj
an t > T1 (Sq ma 3.12a) gia to opo o upojètoume ìti uprqei to anptugma Fourier.
86 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
T0 ak
2T1
Ô0 = 4Ô1 2ù0
Ô0 = 4Ô1
t ð
0 ð ù
T1 T1
( a1) ( â1)
T0 ak
2T1
Ô0 = 8Ô1 4ù0
t ð
0 ð ù
Ô0 = 8Ô1 T1 T1
( a2) ( â2)
T0 ak
2T1
Ô0 = 16Ô1 8ù0
Ô0 = 16Ô1
t ð
0
ð ù
T1 T1
( a3) ( â3)
Sq ma 3.11 (a) To periodikì orjog¸nio kÔma kai (b) ta ginìmena T0 ak kai h peribllous
tou , gia stajer tim T1 kai gia diaforetikè timè periìdou T0 .
()
Me th bo jeia tou s mato x t dhmiourgoÔme to periodikì s ma x t me per odo ~( )
( 2 ) ()
T0 T0 > T1 , tou opo ou to x t apotele m a per odo (Sq ma 3.12b).
~( )
Epeid to s ma x t e nai periodikì, mpore na anaptuqje se seir Fourier gia
ìla ta t ;2 ( 1 +1
). Oi exis¸sei sÔnjesh kai anlush tou s mato x t e nai ~( )
1
X
x~(t) = ak ejk!0 t (3.3.5)
k= 1
T0
ak =
1Z 2
x~(t)e jk!0 t dt (3.3.6)
T0 2
T0
Epeid x~(t) = x(t) gia jtj T0 , oi suntelestè th seir Fourier d nontai apì thn
2
T0
ak =
1Z 2
x(t)e jk!0 t dt (3.3.7)
T0 T0
2
Enìthta 3.3 Metasqhmatismì Fourier 87
x(t)
T1 0 T1 t
(á)
x(t)
T0 T1 0 T1 T0 2T0 t
( â)
Sq ma 3.12 (a) Mh periodikì s ma x(t) kai (b) x~(t) h periodik epèktash tou x(t).
kai, afoÔ x t ( )=0
èxw apì to disthma olokl rwsh , èqoume telik gia tou sunte-
lestè th seir Fourier
ak =
1 Z 1
x(t)e jk!0 t dt (3.3.8)
T0 1
Or zoume th migadik sunrthsh X (!) th pragmatik metablht !
Z +1
X (!) = x(t)e j!t dt (3.3.9)
1
Me th bo jeia th sunrthsh aut oi suntelestè ak mporoÔn na ekfrastoÔn w
ak =
1 X (k!0 ) (3.3.10)
T0
kai apì thn (3.3.5) to s ma x~(t), dhlad , h periodik epanlhyh tou x (t), d netai apì
thn
+1
X 1 X (k!0 )ej!0t
x~(t) = (3.3.11)
T
k= 1 0
kai epeid !0 = 2=T0
1 +1
X
x~(t) =
2 k= 1 X (k!0 )e !0
jk!0 t (3.3.12)
A jewr soume t¸ra ìti to disthma T0 auxnei suneq¸ , me T0 . H apìstash !1
loipìn metaxÔ twn diadoqik¸n armonik¸n, !0 =2
=T0 , suneq¸ elatt¸netai kai te nei
sto mhdèn !0 ( ! )
d! kai to k!0 g netai h suneq metablht ! k!0 ! . àtsi, ( ! )
to fsma g netai suneqè kai to jroisma sto deÔtero mèlo th (3.3.12) grfetai w
88 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
olokl rwma. Ep sh , to s ma x~(t) prosegg zei to s ma x(t) kai to s ma x(t) d netai
apì thn ex swsh
Z +1
x(t) =
1
2 1 X (!)e d!
j!t (3.3.13)
H ex swsh (3.3.13) apotele thn ex swsh sÔnjesh kai anasunjètei to s ma sto ped o
tou qrìnou. H sunrthsh
Z +1
X (! ) = x(t)e j!t dt (3.3.14)
1
apotele thn ex swsh anlush kai e nai o Metasqhmatismì Fourier (MF) tou s mato
()
x t . Akribèstera, metasqhmatismì Fourier e nai o kanìna eÔresh th X ! apì ()
() ()
thn x t , dhlad , h (3.3.14). H sunrthsh X ! (pou e nai m a apeikìnish X IR : !
Z ) lègetai metasqhmatismì Fourier. Suqn anaferìmaste se autìn kai w to fsma
tou s mato . O metasqhmatismì Fourier èqei nìhma gia ìlo to disthma ; ( 1 +1)
kai anaparist mh periodik s mata me th bo jeia ekjetik¸n shmtwn kai me ton
trìpo autì anadeiknÔetai to fasmatikì tou perieqìmeno.
Parathr sei
1. Sto anptugma se seir Fourier, h ex swsh anlush analÔei èna s ma x t sto ()
[
disthma t0 ; t0 T + ℄
sto disthma ; ( 1 +1)
an to s ma e nai periodikì se
èna diakritì fsma periodik¸n ekjetik¸n shmtwn me armonik susqetizìmene
suqnìthte , pollaplsie th jemeli¸dou kuklik suqnìthta !0 , sto opo o
h armonik k txh èqei “plto " ak . An, gia pardeigma, to s ma x t e nai ()
s ma tsh , h monda mètrhsh twn suntelest¸n ak e nai “Volts".
2. Sto metasqhmatismì Fourier, h ex swsh anlush analÔei èna mh periodikì s -
()
ma x t sto disthma ; ( 1 +1)
se èna suneqè fsma periodik¸n ekjetik¸n
shmtwn. To fasmatikì perieqìmeno sto apeirostì disthma suqnot twn !; ! [ +
℄ () [
d! e nai X ! . H suneisfor twn suqnot twn !; ! d! èqei “plto " X ! + ℄ ()
( 2) ()
d!= . An, gia pardeigma, to x t e nai s ma tsh , tìte o X ! èqei monda ()
mètrhsh “Volts an monda suqnìthta ". O metasqhmatismì Fourier den e nai,
loipìn, èna fsma pltou , all fasmatik puknìthta pltou .
An ant th ! qrhsimopoi soume th suqnìthta f = !=2, oi exis¸sei anlush
kai sÔnjesh pa rnoun th morf
Z +1
X (f ) = x(t)e j 2ft dt (3.3.15)
1
Z +1
x(t) = X (f )ej 2ft df (3.3.16)
1
Enìthta 3.3 Metasqhmatismì Fourier 89
()
O metasqhmatismì Fourier X ! , gia kje tim th suqnìthta ! , e nai migadik
sunrthsh kai, epomènw , mpore na anaparastaje se polik morf
X (!) = jX (!)j ej arg X (!) (3.3.17)
se kartesian morf
X (!) = <e [X (!)℄ + j =m [X (!)℄ = R(!) + jI (!) (3.3.18)
Pardeigma 3.3.1
Na upologiste o metasqhmatismì Fourier tou orjog¸niou palmoÔ dirkeia T1
x(t) = 1; jtj < T1
0; alli¸ (3.3.19)
LÔsh Epeid to s ma e nai mhdèn gia t < T1 kai t > T1 , o metasqhmatismì Fourier
e nai
Z +T1 1 Z +T1
j j!t +T1
X (! ) = e j!t dt = j! e j!t d ( j!t) = e T1
T1 T1 !
= 2 sin(!!T1) (3.3.20)
Sto Sq ma 3.13 d netai h grafik parstash tou s mato x(t) kai o metasqhmatismì
Fourier tou.
x(t) X(ù)
2T1
1
ð ð
T1 T1
T1 0 T1 t 0 ù
(á) ( â)
Sq ma 3.13 (a) O orjog¸nio palmì kai (b) O metasqhmatismì Fourier tou.
Parathr sei
1. O metasqhmatismì Fourier, X (!), tou orjog¸niou palmoÔ e nai pragmatik
sunrthsh.
2. H tim tou metasqhmatismoÔ Fourier sto mhdèn e nai
X (0) = !lim
2sin(!T1 ) (3.3.21)
!0 !
90 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
0 1,
Efarmìzoume ton kanìna L’ Hospital gia ti aprosdiìriste morfè 0 1
sÔmfwna me ton opo o
f (x) 0 1 f (x) f 0 (x)
an lim
x!1 g (x)
= 0 1 tìte lim
x!1 g (x)
= lim
x!1 g 0 (x)
(3.3.22)
kai èqoume
X (0) = lim 2T1 os(!T1) = 2T1 (3.3.23)
!!0
3. ()
Oi timè sti opo e mhden zetai to X ! e nai ta fasmatik mhdenik, d non-
tai apì thn ex swsh sin(
!T1 )=0
kai e nai oi suqnìthte ! k=T1 , k = =
1 2
; ; :::.
4. To fsma te nei sto mhdèn kaj¸ pernme se polÔ uyhlè suqnìthte , dhlad ,
j j!1
! .
5. An jewr soume to olokl rwma sÔnjesh se peperasmèno disthma suqnot twn
Z +W
x^W (t) =
1 2 sin(!T1) ej!t d!
2 W !
(3.3.24)
parousizetai to fainìmeno Gibbs. Dhlad , to x^W (t) parousizei kumnsei
gÔrw apì to shme o asunèqeia , to plto twn opo wn den elatt¸netai kaj¸
to W auxnei all sumpièzontai gÔrw apì thn asunèqeia kai h enèrgei tou
te nei sto mhdèn, ìtan W . !0
6. Sto ìrio W ! 1, h (3.3.24) pa rnei th morf
Z +1
x^W (t) =
1 2 sin(!T1) ej!t d!
2 1 !
(3.3.25)
äpw kai sto pardeigma tou periodikoÔ orjog¸niou s mato (Sq ma 3.8), e nai
^( ) = ( )
x t x t , ektì apì ta shme a asunèqeia t T1 ìpou x t 12 , pou e nai
= ^( ) =
()
h mèsh tim twn tim¸n tou x t sti dÔo pleurè th asunèqeia .
Pardeigma 3.3.2
Na upologiste o metasqhmatismì Fourier th sunrthsh x(t) = Æ(t)
LÔsh O metasqhmatismì Fourier th sunrthsh dèlta e nai
Z 1
X (! ) = Æ(t)e j!t dt = 1 (3.3.26)
1
R1
ìpou qrhsimopoi jhke h
1 x(t) Æ(t t0 ) dt = x(t0 ) ParathroÔme ìti to fsma th
Æ(t) kalÔptei ìlo to eÔro suqnot twn.
Enìthta 3.3 Metasqhmatismì Fourier 91
3.3.1 Ìparxh tou metasqhmatismoÔ Fourier
Sthn prohgoÔmenh enìthta or same to metasqhmatismì Fourier èqonta upojèsei ìti ta
oloklhr¸mata (3.3.13) kai ( 3.3.14) uprqoun. Ta oloklhr¸mata aut den uprqoun
pnta e nai dunatì na uprqei to èna kai na mhn uprqei to llo. Oi sunj ke
Dirichlet e nai ikanè sunj ke gia na uprqoun kai ta dÔo oloklhr¸mata, ta opo a
apoteloÔn to zeÔgo metasqhmatism¸n Fourier.
()
Ikan Sunj kh 1. H sunrthsh (s ma) x t na e nai apìluta oloklhr¸simh,
dhlad , Z 1
jx(t)j dt < 1 (3.3.27)
1
H sunj kh aut exasfal zei thn Ôparxh tou olokl rwmato sth (3.3.14). Prgmati,
Z 1 Z 1
jX (!)j = x(t)e j!t dt jx(t)j dt < 1 (3.3.28)
1 1
Ikan Sunj kh 2. H sunrthsh (s ma) x t e nai suneq ()
perièqei peperasmèno
arijmì asuneqei¸n, kje m a apo ti opo e na e nai peperasmènou Ôyou .
()
Ikan Sunj kh 3. H sunrthsh (s ma) x t e nai fragmènh kÔmansh .
Pardeigma 3.3.3
Na upologiste o metasqhmatismì Fourier tou aitiatoÔ ekjetikoÔ s mato
x(t) = e at u(t) a 2 R (3.3.29)
LÔsh Epeid to s ma e nai so me mhdèn gia t < 0, o metasqhmatismì Fourier tou
e nai
Z +1 Z +1
X (!) = e at e j!t dt = e (a+j!)t dt
0 0
= 1 e (a+j!)t 1 = 1 h lim e (a+j!)t e0
i
a + j! 0 a + j! t!1
kai epeid
lim e (a+j!)t = lim e = tlim
!1 e [ os(!t) j sin(!t)℄ = 0 ìtan a > 0
at e j!t at
t!1 t!1
o metasqhmatismì Fourier uprqei gia a > 0 kai e nai
X (!) =
1
a + j!
(3.3.30)
Sto Sq ma 3.14 d netai h grafik parstash tou s mato x(t) kai oi grafikè parast-
sei tou mètrou kai th fsh tou metasqhmatismoÔ Fourier. ParathroÔme ìti to mètro
aposbènei sti uyhlè suqnìthte , dhlad , limj!j!1 jX (!)j = 0.
92 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
x(t) X(ù) arg X(ù)
ð
1 2
á
1 1
ð
4
á2
á
-á t
ð
4
0 t -á 0 á t ð
2
(á) ( â) (ã)
Sq ma 3.14 H grafik parstash (a) tou s mato x(t) = e t u(t), > 0, (b) tou mètrou
kai (g) th fsh tou metasqhmatismoÔ Fourier tou.
Pardeigma 3.3.4
Na upologiste to s ma, tou opo ou o metasqhmatismì Fourier e nai parjuro suqnot -
twn me plto W , dhlad ,
X (!) = 1; j!j < W
0; alli¸ (3.3.31)
LÔsh Epeid o metasqhmatismì Fourier tou s mato e nai so me mhdèn gia !< W
kai ! > W , to s ma ja e nai
Z +W
x(t) =
1 1 j!t +W
2 W e d! = 2jt e W
j!t
= 2jt1 ejW t e jW t = 1 2j sin(W t)
2jt
= sin(tW t) (3.3.32)
Sto Sq ma 3.15 d netai to grfhma tou s mato x(t) sto ped o suqnot twn kai sto
ped o tou qrìnou.
x(t)
X(ù) W
ð
1
ð ð
W W
W 0 W ù 0 t
(á) ( â)
Sq ma 3.15 Perigraf tou s mato x(t) (a) sto ped o suqnot twn kai (b) sto ped o tou
qrìnou.
Enìthta 3.3 Metasqhmatismì Fourier 93
Oi exis¸sei X ! ( )= 2 sin(!T )
1 kai x t ()= sin(W t)
! t , ti opo e sunant same sta
Parade gmata 3.3.1 kai 3.3.4, mporoÔn na ekfrastoÔn me enia o trìpo me th bo jeia
th sunrthsh sin sin(t) ; t 6= 0
sin (t) = 1; t t=0
(3.3.33)
kai e nai gnwst w sunrthsh deigmatolhy a , h grafik parstash th opo a
sinc(t)
1
-5 -4 -3 -2 -1 0 1 2 3 4 5 t
Sq ma 3.16 H sunrthsh sin (x).
fa netai sto Sq ma 3.16. ParathroÔme ìti h sunrthsh dièrqetai periodik apì to
mhdèn kai ìti to Ôyo twn deutereuìntwn lob¸n mei¸netai asumptwtik sto mhdèn. H
sunrthsh aut e nai idia terh shmas a kai thn sunantme suqn tìso sthn epexer-
gas a shmtwn (anlush Fourier, melèth GQA susthmtwn) ìso kai sti epikoinwn e .
Me th bo jeia th sunrthsh deigmatolhy a , o metasqhmatismì Fourier tou Pa-
()
rade gmato (3.3.1) kai to s ma x t tou Parade gmato (3.3.4) grfontai w
X (!) =
2sin(!T1 ) = 2T1 sin !T 1 = 2T1 sin
!T1
(3.3.34)
! !T1
x(t) =
sin(W t) = W sin W t
= W
sin
Wt
(3.3.35)
t W t
3.3.2 Idiìthte tou metasqhmatismoÔ Fourier
Sthn enìthta aut ja parousisoume ti basikè idiìthte pou èqei o metasqhmatismì
Fourier. Gia eukol a, o metasqhmatismì Fourier tou s mato x t merikè forè sum- ()
bol zetai w F [ ( )℄
x t kai h sqèsh metaxÔ tou x t kai tou metasqhmatismoÔ Fourier ()
tou upodeiknÔetai w
x(t) F! X (!) (3.3.36)
(1) Suzug a
An to s ma x(t) èqei metasqhmatismì Fourier X (!), tìte
x? (t) F! X ? ( !) (3.3.37)
94 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
Apìdeixh
O metasqhmatismì Fourier tou suzugoÔ s mato e nai
Z 1 Z 1 ?
F [ (t)℄ =
x? x? (t)e j!t dt = x(t) ej!t dt = X ? ( !)
1 1
IsqÔei ep sh
x? ( t) F! X ? (!) (3.3.38)
(2) Grammikìthta
An x1 (t) F! X1 (!) kai x2 (t) F! X2 (!), tìte
F! X (!) + X (!)
1 x1 (t) + 2 x2 (t) 1 1 2 2 (3.3.39)
H apìdeixh e nai mesh sunèpeia th grammikìthta tou oloklhr¸mato .
(3) rtio-perittì mèro s mato . Pragmatikì-fantastikì mèro fsmato
()
äpw e nai gnwstì, (blèpe 1.2.7) kje s ma x t mpore na ekfraste w jroisma
() ()
enì rtiou, xe t , kai enì perittoÔ s mato , xo t . An X ! e nai o metasqhmatismì ()
()
Fourier tou s mato x t , tìte èqoume
xe (t) F! <e[X (!)℄ (3.3.40)
xo (t) F! j =m[X (!)℄ (3.3.41)
H apìdeixh apotele meso epakìloujo th grammikìthta kai th suzug a .
(4) Ol sjhsh sto qrìno
An x(t) F! X (!), tìte gia kje pragmatikì arijmì t0 isqÔei
x(t t0 ) F! e j!t0 X (!) (3.3.42)
Apìdeixh
O metasqhmatismì Fourier tou s mato x(t t0 ) e nai
Z 1
F [x(t t0)℄ = x(t t0 )e j!t dt
1
jètw =t t0 , opìte èqw
Z 1 Z 1
F [x(t t0)℄ = x( )e j!( +t0 ) d =e j!t0 x( )e j! d =e j!t0 X (!)
1 1
Enìthta 3.3 Metasqhmatismì Fourier 95
ParathroÔme ìti, an to s ma metatopiste sto ped o tou qrìnou kat t0 , to fsma
tou pollaplasizetai me to fasmatikì pargonta e j!t0 . àtsi, to fsma enì s -
mato olisjhmènou sto qrìno èqei to dio mètro me to arqikì s ma, en¸ h fsh tou
metablletai grammik. Prgmati, an xt X! X ! ej(!) , tìte
F [ ( )℄ = ( ) = j ( )j
F [x(t t0 )℄ = e j!t0 X (!) = jX (!)j ej[(!) !t0 ℄
(5) Ol sjhsh suqnìthta
An x(t) F! X (!), tìte gia kje pragmatikì arijmì !0 isqÔei
ej!0 t x(t) F! X (! !0 ) (3.3.43)
Apìdeixh
Me th bo jeia tou antistrìfou metasqhmatismoÔ Fourier br skoume ìti to s ma,
( )
pou èqei metasqhmatismì Fourier X ! !0 , e nai to ej!0 t x t . Prgmati, ()
1 Z 1 X (! !0 )ej!t d! !0 =! !0
= 1 Z 1 X (!0 )ej(!0 +!0)t d!0
2 1 2 Z1
ej!0 t 1 0
= 2 1 X (!0 )ej! t d!0
= e 0 t x(t)
j!
Pardeigma 3.3.5 (H bsh th diamìrfwsh ).
Na upologiste o metasqhmatismì Fourier tou s mato
z (t) = x(t) os(!0 t) (3.3.44)
LÔsh Me th bo jeia th sqèsh tou Euler to s ma z (t) grfetai
z (t) = x(t) os(!0t) = x(t) 12 [ej!0 t + e j!0 t ℄ = 12 x(t)ej!0 t + 21 x(t)e j!0 t
Me th bo jeia th idiìthta th grammikìthta kai th ol sjhsh suqnìthta , o metasqh-
matismì Fourier tou z (t) e nai
1
Z (!) = F x(t)e j!0 t + F 1 x(t)e j!0 t = 12 [X (! !0 ) + X (! + !0 )℄
2 2
H idiìthta aut apotele th bsh th diamìrfwsh pou qrhsimopoie tai eurèw
()
sti thlepikoinwn e . Kat th diamìrfwsh, èna s ma x t pou metafèrei sugkekrimè-
nh plhrofor a pollaplasizetai me èna s ma apl suqnìthta !0 t , h opo a os( )
96 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
onomzetai fèrousa, me skopì thn ekpomp tou se èna mèso metdosh , p.q., zeÔgo
surmtwn, atmìsfaira, klp.
ParathroÔme ìti o pollaplasiasmì , tou s mato x t me to !0 t den al- () os( )
loi¸nei th morf tou metasqhmatismoÔ Fourier X ! (me thn pro pìjesh ìti to !0 ()
()
e nai arket meglo kai to X ! e nai mhdèn èpeita apì m a orismènh suqnìthta,
ìpw sto Sq ma 3.17a), all metafèretai sthn perioq twn suqnot twn !0 , ìpw
perigrfetai sto Sq ma 3.17b.
X(ù)
A
W 0 W ù
(á)
F x(t) cos (ù0 t)
2W 2W
A
2
ù W 0 ù0 ù W0 0 ù W
0 ù 0 ù W
0
ù
( â)
Sq ma 3.17 H diamìrfwsh pltou (a) to fsma tou s mato mhnÔmato gia èna auja reto
s ma x(t) kai (b) to fsma tou diamorfwmènou s mato .
(6) Allag kl maka sto qrìno kai th suqnìthta - Anklash
An x(t) F! X (!), tìte gia kje pragmatikì arijmì a isqÔei
x(at) F! 1 X ! kai 1 x t F! X (a!) (3.3.45)
jaj a jaj a
Apìdeixh
O metasqhmatismì Fourier tou s mato x(at) e nai
Z 1
F [x(at)℄ = x(at)e j!t dt
1
jètoume =
at kai diakr noume dÔo peript¸sei
0
an a > e nai
1
F [x(at)℄ = a
Z 1
x( )e j !a d = a1 X
!
1 a
an a < 0 e nai
1
F [x(at)℄ = a
Z 1
x( )e j !a d =
1 Z 1
x( )e j !a d = a1 X !a
+1 a 1
Enìthta 3.3 Metasqhmatismì Fourier 97
H idiìthta th allag kl maka parousizetai sto Sq ma 3.18. ParathroÔme
1
ìti, an a > , to s ma sumpièzetai sto ped o tou qrìnou, me sunèpeia na metablletai
pio gr gora sth monda tou qrìnou. An analogistoÔme ìti oi gr gore metabolè sto
qrìno antistoiqoÔn se suneisfor apì uyhlìtere suqnìthte sto ped o suqnot twn,
sumpera noume ìti to fsma tou diastèlletai sto ped o suqnot twn (Sq ma 3.18b).
0 1
Ant jeta, an < a < , to s ma diastèlletai sto ped o tou qrìnou, me sunèpeia na
metablletai pio arg sth monda tou qrìnou kai, epeid èna s ma qamhl suqnìth-
ta metablletai me argoÔ rujmoÔ , to fsma tou sumpièzetai (Sq ma 3.18g). An
x(t) X(ù)
2T1
1
ð ð
T1 T1
T1 0 T1 t 0 ù
(á) ÓÞìá x(t) êáé ôï öÜóìá ôïõ X(ù).
x(t) X(ù)
0 t 0 ù
(â) ÓÞìá x1(t) = x(at) ìå á > 1 êáé ôï öÜóìá ôïõ X1(ù).
x(t) X(ù)
0 t 0 ù
( ã) ÓÞìá x2(t) = x(at) ìå 0 < á < 1 êáé ôï öÜóìá ôïõ X2(ù).
Sq ma 3.18 Apeikìnish th allag kl maka .
a= 1, prokÔptei h idiìthta th Anklash
x( t) F! X ( !) (3.3.46)
(7) Je¸rhma th Sunèlixh
M a apì ti shmantikè idiìththte tou metasqhmatismoÔ Fourier, ìson afor
th qr sh tou sta grammik qronik anallo wta sust mata, e nai h ep dras tou
sth leitourg a th sunèlixh . Gnwr zoume ìti h èxodo y t enì GQA sust mato , ()
98 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
me kroustik apìkrish h (t), ìtan h e sodì tou e nai to s ma x(t), d netai apì to
olokl rwma th sunèlixh
Z 1
y(t) = x( )h(t ) d (3.3.47)
1
Sthn pargrafo aut ja anade xoume th sqèsh pou sundèei tou metasqhmatismoÔ
twn antisto qwn shmtwn. O metasqhmatismì Fourier th exìdou tou sust mato
e nai
Z 1
Y (!) = y(t)e j!t dt
1
Z 1 Z 1
= x( )h(t ) d e j!t dt
Z
1 1Z
1 1
= x( ) h(t )e d j!t dt
1 1
Me allag metablht = t èqoume
Z 1 Z 1
Y (!) = x( ) h( )e j! ( + ) d d
Z
1 1Z
1 1
= x( )e j! h( )e j! d d
1 1
To perieqìmeno th agkÔlh [℄ e nai o metasqhmatismì Fourier th h(t) (h(t) F!
( ))
H ! , ètsi Z 1
Y (!) = H (!) x( )e j! d
1
an X (!) e nai o metasqhmatismì Fourier tou s mato x(t), èqoume
y(t) = h(t) ? x(t) F! Y (!) = H (!) X (!) (3.3.48)
ParathroÔme ìti h upologistik polÔplokh sqèsh th sunèlixh metasqhmatizìme-
nh kat Fourier katal gei se èna aplì ginìmeno sunart sewn.
To je¸rhma th sunèlixh ma d nei th dunatìthta na upolog soume to fsma tou
s mato exìdou enì GQA sust mato an gnwr zoume to fsma tou s mato eisìdou
()
X ! kai to fsma th kroustik apìkrish H ! tou sust mato . ()
Anlogh sqèsh isqÔei kai gia th sunèlixh twn metasqhmatism¸n Fourier X ! kai ()
() () ()
Y ! twn shmtwn x t kai y t ant stoiqa, dhlad ,
F 1
x(t) y(t) ! X (!) ? Y (!) =
1 Z 1
2 2 X ()Y (! 1
) d (3.3.49)
Enìthta 3.3 Metasqhmatismì Fourier 99
Pardeigma 3.3.6
àstw s ma x(t) me metasqhmatismì Fourier X (!). Ja upolog soume to s ma pou èqei
metasqhmatismì Fourier jX (! )j2 = X (! ) X ? (! ).
LÔsh Me th bo jeia th idiìthta th suzug a , th anklash kai tou jewr mato
th sunèlixh , br skoume ìti to s ma, Rx( ), to opo o èqei metasqhmatismì Fourier
jX (!)j2 , d netai apì thn
Z 1 Z 1
Rx ( ) = x( ) ? x? ( ) = x(t) x? (t ) dt = x(t + )x? (t) dt
1 1
()
To s ma Rx kale tai sunrthsh autosusqètish tou x t kai parèqei èna mètro ()
()
tou susqetismoÔ twn tim¸n tou s mato x t gia dÔo qronik stigmiìtupa pou diafè-
roun kat . Th sunrthsh autosusqètish ja th sunant soume kai sthn Enìthta 3.4.
(8) Je¸rhma tou Parseval
To je¸rhma tou Parseval ekfrzei th dunatìthta eÔresh th enèrgeia enì s -
mato e te sto ped o tou qrìnou e te sto ped o suqnot twn.
Ex =
Z 1
jx(t)j2 dt = 1 Z 1 jX (!)j2 d!
1 2 1 (3.3.50)
SÔmfwna me to je¸rhma tou Parseval h olik enèrgeia enì s mato mpore na upo-
logiste e te a) upolog zonta thn enèrgeia an monda qronikoÔ diast mato xt 2 j ( )j
kai oloklhr¸nonta gia ìlo to qrìno e te b) upolog zonta thn enèrgeia an monda
jX (!)j2
kuklik suqnìthta 2 kai oloklhr¸nonta gia ìle ti suqnìthte .
Apìdeixh
Gia to pr¸to mèlo th isìthta èqoume
Z 1 Z 1
jx(t)j2 dt = x(t)x? (t) dt
1 1
=
Z 1
x(t)
1 Z 1
2 1 X (!)e
? j!t d! dt
1
allzonta th seir olokl rwsh èqoume
Z 1
Z 1
jx(t)j2 dt 1 Z 1
= 2 X (!)
? x(t)e j!t dt d!
1 1 1
= 21 X ? (!)X (!) d!
Z 1
1
1 Z 1
= 2 jX (!)j2 d!
1
100 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
() =1
An x t e nai h tsh sta kra ant stash R R , tìte h enèrgeia pou parèqetai
1 x2 t dt. Apì to dexiì mèlo
sthn ant stash d netai apì to olokl rwma x 1 E = ()
th (3.3.50) èqoume ìti h enèrgeia x isoÔtai me to 21 tou embadoÔ pou perikle ei h
E
kampÔlh X ! 2 . H posìthta loipìn X f 2 ekfrzei thn katanom th enèrgeia
j ( )j j ( )j
an monda suqnìthta kai onomzetai fasmatik puknìthta enèrgeia tou s mato
() E
x t . Me lla lìgia h stoiqei¸dh enèrgeia d pou suneisfèroun oi suqnìthte pou
br skontai sthn perioq f; f (
df + ) (
!; ! d! e nai + )
dE = jX (f )j2 df ) ddfE = jX (f )j2
(9) Parag¸gish
An x(t) F! X (!), kai uprqei o metasqhmatismì Fourier th parag¸gou dxdt(t) ,
tìte
dx(t) F
dt
! j!X (!) (3.3.51)
Apìdeixh
MporoÔme na apode xoume thn idiìthta an proume to metasqhmatismì Fourier th
parag¸gou, dhlad ,
dx(t) 1 dx(t)
Z
F dt
= dt
e j!t dt (3.3.52)
1
kai oloklhr¸soume kat pargonte
dx(t)
Z 1
1 de j!t
F dt
= x(t)e j!t
1 x (t) dt
dt
1
Z 1
1
= x(t)e j!t
1 + j! 1 x(t)e
j!t dt
Gia thn apìdeixh upojètoume ìti, ìtan t ! 1 to s ma x(t) ! 0, opìte e nai kai
limjtj!1 x(t)ej!t = 0. àtsi èqoume
dx(t)
F dt
= j!X (!) (3.3.53)
Epanalhptik efarmog th parapnw idiìthta d nei th genik èkfrash th idiìthta
parag¸gish sto qronikì ped o
dn x(t) F
dtn
! (j!)n X (!) (3.3.54)
Enìthta 3.3 Metasqhmatismì Fourier 101
Me parìmoio trìpo skèyh èqoume gia thn parag¸gish sto ped o suqnot twn
n X (! )
( jt)n x(t) F! d d! n (3.3.55)
Pardeigma 3.3.7
Na upologiste o metasqhmatismì Fourier th sunrthsh pros mou sgn(t)
sgn(t) = 1;1; t>0
t<0 (3.3.56)
LÔsh ParathroÔme ìti
dsgn(t)
dt
= 2Æ(t)
Lambnonta to metasqhmatismì Fourier sta dÔo mèlh th parapnw ex swsh èqoume
F dsgn(
dt
t)
= F [2Æ(t)℄
E nai ìmw F [Æ(t)℄ = 1 (Pardeigma 3.3.2) kai lìgw th idiìthta th parag¸gish
èqoume
j!F [sgn(t)℄ = 2 ) F [sgn(t)℄ =
2 ; ! 6= 0 (3.3.57)
j!
(10) Olokl rwsh
An x(t) F! X (!), tìte
Z t
x( ) d F!
1 X (!) + X (!)Æ(!) (3.3.58)
1 j!
Apìdeixh
An y t ( ) = Rt ()
1 x d , tìte h y (t) mpore na jewrhje w h sunèlixh th x(t) kai
th sunrthsh monadia ou b mato u(t), dhlad ,
Z 1
y(t) = x(t) ? y(t) = x( )u(t ) d
1
Me th bo jeia tou jewr mato th sunèlixh prokÔptei ìti
Y (!) = X (!)U (!) = X (!) Æ(!) +
1 = 1 X (!) + X (!)Æ(!)
j! j!
M a epipìlaia efarmog th idiìthta th parag¸gish ja mporoÔse na ma odhg sei
se esfalmèna sumpersmata. Prgmati,
) dydt(t) = x(t) ) j!Y (!) = X (!)
Z t
y(t) = x( ) d
1
102 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
apì thn teleuta a sqèsh den sunepgetai ìti Y ! ( ) = Xj!(! all Y (!) = Xj!(! + CÆ(!)
ìpou C m a stajer, diìti isqÔei j!Æ ! j Æ! ( ) = 0 ( ) = 0.
(11) Summetr e gia pragmatik s mata
() ()
àstw x t pragmatikì s ma kai X ! o metasqhmatismì Fourier, o opo o den e -
nai apara thta kai autì pragmatikì arijmì . Ja de xoume ìti isqÔoun oi summetr e
X ( !) = X ? (!)
<e[X ( !)℄ = <e[X (!)℄ (3.3.59)
=m[X ( !)℄ = =m[X (!)℄
Apìdeixh
()
Epeid to s ma x t e nai pragmatikì, ja e nai x? (t) = x(t). àtsi, apì thn idiìthta
th summetr a èqoume
x(t) = x? (t) F! X (!) = X ? ( !)
dhlad , to fsma e nai suzug rtia sunrthsh th suqnìthta .
Me th bo jeia th sqèsh tou Euler èqoume
Z 1
X (!) = x(t)e j!t dt
Z 1
1
= x(t)[ os(!t) j sin(!t)℄ dt
Z
1 Z
1 1
= x(t) os(!t) dt j x(t)sin(!t) dt
1 1
kai epeid to s ma e nai pragmatikì èqoume ìti
Z 1
<e[X (!)℄ = x(t) os(!t) dt kai
Z1 1
=m[X (!)℄ = x(t)sin(!t) dt (3.3.60)
1
Apì ti teleuta e sqèsei sumpera noume
<e[X (!)℄ = <e[X ( !)℄ kai
jX (!)j = jX ( !)j (3.3.61)
dhlad , to pragmatikì mèro kai to mètro th e nai rtie sunart sei , kai
=m[X (!)℄ = =m[X ( !)℄ kai
arg[X (!)℄ = arg[jX ( !)℄ (3.3.62)
Enìthta 3.3 Metasqhmatismì Fourier 103
dhlad , to fantastikì mèro kai h fsh apoteloÔn perittè sunart sei .
Mpore eÔkola na apodeiqje ìti oi sqèsei (3.3.59 ) apoteloÔn kai anagka e
()
sunj ke gia na e nai to s ma x t pragmatikì. Prgmati, èstw
X (!) = R(!) + jI (!)
ìpou R(! ) kai I (! ) to pragmatikì tm ma kai to fantastikì tm ma th X (!). To s ma
x(t) ja e nai
x(t) =
1 Z 1
2 Z 1 X (!)e d!
j!t
= 21 [R(!) + jI (!)℄[ os(!t) + j sin(!t)℄ d!
1
1
1 Z 1
= 2 [R(!) os(!t) I (!)sin(!t)℄ d!
1
1 Z 1
+j 2 [R(!)sin(!t) + I (!) os(!t)℄ d!
1
An diaspsoume to deÔtero olokl rwma se dÔo oloklhr¸mata, me ìrio olokl rwsh
apì 1èw 0 to pr¸to kai apì 0 èw 1
to deÔtero, èqoume gia to fantastikì mèro
()
tou x t
=m[x(t)℄ = 1 Z 0 [R(!)sin(!t) + I (!) os(!t)℄ d!
2 Z1
+ 21 [R(!)sin(!t) + I (!) os(!t)℄ d!
1
0
= 1 Z 1[R( !)sin( !t) + I ( !) os( !t)℄ d!
2 0Z
+ 21 [R(!)sin(!t) + I (!) os(!t)℄ d!
1
0
= 1 Z 1[ R(!)sin(!t) I (!) os(!t)℄ d!
2 0Z
+ 21 [R(!)sin(!t) + I (!) os(!t)℄ d!
1
0
= 0
àtsi, to s ma x(t) e nai pragmatikì kai d netai apì th sqèsh
x(t) =
1 Z 1
2 1[R(!) os(!t) I (!)sin(!t)℄ d! (3.3.63)
104 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
Apì thn trigwnometr a, ep sh , gnwr zoume
R(!) os(!t) I (!)sin(!t) = A(!) os[!t + (!)℄ (3.3.64)
p
ìpou A(! ) = R2 (! ) + I 2 (! ) e nai to plto tou X (! ) kai (! ) = tan 1 I (!) e nai
R(!)
h fsh tou, ètsi èqoume gia to pragmatikì s ma x(t)
x(t) =
1 Z 1
2Z 1 A(!) os[!t + (!)℄ d!
= 1 A(!) os[!t + (!)℄ d!
1
(3.3.65)
0
Apì thn (3.3.65 ) parathroÔme ìti me th bo jeia tou metasqhmatismoÔ Fourier X ! ( )=
( )+ ( ) ()
R ! jI ! èna pragmatikì s ma x t anaptÔssetai se èna peiro (mh arijm simo)
pl jo shmtwn apl¸n suqnot twn. Kajem a apì ti aplè autè suqnìthte upei-
[ () ℄ ()
sèrqetai me plto A ! = d! kai fsh ! , ìpou ! e nai h kuklik suqnìthta.
Autì e nai kai o lìgo pou h metablht ! tou metasqhmatismoÔ Fourier anafèretai
kai w kuklik suqnìthta. Apìrroia autoÔ e nai kai h onomas a tou metasqhmatismoÔ
Fourier w fsma suqnot twn, kat' analog a th anlush pou uf statai to leukì fw
sti epimèrou suqnìthte pou to apart zoun.
Shmei¸noume ìti, an èna s ma e nai fantastikì, dhlad x t ()= ()
jy t ìpou y t ()
e nai èna s ma pragmatikì, tìte eÔkola apodeiknÔetai ìti
X ( !) = X ? (!)
<e[X ( !)℄ = <e[X (!)℄ (3.3.66)
=m[X ( !)℄ = =m[X (!)℄
(12) Duðsmì
Thn idiìthta tou duðsmoÔ tou metasqhmatismoÔ Fourier thn èqoume dh sunant sei
sta Parade gmata 3.3.1 kai 3.3.4, ìpou e dame ìti o metasqhmatismì Fourier enì
orjog¸niou palmoÔ èqei th morf mia sunrthsh sin
kai o metasqhmatismì Fourier
mia sunrthsh sin èqei th morf enì orjog¸niou palmoÔ. àstw x(t) F! X (!),
( ) = X (t) èqei metasqhmatismì Fourier
tìte to s ma y t
Y (!) = 2x( !) (3.3.67)
Apìdeixh
Apì thn ex swsh anlush èqoume
X (!) =
Z 1
x(t)e j!t dt 1 Z 1
= 2 2x(t)e j!t dt
1 1
Enìthta 3.3 Metasqhmatismì Fourier 105
Me antikatstash tou t me t èqoume
X (!) =
1 Z 1
2 1 2x( t)e dt
j!t
En enallxoume to t me !, èqoume
X (t) =
1 Z 1
2 1 2x( !)e d!
j!t
An sugkr noume thn teleuta a ex swsh me th ex swsh sÔnjesh , èqoume
X (t) F! 2x( !)
Sto Sq ma 3.19 perigrfetai sqhmatik h idiìthta tou duðsmoÔ.
x(t) X(ù)
2T1
1
F
ð ð
T1 T1
T1 0 T1 t 0 ù
x(t)
W X(ù)
ð
1
F
ð ð
W W
0 t W 0 W ù
Sq ma 3.19 H idiìthta duðsmoÔ tou metasqhmatismoÔ Fourier.
Efarmogè
1) Sthn skhsh 3.8 èqoume de xei
x(t) = e ajtj F! X (!) = 2
2a
a + !2
àtsi, to s ma
y(t) = 2
2
t +1
(3.3.68)
èqei metasqhmatismì Fourier
Y (!) = 2e j!j (3.3.69)
Ston P naka 3.2 parousizontai oi idiìthte tou metasqhmatismoÔ Fourier.
106 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
PINAKAS 3.2 Idiìthte tou metasqhmatismoÔ Fourier
Idiìthta Ped o qrìnou Ped o suqnìthta
Suzug a sto qrìno x? (t) X ? ( !)
Suzug a sth suqnìthta x? ( t) X ? (!)
Anklash x( t) X ( !)
Grammikìthta ax1 (t)+bx2 (t) aX1 (!)+bX2 (!)
rtio mèro s mato xe (t)= 12 [x(t)+x? ( t)℄ <e[X (!)℄=R(!)
Pragmatikì mèro fsmato
Perittì mèro s mato xo (t)= 12 [x(t) x? ( t)℄ j =m[X (!)℄=jI (!)
Fantastikì mèro fsmato
Qronik metatìpish x(t t0 ) e j!t0 X (!)
Ol sjhsh suqnìthta ej!0 t x(t) X (! !0 )
Rt
1 x( ) d
Olokl rwsh 1 X (!)+X (!)Æ(!)
j!
X (!)=X ? (!)
<e[X (!)℄=<e[X ( !)℄
Pragmatikì s ma x(t)=x? (t) =m[X (!)℄= =m[X ( !)℄
jX (!)j=jX ( !)j
arg X (!)= arg X ( !)
Sunèlixh x(t)?y(t) X (!)Y (!)
Diamìrfwsh x(t)y(t) 2 [X (!)?Y (!)℄
1
Parag¸gish sto
dx(t) j!X (!)
dt
qronikì ped o
Parag¸gish sto tx(t) j dXd!(!)
ped o suqnot twn
x(at) jaj X ( a )
Allag kl maka 1 !
Duðsmì an F [x(t)℄ = X (!) y(t)=X (t) Y (!)=2x( !)
R1
Je¸rhma Parseval Ex = 1 jx(t)j2 dt Ex = 21 R 11 jX (!)j2 d!
2) Sto Pardeigma 3.3.2 èqoume de xei ìti Æt F [ ( )℄ = 1
. Lìgw th idiìthta tou
duðsmoÔ kai, epeid h kroustik sunrthsh e nai rtia, sunepgetai ìti èna suneqè
s ma èqei m a fasmatik sunist¸sa gia ! =0
, dhlad ,
1 F! 2Æ( !) = 2Æ(!) (3.3.70)
Enìthta 3.3 Metasqhmatismì Fourier 107
Pardeigma 3.3.8
Na upologiste o metasqhmatismì Fourier tou monadia ou b mato .
LÔsh H sunrthsh u(t) mpore na grafe w
1 + 1 sgn(t)
u(t) =
2 2
1 F! Æ(!), sgn(t) F! 2 kai me th bo
Apì ta zeÔgh Fourier 2 j! jeia th idiìthta
th grammikìthta sunepgetai ìti o metasqhmatismì Fourier tou monadia ou b mato
e nai
u(t) F! Æ(!) +
2 (3.3.71)
j!
Pardeigma 3.3.9
Na upologiste o metasqhmatismì Fourier tou trigwnikoÔ s mato
jtj
t
T1
= 01; T1 ; jtj < T1
alli¸
(3.3.72)
LÔsh Paragwg zonta to trigwnikì s ma dÔo forè , èqoume
d2
t
dt2 T1
= T11 Æ(t + T1) T21 Æ(t) + T11 Æ(t T1 ) (3.3.73)
Sto Sq ma 3.20 eikon zontai to trigwnikì s ma, h pr¸th kai h deÔterh pargwgì tou.
Apì thn idiìthta th parag¸gish sto qronikì ped o èqoume
d t d2 t
Ë ( Ôt (
1 dt Ë Ô1( ( ( (
dt 2 Ë Ô1
1 1
Ô1 1 1
T1 Ô1 Ô1
T1 0 T1 t T1 0 t T1 T1 t
2
1 Ô1
Ô1
(á) (â) (ã)
Sq ma 3.20 H grafikè parastsei (a) tou trigwnikoÔ palmoÔ, (b) th pr¸th kai (g) th
deÔterh parag¸gou tou.
2
F d dtx(2t) = (j!)2 X (!) (3.3.74)
Gnwr zoume ìti F [Æ(t)℄ = 1 ètsi, lìgw th idiìthta th qronik metatìpish , èqoume
F [Æ(t T1 )℄ = ej!T1 (3.3.75)
108 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
Lambnonta to metasqhmatismì Fourier kai sta dÔo mèlh th (3.3.73) lìgw twn (3.3.74)
kai (3.3.75) èqoume
(j!)2F Tt1 = T1 ej!T1 2 + e j!T1
1
F Tt = os(
2 !T1 ) 2
T1 (j!)2
1
2 1=2)
= T1 sin(!T(!T
1 =2)2
= T1sin 2 !T1
2 (3.3.76)
Pardeigma 3.3.10
Na upologiste o metasqhmatismì Fourier tou s mato x(t) = os(!0t).
LÔsh An ekfrsoume to sunhm tono me th bo jeia th sqèsh tou Euler, w jroisma
migadik¸n ekjetik¸n ìrwn èqoume
1 1
x(t) = ej!0 t + e j!0 t
2 2 (3.3.77)
Gnwr zoume ìti 1 F! 2Æ(!). àtsi, efarmìzonta thn idiìthta th ol sjhsh suqnìth-
ta br skoume to metasqhmatismì Fourier tou s mato .
F [x(t)℄ = F [ os(!0 t)℄ = [Æ(! !0 ) + Æ(! + !0 )℄ (3.3.78)
Pardeigma 3.3.11
Na upologiste o metasqhmatismì Fourier tou s mato x(t) = os(!0t)u(t).
LÔsh Me th bo jeia th sqèsh tou Euler to s ma grfetai
1 1
x(t) = ej!0 t u(t) + e j!0 t u(t)
2 2 (3.3.79)
Lìgw th idiìthta th ol sjhsh th suqnìthta kai epeid F [u(t)℄ = j!1 + Æ(!) ,
èqoume
F [x(t)℄ = 21 j (! 1 ! ) + Æ(! !0) + 12 j (! +1 ! ) + Æ(! + !0 )
0 0
= 2 [Æ(! !0 ) + Æ(! + !0 )℄ + 2
j!
!0 !2
(3.3.80)
Ston P naka 3.3 uprqoun basik s mata kai oi ant stoiqoi metasqhmatismo
Fourier tou .
Enìthta 3.3 Metasqhmatismì Fourier 109
PINAKAS 3.3 Metasqhmatismo Fourier merik¸n basik¸n sunart sewn
A/A Ped o qrìnou Ped o suqnìthta
1 Æ(t) 1
2 1 2Æ(!) Æ(f )
u(t) 1 + Æ(!) 1 + 1 Æ(f )
3 j! j 2f 2
4 Æ(t t0 ) e j!t0
5 ej!0 t 2Æ(! !0 )
6 os(!0 t) [Æ(! !0 ) + Æ(! + !0 )℄
7 sin(!0 t) [Æ (! ! ) Æ (! + ! )℄
j P 0 0
P1 1
8 k = 1 ak e
jk!0 t 2 k= 1 ak Æ(! k!0 )
P1 2 P1 Æ ! 2k
9 k= 1 Æ (t nT ) T k= 1 T
10 2T1 = 10;; jjttjj<T
t
>T1
1
2T1 sin !T 1 = 2 sin(!!T1 )
11 W sin W t = sin(W t) X (! ) = 1; j!j<W
t 0; j!j>W
8
< 1 jtj ; jtjT1
12 t
T1 =: T1 T1 sin 2 !T
2
1
0; jtj>T1
sin(W t)2 1 2j!Wj ; j!j2W
13 W
Wt X (! ) = 0; j!j>2W
e at u(t);
<e[a℄ > 0 1
14 a+j!
te at u(t);
<e[a℄ > 0 1
15 (a+j!)2
1
(n 1)! e u(t); <e[a℄ > 0
tn 1 at
16 (a+j!)n
17 os(!0t)u(t) [Æ (! ! ) + Æ (! + ! )℄ + j!
2 0 0 !02 !2
18 sin(!0t)u(t) 2j [Æ(! !0 ) Æ(! + !0 )℄ + !02 !2
!0
e ajtj ; <e[a℄ > 0 2a
19 a2 +!2
1
20 t jsgn 2! jsgn(f )
21 sgn(t) = 1; t>0 1 2
1; t<0 jf j!
3.3.3 Metasqhmatismì Fourier periodik¸n shmtwn
()
H qr sh th sunrthsh Æ t , ma epitrèpei na prosdior soume to metasqhmatismì
Fourier kai gia periodik s mata.
110 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
O metasqhmatismì Fourier th sunrthsh dèlta e nai
Æ(t) F! 1 (3.3.81)
Ep sh ,
1 F! 2Æ( !) = 2Æ(!) (3.3.82)
Apì thn (3.3.81) kai (3.3.82) lìgw th idiìthta ol sjhsh èqoume ant stoiqa
Æ(t t0 ) F! e j!t0 (3.3.83)
ej!0 t F! 2Æ(! !0 ) (3.3.84)
äpw gnwr zoume, èna periodikì s ma anaptÔssetai se seir Fourier
1
X
x(t) = ak ejk!0 t
k= 1
Me th bo jeia th (3.3.84) kai lìgw th idiìthta th grammikìthta mporoÔme na
ekfrsoume to metasqhmatismì Fourier twn periodik¸n shmtwn w ex
1
X 1
X
x(t) = ak ejk!0 t F! X (!) = 2ak Æ(! k!0 ) (3.3.85)
k= 1 k= 1
An efarmìsoume ta parapnw sta Parade gmata 3.2.4 kai 3.2.6 èqoume ta ant stoiqa
fsmata sta Sq mata 3.21 kai 3.22.
ParathroÔme ìti o metasqhmatismì Fourier epekte netai kai sta periodik s -
()
mata. àtsi, to fsma X ! enì periodikoÔ s mato me periìdo T0 apotele tai apì
sunart sei dèlta omoiìmorfa katanemhmène se apìstash !0 =T0 , me Ôyo =2 2
forè ton ant stoiqo suntelest th seir Fourier.
X(ù)
ð ð X(ù)=ð[ä(ù−ù0)+ä(ù+ù0)]
ù0 0 ù0 ù
Sq ma 3.21 O metasqhmatismì Fourier gia to s ma x(t) = os(!0 t).
3.4 Enèrgeia kai IsqÔ
Oi ènnoie th enèrgeia kai th isqÔo enì s mato parousisthkan sto pr¸to ke-
flaio. Sthn enìthta aut ja epekte noume ti ènnoie autè tìso sto ped o tou
qrìnou, ìso kai sto ped o suqnot twn.
Enìthta 3.3 Enèrgeia kai IsqÔ 111
∞
X(ù) X(ù)= Ó 2ða ä(ù−kù )
k 0
ð k=−∞
2 2
ù0 0 ù0 ù
Sq ma 3.22 O metasqhmatismì Fourier gia to periodikì orjog¸nio kÔma.
3.4.1 Energeiak s mata
Gia èna energeiakì s ma x (t), ìpw èqoume dei kai sto Pardeigma 3.3.6, or zetai h
sunrthsh autosusqètish Rx ( ) w
Z 1 Z 1
Rx( ) = x( ) ? x? ( ) = x(t)x? (t ) dt = x(t + )x? (t) dt (3.4.1)
1 1
Sth sunèqeia ja doÔme merikè basikè idiìthte th sunrthsh autosusqètish .
a) H enèrgeia tou s mato e nai sh me thn tim th sunrthsh autosusqètish tou
()
x t , gia =0
. Prgmati,
Z 1
Rx(0) = jx(t)j2 dt = Ex (3.4.2)
1
b) O metasqhmatismì Fourier th sunrthsh autosusqètish enì s mato isoÔtai
me th fasmatik puknìthta enèrgeia tou s mato . H sunrthsh fasmatik puknìth-
ta enèrgeia perigrfei ton trìpo me ton opo o katanèmetai h enèrgeia tou s mato
sto q¸ro suqnot twn. Prgmati, lìgw tou jewr mato th sunèlixh tou metasqhma-
tismoÔ Fourier èqoume
Rx ( ) = x( ) ? x? ( ) ) F [Rx ( )℄ = jX (!)j2 (3.4.3)
kai apì to je¸rhma tou Parseval èqoume
Ex = Rx(0) =
Z 1
jx(t)j2 dt = 1 Z 1 jX (!)j2 d!
1 2 1 (3.4.4)
Sth sunèqeia, ja prosdior soume th sqèsh pou sundèei th sunrthsh autosusqè-
tish tou s mato eisìdou kai tou s mato exìdou enì GQA sust mato .
An s ma x(t) efarmoste sthn e sodo enì GQA sust mato me kroustik apìkri-
sh h(t) kai apìkrish suqnìthta H (!), tìte h èxodo tou sust mato e nai y(t) =
112 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
x(t) ? h(t) sto ped o suqnot twn Y (!) = X (!) H (!). H enèrgeia tou s mato
y(t) e nai
Ey =
Z 1
jy(t)j2 dt 1 Z 1
= 2 jY (!)j2 d!
1 1
1 Z 1
= 2 jX (!)j2 jH (!)j2 d! = Ry (0) (3.4.5)
1
( )= ( ) ( )
ìpou Ry y ?y? e nai h sunrthsh autosusqètish th exìdou tou sust -
mato . Qrhsimopoi¸nta ton ant strofo metasqhmatismì Fourier gia to Y ! 2 è- j ( )j
qoume
Ry ( ) = F 1 [jY (!)j2 ℄ = F 1 [jX (!)j2 jH (!)j2 ℄
= F 1 [jX (!)j2 ℄ ? F 1 [jH (!)j2 ℄
= Rx( ) ? Rh( ) (3.4.6)
Pardeigma 3.4.1
Na upologistoÔn h sunrthsh autosusqètish , h fasmatik puknìthta enèrgeia kai
h enèrgeia tou s mato x(t) = e at u(t); a > 0
LÔsh Gnwr zoume ìti
x(t) = e at u(t) F! X (!) =
1
a + j!
H fasmatik puknìthta enèrgeia tou s mato e nai
jX (!)j2 = a2 +1 !2
H sunrthsh autosusqètish isoÔtai me ton ant strofo metasqhmatismì Fourier th
fasmatik puknìthta enèrgeia tou s mato
Rx ( ) = F 1 [jX (!)j2 ℄ =
1e aj j
2a
ìpou qrhsimopoi jhke to zeÔgo MF 19 tou P naka 3.3. H enèrgeia isoÔtai me thn tim
pou èqei h sunrthsh autosusqètish sto mhdèn, ètsi
Ex = Rx(0) = 21a
Shmei¸netai ìti h enèrgeia tou s mato mpore na breje kai apì thn (1.2.13).
Enìthta 3.3 Enèrgeia kai IsqÔ 113
3.4.2 S mata isqÔo
Sthn enìthta aut ja or soume th mèsh qronik sunrthsh autosusqètish enì s -
mato isqÔo kai ja diatup¸soume ti basikè th idiìthte .
H mèsh qronik sunrthsh autosusqètish gia èna s ma isqÔo x t or zetai w ()
Rx( ) = Tlim 1 Z T
x(t)x? (t ) dt
!1 2T
(3.4.7)
T
H isqÔ tou s mato Px e nai sh me th mèsh qronik sunrthsh autosusqètish gia
=0
. Prgmati,
Px = Tlim 1 Z T
jx(t)j2 dt = Rx(0)
!1 2T T
(3.4.8)
()
àstw Sx ! o metasqhmatismì Fourier th mèsh qronik sunrthsh autosusqèti-
sh , tìte èqoume
Rx( ) = 21 Z 1
Sx (!)e d! ) Rx (0) =
1 Z 1
2 1 Sx(!) d!
j!
1
àtsi mporoÔme na ekfrsoume thn isqÔ tou s mato x(t) me th bo jeia th Sx (! ).
Prgmati,
Px = Rx(0) = 2 1 Z 1
Sx(!) d! (3.4.9)
1
H sunrthsh Sx (! ) perigrfei ton trìpo me ton opo o katanèmetai h isqÔ tou s ma-
to sto q¸ro twn suqnot twn kai onomzetai fasmatik tou s mato
puknìthta isqÔo
x(t).
An to s ma x(t), efarmoste sthn e sodo enì GQA sust mato me kroustik
apìkrish h(t) kai apìkrish suqnìthta H (! ), tìte h mèsh qronik sunrthsh auto-
susqètish th exìdou d netai apì th sqèsh
Ry ( ) = Rx( ) ? h( ) ? h( ) (3.4.10)
H apìdeixh th (3.4.10) e nai pèra apì ta pla sia tou parìnto egqeirid ou. O endi-
aferìmeno anagn¸sth parapèmpetai sto bibl o [6℄ sthn Anafor.
Lambnonta to metasqhmatismì Fourier kai twn dÔo pleur¸n th (3.4.10) br skou-
me th sqèsh pou sundèei th fasmatik puknìthta isqÔo eisìdou kai exìdou enì GQA
sust mato , w ex
Sy (! ) = Sx(!) H (!) H ? (!)
= Sx(!) jH (!)j2 (3.4.11)
H jH (!)j2 onomzetai apìkrish isqÔo tou sust mato .
114 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
Pardeigma 3.4.2
D netai to s ma x(t) = u(t). Na deiqje , ìti to s ma e nai s ma isqÔo kai na upo-
logistoÔn a) h isqÔ tou, b) h mèsh qronik sunrthsh autosusqètish tou kai g) h
fasmatik puknìthta isqÔo tou.
LÔsh H enèrgeia tou s mato e nai
Z T Z T
Ex = Tlim
!1
ju(t)j2 dt = Tlim
!1
dt = Tlim
!1
T =1
T 0
To s ma den e nai energeiakì s ma. H isqÔ tou s mato e nai
Px = Tlim 1 Z T
ju(t)j2 dt = Tlim 1 Z T
dt = lim 12 = 12
!1 2T !1 2T
(3.4.12)
T 0 T !1
To s ma e nai, loipìn, s ma isqÔo . H mèsh qronik sunrthsh autosusqètish tou
s mato e nai
Rx ( ) = Tlim 1 Z T
u(t)u(t ) dt
!1 2T T
= Tlim 1 Z T dt = lim 1 (T ) =
1
!1 2T T !1 2T 2 (3.4.13)
H fasmatik puknìthta isqÔo tou s mato e nai o metasqhmatismì Fourier th mèsh
qronik sunrthsh autosusqètish . àtsi, me th bo jeia tou zeÔgou 2 metasqhma-
tismoÔ Fourier tou P naka 3.3 èqoume
Sx (!) = F [Rx ( )℄ = Æ(!) (3.4.14)
SÔnoyh Kefala ou
Sthn arq tou kefala ou or same to eswterikì ginìmeno dÔo shmtwn kai to mètro
enì s mato . E dame pìte m a oikogèneia shmtwn apotele orjokanonikì sÔnolo sto
q¸ro twn shmtwn. Parathr same ìti ta armonik migadik ekjetik s mata kai ta
trigwnometrik s mata se èna peperasmèno disthma, sunistoÔn orjog¸nio sÔnolo
ston ant stoiqo q¸ro shmtwn.
Me bsh ta anwtèrw perigryame to anptugma se seir Fourier, me th bo jeia
tou opo ou analÔoume èna s ma se seir apì armonik migadik ekjetik s mata
se jroisma (sun)hmitìnwn, dhlad se s mata sugkekrimènh suqnìthta . Peri-
gryame th mèjodo prosdiorismoÔ twn suntelest¸n tou anaptÔgmato kai d¸same th
fusik tou shmas a. DieurÔname ta parapnw apotelèsmata kai ètsi perigryame to
metasqhmatismì Fourier enì s mato . Parathr same ìti, ìpw to anptugma Fouri-
er twn periodik¸n shmtwn, ètsi kai o metasqhmatismì Fourier twn mh periodik¸n
shmtwn anaparist mh periodik s mata me ekjetik s mata kai me ton trìpo autì
apokalÔptei to fasmatikì tou perieqìmeno.
Enìthta 3.5 Probl mata 115
Perigryame ti basikè idiìthte pou èqei o metasqhmatismì Fourier. Parousi-
same leitourg e , ìpw h diamìrfwsh, h opo a apotele basik leitourg a sthn ekpom-
p enì s mato apì èna shme o se llo mèsa apì èna kanli (zeÔgo surmtwn
th atmìsfaira ), to je¸rhma th sunèlixh , me th bo jeia tou opo ou h upologi-
stik polÔplokh sqèsh th sunèlixh metasqhmatizìmenh kat Fourier katal gei se
èna aplì ginìmeno sunart sewn. Me th bo jeia tou jewr mato tou Parseval e dame
ìti mporoÔme na upolog soume thn enèrgeia enì s mato e te sto ped o tou qrìnou
e te sto ped o twn suqnot twn.
E dame ìti o metasqhmatismì Fourier uprqei kai gia ta periodik s mata kai
shmei¸same ìti ta periodik s mata èqoun fsma diakritì, en¸ ta mh periodik èqoun
fsma suneqè .
Or same th sunrthsh autosusqètish enì energeiakoÔ s mato kai th mèsh
qronik sunrthsh autosusqètish enì s mato isqÔo . Parathr same ìti oi metasqh-
matismo Fourier twn dÔo aut¸n sunart sewn e nai h sunrthsh fasmatik puknìth-
ta enèrgeia kai h sunrthsh fasmatik puknìthta isqÔo .
Sto tèlo tou kefala ou parousisthkan dÔo p nake . Ston P naka 3.2 uprqoun
oi idiìthte tou metasqhmatismoÔ Fourier, en¸ ston P naka 3.3 oi metasqhmatismo
Fourier merik¸n basik¸n sunart sewn. Ja prèpei, telei¸nonta to dibasma tou ke-
fala ou, na gnwr zete kal ti idiìthte kai na mpore te, basizìmenoi sta parade g-
mata tou kefala ou kai sti idiìthte , na br skete tou metasqhmatismoÔ Fourier twn
basik¸n sunart sewn pou uprqoun sto deÔtero p naka.
3.5 PROBLHMATA
3.1 Na upologistoÔn kai na sqediastoÔn to mètro kai h fsh twn suntelest¸n th
ekjetik seir Fourier tou s mato .
x(t) = 1 + 2sin(!0 t) + os 2!0 t + 4
3.2 Na upologiste h mèsh isqÔ tou periodikoÔ orjog¸niou s mato tou Parade g-
mato 3.2.6. Ep sh , na upologiste h isqÔ twn suqnot twn pou perièqei o ken-
trikì lobì (dhlad , h sunolik isqÔ tou kentrikoÔ loboÔ). O kentrikì lobì
perièqei ìle ti suqnìthte metaxÔ tou pr¸tou arister mhdenismoÔ kai tou
pr¸tou dexi mhdenismoÔ.
3.3 Na upologistoÔn oi suntelestè th ekjetik seir Fourier gia to s ma
x(t) = os(4t) os(6t)
116 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
3.4 Na upologistoÔn oi suntelestè th ekjetik seir Fourier gia to s ma
x(t) = os2 (2t)
3.5 Na upologistoÔn oi ekjetikè seirè Fourier gia ta s mata.
1. x(t) = ej 200t
2. x(t) = os 4 (t 1)
3. x(t) = os(4t) + sin(8t)
4. x(t) = os(4t) + sin(6t)
5. x(t) e nai periodikì me per odo sh me 2 kai x(t) = e t gia 1<t<1
6. x(t) = [1 + os(2t)℄[ os 10t 4
3.6 Na upolog sete to mètro kai th fsh tou metasqhmatismoÔ Fourier tou aitia-
toÔ ekjetikoÔ s mato x t ()= ()
e at u t . Na parathr sete ìti X 1 kai j (0)j =
jX (a)j = ap1 2
3.7 Na upologistoÔn oi suntelestè th ekjetik seir Fourier kai th trigwno-
metrik seir gia to periodikì s ma to opo o perigrfetai sto Sq ma 3.23.
x(t)
1
-1 1 t
-1 Sq ma 3.23 To s ma tou Probl mato 3.7.
3.8 Na upologiste o metasqhmatismì Fourier tou s mato
x(t) = e ajtj ; a>0
kai na knete th grafik parstash tou s mato kai tou mètrou tou metasqh-
matismoÔ Fourier.
3.9 Na upologiste o metasqhmatismì Fourier gia kajèna apì ta s mata
1. x(t) = [eat os(!0 t)℄u(t); a > 0
2. x(t) = e ajtj sin(bt); a 6= b
3. x(t) = e2+t u( t + 1)
4. x(t) = e 3t [u(t + 2) u(t 3)℄
5. x(t) = 10;+ os(t); jjttjj >1
1
Enìthta 3.5 Probl mata 117
P
6. x(t) = 1 k=0 a Æ (t kT ); jaj < 1
k
7. x(t) = [t e 2t sin(4t)℄u(t)
8. x(t) = u(t) + 2Æ(3 2t)
3.10 Na upologiste o metasqhmatismì Fourier tou s mato x(t) = sin(!0 t) .
3.11 Na upologiste o metasqhmatismì Fourier tou s mato x(t) = sin(!0 t)u(t) .
3.12 Na upologiste o metasqhmatismì Fourier tou summetrikoÔ orjog¸niou palmoÔ
( )
t , o opo o èqei qronik dirkeia sh me m a qronik monda kai plto ep sh
so me m a monda m kou , dhlad ,
1 1
(t) = 01;; 2 <t< 2
alli¸
3.13 Na upologiste o metasqhmatismì Fourier gia to s ma to opo o perigrfetai
sto Sq ma 3.24.
x(t)
2
1
3
-2 -1 1 2 4 t
-1 Sq ma 3.24 To s ma tou Probl mato 3.13.
3.14 Na upologiste o metasqhmatismì Fourier gia ta periodik s mata
1. x1 (t) = os(4t) os(6t)
2. x2 (t) = sin2 (2t)
3.15 Na upologiste o ant strofo metasqhmatismì Fourier gia kje èna apì ta
akìlouja fsmata
1. X (!) = 2 sin[3(
! 2
! 2)℄
2. X (!) = os 4! + 3
3. X (!) = 2[Æ(! 1) Æ(! + 1)℄ + 3[Æ(! 2) Æ(! + 2)℄
3.16 Na upologiste o ant strofo metasqhmatismì Fourier gia to s ma tou opo ou
to mètro kai h fsh perigrfetai sto Sq ma 3.25
3.17 Na de xete ìti to s ma x t ( ) = os( )
!0 t e nai s ma isqÔo kai na upolog sete
thn isqÔ tou me th bo jeia tou jewr mato Parseval, dhlad e te sto ped o tou
qrìnou e te sto ped o suqnot twn.
3.18 An to s ma, x t ()= () 0
eat u t ; a > efarmoste sthn e sodo enì GQA sust -
mato , to opo o èqei kroustik apìkrish h t ()=
e t u t ; > kai ()
a, na 0 6=
brejoÔn h sunrthsh autosusqètish , h fasmatik puknìthta enèrgeia kai h
enèrgeia tou s mato exìdou.
118 Anptugma - Metasqhmatismì Fourier Analogik¸n Shmtwn Keflaio 3
X(ù) arg X(ù)
1 3
1
-1 ù
ù -3
-1 0 1
Sq ma 3.25 To mètro kai h fsh tou MF tou s mato sto Prìblhma 3.16.
3.19 An to s ma x t ( )= ( )
u t efarmoste sthn e sodo enì GQA sust mato , to opo o
èqei kroustik apìkrish h t ( ) = sin (6 )
t , na brejoÔn h fasmatik puknìthta
isqÔo kai h isqÔ tou s mato exìdou.
3.20 Na upologiste o metasqhmatismì Fourier gia to s ma to opo o perigrfetai
sto Sq ma 3.26.
x(t)
1
0 1 t Sq ma 3.26 To s ma tou Probl mato 3.20.
3.21 Qrhsimopoi¸nta ti idiìthte tou metasqhmatismoÔ Fourier, na breje o metasqh-
matismì Fourier tou s mato x t t ( ) = (2
ìpou 3) ( )
t e nai o trigwnikì
palmì kai na g nei h grafik tou parstash se sunrthsh me th suqnìthta.
Bibliograf a
3.1 S. Jeodwr dh , K. Mpermper dh , L. Kof dh , “Eisagwg sth Jewr a Shmtwn
kai Susthmtwn”, Tupwj tw - Gi¸rgo Dardanì , Aj na 2003.
3.2 N. Kaloupts dh , “S mata Sust mata kai Algìrijmoi”, D aulo , Aj na, 1994.
3.3 A. Mrgarh , “S mata kai Sust mata SuneqoÔ kai DiakritoÔ Qrìnou ”, Ekdì-
sei Tziìla 2012.
3.4 S. Haykin, B. Veen, “Signal and Systems”, John & Wiley Sons, Inc. 2003
3.5 A. V. Oppenheim, R. S. Willsky, I. T. Young, “Signal and Systems”, Prentice -
Hall Inc., N. Y., 1983.
3.6 R. E. Siemer, W. H. Tranter, D. R. Fannin, “Signals & Systems Continuous and
Discrete”, Prentice Hall, 1998.
3.7 A. Papoulis, “The Fourier integral and its Applications”, McGraw Hill., 1962.
3.8 J. G. Proakis, M. Salehi, “Communication System Engineering”, Prentice Hall
1994.
ÊÅÖÁËÁÉÏ 4
ÅÖÁÑÌÏÃÅÓ ÔÏÕ ÌÅÔÁÓ×ÇÌÁÔÉÓÌÏÕ FOURIER
Skopì tou kefala ou e nai na parousisei merikè efarmogè tou Metasqhma-
tismoÔ Fourier (MF). Eidikìtera, sto keflaio autì ja perigrafoÔn èmmesoi trìpoi
upologismoÔ tou ant strofou MF, trìpoi oi opo oi e nai idia tera qr simoi, an h mor-
f tou MF den e nai apl . Se aut thn per ptwsh, o apeuje a upologismì tou
ant strofou me thn ex swsh sÔnjesh g netai m a dÔskolh diadikas a. Ep sh , ja
perigrafe m a eÔkolh mèjodo eÔresh th apìkrish suqnìthta , th kroustik
apìkrish kai th exìdou enì sust mato , tou opo ou gnwr zoume th diaforik ex sw-
sh pou susqet zei ta s mata eisìdou-exìdou tou sust mato .
Eisagwg
Sto Keflaio 3 or same to MF, o opo o parèqei th dunatìthta metbash apì
to ped o tou qrìnou sto ped o suqnìthta . O MF th sunèlixh dÔo shmtwn up-
olog zetai me èna aplì ginìmeno twn ant stoiqwn metasqhmatism¸n. Me ton trìpo
autì, upolog zetai pr¸ta o MF th exìdou kai sth sunèqeia, me ènan ant strofo MF,
prosdior zetai h èxodo tou sust mato sto ped o tou qrìnou. Sto
keflaio autì ja parousiastoÔn merikè akìma efarmogè tou MF sth melèth gram-
mik¸n susthmtwn.
4.1 APOKRISH SUQNOTHTAS SUSTHMATOS
Sto Keflaio 2 e dame ìti èna grammikì qronik anallo wto sÔsthma perigrfetai
() ()
pl rw apì thn kroustik tou apìkrish, h t , kai ìti h e sodo , x t , kai h èxodo ,
()
y t , tou GQA sust mato sundèontai me to olokl rwma th sunèlixh
y(t) = Zx(t) ? y(t)
1
= x( )h(t ) d (4.1.1)
1
() ()
O metasqhmatismì Fourier H ! , th kroustik apìkrish h t , ìpw èqoume dei
sthn Enìthta 2.5, apotele thn apìkrish suqnìthta tou sust mato kai d netai w
120 Efarmogè tou MetasqhmatismoÔ Fourier Keflaio 4
to phl ko twn metasqhmatism¸n Fourier eisìdou-exìdou, w ex
Y (!)
H (!) =
X (! )
(4.1.2)
()
ParathroÔme ìti h sunrthsh H ! mpore na breje e te upolog zonta to metasqh-
() ()
matismì Fourier th h t , afoÔ pr¸ta upologiste h h t , e te w phl ko twn metasqh-
matism¸n Fourier eisìdou-exìdou. O deÔtero trìpo upologismoÔ th H ! e nai ()
saf¸ eukolìtero tou pr¸tou kai gia to lìgo autì o metasqhmatismì Fourier
apotele èna isqurì majhmatikì ergale o gia th melèth grammik¸n susthmtwn.
4.1.1 H apìkrish suqnìthta gia sust mata pou perigrfontai
apì diaforikè exis¸sei me stajeroÔ suntelestè
Sto Keflaio 2 or same w sÔsthma thn ontìthta eke nh h opo a metatrèpei mia
fusik posìthta pou perigrfetai apì to s ma eisìdou x t se mia llh pou peri- ()
()
grfetai apì to s ma exìdou y t . H diadikas a aut metasqhmatismoÔ ekfrzetai me
th bo jeia m a diaforik ex swsh pou susqet zei ta s mata eisìdou-exìdou. ätan
to sÔsthma e nai grammikì qronik anallo wto, ìpw èqoume sta Parade gmata 2.1.1
kai 2.1.2, h ant stoiqh diaforik ex swsh e nai grammik me stajeroÔ suntelestè ,
dhlad èqei th genik morf
N
X dk y(t) M
X dk x(t)
ak
dtk
= bk
dtk
(4.1.3)
k=0 k=0
Sth sunèqeia ja perigryoume ton trìpo me ton opo o prosdior zoume thn apìkrish
()
suqnìthta H ! apì thn (4.1.3) me th bo jeia tou metasqhmatismoÔ Fourier kai twn
idiot twn tou. Efarmìzoume to metasqhmatismì Fourier kai sta dÔo mèlh th ex swsh
(4.1.3) kai pa rnoume
" # " #
N
X dk y(t) M
X dk x(t)
F ak k
dt
=F bk k
dt
k=0 k=0
Lìgw th idiìthta th grammikìthta , pou qarakthr zei to metasqhmatismì Fourier,
èqoume
k
N
X
ak F
dy (t) = X
M
bk F
k
d x(t)
k=0
dtk k=0
dtk
lìgw th idiìthta th diafìrish , èqoume thn ex swsh
N
X M
X
ak (j!)k Y (!) = bk (j!)k X (!)
k=0 k=0
Enìthta 4.1 Upologismì tou Ant strofou MF 121
isodÔnama
N
X M
X
Y (!) ak (j!) k = X (! ) bk (j!)k
k=0 k=0
kai lìgw th (4.1.2), èqoume
PM
Y (!)
= PNk=0 bk (j!)k
k
H (!) =
X (!)
(4.1.4)
k=0 ak (j! )
ParathroÔme ta ex
()
H apìkrish suqnìthta H ! , enì GQA sust mato e nai m a rht sunrthsh,
dhlad mpore na ekfraste w lìgo dÔo poluwnÔmwn th metablht j! . ( )
Ston upologismì th apìkrish suqnìthta tou sust mato den upeisèrqontai
oi arqikè sunj ke sti opo e br sketai to sÔsthma, se ant jesh me th lÔsh
th (4.1.3), h opo a exarttai apì ti arqikè sunj ke tou sust mato . Autì
ofe letai sto gegonì ìti o metasqhmatismì Fourier pro pojètei ènarxh th
diadikas a sto 1 , pou e nai to ktw ìrio tou oloklhr¸mato ston tÔpo
orismoÔ tou (3.3.14) kai apì sÔmbash deqìmaste ìti sto oi arqikè sunj ke 1
e nai pnta mhdèn.
Pardeigma 4.1.1 (SÔsthma pr¸th txh ).
Na upologiste h apìkrish suqnìthta kai h kroustik apìkrish tou GQA sust mato
pr¸th txh , to opo o, ìpw e nai gnwstì, perigrfetai apì th diaforik ex swsh
dy(t)
dt
+ ay(t) = bx(t); a > 0 (4.1.5)
LÔsh Efarmìzonta to metasqhmatismì Fourier kai sta dÔo mèlh th ex swsh , lìgw
twn idiot twn th grammikìthta kai diafìrish , èqoume diadoqik
dy(t)
F dt
+ F [ay(t)℄ = F [bx(t)℄
(j!)Y (!) + aY (!) = bX (!)
b
H (!) =
j! + a
(4.1.6)
ìpou sthn teleuta a isìthta qrhsimopoi jhke to je¸rhma th sunèlixh Y (!) = H (!)
X (!) Sto Pardeigma 3.3.3 èqoume de xei
x(t) = e at u(t) F! X (!) =
1
j! + a
(4.1.7)
Epomènw , h kroustik apìkrish tou sust mato pr¸th txh e nai
h(t) = be (t)
at u (4.1.8)
122 Efarmogè tou MetasqhmatismoÔ Fourier Keflaio 4
4.2 UPOLOGISMOS TOU ANTISTROFOU METASQHMATISMOU
FOURIER
E nai profanè ìti, an gnwr zoume thn apìkrish suqnìthta enì sust mato , tìte me
th bo jeia th (3.3.13), h opo a perigrfei ton ant strofo metasqhmatismì Fourier,
mporoÔme na upolog soume thn kroustik apìkrish tou sust mato . An h morf th
() ()
H ! den e nai apl , tìte o apeuje a upologismì th h t apì thn (3.3.13) mpore na
apodeiqje m a ep ponh diadikas a. Gia to lìgo autì, sun jw , akoloujoÔntai èmmesoi
trìpoi upologismoÔ tou ant strofou metasqhmatismoÔ Fourier. An h H ! èqei apl ()
morf , ìpw sto Pardeigma 4.1.1, e nai dunatìn me th bo jeia tou P naka 3.3 na
prosdior zoume eÔkola thn kroustik apìkrish tou sust mato . Genikìtera, an h
apìkrish suqnìthta mpore na ekfraste w jroisma epimèrou stoiqeiwd¸n ìrwn,
tìte me th bo jeia tou P naka 3.3 kai twn idiot twn tou metasqhmatismoÔ Fourier
mporoÔme na upolog soume thn kroustik apìkrish tou sust mato . Sto Parrthma
B perigrfetai h diadikas a anptuxh m a rht sunrthsh , se jroisma apl¸n
klasmtwn. Paraktw efarmìzoume th mejodolog a aut se èna pardeigma.
Pardeigma 4.2.1 (SÔsthma deÔterh txh ).
D netai to GQA sÔsthma deÔterh txh , to opo o arqik br sketai se hrem a, kai
qarakthr zetai apì th diaforik ex swsh
d2 y(t)
dt2
+ 4 dydt(t) + 3y(t) = dxdt(t) + 2x(t) (4.2.1)
Na upologiste h kroustik apìkrish tou sust mato .
LÔsh Efarmìzoume to metasqhmatismì Fourier kai sta dÔo mèlh th ex swsh (4.2.1)
kai me th bo jeia th idiìthta th grammikìthta kai th diafìrish , pou èqei o
metasqhmatismì Fourier, br skoume ìti h apìkrish suqnìthta tou sust mato e nai
H (!) =
(j!) + 2
(j!) + 4(j!) + 3
2 (4.2.2)
AnalÔoume ta polu¸numa tou arijmht kai tou paranomast se ginìmena poluwnÔmwn
pr¸tou deÔterou bajmoÔ w pro (j!) kai sth sunèqeia anaptÔssoume thn apìkrish
suqnìthta se apl klsmata. àtsi, pa rnoume
H (!) =
(j!) + 2 C1 C2
(j! + 1)(j! + 3) = j! + 1 + j! + 3
Sth sunèqeia upolog zoume ti stajerèC1 kai C2
j! + 2 1
C1 = [(j! + 1)H (!)℄jj!= 1 = =
j! + 3 j!= 1 2
j! + 2 1
C2 = [(j! + 3)H (!)℄jj!= 3 = ) C2 =
j! + 1 j!= 3 2
Enìthta 4.1 Upologismì tou Ant strofou MF 123
Epomènw , h apìkrish suqnìthta pa rnei th morf
H (!) =
1 1 1 1
2 j! + 1 + 2 j! + 3 (4.2.3)
Me th bo jeia th idiìthta th grammikìthta tou metasqhmatismoÔ Fourier kai th
(4.1.7), h kroustik apìkrish tou sust mato e nai
1 1
h(t) = e t u(t) + e 3t u(t)
2 2 (4.2.4)
Pardeigma 4.2.2
An h e sodo tou sust mato sto Pardeigma 4.2.1 e nai
x(t) = e t u(t) (4.2.5)
na upologiste h èxodo tou sust mato .
LÔsh O metasqhmatismì Fourier tou s mato eisìdou x(t) = e t u(t) e nai X (!) =
1=(j! + 1). Me th bo jeia tou jewr mato th sunèlixh , o metasqhmatismì Fourier
th exìdou e nai
j! + 2
1
Y (!) = H (!)X (!) = = (j! +j!1)2+(j!2 + 3)
(j! + 1)(j! + 3)
j! + 1
Sthn per ptwsh aut h anptuxh se apl klsmata tou Y (! ) èqei th morf
C C12 C21
Y (!) = 11 + +
j! + 1 (j! + 1) j! + 3
2
Sth sunèqeia upolog zoume ti C11 , C12 kai C21
stajerè
C11 =
1 d 2 d j! + 2 1
(2 1)! d(j!) (j! + 1) Y (!) j!= 1 = d(j!) j! + 3 j!= 1 ) C11 = 4
j! + 2 1
C12 = (j! + 1)2 Y (!) j!= 1 = ) C11 =
j! + 3 j!= 1 2
j! + 2 1
C21 = [(j! + 3)Y (!)℄jj!= 3 = ) C21 =
(j! + 1) j!= 3
2 4
Epomènw , o metasqhmatismì Fourier th exìdou pa rnei th morf
Y (!) =
1 1 +1 1 1 1
4 j! + 1 2 (j! + 1) 4 j! + 3
2 (4.2.6)
An efarmìsoume thn idiìthta th diafìrish sthn (4.1.7) sto ped o suqnot twn, èqoume
t x(t) F! j X (!)
d!
t e at u(t) F! 1
(j! + a)2 (4.2.7)
124 Efarmogè tou MetasqhmatismoÔ Fourier Keflaio 4
Me th bo jeia tou ant strofou metasqhmatismoÔ Fourier, th idiìthta th grammikìth-
ta , th (4.1.7) kai th (4.2.7) br skoume ìti h èxodo tou sust mato e nai
y(t) =
1 e t + 1 te t 1e 3t u(t)
4 2 4 (4.2.8)
Prèpei na toniste sta Parade gmata 4.2.1 kai 4.2.2 ìti, an zhte tai mìno h y(t) kai ìqi h
Y (!), tìte o eukolìtero trìpo eÔres th e nai h apeuje a ep lush th diaforik
ex swsh .
Pardeigma 4.2.3 (Prosdiorismì sust mato apì thn e sodì tou kai èxodì tou).
H èxodo enì GQA sust mato se s ma eisìdou x(t) = e 2t u(t) e nai y(t) = e t u(t)
Na breje h apìkrish suqnìthta tou sust mato kai h kroustik apìkrish.
LÔsh O metasqhmatismì Fourier tou s mato eisìdou x(t) e nai
X (!) =
1
2 + j!
kai o metasqhmatismì Fourier tou s mato exìdou y(t) e nai
Y (! ) =
1
1 + j!
H apìkrish suqnìthta tou sust mato upolog zetai me th bo jeia th 4.1.2
Y (!)
H (! ) =
X (!)
= 21 ++ j!
j!
(4.2.9)
H apìkrish suqnìthta tou sust mato grfetai w
H (!) = 1 +
1
1 + j!
H kroustik apìkrish tou sust mato e nai
h(t) = Æ(t) + e tu(t) (4.2.10)
Shmei¸netai ìti, ìtan to s ma eisìdou e nai s ma m a suqnìthta , ja prèpei kai to s ma
exìdou na e nai s ma th dia suqnìthta kai sthn per ptwsh aut prosdior zetai mìno
h tim th apìkrish suqnìthta sth suqnìthta tou s mato eisìdou.
4.3 DIAGRAMMATA BODE
Apì to je¸rhma th sunèlixh gnwr zoume ìti o metasqhmatismì Fourier th eisìdou
X (!) kai th ()
exìdou Y ! enì grammikoÔ qronik anallo wtou sust mato , to opo o
()
èqei apìkrish suqnìthta H ! , sundèontai me th sqèsh
Y (!) = H (!) X (!) (4.3.1)
Enìthta 4.2 Diagrmmata Bode 125
ParathroÔme ìti o metasqhmatismì Fourier th exìdou prokÔptei w ginìmeno tou
metasqhmatismoÔ Fourier th eisìdou me thn apìkrish suqnìthta . Apì thn (4.3.1)
èqoume gia ta mètra kai ti fsei twn antisto qwn posot twn ti exis¸sei
jY (!)j = jH (!)j jX (!)j kai argY (!) = argH (!) + argX (!) (4.3.2)
H ajroistik morf th deÔterh ex swsh epitrèpei ton prosdiorismì th grafik
parstash th fsh exìdou me apl prìsjesh twn grafhmtwn th fsh eisìdou
me th fsh th apìkrish suqnìthta . Gia na petÔqoume anlogh sumperifor gia to
mètro, logarijm zoume thn pr¸th ex swsh kai èqoume
logjY (!)j = logjH (!)j + logjX (!)j (4.3.3)
Suqn, gia th grafik anaparstash tou mètrou, qrhsimopoioÔme logarijmik kl ma-
ka kai w monda mètrou to decibel (dB). H kl maka twn dB bas zetai sthn antistoiq a
dB = 20log10 jH (!)j (4.3.4)
Endeiktik èqoume ti akìlouje timè
0 dB antistoiqe se jH (!)j = 1;
20 dB antistoiqe se jH (!)j = 10; 20 dB antistoiqe se jH (!)j = 0; 1
40 dB antistoiqe se jH (!)j = 100; 40 dB antistoiqe se jH (!)j = 0; 01
ParathroÔme epiplèon ìti
1 dB antistoiqe se jH (!)j 1; 12 kai 6 dB antistoiqe se jH (!)j 2
Diagrmmata pou apeikon zoun th fsh kai to mètro se dB, se sunrthsh me th
suqnìthta, onomzontai diagrmmata Bode. Epeid o logrijmo ekte nei thn kl maka,
h qrhsimopo hsh logarijmik kl maka exasfal zei kalÔterh eukr neia ìtan to eÔro
twn suqnot twn pou ma endiafèrei e nai meglo perior zetai se mikrè timè kont
sto mhdèn. Efarmìzoume ta parapnw sto Pardeigma 4.3.1.
Pardeigma 4.3.1
Na upologiste h apìkrish tou sust mato pr¸th txh ìtan h e sodo e nai h sunrth-
sh monadia ou b mato .
LÔsh O metasqhmatismì Fourier tou monadia ou b mato e nai (blèpe P naka 3.3)
U (!) =
1 + Æ(!) (4.3.5)
j!
Apì to je¸rhma th sunèlixh prokÔptei ìti o metasqhmatismì Fourier th exìdou
1 + Æ(!)
e nai
b
Y (!) = H (!) U (!) =
j! + j!
126 Efarmogè tou MetasqhmatismoÔ Fourier Keflaio 4
gia ! 6= 0 epeid Æ(!) = 0, èqoume
b b 1 b 1
Y (! ) =
(j! + a)j! = +
a j! + a a j!
Me th bo jeia tou ant strofou metasqhmatismoÔ Fourier, th idiìthta th grammikìth-
ta , th (4.1.7) kai tou metasqhmatismoÔ Fourier tou monadia ou b mato upolog zoume
thn èxodo tou sust mato
b b at
y(t) = e u(t) (4.3.6)
a a
H kroustik apìkrish tou sust mato pr¸th txh kai h apìkris tou sto monadia o
b ma paristnontai grafik sto Sq ma 4.1.
h(t) y(t)
b
b a
b( 1
a 1 e (
b
e
0 ô t 0 ô t
(a) (â)
Sq ma 4.1 Apokr sei sust mato pr¸th txh (a) kroustik apìkrish (b) apìkrish sto
monadia o b ma.
H asumptwtik katstash th apìkrish sto monadia o b ma e nai b=a. H parmetro
=1=a onomzetai qronik stajer kai shmatodote to rujmì me ton opo o to sÔsth-
ma apokr netai. Th qronik stigm t =
, h kroustik apìkrish mei¸nei thn tim pou
1
e qe arqik =e forè , en¸ h apìkrish sto monadia o b ma apèqei =e forè apì thn 1
telik th tim (Sq ma 4.1). ParathroÔme ìti, kaj¸ a , h qronik stajer ! +1
mikra nei kai h ptwtik tsh th kroustik apìkrish g netai pio apìtomh.
H apìkrish suqnìthta , h kroustik apìkrish kai h apìkrish sto monadia o b ma
tou sust mato pr¸th txh , ìtan a b, e nai =
H (! ) =
1 ; h(t) = 1 e t
u (t) kai y(t) = 1
e
t
u (t)
j! + 1
(4.3.7)
To mètro th apìkrish suqnìthta h apìkrish pltou tou sust mato e nai
r
jH (!)j = H (!)H ? (!) = j! 1+ 1 j! 1 + 1 = p 1 2 2
p
(4.3.8)
1+ !
kai se dB
20log10 jH (!)j = 10log10 1 + ( !)2 (4.3.9)
ParathroÔme ìti,
Enìthta 4.2 Diagrmmata Bode 127
1
An ! << , isqÔei log 1 + ( ) log 1 = 0
10 ! 2 10 . Sunep¸ , sti qamh-
lè suqnìthte to mètro se dB th apìkrish suqnìthta e nai per pou mhdèn,
20log j ( )j 0
efìson 10 H ! gia ! << = .1
An ! >> 1, isqÔei log10 1 + ( !) log10 ( !)2 = 20log10 + 20log10 !.
2
Sunep¸ , sti uyhlè suqnìthte to mètro se dB prosegg zetai apì grammik
sunrthsh tou log10 (! ), h opo a èqei kl sh -20,
20log jH (!)j 20log 20log ! gia ! >> 1
10 10 10
0
-3dB
20log10 H(ù)
-3
-10
-20
log10( 1 ( log10(1ô( log10(10
ô(
10 ô
log10(ù)
arg H(ù)
ð
4
ð
2
log10( 1 ( log10(1ô( log10(10
ô(
10 ô
log10(ù)
Sq ma 4.2 Ta diagrmmata Bode tou sust mato pr¸th txew .
Sto Sq ma 4.2 fa nontai ta diagrmmata Bode sust mato pr¸th txew . Gia to
shme o tom twn asÔmptwtwn eujei¸n pou prosegg zoun to mètro sti qamhlè kai
uyhlè suqnìthte isqÔei 10 ! log ( ) = log ( )
10 kai antistoiqe sth suqnìthta ! =
1= . Sth suqnìthta aut to mètro se dB e nai
" #
20log10 H
1 = 10log 1 2 + 1 = 10log (2) 3dB
10 10
Gia to lìgo autì, h kuklik suqnìthta ! =1
= onomzetai kuklik suqnìthta -3 dB.
1 p2
Genik, h kuklik suqnìthta -3dB e nai h kuklik suqnìthta gia thn opo a to mètro
th apìkrish suqnìthta enì sust mato apokt to = th mègisth tim tou
128 Efarmogè tou MetasqhmatismoÔ Fourier Keflaio 4
mètrou th apìkrish suqnìthta tou sust mato .
jH (!)jj! 3dB = p12 jH (!)jmax (4.3.10)
Se anloga sumpersmata katal goume gia th fsh, ìpou sthn per ptwsh aut
uprqoun trei asÔmptwte euje e (Sq ma 4.2).
4.4 IDANIKO FILTRO BASIKHS ZWNHS - BAJUPERATO FILTRO
àna sÔsthma to opo o enisqÔei apodunam¸nei to mètro th apìkrish suqnìthta
anloga me thn tim to disthma tim¸n th suqnìthta ! , onomzetai f ltro. Idanikì
bajuperatì f ltro idanikì f ltro basik z¸nh
onomzetai to GQA sÔsthma to
opo o èqei apìkrish suqnìthta
H (! ) = e j!t0 ; j!j < !
0; j!j > ! (4.4.1)
ìpou t0 e nai stajer
posìthta. H stajer kuklik suqnìthta ! qarakthr zetai w
kuklik tou f ltrou. Sto Sq ma 4.3 d netai h grafik parstash
suqnìthta apokop
tou mètrou kai th fsh tou idanikoÔ f ltrou basik z¸nh . To mètro th apìkrish
H(ù) arg H(ù)
1
ùc
-ùc 0 ù
-ùc 0 ùc ù
Æþíç Æþíç êëßóç =-t0
áðïêïðÞò Æþíç äéÝëåõóçò áðïêïðÞò
(a) ( â)
Sq ma 4.3 (a) To mètro kai (b) h fsh th apìkrish suqnìthta tou idanikoÔ f ltrou
basik z¸nh . H kl sh th euje a prosdior zetai apì to t0 .
suqnìthta tou bajuperatoÔ f ltrou e nai so me 1 gia ! < ! < ! . Dedomènou
ìti Y ! ( )= ( ) ( )
H ! X ! , e nai profanè ìti oi suqnìthte tou s mato eisìdou pou
br skontai se autì to disthma dièrqontai apì to f ltro me ametblhto plto . Gia
to lìgo autì, to disthma autì kale tai kai z¸nh dièleush tou f ltrou. Ep sh ,
epeid to mètro th apìkrish suqnìthta tou bajuperatoÔ f ltrou e nai so me 0
gia ! < ! kai ! < ! , e nai profanè ìti to bajuperatì f ltro aporrof ti
suqnìthte eke ne tou fsmato tou s mato eisìdou pou e nai megalÔtere apì th
suqnìthta apokop . To disthma autì apotele th z¸nh apokop tou f ltrou.
emfan zetai kai w katwperatì f ltro kai katwdiabatì f ltro
Enìthta 4.3 Idanikì F ltro Basik Z¸nh - Bajuperatì F ltro 129
A upojèsoume t¸ra, ìti sthn e sodo tou bajuperatoÔ f ltrou to s ma apotele -
()
tai apì dÔo sunist¸se , thn xep t pou e nai h epijumht sunist¸sa tou s mato kai
()
thn xan t pou e nai h anepijÔmhth sunist¸sa, p.q. èna s ma parembol jìrubo .
j
àstw de ìti Xep ! ( )j = 0 j j
gia ! > ! se ant jesh me thn anepijÔmhth sunist¸sa,
th opo a o metasqhmatismì Fourier den ikanopoie ant stoiqh sqèsh. Gia mia tè-
toia per ptwsh, to idanikì bajuperatì f ltro ja af nei thn epijumht sunist¸sa na
dièrqetai en¸ ja aporrof sei to tm ma th anepijÔmhth sunist¸sa to opo o periè-
qei suqnìthte megalÔtere apì th suqnìthta apokop , me apotèlesma th belt wsh
th poiìthta tou s mato xep t . ()
A upojèsoume t¸ra ìti to mh mhdenikì mèro tou metasqhmatismoÔ Fourier, X ! , ()
()
tou s mato eisìdou, x t , entop zetai sth z¸nh dièleush . Tìte o metasqhmatismì
Fourier th exìdou tou sust mato e nai
Y (!) = X (!)H (!) = X (!)e j!t0
sto ped o tou qrìnou
y(t) = x(t t0 ) (4.4.2)
Me lla lìgia, to gegonì ìti h fsh th apìkrish suqnìthta e nai grammik
sunrthsh th suqnìthta , h ep drash tou f ltrou se èna s ma eisìdou, me fasmatikì
perieqìmeno entopismèno sth z¸nh dièleush , e nai m a qronik kajustèrhsh t0 .
Apì to metasqhmatismì Fourier tou orjog¸niou palmoÔ kai me th bo jeia th
idiìthta th qronik metatìpish upolog zetai h kroustik apìkrish tou idanikoÔ
katwperatoÔ filtroÔ
h(t) =
sin[! (t t0)℄ = ! sin ! (t t0 )
(t t0 )
(4.4.3)
!
H grafik parstash th kroustik apìkrish e nai sto Sq ma 4.4.
h(t)
ùc
ð
t ð ð
0 ùc t
0 ùc
0 t
0 t
Tc 2ð
ùc
Sq ma 4.4 H kroustik apìkrish tou idanikoÔ bajuperatoÔ f ltrou.
Parathr sei
1. äso mikrìterh e nai h suqnìthta apokop , tìso megalÔterh e nai h dirkeia
th kroustik apìkrish .
130 Efarmogè tou MetasqhmatismoÔ Fourier Keflaio 4
2. To idanikì katwperatì f ltro e nai mh aitiatì, efìson h kroustik apìkris
tou e nai mh mhdenik gia arnhtikè timè tou qrìnou kai, epomènw , e nai mh
pragmatopoi simo. An h tim th stajer t0 e nai arket meglh, oi timè th
kroustik apìkrish ti arnhtikè qronikè stigmè mporoÔn na jewrhjoÔn
amelhtèe . Kai w ek toÔtou mporoÔme na prosegg soume to f ltro me èna
aitiatì sÔsthma.
3. äso megalÔterh e nai h suqnìthta apokop , tìso taqÔtera to f ltro èqei th du-
natìthta na parakolouje apìtome metabolè tou s mato eisìdou. Autì e nai
logikì, afoÔ gr gore qronikè metabolè antistoiqoÔn sti uyhlè suqnìthte ,
oi opo e dièrqontai apì to f ltro an epilèxoume meglh suqnìthta apokop .
H grafik parstash th apìkrish pltou twn pragmatik¸n f ltrwn basik
z¸nh ta opo a sunantme sthn prxh, fa netai sto Sq ma 4.5. Sta pragmatik f ltra
ektì apì ti z¸ne dièleush kai apokop uprqei kai h z¸nh metbash . Ep sh ,
sta pragmatik f ltra h suqnìthta apokop e nai sh me th suqnìthta 3dB
H(ù)
A
A
2
-ùc ùc ù Sq ma 4.5 To mètro th apìkrish
0
Æþíç Æþíç Æþíç Æþíç Æþíç suqnìthta enì pragmatikoÔ f ltrou
áðïêïðÞò ìåôÜâáóçò äéÝëåõóçò ìåôÜâáóçò áðïêïðÞò
basik z¸nh .
Anloga me thn perioq twn suqnot twn pou dièrqontai apì to f ltro autì qarak-
thr zetai w
1. bajuperatì f ltro f ltro basik z¸nh f ltro dièleush qamhl¸n suqnot -
twn (lowpass filter (LPF)) Sq ma 4.6a.
2. uyiperatì f ltro f ltro dièleush uyhl¸n suqnot twn (highpass filter (HPF))
Sq ma 4.6b.
3. zwnoperatì f ltro zwnodiabatì f ltro f ltro dièleush z¸nh suqnot twn
(bandpass filter (BPF)) Sq ma 4.6g..
4. zwnofraktikì f ltro f ltro apokop z¸nh suqnot twn (bandstop filter (B-
SF)) Sq ma 4.6d.
Oi apokr sei pltou twn antisto qwn idanik¸n f ltrwn fa nontai sto Sq ma 4.6 kai
twn pragmatik¸n f ltrwn sto Sq ma 4.7.
Pardeigma 4.4.1
D netai to sÔsthma to opo o perigrfetai sto Sq ma 4.8.
1. Na breje h kroustik apìkrish tou sust mato .
Enìthta 4.3 Idanikì F ltro Basik Z¸nh - Bajuperatì F ltro 131
H(ù) H(ù)
LPF HPF
1 1
-ùc 0 ùc ù -ùc 0 ùc ù
(á) ( â)
H(ù) H(ù)
ÂPF ÂSF
1 1
-ù2 -ù1 0 ù1 ù2 ù -ù2 -ù1 0 ù1 ù2 ù
( ã) ( ä)
Sq ma 4.6 Oi apokr sei pltou idanikoÔ (a) bajuperatoÔ (b) uyiperatoÔ (g) zwnoperatoÔ
kai (d) zwnofraktikoÔ f ltrou.
H(ù) H(ù) HPF
A LPF A
A A
2 2
-ùc 0 ùc ù -ùc 0 ùc ù
(á) ( â)
H(ù) H(ù) ÂSF
A ÂPF A
A A
2 2
-ù2 -ù1 0 ù1 ù2 ù -ù2 -ù1 0 ù1 ù2 ù
( ã) ( ä)
Sq ma 4.7 Oi apokr sei pltou pragmatikoÔ (a) bajuperatoÔ (b) uyiperatoÔ (g) zwnop-
eratoÔ kai (d) zwnofraktikoÔ f ltrou.
2. Na g nei h perigraf tou sust mato sto ped o suqnìthta .
3. Na breje h apìkrish suqnìthta tou sust mato .
LÔsh
1. Gnwr zoume ìti h kroustik apìkrish enì sust mato e nai sh me thn èxodì tou
ìtan to s ma eisìdou e nai h kroustik sunrthsh Æ(t), ètsi èqoume
Z t
h(t) = (Æ() Æ( )) d
1
= u(t) u(t )
= t =2 (4.4.4)
132 Efarmogè tou MetasqhmatismoÔ Fourier Keflaio 4
Åßóïäïò Óýóôçìá ¸îïäïò
x(t) ïëïêëÞñùóçò y(t)
Óýóôçìá ∫ (⋅) dt
êáèõóôÝñçóçò
êáôÜ ô
Sq ma 4.8 Perigraf tou sust mato tou Parade gmato 4.4.1 sto ped o tou qrìnou.
2. Me th bo jeia twn idiot twn th qronik metatìpish kai th olokl rwsh
tou metasqhmatismoÔ Fourier èqoume thn perigraf tou sust mato sto ped o
suqnìthta tou Sq mato 4.9.
3. Me th bo jeia th prosetairistik kai th epimeristik idiìthta th sunèlixh
èqoume gia thn apìkrish suqnìthta tou sust mato
H (!) = [H1 (!) H2 (!)℄H3 (!) = 1 j! e j!
2
= !
ej! 2 e j! 2
e j! 2
2j
!
= sin 2 e j! 2 (4.4.5)
H apìkrish suqnìthta mpore na breje , ep sh , upolog zonta to metasqhma-
tismì Fourier th kroustik apìkrish tou sust mato .
Åßóïäïò ¸îïäïò
X(ù) 1
Ç1(ù)=1 Ç3(ù)= jù Y(ù)
Ç2(ù)=e- jùô
Sq ma 4.9 Perigraf tou sust mato tou Parade gmato 4.4.1 sto ped o suqnìthta .
SÔnoyh Kefala ou
Sto keflaio autì upolog same thn apìkrish suqnìthta enì GQA sust mato
apì th diaforik ex swsh pou susqet zei ta s mata eisìdou-exìdou, ekmetalleuìmenoi
ti idiìthte th grammikìthta , th diafìrish kai to je¸rhma th sunèlixh . ätan
h apìkrish suqnìthta èqei apl morf , tìte e nai dunatì me th bo jeia twn gnw-
st¸n zeug¸n MF, pou anafèrontai ston P naka 3.3, na upolog soume thn kroustik
apìkrish tou sust mato . Sto keflaio autì parousisthkan,ep sh , èmmesoi trìpoi
upologismoÔ tou ant strofou MF, oi opo oi e nai idia tera qr simoi ìtan o MF den
èqei apl morf . Eidikìtera, sti perissìtere praktikè efarmogè , o MF e nai m a
rht sunrthsh. Sthn per ptwsh aut analÔoume th sunrthsh se jroisma apl¸n
klasmtwn kai me th bo jeia tou P naka 3.3 upolog zoume ton ant strofo MF. H
Enìthta 4.5 Probl mata 133
parapnw mejodolog a efarmìzetai kai gia ton upologismì th exìdou, sto ped o tou
qrìnou, enì GQA sust mato , en èqoume upolog sei pr¸ta ton MF th exìdou me
th bo jeia tou jewr mato th sunèlixh .
Ep sh , sto keflaio autì parousisthkan ta diagrmmata Bode. Ta diagrmmata
Bode, epeid o logrijmo ekte nei thn kl maka, exasfal zoun perissìterh eukr neia
an to eÔro twn suqnot twn pou ma endiafèrei e nai meglo , ep sh , an perior zetai
se mikrè timè kont sto mhdèn. Tèlo , parousisthkan oi ènnoie twn idanik¸n kai
pragmatik¸n f ltrwn.
4.5 PROBLHMATA
4.1 D netai to kÔklwma pou apotele tai apì ant stash R K , phn o L ; H= 10 =0 1
kai puknwt C = 10
F se seir, to opo o perigrfetai sto Sq ma 4.10. An h
e sodo tou sust mato e nai h efarmozomènh tsh in t kai èxodo h èntash ()
()
tou reÔmato i t , na upologiste h apìkrish suqnìthta tou sust mato .
A B
L
õin(t) i(t)
R
C
Ä Ã Sq ma 4.10 To kÔklwma tou Probl mato 4.1.
4.2 Na upologiste h èntash tou reÔmato pou diarrèei to kÔklwma tou Sq mato
4.8 ìtan h tsh eisìdou e nai
in (t) = 10 os 2t + 3
4.3 D netai to GQA sÔsthma to opo o èqei kroustik apìkrish
h(t) =
sin(4t)
t
Me th bo jeia tou jewr mato th sunèlixh na prosdioriste h èxodo tou
sust mato ìtan h e sodo tou sust mato e nai to s ma
x(t) = os(2t) + sin(6t)
4.4 H diaforik ex swsh h opo a sundèei thn e sodo kai thn èxodo enì GQA sust -
mato e nai
dy(t)
dt
+ 2y(t) = x(t)
134 Efarmogè tou MetasqhmatismoÔ Fourier Keflaio 4
1. Na prosdioriste h apìkrish suqnìthta tou sust mato kai na g noun ta
diagrmmata Bode.
2. An h e sodo tou sust mato e nai to s ma x (t) = e t u(t), na upologiste
o metasqhmatismì Fourier th exìdou.
3. ()
AfoÔ anaptuqje se apl klsmata h Y ! na upologiste h èxodo y(t)
tou sust mato , ìtan h e sodo e nai to s ma x t . ()
4.5 An h phg tsh th morf tou Sq mato 4.11 efarmìzetai sthn e sodo enì
idanikoÔ katwperatoÔ f ltrou, me apìkrish suqnìthta
H (!) = e j! ; j!j < 4
0; j!j > 4
Na upologiste h èxodo tou f ltrou.
õ(t)
V
-2ð -ð ð 2ð t
-V Sq ma 4.11
4.6 àstw ìti h phg tsh th morf tou Sq mato 4.11 efarmìzetai sthn e sodo
enì GQA sust mato pr¸th txh , to opo o qarakthr zetai apì thn ex swsh
dy(t)
dt
+ y(t) = x(t)
Na upologiste h èxodo tou sust mato .
4.7 To aitiatì ekjetikì s ma tou Sq mato 4.12a efarmìzetai sthn e sodo tou ida-
nikoÔ f ltrou basik z¸nh tou Sq mato 4.12b. Na upologiste h suqnìthta
apokop ! , ètsi ¸ste to f ltro na epitrèpei th dièleush
R 1 th mis1 enèrgeia
1 x . = tan
tou s mato eisìdou. D netai to aìristo olokl rwma a2 + x2 dt a a
x(t)
2 t Ç(ùù
ùó )
x(t)=2e 5 1
x(t) y(t)
0 -ùc 0 ùc ù
t
(á) ( â)
Sq ma 4.12 (a) To s ma eisìdou kai (b) to idanikì f ltro sto Prìblhma 4.7.
Enìthta 4.5 Probl mata 135
4.8 D netai to kÔklwma to opo o perigrfetai sto Sq ma 4.13. Na upologiste h
apìkrish suqnìthta tou sust mato kai na g nei h grafik parstash th
apìkrish pltou se sunrthsh me th suqnìthta.
R=1 Ù L= 12 H
Åßóïäïò ¸îïäïò
x(t) y(t)
C= 2 F
Sq ma 4.13 To kÔklwma tou Probl mato 4.8.
4.9 Kat th diamìrfwsh, ()
èna s ma m t periorismènou eÔrou z¸nh , dhlad ,
M (!) = 0 gia j!j > W (Sq ma 4.12b), pou metafèrei sugkekrimènh plhrofo-
r a, pollaplasizetai me èna s ma apl suqnìthta !0 t , h opo a onomze-os( )
tai fèrousa, me skopì thn ekpomp tou se èna mèso metdosh , p.q., zeÔgo
surmtwn, atmìsfaira, klp. Sto Sq ma 4.14a perigrfetai èna aplopoihmèno
sÔsthma epikoinwn a . An jewr soume ìti kat th metdosh tou s mato den
alloi¸netai apì to mèso kai ìti o jìrubo tou kanalioÔ e nai amelhtèo , tìte
()
to lambanìmeno s ma r t e nai so me to ekpempìmeno. JewroÔme ìti h z¸nh
dièleush tou idanikoÔ katoperatoÔ f ltrou sto dèkth e nai sh me to eÔro
()
z¸nh W tou s mato mhnÔmato m t . Na melethje to sÔsthma sto ped o
suqnìthta .
M(ù)
u(t) êáíÜëé r(t) z(t) y(t)
m(t) ìåôÜäïóçò
Êáôùðåñáôü A
ößëôñï
cos(ùc t) cos(ùc t) ù
W 0 W
Ðïìðüò ÄÝêôçò
(á) (â)
Sq ma 4.14 (a) To aplopoihmèno sÔsthma epikoinwn a kai (b) to periorismènou eÔrou
z¸nh fsma tou s mato mhnÔmato gia èna auja reto s ma m(t).
4.10 ätan to s ma eisìdou se èna grammikì qronik analloi¸to sÔsthma e nai x (t) =
e at , to s ma exìdou e nai y t ()=
e bt u t . Na brejoÔn ()
1. h apìkrish suqnìthta kai
2. h kroustik apìkrish tou sust mato .
4.11 Me th bo jeia th (3.3.83) na breje o metasqhmatismì Fourier tou periodikoÔ
()
orjog¸niou s mato x t tou Parade gmato 3.2.6.
()
An to s ma x t efarmoste sthn e sodo grammikoÔ qronik anallo wtou sust -
mato me kroustik apìkrish h t 1 ( ) = sin( )
t , na de xete, qrhsimopoi¸nta to
t
136 Efarmogè tou MetasqhmatismoÔ Fourier Keflaio 4
je¸rhma th sunèlixh , ìti h èxodo tou sust mato e nai
1
y(t) = + os
t
2 2
4.12 Apì thn idiìthta th ol sjhsh th kroustik sunrthsh
Z 1
x(t)Æ(t t0 ) dt = x(t0 )
1
ParathroÔme ìti h sunèlixh kje sunrthsh me thn kroustik sunrthsh èqei
w apotèlesma m a olisjhmènh èkdosh th arqik sunrthsh , dhlad ,
g(t) Æ(t t0 ) = g(t t0 )
An to s ma x t 2 t efarmoste sthn e sodo grammikoÔ qronik anal-
( ) = sin ( )
()
lo wtou sust mato me kroustik apìkrish h t , na breje kai na sqediasje o
metasqhmatismì Fourier th exìdou tou sust mato ìtan
1. h(t) = 1 + os 2t
P
2. h(t) = 1 k= 1 Æ (t kT )
4.13 Na breje o majhmatikì tÔpo th kroustik apìkrish tou sust mato tou
Sq mato 4.15 kai na g nei h grafik th parstash se sunrthsh me to qrìno.
(Upìdeixh: Na jewr sete to sÔsthma w thn en seir sÔndesh dÔo sust matwn).
Åßóïäïò Óýóôçìá Óýóôçìá ¸îïäïò
ïëïêëÞñùóçò ïëïêëÞñùóçò
x(t) t t y(t)
dt dt
Óýóôçìá −∞ Óýóôçìá −∞
êáèõóôÝñçóçò êáèõóôÝñçóçò
êáôÜ ô êáôÜ ô
Sq ma 4.15 To sÔsthma sto Prìblhma 4.13.
Bibliograf a
4.1 S. Jeodwr dh , K. Mpermper dh , L. Kof dh , “Eisagwg sth Jewr a Shmtwn
kai Susthmtwn”, Tupwj tw - Gi¸rgo Dardanì , Aj na 2003.
4.2 S. Haykin, B. Veen, “Signal and Systems”, John & Wiley Sons, Inc. 2003
4.3 A. V. Oppenheim, R. S. Willsky, I. T. Young, “Signal and Systems”, Prentice -
Hall Inc., N. Y., 1983.
ÊÅÖÁËÁÉÏ 5
ÓÅÉÑÁ - ÌÅÔÁÓ×ÇÌÁÔÉÓÌÏÓ FOURIER
ÄÉÁÊÑÉÔÏÕ ×ÑÏÍÏÕ
Sto keflaio autì ja melet soume majhmatik ergale a, ta opo a ma epitrèpoun
na analÔoume èna sÔnjeto s ma diakritoÔ qrìnou se s mata diakritoÔ qrìnou apl¸n
suqnot twn. Mia tètoia prosèggish ma dieukolÔnei ¸ste na upolog soume thn èxodo
enì sust mato diakritoÔ qrìnou, to opo o diege retai apì èna sÔnjeto s ma, me
th bo jeia twn apokr sewn tou sust mato sti ep mèrou sunist¸se twn apl¸n
suqnot twn sti opo e analÔetai to sÔnjeto s ma. Sth sunèqeia ja efarmìsoume
ti mejìdou autè , ¸ste na analÔsoume ènan arijmì shmtwn ta opo a sunantme
suqn sth prxh. Tèlo , sto keflaio autì ja parousiastoÔn merikè efarmogè tou
metasqhmatismoÔ Fourier diakritoÔ qrìnou.
Eisagwg
Gnwr zoume ìti to migadikì ekjetikì s ma diakritoÔ qrìnou ej (2=N )n e nai pe-
riodikì me jemeli¸dh per odo N . Ta ekjetik s mata pou èqoun kuklik suqnìthta
pollaplsia th 0 =2 = 0 1 2
=N (ejk 0 n , me k ; ; ; ::: ) e nai ep sh periodik. Ta
ekjetik s mata e jk 0 n kaloÔntai armonik susqetizìmena ekjetik s mata diakritoÔ
qrìnou epeid oi jemeli¸dei suqnìthtè tou e nai akèraia pollaplsia th kuklik
suqnìthta 0 . Ta migadik ekjetik s mata diakritoÔ qrìnou twn opo wn oi kuklikè
2
suqnìthte diafèroun kat pollaplsio tou e nai dia. Prgmati:
ej ( +2)n = ej n ej 2n = ej n
Uprqoun N to pl jo diaforetik migadik ekjetik s mata diakritoÔ qrìnou ta
opo a sqhmat zoun èna orjog¸nio sÔnolo dhlad , e nai an dÔo orjog¸nia. Prgmati
to eswterikì ginìmeno twn ekjetik¸n shmtwn ejk 0 n kai ejm 0 n e nai
X1
hejk 0n; ejm 0ni =
N
ej (k m) 0 n = N; k=m = NÆ(k m)
n=0
0; k 6= m
138 Seir - Metasqhmatismì Fourier DiakritoÔ Qrìnou Keflaio 5
5.1 PARASTASH PERIODIKWN SHMATWN -
SEIRA FOURIER DIAKRITOU QRONOU
Ta periodik s mata diakritoÔ qrìnou paristnontai me peperasmèna ajro smata
NX1
2
x(n) = ak ejk N n (5.1.1)
k=0
H ex swsh aut d nei thn parstash ( anptugma) periodik¸n shmtwn diakritoÔ
qrìnou se seir Fourier diakritoÔ qrìnou.
5.1.1 Prosdiorismì th seir Fourier diakritoÔ qrìnou.
()
An x n e nai m a akolouj a h opo a e nai periodik me per odo N , o prosdiorismì
twn suntelest¸n ak ja mporoÔse na g nei apì th lÔsh tou grammikoÔ sust mato
X1
N
x(0) = ak
k=0
X1
N
2
x(1) = ak ejk N
k=0
.. ..
. .
X1
N
ak ejk N (N 1)
2
x(N 1) = (5.1.2)
k=0
àna llo trìpo prosdiorismoÔ e nai o pollaplasiasmì kai twn dÔo mel¸n th
(5.1.1) me e jm(2=N )n kai ajro soume w pro n, dhlad ,
NX1 NX1 N
X1
x(n)e jm 2N n = ak ej (k m) 2N n
n=0 n=0 k=0
NX1 NX1
ej (k m) N n
2
= ak
k=0 n=0
kai lìgw th hejk 0n; ejm 0ni = NÆ(k m) e nai
NX1
x(n)e jm 2N n = Nam
n=0
X1
= N1 x(n)e
N
am jm 2N n
n=0
Enìthta 5.1 Parstash Periodik¸n Shmtwn - Seir Fourier DiakritoÔ Qrìnou 139
àtsi èqoume ti exis¸sei :
X1
N
2
x(n) = ak ejk N n ; Ex swsh sÔnjesh (5.1.3)
k=0
NX1
ak =
1 x(n)e jk 2N n Ex swsh anlush (5.1.4)
N n=0
To zeÔgo twn exis¸sewn aut¸n or zoun th seir Fourier diakritoÔ qrìnou (discrete-
time Fourier series (DTFS) tou periodikoÔ s mato diakritoÔ qrìnou x n . Oi sunte- ()
lestè ak kaloÔntai suntelestè Fourier , ìpw ja doÔme, fasmatikè grammè .
Pardeigma 5.1.1
Na breje h parstash se seir Fourier tou s mato diakritoÔ qrìnou x(n) = sin( 0 n).
LÔsh Gnwr zoume ìti to s ma e nai periodikì an 2= 0 e nai akèraio rhtì ari-
jmì , ètsi mìno tìte mporoÔme na èqoume anptugma se seir Fourier diakritoÔ qrìnou.
Diakr noume ti peript¸sei :
1. To s ma e nai periodikì me jemeli¸dh per odo N kai 0 = 2=N . Me th bo jeia
th sqèsh tou Euler to s ma grfetai
x(n) = sin( 0 n)
= 21j ej 2N n 21j e j 2N n (5.1.5)
Sugkr nonta thn (5.1.5) me thn ex swsh sÔnjesh (5.1.3), parathroÔme ìti oi
suntelestè e nai a1 = 1=(2j ) kai a 1 = 1=(2j ) kai ak = 0 gia thn upìloiph
per odo. Oi suntelestè auto epanalambnontai me per odo sh me N ètsi èqoume:
akN +1 =
1 kai akN 1 = 1 ; k = 0; 1; 2; : : :
2j 2j (5.1.6)
Sto Sq ma 5.1 èqoun sqediaste to s ma x(n) kai oi suntelestè th seir Fourier
diakritoÔ qrìnou me N = 5 oi opo oi epanalambnontai. Prosoq ìmw sthn
ex swsh sÔnjesh uprqoun mìno oi suntelestè mia periìdou.
2. An 2= 0 = N=m, dhlad , rhtì 0 = (2m)=N . Upojètoume ìti
arijmì , tìte
ta m kai N den èqoun koinì pargonta ètsi to x(n) èqei jemeli¸dh per odo sh
me N . Me th bo jeia th sqèsh tou Euler to s ma grfetai
x(n) = sin
2 m
n
N
= 21j ejm 2N n 21j e jm 2N n (5.1.7)
apì ìpou èqoume: am = 1=(2j ) a m = 1=(2j ) kai ak = 0 gia thn upìloiph
per odo. Sto Sq ma 5.2 èqoun sqediaste to s ma x(n) kai oi suntelestè Fourier
ìtan m = 3 kai N = 5. Lìgw periodikìthta (N = 5) e nai :::a7 = a2 = a 3 =
a 8 = ::: = 1=(2j ), en¸ h ex swsh sÔnjesh èqei mìno dÔo ìrou .
140 Seir - Metasqhmatismì Fourier DiakritoÔ Qrìnou Keflaio 5
3. ätan to s ma e nai mh periodikì, den anaptÔssetai se seir Fourier diakritoÔ
qrìnou.
x(n)
2ð
N x(t)= sin(ù0 t )
Ù0
••• •••
3 4 8 9
0 1 2 5 6 7 10 n
(a)
ak
1 N
••• 2j •••
6 1 4 9
10 9 8 7 5 4 3 2 0 1 2 3 5 6 7 8 10 k
1
2j
(â)
Sq ma 5.1 (a) To s ma x(n) = sin 2N n , ìpou N = 5 kai (b) oi suntelestè Fourier.
x(n)
N m 2ð x(t)= sin(ù0 t )
Ù0
••• •••
1 3 6 8
0 2 4 5 7 9 10 n
(a)
ak
N 1
•••
7
2j •••
8 3 2
10 9 7 6 5 4 2 1 0 1 3 4 5 6 8 9 10 k
1
(â) 2j
Sq ma 5.2 (a) To s ma x(n) = sin m 2N n , ìpou N = 5, m = 3 kai (b) oi suntelestè
Fourier.
Pardeigma 5.1.2
Na breje h parstash se seir Fourier diakritoÔ qrìnou tou periodikoÔ orjog¸niou
kÔmato
x(n) = 1; jnj N1
0; N1 < jnj < N=2 (5.1.8)
me per odo sh me N.
LÔsh To periodikì orjog¸nio s ma diakritoÔ qrìnou fa netai sto Sq ma 5.3. Gia na
upolog soume tou suntelestè Fourier qrhsimopoioÔme thn ex swsh anlush
NX1
= N1 = N1
XN1
2 2
k ()
x n e jk N n e jk N n (5.1.9)
n=0 n= N1
Enìthta 5.1 Parstash Periodik¸n Shmtwn - Seir Fourier DiakritoÔ Qrìnou 141
x(n)
-N -N1 0 N1 N n
Sq ma 5.3 To periodikì orjog¸nio kÔma tou Parade gmato 5.1.2.
An jèsoume m = n + N1 èqoume:
2N1
= N1
X
e jk N (m N1 )
2
k
m=0
2N1
= N1 ejk 2N N1 e
X
jk 2N m (5.1.10)
m=0
dhlad , èqoume jroisma twn 2N1 +1 pr¸twn ìrwn gewmetrik proìdou, gia thn opo a
gnwr zoume ìti
(
NX1 N; =1
n = 1 N
; 6= 1 (5.1.11)
n=0 1
àtsi, gia k 6= 0; N; 2N; ::: èqoume
jk 2 (2N1 +1)
k = N1 ejk 2N N1 1 1e e Njk 2N
h i
jk 2N (N1 + 12 ) ejk 2N (N1 + 12 ) e jk N (N1 + 2 )
2 1
1 2
= ejk N N1 e
h i
N e jk 22N ejk 22N e jk 22N
To ginìmeno twn suntelest¸n ak ep to pl jo twn deigmtwn N e nai
1
k 2N N1 + 2
sin
N k=
sin k 22N ; k = 1; 2; :::N 1 k 6= 0; N; 2N; ::: (5.1.12)
en¸ ìtan k = 0; N; 2N; ::: èqoume
N k = 2N1 + 1 (5.1.13)
Sto Sq ma 5.4 èqoume thn akolouj a tou ginomènou twn suntelest¸n th seir Fourier
diakritoÔ qrìnou ep to pl jo twn deigmtwn tou periodikoÔ orjog¸niou kÔmato gia
difore timè tou N.
H èkfrash twn suntelest¸n th seir Fourier diakritoÔ qrìnou, ìpw aut perigrfe-
tai apì thn (5.1.12), ma epitrèpei na jewr soume to ginìmeno twn suntelest¸n ep to
pl jo twn deigmtwn w de gmata th sunrthsh
sin[(2N1 + 1)( =2)℄
sin( =2) (5.1.14)
142 Seir - Metasqhmatismì Fourier DiakritoÔ Qrìnou Keflaio 5
Na0
Na1
ðåñéâÜëëïõóá
N=10
0 2ð ð 2ð
10
Na0 Na1
N=20
0 2ð ð 2ð
20
Na0 Na1
N=40
0 2ð ð 2ð
40
Sq ma 5.4 To ginìmeno twn suntelest¸n th seir Fourier diakritoÔ qrìnou ep to pl jo
twn deigmtwn tou periodikoÔ orjog¸niou kÔmato gia N1 = 2 kai N = 10; 20 kai 40.
dhlad ,
N k=
sin [(2N1 + 1)( =2)℄
sin( =2) =k(2=N )
H sunrthsh (5.1.14) e nai h peribllousa twn suntelest¸n th seir Fourier diakri-
toÔ qrìnou tou periodikoÔ orjog¸niou kÔmato .
Sto Sq ma 5.5 eikon zetai to periodikì orjog¸nio kÔma diakritoÔ qrìnou ìpw
upolog zetai apì to merikì jroisma
M
X
x^(n) = jk 2 n
ke N (5.1.15)
k= M
Gia M =4
parathroÔme ìti to merikì jroisma (5.1.15) d nei to s ma x n , dhlad , ()
se ant jesh me th suneq per ptwsh, den emfan zetai fainìmeno Gibbs. Genik den em-
fan zetai fainìmeno Gibbs sthn seir Fourier diakritoÔ qrìnou. Autì ofe letai sto
()
gegonì ìti kje periodik akolouj a x n e nai pl rw orismènh apì èna pepera-
smèno arijmì N tim¸n, dhlad , ton arijmì twn tim¸n th akolouj a se m a per odo.
H (5.1.4) apl metasqhmat zei th seir twn N tim¸n se mia isodÔnamh seir N sunte-
lest¸n Fourier kai h (5.1.3) d nei to trìpo ankthsh twn tim¸n th akolouj a x n ()
me th bo jeia m a peperasmènh seir . An N e nai perittì arijmì kai jèsoume
M =(N 1) 2
= sthn (5.1.15), to jroisma apotele tai akrib¸ apì N ìrou . E-
pomènw apì thn ex swsh sÔnjesh èqoume x n ^( ) = ( )
x n . En N e nai rtio arijmì ,
Enìthta 5.2 Metasqhmatismì Fourier DiakritoÔ Qrìnou 143
x(n)
M=1
-18 -9 0 9 18 n
x(n)
M=2
-18 -9 0 9 18 n
x(n)
M=3
-18 -9 0 9 18 n
x(n)
M=4
-18 -9 0 9 18 n
Sq ma 5.5 To periodikì orjog¸nio kÔma diakritoÔ qrìnou, ìpw upolog zetai apì to merikì
jroisma (5.1.15), ìtan N = 9 kai 2N1 + 1 = 5 gia M = 1; 2; 3 kai 4.
PM jk 2 n
tìte to jroisma x n ^( ) = k= M +1 k e N me M = 2
N= perièqei N ìrou kai
katal goume sthn ex swsh sÔnjesh (5.1.3). àtsi x n xn. ^( ) = ( )
Ant jeta, èna periodikì s ma suneqoÔ qrìnou kat th dirkeia mia periìdou lam-
bnei peire suneqe timè epomènw , apaite tai peiro arijmì suntelest¸n Fouri-
er
PN
gia thn anaparstas tou. Genik ìle oi peperasmènou m kou seirè xN t ()=
k= N k e ()
jk!0 t den d noun akrib¸ ti timè tou x t kai parousizoun fainìmena
sÔgklish .
5.2 METASQHMATISMOS FOURIER DIAKRITOU QRONOU
()
Lambnoume s ma diakritoÔ qrìnou x n peperasmènh dirkeia epomènw , uprqei
akèraio N1 tètoio ¸ste x n ( )=0 jj
gia kje n > N1 . àstw N > N1 . Sqhmat zoume2
()
thn periodik epèktash tou x n , blèpe Sq ma 5.6.
1
X
x~(n) = x(n rN ) (5.2.1)
r= 1
To s ma x~(n) èqei per odo N , sump ptei me to x(n) sto disthma N=2 n N=2
144 Seir - Metasqhmatismì Fourier DiakritoÔ Qrìnou Keflaio 5
x(n)
-N1 0 N1 n
x(n)
N -N1 0 N1 N n
Sq ma 5.6 (a) To s ma x(n) kai (b) to s ma x~(n), h periodik epèktash tou x(n).
kai èqei anptugma se seir Fourier diakritoÔ qrìnou
X 2
x~(n) = ak ejk N n (5.2.2)
k=hN i
Oi suntelestè th seir Fourier diakritoÔ qrìnou d nontai apì th sqèsh:
= N1
X
ak x~(n)e jk 2N n
n=hN i
= N1
XN1
x(n)e jk 2N n
n= N1
1
= N1
X
x(n)e jk 2N n (5.2.3)
n= 1
Or zoume th migadik sunrthsh X( ) th pragmatik metablht
1
X
X( )= x(n)e j n (5.2.4)
n= 1
h opo a e nai periodik me per odo 2, opìte parathroÔme ìti oi suntelestè ak
mporoÔn na ekfrastoÔn w
1
ak = X (k 0 ) (5.2.5)
N
dhlad , oi suntelestè ak lambnontai apì deigmatolhy a th sunrthsh X( ) me
per odo deigmatolhy a 0 = 2=N .
Enìthta 5.2 Metasqhmatismì Fourier DiakritoÔ Qrìnou 145
àtsi, to s ma x ~(n), dhlad , h periodik epèktash tou x(n), d netai apì th sqèsh
x~(n) =
X 1 X (k )ejk 0n
N 0 (5.2.6)
k=hN i
kai, epeid 0 = 2=N 1=N = 0 =2, èqoume
x~(n) =
1 X X (k 0 )ejk 0n 0
2 k=hN i (5.2.7)
ParathroÔme ìti, ìtan N ! 1, tìte x~(n) = x(n), dhlad ,
x(n) = lim x~(n) = lim
1 X X (k )ejk 0n
N !1 N !1 2
0 0
k=hN i
= 21 X ( )ej n d
Z
(5.2.8)
2
ìpou qrhsimopoi jhke to Sq ma 5.7, gia na èqoume thn teleuta a isìthta. àtsi èqoume
ti exis¸sei sÔnjesh kai anlush gia to metasqhmatismì Fourier diakritoÔ qrìnou
(discrete time Fourier transform (DTFT)).
x(n) =
1 Z
2 2 X ( )e
j nd ; Ex swsh sÔnjesh (5.2.9)
1
X
X( )= x(n)e j n Ex swsh anlush (5.2.10)
n= 1
X(Ù)e jÙn
X(kÙ0)e jkÙ n0
kÙ0
-2ð -ð 0 ð 2ð Ù Sq ma 5.7 H grafik ermhne a tou
P
k=hN i X (k 0 )e
Ù0 ajro smato
jk 0 n 0 .
H exis¸sh (5.2.9) ekfrzei thn anlush tou s mato diakritoÔ qrìnou x n se ek- ()
jetik s mata ej n , ta opo a ekte nontai se èna suneqè fsma kuklik¸n suqnot twn
[0 2 )
periorismèno sto disthma ; , gegonì pou ofe letai sthn periodikìthta th
sunrthsh X . ( )
H sunrthsh X ( )
h opo a or zetai apì thn (5.2.10) e nai o metasqhmatismì
Fourier diakritoÔ qrìnou suqn anafèretai kai w fsma tou x n , giat perièqei thn ()
146 Seir - Metasqhmatismì Fourier DiakritoÔ Qrìnou Keflaio 5
()
plhrofor a pw to x n sunt jetai apì ekjetik s mata diaforetik¸n suqnot twn.
To fasmatikì perieqìmeno sto apeirostì disthma suqnot twn ; d e nai [ + ℄
( )
X kai h suneisfor twn suqnot twn P;
1
[
d èqei plto X + ℄ d =. ( )( 2 )
Shmei¸netai ìti to jroisma X ( )=
n= 1 x n e ()
j n uprqei ìtan
1
X 1
X
jx(n)j < 1 jx(n)j2 < 1 (5.2.11)
n= 1 n= 1
dhlad , h akolouj a èqei peperasmènh enèrgeia.
O metasqhmatismì Fourier diakritoÔ qrìnou èqei dÔo diaforè apì to metasqh-
matismì Fourier suneqoÔ qrìnou, oi opo e ofe lontai sto gegonì ìti ta ekjetik
s mata diakritoÔ qrìnou e nai periodik me per odo . 2
1. OX ( ) ()
e nai periodikì en¸ o X ! ìqi. àtsi to olokl rwma sthn ex swsh
sÔnjesh (5.2.9) èqei peperasmèno disthma olokl rwsh .
2. Sthn per ptwsh tou suneqoÔ qrìnou, oi qamhlè suqnìthte perigrfontai apì
diast mata mikroÔ eÔrou kentrarismèna sthn arq twn suntetagmènwn, en¸ oi
uyhlè suqnìthte e nai topojethmène makri apì thn arq twn axìnwn pro ta
arister pro ta dexi tou xona suqnot twn. Sthn per ptwsh tou diakritoÔ
qrìnou h periodikìthta tou metasqhmatismoÔ Fourier epibllei mia diaforetik
eikìna. Oi qamhlè suqnìthte antistoiqoÔn se diast mata gÔrw apì th jèsh
=0 , , lìgw th periodikìthta , gÔrw apì ti jèsei = 2
k. Oi uyhlè
suqnìthte topojetoÔntai kont se perioqè ìpou , = k = (2 + 1)
,
blèpe Sq ma 5.8.
Pardeigma 5.2.1
Na upologiste o metasqhmatismì Fourier diakritoÔ qrìnou tou aitiatoÔ ekjetikoÔ
s mato diakritoÔ qrìnou
x(n) = an u(n); jaj < 1 kai a 2 C (5.2.12)
LÔsh Me th bo jeia th (5.2.10) o metasqhmatismì Fourier diakritoÔ qrìnou e nai
1
X X1 n
X( )= an u(n)ej n = ae j (5.2.13)
n= 1 n=0
To jroisma apotele gewmetrik seir h opo a sugkl nei, epeid
ae j = jaj e j = jaj < 1
O metasqhmatismì Fourier diakritoÔ qrìnou tou aitiatoÔ ekjetikoÔ s mato e nai
X( ) = 1 ae1 j (5.2.14)
Enìthta 5.2 Metasqhmatismì Fourier DiakritoÔ Qrìnou 147
x1(n) X1(Ù)
0 n -3ð -2ð -ð 0 ð 2ð 3ð Ù
(a) (â)
x2(n)
X2(Ù)
0 n -3ð -2ð -ð 0 ð 2ð 3ð Ù
(ã) (ä)
Sq ma 5.8 (a) S ma diakritoÔ qrìnou x1 (n) pou èqei (b) MF diakritoÔ qrìnou X1 ( ) me
qamhlè suqnìthte . (g) S ma diakritoÔ qrìnou x2 (n) pou èqei (d) MF diakritoÔ qrìnou X2 ( )
me uyhlè suqnìthte .
Sto Sq ma 5.9 èqoun sqediaste to mètro kai h fsh tou metasqhmatismoÔ Fourier
diakritoÔ qrìnou tou aitiatoÔ ekjetikoÔ s mato diakritoÔ qrìnou gia pragmatikè
timè tou a, me 0 < a < 1 kai 1 < a < 0.
X(Ù) 0<a <1 X(ù) -1< a<0
1 1
1 a 1 a
1 1
1 a 1 a
-2ð -ð 0 ð 2ð Ù -2ð -ð 0 ð Ù
arg X(Ù) arg X(Ù)
tan1 a tan1 a
1 a
2
1 a2
-2ð 2ð
-ð 0 ð 2ð Ù -2ð -ð 0 ð Ù
tan1 a tan1 a
1 a 2 1 a
2
Sq ma 5.9 To mètro kai h fsh tou metasqhmatismoÔ Fourier diakritoÔ qrìnou tou aitiatoÔ
ekjetikoÔ s mato x(n) = an u (n); jaj < 1; a 2 R.
148 Seir - Metasqhmatismì Fourier DiakritoÔ Qrìnou Keflaio 5
Pardeigma 5.2.2
Na upologiste o metasqhmatismì Fourier diakritoÔ qrìnou tou s mato
x(n) = ajnj ; jaj < 1 kai a 2 R (5.2.15)
LÔsh O metasqhmatismì Fourier diakritoÔ qrìnou e nai
1
X X1 X1
X( )= ajnj e j n = a n e j n + an e j n
n= 1 n= 1 n=0
X1 X1
= an ej n + an e j n
n=1 n=0
X1 X1
= an ej n 1 + an e j n
n=0 n=0
X1 n X1 n
= aej + ae j 1
n=0 n=0
= 1 1aej + 1 ae1 j 1
2
= 1 + a21 2aa os( ) (5.2.16)
Sto Sq ma 5.10 eikon zetai to s ma kai o metasqhmatismì Fourier diakritoÔ qrìnou
tou. ParathroÔme ìti h X( ) e nai pragmatik sunrthsh tou , afoÔ a e nai prag-
matikì arijmì . Genik ta rtia s mata èqoun pragmatikoÔ metasqhmatismoÔ Fourier
diakritoÔ qrìnou .
x ( n)
-20
- -15
- -10
- -5
- 0 5 10 15 20 n
X(Ù)
1 a
1 a
Sq ma 5.10 To s ma x(n) = ajnj
-2ð -ð 0 ð 2ð Ù kai to fsma tou.
Pardeigma 5.2.3
Na upologiste o metasqhmatismì Fourier diakritoÔ qrìnou tou orjog¸niou palmoÔ
x(n) = 1; jnj N1
0; jnj > N1 (5.2.17)
Enìthta 5.2 Metasqhmatismì Fourier DiakritoÔ Qrìnou 149
LÔsh O metasqhmatismì Fourier diakritoÔ qrìnou tou orjog¸niou palmoÔ diakritoÔ
qrìnou e nai
N1 2N1 2N1
X m=n+N1 X
j (m N1 )
X
X( )= e j n = e = ej N1 e j m
n= N1 m=0 m=0
An efarmìsoume thn (5.1.11) gia N 1 = 2N1 N = 2N1 + 1 kai gia thn per ptwsh
pou 6= 0, o metasqhmatismì Fourier diakritoÔ qrìnou apokt th morf
X( ) = ej N1
1 e j (2N1 +1)
1 ej
e (N1 + 2 ) ej (N1 + 2 ) e j (N1 + 2 )
j 1 1 1
= ej N1
e j 12 ej 12 e j 21
= sin N1 + 12
sin 12 (5.2.18)
Sto Sq ma 5.11 èqei sqediaste o orjog¸nio palmì kai to fsma tou.
X(Ù)
x(n) 2Í+1
1
2ð
2Í+1
1
-N1 0 N1 n 2ð ð 0 ð 2ð Ù
Sq ma 5.11 O orjog¸nio palmì diakritoÔ qrìnou kai to fsma tou.
Parathr sei
1. Gia th tim =0
o metasqhmatismì Fourier diakritoÔ qrìnou e nai X (0) =
2 +1
N1 .
2. Oi timè pou mhden zoun ton X ( )
e nai oi suqnìthte gia ti opo e N1 sin[ ( +
1 2)℄ = 0
= , dhlad , N1 = ( + 1 2) = ) =
2N21+1 ; Z . 2
3. äso megalÔtero e nai to eÔro N1 2 +1 tou palmoÔ, tìso megalÔtero e nai o
( )
arijmì twn suqnot twn pou mhden zoun ton X , tìso mikrìtero to eÔro tou
kentrikoÔ loboÔ kai tìso megalÔterh h tim tou sto mhdèn.
4. Gia N1 !1 o metasqhmatismì Fourier diakritoÔ qrìnou te nei pro th sunrth-
sh dèlta.
5. H anakataskeu tou s mato apì suqnìthte j j
W d nei to s ma:
x^(n) =
1 Z W
2 W X ( )e
j nd
150 Seir - Metasqhmatismì Fourier DiakritoÔ Qrìnou Keflaio 5
6. gia W = , dhlad , ìtan qrhsimopoioÔntai ìle oi suqnìthte , e nai x^(n) =
x(n). ParathroÔme ìti den èqoume emfnish fainomènwn Gibbs.
Pardeigma 5.2.4
Na upologiste to s ma diakritoÔ qrìnou x(n), tou opo ou o metasqhmatismì Fourier
diakritoÔ qrìnou e nai orjog¸nio periodikì kÔma (Sq ma 5.12a), dhlad ,
X( ) = 01;; W
j jW
<j j (5.2.19)
X(Ù)
1
2ð ð W 0 W ð 2ð Ù
x(n)
W
ð
ð 0 ð n
W W
Sq ma 5.12 O metasqhmatismì Fourier diakritoÔ qrìnou tou s mato sto Pardeigma 5.2.4
kai h apokatstas tou sto qrìno.
LÔsh To s ma x(n) lambnetai apì ton X( ) mèsw tou antistrìfou metasqhma-
tismoÔ Fourier diakritoÔ qrìnou,
x(n) =
1 Z W ej nd = sin(n
W n)
2 W (5.2.20)
ParathroÔme anlogh sumperifor me aut tou Parade gmato 3.3.4.
Pardeigma 5.2.5
1. Na upologiste o metasqhmatismì Fourier diakritoÔ qrìnou tou s mato x(n) =
Æ(n) kai tou olisjhmènou monadia ou de gmato .
2. Na g nei h anakataskeu tou s mato x(n) apì to fsma tou.
LÔsh
1. Epeid x(n) = 0 gia kje n 6= 0, br skoume X ( ) = x(0)ej0 = 1, dhlad , to
fsma tou monadia ou de gmato ekte netai se ìle ti suqnìthte . Genikìtera to
fsma tou olisjhmènou monadia ou de gmato e nai
x(n) = Æ(n k) F! X ( )=e jk (5.2.21)
Enìthta 5.3 Idiìthte tou MetasqhmatismoÔ Fourier DiakritoÔ Qrìnou 151
To mètro tou fsmato e nai
jX ( )j = 1 (5.2.22)
H fsh tou fsmato e nai grammik se m a per odo.
arg X ( ) = k (5.2.23)
Sto Sq ma 5.13 èqoun sqediaste to mètro kai h fsh tou MF diakritoÔ qrìnou
tou olisjhmènou kat k de gmata monadia ou de gmato .
X(Ù)
1
-2ð -ð 0 ð 2ð Ù
arg X(Ù)
Sq ma 5.13 To mètro kai h fsh tou MF
-2ð -ð 0 ð 2ð Ù diakritoÔ qrìnou tou monadia ou ol sjh-
mènou de gmato .
2. H anakataskeu tou x(n) apì to fsma tou e nai
= 21
Z W
x^(n) ej nd
W
= sin( W n)
(5.2.24)
n
Sto Sq ma 5.14 uprqei to x^(n) gia diaforetikè timè tou W . ParathroÔme ìti
x^(n) = Æ(n) ìtan W = .
5.3 IDIOTHTES TOU METASQHMATISMOU FOURIER DIAKRITOU
QRONOU
O metasqhmatismì Fourier diakritoÔ qrìnou parousizei dÔo idiìthte diaforetikè
apì to metasqhmatismì Fourier suneqoÔ qrìnou, ti opo e parajètoume ston P naka
5.1.
H pr¸th diafor uprqei sthn idiìthta diafìrish sto ped o tou qrìnou, ìpou èna
s ma diakritoÔ qrìnou den e nai diafor simo sunart sei tou qrìnou, epeid lambnei
diakritè timè .
H deÔterh diafor br sketai sthn idiìthta th allag kl maka . Sthn per ptwsh
( )
tou suneqoÔ qrìnou ìpou to s ma x at èqei pntote ènnoia kai antiproswpeÔei
1 1
sump esh en a > kai diastol en a < . To ant stoiqo s ma x dn den or zetai ( )
en o d den e nai akèraio . Ep sh , ìpw ja doÔme, uprqei diafor kai sthn idiìthta
th olokl rwsh .
152 Seir - Metasqhmatismì Fourier DiakritoÔ Qrìnou Keflaio 5
x(n) x(n)
1 3
4 W= ð4 8 W= 38ð
x(n)
3
x(n) 4 W= 3ð
1 4
2 W= ð2
x(n)
x(n) 1
7 W= 78ð
8 W=ð
Sq ma 5.14 H prosèggish tou monadia ou de gmato apì thn x^(n) = sin(W n)=n gia
difore timè th paramètrou W.
5.3.1 Apodektish sto qrìno
M a kathgor a apì sust mata apaitoÔn antiproswpeutik apoj keush kai metdosh
shmtwn me diaforetikoÔ rujmoÔ deigmatolhy a .
()
àstw ìti to s ma diakritoÔ qrìnou x n prokÔptei apì èna analogikì s ma s t ()
me rujmì deigmatolhy a Fs =1 ( )= ( )
=Ts , dhlad , x n s nTs . An o rujmì elattwje
~ =1 ~
se Fs =Ts , tìte to yhfiakì s ma e nai
!
T~
s(nT~s ) = s n s Ts = s(nMTs) = x(nM ) = xM (n) (5.3.1)
Ts
ìpou o akèraio M or zetai w o lìgo twn rujm¸n deigmatolhy a M Ts =Ts =~ =
~ 1 ()
Fs =Fs > . ParathroÔme ìti to s ma xM n lambnetai me deigmatolhy a tou
()
x n gia kje M de gmata. H elttwsh tou rujmoÔ deigmatolhy a antistoiqe se
apodektish sthn anaparstash diakritoÔ qrìnou. Sto Sq ma 5.15 fa netai o trìpo
()
me ton opo o prokÔptei h apodekatismènh morf x2 n tou s mato x n . Gia na ka- ()
jor soume ta apotelèsmata th apodektish sto ped o suqnot twn or zoume to s ma
() ()
z n to opo o sump ptei me to x n gia kje tim tou n pou e nai pollaplsio tou M
Enìthta 5.3 Idiìthte tou MetasqhmatismoÔ Fourier DiakritoÔ Qrìnou 153
x(n)
-8 -6 -4 -2 0 2 4 6 8 n
x2(n)
-8 -6 -4 -2 0 2 4 6 8 n
z(n)
-8 -6 -4 -2 0 2 4 6 8 n
Sq ma 5.15 Apodektish sto qrìno (a) to s ma x(n), (b) h apodekatismènh morf x2 (n)
kai (g) to bohjhtikì s ma z (n).
kai e nai mhdèn opoud pote alloÔ, dhlad ,
z (n) = x(n); an n = 0modM
0; alli¸
(5.3.2)
O metasqhmatismì Fourier diakritoÔ qrìnou XM ( ) tou s mato xM (n) e nai
1
X
XM ( ) = xM (n)e j n
n= 1
1
X
= x(Mn)e j n
n= 1
m=Mn X
= x(m)e
m
j M
m=0modM
E nai ìmw
1
X
Z
M
= z (m)e jMm
m= 1
1
X
= x(m)e jMm
m=0modM
154 Seir - Metasqhmatismì Fourier DiakritoÔ Qrìnou Keflaio 5
ètsi
XM ( )=Z (5.3.3)
M
Ja ekfrsoume to fsma tou apodekatismènou s mato se sqèsh me to arqikì. E nai
gnwstì ìti
1 X1
M
2
1; an n = 0modM
ej M kn = 0; (5.3.4)
M k=0 alli¸
àtsi to s ma z (n) mpore na grafe w
1 X1
M
2
z (n) = x(n) ej M kn
M k=0
kai o metasqhmatismì Fourier diakritoÔ qrìnou e nai:
1
X 1
X 1 X1
M
2
Z( )= z (n)e j n = x(n) ej M kne j n
n= 1 n= 1 M k=0
X1 X
1
= M1
M
x(n)e j ( M )
2 k n
k=0 n= 1
X1 2 k
= M1
M
X (5.3.5)
k=0
M
Sunduzonta thn teleuta a ex swsh me thn XM ( ) = Z M èqoume:
1 X1
M
2 k
XM ( ) = X (5.3.6)
M M k=0
M
To apodekatismèno fsma lambnetai me dieÔrunsh tou fsmato X kat M , ( )
2 (2
metatìpish kat =M , =M , ..., =M M )2 (2 )(
kai upèrjesh. H diadikas a 1)
apeikon zetai sto Sq ma 5.16 gia M . =2
5.3.2 Parembol
H mèjodo th parembol pragmatopoie aÔxhsh tou rujmoÔ deigmatolhy a . àstw
() ~( )
ta s mata x n kai x n ta opo a prokÔptoun apì to analogikì s ma s t me diafore- ()
tikoÔ rujmoÔ deigmatolhy a , dhlad ,
x(n) = s(nTs)
x~(n) = s(nT~s) (5.3.7)
Enìthta 5.3 Idiìthte tou MetasqhmatismoÔ Fourier DiakritoÔ Qrìnou 155
X(Ù)
-3ð -2ð -ð 0 ð 2ð 3ð Ù
(á)
X2(Ù)
2(
×( Ù × ( Ù-2ð
2 (
1 1
2 2
-3ð -2ð -ð 0 ð 2ð 3ð Ù
(â)
Sq ma 5.16 (a) To arqikì fsma X( ) kai (b) to fsma tou apodekatismènou s mato
XM ( ) gia M = 2.
ìpou Fs =1 ~ =1 ~ ~
=Ts kai Fs =Ts me Fs < Fs Ts > Ts kai èqoun lìgo Fs =Fs L > .~ ~ = 1
H aÔxhsh tou rujmoÔ deigmatolhy a kat èna pargonta L apaite upologismì tou
()
x n sti timè n=L h opo a e nai efikt mìno en to n e nai pollaplsio tou L. Oi
upìloipe timè ja prèpei na paremblhjoÔn. àtsi or zoume to s ma
x n = 0modL
xL (n) = L ; an n
0; alli¸
(5.3.8)
()
Sto Sq ma 5.17 pargoume to s ma x2 n pargetai apì to s ma x n parembllon- ()
()
ta èna mhdenikì anmesa se diadoqikè timè tou x n . Me anlogo trìpo pargetai
()
kai to s ma x3 n .
To fsma tou s mato xL n e nai ()
1
X 1
X
XL ( )= xL (n)e j n = xL (kL)e j kL
n= 1 k= 1
1
X
kL
= x
L
e j kL
k= 1
1
X
= x(k)e j (L )k
k= 1
= X (L ) (5.3.9)
ParathroÔme ìti to fsma XL ( ) lambnetai apì to arqikì fsma X ( ) me allag
kl maka suqnot twn kat L.
156 Seir - Metasqhmatismì Fourier DiakritoÔ Qrìnou Keflaio 5
X(Ù)
x(n) 5
2ð
5
0 n 2ð ð 0 ð 2ð Ù
(á)
X2(Ù)=×(2Ù)
x2(n) 5
0 n Ù
2ð ð 0 ð 2ð
(â)
X3(Ù)=×(3Ù)
x3(n) 5
0 n Ù
2ð 4ð 2ð 0 2ð 4ð 2ð
(ã) 3 3 3 3
Sq ma 5.17 (a) To s ma x(n) kai to fsma tou. To s ma xL (n) pou pargetai apì to
x(n) parambllonta L 1 mhdenik anmesa se diadoqikè timè tou x(n) kai to fsma tou
XL( ) gia (b) L = 2 kai (g) L = 3.
5.3.3 jroisma
An x t( ) F! X (!), h idiìthta th olokl rwsh tou metasqhmatismoÔ Fourier suneqoÔ
qrìnou e nai
Z t
x( ) d F!
1 X (!) + X (0)Æ(!)
1 j!
()
Me th bo jeia tou s mato x n or zoume to s ma y n ( ) = Pn
m= 1 x m kai parath- ( )
roÔme ìti y n() yn ( 1) = ( )
x n . Efarmìzoume metasqhmatismì Fourier diakritoÔ
qrìnou kai sta dÔo mèlh th teleuta a ex swsh kai, lìgw th idiìthta th gram-
mikìthta kai th qronik metatìpish , èqoume:
F [y(n) y(n 1)℄ = F [x(n)℄ Y ( ) e j Y ( ) = X ( )
An 6= 0 èqoume
Y( ) = 1 1e j X( )
Enìthta 5.3 Idiìthte tou MetasqhmatismoÔ Fourier DiakritoÔ Qrìnou 157
ApodeiknÔetaiy ìti h pl rh sqèsh h opo a ekfrzei thn idiìthta th jroish e nai:
n
X 1 1
X
x(m) F! X( ) + X (0) Æ( 2k)
m= 1 1 e j
k= 1
(5.3.10)
5.3.4 Idiìthta th Diamìrfwsh
() ()
Me th bo jeia twn shmtwn x1 n kai x2 n , pou èqoun metasqhmatismoÔ Fourier
diakritoÔ qrìnou X1 kai X2 ( ) ( )
ant stoiqa, sqhmat zoume to s ma y n x1 n ( ) = ( )
()
x2 n . O metasqhmatismì Fourier diakritoÔ qrìnou tou s mato y n ja e nai: ()
1
X 1
X
Y( )= y(n)e j n = x1 (n) x2 (n)e j n
n= 1 n= 1
1
X 1 Z
= x2 (n)
2 2 X1()e d e
jn j n
n= 1 " #
1
= 21 X1 ()
Z X
x2 (n) ej ( )n d
2 n= 1
1 Z
= 2 X1 ()X2 ( ) d (5.3.11)
2
H teleuta a ex swsh apotele thn twn X1 ( ) kai X2 ( ). H di-
periodik sunèlixh
amìrfwsh s mato diakritoÔ qrìnou me th bo jeia th idiìthta aut epexhge tai
sto pardeigma pou akolouje .
Pardeigma 5.3.1
àstw x1 (n) h periodik akolouj a, me per odo 2
x1 (n) = ejn = ( 1)n (5.3.12)
kai to s ma x2 (n); tou opo ou o metasqhmatismì Fourier diakritoÔ qrìnou X2 ( )
fa netai sto Sq ma 5.18b. Na prosdioriste grafik o metasqhmatismì Fourier di-
akritoÔ qrìnou tou s mato y(n) = x1 (n) x2 (n).
LÔsh O MF diakritoÔ qrìnou th periodik akolouj a x1 (n) e nai
1
X
X1 ( ) = 2 Æ( (2r + 1)) (5.3.13)
r= 1
H grafik parstash tou X1 ( ) e nai sto Sq ma 5.18a. Sto Sq ma 5.18g èqoume
sqedisei ta X1 () kai X2 ( ). ParathroÔme ìti
X1 () X2 ( ) = 2X2 ( )Æ( ) gia 0 < < 2
y Anafor 5.4 kai 5.5 th proteinìmenh bibliograf a
158 Seir - Metasqhmatismì Fourier DiakritoÔ Qrìnou Keflaio 5
X1(Ù)
2ð
(á)
-3ð -2ð -ð 0 ð 2ð 3ð Ù
X2(Ù)
1
(â)
-3ð -2ð -ð 0 ð 2ð 3ð Ù
X1(è)
2ð
-3ð -2ð -ð 0 ð 2ð 3ð è
X2(Ù-è) (ã)
1
-3ð -ð ð 3ð
-2ð 0 Ù 2ð è
ðåñéï÷Þ
ïëïêëÞñùóçò
Y(Ù)
1
(ä)
-3ð -2ð -ð 0 ð 2ð 3ð Ù
Sq ma 5.18 H idiìthta th diamìrfwsh diakritoÔ qrìnou. (a) O DTFT tou s mato
x1 (n) = ( 1)n, (b) o DTFT tou s mato x2 (n), (g) oi posìthte oi opo e qreizontai gia ton
prosdiorismì tou periodikoÔ sugkerasmoÔ kai (d) o DTFT tou s mato y(n) = x1 (n)x2 (n) =
( 1)nx2(n)
ètsi èqoume
Z 2
Y( )= X2 ( )Æ( ) = X2 ( ) (5.3.14)
0
O Y( ) èqei sqediaste sto Sq ma 5.18d.
Ja mporoÔsame na broÔme thn teleuta a sqèsh me thn bo jeia th idiìthta th
ol sjhsh fsh tou metasqhmatismoÔ Fourier diakritoÔ qrìnou
ejn x(n) = ( 1)n x(n) F! X ( )
ParathroÔme ìti o pollaplasiasmì enì s mato me ( 1)n èqei w apotèlesma thn al-
lag tou pros mou stou perittoÔ ìrou th akolouj a . Sto de q¸ro twn suqnot twn
èqei w apotèlesma thn ol sjhsh tou periodikoÔ fsmato X2 ( ) kat mis per odo
Enìthta 5.3 Idiìthte tou MetasqhmatismoÔ Fourier DiakritoÔ Qrìnou 159
PINAKAS 5.4 Idiìthte tou metasqhmatismoÔ Fourier diakritoÔ qrìnou
Idiìthta Ped o qrìnou Ped o suqnìthta
Suzug a sto qrìno x? (n) X?( )
Suzug a sth suqnìthta x? ( n) X?()
Anklash x( n) X( )
Grammikìthta ax1 (n)+bx2 (n) aX1 ( )+bX2 ( )
rtio mèro s mato xe (n)= 12 [x(n)+x? ( n)℄ <e[X ( )℄=R( )
Pragmatikì mèro fsmato
Perittì mèro s mato xo (n)= 12 [x(n) x? ( n)℄ j =m[X ( )℄=jI ( )
Fantastikì mèro fsmato
Qronik metatìpish x(n n0 ) e j n0 X ( )
Ol sjhsh suqnìthta ej 0 n x(n) X( 0)
X ( )=X ? ( )
<e[X ( )℄=<e[X ( )℄
Pragmatikì s ma x(n)=x? (n) =m[X ( )℄= =m[X ( )℄
jX ( )j=jX ( )j
arg X ( )= arg X ( )
Pn P1
m= 1 x(m) 1+e j X ( )+X (0) k= 1 Æ( 2k)
jroisma 1
x(n)y(n) 1 R
Diamìrfwsh 2 2 X ( )Y () d
Sunèlixh x(n)?y(n) X ( )Y ( )
Apodektish xM (n)=x(Mn) XM ( )= M1 M
P 1
k=0 X M M
2k ( )
Diafìrish sto ( j )k nk x(n) dk X ( )
d k
ped o suqnot twn
Diafor x(n) x(n 1) (1 e j X( ) )
Parembol xM (n)=x Mn( ) X (M )
Je¸rhma Parseval Ex =P1
n 1 jx(n)j
2 Ex = 21 R2 jX ( )j2 d
(dhlad kat p).
Lìgw th periodikìthta thn opo a parousizoun ta fsmata twn
diakrit¸n shmtwn, autì èqei w apotèlesma thn enallag twn uyhl¸n kai qamhl¸n
suqnot twn.
160 Seir - Metasqhmatismì Fourier DiakritoÔ Qrìnou Keflaio 5
5.3.5 O metasqhmatismì Fourier diakritoÔ qrìnou gia periodik s mata
Ja prosdior soume to s ma x (n) tou opo ou to fsma e nai h periodik epèktash th
sunrthsh dèlta, dhlad ,
1
X
X( )= Æ( 2k) (5.3.15)
k= 1
Me th bo jeia tou ant strofou metasqhmatismoÔ Fourier diakritoÔ qrìnou èqoume
x(n) =
1 Z
2 X ( )e d
j n
1 Z X 1
= 2 Æ( 2k)ej nd (5.3.16)
k= 1
2
Epeid kje disthma me m ko perièqei m a mìno sunrthsh dèlta apì ton kroustikì
surmì tou ajro smato , (dhlad , sto disthma ( )
; uprqei m a sunrthsh dèlta
sth jèsh =0), èqoume:
x(n) =
1 Z
= 21 ej0n = 21
2 Æ( )e
j nd
àtsi èqoume to zeÔgo metasqhmatismoÔ Fourier:
1
X
x(n) = 1 F! X ( ) = 2 Æ( 2k) (5.3.17)
k= 1
Lìgw th idiìthta th ol sjhsh sto ped o twn suqnot twn èqoume:
1
X
x(n) = ej 0 n F! X ( ) = 2 Æ( 2k)
0 (5.3.18)
k= 1
Sto Sq ma 5.19 èqei sqediaste o metasqhmatismì Fourier diakritoÔ qrìnou tou s -
mato x n( )= ej 0 n .
X(Ù)
2ð
Ù0-4ð Ù0-2ð 0 Ù0 Ù0+2ð Ù0+4ð Ù
Sq ma 5.19 O metasqhmatismì Fourier diakritoÔ qrìnou tou s mato x(n) = ej 0 n .
Enìthta 5.3 Idiìthte tou MetasqhmatismoÔ Fourier DiakritoÔ Qrìnou 161
ParathroÔme ìti, epeid to s ma suneqoÔ qrìnou x t ej!0 t èqei metasqhma- ()=
tismì Fourier m a sunrthsh dèlta sth suqnìthta ! !0 , to s ma diakritoÔ qrìnou =
( )=
xn j n
e , lìgw th periodikìthta e
0 j 0 n e +2r)n , èqei metasqhmatismì
j ( 0 =
Fourier diakritoÔ qrìnou èna kroustikì surmì ston opo o ta kroustik s mata e nai
topojethmèna sti suqnìthte 0 ; 0 ; 0 ; ::: 2 4
Gnwr zoume ìti kje periodikì s ma diakritoÔ qrìnou me per odo N anaptÔssetai
se seir Fourier diakritoÔ qrìnou
NX1
2
x(n) = am ejm N n (5.3.19)
m=0
x(n) = a0 + a1 ej N n + a2 ej 2 N n + ::: 1 ej (N 1) N n
2 2 2
+ aN (5.3.20)
kai o metasqhmatismì Fourier diakritoÔ qrìnou e nai:
1
X
X( ) = a0 2Æ( 2k)
k= 1
1
X
2 2k
+a1 2Æ N
k= 1
..
.
1
X
2
+aN 1 2Æ (N 1) N 2k
k= 1
Efarmog twn parapnw sqèsewn g netai sto Sq ma 5.20 gia N . Lìgw th peri- =4
odikìthta twn suntelest¸n a0 ; a1 ; :::; aN 1 , dhlad , al aNk+l a Nk+l , 2 =2 =2
= 0 1 2
k ; ; ; ::: kai l ; ; :::; N =0 1 1
apì thn teleuta a sqèsh èqoume
NX1 1
am
2
ejm N n F! 2 X
al Æ l
2
(5.3.21)
m=0 l= 1 N
Pardeigma 5.3.2
Oi suntelestè Fourier th periodik epèktash tou monadia ou de gmato periìdou N
(Sq ma 5.21a)
1
X
x(n) = Æ(n kN ); k = 0; 1; 2; ::: (5.3.22)
k= 1
162 Seir - Metasqhmatismì Fourier DiakritoÔ Qrìnou Keflaio 5
∞
2ðá0 Ó ä(Ù-2ðk)
k=-∞
2ðá-4=2ðá0 2ðá0 2ðá4=2ðá0
-2ð 0 2ð Ù
∞
2ðá1 Ó ä(Ù- 2ð
k=-∞ Í -2ðk
(
2ðá-3=2ðá1 2ðá1 2ðá5=2ðá1
2ð 2ð 2ð Ù
(-N+1) Í (N+1) Í
Í
∞
2ðá2 Ó ä( Ù-2 2ð
k=-∞ Í -2ðk
(
2ðá-6=2ðá2 2ðá-2=2ðá2 2ðá2 2ðá6=2ðá2
2ð 2ð 2ð 2ð Ù
(-2N+2) Í (-N+2) Í 2Í (N+2) Í
2ðá3 Ó ä( Ù-3 Í -2ðk (
∞ 2ð
k=-∞
2ðá-5=2ðá3 2ðá-1=2ðá3 2ðá3 2ðá7=2ðá3
2ð 2ð 2ð 2ð Ù
(-2N+3) Í (-N+3) Í 3Í (N+3) Í
N-1
2ð Ó ál ä(Ù-l 2ð
Í(
l=0
2ðá0
2ðá1 2ðá 2ðá3
2
-2ð 0 2ð 2ð 2ð 2ð Ù
Í 2Í 3Í
x(n) =
PN 1 jm 2 n
Sq ma 5.20 O DTFT enì periodikoÔ s mato diakritoÔ qrìnou m=0 am e N
ìpou N = 4.
e nai
= N1
X 2
ak x(n)e jk N n
n=hN i
NX1 X 1
= N1 2
Æ(n kN )e jk N n
n=0 k=1
= 1 N
(5.3.23)
Enìthta 5.4 Deigmatolhpthmèna S mata sto Ped o Suqnot twn 163
Epomènw , h periodik epèktash tou monadia ou de gmato anaptÔssetai se seir Fouri-
er diakritoÔ qrìnou w
1 1
X
x(n) = ejk(2=N )n (5.3.24)
N k= 1
O metasqhmatismì Fourier diakritoÔ qrìnou th periodik epèktash tou monadia ou
de gmato lìgw th (5.3.21) e nai
1
2k
) = 2N
X
X( Æ (5.3.25)
k= 1 N
Sto Sq ma 5.21b èqei sqediaste o metasqhmatismì Fourier diakritoÔ qrìnou tou s -
mato x(n).
x(n)
1
-N 0 N n
(a)
X(Ù)
2ð
Í
4ð 2ð 0 2ð 4ð Ù
Í Í Í Í
(â)
Sq ma 5.21 (a) H periodik epèktash tou monadia ou de gmato kai (b) to fsma th .
Pardeigma 5.3.3
Na upologiste o metasqhmatismì Fourier diakritoÔ qrìnou th monadia a bhmatik
akolouj a .
LÔsh Gnwr zoume ìti x(n) = Æ(n) F! X ( ) = 1. Me th bo jeia th idiìthta tou
ajro smato o metasqhmatismì Fourier diakritoÔ qrìnou th bhmatik akolouj a
u(n) e nai
n
X 1 1
X
Æ(m) = u(n) F! + Æ( 2k)
m= 1 1 e j k= 1
(5.3.26)
Ston P naka 5.2 parousizontai merik zeÔgh metasqhmatism¸n Fourier diakritoÔ
qrìnou.
5.4 SHMATA APO DEIGMATOLHYIA STO PEDIO SUQNOTHTWN
()
D netai to analogikì s ma xa t to opo o èqei metasqhmatismì Fourier X ! . An ()
()
lboume de gmata tou xa t me per odo Ts , dhlad , an metr soume ti timè tou s -
164 Seir - Metasqhmatismì Fourier DiakritoÔ Qrìnou Keflaio 5
PINAKAS 5.5 DTFT basik¸n sunart sewn diakritoÔ qrìnou
A/A Ped o qrìnou Ped o suqnìthta
1
P
k=<N> ak
( )
2
ejk N n
P1
2 k= 1 ak Æ 2k
N
2 ej 0 n 2 P1l= 1 Æ ( 0 2l)
3 os( 0n) P
1l= 1 [Æ( 0 2l)+Æ( + 0 2l) ℄
4 sin( 0n) P
1
j l= 1 [Æ(
P1
0 2l) Æ( + 0 2l) ℄
5 x(n) = 1 2 k= 1 Æ( 2k)
8
< 1; jnjN1 2k
6 x(n+N )=x(n)=:
0; N1 <jnjN=2
2 P1k= 1 ak Æ N
P1 2 P1 Æ 2k
7 k= 1 Æ (n kN ) N k= 1 N
an u(n); jaj < 1 1 j
8
1 ae
x(n) = 10;; jjnnj N1 sin[ (N1 + 12 )℄
9
j>N1 sin( 2 )
8
sin(W n) = W sin Wn ; < 1; 0j jW
10 n 0<W < X ( +2k)=X ( )=:
0; W <j j
11 Æ(n) 1
u(n) 1j + P1
12 1 e k= 1 Æ( 2k)
13 Æ(n n0 ) e j n0
14 (n + 1)an u(n); jaj < 1 1
(1 ae j )2
(n+r+1)! an u(n); jaj < 1 1
15 n!(r 1)! (1 ae j )r
mato suneqoÔ qrìnou se diakritè qronikè stigmè pou e nai pollaplsia th Ts ,
pa rnoume to s ma diakritoÔ qrìnou
xs (n) = xa (nTs ) (5.4.1)
To qronikì disthma Ts e nai gnwstì w per odo deigmatolhy a kai fs 1 =
Ts e nai
h suqnìthta deigmatolhy a . H epilog th Ts ja prèpei na g netai ètsi, ¸ste na
mh qaje plhrofor a pou perièqetai sto analogikì s ma, (dhlad , na e nai dunat h
anakataskeu tou apì ta de gmat tou), all oÔte na auxhje qwr lìgo h apaitoÔmh
mn mh.
Ja prosdior soume th sqèsh anmesa sto metasqhmatismì Fourier diakritoÔ qrìnou
( ) ()
Xs tou s mato xs n , pou èqei proèljei apì deigmatolhy a, kai tou antisto qou
Enìthta 5.4 Deigmatolhpthmèna S mata sto Ped o Suqnot twn 165
tou suneqoÔ qrìnou Xa (!). O metasqhmatismì Fourier diakritoÔ qrìnou tou s ma-
()
to xs n e nai
1
X
Xs ( )= xs (n)e j n (5.4.2)
n= 1
H ex swsh sÔnjesh tou xs(n) e nai
xs (n) =
1 Z
2 Xs( )e d
j n (5.4.3)
H ex swsh sÔnjesh tou xa (t) e nai
xa (t) =
1 Z 1
2 1 Xa (!)e d!
j!t (5.4.4)
Efarmìzoume thn (5.4.4) gia t = nTs
xa (nTs ) =
1 Z 1
2 1 Xa (!)e d!
j!nTs (5.4.5)
H sÔgkrish twn (5.4.5) kai (5.4.3) ma kajodhge sto metasqhmatismì !Ts , pou =
e nai h gnwst sqèsh metaxÔ analogik kai yhfiak kuklik suqnìthta (1.4.11).
àtsi h (5.4.5) d nei
xa (nTs) =
1 Z 1
j nd
2Ts 1 Xa Ts e (5.4.6)
H (5.4.6) diafèrei apì thn (5.4.3) w pro ta ìria tou oloklhr¸mato . Qwr zoume to
olokl rwma se jroisma oloklhrwmtwn se diast mata m kou , dhlad , 2
1 X1 Z (2k+1)
xa (nTs ) = j nd
2Ts k= 1 (2k 1) Xa Ts e (5.4.7)
Allag th metablht = 2k odhge sthn
1 Z X1
+ 2
k j
xa (nTs) = nd
2Ts k= 1 Xa Ts e (5.4.8)
H sÔgkrish th (5.4.3) me thn (5.4.8) d nei thn
1 1
X
+ 2k
Xs ( ) = Xa (5.4.9)
Ts k= 1 Ts
166 Seir - Metasqhmatismì Fourier DiakritoÔ Qrìnou Keflaio 5
, w pro thn analogik kuklik suqnìthta,
1
xs (n) = xa (nTs ) F! X (!T ) = 1 X X ! + 2k (5.4.10)
s s
Ts k = 1 a Ts
O metasqhmatismì Fourier diakritoÔ qrìnou tou s mato pou èqei proèljei apì
deigmatolhy a prokÔptei apì to fsma tou analogikoÔ s mato me ti ex diadikas e :
(blèpe Sq ma 5.22)
xa(t) Xá(ù) Õ(Ù)= 1 Xá Tù ((
Ts s
1
1 Ts
0 t ù 0 ù ù ù 0 ù Ù
2 Ts 2 Ts
0 0 0 0
2 2
(a1) (a2) (a3)
Xs(Ù)
xs(n)
1
Ts
0 n -2ð -ð ù 0 ù ð 2ð Ù
2 Ts 2 Ts
0 0
(â1) (â2)
Xs(Ù)
xs(n)
1
Ts
0 n -4ð -2ð ù 0 ù 2ð 4ð Ù
2 Ts 2 Ts
0 0
(ã1) (ã2)
Sq ma 5.22 ( 1) Analogikì s ma, ( 1)
kai ( 1 ) s mata apì deigmatolhy a me diafore-
tikì rujmì deigmatolhy a . Ta fsmata ( 2 ) tou analogikoÔ s mato , ( 3 ) met thn allag
kl maka , ( 2) ol sjhsh kai upèrjesh qwr epikluyh kai ( 2 ) ol sjhsh kai upèrjesh me
epikluyh fasmtwn.
- Allag kl maka Xs ( ) ! Y ( ) = Xs( =Ts )
- Olisj sei Y ( ) ! Y ( + 2k)
- Upèrjesh twn olisjhmènwn fasmtwn kai diabjmish me 1=Ts
Enìthta 5.4 Deigmatolhpthmèna S mata sto Ped o Suqnot twn 167
()
Diapist¸noume ìti, an to analogikì s ma xa t èqei periorismèno eÔro suqnot twn,
dhlad , Xa ! ( )=0 j j 2
gia ! > !0 = , kai an g nei deigmatolhy a me rujmì deigmato-
lhy a megalÔtero apì to eÔro z¸nh tou s mato !0 , to fsma Xa ! diathre tai ()
anallo wto mèsa sto fsma Xs ( )
twn deigmtwn tou. Ktw apì ti pro pojèsei
autè , anamènoume ìti to analogikì s ma mpore na anakataskeuaste apì ta de gmat
tou. Prgmati,
1 1
X
2 k
Xs (!Ts ) = Xa !+ (5.4.11)
Ts k= 1 Ts
apì thn opo a èqoume gia =Ts < ! =Ts
Xs (!Ts ) =
1 Xa (!) (5.4.12)
Ts
àtsi èqoume gia to analogikì s ma xa (t)
xa (t) =
1 Z 1 Z
Ts =Ts
2 1 Xa (!)e d! = 2 =Ts Xs(!Ts )e d!
j!t j!t (5.4.13)
Lìgw th (5.4.2)
Ts
Z =Ts 1
X
xa (t) = xs (n)e j!Ts n ej!t d!
2 =Ts n= 1
1
X T
Z =Ts
= xa (nTs ) s ej!(t Ts n) d!
n= 1 2 =Ts
=
1
X
xa (nTs )
sin Ts (t nTs)
Ts (t nTs )
n= 1
1
X
t nTs
= xa (nTs )sin
Ts
(5.4.14)
n= 1
Apì th (5.4.14) parathroÔme ìti to anakataskeuasmèno s ma sump ptei me to xa t ()
sti stigmè deigmatolhy a (blèpe Sq ma 5.23). Shmei¸netai ìti to jroisma sump ptei
me to arqikì s ma gia ìle ti endimese qronikè stigmè .
5.4.1 Je¸rhma deigmatolhy a
An xa (t) analogikì s ma periorismènou eÔrou suqnot twn, dhlad ,
!
Xa (!) = 0 gia j!j > 0
2 (5.4.15)
168 Seir - Metasqhmatismì Fourier DiakritoÔ Qrìnou Keflaio 5
kai o rujmì deigmatolhy a ikanopoie to krit rio Nyquist
!0
2 (5.4.16)
Ts
()
tìte to s ma xa t anakataskeuzetai apì ta de gmat tou sÔmfwna me thn (5.4.14).
2
H suqnìthta =Ts onomzetai suqnìthta Nyquist.
xa(t)
0 t
(á)
xs(n)
0 Ts 2Ts 3Ts 4Ts 5Ts 6Ts 7Ts 8Ts 9Ts t
(â)
x(t)
x (Ts( sinc t-Ts
( ( x (2Ts ( sinc t-2Ts
( ( x (3Ts ( sinc t-3Ts
( (
Ts Ts Ts
t
0 Ts 2Ts 3Ts 4Ts 5Ts 6Ts 7Ts 8Ts 9Ts
( ã)
Sq ma 5.23 H anakataskeu tou analogikoÔ s mato apì ta de gmat tou. (a) To analogikì
s ma xa (t), (b) to s ma diakritoÔ qrìnou xs (n) kai (g) h anakataskeu tou analogikoÔ
s mato apì ta de gmat tou sÔmfwna me thn ex swsh (5.4.14) gia !0 = 2=Ts.
5.4.2 Oi Suntelestè Fourier w de gmata se m a per odo tou metasqhmatismoÔ
Fourier
àstw to periodikì s ma x ~(n) me per odo N kai to s ma x(n), pou antistoiqe se m a
~( )
per odo tou x n , dhlad ,
x(n) = 0x~;(n); M nM +N 1
alli¸
, ìpou M pragmatikì akèraio (5.4.17)
Gnwr zoume (5.2.5)
Nak = X k
2
(5.4.18)
n
Enìthta 5.4 Deigmatolhpthmèna S mata sto Ped o Suqnot twn 169
ìpou ak e nai oi suntelestè th seir Fourier diakritoÔ qrìnou tou x n , kai X ~( ) ( )
()
e nai o metasqhmatismì Fourier diakritoÔ qrìnou tou x n . Me th bo jeia tou Pa-
rade gmato 5.4.1 pou akolouje , ja de xoume ìti o X ( )
exarttai apì thn tim tou
(2 )
M , en¸ oi timè tou sti suqnìthte deigmatolhy a k =N den exart¸ntai apì thn
tim tou M .
Pardeigma 5.4.1
An or soume to s ma x1 (n) (Sq ma 5.24b), to opo o antistoiqe se mia per odo th
x(n)
1
-N 0 N 2N n
(a)
x1(n)=ä(n)
1
-N 0 N 2N n
(â)
x2(n)=ä(n-N)
1
-N 0 N 2N n
(ã)
Sq ma 5.24 (a) H periodik epèktash tou monadia ou de gmato kai (b), (g) dÔo mh periodikè
akolouj e , h kje mia apì ti opo e e nai sh me to x(n) sth dirkeia mia periìdou.
periodik epèktash tou monadia ou de gmato periìdou N x(n) (Sq ma 5.24a) w :
x1 (n) = x(n); 0nN 1 x1 (n) = Æ(n)
0; alli¸
, dhlad , (5.4.19)
apì thn ex swsh anlush ) = P1n= 1 x1(n)e j n upolog zoume to metasqh-
X1 (
matismì Fourier diakritoÔ qrìnou tou x1 (n) w
X1 ( ) = 1 (5.4.20)
en¸, an or soume to s ma x2 (n) (Sq ma 5.24g),
x2 (n) = x0;(n); M n M + N 1 ìpou 0 < M < N
alli¸
(5.4.21)
170 Anptugma - Metasqhmatismì Fourier Diakrit¸n Shmtwn Keflaio 5
dhlad , x2 (n) = Æ(n N ), o metasqhmatismì tou x2 (n) e nai
X2 ( )=e j N (5.4.22)
ParathroÔme ìti X1 ( ) 6= X2( ) all gia ti suqnìthte = k(2=N ) e nai
X1 k
2 2
= X2 k = 1 (5.4.23)
N N
5.5 DIAKRITOS METASQHMATISMOS FOURIER
O metasqhmatismì Fourier diakritoÔ qrìnou e nai suneq periodik sunrthsh me
2
per odo . Gia na epexergastoÔme to metasqhmatismì Fourier me yhfiak mèsa a-
paite tai h metatrop tou se akolouj a arijm¸n peperasmènh akr beia . Ja prèpei,
loipìn na g nei katllhlh deigmatolhy a tou metasqhmatismoÔ Fourier ètsi, ¸ste na
e nai dunat h anakataskeu tou apì ta de gmat tou.
()
D netai h peperasmènou m kou N akolouj a x n , dhlad , h x n gia n N. ( )=0
O metasqhmatismì Fourier diakritoÔ qrìnou th akolouj a x n , ìpw e nai gnwstì,()
e nai
X1
N
X( )= x(n)e j n; 0 < 2 (5.5.1)
n=0
En g nei deigmatolhy a th suneqoÔ sunrthsh X ( ) se M diakritè kuklikè
suqnìthte pou e nai pollaplsie th s sto disthma 0 < 2, pa rnoume ta
de gmata:
NX1
XM (k) = X ( )j =k s = x(n)ejk sn; k = 0; 1; : : : ; M 1 (5.5.2)
n=0
O arijmì twn deigmtwn pou ja lhfjoÔn ja prèpei na e nai katllhlo ètsi ¸ste
afenì na e nai dunat h ankthsh tou metasqhmatismoÔ Fourier diakritoÔ qrìnou
gia kje tim th kuklik suqnìthta , afetèrou na mhn auxhjoÔn h apaitoÔmenh
mn mh kai h taqÔthta epexergas a .
To je¸rhma deigmatolhy a sto ped o tou qrìnou anafèrei ìti gia èna analogikì
()
s ma xa t me periorismèno eÔro z¸nh qamhl¸n suqnot twn W , dhlad , X ! ( )=
0 j j
gia ! W e nai dunat h anakataskeu tou arqikoÔ s mato xa t apì thn ()
f (
akolouj a twn deigmtwn tou xa nTs 1 )g
n= 1 , ìtan h per odo deigmatolhy a Ts
plhro to krit rio tou Nyquist, dhlad , Ts 1
2W .
Me anlogo trìpo to je¸rhma deigmatolhy a sto ped o suqnot twn anafèrei
ìti o metasqhmatismì Fourier diakritoÔ qrìnou mpore na anakthje apì ta de gmat
()
tou XM k ; k =01
; ; :::; M 1
efìson to s ma diakritoÔ qrìnou x n e nai ()
Enìthta 5.5 Diakritì Metasqhmatismì Fourier 171
peperasmènh dirkeia N kai isqÔei M N sthn per ptwsh aut isqÔei h sunj kh
Nyquist, dhlad ,
s 2N (5.5.3)
Gia thn oriak per ptwsh ìpou s =2 , dhlad , ìtan g netai deigmatolhy a tou
N
( )
X sti suqnìthte k k s = = k 2N ; k = 0; 1; : : : ; N 1, h (5.5.2) grfetai
XN (k) = X ( )j = X k
2
=k 2N N
NX1
2
= x(n)ejk N n ; k = 0; 1; : : : ; N 1 (5.5.4)
n=0
Ta de gmata XN k () ()
apl X k tou metasqhmatismoÔ Fourier diakritoÔ qrìnou X ( )
apoteloÔn to diakritì metasqhmatismì Fourier (Disctete Fourier Transform, DFT) th
akolouj a x n . ()
ApodeiknÔetaiz ìti mporoÔme na anakataskeusoume thn akolouj a x n apì ta ()
()
de gmata XN k tou metasqhmatismoÔ Fourier diakritoÔ qrìnou me thn
x(n) =
1 NX1 X (k)e jk 2N n ; n = 0; 1; : : : ; N 1 (5.5.5)
N
N k=0
H (5.5.5) apotele ton ant strofo diakritì metasqhmatismì Foureir (inverse DFT, IDFT).
Oi exis¸sei (5.5.4) kai (5.5.5) apoteloÔn tou zeÔgo diakritoÔ metasqhmatismoÔ
FourierN -shme wn kai ja to sumbol zoume w x(n) DFT!N XN (k).
Oi akolouj e x(n) kai XN (k ) èqoun dio m ko N kai e nai periodikè me per o-
do N .
Pardeigma 5.5.1
D netai h 4-shme wn akolouj a x(n)
x(n) = 1; 0 n 3
0; alli¸ (5.5.6)
1. Na breje o metasqhmatismì Fourier diakritoÔ qrìnou X ( ) kai na g nei h grafik
parstash tou mètrou tou se sunrthsh me thn kuklik suqnìthta .
2. Na breje o diakritì metasqhmatismì Fourier 4-shme wn th akolouj a x(n).
z Anafor 5.4 kai 5.5 th proteinìmenh bibliograf a
172 Anptugma - Metasqhmatismì Fourier Diakrit¸n Shmtwn Keflaio 5
LÔsh
1. O metasqhmatismì Fourier diakritoÔ qrìnou th akolouj a x(n) e nai
3
X
X( ) = x(n)e j n
n=0
= 1 + e j + e j2 + e j3
j4
= 11 ee j
sin(2 ) e j3 =2
= sin( =2)
Sto Sq ma 5.25 èqei g nei h grafik parstash tou mètrou tou metasqhmatismoÔ
Fourier diakritoÔ qrìnou jX ( )j se sunrthsh me thn .
2. Gia to diakritì metasqhmatismì Fourier 4-shme wn th akolouj a x(n) èqoume
3
X 2
X4 (k) = x(n)ejk N n ; k = 0; 1; 2; 3:
n=0
Gia k = 0 èqoume
3
X 3
X
x(n)ej0 N n =
2
X4 (0) = x(n) = x(0) + x(1) + x(2) + x(3) = 4
n=0 n=0
kai gia k = 1; 2; kai 3 èqoume
3
X 3
X
x(n)ej1 N n =
2
X4 (1) = x(n)( j )n = 0
n=0 n=0
3
X 3
X
x(n)ej2 N n = x(n)( j )2n = 0
2
X4 (2) =
n=0 n=0
3
X 3
X
x(n)ej2 N n = x(n)( j )3n = 0
3
X4 (3) =
n=0 n=0
O diakritì metasqhmatismì Fourier 4-shme wn th akolouj a x(n) e nai, loipìn
X4 (k) = [ 4; 0; 0; 0℄ (5.5.7)
"
O diakritì metasqhmatismì Fourier 4-shme wn th akolouj a x(n) br sketai
kai en g nei deigmatolhy a tou metasqhmatismoÔ Fourier diakritoÔ qrìnou X ( )
Enìthta 5.5 Diakritì Metasqhmatismì Fourier 173
se 4 isapèqouse kuklikè suqnìthte pou apèqoun an dÔo kat = 2 . àqou-
me, loipìn, ta de gmata
X4 (k ) = X ( )j =k 2
= sin(2 k 2 ) j3k 4
sin(k 4 ) e
= [ 4; 0; 0; 0 ℄ (5.5.8)
"
Sto Sq ma 5.25 èqoume kai grafik parstash tou diakritoÔ metasqhmatismoÔ
Fourier 4-shme wn th akolouj a x(n).
X(Ù)
4
0 ð 2ð Ù
X4(k)
4
Sq ma 5.25 H grafik parstash tou
MF diakritoÔ qrìnou tou s mato x(n)
sto Pardeigma 5.5.1 kai o diakritì MF
0 1 2 3 4 k 4-shme wn tou.
Shmei¸netai ìti to de gma gia mhdenik kuklik suqnìthta XN (0) e nai pntote so
me to jroisma twn stoiqe wn th akolouj a x n . ()
5.5.1 Kuklik anklash akolouj a
()
H anklash mia akolouj a N -shme wn, x n , d nei thn akolouj a x n , h opo a ( )
den e nai akolouj a N -shme wn, kai ètsi den e nai dunatì na upologiste o diakritì
metasqhmatismì Fourier. H kuklik anklash mia akolouj a mpore na parastaje
me th bo jeia twn upolo pwn (modulo) w x (( ))
n N , ìpou o sumbolismì m N (( ))
diabzetai w m modulo N kai shma nei to upìloipo th dia resh tou m dia tou N
kai e nai
x(0);
x(( n))N = n=0
x(N n); 1 n N 1 (5.5.9)
Sto Sq ma 5.26 fa netai h akolouj a 11-shme wn x(n) = n ìpou 0 n 10 kai
0 < < 1 h anklas th h opo a den e nai akolouj a 11-shme wn kai h kuklik
anklas th h opo a e nai akolouj a 11-shme wn.
174 Anptugma - Metasqhmatismì Fourier Diakrit¸n Shmtwn Keflaio 5
x(n)=an 0 ≤ n ≤ 10
0<a<1
-10 -5 0 5 10 15 n
(a)
x(-n)
-10 -5 0 5 10 15 n
(â)
x((-n))11
Sq ma 5.26 KÔklik anklash akolouj -
a (a) h akolouj a N -shme wn x(n), (b) h
-10 -5 0 5 10 15 n anklash th akolouj a kai (g) h kuklik
(ã) anklash th akolouj a .
5.5.2 Kuklik ol sjhsh akolouj a
H periodik epèktash an N de gmata th peperasmènou m kou akolouj a x(n) pou
èqei N de gmata sto disthma ; ; : : : ; N 01 1
e nai h periodik akolouj a
1
X
x~(n) = x(n kN ) (5.5.10)
k= 1
H ol sjhsh (metatìpish) th periodik akolouj a x~(n) kat m de gmata pro ta
dexi d nei thn ep sh periodik akolouj a
1
X
x~(n m) = x(n m + kN ) (5.5.11)
k= 1
H peperasmènou m kou akolouj a
x~(n m)RN (n) = x~(n m); 0nN 1
0; alli¸
(5.5.12)
ìpou RN (n) e nai to orjog¸nio parjuro m N , dhlad , kou
RN (n) = 10;; 0alli¸
nN 1 (5.5.13)
apotele thn kuklik ol sjhsh M -shme wn th akolouj a x(n). ParathroÔme ìti h
kuklik ol sjhsh m shme wn mia akolouj a N shme wn proèrqetai apì thn para-
jÔrwshx th grammik olisjhmènh kat m shme a periodik epèktash th akolou-
x O pollaplasiasmì mia sunrthsh me m a llh, me skopì to mhdenidmì th pr¸th èxw apì èna disthma,
e nai gnwstì w parajÔrwsh
Enìthta 5.5 Diakritì Metasqhmatismì Fourier 175
j a . H periodik epèktash mia akolouj a mpore na parastaje me th bo jeia twn
~
x(n)=an 0 ≤ n ≤ 10 x(n-3)= x((n-3))11
0<a<1
-5 0 5 10 15 n -5 0 5 10 15 20 n
(a) (ã)
~
x(n)= x((n))11 x((n-3))11R11(n)
x(n)
-5 0 5 10 15 n -5 0 5 10 15 20 n
(â) (ä)
Sq ma 5.27 Kuklik ol sjhsh akolouj a (a) arqik akolouj a x(n), (b) periodik epèktash
th x(n) (g) grammik ol sjhsh kat tr a de gmata th periodik epèktash , kai d) kuklik
olisjhmènh akolouj a kat tr a de gmata.
upolo pwn w
x~(n m) = x((n m))N (5.5.14)
opìte h kuklik ol sjhsh ekfrzetai kai w
x~(n m)RN (n) = x((n m))N RN (n) (5.5.15)
Sto Sq ma 5.27 fa netai h akolouj a 11-shme wn x n n (ìpou n( )= kai 0 10
0 1
< < ), h periodik epèktash th akolouj a kat 11 de gmata, h grammik
ol sjhsh th periodik epèktash kat tr a de gmata pro ta dexi kai h kuklik
ol sjhsh kat tr a de gmata pro ta dexi th akolouj a x n , h opo a e nai ep sh ()
akolouj a 11-shme wn.
5.5.3 Kuklik sunèlixh akolouji¸n
H kuklik sunèlixh dÔo akolouji¸n x1 (n) kai x2 (n), n = 0; 1; : : : ; N 1 dhl¸netai
x1 (n) x2 (n) kai or zetai apì th sqèsh
NX1
y(n) = x1 (n) x2 (n) = x1 (m) x2 ((n m))N (5.5.16)
m=0
176 Anptugma - Metasqhmatismì Fourier Diakrit¸n Shmtwn Keflaio 5
()
H akolouj a y n èqei m ko N , ìso, dhlad , kai to m ko kajemi apì ti arqikè
akolouj e , kai ìqi m ko N 2 1
, ìpw sumba nei sthn per ptwsh th grammik
sunèlixh twn dÔo aut¸n akolouji¸n.
Ta b mata gia ton upologismì th kuklik sunèlixh dÔo akolouji¸n e nai:
1. kuklik anklash (katoptrismì ) th mia akolouj a ,
2. kuklik ol sjhsh (metatìpish) th katoptrik akolouj a ,
3. pollaplasiasmì th metatopismènh katoptrik akolouj a me th llh akolou-
j a shme o pro shme o, kai
4. jroish twn ginomènwn.
Ta b mata aut epanalambnontai.
Pardeigma 5.5.2
Na upologiste h kuklik sunèlixh 4-shme wn twn akolouji¸n x1 (n) = [ 3; 2; 1 ℄ kai
x2 (n) = [ 1; 2; 3; 4℄
LÔsh H kuklik sunèlixh 4-shme wn d netai apì th
3
X
x1 (n) 4 x2 (n) = x1 (m)x2 ((n m))4 (5.5.17)
m=0
gia n = 0 èqoume
P3 P3
m=0 x1 (m)x2 ((0 m))4 = m=0 [f3; 2; 1; 0gf1; 4; 3; 2g℄ = P3m=0f3; 8; 3; 0g = 14;
gia n = 1 èqoume
P3 P3
m=0 x1 (m)x2 ((1 m))4 = m=0 [f3; 2; 1; 0gf2; 1; 4; 3g℄ = P3m=0f6; 2; 4; 0g = 12;
gia n = 2 èqoume
P3 P3
m=0 x1 (m)x2 ((2 m))4 = m=0 [f3; 2; 1; 0gf3; 21; 4g℄ = P3m=0f9; 4; 1; 0g = 14;
gia n = 3 èqoume
P3 P3
m=0 x1 (m)x2 ((3 m))4 = m=0 [f3; 2; 1; 0gf4; 3; 2; 1g℄ = P3m=0f12; 6; 2; 0g = 20;
epomènw h kuklik sunèlixh e nai
x1 (n) 4 x2 (n) = [14; 12; 14; 20℄ (5.5.18)
Pardeigma 5.5.3
Me th bo jeia th idiìthta th kuklik sunèlixh na upologiste h kuklik sunèlixh
4-shme wn twn akolouji¸n x1 (n) kai x2 (n) tou Parade gmato 5.5.2
X (k) = 1 x(n)e j 2N kn
PN
LÔsh Me th bo jeia th ex swsh anlush n=0 pros-
dior zetai o diakritì metasqhmatismì Fourier twn shmtwn x1 (n) kai x2 (n)
h p p i
X1 (k) = 6; 2 2e j 4 ; 2; 2 2 ej 4 = [6; 2 j 2; 2; 2 + j 2℄
Enìthta 5.5 Diakritì Metasqhmatismì Fourier 177
h p p i
X2 (k) = 10; 2 2 ej 34 ; 2; 2 2 e j 4 = [10; 2 + j 2; 2; 2 j 2℄
To ginìmeno twn diakrit¸n metasqhmatism¸n Fourier e nai
X1 (k) X2 (k) =60; 8 ej 2 ; 4ej ; 4e j 2 = [60; j 8; 4; j 8℄ (5.5.19)
Me th bo jeia th ex swsh sÔnjesh x(n) =
1 PN 1
N k=0 X (k )e N prosdior zetai to
j 2 kn
s ma x(n), dhlad , h kuklik sunèlixh twn shmtwn x1 (n) kai x2 (n)
x1 (n) 4 x2 (n) = [14; 12; 14; 20℄
h opo a e nai dia me thn kuklik sunèlixh pou upolog sthke sto Pardeigma 5.5.2, sto
opo o melet same to prìblhma sto ped o tou qrìnou.
5.5.4 Idiìthte tou diakritoÔ metasqhmatismoÔ Fourier
Sthn enìthta aut ja parousiasjoÔn oi basikè idiìthte pou èqei o diakritì metasqh-
matismì Fourier.
Merikè idiìthte tou diakritoÔ metasqhmatismoÔ Fourier e nai anloge me ti an-
t stoiqe idiìthte tou metasqhmatismoÔ Fourier diakritoÔ qrìnou. Uprqoun ìmw kai
diaforè , oi opo e ofe lontai sto peperasmèno m ko tìso twn diwn twn akolouji¸n,
ìso kai tou diakritoÔ metasqhmatismoÔ Fourier tou . Ston P naka 5.3 parousizontai
oi idiìthte tou diakritoÔ metasqhmatismoÔ Fourier.
5.5.5 H grammik sunèlixh me th bo jeia tou diakritoÔ metasqhmatismoÔ Fourier
Gnwr zoume ìti, ìtan èna grammikì qronik anallo wto sÔsthma diakritoÔ qrìnou me
() ()
kroustik apìkrish h1 n diegerje apì thn akolouj a x1 n , h èxodì tou y n e nai ()
()
h grammik sunèlixh twn x1 n kai h1 n , dhlad , ()
1
X
y(n) = h1 (n) ? x1 (n) = x1 (k)h1 (k n)
k= 1
An h akolouj a eisìdou e nai akolouj a N1 -shme wn, kai h kroustik apìkrish e nai
akolouj a N2 -shme wn, tìte h èxodo tou sust mato e nai akolouj a N1 N2 - ( + 1)
shme wn.
Apì ti idiìthte tou metasqhmatismoÔ Fourier diakritoÔ qrìnou gnwr zoume ìti
y(n) = h1 (t) ? x1 (n) F! H1 ( ) X1 ( ) (5.5.20)
ParathroÔme ìti, gia na anakataskeusoume thn akolouj a exìdou apì ta de g-
mata tou Y ( )
, dhlad , apì to diakritì metasqhmatismì Fourier Y k , ja prèpei na ()
g nei deigmatolhy a th Y ( )
se toulqiston N1 N2 + 1
shme a sth diakrit kuklik
178 Anptugma - Metasqhmatismì Fourier Diakrit¸n Shmtwn Keflaio 5
PINAKAS 5.6 Idiìthte tou diakritoÔ metasqhmatismoÔ Fourier
Idiìthta Ped o qrìnou Ped o suqnìthta
x1 (n) X1 (k)
x2 (n) X2 (k)
Grammikìthta ax1 (n)+bx2 (n) aX1 (k)+bX2 (k)
Anklash sto qrìno x(( n))N X (( k))N
Suzug akolouj a x? (n) X ? (( k))
<e[X (k)℄=<e[X (( k))N ℄
Summetrikè idiìthte x(n)=x? (n) =m[X (k)℄= =m[X ((N k))N ℄
gia pragmatikè akolouj e jX (k)j=jX (( k))N j
arg X (k)= arg X (( k))N
2
KÔklik ol sjhsh sto qrìno x((n n0 ))N e jk N n0 X ( k)
2
KÔklik ol sjhsh sth suqnìthta ejk0 N n x(n) X ((k k0 ))N
KÔklik sunèlixh x1 (n) x2 (n) X1 (k)X2 (k)
Pollaplasiasmì x1 (n)x2 (n) 1 X1 (k) N X2 (k)
N
Je¸rhma tou Parseval Ex =PNn=01 jx(n)j2 Ex = N1 PNk=01 jX (k)j2
suqnìthta. àpomènw , o diakritì metasqhmatismì Fourier ja prèpei na èqei m ko
N N1 N2 . + 1
() ()
Oi akolouj e x1 n kai h1 n èqoun m ko mikrìtero tou N prèpei, loipìn, na
prostejoÔn stoiqe a mhdenik tim se kje m a apì autè ètsi, ¸ste to m ko tou na
() ()
g nei so me N sqhmat zonta ti akolouj e x n kai h n . H prìsjesh mhdenik¸n se
kje m a apì ti akolouj e autè den ephrezei to metasqhmatismì Fourier diakritoÔ
qrìnou, all èqei w apotèlesma na auxhjoÔn ta shme a deigmatolhy a pèra apì to
elqisto m ko N1 kai N2 ant stoiqa, pou e nai o elqisto arijmì .
Pollaplasizonta tou diakritoÔ metasqhmatismoÔ Fourier N -shme wn twn
() ()
akolouji¸n x1 n kai h1 n br sketai o ep sh N shme wn diakritì metasqhma-
()
tismì Fourier Y k . Sth sunèqeia me ant strofo diakritì metasqhmatismì Fourier
N -shme wn br sketai h akolouj a exìdou tou sust mato .
H kuklik sunèlixh twn akolouji¸n x(n) kai h(n) e nai isodÔnamh me th grammik
sunèlixh twn akolouji¸n x1 (n) kai h1 (n). Me lla lìgia o diakritì
metasqhmatismì
Fourier mpore na qrhsimopoihje gia ton upologismì th grammiksunèlixh , an oi
akolouj e èqoun katllhla epimhkunje me thn prìsjesh mhdenik¸n stoiqe wn sth
kje m a apì autè .
Enìthta 5.5 Diakritì Metasqhmatismì Fourier 179
Pardeigma 5.5.4
H kroustik apìkrish enì GQA sust mato e nai h1 (n) = [ 3; 2; 1 ℄: Me th bo jeia
tou diakritoÔ metasqhmatismoÔ Fourier na upolog sete thn èxodo tou sust mato , ìtan
h e sodo x1 (n) = [ 1; 2; 3; 4 ℄.
e nai to s ma
LÔsh H èxodo tou sust mato y (n) e nai akolouj a N1 + N2 1 = 3 + 4 1 = 6-
shme wn. Sti akolouj e h1 (n) kai x1 (n) ja prèpei na prostejoÔn tr a kai dÔo mh-
denik ant stoiqa ¸ste na g noun akolouj e 6-shme wn. Oi diakrito metasqhmatismo
Fourier twn akolouji¸n h(n) = [ 3; 2; 1; 0; 0; 0 ℄ kai x(n) = [ 1; 2; 3; 4; 0; 0 ℄ e nai
5
X
H (k ) = 2
h(n) ejk 6 n = h(0) + h(1) ejk 6
2
+ h(2) ejk 26 2
n=0
kai
5
X
X (k ) = 2
x(n) ejk 6 n = x(0) + x(1) ejk 6
2
+ x(2) ejk 26 2 + x(3) ejk 26 3
n=0
ant stoiqa. O diakritì metasqhmatismì Fourier th akolouj a exìdou br sketai me
pollaplasismì twn H (k) kai X (n)
Y (k )
= h(0) x(0)
+ [h(1)x(0) + h(0)x(1)℄ ejk 26 1
+ [h(2)x(0) + h(1)x(1) + h(0)x(2)℄ ejk 26 2
+ [h(2)x(1) + h(1)x(2) + h(0)x(3)℄ ejk 36 3
+ [h(2)x(2) + h(1)x(3)℄ ejk 26 4
+ [h(2)x(3)℄ ejk 26 5
Y (k) = 3 + 8 ejk 6 1 + 14 ejk 6 2 + 20 ejk 6 3 + 11 ejk 6 4 + 4 ejk 6 5
2 2 3 2 2
(5.5.21)
h akolouj a exìdou tou sust mato br sketai me ant strofo diakritì metasqhmatismì
Fourier 6-shme wn kai e nai
y(n) = [ 3; 8; 14; 20; 11; 4 ℄ (5.5.22)
Sto Prìblhma 2.12 h èxodo tou sust mato brèjhke me th bo jeia tou ajro sma-
to th sunèlixh . An prosdior soume thn èxodo tou sust mato qrhsimopoi¸nta
kuklik sunèlixh 5-shme wn, tìte prosdior zetai h akolouj a ; ; ; ; , èn- [7 8 14 20 11℄
w, an prosdior soume thn èxodo tou sust mato qrhsimopoi¸nta kuklik sunèlixh
4-shme wn tìte prosdior zetai h akolouj a [14 12 14 20℄
; ; ; , h opo a e nai sh me thn
kuklik sunèlixh pou upolog sthke sta Pardeigma 5.5.2 kai 5.5.3. ParathroÔme
ìti h akolouj a [14 12 14 20℄
; ; ; èqei proèljei apì thn y n ( ) = [3 8 14 20 11 4℄
; ; ; ; ;
me anad plwsh twn stoiqe wn 11
kai prgmati, 4; ; ; [14 12 14 20℄ = [3 + 11 8 +
;
4 14 20℄
; ; .
180 Anptugma - Metasqhmatismì Fourier Diakrit¸n Shmtwn Keflaio 5
5.5.6 O diakritì metasqhmatismì Fourier se morf pinkwn
An efarmìsoume thn ex swsh anlush tou diakritoÔ metasqhmatismoÔ Fourier
NX1
XN (k ) = x(n)e jk 2N n ; k = 0; 1; : : : ; N 1 (5.5.23)
n=0
gia k = 0; 1; 2; : : : ; N 1, èqoume ti exis¸sei :
XN (0) =e0 x(0) +e0 x(1) +e0 x(2) : : : +e0 x(N 1)
XN (1) =e0 x(0) +e j 2N x(1) +e j 2N 2 x(2) : : : +e j 2N (N 1) x(N 1)
XN (2) =e0 x(0) +e j2 2N x(1) +e j2 2N 2 x(2) : : : +e j2 2N (N 1) x(N 1)
.. .. .. .. .. ..
. . . . . .
XN (N ) =e0 x(0) +e j (N 1) 2N x(1) +e j (N 1) 2N 2 x(2) : : : +e j (N 1) 2N (N 1) x(N 1)
An x e nai to dinusma twn N stoiqe wn th akolouj a x(n), dhlad ,
x = [x(0); x(1); x(2); : : : ; x(N 1)℄T (5.5.24)
ìpou o ekjèth T dhl¸nei ton anstrofo p naka, X e nai to dinusma twn N sunte-
()
lest¸n X k dhlad ,
X = [XN (0); XN (1); XN (2); : : : ; XN (N 1)℄T (5.5.25)
kai W o N N DFT p naka
2
1 1 1 ::: 1 3
6 1 WN WN2 ::: WNN 1 7
W=6 6
. .. .. .. ..
7
7 (5.5.26)
4 .. . . . . 5
1 WNN 1 WN2(N 1) : : : WN(N 1)(N 1)
2
ìpou WN =
e j N e nai h Nost r za th monda . Me th bo jeia twn orism¸n aut¸n
oi exis¸sei pou d noun to diakritì metasqhmatismì Fourier ekfrzontai se morf
pinkwn w
X W x = (5.5.27)
En o ant strofo tou W uprqei kai e nai o W 1 , èqoume apì thn (5.5.27)
x=W 1X (5.5.28)
Oi sqèsei (5.5.5) ekfrzetai se morf pinkwn w
x=
1 W? X (5.5.29)
N
Enìthta 5.5 Diakritì Metasqhmatismì Fourier 181
ìpou W? e nai o suzug migadikì tou p naka W. SÔgkrish twn dÔo teleuta wn
exis¸sewn odhge sto sumpèrasma
W 1=
1 W? (5.5.30)
N
apì thn opo a èqoume
W W? = N I (5.5.31)
ìpou I e nai o monadia o p naka diastsewn N N . ParathroÔme ìti o p naka W
e nai summetrikì kai orjog¸nio p naka .
5.5.7 TaqÔ metasqhmatismì Fourier
O diakritì metasqhmatismì Fourier mia akolouj a N -shme wn x(n); n = 0; 1; : : : ;
N 1
, or zetai w h akolouj a N ìrwn
X1
N
X (k) = x(n)WNnk ; k = 0; 1; 2; : : : ; N 1 kai WN = e j 2N (5.5.32)
n=0
Gia na upologisje kje ìro th akolouj a tou diakritoÔ metasqhmatismoÔ Foureir,
apaitoÔntai N pollaplasiasmo kai N 1
prosjèsei . Gia na upologiste , epomè-
nw , olìklhrh h akolouj a X k , qreizontai N 2 pollaplasiasmo kai N N
() ( 1)
prosjèsei . Gia pardeigma, o upologismì tou diakritoÔ metasqhmatismoÔ Fourier
mia akolouj a me m ko N apaite N 2
= 512 = 262144
pollaplasiasmoÔ kai
NN ( 1) = 261632
prosjèsei . O arijmì twn prxewn auxnetai lìgw tou gegonìto
ìti uprqoun kai prxei metaxÔ migadik¸n arijm¸n.
O p naka W, o opo o qrhsimopoie tai kat ton upologismì tou diakritoÔ metasqh-
matismoÔ Fourier, e nai summetrikì . Axiopoi¸nta th summetr a kai thn periodikìthta
twn tim¸n tou p naka katal goume se mejìdou upologismoÔ tou diakritoÔ metasqh-
matismoÔ Foureir me arket ligìtere prxei .
àqoun anaptuqje èna pl jo apì diaforetikoÔ algìrijmou pou epitugqnoun
to skopì autì. Oi diaforè tou br skontai sto pl jo kai to e do twn prxewn
kaj¸ kai sto mègejo th apaitoÔmenh mn mh . Ja anafèroume ton algìrijmo twn
Cooley-Tukey, o opo o protjhke to 1965. O algìrijmo autì mpore na efarmoste
se akolouj e N =2
n -shme wn. Me to pardeigma pou akolouje ja parousiaste
h dunatìthta periorismoÔ twn apaitoÔmenwn prxewn lìgw twn idiot twn th sum-
metr a kai th periodikìthta pou parousizei o p naka W.
Pardeigma 5.5.5
Na breje o diakritì metasqhmatismì Fourier 4-shme wn th akolouj a
[ x(0); x(1); x(2); x(3) ℄
182 Anptugma - Metasqhmatismì Fourier Diakrit¸n Shmtwn Keflaio 5
LÔsh An efarmìsoume thn ex swsh anlush tou diakritoÔ metasqhmatismoÔ Fourier
NX1
2
XN (k) = x(n)W4nk ; 0 k 3; W4 = ej 4 = j (5.5.33)
n=0
gia k = 0; 1; 2; 3 kai ekfrsoume ti exis¸sei se morf pinkwn, èqoume
2 3 2 3 2 3
X4 (0) W40 W40 W40 W40 x(0)
6 7 6 W0
X4 (1) W41 W42 W43 7 6 x(1) 7
6 7=6 4 76 7
4 5 4 W40
X4 (2) W42 W44 W46 5 4 x(2) 5 (5.5.34)
X4 (3) W40 W43 W46 W49 x(3)
Epeid W40 = W44 = 1, W41 = W49 = j , W42 = W46 = 1 kai W43 = j , èqoume
2 3 2
X4 (0) 1 1 1 1 3 2 x(0)
3
6 X4 (1) 7 6 1
7=6 j 1 j 7 6 x(1) 7
1 1 1 54
6 7 6 7
4 X4 (2) 5 4 1 x(2) 5 (5.5.35)
X4 (3) 1 j 1 j x(3)
Ekmetalleuìmenoi th summetr a èqoume
X4 (0) = x(0) + x(1) + x(2) + x(3) = [x| (0) {z+ x(2)}℄ + [|x(1) {z+ x(3)℄}
g1 g2
X4 (1) = x(0) jx(1) x(2) + jx(3) = [|x(0) {z x(2)℄} j [|x(1) {z x(3)℄}
h1 h2
X4 (2) = x(0) x(1) + x(2) x(3) = [|x(0) {z
+ x(2)℄} [|x(1) {z+ x(3)℄}
g1 g2
X4 (3) = x(0) + jx(1) x(2) jx(3) = [|x(0) {z x(2)℄} +j [|x(1) {z x(3)℄}
h1 h2
Oi sqèsei autè odhgoÔn se ènan apotelesmatikì algìrijmo pou èqei dÔo b mata
B ma I B ma II
g1 = x(0) + x(2) XN (0) = g1 + g2
g2 = x(1) + x(3) XN (1) = h1 jh2 (5.5.36)
h1 = x(0) x(2) XN (2) = g1 g2
h2 = x(1) x(3) XN (1) = h1 + jh2
O algìrijmo autì qreizetai mìno dÔo migadikoÔ pollaplasiasmoÔ . Sto Sq ma 5.28
d netai to digramma ro tou algìrijmou.
O algìrijmo (5.5.36) mpore na ulopoihje me diaforetikì trìpo. Arqik h akolouj a
4-shme wn x(n) diaire tai se dÔo akolouj e 2-shme wn, oi opo e dieujetoÔntai se dÔo
dianÔsmata st lh w
x(0) ; x(1) = x(0) x(2)
x(2) x(3) x(1) x(3)
Enìthta 5.5 Diakritì Metasqhmatismì Fourier 183
x(0) X(0)
g1
x(1) X(1)
-1 h1 -j
x(2) -1 X(2)
g2
x(3) X(3)
-1 h2 j
Sq ma 5.28 Digramma ro sto Pardeigma 5.5.5.
Sth sunèqeia br sketai o mikrìtero 2-shme wn diakritì metasqhmatismì Fourier
W2 x(0) x(1)
x(2) x(3)
1
= 1 1 1) x(0) x(1)
x(2) x(3)
= xx(0) + x(2)
(0) x(2)
x(1) + x(3)
x(1) x(3)
= g1 g2
h1 h2
Katìpin, kje stoiqe o tou p naka pou prokÔptei pollaplasizetai me fW4pq g, ìpou p
e nai o de kth gramm kai q e nai o de kth st lh . Dhlad , ektele tai to eswterikì
ginìmeno
1 1 ? g1 g2
=
g1 g2
1 j h1 h2 h1 jh2
g1
h1
g2
jh2 W2 = g1 g2
h1 jh2 1 j
1 1
= g1 + g2 g1 g2
h1 jh2 h1 + jh2
= X4 (0) X4 (2)
X4 (1) X4 (3) (5.5.37)
An kai o diaforetikì autì trìpo ulopo hsh apaite perissìterou pollaplasias-
moÔ apì ton apotelesmatikì algìrijmo (5.5.36), upodeiknÔei m a susthmatik prosèg-
gish prosdiorismoÔ enì diakritoÔ metasqhmatismoÔ Foureir meglh txh me th bo -
jeia diakrit¸n metasqhmatism¸n Foureir mikrìterh txh .
Sth sunèqeia ja genikeÔsoume ta sumpersmata tou parade gmato dhlad , ja
de xoume ìti, gia rtio N , o diakritì metasqhmatismì Fourier m kou N upolog zetai
me katllhlo sunduasmì dÔo akolouji¸n diakrit¸n metasqhmatism¸n Foureir m kou
2 2
N= . An kai o N= e nai rtio , tìte to dio mpore na g nei gia kje m a apì ti
dÔo akolouj e dhlad , na upologisjoÔn me katllhlo sunduasmì dÔo diakrit¸n
4
metasqhmatism¸n Foureir m kou N= . An o N e nai dÔnamh tou 2 (N p ), h =2
184 Anptugma - Metasqhmatismì Fourier Diakrit¸n Shmtwn Keflaio 5
diadikas a aut suneq zetai mèqri na ftsoume se diakritì metasqhmatismì Foureir
2-shme wn, poÔ e nai eÔkolo na upologiste .
()
Arqik, h akolouj a twn N ìrwn x n qwr zetai se dÔo akolouj e m kou N= 2
h kje m a, ti g1 n ( ) = (2 )
x n kai g2 n x n ( ) = (2 + 1)
, gia n ; ; : : : ; N2 , oi =0 1 1
opo e apoteloÔntai apì tou ìrou me rtiou kai perittoÔ de kte ant stoiqa. H
(5.5.32) grfetai
X1
N
X (k) = x(n)WNnk
n=0
2 1 2 1
N N
X X
= x(2n)WN2nk + x(2n + 1)WN(2n+1)k
n=0 n=0
2 1 2 1
N N
X X
= x(2n)WN2nk + WNk x(2n + 1)WN2nk (5.5.38)
n=0 n=0
j 2N 2nk 2 nk
àpeid WN2nk = e =e j N=2 = WN=
nk h (5.5.38) grfetai
2
2 1 2 1
N N
X X
X (k ) = x(2n) nk
WN=2 +WNk x(2n + 1)WN=
nk
2
n=0 n=0
| {z } | {z }
G1 (k) G2 (k)
X (k) = G1 (k) + WNk G2 (k); k = 0; 1; 2; : : : ; N 1 (5.5.39)
Epiplèon, epeid
WNk+ 2 = e j 2N (k+ N2 ) = e N
N
j e j 2N k = WNk ; k = 0; 1; : : : ;
2
h (5.5.39) an k = k + N
2 d nei
N N
X (k) = X k + = G1 (k) + WNk G2 (k); k = 0; 1; 2; : : : ;
2 2 1 (5.5.40)
ParathroÔme ìti o upologismì tou X (k ) èqei ekfraste me th bo jeia dÔo diakrit¸n
metasqhmatism¸n Fourier me pl jo shme wn N=2 o kajèna .
H diadikas a anlush pou akolouj jhke prohgoumènw mpore na suneqiste
kai gia ton upologismì twn dÔo nèwn diakrit¸n metasqhmatism¸n Fourier G1 (k ) kai
G2 (k). H diadikas a aut suneq zetai mèqri na ftsoume se diakritì metasqhmatismì
Fourier 2-shme wn pou e nai eÔkolo na upologiste .
Enìthta 5.5 Diakritì Metasqhmatismì Fourier 185
x(0) X(0)
0
WN
0
WN
0
WN
x(4) 4 X(1)
WN WN
2
WN
1
x(2) 4 X(2)
0 WN WN
2
W N
x(8) 4 6 X(3)
WN WN WN
3
x(1) 4 X(4)
0 0 WN
WN WN
x(5) 4 5 X(5)
WN 2 WN
WN
x(3) 4 6 X(6)
0 WN WN
WN
x(7) 4 6 7 X(7)
WN WN WN
Sq ma 5.29 Digramma ro diakritoÔ metasqhmatismoÔ Fourier okt¸ shme wn.
Sto Sq ma 5.29 fa netai to digramma ro tou diakritoÔ metasqhmatismoÔ Fourier
okt¸ shme wn.
H ditaxh twn deigmtwn tou diakritoÔ metasqhmatismoÔ Fourier sthn èxodo e -
nai kanonik dhlad , X (0) (1) (7)
; X ; : : : ; X . Ant jeta, h ditaxh twn deigmtwn
(0) (4) (2) (6) (1) (5) (3) (7)
eisìdou e nai mh kanonik : x ; x ; x ; x ; x ; x ; x ; x . H ditaxh
aut prokÔptei apì thn kanonik ditaxh twn deigmtwn me antistrof th seir
twn duadik¸n yhf wn sth duadik anaparstash twn deikt¸n bit reversal.
x(0); x(1); x(2); x(3); x(4); x(5); x(6); x(7); !
x(000); x(001); x(010); x(011); x(100); x(101); x(110); x(111); !
x(000); x(100); x(010); x(110); x(001); x(101); x(011); x(111); !
x(0); x(4); x(2); x(6); x(1); x(5); x(3); x(7):
Apì to sq ma parathroÔme ìti se kje stdio oi èxodoi mporoÔn na apojhkeÔontai
sti die jèsei mn mh , sti opo e tan apojhkeumène oi ant stoiqe e sodoi tou
stad ou.
Shmei¸netai ìti o taqÔ metasqhmatismì Fourier den apotele nèo metasqhma-
tismì Fourier, all apotele m a apodotik algorijmik mèjodo, me thn ènnoia ìti
elatt¸nei thn upologistik poluplokìthta, dhlad , to sunolikì pl jo prxewn
(pollaplasiasm¸n kai prosjèsewn). Prgmati, h upologistik poluplokìthta tou
taqèo metasqhmatismoÔ Fourier e nai th txew N 2 N , kai ìqi N 2 tou diakritoÔ
log
metasqhmatismoÔ Fourier.
186 Anptugma - Metasqhmatismì Fourier Diakrit¸n Shmtwn Keflaio 5
5.6 EFARMOGES TOU METASQHMATISMOU FOURIER DIAKRI-
TOU QRONOU
äpw kai sthn per ptwsh twn susthmtwn suneqoÔ qrìnou, me th bo jeia th idiìth-
ta th sunèlixh mporoÔme na upolog soume thn èxodo, y n enì GQA sust mato ()
()
diakritoÔ qrìnou to opo o èqei kroustik apìkrish h n , ìtan gnwr zoume thn e sodì
()
tou x n .
Pardeigma 5.6.1
D netai to grammikì qronik anallo wto sÔsthma to opo o èqei kroustik apìkrish
h(n) = Æ(n n0 ) (5.6.1)
Na upologiste h sqèsh metaxÔ th akolouj a eisìdou-exìdou tou sust mato .
LÔsh H apìkrish suqnìthta tou sust mato e nai
1
X
H( )= Æ(n n0 )e j n = e j n0 (5.6.2)
n= 1
An h e sodo tou sust mato e nai to s ma x(n), to opo o èqei MF diakritoÔ qrìnou
X( ), o MF diakritoÔ qrìnou th exìdou ja e nai
Y ( ) = H ( ) X ( ) = e j n0 X ( ) (5.6.3)
Me th bo jeia th idiìthta th qronik metatìpish parathroÔme ìti h èxodo tou
sust mato e nai sh me thn e sodo tou sust mato metatopismènh qronik kat n0 ,
dhlad ,
y(n) = x(n n0 ) (5.6.4)
Pardeigma 5.6.2
D netai to grammikì qronik anallo wto sÔsthma me kroustik apìkrish
h(n) = nu(n) (5.6.5)
An h e sodo tou sust mato e nai to s ma:
x(n) = n u(n) (5.6.6)
na prosdioriste h èxodo tou sust mato .
LÔsh Oi MF diakritoÔ qrìnou th kroustik apìkrish kai tou s mato eisìdou
tou sust mato e nai
H( ) = 1 1e j kai X( ) = 1 1e j (5.6.7)
Me th bo jeia th idiìthta th sunèlixh prosdior zetai o MF diakritoÔ qrìnou th
exìdou tou sust mato
Y( ) = H ( ) X ( ) = (1 1
e j ) (1 e j ) (5.6.8)
Enìthta 5.6 Efarmogè tou DiakritoÔ MetasqhmatismoÔ Fourier 187
An 6= , h anlush th Y( ) se apl klsmata d nei
C1 C2
Y( )= 1 +1 (5.6.9)
e j e j
Oi timè twn stajer¸n C1 kai C2 e nai
C1 = kai C2 = (5.6.10)
Me ant strofo MF diakritoÔ qrìnou prosdior zetai h èxodo tou sust mato
y(n) = nu (n) nu (n)
= 1 n+1 n+1 u(n) (5.6.11)
An = , o MF diakritoÔ qrìnou th exìdou e nai
2
Y( ) = 1 1e j (5.6.12)
Me th bo jeia tou zeÔgou 14 MF diakritoÔ qrìnou ston P naka 5.2 prosdior zetai to
s ma exìdou tou sust mato
y(n) = (n + 1) n u(n + 1) (5.6.13)
H èxodo tou sust mato mpore na grafe kai w
y(n) = (n + 1) n u(n) (5.6.14)
kaj¸ oi dÔo ekfrsei d noun thn dia akolouj a :::; y( 1) = 0; y(0) = 1; y(1) =
2 ; y(2) = 3 2 ; :::.
5.6.1 H apìkrish suqnìthta gia sust mata ta opo a qarakthr zontai
apì grammikè exis¸sei diafor¸n me stajeroÔ suntelestè
M a meglh kathgor a apì grammik qronik anallo wta (GQA) sust mata diakritoÔ
qrìnou e nai aut sta opo a h e sodo kai h èxodo ikanopoioÔn m a grammik ex swsh
diafor¸n me stajeroÔ suntelestè th morf
N
X M
X
ak y(n k) = bk x(n k) (5.6.15)
k=0 k=0
Efarmìzoume MF diakritoÔ qrìnou kai sta dÔo mèlh th ex swsh
" N # "M #
X X
F ak y(n k) =F bk x(n k) (5.6.16)
k=0 k=0
188 Anptugma - Metasqhmatismì Fourier Diakrit¸n Shmtwn Keflaio 5
Lìgw th idiìthta th grammikìthta pou èqei o MF diakritoÔ qrìnou èqoume
N
X M
X
ak F [y(n k)℄ = bk F [x(n k)℄ (5.6.17)
k=0 k=0
kai, lìgw th idiìthta th qronik metatìpish pou èqei o MF diakritoÔ qrìnou,
èqoume thn ex swsh
N
X M
X
Y( ) ak e jk = X( ) bk e jk (5.6.18)
k=0 k=0
Qrhsimopoi¸nta thn idiìthta th sunèlixh èqoume
PM
Y( )=P k=0 bk e
jk
H( )= Y( ) N (5.6.19)
jk
k=0 ak e
ParathroÔme ìti h apìkrish suqnìthta enì GQA sust mato e nai rht sunrthsh
dhlad , mpore na ekfraste w lìgo dÔo poluwnÔmwn th metablht e j .
Pardeigma 5.6.3 (SÔsthma pr¸th txh ).
D netai to GQA sÔsthma diakritoÔ qrìnou, to opo o arqik br sketai se hrem a, kai
qarakthr zetai apì thn ex swsh diafor¸n
y(n) ay(n 1) = x(n) me jaj < 1 (5.6.20)
Na brejoÔn h apìkrish suqnìthta , h kroustik apìkrish tou sust mato kai h apì-
krish tou sust mato sto monadia o b ma.
LÔsh Efarmìzonta MF diakritoÔ qrìnou kai sta dÔo mèlh th ex swsh èqoume,
lìgw twn idiot twn th grammikìthta kai th qronik metatìpish ,
F [y(n) ay(n 1)℄ = F [x(n)℄
F [y(n)℄ aF [y(n 1)℄ = F [x(n)℄
Y( ) ae j Y ( ) = X( )
H( ) =
1
1 ae j (5.6.21)
H kroustik apìkrish tou sust mato e nai
h(n) = an u(n) (5.6.22)
Sto Sq ma 5.30 èqei sqediaste h kroustik apìkrish tou sust mato pr¸th txh gia
difore timè th stajer a. ParathroÔme ìti h(n) sugkl nei sthn telik th tim
me rujmì o opo o exarttai apì to rujmì me ton opo o h jajn sugkl nei sto mhdèn. H
Enìthta 5.6 Efarmogè tou DiakritoÔ MetasqhmatismoÔ Fourier 189
h(n) h(n)
a 1 a 1
4 4
0 n 0 n
h(n) h(n)
1 1
a 2 a 2
0 n 0 n
h(n) h(n)
a 3 a 3
4 4
0 n 0 n
h(n) h(n)
a 7 a 7
8 8
0 n 0 n
Sq ma 5.30 H kroustik apìkrish tou sust mato pr¸th txh , gia difore timè th
stajer a.
stajer a èqei parìmoio rìlo me th stajer qrìnou tou sust mato pr¸th txh
suneqoÔ qrìnou.
Sto Sq ma 5.31 èqei sqediaste h apìkrish pltou tou sust mato pr¸th txh gia
difore timè th stajer a. ätan h stajer a > 0, to sÔsthma prokale exasjènhsh
sti uyhlè suqnìthte
dhlad , to jH ( )j èqei mikrè timè , ìtan to lambnei timè
sthn perioq tou se sÔgkrish me th timè pou èqei, ìtan to lambnei timè sthn
perioq tou 0. To ant jeto sumba nei, ìtan a < 0.
ParathroÔme ep sh ìti gia mikrè timè th stajer jaj h mègisth tim tou mètrou
jH ( )j pou e nai 1=(1 + a) kai h elqisth tim tou 1=(1 a) èqoun mikr diafor gia
a < 0, me apotèlesma to mètro jH ( )j na e nai sqetik stajerì, se ant jesh me thn
per ptwsh sthn opo a to jaj èqei megle timè .
190 Anptugma - Metasqhmatismì Fourier Diakrit¸n Shmtwn Keflaio 5
20 log 10 H(Ù)
a 7
8
3 20
a 4
a 1
2 10
a 1
4
-ð ð
-2ð 0 2ð Ù
20 log 10 H(Ù)
a 7
3 20 8
a 4
a 1
a 1 10 2
4
-2ð 0 2ð
-ð ð Ù
Sq ma 5.31 H apìkrish pltou tou sust mato pr¸th txh , gia difore timè th sta-
jer a.
H apìkrish tou sust mato pr¸th txh sto monadia o b ma e nai
1 an+1
y(n) = h(n) ? u(n) = u(n)
1 a
(5.6.23)
Pardeigma 5.6.4
D netai to GQA sÔsthma diakritoÔ qrìnou, to opo o arqik br sketai se hrem a, kai
qarakthr zetai apì thn ex swsh diafor¸n
y(n)
3 y(n 1) + 1 y(n 2) = 2x(n)
4 8 (5.6.24)
Na brejoÔn h apìkrish suqnìthta kai h kroustik apìkrish tou sust mato .
An h e sodo tou sust mato e nai
n
x(n) =
1 u(n)
4 (5.6.25)
na breje to s ma exìdou tou sust mato .
LÔsh H apìkrish suqnìthta tou sust mato e nai
H( )= 1 2
3 j + 18 e j2 (5.6.26)
4e
Enìthta 5.6 Efarmogè tou DiakritoÔ MetasqhmatismoÔ Fourier 191
AnalÔoume ton paranomast se ginìmeno poluwnÔmwn pr¸tou deutèrou bajmoÔ w
pro e j ètsi èqoume
H( )= 1 2
1 j 1 1e j (5.6.27)
2e 4
h anlush se apl klsmata d nei
H( ) = 1 14e j 1 1
2
j (5.6.28)
2 4e
Me th bo jeia antistrìfou MF diakritoÔ qrìnou prosdior zetai h kroustik apìkrish
tou sust mato
n n
h(n) = 4
1 u(n) 2 14 u(n)
2 (5.6.29)
ma x(n) =
1 n u(n), an qrhsimopoihje to
ätan h e sodo tou sust mato e nai to s 4
zeugri MF diakritoÔ qrìnou
n
x(n) =
1 u(n) F! X ( ) = 1 11e
4 4 j
o metasqhmatismì Fourier diakritoÔ qrìnou th exìdou tou sust mato ja e nai
) = H ( ) X ( ) = 1 1 e j 2 1 1 e j 1 11e
Y( j
2 4 4
H anlush se apl klsmata tou Y ( ) èqei th morf
Y( )=
4 2 + 8
1 14 e j 1 4 e j 2 1 12 e j
1 (5.6.30)
Me th bo jeia antistrìfou MF diakritoÔ qrìnou an qrhsimopoihjoÔn ta zeugria MF
diakritoÔ qrìnou 8 kai 14, prosdior zetai h èxodo tou sust mato
n n n
y(n) = 4 41 u(n) 2(n + 1) 14 u(n) + 8 12 u(n) (5.6.31)
Pardeigma 5.6.5
D netai sÔsthma diakritoÔ qrìnou tou opo ou h sqèsh metaxÔ twn shmtwn eisìdou
exìdou perigrfetai apì thn ex swsh diafor¸n
y(n) =
1 (x(n) + x(n 1))
2 (5.6.32)
Na breje h kroustik apìkrish, h apìkrish suqnìthta tou sust mato kai na g nei
h grafik parstash th apìkrish pltou .
192 Anptugma - Metasqhmatismì Fourier Diakrit¸n Shmtwn Keflaio 5
LÔsh H kroustik apìkrish tou sust mato e nai
1
h(n) = SfÆ(n)g = Æ(n) + Æ(n
1 1)
2 2 (5.6.33)
H apìkrish suqnìthta tou sust mato e nai o MF diakritoÔ qrìnou th kroustik
apìkrish
ètsi h (5.6.33) d nei
H( ) = 12 + 12 e j (5.6.34)
an qrhsimopoi je to zeugri 11 MF diakritoÔ qrìnou kai h idiìthta th qronik
metatìpish . H kroustik apìkrish tou sut mato mpore na grafe w
H( )=e j2 ej 2 +e j2
=e j2 os 2
2 (5.6.35)
apì thn opo a parathroÔme ìti h apìkrish pltou e nai
jH ( )j = os 2 (5.6.36)
kai h apìkrish fsh e nai
argfH ( )g = 2 (5.6.37)
Sto Sq ma 5.32 èqei g nei h grafik parstash th apìkrish pltou se sunrthsh
me th . ParathroÔme ìti to sÔsthma e nai èna qamhloperatì f ltro.
H(Ù)
1
Sq ma 5.32 H grafik parstash th
apìkrish pltou se sunrthsh me th
-ð 0 ð Ù tou sust mato sto Pardeigma 5.6.5.
Pardeigma 5.6.6
H sqèsh metaxÔ twn shmtwn eisìdou x(t) kai exìdou y(t) se èna thlepikoinwniakì
kanli sto opo o parousizetai to fainìmeno twn pollapl¸n diadrom¸n perigrfetai
apì thn ex swsh
NX1
y(t) = ak x(t tk ) (5.6.38)
k=0
ìpou ak e nai o pargonta exasjènhsh o sqetikì me thn k-sth diadrom didosh ,
kai tk e nai h ant stoiqh qronik kajustèrhsh didosh .
Na breje h kroustik apìkrish kai h apìkrish pltou thlepikoinwniakoÔ kanalioÔ
sto opo o parousizontai dÔo diadìsei , dhlad , perigrfetai apì thn ex swsh di-
afor¸n,
y(n) = x(n) + ax(n 1) (5.6.39)
Enìthta 5.6 Efarmogè tou DiakritoÔ MetasqhmatismoÔ Fourier 193
LÔsh H kroustik apìkrish tou sust mato e nai
h(n) = SfÆ(n)g = Æ(n) + aÆ(n 1) (5.6.40)
H apìkrish suqnìthta tou sust mato e nai o MF diakritoÔ qrìnou th kroustik
apìkrish
ètsi h (5.6.40) d nei
H( ) = 1 + ae j (5.6.41)
ìpou a = jajej argfag . H apìkrish suqnìthta tou sut mato mpore na grafe w
H( ) = 1 + jaje j( argfag) (5.6.42)
Qrhsimopoi¸nta thn tautìthta Euler èqoume
H( ) = 1 + jaj os( argfag) j jaj sin( argfag) (5.6.43)
H apìkrish pltou e nai
jH ( )j = (1 + jaj os( argfag))2 + jaj2 sin2 ( argfag)1=2
= (1 + jaj2 + 2jaj os( argfag))1=2 (5.6.44)
lìgw th trigonwmetrik tautìthta os2 + sin2 = 1.
Sto Sq ma 5.33 èqoun g nei oi grafikè parastsei th apìkrish pltou gia ti
peript¸sei ìpou a) a = 0; 5ej=3 , dhlad , ìtan sto dèkth ftnei ektì apì to s ma
aupeuje a didosh kai èna s ma me exasjènhsh sh me 0,5 kai qronik kajustèrhsh
sh me t = T=6 kai b) a = 0; 9ej2=3.
H(Ù) H(Ù)
2 2
1,5 1,5
1 1
0,5 0,5
-ð 0 ð Ù -ð 0 ð Ù
(a) (â)
Sq ma 5.33 H grafik parstash th apìkrish pltou se sunrthsh me th tou sust -
mato sto Pardeigma 5.6.5. a) ìtan a = 0; 5 ej=3 kai b) a = 0; 9 ej2=3 .
Se pollè efarmogè pragmatikoÔ qrìnou h akolouj a eisìdou enì FIR f ltrou
èqei meglo m ko parade gmato qrin, h akolouj a pou proèrqetai apì s ma omil a
enì mikrof¸nou, h opo a mpore na jewrhje w m a akolouj a ape rou m kou . Upo-
log zoume thn èxodo tou f ltrou me th bo jeia grammik sunèlixh qrhsimopoi¸nta
194 Anptugma - Metasqhmatismì Fourier Diakrit¸n Shmtwn Keflaio 5
taqÔ metasqhmatismì Fourier, o opo o ja èqei, fusik, meglo m ko . Epiplèon den
e nai dunatì o upologismì th exìdou, prin epexergastoÔme ìla ta de gmata th
eisìdou, kai autì dhmiourge meglh kajustèrhsh.
Sti peript¸sei autè me th bo jeia taqÔ metasqhmatismoÔ Fourier pou t¸ra èqei
mikrì m ko upolog zontai oi epimèrou èxodoi tou sust mato , ìtan e nai gnwstì
èna tm ma (mplok) th akolouj a eisìdou. Sth sunèqeia upolog zetai h èxodo tou
f ltrou me th bo jeia twn epimèrou exìdwn tou f ltrou. Ta parapnw epexhgoÔntai
sto pardeigma pou akolouje .
Pardeigma 5.6.7
D netai to GQA sÔsthma diakritoÔ qrìnou pou èqei kroustik apìkrish
h(n) = [ 1; 0; 1℄
"
An h e sodo tou sust mato e nai h akolouj a x(n) = n + 1; 0 n 9, na breje h
èxodo tou sust mato me th bo jeia kuklik sunèlixh 6-shme wn.
LÔsh H kroustik apìkrish tou f ltrou e nai akolouj a N2 = 3-shme wn. An h
akolouj a eisìdou katatmhje se akolouj e N1 = 6-shme wn, tìte e nai gnwstì ìti h
grammik sunèlixh kje upoakolouj a me thn kroustik apìkrish ja e nai akolouj a
N1 + N2 1 = 8-shme wn. An qrhsimopoihje kuklik sunèlixh N = 6-shme wn, tìte
ta pr¸ta N1 + N2 + 1 N = 2 stoiqe a kje akolouj a ja e na esfalmèna lìgw tou
fainomènou th epikluyh .
H akolouj a eisìdou x(n) = [ 1; 2; 3; 4; 5; 6; 7; 8; 9; 10 ℄ e nai akolouj a 10-shme wn.
Sthn akolouj a aut prost jentai sthn arq dÔo mhdenik, kai sqhmat zontai oi upoa-
kolouj e
x1 (n) = [ 0; 0; 1; 2; 3; 4 ℄
x2 (n) = [ 3; 4; 5; 6; 7; 8 ℄
x3 (n) = [ 7; 8; 9; 10; 0; 0 ℄
Kje upoakolouj a epikalÔptetai apì thn prohgoÔmen th stou dÔo pr¸tou ìrou .
Sthn teleuta a upoakolouj a èqoun prosteje mhdenik, ¸ste na g nei akolouj a 6-
shme wn.
H kuklik sunèlixh 6-shme wn kje upoakolouj a xk (n); k = 1; 2; 3 me thn kroustik
apìkrish tou sust mato d nei ti akolouj e
y1 (n) = x1 (n) 4 h(n) = [ 3; 4; 1; 2; 2; 2 ℄
y2 (n) = x2 (n) 4 h(n) = [ 4; 4; 2; 2; 2; 2 ℄
y3 (n) = x3 (n) 4 h(n) = [ 7; 8; 2; 2; 9; 10 ℄
Apì ti akolouj e yk (n); k = 1; 2; 3 diagrfoume tou dÔo pr¸tou ìrou , oi opo oi
lìgw th epikluyh e nai esfalmènoi, kai sqhmat zetai h akolouj a
y(n) = [ 1; 2; 2; 2; 2; 2; 2; 2; 2; 2; 9; 10 ℄ (5.6.45)
Enìthta 5.6 Probl mata 195
H akolouj a aut e nai sh me thn grammik sunèlixh
x(n) ? h(n) = [ 1; 2; 2; 2; 2; 2; 2; 2; 2; 2; 9; 10 ℄ (5.6.46)
SÔnoyh Kefala ou
Sto keflaio autì perigryame to anptugma se seir Fourier diakritoÔ qrìnou
periodik¸n akolouji¸n, me th bo jeia tou opo ou analÔoume èna periodikì s ma di-
akritoÔ qrìnou se seir apì armonik migadik ekjetik s mata diakritoÔ qrìnou,
dhlad , se s mata apl suqnìthta . Perigryame th mèjodo prosdiorismoÔ twn
suntelest¸n tou anaptÔgmato kai d¸same th fusik tou shmas a. DieurÔname ta
parapnw apotelèsmata kai ètsi perigryame to MF diakritoÔ qrìnou enì s mato
diakritoÔ qrìnou. Parathr same, ìti, ìpw to anptugma se seir Fourier twn peri-
odik¸n shmtwn diakritoÔ qrìnou, ètsi kai o MF diakritoÔ qrìnou twn mh periodik¸n
shmtwn diakritoÔ qrìnou anaparist mh periodik s mata me ekjetik s mata kai
me ton trìpo autì apokalÔptei to fasmatikì tou perieqìmeno.
Perigryame ti basikè idiìthte sti opo e diafèroun o MF diakritoÔ qrìnou
apì ton MF suneqoÔ . Parousisame leitourg e , ìpw h diamìrfwsh kai to je¸rhma
th sunèlixh , me th bo jeia tou opo ou h upologistik polÔplokh sqèsh th sunèli-
xh metasqhmatizìmenh kat Fourier katal gei s' èna aplì ginìmeno sunart sewn.
Me th bo jeia tou jewr mato tou Parseval e dame ìti mporoÔme na upolog soume thn
enèrgeia enì s mato e te sto ped o tou qrìnou e te sto ped o twn suqnot twn.
Gia na epexergastoÔme to MF diakritoÔ qrìnou me yhfiak mèsa prob kame se
katllhlh deigmatolhy a tou MF kai sqhmat same to diakritì MF. Parousisjhkan
oi idiìthte tou diakritoÔ MF. Sth sunèqeia perigrfhke o taqÔ MF, me ton opo o
epiteÔqjhke shmantik upologistik elttwsh.
Sto kefala o parousizontai trei p nake . Ston pr¸to uprqoun oi idiìthte
tou MF diakritoÔ qrìnou, en¸ ston deÔtero oi MF diakritoÔ qrìnou merik¸n basik¸n
akolouji¸n kai ston tr to p naka oi idiìthte tou diakritoÔ MF. Ja prèpei, telei¸non-
ta to dibasma tou kefala ou, na gnwr zete kal ti idiìthte kai na mpore te, ba-
sizìmenoi sta parade gmata tou kefala ou kai sti idiìthte , na br skete tou MF
twn basik¸n akolouji¸n pou uprqoun sto deÔtero p naka.
PROBLHMATA
5.1 Na upologiste o metasqhmatismì Fourier diakritoÔ qrìnou gia ta s mata
x(n) = 1; 0 n 3
1.
0; alli¸
2. x(n) = 3n u( n)
3. x(n) = 31 n u(n)
196 Anptugma - Metasqhmatismì Fourier Diakrit¸n Shmtwn Keflaio 5
x(n) = 1 n
4.
2 os( 0 n)u(n)
x(n) = 1 n [u(n + 2) u(n 3)℄
5.
3
x(n) = n 13 jnj
6.
x(n) = n 12 jnj os( 0 n)
7.
8. x(n) = os 125 n + sin(3n)
5.2 Na upologistoÔn oi akolouj e twn opo wn o metasqhmatismì Fourier diakritoÔ
qrìnou e nai
1. X( ) = 10;; W <j j
0j jW
2. X( ) = 1 4e j3 + 2e j2 + 5e j6
3. X( ) = P1m= 1( 1)m Æ m
2
4. X( ) = sin2
5. X( ) = ; 0 j jj j < 0
6. X( ) = 6+e 6je j e j2
5.3 D netai GQA sÔsthma diakritoÔ qrìnou to opo o èqei kroustik apìkrish
h(n) =
sin(n=6)
n
Na upologiste h èxodo tou sust mato , ìtan h e sodì tou e nai to s ma di-
akritoÔ qrìnou n
n
x(n) = sin 2 4
8
5.4 D netai GQA sÔsthma diakritoÔ qrìnou to opo o qarakthr zetai apì thn ex swsh
diafor¸n
1
y(n) + y(n 1) = x(n)
2
1. Na upologiste h apìkrish suqnìthta tou sust mato .
2. An h e sodo tou sust mato e nai to s ma
n
x(n) =
1 u(n)
2
na upologiste h èxodo tou sust mato .
Enìthta 5.6 Probl mata 197
5.5 D nontai dÔo GQA sust mata diakritoÔ qrìnou ta opo a èqoun apìkrish suqnìth-
ta
) = 11+ 1ee j 1
j
H1 ( H2 ( )= 1 1
2
kai
2e j + 14 e j2
ant stoiqa. Ta dÔo sust mata sundèontai se seir. Na upologiste h ex swsh
diafor¸n h opo a qarakthr zei to sunolikì sÔsthma.
5.6 D netai GQA sÔsthma diakritoÔ qrìnou to opo o qarakthr zetai apì thn ex swsh
diafor¸n
y(n)
1
9 y(n 1) = x(n)
1. Na upologiste h apìkrish suqnìthta tou sust mato .
2. An h e sodo tou sust mato e nai to s ma
n
x(n) = (n + 1)
1 u(n)
3
na upologiste h èxodo tou sust mato .
5.7 D netai GQA sÔsthma diakritoÔ qrìnou to opo o èqei kroustik apìkrish
n
h(n) =
1 u(n) +
1 1 n u(n)
2 2 4
Na upologiste h ex swsh diafor¸n h opo a sundèei thn e sodo kai thn èxodo
tou sust mato .
5.8 D netai èna GQA sÔsthma diakritoÔ qrìnou me kroustik apìkrish
h(n) = [ 1; 2; 1℄
"
1. Na g noun oi grafikè parastsei tou mètrou kai th fsh th apìkrish
suqnìthta tou sust mato se sunrthsh me th kuklik suqnìthta.
2. Na breje h èxodo tou sust mato sth mìnimh katstash, an to s ma
eisìdou e nai
x(n) = 1 + 2 os 3 n 6
5.9 D netai èna GQA sÔsthma diakritoÔ qrìnou me kroustik apìkrish
h(n) = [ 1; 0; 1℄
"
H e sodo tou sust mato e nai to s ma diakritoÔ qrìnou
x(n) = 2(n 1); 0 n 3
Qrhsimopoi¸nta kuklik sunèlixh na prosdior sete thn èxodo tou sust mato .
198 Anptugma - Metasqhmatismì Fourier Diakrit¸n Shmtwn Keflaio 5
x(n) = 1; 0 n 4
5.10 D netai to s ma
0; alli¸
1. Na apodeiqje ìti o metasqhmatismì Fourier diakritoÔ qrìnou X( ) e nai
X(
sin(2 ) e
) = sin( j 32
=2)
2. An g nei omoiìmorfh deigmatolhy a tou metasqhmatismoÔ Fourier diakri-
toÔ qrìnou se suqnìthte pou apèqoun metaxÔ tou kat 2 , na =
N
sqedisete to anakataskeuasmèno s maapì ta de gmata tou metasqhma-
tismoÔ Fourier diakritoÔ qrìnou X k 2N gia N kai N . =4 =8
5.11 Na upologiste kai na sqediaste h apìkrish pltou tou sust mato pou peri-
grfetai apì thn ex swsh diafor¸n
y(n) =
1 [x(n + 1) + x(n) + x(n 1)℄
3
5.12 D netai to analogikì s ma
x(t) = os(200t) + 0; 6 os(624t)
G netai deigmatolhy a tou s mato me suqnìthta 512 Hz. Me th gnwst sqèsh
1
X
t nTs
xa (t) = xa (nT s)sin
n= 1 Ts
()
anakataskeuzetai to analogikì s ma xa t . Na sugkr nete ta arqikì analogikì
() ()
s ma x t kai to anakataskeuasmèno s ma xa t . Poie e nai oi parathr sei
sa kai pw autè dikaiologoÔntai;
Bibliograf a
5.1 S. Jeodwr dh , K. Mpermper dh , L. Kof dh , “Eisagwg sth Jewr a Shmtwn
kai Susthmtwn”, Tupwj tw - Gi¸rgo Dardanì , Aj na 2003.
5.2 N. Kaloupts dh , “S mata Sust mata kai Algìrijmoi”, D aulo , Aj na, 1994.
5.3 S. Haykin, B. Veen, “Signal and Systems”, John & Wiley Sons, Inc. 2003
5.4 J. G. Proakis, D. G. Manolakis, “Introduction to Digital Signal Processing”,
MacMillan Publishing Company, 1994.
5.5 A. V. Oppenheim, R. W. Schafer, “Digital Signal Processing”, Prentice - Hall Inc.,
N. Y., 1975.
5.6 A. V. Oppenheim, R. W. Schafer, J. R.Buck “Discrete-Time Signal Processing”,
2nd ed. Prentice - Hall Inc., N. Y., 1999.
ÊÅÖÁËÁÉÏ 6
ÌÅÔÁÓ×ÇÌÁÔÉÓÌÏÓ LAPLACE
Skopì tou kefala ou e nai na or sei ton amf pleuro metasqhmatismì Laplace
, apl¸ , metasqhmatismì Laplace (ML) kai to monìpleuro metasqhmatismì Laplace
(MML), na perigryei ti basikè tou idiìthte kai na upolog sei tou ant stoiqou
metasqhmatismoÔ stoiqeiwd¸n shmtwn, pou antimetwp zoume sth melèth grammik¸n
susthmtwn. Ep sh , sto keflaio autì ja parousisoume th dunatìthta pou èqei
o MML na epilÔei diaforikè exis¸sei , oi opo e èqoun mh mhdenikè arqikè sun-
j ke kai sth sunèqeia ja ekmetalleutoÔme th dunatìthta aut gia th melèth GQA
susthmtwn. Tèlo , skopì tou kefala ou e nai na anade xei th sqèsh pou uprqei
metaxÔ th aitiìthta , th eustjeia enì GQA sust mato , tou ped ou sÔgklish
th sunrthsh metafor tou kai th jèsh twn pìlwn aut sto migadikì ep pedo,
ìpou or zetai o metasqhmatismì Laplace.
EISAGWGH
Sto Keflaio 2, e dame ìti h e sodo kai h èxodo enì analogikoÔ GQA sust -
mato sundèontai me m a diaforik ex swsh me stajeroÔ suntelestè . àtsi, gia na
prosdior soume thn èxodo enì sust mato an gnwr zoume thn e sodì tou, prèpei na
epilÔoume thn ant stoiqh diaforik ex swsh. Sto dio Keflaio parathr same ìti
mporoÔme na upolog soume thn èxodo enì sust mato an gnwr zoume thn e sodì tou,
me th bo jeia tou oloklhr¸mato th sunèlixh . Sto Keflaio 3 or same to MF,
o opo o parèqei th dunatìthta metbash apì to ped o tou qrìnou sto ped o th
suqnìthta . H idiìthta th sunèlixh tou MF metatrèpei to olokl rwma th sunèli-
xh se èna aplì ginìmeno twn antisto qwn metasqhmatism¸n, me th bo jeia tou opo ou
upolog zetai o MF th exìdou kai sth sunèqeia me èna ant strofo MF prosdior zetai
h èxodo tou sust mato sto ped o tou qrìnou. O MF, loipìn, èdwse m a eÔkolh lÔsh
sto prìblhma eÔresh th exìdou enì sust mato , sthn per ptwsh pou gnwr zoume
thn e sodì tou kai thn kroustik tou apìkrish. Dustuq¸ , ìmw , uprqoun poll
s mata, ta opo a suqn sunantme sthn prxh, gia ta opo a den uprqei o MF.
Sto keflaio autì ja perigryoume to Metasqhmatismì Laplace, o opo o meta-
trèpei èna s ma suneqoÔ qrìnou se m a analutik sunrthsh migadik metablht .
äpw ja doÔme, poll apì ta s mata me praktik spoudaiìthta, gia ta opo a den
200 METASQHMATISMOS LAPLACE Keflaio 6
uprqei o MF, uprqei o ML kai ètsi dieurÔnetai to sÔnolo twn shmtwn gia ta opo a
mpore na epiteuqje metbash apì to ped o tou qrìnou sto ped o suqnìthta .
Sto Keflaio 4, me th bo jeia tou MF upolog same thn èxodo enì GQA sust -
mato to opo o br sketai arqik se katstash hrem a . Sto keflaio autì ja doÔme
ìti ìtan to sÔsthma de br sketai se katstash hrem a , o MML ma epitrèpei na
sumperilboume ti arqikè sunj ke sth diaforik ex swsh pou sundèei to s ma
eisìdou kai exìdou tou sust mato kai na prosdior soume thn èxodo tou sust mato .
Tèlo , sto keflaio autì ja doÔme ìti h qr sh tou migadikoÔ ped ou suqnìthta
kai h jèsh twn pìlwn se autì ma epitrèpei na exgoume basikè idiìthte twn susth-
mtwn, ìpw h aitiìthta kai h eustjeia. Gia ìlou tou parapnw lìgou , o ML
apotele èna akìma basikì majhmatikì ergale o gia th melèth GQA susthmtwn.
6.1 ORISMOI
Sthn Enìthta 2.5.1 èqoume dei ìti an h e sodo enì grammikoÔ qronik anallo wtou
sust mato e nai to s ma x t ()=
Aest , tìte to s ma exìdou e nai
y(t) = H (s) Aest (6.1.1)
ìpou Z 1
H (s) = h(t)e st dt (6.1.2)
1
e nai o metasqhmatismì Laplace th kroustik apìkrish tou sust mato kai e nai
h sunrthsh metafor tou sust mato .
O metasqhmatismì Laplace antistoiqe sto s ma suneqoÔ qrìnou x t th sunrth- ()
sh Z 1
L[x(t)℄ = X (s) x(t)e st dt (6.1.3)
1
H X (s) e nai migadik sunrthsh th migadik metablht + j! kai onomzetai
Metasqhmatismì ()
Laplace (ML) tou s mato x t . Merikè forè anafèretai kai w
amf pleuro metasqhmatismì Laplace gia na toniste h diafor tou apì to monìpleuro
metasqhmatismì Laplace pou ja or soume sthn Enìthta 6.3. To sÔnolo twn migadik¸n
arijm¸n + ()
j!, gia to opo o uprqei h X s , dhlad to ant stoiqo olokl rwma
orismoÔ th sugkl nei, onomzetai perioq sÔgklish (PS) th X s . Gia eukol a, o ()
() [ ( )℄
ML tou s mato x t merikè forè sumbol zetai w L x t kai h sqèsh metaxÔ tou
()
x t kai tou ML upodeiknÔetai w
x(t) ! X (s)
L
(6.1.4)
h de perioq sÔgklish dhl¸netai w R.
Enìthta 6.1 Orismo 201
Parathr sei
1. An o ML uprqei kai gia timè me
R , dhlad s j!, tìte X j!
1 x t e j!t dt pou den e nai t pote llo apì to MF th sunrthsh x
=0 = ( )=
1 () (t),
dhlad
X (s)js=j! = F [x(t)℄ (6.1.5)
2. O ML sqet zetai me to MF kai sthn per ptwsh ìpou h metablht s den e nai
fantastikì arijmì ( ). Prgmati, 6= 0
Z 1 Z 1 t e j!t dt
X ( + j!) = x(t)e (+j!)t dt = x(t)e (6.1.6)
1 1
()
O ML th x t mpore na ermhneuje kai w o MF th sunrthsh x t () =
()
x t e t . Epomènw , gia na uprqei o metasqhmatismì Laplace tou s mato
()
x t prèpei na uprqei o metasqhmatismì Fourier tou s mato x t e t , dhlad ()
()
to s ma x t e t na e nai apolÔtw oloklhr¸simo
Z 1
x(t)e t dt < 1
1
H parous a tou ìrou e t parèqei th dunatìthta sÔgklish tou oloklhr¸ma-
to kai kat sunèpeia thn Ôparxh tou ML akìma kai an den uprqei o MF th
()
x t . Gia pardeigma, h ekjetik aÔxousa sunrthsh (s ma) x t eat u t () = ()
gia a jetik pragmatik stajer den e nai apolÔtw oloklhr¸simh kai w
ek toÔtou den uprqei o metasqhmatismì Fourier. An epilege > a, tìte
h sunrthsh x t e t e(a )t u t e nai apolÔtw oloklhr¸simh, epomè-
() = ()
nw uprqei o metasqhmatismì Laplace. Sto Sq ma 6.1 uprqoun oi grafikè
()
parastsei twn shmtwn x t , e t kai x t e t se sunrthsh me to qrìno. ()
x(t) e- ó t x(t) e -ó t
1 ó>á
e atu(t) 1 1 ó>á
0 t 0 t 0 t
(á) ( â) (ã)
Sq ma 6.1 x(t) = eat u(t) gia to opo o den uprqei o MF (b) o pargonta exas-
(a) To s ma
jènish e t kai (g) to s ma x(t)e t = e(a )t u(t) to opo o e nai apolÔtw oloklhr¸simo.
202 METASQHMATISMOS LAPLACE Keflaio 6
6.1.1 Metasqhmatismì Laplace stoiqeiwd¸n shmtwn
Sthn pargrafo aut ja upolog soume tou ML orismènwn stoiqeiwd¸n shmtwn.
Pardeigma 6.1.1 (Migadikì aitiatì ekjetikì s ma)
Na upologiste o ML tou s mato x(t) = e at u(t), ìpou a migadikì arijmì .
LÔsh Apì ton orismì tou ML èqoume
Z 1 Z T
X (s) = e at e st dt = Tlim e (a+s)t dt
0 !1 0
= Tlim 1 he (a+s)T 1i
!1 a + s
(6.1.7)
All limT !1 e
(a+s)T = 0, en <e[a + s℄ > 0, sunep¸ pa rnoume
x(t) = e atu(t) L! X (s) =
1 me perioq sÔgklish <e[s℄ > <e[a℄
s+a
(6.1.8)
ParathroÔme ìti h perioq sÔgklish R tou migadikoÔ aitiatoÔ ekjetikoÔ s mato e nai
to dexiì hmiep pedo me sÔnoro th gramm pou e nai kjeth ston pragmatikì xona sth
jèsh <e[a℄, (blèpe Sq ma 6.2).
x(t) ℑm
1
ℜe a ℜe
0 (á) t
( â)
Sq ma 6.2 (a) To s ma x(t) = e at u(t) kai (b) h perioq sÔgklish tou ML.
Parathr sei
1. An a = 0, tìte x(t) e nai h sunrthsh monadia ou b mato , x(t) = u(t) kai o
ML e nai
X (s) = L[u(t)℄ =
1 me perioq sÔgklish <e[s℄ > 0 (6.1.9)
s
2. An <[℄ 0
e a < , mporoÔme na upolog soume to X (s) gia = 0, dhlad uprqei
kai o MF kai e nai
X (0 + j!) =
1
j! + a
(6.1.10)
3. An <e[a℄ > 0, o MF den uprqei, en¸ profan¸ uprqei o ML.
Enìthta 6.1 Orismo 203
Pardeigma 6.1.2 (Austhr mh aitiatì ekjetikì s ma)
Na upologiste o ML tou s mato x(t) = e atu( t), ìpou a migadikì arijmì
LÔsh O ML tou s mato e nai
Z 0 Z 0
X (s) = e lim e (a+s)t dt
at e st dt
T !1 T
=
1
= s +1 a 1 Tlim
h i
e(a+s)T (6.1.11)
!1
E nai limT !1 e
(a+s)T = 0 an <e[s + a℄ < 0. Sunep¸
X (s) =
1 me perioq sÔgklish <e[s℄ < <e[a℄
s+a
(6.1.12)
ParathroÔme ìti h perioq sÔgklish R tou austhr mh aitiatoÔ ekjetikoÔ s mato
e nai to aristerì hmiep pedo me sÔnoro th gramm pou e nai kjeth ston pragmatikì
xona sth jèsh <e[a℄ (blèpe Sq ma 6.3).
x(t) ℑm
0
t
ℜe a ℜe
-1
(á) ( â)
Sq ma 6.3 (a) To s ma x(t) = e at u( t) kai (b) h perioq sÔgklish tou ML.
ParathroÔme ìti ta s mata x t () = ()
e at u t (Pardeigma 6.1.1) kai x t () =
at ( )
e u t , (Pardeigma 6.1.2) èqoun thn dia sunrthsh w ML all diaforetik
perioq sÔgklish . Gia to lìgo autì, pnta ektì apì thn X s ja prèpei na d netai ()
kai h ant stoiqh perioq sÔgklish ¸ste na prosdior zetai monos manta to s ma x t . ()
Pardeigma 6.1.3
Na upologiste o ML tou s mato x(t) = e t u(t) + e 2t u(t).
LÔsh O ML tou s mato e nai
Z 1
Z 1 Z 1
X (s) = e tu (t) + e 2tu(t) e st dt = e t e st dt + e 2t e st dt
1 0 0
(6.1.13)
Kje èna apì ta oloklhr¸mata sthn (6.1.13) èqoun thn dia morf me to olokl rwma
sthn (6.1.7) ètsi, an qrhsimopoi soume to apotèlesma tou Parade gmato 6.1.1 èqoume
X (s) =
1 + 1 = 2s + 3
s+1 s + 2 s2 + 3s + 2
(6.1.14)
204 METASQHMATISMOS LAPLACE Keflaio 6
To s ma x(t) e nai jroisma dÔo pragmatik¸n ekjetik¸n shmtwn kai apì thn (6.1.14)
parathroÔme ìti o X (s) e nai so me to jroisma twn ML twn epimèrou shmtwn.
O pr¸to ìro e nai o ML tou e u(t) me PS <e[s℄ > 1 kai o deÔtero e nai o ML
t
tou e
2 t u(t) me PS <e[s℄ > 2. Oi koinè timè tou s gia ti opo e kai oi dÔo ML
sugkl noun e nai autè gia ti opo e <e[s℄ > 1. àqoume, epomènw ,
e t u(t) + e 2t u(t) L! 2
2s + 3 ; <e[s℄ > 1
s + 3s + 2
me PS (6.1.15)
Se kje èna apì ta tr a parapnw parade gmata o ML e nai rht sunrthsh, dhlad
e nai lìgo dÔo poluwnÔmwn th migadik metablht s, ètsi
N (s)
X (s) =
D(s)
(6.1.16)
()
M a sunrthsh X s onomzetai analutik sthn perioq R tou migadikoÔ epipèdou-s,
en (a) e nai monìtimh sunrthsh sthn R kai (b) e nai paragwg simh se kje shme o
()
th R. An h X s den e nai analutik se èna shme o s0 , tìte to shme o autì lègetai
()
shme o anwmal a . Oi r ze tou arijmht N s onomzontai mhdenik th X s kai ()
Æ
paristnontai me “ ” sto migadikì ep pedo. Sta shme a aut h X s mhden zetai. Oi ()
() ()
r ze tou paronomast D s , ìpou h X s den or zetai, onomzontai pìloi th X s ()
kai paristnontai me “ ” sto migadikì ep pedo. Sto Sq ma 6.4 fa netai h perioq
sÔgklish , oi pìloi kai to mhdenikì tou ML tou s mato sto Pardeigma 6.1.3.
ℑm
Sq ma 6.4 H perioq sÔgklish , oi
-2 -1 ℜe
pìloi kai to mhdenikì tou ML tou s -
mato x(t) sto Pardeigma 6.1.3.
Pardeigma 6.1.4
Na upologiste o ML th sunrthsh Æ(t).
LÔsh
Z 1
L[Æ(t)℄ = Æ(t)e st dt = 1 me perioq sÔgklish <e[s℄ > 1 (6.1.17)
1
ìpou qrhsimopoi jhke h (1.4.19).
Pardeigma 6.1.5
D netai to s ma x(t) = e bjtj (blèpe Sq ma 6.5a). Na upologiste o ML.
Enìthta 6.1 Orismo 205
x(t) x(t)
1
1
0 t 0 t
(á) ( â)
Sq ma 6.5 H grafik parstash tou s mato x(t) = e bjtj gia (a) b > 0 kai (b) b < 0.
LÔsh To s ma grfetai
x(t) = e bt u(t) + ebt u( t) (6.1.18)
Gnwr zoume ìti
1 ; me PS <e[s℄ > b (Pardeigma 6.1.1)
e bt u(t) L!
s+b
ebt u( t) L!
1 ; me PS <e[s℄ < b (Pardeigma 6.1.2)
s b
ParathroÔme ìti, an b < 0, oi dÔo epimèrou ìroi den èqoun koin perioq sÔgklish
kai to s ma x(t) den èqei ML. An b > 0, èqoume
e bjtj L!
1 1 = 2b me PS b < <e[s℄ < b
s+b s b s2 b2
(6.1.19)
Sto Sq ma 6.6 fa nontai h perioq sÔgklish kai oi pìloi tou ML tou s mato x(t).
ℑm
ℜe Sq ma 6.6 H perioq sÔgklish tou ML tou s mato
x(t) = e bjtj
-b b
sto Pardeigma 6.1.5 kai oi pìloi tou gia
b>0.
Pardeigma 6.1.6 (Poluwnumikì ekjetikì s ma).
Na upologiste o ML tou ekjetikoÔ poluwnumikoÔ s mato txh m, pou or zetai w
tm at
x(t) = e u(t)
m!
(6.1.20)
LÔsh Sto Pardeigma 6.1.1 èqoume de xei
Z 1 1
e at e st dt = <e[s℄ > <e[a℄
s+a
gia (6.1.21)
0
206 METASQHMATISMOS LAPLACE Keflaio 6
Paragwg zonta w a kai ta dÔo mèlh th (6.1.21) èqoume
pro
d
Z 1 1 Z 1 te ate st dt = 1
e at e st dt =
da 0 (s + a)2 0 (s + a)2 (6.1.22)
H teleuta a isìthta dhl¸nei ìti to s ma x(t) = te
at u(t) èqei ML
x(t) = te at u(t) L! X (s) =
1
(s + a)2 me <e[s℄ > <e[a℄ (6.1.23)
An a = 0, èqoume
L [tu(t)℄ = L [r(t)℄ = 2
1 me <e[s℄ > 0 (6.1.24)
s
ìpou r(t) e nai h sunrthsh kl sh . Nèa parag¸gish th (6.1.22) w pro a d nei
Z 1
t2 e at e st dt =
2
0 (s + a)3
t2 at 1
opìte
e u(t) L!
2 (s + a)3 me PS <e[s℄ > <e[a℄ (6.1.25)
Genik mpore na deiqje epagwgik ìti
tm at
e u(t) L!
1 <e[s℄ > <e[a℄
m! (s + a)m+1 me PS (6.1.26)
6.1.2 Idiìthte th perioq sÔgklish - Ìparxh metasqhmatismoÔ Laplace
Apì ta prohgoÔmena parade gmata parathr same ìti o metasqhmatismì Laplace den
prosdior zei monos manta to s ma ektì an èqei orisje h perioq sÔgklish . Ep sh ,
parathr same ìti h morf th perioq sÔgklish tou metasqhmatismoÔ Laplace
exarttai apì ta qarakthristik tou s mato . Sthn enìthta aut ja broÔme ton
trìpo me to opo o sundèetai h perioq sÔgklish me ta qarakthristik tou s ma-
()
to x t . Ja parousisoume ti idiìthte qrhsimopoi¸nta diaisjhtik epiqeir mata
par austhrè majhmatikè apode xei . Gnwr zonta ti idiìthte e nai efiktì o pros-
diorismì th perioq sÔgklish apì to metasqhmatismì Laplace X s kai èqonta ()
periorismènh gn¸sh twn qarakthristik¸n tou s mato x t . ()
àna pr¸to sumpèrasma e nai ìti h perioq sÔgklish tou metasqhmatismoÔ Laplace
pou e nai rht sunrthsh th metablht s den perièqei pìlou .
äpw parathr same sthn prohgoÔmenh enìthta, gia na uprqei o metasqhmatismì
Laplace tou s mato ()
x t prèpei
Z 1
I () = jx(t)j e t dt < 1 (6.1.27)
1
Enìthta 6.1 Orismo 207
Oi timè th pragmatik metablht pou ikanopoioÔn thn parapnw anisìthta or -
zoun thn perioq sÔgklish tou metasqhmatismoÔ. H posìthta e nai to pragmatikì
mèro tou migadikoÔ arijmoÔ s s ( = + )
j! , epomènw h perioq sÔgklish exarttai
apì to pragmatikì mèro , en¸ to fantastikì mèro den ephrezei th sÔgklish. Gia
to lìgo autì h perioq sÔgklish e nai z¸ne parllhle sto fantastikì xona tou
epipèdou-s.
()
Gia èna peperasmèno s ma x t (uprqei jetikì arijmì M gia ton opo o e nai
j ( )j
xt M ), to opo o e nai peperasmènh dirkeia (x t gia t < T1 kai t > T2 ), ( )=0
tìte
Z 1 Z T2 W T2 T1 ; 6= 0
I () = jx(t)j e t dt Me t dt = e e
1 T1 M (T2 T1 ); =0
()
Sthn per ptwsh aut to olokl rwma I e nai peperasmèno gia ìle ti peperasmène
timè th metablht . Sumpera noume, loipìn, ìti:
H perioq sÔgklish enì peperasmènou s mato kai peperasmènh dirkeia
sumperilambnei olìklhro to ep pedo-s.
Sth sunèqeia ja exetsoume th genik per ptwsh kat thn opo a to s ma x t ()
den e nai periorismènh dirkeia kai e nai mh peperasmèno. To s ma autì onomzetai
amf pleuro s ma. Sthn per ptwsh aut diaqwr zoume to I se dÔo tm mata. Sto ()
()
pr¸to tm ma I ta ìria olokl rwsh e nai kai 1 0
Z 0
I () = jx(t)je t dt
1
Sto deÔtero tm ma I+ () ta ìria olokl rwsh e nai 0 kai 1
Z 1
I+ () = jx(t)je t dt
0
dhlad ,
Z 0 Z 1
I () = jx(t)j e t dt + jx(t)j e t dt = I () + I+()
1 0
()
Gia na e nai to I peperasmèno prèpei kai ta dÔo epimèrou oloklhr¸mata na e nai
j ( )j
peperasmèna. Autì sunepgetai ìti to x t prèpei na e nai fragmèno kai gia ti
jetikè kai gia ti arnhtikè timè tou qrìnou.
A upojèsoume ìti mporoÔme na frxoume to x t gia ti jetikè kai gia ti j ( )j
arnhtikè timè tou qrìnou br skonta ti elqiste stajerè M > kai # tètoie 0
¸ste
xt j ( )j
Me# t ; t > ; 0
208 METASQHMATISMOS LAPLACE Keflaio 6
kai th mègisth stajer tètoia ¸ste
jx(t)j Me t; t < 0;
()
To s ma x t to opo o ikanopoie ta frgmata aut qarakthr zetai w s ma ekjetik
j ( )j
txh . Ta frgmata upodhl¸noun ìti to x t den auxnetai taqÔtera apì to ekjetikì
s ma e# t gia ti jetikè timè tou qrìnou kai e t gia ti arnhtikè timè tou qrìnou.
Shmei¸netai ìti uprqoun s mata ta opo a den e nai ekjetik txh , gia pardeigma
ta s mata et kai t3t . Ta s mata aut den emfan zontai se fusikè efarmogè kai ètsi
2
den ma dhmiourgoÔn probl mata.
Upojètonta ìti to s ma x(t) e nai ekjetik txh èqoume gia to olokl rwma
I ()
Z 0 Z 0
I () = jx(t)j e t dt M e( )t dt
1 1
M 0
= e( )t (6.1.28)
1
kai gia to olokl rwma I+ ()
Z 1 Z 1
I+ () = jx(t)j e t dt M e(# )t dt
0 0
M h 1i
= e(# )t (6.1.29)
# 0
()
ParathroÔme ìti to I e nai peperasmèno ìtan < kai to I+ e nai peperas- ()
()
mèno ìtan > # . To olokl rwma I sugkl nei gia ti timè tou gia ti opo e kai
() ()
ta dÔo oloklhr¸mata I kai I+ sugkl noun. ra, o metasqhmatismì Laplace
uprqei gia ti timè tou oi opo e ikanopoioÔn th # < < .
H perioq sÔgklish enì amf pleurou s mato e nai m a z¸nh sto migadikì ep pedo
me ìria ti euje e =
kai # . =
To s ma x t ()=ebjtj , tou opo ou o metasqhmatismì Laplace brèjhke sto Pardeigma
6.1.5, e nai èna amf pleuro s ma.
Shmei¸netai ìti sthn per ptwsh ìpou # > , den uprqoun timè tou gia ti
opo e o metasqhmatismì Laplace sugkl nei.
Sth sunèqeia ja exetsoume thn per ptwsh kat thn opo a to s ma x t ( ) = 0 gia
t > T2 . àna s ma pou ikanopoie thn idiìthta aut onomzetai aristerìpleuro s ma.
Enìthta 6.1 Orismo 209
Sthn per ptwsh to I () èqoume
Z 0 Z T2
I () = jx(t)je dt + jx(t)jet dt
t
1 0 ( h i
0 0 ; 6= 0
t jT2
M h i W e
e ( ) t + (6.1.30)
1 M (T2 0); = 0
Apì thn parapnw parathroÔme ìti h perioq sÔgklish enì aristerìpleurou s -
mato apotele tai apì ti timè th metablht s gia ti opo e to pragmatikì mèro
ikanopoie th < , dhlad
H perioq sÔgklish enì aristerìpleurou s mato e nai to aristerì hmiep pedo
me ìrio thn euje a . =
To s ma x t ()=e at u ( t), to opo o melet same sto Pardeigma 6.1.2, e nai
èna aristerìpleuro s ma.
Tèlo , ja exetsoume thn per ptwsh kat thn opo a to s ma x t gia t < T2 .( )=0
àna s ma pou ikanopoie thn idiìthta aut onomzetai dexiìpleuro s ma. Me ìmoio
trìpo parathroÔme ìti h perioq sÔgklish enì dexiìpleurou s mato apotele tai
apì ti timè th metablht s gia ti opo e to pragmatikì mèro ikanopoie th > # .
H perioq sÔgklish enì dexiìpleurou s mato e nai to dexiì hmiep pedo me ìrio
thn euje a =
# .
To s ma x(t) = e (t), to opo o melet
at u same sto Pardeigma 6.1.1, e nai èna
dexiìpleuro s ma.
6.1.3 Idiìthte tou metasqhmatismoÔ Laplace
Sthn pargrafo aut ja parousisoume merikè idiìthte tou ML. Oi idiìthte autè
ja ma bohj soun ston upologismì tou ML shmtwn, qwr na qreiaste na upo-
log soume to olokl rwma orismoÔ. Arketè apì ti idiìthte jum zoun ti idiìthte
tou MF. Gia ti apode xei twn idiot twn, ìpou autè den e nai profane , o endiafe-
rìmeno anagn¸sth parapèmpetai sto [1℄.
(1) Grammikìthta
An x1 t ( ) L! X1 (s) me PS R1 kai x2(t) L! X2 (s) me PS R2, tìte gia opoies-
d pote stajerè a kai b e nai
ax1 (t) + bx2 (t) L
! aX1 (s) + bX2(s) me PS R = R1 \ R2
dhlad h perioq sÔgklish tou grammikoÔ sunduasmoÔ e nai h tom twn epimèrou
() ()
perioq¸n sÔgklish twn X1 s kai X2 s . Se orismène peript¸sei e nai dunatìn h
210 METASQHMATISMOS LAPLACE Keflaio 6
perioq sÔgklish na e nai megalÔterh, ìtan kpoia mhdenik th m a sunrthsh
akur¸noun kpoiou pìlou th llh .
(2) Metatìpish sto qrìno
An x(t) ! X (s) me PS R tìte gia opoiad
L
pote qronik tim t0 isqÔei
x(t t0 ) L
!e st0 X (s) me thn dia PS R
(3) Metatìpish sth migadik suqnìthta
An x(t) ! X (s) me PS R, tìte
L
es0 t x(t) ! X (s s0) me PS R + <e[s0℄
L
ìpou h stajer s0 , sth genik per ptwsh, e nai migadik posìthta. H PS tou X (s s0)
()
e nai h PS tou X s metatopismènh kat e s0 . <[ ℄
(4) Klimkwsh sto qrìno kai sth suqnìthta
An x(t) ! X (s) me PS 1 < <e[s℄ < 2, tìte gia opoiad
L
pote stajer a
! ja1j X as
1
x(at) L
me PS < <e[s℄ < 2
a a
(5) Parag¸gish sth suqnìthta
An x(t) ! X (s) me PS R, tìte
L
( t)nx(t) L! d dsXn(s) me PS R
n
(6) Olokl rwsh sth suqnìthta
An x(t) ! X (s) me PS R, tìte
L
x(t)
Z 1
L
! X ( ) d me PS R
t s
(7) Metasqhmatismì Laplace parag¸gou
An x(t) ! X (s) me PS R, tìte
L
dx(t)
dt
! sX (s) me thn dia PS R
L
Enìthta 6.1 Ant strofo Metasqhmatismì Laplace 211
()
H PS tou sX s mpore na e nai megalÔterh apì thn R, an h X s èqei w aplì pìlo ()
ton s =0o opo o akur¸netai me ton pollaplasiasmì me s. Epagwgik mporoÔme na
genikeÔsoume thn parapnw idiìthta
dn x(t)
dtn
! snX (s) me thn dia PS R
L
(8) Metasqhmatismì Laplace oloklhr¸mato
An x(t) ! X (s) me PS R, tìte
L
! 1s X (s) me PS thn R \ f<e[s℄ > 0g
Z t
x( ) d L
1
(1 ) ( )
H PS tou =s X s mpore na e nai megalÔterh apì thn R \f<e[s℄ > 0g, an o pìlo
sto mhdèn akur¸netai me ant stoiqo mhdenikì th X s . ()
(9) Je¸rhma th sunèlixh sto qrìno
An x1 (t) L
! X1 (s) me PS R1 kai x2(t) L! X2 (s) me PS R2 , tìte
y(t) = x1 (t) ? x2 (t) L
! Y (s) = X1 (s) X2(s)
\
me perioq sÔgklish thn R1 R2 , dhlad h perioq sÔgklish tou ginomènou X1 s ( )
()
X2 s e nai h tom twn epimèrou perioq¸n sÔgklish twn X1 s kai X2 s . E nai () ()
dunatìn na or zetai kai megalÔterh perioq sÔgklish , an kpoia mhdenik th mia
sunrthsh akur¸noun kpoiou pìlou th llh . ParathroÔme, dhlad , ìti ìpw
kai sthn per ptwsh tou MF, h sunèlixh metasqhmat zetai se ginìmeno.
Pèra apì ta stoiqei¸dh s mata, ta opo a melet jhkan sta Parade gmata 6.1.1 -
6.1.6, uprqoun kai arket lla pou ep sh emfan zontai w sustatik mèrh llwn
shmtwn, pou sunantme sth melèth grammik¸n susthmtwn. Oi ML twn shmtwn
aut¸n upolog zontai me th bo jeia tou orismoÔ kai twn idiot twn tou metasqhmatismoÔ
Laplace. Ston P naka 6.1 uprqoun oi idiìthte tou ML, en¸ ston P naka 6.2 uprqoun
oi ML kai oi ant stoiqe perioqè sÔgklish gia ti plèon sunhjismène kai qr sime
peript¸sei .
6.2 ANTISTROFOS METASQHMATISMOS LAPLACE
An e nai èna pragmatikì arijmì , èqoume parathr sei ìti o ML X s tou s mato ()
() ()= ()
x t sump ptei me to MF tou s mato x t x t e t se ìla ta shme a tou migadikoÔ
212 METASQHMATISMOS LAPLACE Keflaio 6
PINAKAS 6.7 Oi idiìthte tou MetasqhmatismoÔ Laplace
Idiìthta S ma ML Ped o sÔgklish
x(t) X (s) R=fs:1 <<e[s℄<2 g
x1 (t) X1 (s) R1
x2 (t) X2 (s) R2
Grammikìthta ax1 (t) + bx2 (t) aX1 (s) + bX2 (s) R1 \ R2
Metatìpish sto qrìno x(t t0 ) e st0 X (s) R
Metatìpish sth es0 t x(t) X (s s0 ) R + <e[s0 ℄
migadik suqnìthta
x(at); a > 0 1 s R
Klimkwsh sto qrìno
aX a a
kai sth suqnìthta
Parag¸gish sth ( t)nx(t) dn
dsn X (s) R
suqnìthta
x(t) R1
X ( ) d R
Olokl rwsh sth
t s
suqnìthta
dn x(t) sn X (s) R
ML parag¸gou dtn
Rt 1 X (s)
ML oloklhr¸mato 1 x( ) d s R
H idiìthta th y(t)=x1 (t)?x2 (t) Y (s)=X1 (s)X2 (s) R1 \ R2
sunèlixh
Periodik s mata x(t)=x(t+T )
R
X (s)= 1 e1 sT 0T x(t)e st dt <e[s℄ > 0
epipèdou-s pou an koun sthn euje a <e[s℄ = . Prgmati,
Z 1
X (s) = x(t)e st dt,
Z
1
1
X ( + j!) = x(t)e (+j!)t dt
Z
1
1
= x (t)e j!t dt (6.2.1)
1
àtsi, an qrhsimopoi soume ton tÔpo antistrof tou MF, pa rnoume
= F 1 [X ( + j!)℄ = 21
Z 1
x(t)e t X ( + j!)ej!t d! (6.2.2)
1
pollaplasizonta kai ta dÔo mèlh me et èqoume
x(t) =
1 Z 1
(+j!)t d!
2 1 X ( + j!)e (6.2.3)
Enìthta 6.1 Ant strofo Metasqhmatismì Laplace 213
PINAKAS 6.8 Metasqhmatismo Laplace merik¸n basik¸n shmtwn
S ma Metasqhmatismì Laplace Perioq sÔgklish
1 Æ(t) 1 Gia kje s
2 u(t) 1=s <e[s℄ > 0
3 u( t) 1=s <e[s℄ < 0
e at u(t) 1 <e[s℄ > <e[a℄
4 s+a
e at u( t) 1 <e[s℄ < <e[a℄
5 s+a
tm e at u(t) 1 <e[s℄ > <e[a℄
6 m! (s+a)m+1
tm e at u( t) 1 <e[s℄ < <e[a℄
7 m! (s+a)m+1
8 Æ(t T ); T 0 e sT Gia kje s
9 [ os(!0t)℄u(t) s
s2 +!02 <e[s℄ > 0
10 [sin(!0t)℄u(t) !0
s2 +!02 <e[s℄ > 0
11 [e at os(!0t)℄u(t) s+a <e[s℄ > <e[a℄
(s+a)2 +!02
11 [e at sin(!0t)℄u(t) !0
(s+a)2 +!02 <e[s℄ > <e[a℄
un (t) = d dtÆn(t)
n
13 sn Gia kje s
+
me allag metablht apì j! se s odhgoÔmaste sthn ex swsh antistrof tou
=
ML (e nai ds jd! afoÔ h e nai stajer)
Z +j 1
x(t) =
1 X (s)est ds
2j j1
(6.2.4)
To olokl rwma èqei thn ènnoia ìti h olokl rwsh ektele tai pnw sthn euje a <e[s℄ =
, h opo a prèpei na perièqetai sto ped o sÔgklish tou X s . ()
6.2.1 Upologismì tou Ant strofou MetasqhmatismoÔ Laplace
O apeuje a upologismì tou ant strofou ML mèsw th ep lush tou oloklhr¸ma-
to th (6.2.4) apaite efarmog teqnik¸n olokl rwsh migadik¸n sunart sewn. H
mèjodo aut mpore na apodeiqje ep ponh diadikas a kai gia to lìgo autì sun jw
akoloujoÔntai èmmesoi trìpoi upologismoÔ tou ant strofou ML.
()
An h morf th sunrthsh X s e nai apl kai mpore eÔkola na ekfraste w
jroisma epimèrou stoiqeiwd¸n ìrwn, tìte me th qr sh twn gnwst¸n ML (P naka
214 METASQHMATISMOS LAPLACE Keflaio 6
6.2) kai twn idiot twn tou ML mporoÔme apeuje a na upolog soume ton L 1 X s [ ( )℄ =
()
x t . äpw èqoume parathr sei kai sta ant stoiqa parade gmata, sta perissìtera
probl mata pou antimetwp zoume sth jewr a twn susthmtwn, o ML èqei th morf
rht sunrthsh . Sti peript¸sei autè o ML mpore na ekfraste w jroisma
apl¸n klasmtwn, gia kajèna apì ta opo a upolog zoume ton ant strofo ML, me th
bo jeia tou P naka 6.2. Sto Parrthma B perigrfontai oi trìpoi anlush rht¸n
sunart sewn se apl klsmata. Sta parade gmata pou akoloujoÔn efarmìzoume th
mèjodo aut .
Pardeigma 6.2.1 (Oi r ze tou paronomast e nai aplè kai pragmatikè )
Na upologiste o ant strofo ML th sunrthsh X (s), ìpou
X (s) = 2
3s + 7 ; <e[s℄ > 1
s + 4s + 3
me PS (6.2.5)
LÔsh Oi r ze tou paronomast e nai 1 = 1 kai 2 = 3. H X (s) mpore na
ekfraste w jroisma apl¸n klasmtwn
X (s) = 2
3s + 7 = sC+1 1 + sC+2 3
s + 4s + 3
(6.2.6)
kai upolog zoume ti stajerè C1 kai C2 w ex :
C1 = (s + 1)X (s)js= 1 =
3s + 7 =2
s + 3 s= 1
kai
C2 = (s + 3)X (s)js= 3 =
3s + 7 =1
s + 1 s= 3
ra
X (s) =
2 + 1
s+1 s+3
(6.2.7)
O L 1 [X (s)℄ e nai so me to jroisma twn ant strofwn ML twn apl¸n klasmtwn
oi opo oi br skontai eÔkola me th bo jeia tou zeÔgou (4) tou P naka 6.2. Telik o
zhtoÔmeno ant strofo ML e nai
x(t) = 2e t + e 3t u(t) (6.2.8)
Pardeigma 6.2.2 (Ìparxh pollapl pragmatik r za ston paronomast )
Na upologiste o ant strofo ML th sunrthsh X (s), ìpou
s2 3s + 1
X (s) =
(s 1)2(s 2) ; me PS <e[s℄ > 2 (6.2.9)
Enìthta 6.1 Ant strofo Metasqhmatismì Laplace 215
LÔsh O paronomast èqei m a dipl pragmatik r za thn 1 = 1 kai m a apl thn
2 = 2. H X (s) mpore na ekfraste w jroisma apl¸n klasmtwn
C11
X (s) = + C12
s 1 (s 1)2
+ sC212 (6.2.10)
kai upolog zoume ti stajerè C11 , C12 kai C21
d s2 3s + 1
C11 =
1 d
(s 1)2 X (s) = =2
(2 1)! ds s=1 ds s 2 s=1
s2
kai
C12 = (s 1)2X (s) s=1 = 3s + 1 = 1
s 2 s=1
2
C21 = (s 2)X (s)js=2 = s (s 3s1)+2 1 = 1
s=2
ra
X (s) =
2 + 1 1
s 1 (s 1)2 s 2 (6.2.11)
Me th bo jeia twn zeug¸n ML (4) kai (6) tou P naka 6.2, o L 1 [X (s)℄ prokÔptei
x(t) = tet + 2et e2t u(t) (6.2.12)
Pardeigma 6.2.3 (Ìparxh migadik¸n riz¸n ston paronomast )
Na upologiste o ant strofo ML th sunrthsh X (s), ìpou
s+2
X (s) = 2 ; <e[s℄ > 2
s + 4s + 13
me PS (6.2.13)
LÔsh Oi r ze tou paronomast e nai 1 = 2+3j kai 2 = 2 3j . H X (s) mpore
na ekfraste w jroisma apl¸n klasmtwn
C1
X (s) =
s + 2 3j
+ s + C2 2+ 3j (6.2.14)
Upolog zoume ti stajerè C1 kai C2
s+2 1
C1 = (s + 2 3j )X (s)js= 2+3j = s + 2 + 3j =
s= 2+3j 2
= s +s 2+ 2 3j = 12
kai
C2 = (s + 2 + 3j )X (s)js= 2 3j
s= 2 3j
216 METASQHMATISMOS LAPLACE Keflaio 6
ra
X (s) =
1 1 +1 1
2 s + 2 + 3j 2 s + 2 3j (6.2.15)
O ant strofo ML lìgw th (4) tou P naka 6.2 e nai
x(t) = 1 he (2+3j)t + e (2 3j)t i u(t)
2
= 1 e 2t e 3jt + e3jt u(t)
2
= e 2t os(3t)u(t) (6.2.16)
Sthn per ptwsh pou èqoume migadikè r ze , mporoÔme na akolouj soume ènan enal-
laktikì trìpo, o opo o bas zetai sta zeÔgh ML (9), (10), (11) kai (12) tou P naka 6.2.
H X (s) grfetai
s+2
X (s) = 2
s + 4s + 13
= (s +s2)+22+ 32 (6.2.17)
Me th bo jeia tou zeÔgou ML (11) sto P naka 6.2 parathroÔme ìti
x(t) = L 1 [X (s)℄ = e 2t os(3t)u(t) (6.2.18)
Sto Sq ma 6.7 fa nontai oi suzuge migadiko pìloi, oi opo oi br skontai sto arnhtikì
hmiep pedo, to mhdenikì kai h perioq sÔgklish tou X (s), h opo a perièqei to fan-
tastikì xona. ParathroÔme ìti to s ma x(t) e nai aitiatì afoÔ to ped o sÔgklish
e nai to dexiì hmiep pedo kai e nai èna fj non sunhmitonoeidè s ma.
ℑm x(t) = e-2t cos(3t)u(t)
3j 1
e-2t
-2 ℜe 0 t
-2t
e
-3j
-1
(á) (â)
Sq ma 6.7 (a) H perioq sÔgklish , oi pìloi kai to mhdenikì tou ML (b) tou s mato x(t)
sto Pardeigma 6.2.3.
6.3 O MONOPLEUROS METASQHMATISMOS LAPLACE
Sthn Pargrafo 6.1 or same ton ML. Sthn pargrafo aut ja or soume to Monì-
()
pleuro Metasqhmatismì Laplace (MML) tou s mato x t . Ja estisoume sta basik
shme a tou MML kai kur w se aut pou ton diaforopoioÔn apì to ML.
Enìthta 6.2 O monìpleuro Metasqhmatismì Laplace 217
H diafor metaxÔ twn dÔo metasqhmatism¸n entop zetai sta ìria olokl rwsh
tou orismoÔ. An to ktw ìrio sto olokl rwma sth sqèsh (6.1.3) e nai to mhdèn, tìte
or zetai o monìpleuro metasqhmatismì Laplace
Z 1
L [x(t)℄ = X (s) = X +(s) x(t)e st dt (6.3.1)
0
To sÔnolo twn migadik¸n arijm¸n s = +
j! pnw sto opo o uprqei kai or zetai h
X( )
s , ìpou dhlad to ant stoiqo olokl rwma orismoÔ sugkl nei, onomzetai perioq
sÔgklish (PS) th s. X( )
Profan¸ , an dÔo s mata e nai diaforetik gia t < kai sa gia t , tìte èqoun 0 0
ton dio MML kai diaforetikì ML. Gia aitiat s mata, x t gia t < , o ML kai o ( )=0 0
MML sump ptoun. Me lla lìgia o monìpleuro metasqhmatismì Laplace tou s ma-
()
to x t taut zetai me ton amf plero metasqhmatismì Laplace tou s mato x t u t . () ()
() ()
Efìson to s ma x t u t e nai aitiatì s ma, h perioq sÔgklish tou monìpleurou
metasqhmatismoÔ Laplace e nai pnta to mègisto dexiì hmiep pedo pou den perièqei
pìlou tou sust mato . H diafor metaxÔ twn dÔo orism¸n e nai ousiastik , kai
ìpw ja doÔme, parèqei sto MML th dunatìthta ep lush diaforik¸n exis¸sewn, oi
opo e èqoun mh mhdenikè arqikè sunj ke . ätan to s ma perièqei sunart sei Æ t , ()
o orismì tou or ou sto mhdèn apaite prosoq , diìti to apotèlesma ja exarttai
en prosegg zoume to mhdèn apì arister apì dexi. Sthn per ptws ma jew-
roÔme to ìrio apì arister t ( !0 )
kai epomènw h sunrthsh Æ t emperièqetai ()
()
sto olokl rwma. En den uprqoun sunart sei Æ t , to parapnw sqìlio e nai neu
shmas a .
O monìpleurou metasqhmatismì Laplace èqei parìmoie idiìthte me to metasqh-
matismì Laplace ti opo e parajètoume ston P naka 6.3. M a diafor uprqei sthn
idiìthta parag¸gish sto qrìno.
(1) Parag¸gish sto ped o tou qrìnou
An x t ( ) L! X ( )
s me PS R kai h sunrthsh (s ma) x t e nai paragwg simh gia ()
t0 , tìte o metasqhmatismì Laplace th parag¸gou th e nai
dx(t) L
dt
! sX (s) x(0 ) me thn dia PS R
Apìdeixh
O monìpleuro metasqhmatismì Laplace th parag¸gou e nai
dx(t) 1 dx(t)
Z
L dt
= dt
e st dt
0 Z 1
= [x(t)e st ℄1+
0 sx(t)e st dt
0 Z 1
= tlim
!1
[x(t)e ℄ tlim
!0
[x(t)e ℄ + s
st st x(t)e st dt
0
218 METASQHMATISMOS LAPLACE Keflaio 6
ìpou qrhsimopoi jhke h mèjodo olokl rwsh kat pargonte . Dedomènou ìti h
x(t) e nai ekjetiktxh ja uprqei M > kai t0 gia ta opo a 0
= jx(t)j e t < Meat e t = Me( a)t
x(t)e st
gia kje t
R1t0 . àtsi, limt!1 [x(t)e st ℄ = 0 ìtan <e[s℄ > a. Ep sh , e nai limt!0 [x(t)e st ℄ =
x(0 ) kai 0 x(t)e st dt = X (s). Epomènw , èqoume
dx(t) L
! sX (s) x(0 ); <e[s℄ > a
dt
ParathroÔme ìti, epeid to ktw ìrio tou oloklhr¸mato orismoÔ tou monìpleurou
metasqhmatismoÔ Laplace e nai to mhdèn, sto monìpleuro metasqhmatismì Laplace th
()
parag¸gou tou x t uprqei h arqik sunj kh x . (0 )
O monìpleuro metasqhmatismì Laplace th parag¸gou txh n e nai
dn x(t) L n dx(t) n 1
dtn
! s X (s) sn 1x(0 ) sn 2
dt t=0
d dtnx(1t)
t=0
M a deÔterh diafor uprqei sthn idiìthta th olokl rwsh sto qrìno.
(2) Olokl rwsh sto ped o tou qrìnou
An x(t) L! X (s) me PS R, tìte
y(t) =
Z t
x( ) d L!
1 X (s) + 1 Z 0 x( ) d = X (s) + y(0 )
1 s s 1 s s
Apìdeixh
ParathroÔme ìti dt
dy(t) = () ()
x t . H y t e nai paragwg simh, an h x t e nai suneq . ()
Ep sh e nai ekjetik txh . Qrhsimopoi¸nta thn idiìthta th parag¸gish sto
ped o tou qrìnou èqoume
dy(t)
L dt
= L[x(t)℄ ) sY (s) y(0 ) = X (s)
Epomènw , èqoume
Y (s) = X s(s) + y(0s )
ParathroÔme ìti sto monìpleuro metasqhmatismì Laplace tou oloklhr¸mato tou
()
x t uprqei h arqik sunj kh y . (0 )
Tèlo , o monìpleuro metasqhmatismì Laplace parèqei th dunatìthta na prosdio-
risje h arqik tim x + kai to (0 ) lim ()
t!1 x t me to je¸rhma th arqik kai telik
tim kai ìqi mèsw tou id ou tou s mato x t . ()
Enìthta 6.3 Efarmogè twn Metasqhmatism¸n Laplace 219
(3) Je¸rhma arqik tim
()
àstw to s ma x t , to opo o den perièqei kroustikè sunart sei sto t = 0, me
MML X( ) <[℄
s kai perioq sÔgklish e s > 0 . Tìte isqÔei
x(0+ ) = slim
!1 sX (s) (Arqik tim )
ìpou x + e nai h tim tou s mato
(0 ) x(t) ìtan h metablht t plhsizei to mhdèn apì
jetikè timè .
(4) Je¸rhma telik tim
()
àstw to s ma x t , me MML X( ) <[℄
s kai perioq sÔgklish e s > 0 . An h s s X( )
e nai analutik sunrthsh sto fantastikì xona kai sto dexiì migadikì hmiep pedo,
tìte isqÔei
xt lim ( ) = slim
t!1
X( )
s s (Telik tim )
!0
To je¸rhma th telik tim qrhsimopoie tai sth melèth grammik¸n susthmtwn
gia ton prosdiorismì twn tim¸n isorrop a kai th mìnimh katstash tou .
Oi idiìthte autè apoteloÔn th dÔnamh tou monìpleurou metasqhmatismoÔ Laplace
giat èqoume th dunatìthta na epilÔoume grammikè diaforikè exis¸sei me stajeroÔ
suntelestè kai na analÔoume GQA sust mata ta opo a den br skontai se hrem a.
6.4 EFARMOGES TWN METASQHMATISMWN LAPLACE
Sthn enìthta aut ja anaptÔxoume ti efarmogè twn metasqhmatism¸n Laplace. Ei-
dikìtera ja axiopoi soume th dunatìthta pou parèqei o MML gia thn ep lush di-
aforik¸n exis¸sewn oi opo e èqoun mh mhdenikè arqikè sunj ke kai ja thn efar-
mìsoume sth melèth GQA susthmtwn. Telei¸nonta , ja exetsoume th sqèsh pou
uprqei metaxÔ th jèsh twn pìlwn th sunrthsh metafor sto migadikì ep pedo
me ti idiìthte th aitiìthta kai th eustjeia enì GQA sust mato .
6.4.1 Ep lush grammik diaforik ex swsh me th bo jeia MML
Lìgw th idiìthta tou MML pou anafèretai sthn pargwgo kai to olokl rwma
mia sunrthsh , èqoume th dunatìthta na epilÔoume grammikè diaforikè exis¸sei
me stajeroÔ suntelestè . H genik morf mia grammik diaforik ex swsh me
stajeroÔ suntelestè e nai
dn x(t) dn 1 x(t)
an
dtn
+ a n 1
dtn 1
+ ::: + a1 dxdt(t) + a0 x(t) = g(t) (6.4.1)
me arqikè sunj ke
dx(t) dn 1 x(t)
x(t)jt=0 = b0 ; ; = b1 ; ::: ; =b
dtn 1 t=0 n 1
(6.4.2)
dt t=0
220 METASQHMATISMOS LAPLACE Keflaio 6
PINAKAS 6.9 Oi idiìthte tou Monìpleurou MetasqhmatismoÔ Laplace
Idiìthta S ma Monìpleuro ML
x(t) X (s)
x1 (t) X1(s)
x2 (t) X2(s)
Grammikìthta ax1 (t) + bx2 (t) aX1 (s) + bX2 (s)
Metatìpish sth suqnìthta es0 t x(t) X (s s0)
x(at); a > 0 1 s
Klimkwsh sto qrìno aX a
Sunèlixh
x1 (t) = x2 (t) = 0 gia t < 0 x1 (t) ? x2 (t) X1(s) X2 (s)
dt x(t) sX (s) x(0 )
Parag¸gish sto qrìno? d
Parag¸gish sth suqnìthta tx(t) ds X (s)
d
Rt X (s) + y(0 )
Olokl rwsh sto qrìno? 1 x( ) d s s
Arqik tim x(0+ ) = lims!1 sX (s)
Telik tim limt!1 x(t) = lims!0 sX (s)
? ätan oi arqikè sunj ke e nai mhdèn, oi idiìthte th parag¸gish kai ti olo-
kl rwsh tou MML e nai oi die me ti ant stoiqe tou ML.
Ta b mata pou akoloujoÔme gia thn ep lush th e nai:
1. Pa rnoume to MML kai sta dÔo mèlh th ex swsh . Lìgw th grammikìthta ,
o MML tou aristeroÔ mèrou isoÔtai me to jroisma twn MML twn epimèrou
ìrwn.
2. LÔnoume thn ex swsh pou prokÔptei w pro ton MML X (s) th sunrthsh
()
xt.
3. Br skoume ton L 1 [X (s)℄, dhlad th lÔsh x(t).
Efarmìzoume ta parapnw sto pardeigma pou akolouje .
Pardeigma 6.4.1
Na epiluje h diaforik ex swsh
dx(t)
Z t
dt
+ 3 x(t) + 2 x( ) d = u(t) (6.4.3)
1
Enìthta 6.3 Efarmogè twn Metasqhmatism¸n Laplace 221
me arqikè sunj ke x(0) = 2 kai R 01 x() d = 0.
LÔsh Efarmìzonta MML kai sta dÔo mèrh th (6.4.3) kai qrhsimopoi¸nta ti
idiìthte pou anafèrontai sthn pargwgo, to olokl rwma mia sunrthsh kai ti
arqikè sunj ke èqoume
2 ( s) 2 0 1
Z
sX (s) x(0 ) + 3X (s) +
X +s x( ) d =
s s 1
sX (s) 2 + 3X (s) + 2Xs(s) = 1s
X (s) = s2 2+s 3+s 1+ 2 = s +1 1 + s +3 2
kai telik
(6.4.4)
Efìson x(t) = 0, gia t < 0, oi X (s) kai X (s) taut zontai. ra, o ant strofo MML
th X (s) d nei th lÔsh th diaforik ex swsh pou e nai
x(t) = L 1 [X (s)℄ = e t + 3e 2t u(t) (6.4.5)
6.4.2 H qr sh tou metasqhmatismoÔ Laplace sthn anlush GQA susthmtwn
Sthn pargrafo aut ja estisoume se efarmogè tou ML sth melèth GQA susth-
mtwn basismènoi sth gn¸sh twn ant stoiqwn shmtwn eisìdou kai exìdou. Sto Ke-
()
flaio 2, e dame ìti h èxodo y t enì GQA sust mato sundèetai me thn e sodì tou
()
x t me to olokl rwma th sunèlixh
Z 1 Z 1
y(t) = x(t) ? h(t) = h( )x(t ) d = h(t )x( ) d (6.4.6)
1 1
Lìgw th idiìthta th sunèlixh tou ML èqoume
Y (s) = H (s) X (s) me PS toulqiston thn RH \ RX (6.4.7)
()
ìpou H s e nai h sunrthsh metafor tou sust mato (Enìthta 2.5).
()
ParathroÔme ìti h H s d netai w phl ko twn ML th exìdou tou sust mato
pro ton ML th eisìdou tou sust mato , dhlad H s Y s =X s . ( )= ( ) ( )
Sto Keflaio 2, ep sh , èqoume dei ìti an ekmetalleujoÔme ti fusikè sqèsei pou
uprqoun metaxÔ twn stoiqe wn enì sust mato GQA, katal goume se mia grammik
diaforik ex swsh me stajeroÔ suntelestè , h opo a èqei th genik morf
N
X dk y(t) M
X dk x(t)
ak
dtk
= bk
dtk
(6.4.8)
k=0 k=0
222 METASQHMATISMOS LAPLACE Keflaio 6
ìpou ak ; k =012
; ; ; :::; N kai bk ; k =012
; ; ; :::; M pragmatikè stajerè , oi
opo e perigrfoun to sÔsthma. Efarmìzonta to ML kai sta dÔo mèlh th (6.4.8),
katal goume sth sqèsh
PM
Y (s) k
H (s) =
X (s)
= PNk=0 bk sk (6.4.9)
k=0 ak s
Apì thn (6.4.9) upolog zoume th sunrthsh metafor tou sust mato me th bo jeia
twn suntelest¸n ak ; k =0 1 2
; ; ; :::; N kai bk ; k ; ; ; :::; M . =0 1 2
Parat rhsh: Tìso h (6.4.7) ìso kai h (6.4.9), pro pojètoun ìti gnwr zoume ti
emplekìmene sunart sei gia t > 1
, me lla lìgia gia mhdenikè arqikè sunj ke .
En oi emplekìmene sunart sei e nai aitiatè kai oi arqikè sunj ke e nai mhdèn,
ja ftsoume sti (6.4.7) kai (6.4.9) kai me ton MML, pou se aut n thn per ptwsh
taut zetai me ton ML. Dhlad , h sunrthsh metafor H s èqei nìhma mìno ktw ()
apì mhdenikè arqikè sunj ke .
()
Ton zetai ìti, epeid h sunrthsh H s qarakthr zei to sÔsthma, ja prèpei loipìn
na mhn exarttai apì ti arqikè sunj ke sti opo e br sketai to sÔsthma.
Pardeigma 6.4.2
Na upologiste h sunrthsh metafor kai h kroustik apìkrish tou kukl¸mato tou
Sq mato 6.8.
L
Ä
i(t)
õin(t)=V R õR(t)
Sq ma 6.8 To kÔklwma tou
Parade gmato 6.4.2.
LÔsh Efarmìzoume to deÔtero kanìna Kirchhoff sto kÔklwma kai pa rnoume
di(t)
L
dt
+ Ri(t) = in(t) (6.4.10)
kai epeid R = iR; d dtR = R dtdi èqoume
d R (t) R
dt
+ R(t) = R in (t)
L L
(6.4.11)
àtsi, efarmìzonta metasqhmatismì Laplace kai sta dÔo mèlh th (6.4.11) pa rnoume
R R
sVR (s) + VR (s) = Vin (s)
L L
apì thn opo a prosdior zetai h sunrthsh metafor tou kukl¸mato
R=L
H (s) =
s + R=L
(6.4.12)
Enìthta 6.3 Efarmogè twn Metasqhmatism¸n Laplace 223
To sÔsthma èqei èna pìlo sto shme o R=L, me apotèlesma to PS e nai <e[s℄ > R=L,
afoÔ to sÔsthma w fusikì sÔsthma prèpei na e nai aitiatì. H kroustik apìkrish
tou sust mato e nai
R
h(t) = L 1 [H (s)℄ = e (t)
Rt
L u (6.4.13)
L
An den èqoume arqikè sunj ke , tìte h èxodo sust mato prosdior zetai me th bo -
jeia tou jewr mato th sunèlixh sto qrìno gnwr zonta th sunrthsh metafor
tou sust mato kai to ML tou s mato eisìdou.
Pardeigma 6.4.3
D netai to kÔklwma tou Sq mato 6.8. An to kÔklwma arqik hreme kai sthn e sodì tou,
kat th qronik stigm t0 = 0, efarmìsoume phg stajer tsh V , na prosdioriste
h tsh sta kra th ant stash , R (t), se sunrthsh me to qrìno.
LÔsh Epeid h e sodo e nai phg stajer tsh , h opo a efarmìzetai th qronik
stigm t0 = 0, èqoume
V
in (t) = V u(t) L! Vin (s) = ; <e[s℄ > 0 (6.4.14)
s
To sÔsthma br sketai se hrem a kai o ML th exìdou VR (s) upolog zetai me th bo jeia
tou jewr mato th sunèlixh pou d nei
VR (s) = H (s)Vin (s)= s[sV+(R=L )
(R=L)℄
R C1 C2
= V L s + s + (R=L)
1 1
= V s s + (R=L) ; <e[s℄ > 0 (6.4.15)
ìpou h VR (s) èqei analuje se apl klsmata kai oi stajerè C1 = L=R kai C2 =
L=R èqoun prokÔyei me to gnwstì trìpo. H èxodo R (t) upolog zetai me ton an-
t strofo ML th VR (s), dhlad
R (t) = V u(t) V e (t) = V 1 u(t)
Rt Rt
L u e L (6.4.16)
An èqoume arqikè sunj ke , tìte sth diaforik ex swsh (6.4.8), lìgw twn idiot twn
th parag¸gish sto qrìno kai th olokl rwsh sto qrìno pou èqei o monìpleuro
metasqhmatismì Laplace, sumperilambnoume ti arqikè sunj ke kai sthn Y s ( )=
() ()
H s X s emfan zetai kai èna epiplèon ìro o opo o proèrqetai apì ti arqikè
sunj ke .
224 METASQHMATISMOS LAPLACE Keflaio 6
Pardeigma 6.4.4
D netai to kÔklwma tou Sq mato 6.8. An h tsh sta kra th ant stash arqik e nai
R (0 ) kai kat th qronik stigm t0 = 0, efarmìsoume phg stajer tsh V , na
prosdioriste h tsh sta kra th ant stash , R (t), se sunrthsh me to qrìno.
LÔsh Epeid to sÔsthma èqei arqikè sunj ke , efarmìzonta to monìpleuro metasqh-
matismì Laplace kai sta dÔo mèlh th diaforik ex swsh (6.4.11), pou qarakthr zei
to kÔklwma, enswmat¸noume ti arqikè sunj ke kai èqoume
sVR (s) R (0 ) + RL VR(s) = R
V (s)
L in
R R
s+ VR (s) = V (s) + R (0 )
L L in
VR (s) = R=L
Vin (s) +
1 R(0 )(6.4.17)
s + R=L s + R=L
Epeid to s ma eisìdou e nai aitiatì, o MML tou e nai so me to ML
V
in (t) = V u(t) L! Vin (s) = (6.4.18)
s
èqoume
VR (s) = s +R=L V
+ 1 (0 )
R=L s s + R=L R
VR (s) = V 1s s + 1R=L + R (0 ) s + 1R=L (6.4.19)
Sugkr nonta thn (6.4.15) me thn (6.4.19) parathroÔme ìti o VR (s) perièqei ènan epi-
plèon ìro o opo o exarttai apì thn arqik sunj kh sthn opo a br sketai to sÔsthma.
H èxodo R (t) upolog zetai me ton ant strofo ML th VR (s), dhlad ,
R (t) = V 1 u(t) + R (0 )e (t)
Rt Rt
e L L u (6.4.20)
Pardeigma 6.4.5
D netai to kÔklwma tou Sq mato 6.8. Na breje o metasqhmatismì Laplace th exìdou
ìtan to s ma eisìdou e nai x(t) = te2t u(t). D netai ìti h stajer qrìnou tou kukl¸-
mato e nai = L=R = 0; 2se kai to sÔsthma br sketai se hrem a.
LÔsh Me th bo jeia twn (6.4.12) kai (6.1.23) br skontai oi ML th kroustik
apìkrish tou sust mato kai tou s mato eisìdou x(t)
H (s) =
5 ; <e[s℄ > 5
s+5
me
kai
X (s) =
1
(s 2)2 ; me <e[s℄ > 2
Enìthta 6.3 Efarmogè twn Metasqhmatism¸n Laplace 225
ìpou qrhsimopoi jhke ìti h stajer qrìnou tou sust mato e nai 0; 2se . Me th bo -
jeia tou jewr mato th sunèlixh br sketai o ML th exìdou tou sust mato
Y (s) =
5
(s 2)2(s + 5) ; me <e[s℄ > 2 (6.4.21)
ParathroÔme ìti to s ma eisìdou den e nai apolÔtw oloklhr¸simo, epomènw den
uprqei o MF tou kai w ek toÔtou den e nai dunat h qr sh twn mejìdwn pou qrhsi-
mopoioÔn metasqhmatismoÔ Fourier gia th lÔsh tou probl mato .
Pardeigma 6.4.6
Na prosdiorisje h arqik kai h telik tim tou s mato x(t) tou opo ou o monìpleuro
metasqhmatismì Laplace e nai
X (s) = s7(ss++102)
LÔsh Efarmìzonta to je¸rhma th arqik tim br skoume ìti
x(0+ ) = slim 7s + 10
!1 s s(s + 2)
= slim 7s + 10
!1 (s + 2)
= 7 (6.4.22)
Efarmìzonta to je¸rhma th telik tim br skoume ìti
x(1) = slim s
7s + 10
!0 s(s + 2)
= slim 7s + 10
!0 (s + 2)
= 5 (6.4.23)
Shmei¸netai ìti h arqik kai h telik tim mpore na breje afoÔ prosdiorisje h
analutik morf tou s mato kai sth sunèqeia prosdiorisje h arqik kai h telik
tim X (s) e nai o MML tou s mato x(t) =
tou s mato . Prgmati, parathroÔme ìti o
5u(t) + 2e 2tu(t) apì thn opo a epalujeÔoume ìti x(0+ ) = 7 kai x(1) = 5.
6.4.3 Parathr sei gia thn perioq sÔgklish tou metasqhmatismoÔ Laplace
äpw gnwr zoume, o ML e nai sunrthsh th migadik metablht s. Upenjum zoume
()
ìti oi r ze tou arijmht N s sthn (6.1.16) onomzontai mhdenik th H s . Pro- ()
()
fan¸ sta shme a aut h H s mhden zetai. Ep sh , oi r ze tou paronomast D s , ()
()
ìpou h H s den or zetai, onomzontai pìloi th H s . ()
Gia na e nai èna sÔsthma aitiatì prèpei h perioq sÔgklish na e nai to dexiì h-
miep pedo tou migadikoÔ epipèdou me sÔnoro th gramm pou e nai kjeth ston prag-
matikì xona sth jèsh < [ ℄j
e ak max , ìpou ak me k =1 2
; ; ::: e nai oi pìloi th H s ()
226 METASQHMATISMOS LAPLACE Keflaio 6
kai max sumbol zei ton pìlo me to mègisto pragmatikì mèro (blèpe Pardeigma
6.1.3). Me lla lìgia, to ped o sÔgklish enì aitiatoÔ sust mato e nai to mègisto
dunatì dexiì hmiep pedo, to opo o den perièqei pìlou th H s . ()
()
An o bajmì tou poluwnÔmou tou N s e nai megalÔtero so apì to bajmì tou
()
poluwnÔmou D s , tìte, prin analÔsoume se apl klsmata, prèpei na ektelèsoume
() () ()
th dia resh N s =D s . Sthn per ptwsh aut h H s perilambnei ìrou th morf
0
sk ; k > . A jewr soume t¸ra èna sÔsthma sth sunrthsh metafor , H s , tou()
()
opo ou uprqei o ìro s. Tìte, an h e sodo tou sust mato e nai h u t , h opo a
1
èqei ML so me =s, h èxodo tou sust mato ja e nai h y t L 1 s =s
( )= [ (1 )℄ = ( )
Æt.
ParathroÔme ìti, h èxodo tou sust mato den e nai fragmènh, se ant jesh me thn
e sodì tou h opo a e nai fragmènh. Me bsh ta parapnw katal goume se èna pr¸to
sumpèrasma:
“ìtan o bajmì tou poluwnÔmou tou N (s) e nai megalÔtero apì to bajmì tou poluwnÔ-
()
mou D s to sÔsthma den e nai FEFE eustajè ”.
Sth sunèqeia ja doÔme ìti pèra apì th sqèsh twn bajm¸n twn poluwnÔmwn N s ()
() ()
kai D s , h jèsh twn pìlwn th H s kajor zei thn eustjeia tou sust mato . Sto
Pardeigma 2.4.1 èqoume dei ìti gia na e nai èna sÔsthma FEFE eustajè prèpei h
kroustik apìkris tou na e nai apìluta oloklhr¸simh. Sthn per ptwsh ìmw aut
()
uprqei o MF th , dhlad h apìkrish suqnìthta tou sust mato H ! . Gnwr zoume,
ep sh , ìti gia na uprqei o MF prèpei to ped o sÔgklish tou ML na perièqei to
fantastikì xona. àtsi:
“gia na e nai to sÔsthma FEFE eustajè , prèpei o fantastikì xona na perièqetai
sto ped o sÔgklish tou ML”.
Sunduzonta t¸ra ta parapnw katal goume ìti gia na e nai èna sÔsthma tautì-
qrona aitiatì kai FEFE eustajè prèpei
1. h perioq sÔgklish na e nai to dexiì hmiep pedo tou migadikoÔ epipèdou me
sÔnoro th gramm pou e nai kjeth ston pragmatikì xona sth jèsh < [ ℄j
e ak max
kai
2. o fantastikì xona na perièqetai sto ped o sÔgklish tou ML.
()
Genikìtera, h jèsh twn pìlwn tou ML X s enì s mato sto migadikì ep pedo- s
prosdior zei th sumperifor tou s mato . Eidikìtera, isqÔoun ta ex
1. <[℄ 0
Aplo pìloi sto arnhtikì hmiep pedo ( e s < kai m s = [ ℄=0 ) tou epipèdou-s
antistoiqoÔn se s mata ta opo a sto ped o tou qrìnou e nai pollaplasiasmèna
me e jajt pou fj nei ekjetik pro to mhdèn, kaj¸ t !1 . Se ant jesh, aplo
Enìthta 6.3 Efarmogè twn Metasqhmatism¸n Laplace 227
pìloi sto dexiì hmiep pedo tou s antistoiqoÔn se s mata pollaplasiasmèna me
ejajt , pou auxnetai ekjetik pro to peiro kaj¸ t ! 1.
2. Aplo pìloi sto fantastikì xona antistoiqoÔn se s mata twn opo wn to plto
paramènei stajerì me to qrìno, en¸ pollaplo pìloi sto fantastikì xona
antistoiqoÔn se s mata pollaplasiasmèna me tn .
3. Migadiko suzuge pìloi antistoiqoÔn se s mata pou uf stantai talntwsh,
toi perièqoun hmitonikoÔ ìrou (Pardeigma 6.2.3). An to pragmatikì mèro
twn suzug¸n pìlwn e nai mhdèn, tìte èqoume ame wte talant¸sei , en¸ an to
pragmatikì mèro e nai mh mhdenikì, èqoume talant¸sei ekjetik aÔxouse , an
oi suzuge pìloi br skontai sto jetikì hmiep pedo, ekjetik fj nouse , an oi
suzuge pìloi br skontai sto arnhtikì hmiep pedo.
Parìmoia isqÔoun kai gia thn kroustik apìkrish enì sust mato anloga me th
()
jèsh twn pìlwn th H s sto migadikì ep pedo. Sto Sq ma 6.9 paristnontai oi
idiìthte enì aitiatoÔ sust mato kai h sumperifor th kroustik apìkrous
tou, ìpw aut prosdior zetai apì th jèsh twn pìlwn tou sto migadikì ep pedo.
jù
ó
ÅõóôáèÝò ÁóôáèÝò
Sq ma 6.9 Oi idiìthte enì sust mato kai h sumperifor th kroustik tou apìkrish
anloga me th jèsh twn pìlwn th sunrthsh metafor tou sto migadikì ep pedo.
Pardeigma 6.4.7
Na deiqje ìti to kÔklwma tou Sq mato 6.8 e nai eustajè sÔsthma.
LÔsh H sunrthsh metafor tou sust mato èqei èna pìlo sth jèsh (R=L) < 0,
dhlad sto arnhtikì hmiep pedo. Epeid to sÔsthma w pragmatikì sÔsthma e nai
aitiatì, to ped o sÔgklish prèpei na e nai <e[s℄ > R.
L ParathroÔme ìti sto ped o
sÔgklish perièqetai o fantastikì xona , me apotèlesma to sÔsthma na e nai eusta-
jè .
228 METASQHMATISMOS LAPLACE Keflaio 6
Pardeigma 6.4.8
àna GQA sÔsthma èqei sunrthsh metafor
s+1
H (s) =
(s + 2)(s 1) (6.4.24)
Na upologiste h kroustik apìkrish tou sust mato .
LÔsh H H (s) analÔetai se apl klsmata th morf
H (s) =
1 1 +2 1
3s+2 3s 1 (6.4.25)
Gia to GQA sÔsthma tou parade gmato den prosdior zetai h perioq sÔgklish
th sunrthsh metafor . Oi pijanè perioqè sÔgklish e nai oi trei , oi opo e
eikon zontai sto Sq ma 6.10. An to sÔsthma e nai aitiatì, h perioq sÔgklish e nai
<e[s℄ > 1 (Sq ma 6.10a) kai h kroustik apìkrish tou sust mato e nai
1 2 2
h(t) = e + e u(t)
t t
3 3 (6.4.26)
Sto Sq ma 6.11a fa netai h kroustik apìkrish tou sust mato ìtan h perioq sug-
klish e nai <e[s℄ > 1. Sthn per ptwsh aut to sÔsthma e nai aitiatì afoÔ to ped o
sÔgklish e nai to mègisto dexiì hmiep pedo to opo o den perièqei pìlou . To sÔsthma
den e nai ìmw eustajè , afoÔ den perièqetai sto ped o sÔgklish o fantastikì xona .
To ìti to sÔsthma den e nai eustajè fa netai kai apì to ìti h(t) ! 1 ìtan t ! 1.
An to sÔsthma e nai eustajè , h perioq sÔgklish e nai h 2 < <e[s℄ < 1 (Sq ma
6.10b) kai h kroustik apìkrish tou sust mato e nai
1
h(t) = e 2t u(t)
2 etu( t)
3 3 (6.4.27)
ℑm ℑm ℑm
-2 1 ℜe -2 1 ℜe -2 1 ℜe
(á) ( â) (ã)
Sq ma 6.10 Oi pijanè perioqè sÔgklish th sunrthsh metafor gia to sÔsthma tou
Parade gmato 6.4.8 (a) aitiatì (b) eustajè kai (g) mh aitiatì mh eustajè sÔsthma.
Enìthta 6.3 Efarmogè twn Metasqhmatism¸n Laplace 229
h(t) h(t) h(t)
1
3
t
-1
1 0 t
0 t
2
(á) 3 (â) (ã)
Sq ma 6.11 Oi pijanè kroustikè apokr sei gia to sÔsthma tou Parade gmato 6.4.8 (a)
aitiatì, (b) eustajè kai (g) mh aitiatì mh eustajè sÔsthma.
Sto Sq ma 6.11b fa netai h kroustik apìkrish tou sust mato ìtan h perioq sug-
klish e nai 2 < <e[s℄ < 1. Parathre ste ìti to sÔsthma t¸ra e nai eustajè , den
e nai ìmw aitiatì.
Tèlo , an h perioq sÔgklish tou sust mato e nai <e[s℄ < 2 (Sq ma 6.10g), to
sÔsthma den e nai oÔte aitiatì, oÔte eustajè kai h kroustik apìkrish tou sust mato
1e 2 et u( t)
e nai
h(t) = 2t
3 3 (6.4.28)
Sto Sq ma 6.11g fa netai h kroustik apìkrish tou sust mato ìtan h perioq sug-
klish e nai <e[s℄ < 2.
Pardeigma 6.4.9
D netai to grammikì qronik anallo wto sÔsthma tou opo ou h sqèsh s mato eisìdou
exìdou perigrfetai apì th diaforik ex swsh
d d2 d
y(t) + 3y(t) = 2 x(t) + x(t) 2x(t) (6.4.29)
dt dt dt
Na breje h sunrthsh metafor tou antistrìfou sust mato . Gia to sÔsthma autì
uprqei ant strofo sÔsthma to opo o na e nai eustajè kai aitiatì;
LÔsh Lambnonta metasqhmatismì Laplace sta mèlh th (6.4.29) èqoume
Y (s)(s + 3) = X (s)(s2 + s 2)
apì thn opo a br sketai h sunrthsh metafor tou sust mato
Y (s) 2
H (s) =
X (s)
= s s++s 3 2 (6.4.30)
H sunrthsh metafor tou ant strofou sust mato e nai
H (s) = 1
H (s)
ant
s+3
= s +s 2
2
s+3
= (s 1)(s + 2) (6.4.31)
230 METASQHMATISMOS LAPLACE Keflaio 6
To ant strofo sÔsthma èqei dÔo pìlou sto shme a s = 1 kai s = 2. Oi dÔo pìloi
br skontai ekatèrwjen tou fantastikoÔ xona, epomènw den e nai dunatìn to mègisto
dexiì hmiep pedo pou den perièqei pìlou (aitiatì sÔsthma) na perièqei to fantastikì
xona (eustajè sÔsthma). ra, gia to sÔsthma (6.4.30) den uprqei ant strofo sÔsth-
ma to opo o na e nai sugqrìnw aitiatì kai eustajè .
SÔnoyh Kefala ou
Sto keflaio autì or same to metasqhmatismì Laplace kai to monìpleuro metasqh-
matismì Laplace, parousisthkan oi idiìthtè tou kai upolog same tou ML orismèn-
wn basik¸n shmtwn, ta opo a sunantme sth melèth grammik¸n susthmtwn. Sth
sunèqeia prosdior same ton ant strofo ML. E dame ìti, an h morf tou ML e nai
apl , tìte mporoÔme na upolog soume ton ant strofo ML me th bo jeia tou P na-
ka 6.2. An o ML den èqei apl morf all e nai rht sunrthsh, tìte analÔoume th
sunrthsh se apl klsmata kai me th bo jeia twn idiot twn tou ML kai tou P naka
6.2 upolog zoume eÔkola to s ma qwr na katafÔgoume sthn ex swsh antistrof .
Ep sh , sto keflaio autì anaptÔxame difore efarmogè tou ML. Eidikìtera
exetsame th dunatìthta pou èqei o MML na epilÔei grammikè diaforikè exis¸-
sei me stajeroÔ suntelestè , oi opo e den èqoun mhdenikè arqikè sunj ke . H
dunatìthta aut ofe letai sti idiìthte tou MML pou anafèrontai sthn pargwgo
kai to olokl rwma mia sunrthsh . Sth sunèqeia parousisthkan oi efarmogè twn
metasqhmatism¸n Laplace se ìti afor th melèth GQA susthmtwn. Prosdior same
th sunrthsh metafor sust mato apì th diaforik ex swsh pou sundèei thn èxodo
kai thn e sodo tou sust mato , upojètonta ìti oi arqikè sunj ke e nai mhdenikè .
Ep sh , me th bo jeia th diaforik ex swsh , prosdior same to MML th exìdou
sust mato , to opo o mpore na mh br sketai se katstash hrem a , kai antistrè-
fonta to MML prosdior same thn èxodo tou sust mato . Tèlo , parousisthkan ta
sumpersmata pou exgoume apì thn perioq sÔgklish kai th jèsh twn pìlwn th
sunrthsh metafor tou sust mato sto migadikì ep pedo aut aforoÔn sthn eu-
stjeia kai sthn aitiìthta tou sust mato kaj¸ kai sth sumperifor th kroustik
apìkris tou.
PROBLHMATA
6.1 Na upologiste to aitiatì s ma pou èqei metasqhmatismì Laplace
a) X (s) = s23+4 s+7
s+3 X (s) = s23s2s5 3
b)
g) X (s) = s2 +4
s s+2
d) X (s) = s2 +4 s+13
e) X (s) = 2ss3 +4
2 +12 2 +5s+15
s st) X (s) = s3 +3s2
s
6.2 Na upologiste h sunèlixh twn shmtwn x1 (t) = u(t) u(t 1) kai x2 (t) =
u(t) u(t 2) me th bo jeia th idiìthta th sunèlixh tou metasqhmatismoÔ
Laplace.
Enìthta 6.4 Probl mata 231
6.3 Na brejoÔn ta aitiat s mata twn opo wn oi metasqhmatismo Laplace e nai
X1 (s) = 2
1 X2 (s) =
s+1
s + 3s + 2 (s + 3)(s2 + 4s + 5)
kai (6.4.32)
6.4 Gia to kÔklwma RLC pou perigrfetai sto Sq ma 6.12.
1. Na prosdioriste h grammik diaforik ex swsh h opo a sundèei thn e sodo
()
tou kukl¸mato in t kai thn èxodì tou o t . ()
2. Na prosdioriste h sunrthsh metafor tou sust mato . E nai to sÔsth-
ma eustajè ;
3. An h e sodo tou kukl¸mato e nai in t e 3t u ()= (t) me th bo jeia tou
metasqhmatismoÔ Laplase, na upolog sete thn èxodo 0 (t) gia t > 0, ìtan
oi arqikè sunj ke e nai kai
d o (t)
(0 ) = 1 = 2.
o dt t=0
R=3Ù
A B
i(t)
õin(t) C=0,5F õï(t)
L=1H
Ä Ã Sq ma 6.12 To kÔklwma tou Probl mato 6.4.
6.5 H sunrthsh metafor , enì GQA aitiatoÔ sust mato , e nai
s+1
H (s) = 2
s + 2s + 2
(6.4.33)
Na upologiste h apìkrish y (t) tou sust mato ìtan to s ma eisìdou x (t) d ne-
tai apì thn
x(t) = e jtj ; 1<t<1 (6.4.34)
Na prosdior sete to ped o sÔgklish kje for pou parousizetai metasqhma-
tismì Laplace.
6.6 H apìkrish enì grammikoÔ qronik anallo wtou sust mato sto s ma
x(t) = u(t)
e nai to s ma
y(t) = (1 e t te t )u(t)
Me th bo jeia tou metasqhmatismoÔ Laplace na breje h e sodo tou sust mato
ìtan to s ma exìdou e nai
y1 (t) = (2 3e t + e 3t )u(t)
232 METASQHMATISMOS LAPLACE Keflaio 6
6.7 Sthn e sodo enì GQA sust mato me kroustik apìkrish
h(t) = e 2t u(t)
Efarmìzetai to s ma
x(t) = e 3t u(t)
Na breje to s ma exìdou y(t), tou sust mato
1. ìtan to sÔsthma arqik hreme kai
2. ìtan y(0 ) = 1.
6.8 D netai to eustajè grammik anallo wto sÔsthma me sunrthsh metafor
H (s) =
2
2 s
1. Na breje h kroustik apìkrish h(t), tou sust mato .
2. Me th bo jeia tou metasqhmatismoÔ Laplace na breje h èxodì tou ìtan
to s ma eisìdou e nai
x(t) = 2e 2t u(t)
6.9 D netai to RC se seir kÔklwma tou Sq mato 6.13. Na upologistoÔn
1. h sunrthsh metafor H (s) tou kukl¸mato ,
2. h kroustik apìkrish h(t) tou kukl¸mato kai
3. h suqnìthta - 3 dB.
-6
C=10 F
6
õin(t) R=10 Ù õï(t)
i(t)
Sq ma 6.13 To kÔklwma tou Probl mato 6.9.
Bibliograf a
6.1 S. Jeodwr dh , K. Mpermper dh , L. Kof dh , “Eisagwg sth Jewr a Shmtwn
kai Susthmtwn”, Tupwj tw - Gi¸rgo Dardanì , Aj na 2003.
6.2 A. Mrgarh , “S mata kai Sust mata SuneqoÔ kai DiakritoÔ Qrìnou ”, Ekdì-
sei Tziìla 2012.
6.3 S. Haykin, B. Veen, “Signal and Systems”, John & Wiley Sons, Inc. 2003
6.4 A. V. Oppenheim, R. S. Willsky, I. T. Young, “Signal and Systems”, Prentice -
Hall Inc., N. Y., 1983.
ÊÅÖÁËÁÉÏ 7
ÌÅÔÁÓ×ÇÌÁÔÉÓÌÏÓ Z
O metasqhmatismì z e nai o ant stoiqo Laplace gia s mata diakritoÔ qrìnou kai
apotele gen keush tou metasqhmatismoÔ Fourier diakritoÔ qrìnou.
Skopì tou Kefala ou e nai na or sei ton amf pleuro metasqhmatismì z, apl
metasqhmatismì z (Mz) kai to monìpleuro metasqhmatismì z (MMz), na perigryei
ti basikè tou idiìthte kai na upolog sei tou ant stoiqou metasqhmatismoÔ stoi-
qeiwd¸n shmtwn pou antimetwp zoume sth melèth grammik¸n susthmtwn diakritoÔ
qrìnou. Ep sh , sto Keflaio autì ja parousiaste h dunatìthta pou èqei o monì-
pleuro metasqhmatismì z na epilÔei exis¸sei diafor¸n, oi opo e èqoun mh mh-
denikè arqikè sunj ke kai sth sunèqeia ja ekmetalleutoÔme th dunatìthta aut
gia th melèth GQA susthmtwn. Tèlo , skopì tou Kefala ou e nai na anade xei th
sqèsh pou uprqei metaxÔ th aitiìthta , th eustjeia enì GQA sust mato , th
perioq sÔgklish th sunrthsh metafor tou kai th jèsh twn pìlwn aut
sto migadikì ep pedo.
Eisagwg
H e sodo kai h èxodo enì GQA sust mato diakritoÔ qrìnou sundèontai me m a
ex swsh diafor¸n me stajeroÔ suntelestè . àtsi gia na prosdior soume thn èxodo
enì sust mato an gnwr zoume thn e sodì tou ja prèpei na epilÔoume thn ant stoiqh
ex swsh diafor¸n. Sto Keflaio 2 parathr same ìti mporoÔme na upolog soume thn
èxodo enì sust mato an gnwr zoume thn e sodì tou, me th bo jeia tou ajro smato
th sunèlixh . Sto Keflaio 5 or same to MF diakritoÔ qrìnou, o opo o parèqei th
dunatìthta metbash apì to ped o tou qrìnou sto ped o th suqnìthta . H idiìthta
th sunèlixh tou MF metatrèpei to jroisma th sunèlixh se èna aplì ginìmeno
twn antisto qwn metasqhmatism¸n, me thn bo jeia tou opo ou upolog zetai o MF th
exìdou kai sth sunèqeia me ant strofo MF prosdior zetai h èxodo tou sust mato
sto ped o tou qrìnou. O MF loipìn, èdwse m a eÔkolh lÔsh sto prìblhma eÔresh
th exìdou enì sust mato , an gnwr zoume thn e sodì tou kai thn kroustik tou
apìkrish. Dustuq¸ , ìmw , uprqoun poll s mata diakritoÔ qrìnou, ta opo a suqn
sunantme sth prxh, gia ta opo a den uprqei o MF.
234 Metasqhmatismì z Keflaio 7
Sto Keflaio autì ja perigryoume ton Metasqhmatismì z, o opo o metatrèpei
èna s ma diakritoÔ qrìnou se m a analutik sunrthsh migadik metablht . äpw
ja doÔme, se poll apì ta s mata diakritoÔ qrìnou me praktik spoudaiìthta, gia
ta opo a den uprqei o MF diakritoÔ qrìnou, uprqei o Mz kai ètsi dieurÔnetai to
sÔnolo twn shmtwn diakritoÔ qrìnou gia ta opo a mpore na epiteuqje metbash
apì to ped o tou qrìnou sto ped o th suqnìthta .
Sto Keflaio 5, me th bo jeia tou MF upolog same thn èxodo enì GQA sust ma-
to diakritoÔ qrìnou to opo o br sketai arqik se katstash hrem a . Sto Keflaio
autì ja doÔme ìti an to sÔsthma de br sketai se katstash hrem a , o monìpleuro
metasqhmatismì z ma epitrèpei na sumperilboume ti arqikè sunj ke sth ex swsh
diafor¸n kai na prosdior soume thn èxodo tou sust mato .
Tèlo , sto Keflaio autì ja doÔme ìti h qr sh tou migadikoÔ ped ou kai h jèsh
twn pìlwn se autì ma epitrèpei na exgoume basikè idiìthte twn susthmtwn di-
akritoÔ qrìnou, ìpw h aitiìthta kai h eustjeia. Gia ìlou tou parapnw lìgou ,
o metasqhmatismì z apotele èna akìma basikì majhmatikì ergale o gia th melèth
GQA susthmtwn diakritoÔ qrìnou.
7.1 ORISMOI
Sthn Enìthta 2.5.2 èqoume dei ìti an h e sodo enì grammikoÔ qronik anallo wtou
()
sust mato diakritoÔ qrìnou, pou èqei kroustik apìkrish h n , e nai z n tìte to
s ma exìdou e nai
( )= ( )
y n H z zn (7.1.1)
ìpou
1
X
H (z ) = h(n)z n (7.1.2)
n= 1
e nai o metasqhmatismì z th kroustik apìkrish h(n) kai e nai h sunrthsh
metafor tou sust mato .
O metasqhmatismì z antistoiqe sthn akolouj a x (n) th sunrthsh
1
X
X (z ) x(n)z n (7.1.3)
n= 1
H X (z ) e nai migadik
sunrthsh th migadik metablht z =
rej kai onomzetai
amf pleuro metasqhmatismì z tou x n() apl metasqhmatismì z . O monìpleuro
metasqhmatismì z or zetai apì th sqèsh
1
X
X (z) x(n)z n (7.1.4)
n=0
Enìthta 7.1 Orismo 235
H perioq tim¸n tou z , gia ti opo e o metasqhmatismì z èqei peperasmènh tim
kale tai perioq sÔgklish (PS) (region of convergence ROC). Gia eukol a o metasqh-
()
matismì z tou x n merikè forè sumbol zetai w Z x n kai h sqèsh metaxÔ tou [ ( )℄
()
x n kai tou metasqhmatismoÔ tou upodeiknÔetai w
x(n) ! X (z)
Z
(7.1.5)
Parathr sei
1. An o Mz uprqei kai gia timè me r
P1
, dhlad , z =1
ej , tìte X ej = =
()
n= 1 x n e
j n pou e nai o MF diakritoÔ qrìnou th akolouj a x (n),
dhlad ,
X (z )jz=ej = F [x(n)℄ (7.1.6)
Sto ep pedo-s o metasqhmatismì Laplace metatrèpetai se metasqhmatismì Fouri-
=
er sto fantastikì xona (dhlad , s j! ). Sto migadikì ep pedo-z o metasqh-
matismì z metatrèpetai se metasqhmatismì Fourier an to mètro th posìthta
metasqhmatismoÔ z e nai monda z ( = )
ej , dhlad , an br sketai sto mona-
dia o kÔklo tou migadikoÔ epipèdou-z Sq ma 7.1. Me lla lìgia o monadia o
kÔklo tou migadikoÔ epipèdou-z èqei ton dio rìlo me to fantastikì xona sto
ep pedo-s tou metasqhmatismoÔ Laplace.
Met thn parat rhsh aut e nai skìpimo na knoume m a apl allag tou sum-
bolismoÔ tou metasqhmatismoÔ Fourier diakritoÔ qrìnou thn opo a sunantme
pollè forè sta enqeir dia
X (z )jz=ej = F [x(n)℄ = X (ej ) ant tou X ( ) (7.1.7)
ℑm z
Ìïíáäéáßïò
êýêëïò
z=e jÙ
Ù
0 1 ℜe z Sq ma 7.1 z
To migadikì ep pedo- . O metasqhma-
tismì z metatrèpetai se metasqhmatismì Fourier gia
ti timè tou z pou br skontai sto monadia o kÔklo.
2. O Mz sqet zetai me to DTFT kai sthn per ptwsh ìpou h metablht r 6= 1.
Prgmati
1
X
X (rej )= x(n)r n e j n = F x(n) r n (7.1.8)
n= 1
236 Metasqhmatismì z Keflaio 7
( )
ParathroÔme ìti X rej e nai o metasqhmatismì Fourier th akolouj a x n ()
pollaplasiasmènh me thn pragmatik ekjetik akolouj a r n . Epomènw gia
na sugkl nei o metasqhmatismì z prèpei o metasqhmatismì Fourier diakritoÔ
( )
qrìnou th akolouj a x n r n na sugkl nei, dhlad , h akolouj a x n z n ()
na e nai arijm simh kat apìluth tim
1
X 1
X
jX (z)j = x(n)z n jx(n)jr n < 1 (7.1.9)
n= 1 n= 1
H parous a tou ìrou r n parèqei th dunatìthta sÔgklish tou ajro smato
kai kat sunèpeia thn Ôparxh tou metasqhmatismoÔ z akìmh kai an den uprqei
()
o DTFT th x n . O DTFT th x n ()= ()
an u n den uprqei ìtan a > , jj 1
()
dedomènou ìti x n e nai m a ekjetik aÔxousa akolouj a ìpw fa netai sto
Sq ma 7.2a. An epilege r > a tìte h akolouj a r n ìpw fa netai sto Sq ma
()
7.2b elatt¸netai taqÔtera apì ìti h x n auxnetai. àtsi h akolouj a x n r n ()
ìpw fa netai sto Sq ma 7.2g e nai arijm simh kat apìluth tim kai uprqei
o metasqhmatismì z.
n
x(n)= á u(n), á>1 r -n x(n) r-n
r >á
1 1
-4 -2 0 2 4 6 8 10 n -4 -2 0 2 4 6 8 10 n -4 -2 0 2 4 6 8 10 n
(á) (â) (ã)
Sq ma 7.2 (a) H akolouj a x(n) = an u(n) gia th opo a den uprqei o DTFT (b) o pargonta
kai (g) h akolouj a x(n)r
exasjènish r n n h opo a e nai arijm simh kat apìluth tim .
7.1.1 Metasqhmatismì z stoiqeiwd¸n akolouji¸n
Sthn pargrafo aut ja upolog soume to Mz orismènwn basik¸n akolouji¸n.
Pardeigma 7.1.1 (S ma peperasmènh èktash )
Na upologiste o metasqhmatismì z tou orjog¸niou parajÔrou dirkeia N + 1
x(n) = 10;; 0alli¸
nN (7.1.10)
LÔsh O metasqhmatismì z e nai
X1 XN XN
X (z ) = x(n)z n = 1z n = z N zn
n= 1 n=0 n=0
Enìthta 7.1 Orismo 237
kai me th bo jeia tou tÔpou me ton opo o upolog zoume to jroisma twn ìrwn gewmetrik
seir èqoume
(
z Nz z 1 1 = zzNN(+1z 1)1 ;
N +1
X (z ) = z 6= 1; z 6= 0
N + 1; z=1
(7.1.11)
H perioq sÔgklish kalÔptei ìlo to migadikì ep pedo ektì apì to mhdèn.
Pardeigma 7.1.2 (To ekjetikì aitiatì s ma)
Na upologiste o metasqhmatismì z gia to s ma
x(n) = an u(n) (7.1.12)
LÔsh O metasqhmatismì z e nai
X1 X1
X (z ) = an u(n)z n = (az 1)n
n=1 n=0
Gia na sugkl nei o X (z ) prèpei to jroisma
P1 twn ape rwn ìrwn th gewmetrik seir ,
me lìgo az 1, n=0
na sugkl nei, dhlad ,j az 1 j n < 1. àtsi h perioq sÔgklish
e nai h perioq tim¸n tou z gia thn opo a jaz
1j < 1 jz j > jaj, opìte
X1 1 = z ; ìtan jzj > jaj
X (z ) = (az 1 )n =
n=0 1 az 1 z a (7.1.13)
O metasqhmatismì z or zetai sto exwterikì kÔklou akt na Rx = jaj blèpe Sq ma 7.3.
ℑm z
Ìïíáäéáßïò
x(n) êýêëïò
Rx
-4 -2 0 2 4 6 8 n 0 a ℜe z
(á)
(â)
Sq ma 7.3 (a) To s ma x(n) = anu(n) ìtan a e nai pragmatikì arijmì < 1 kai (b) h
perioq sÔgklish , o pìlo kai to mhdenikì tou metasqhmatismoÔ z sto Pardeigma 7.1.2.
Gia a = 1 tìte to s ma x(n) e nai h akolouj a monadia ou b mato , h opo a èqei
metasqhmatismì z
u(n) Z! U (z ) =
1 = z z 1 me jzj > 1
1 z 1
(7.1.14)
238 Metasqhmatismì z Keflaio 7
dhlad , me perioq sÔgklish to exwterikì tou monadia ou kÔklou.
Parathr sei
1. ()
O metasqhmatismì z th akolouj a x n sugkl nei gia kje peperasmènh tim
tou a, en¸ o metasqhmatismì Fourier tou x n sugkl nei an a < . () jj 1
2. jj 1
An a > h perioq sÔgklish den perièqei to monadia o kÔklo kai gia ti timè
autè den uprqei o metasqhmatismì Fourier th akolouj a an u n . ()
3. O metasqhmatismì z èqei klasmatik morf , dhlad , èqei mhdenik kai pìlou .
Sto Pardeigma 7.1.2 o metasqhmatismì z èqei èna mhdenikì “ ” gia z kai Æ =0
èna pìlo “ ” gia z =
a (blèpe Sq ma 7.2).
Pardeigma 7.1.3
Na upologiste o metasqhmatismì z gia to s ma
x(n) = nan 1 u(n) (7.1.15)
P1
LÔsh Paragwg zonta thn ex swsh n=0 a
nz n = zza w pro a èqoume
1
X z
nan 1 z n = jz j > jaj
n=1 (z a)2
me
Apì thn parapnw ex swsh prokÔptei
z
x(n) = nan 1 u(n) Z! X (z ) = jz j > jaj
(z a)2
me (7.1.16)
Nèa parag¸gish w pro a d nei
1
X z
n(n 1)an 2z n = 2 (z jz j > jaj
a)3
me
n=2
gia apì thn opo a èqoume
n(n 1) an 2u(n) Z! X (z) = z
x(n) = jz j > jaj
2 (z a)3
me (7.1.17)
An epanalboume thn parag¸gish ja odhghjoÔme se ant stoiqa apotelèsmata.
Pardeigma 7.1.4
Na upologiste o metasqhmatismì z tou monadia ou de gmato .
LÔsh O metasqhmatismì z tou monadia ou de gmato x(n) = Æ(n) e nai
X1
Z [Æ(n)℄ = Æ(n)z n = 1; me z 6= 0 (7.1.18)
n= 1
Enìthta 7.1 Orismo 239
kai tou olisjhmènou kat kb mata monadia ou de gmato Æ(n k), e nai
Z [Æ(n k)℄ = 0; k<0
z k; k 0 (7.1.19)
H perioq sÔgklish kalÔptei ìlo to migadikì ep pedo ektì apì thn arq ,
Pardeigma 7.1.5
Na upologiste o metasqhmatismì z tou s mato
n n
x(n) =
1 u(n) +
1 u(n)
2 3 (7.1.20)
LÔsh O metasqhmatismì z e nai
1 1 n
X 1 1
X
n
X (z ) = + z n z n
n=0 2 n=0 3
1
= 1 1z 1 + 1 1z 11
2 3
2 5z 1
= 1 1 z 16 1 1 z 1
2 3
z 2z 56
Telik
X (z ) =
z 12 z 13
(7.1.21)
Gia na èqoume sÔgklish tou X (z ) prèpei jz j > 1=2 kai jz j > 1=3. Epomènw prèpei
jz j > 1=2. MporoÔme na odhghjoÔme sta dia apotelèsmata an basistoÔme sta dÔo
prohgoÔmena parade gmata kai qrhsimopoi soume thn idiìthta th grammikìthta tou
metasqhmatismoÔ z thn opo a ja doÔme sth sunèqeia.
Pardeigma 7.1.6 (Austhr mh aitiatì ekjetikì s ma)
x(n) = anu( n 1) (7.1.22)
LÔsh O metasqhmatismì z e nai
1
X X1 1
X 1
X
X (z ) = an u( n 1)z n = an z n = a nzn = 1 (a 1z)n
n= 1 n= 1 n=1 n=0
an ja 1 z j < 1 isodÔnama jz j < jaj to jroisma sugkl nei kai èqoume
1 1 a z 1 z 1
X (z ) = 1 1z = 1 a 1z = z a
1 a
(7.1.23)
me perioq sÔgklish to eswterikì kÔklou akt na Rx = jaj Sq ma 7.4.
240 Metasqhmatismì z Keflaio 7
ℑm z
x(n) Ìïíáäéáßïò
êýêëïò
-8 -6 -4 -2 Rx
0 2 4 n
0 a ℜe z
(á) (â)
Sq ma 7.4 (a) To s ma x(n) = an u( n 1) ìtan a pragmatì arijmì > 1 kai (b) h
perioq sÔgklish , o pìlo kai to mhdenikì tou metasqhmatismoÔ z sto Pardeigma 7.1.6.
ParathroÔme ìti o metasqhmatismì z tou austhr mh aitiatoÔ s mato x n ()=
(
an u n 1)
èqei thn dia algebrik parstash me to ekjetikì aitiatì s ma x n ( )=
n ()
a u n , all llh perioq sÔgklish . Autì shma nei ìti h perioq sÔgklish , ìpw
kai sto metasqhmatismì Laplace, apotele anapìspasto tm ma tou metasqhmatismoÔ
z kai prèpei pntote na kajor zetai. An ma doje mìno o metasqhmatismì z qwr
thn ant stoiqh perioq sÔgklis tou ja uprqei abebaiìthta ston prosdiorismì th
akolouj a pou od ghse s' autìn, afoÔ ja uprqoun perissìtere apì m a apant sei .
Pardeigma 7.1.7
Na upologiste o metasqhmatismì z tou s mato
x(n) = ajnj ; 1 < n < 1 kai jaj < 1 (7.1.24)
LÔsh To s ma x(n) grfetai
x(n) = an u(n) + a n u( n 1)
Apì to Pardeigma 7.1.2 èqoume
an u(n) Z!
1 = z z a me jzj > jaj
1 az 1
en¸ apì to Pardeigma 7.1.6 èqoume
a n u( n 1) Z! 1 1 = 1 azaz me jzj < ja1j
a 1z 1
Epeid jaj < 1, e nai kai jaj < jaj 1 . àtsi o metasqhmatismì z tou x(n) e nai
X (z ) =
z
+ az
; me jaj < jz j <
1
a 1 az jaj (7.1.25)
z
Sto Sq ma 7.6 fa nontai h perioq sÔgklish , to mhdenikì kai oi pìloi tou Mz th
akolouj a x(n) = ajnj .
Enìthta 7.1 Orismo 241
ℑm z
x(n)
1
R R
a
a ℜe z
1
0
-8 -6 -4 -2 0 2 4 6 8 n
(á) (â)
Sq ma 7.5 (a) H akolouj a x(n) = ajnj ìtan a pragmatì arijmì < 1 kai (b) h perioq
sÔgklish , oi pìloi kai to mhdenikì tou metasqhmatismoÔ z sto Pardeigma 7.1.7.
Pardeigma 7.1.8
Na upologiste o metasqhmatismì z tou s mato
x(n) = an ; 1<n<1 (7.1.26)
LÔsh To s ma x(n) grfetai
x(n) = an u(n) + an u( n 1)
Apì to Pardeigma 7.1.2 èqoume
an u(n) Z!
1 = z z a me jzj > jaj
1 az 1
en¸ apì to Pardeigma 7.1.6 èqoume
an u( n 1) Z! 1 1az 1 = z z a me jzj < jaj
ParathroÔme ìti en¸ oi metasqhmatismo z th a u(n) kai th a u( n 1) uprqoun,
n n
epeid prèpei tautìqrona na isqÔei jz j > a kai jz j < a, to ekjetikì s ma x(n) = a
n
den èqei amf pleuro metasqhmatismì z.
7.1.2 Idiìthte th perioq sÔgklish -Ôparxh metasqhmatismoÔ z
Apì ta prohgoÔmena parade gmata parathr same ìti o metasqhmatismì z den pros-
dior zei monos manta thn akolouj a (s ma) ektì an èqei orisje h perioq sÔgklish .
Ep sh parathr same ìti h morf th perioq sÔgklish tou metasqhmatismoÔ z
exarttai apì ta qarakthristik th akolouj a . Sthn enìthta aut ja broÔme to
trìpo me to opo o sundèetai h perioq sÔgklish me ta qarakthristik th akolou-
()
j a x n . Ja parousisoume ti idiìthte qrhsimopoi¸nta diaisjhtik epiqeir ma-
ta par austhrè majhmatikè apode xei . Gnwr zonta ti idiìthte e nai efiktì o
242 Metasqhmatismì z Keflaio 7
prosdiorismì th perioq sÔgklish apì to metasqhmatismì z, X (z ), kai èqonta
periorismènh gn¸sh twn qarakthristik¸n th akolouj a x n . ()
M a pr¸th idiìthta e nai ìti h perioq sÔgklish tou metasqhmatismoÔ z, pou
e nai rht sunrthsh th metablht z , den perièqei pìlou .
äpw parathr same sthn prohgoÔmenh enìthta gia na uprqei o metasqhmatismì
z tou s mato ()
x n prèpei
1
X 1
X
jX (z)j = x(n)z n jx(n)jr n < 1 (7.1.27)
n= 1 n= 1
h posìthta r e nai to mètro tou migadikoÔ arijmoÔ z =
r ej kai oi timè th prag-
matik metablht r pou ikanopoioÔn thn parapnw anisìthta or zoun th perioq
sÔgklish tou metasqhmatismoÔ. E nai fanerì ìti h sÔgklish exarttai mìno apì
to mètro r =jjz kai ìqi apì to , to opo o shma nei ìti h perioq sÔgklish ja
kajor zetai apì omìkentrou kÔklou me kèntro thn arq twn axìnwn tou migadikoÔ
epipèdou-z.
Gia m a peperasmènou m kou akolouj a, dhlad , x n ( )=0
an n < N1 kai n > N2
h opo a e nai fragmènh, dhlad , uprqei jetikì arijmì M gia ton opo o x n M j ( )j
o metasqhmatismì z e nai
N2
X
X (z ) = x(n)z n (7.1.28)
n=N1
E nai fanerì ìti h sunrthsh aut sugkl nei gia kje tim tou z , dhlad ,
h perioq sÔgklish mia peperasmènou m kou kai fragmènh akolouj a e nai ìlo
to migadikì ep pedo-z me dÔo pijanè exairèsei
1. 0
an N1 < tìte oi timè tou z me peiro mètro apokle ontai apì thn perioq
sÔgklish .
2. 0
an N1 < kai N2 > 0 tìte h arq twn axìnwn apokle etai apì thn perioq
sÔgklish .
Sth sunèqeia ja exetsoume th genik per ptwsh kat thn opo a h akolouj -
()
a x n den e nai periorismènh dirkeia kai e nai fragmènh. H akolouj a aut
onomzetai amf pleurh akolouj a. Sthn per ptwsh aut diaqwr zoume to jroisma
P1
j ( )j
n= 1 x n r <
n 1
se dÔo tm mata. Sto pr¸to tm ma I r ta ìria tou ajro s- ()
mato e nai 1
kai 1
X1
I (r) = jx(n)jr n
n= 1
Enìthta 7.1 Orismo 243
Sto deÔtero tm ma I+ (r) ta ìria tou ajro smato e nai 0 kai 1
1
X
I+ (r) = jx(n)jr n
n=0
dhlad ,
X1 1
X
jX (z)j jx(n)j r n + jx(n)j r n = I (r) + I+(r)
n= 1 n=0
j ( )j
Gia na e nai to X z fragmèno prèpei kai ta dÔo epimèrou ajro smata na e nai
peperasmèna.
A upojèsoume ìti mporoÔme na frxoume th x n br skonta ti elqiste j ( )j
jetikè stajerè M , M# , r kai r# tètoie ¸ste
jx(n)j M (r )n; n < 0; (7.1.29)
kai
jx(n)j M#(r# )n; n 0 (7.1.30)
()
H akolouj a x n h opo a ikanopoie ta frgmata den auxnetai taqÔtera apì thn
( ) ( )
akolouj a r# n gia ti jetikè timè tou n kai th r n gia ti arnhtikè timè th n.
Shmei¸netai ìti an kai uprqoun akolouj e oi opo e den ikanopoioÔn ta parapnw
2
frgmata, gia pardeigma h akolouj a n , oi akolouj e autè den emfan zontai se
fusikè efarmogè kai ètsi den ma dhmiourgoÔn probl mata.
An to frgma pou prosdior zetai apì thn (7.1.29) ikanopoie tai, tìte
X1 X1
I (r ) = jx(n)j r n M (r )n r n
n= 1 n= 1
r1
X
m
m= n
= M
r
(7.1.31)
m=1
Gia na uprqei to jroisma twn ape rwn ìrwn th gewmetrik proìdou prèpei
o
= r na e nai mikrìtero th monda . Epomènw to rjoisma P1 r m
lìgo r 1 r
uprqei gia ti timè tou z oi opo e ikanopoioÔn thn jzj = r < r . An to frgma pou
prosdior zetai apì thn (7.1.30) ikanopoie tai, tìte
1
X 1
X
I+ (r) = jx(n)j r n M# (r# )nr n
n=0 n=0
1 r m
X
= M# #
r
(7.1.32)
n=0
244 Metasqhmatismì z Keflaio 7
P
1 r# m uprqei gia ti timè tou z oi opo e ikanopoioÔn thn z
To rjoisma 0 r j j=
r > r# .
ParathroÔme ìti to pr¸to jroisma uprqei gia ti timè to z gia ti opo e z < jj
jj
r en¸ to deÔtero gia ti timè to z gia ti opo e z > r# . Sth per ptwsh aut h
perioq sÔgklish e nai h tom twn dÔo parapnw perioq¸n tou epipèdou-z .
H perioq sÔgklish mia amf pleurh akolouj a apotele tai apì ti timè tou
jj
migadikoÔ epipèdou z gia ti opo e isqÔei r# < z < r , dhlad , h perioq sÔgklish
èqei morf daktul ou.
Pardeigma amf pleurh akolouj a e nai h akolouj a x n ( )=
ajnj sto Pardeigma
7.1.7.
An r# r tìte oi dÔo perioqè sÔgklish den allhloepikalÔptontai kai h perioq
()
sÔgklish tou X z e nai to kenì sÔnolo (blèpe Pardeigma 7.1.8).
Sth sunèqeia ja exetsoume th per ptwsh kat thn opo a h akolouj a x n , ()
ikanopoie th sunj kh x n ()=0
gia n < N1 . H akolouj a aut onomzetai dex-
iìpleurh akolouj a. Sthn per ptwsh aut èqoume
1
X 1
X 1 r n
X
jX (z)j jx(n)jr n M# r#n r n = M# #
r
(7.1.33)
n=N1 n=N1 n=N1
ìpou upojèsame ìti to frgma pou prosdior zetai apì thn (7.1.30) ikanopoie tai. Gia
na uprqei to jroisma twn ape rwn ìrwn th gewmetrik proìdou prèpei o lìgo
= ()
rr# na e nai mikrìtero th monda . Epomènw o X z uprqei gia ti timè tou z
oi opo e ikanopoioÔn thn z j j=
r > r# .
H perioq sÔgklish mia dexiìpleurh akolouj a apotele tai apì ti timè tou
jj
migadikoÔ epipèdou z gia ti opo e isqÔei r# < z , dhlad , h perioq sÔgklish mia
dexiìpleurh akolouj a tou Mz ekte netai èxw apì kÔklo akt na r# sto ep pedo-z .
Dedomènou ìti h perioq sÔgklish den perièqei pìlou r# e nai h mikrìterh akt na
kÔklou ston opo o perièqontai ìloi oi pìloi tou X z . ()
H akolouj a x n ( ) = anu(n) thn opo a melet same sto Pardeigma 7.1.2 e nai m a
dexiìpleurh akolouj a.
Sth sunèqeia ja exetsoume th per ptwsh kat thn opo a h akolouj a x(n),
ikanopoie th sunj kh x n ( )=0
gia n > N2 . H akolouj a aut onomzetai aris-
terìpleurh akolouj a. Sthn per ptwsh aut èqoume
N2
X N2
X 1
X r
n
jX (z)j jx(n)jr
n M rn r n =M r
(7.1.34)
n= 1 n= 1 n= N2
Enìthta 7.2 Idiìthte tou MetasqhmatismoÔ z 245
apì thn opo a parathroÔme ìti o X (z) uprqei gia ti timè tou z oi opo e ikanopoioÔn
thn z j j=
r<r .
H perioq sÔgklish mia aristerìpleurh akolouj a apotele tai apì ti timè
jj
tou migadikoÔ epipèdou-z gia ti opo e isqÔei z < r , dhlad , h perioq sÔgk-
lish mia aristerìpleurh akolouj a tou Mz e nai to eswterikì kÔklou akt na
r sto ep pedo-z . Dedomènou ìti h perioq sÔgklish den perièqei pìlou , r e nai h
megalÔterh akt na kÔklou ston opo o den perièqontai oi pìloi tou X z . ()
Pardeigma aristerìpleurh akolouj a e nai h akolouj a x(n) = an u(
n 1)
sto Pardeigma 7.1.6.
Anloga me to e do tou s mato katal xame se trei diaforetikoÔ tÔpou peri-
oq¸n sÔgklish . Shmei¸netai ìti isqÔei kai to ant strofo, dhlad , an h perioq sÔg-
jj
klish e nai to exwterikì kÔklou z > R to s ma e nai aitiatì, an e nai to eswterikì
kÔklou z < R e nai mh aitiatì kai an èqei morf daktul ou R+ < z < R to s ma
jj jj
e nai m a amf pleurh akolouj a.
7.2 IDIOTHTES TOU METASQHMATISMOU Z
Sthn pargrafo aut ja parousiastoÔn basikè idiìthte tou metasqhmatismoÔ z.
Oi idiìthte autè ja ma bohj soun ston upologismì tou metasqhmatismoÔ z mia
dedomènh akolouj a qwr kat' angkh na prosdior zetai to sqetikì jroisma.
(1) Grammikìthta
An x1 (n) Z! X1 (z) me perioq sÔgklish P1 kai x2 n ( ) Z! X2(z) me perioq
sÔgklish P2 tìte gia opoiesd pote stajerè a kai b isqÔei
a x1 (n) + b x2 (n) Z! a X1 (z ) + b X2 (z ) (7.2.1)
H perioq sÔgklish tou Z [a x1 (n) + b x2 (n)℄ e nai toulqiston h P1 \ P2 . H lèxh
“toulqiston” qrhsimopoi jhke gia thn per ptwsh kat thn opo a o grammikì sundu-
asmì e nai tètoio ¸ste kpoia mhdenik na exoudeter¸noun orismènou pìlou . Se
mia tètoia per ptwsh h perioq sÔgklish e nai megalÔterh apì thn tom twn dÔo
epimèrou perioq¸n sÔgklish .
H apìdeixh th grammikìthta apotele meso epakìloujo th grammikìthta tou
ajro smato ston orismì tou metasqhmatismoÔ z.
(2) Idiìthta th qronik ol sjhsh
x(n) me amf pleuro metasqhmatismì z X (z ) kai perioq sÔgklish
àstw s ma
P = fz 2 C : R+ < jz j < R g, tìte gia kje akèraio n0 , jetikì arnhtikì, isqÔei
x(n + n0 ) Z! z n0 X (z ); R+ < jz j < R (7.2.2)
246 Metasqhmatismì z Keflaio 7
Apìdeixh
H apìdeixh aporrèei mesa apì tou orismoÔ . Prgmati, èqoume
1
X
Z [x(n + n0 )℄ = x(n + n0 )z n
n= 1
1
X
i=n+n0
= x(i)z i+n0
i= 1
1
X
= z n0 x(i)z i
i= 1
= z n0 X (z )
Gia n0 = 1 èqoume Z [x(n 1)℄ = z 1 X (z).
To sÔsthma diakritoÔ qrìnou tou Sq mato 7.6 to opo o prokale kajustèrhsh
enì de gmato sumbol zetai me z 1 . Genik èna sÔsthma diakritoÔ qrìnou to opo o
prokale kajustèrhsh kat m de gmata sumbol zetai me z m .
Sq ma 7.6 Sqhmatik perigraf sust -
x(n) z -1 x(n-1)
mato kajustèrhsh enì de gmato .
(3) Idiìthta th sunèlixh sunèlixh sto ped o tou qrìnou
An x1 (n) ! X1 (z) me perioq
Z
sÔgklish P1 kai x2 (n) ! X2(z) me perioq
Z
sÔgklish P2 tìte
x(n) = x1 (n) ? x2 (n) ! X (z) = X1 (z) X2 (z)
Z
(7.2.3)
H perioq sÔgklish tou X1 z ( ) ( )
X2 z e nai toulqiston h P1 P2 . Se merikè \
peript¸sei mpore to ginìmeno na uprqei kai se megalÔterh perioq , gia tou diou
lìgou ìpw kai sth grammikìthta.
Apìdeixh
Epeid
1
X
x(n) = x1 (k)x2 (n k)
k= 1
Enìthta 7.2 Idiìthte tou MetasqhmatismoÔ z 247
èqoume
1 1 " 1 #
X X X
X (z ) = x(n)z n = x1 (k)x2 (n k) z n
n= 1 n= 1 k= 1
1 " 1 #
X X
= x1 (k) x2 (n k)z n
k= 1 n= 1
1 " 1 #
X X
= x1 (k)z k x2 (n k)z (n k)
k= 1 n=1
" #
1
X 1
X
m=n k m)
= x1 (k)z k x2 (m)z
k= 1 m= 1
= X1 (z ) X2 (z ) (7.2.4)
(4) Idiìthta th diamìrfwsh ol sjhsh suqnìthta -klimkwsh sto ped o
tou z
àstw s ma x n ( ) Z! ( )
X z me perioq sÔgklish P z = f 2 C : R+ < jzj < R g,
kai migadikì arijmì . Tìte o metasqhmatismì z tou s mato y(n) = n x(n) e nai
y(n) = n x(n) ! Y (z ) = X z
Z
me j jR+ < jzj < j jR (7.2.5)
Apìdeixh
1
X 1
X
Y (z ) = y(n)z n = nx (n)z n
n= 1 n= 1
1
X z n z
= x(n) =X
n= 1
(5) Idiìthta th parag¸gish sto ped o tou z
àstw s ma x (n) ! X (z) me perioq
Z
sÔgklish P = fz 2 C : R+ < jzj < R g,
tìte
nx(n) Z
! z dXdz(z) me thn dia perioq sÔgklish (7.2.6)
248 Metasqhmatismì z Keflaio 7
Apìdeixh
1
X
X (z ) = x(n)z n
n= 1
dX (z ) 1
X
= nx(n)z (n+1)
dz n= 1
X1
= z 1 nx(n)z n
n= 1
= 1
z Z [nx(n)℄
(6) Idiìthta th suzug a (Suzug akolouj a)
àstw migadikì s ma x(n) me metasqhmatismì z X (z ) me perioq sÔgklish P =
fz 2 C : R+ < jzj < R g, tìte
x? (n) ! X ? (z? ); R+ < jzj < R
Z
<e [x(n)℄ Z! 21 [X (z) + X ? (z? )℄ (7.2.7)
=m [x(n)℄ ! 21j [X (z) X ? (z? )℄
Z
Apìdeixh
An y(n) = x? (n) èqoume
1
X
Y (z ) = x? (n)z n
n= 1
1
X
Y ? (z )= x(n)(z ? ) = X (z ? )
n
n= 1
Y (z ) = Z [x? (n)℄ = X ? (z ? ) (7.2.8)
(7) Katoptrismì sto qrìno
àstw s ma x (n) Z! X (z) me perioq sÔgklish = fz 2 C : R+ < jzj < R g,
P
tìte
x( n) Z
! X (z 1 ); 1 < jzj < 1
R+
me PS (7.2.9)
R
Enìthta 7.2 Idiìthte tou MetasqhmatismoÔ z 249
Apìdeixh
1
X
Z [x( n)℄ = x( n)z n
n= 1
1
X
k= n
= x(k) z 1 k
k= 1
= X (z 1 ) (7.2.10)
(8) Susqètish
An jewr soume dÔo s mata w dianÔsmata, èna arijmì pou metrei thn o-
moiìtht tou e nai to eswterikì tou ginìmeno. Autì g netai mègisto gia dianÔsmata
(s mata) pou sump ptoun, en¸ mhden zetai gia dianÔsmata pou e nai kjeta.
Wstìso, se ìti afor sta s mata, pollè forè aut e nai apl¸ metatopismè-
na to èna se sqèsh me to llo qwr na e nai ousiastik diaforetik. àtsi èna
mìno arijmì (eswterikì ginìmeno) den e nai arketì gia na antimetwp sei ìle ti
pijanè sqetikè metatop sei metaxÔ twn shmtwn. Qreizetai loipìn na oriste èna
nèo s ma (sunrthsh) tou opo ou h anexrthth metablht ja ekfrzei thn metatìpish
metaxÔ twn dÔo shmtwn. Or zetai loipìn h sunrthsh - akolouj a susqètish etero-
susqètish twn s matwn diakritoÔ qrìnou - akolouji¸n x n kai y n w () ()
1
X
rxy (l) = x(n) ? y( n) = x(n)y? (n l); 1<l<1 (7.2.11)
n= 1
Thn anexrthth metablht l ja onomzoume kajustèrhsh (lag).
An x (n) ! Z
X (z ) me perioq sÔgklish P1 kai y(n) Z! Y (z ) me perioq
sÔgklish P2 , tìte o metasqhmatismì z th susqètish twn dÔo shmtwn e nai
rxy (l) Z
! X (z) Y (z 1 ) (7.2.12)
Apìdeixh
H apìdeixh th idiìthta g netai an qrhsimopoihjoÔn oi idiìthte sunèlixh kai
[ ( )℄
tou katoptrismoÔ. H perioq sÔgklish tou Z rxy l e nai toulqiston h P1 P2 . \
H susqètish exarttai apì thn enèrgeia twn dÔo shmtwn. O suntelest susqètish
()
xy l dÔo shmtwn diakritoÔ qrìnou or zetai w
%xy (l) =
(l) ;
prExyp 1<l<1 (7.2.13)
x E y
250 Metasqhmatismì z Keflaio 7
e nai sunrthsh th kajustèrhsh twn dÔo akolouji¸n kai e nai anexrthto apì thn
enèrgei tou .
àna llo polÔ qr simo mègejo e nai h sunrthsh autosusqètish (autocorrelation).
H sunrthsh autosusqètish or zetai apì th sqèsh
1
X
rxx (l) = x(n) ? x( n) = x(n)x? (n l); 1<l<1 (7.2.14)
n= 1
H akolouj a autosusqètish èqei dÔo basikè idiìthte
a) H enèrgeia tou s mato e nai sh me th tim th sunrthsh autosusqètis tou
gia l =0. Prgmati
1
X
rxx (0) = jx(n)j2 dn = Ex (7.2.15)
n= 1
b) O metasqhmatismì Fourier diakritoÔ qrìnou th sunrthsh autosusqètish enì
s mato isoÔtai me th fasmatik puknìthta enèrgeia tou s mato . H sunrthsh
fasmatik puknìthta enèrgeia perigrfei ton trìpo me to opo o katanèmetai h
enèrgeia tou s mato sto q¸ro suqnot twn. Prgmati, lìgw tou jewr mato th
sunèlixh tou metasqhmatismoÔ Fourier èqoume
rxx (l) = x(n) ? x? ( n) ) F [rxx (l)℄ = jX ( )j2 (7.2.16)
kai apì to je¸rhma tou Parseval èqoume
1
X 1 Z jX ( )j2 d
Ex = rxx(0) = jx(n)j2 dn = 2 2 (7.2.17)
n= 1
Pèra apì ta stoiqei¸dh s mata, ta opo a melet jhkan sta parade gmata, uprqoun
kai arket lla pou ep sh sunant¸ntai sth melèth grammik¸n susthmtwn diakritoÔ
qrìnou. Oi metasqhmamo z twn shmtwn aut¸n upolog zontai me th bo jeia tou oris-
moÔ kai twn idiot twn tou metasqhmatismoÔ z. Ston P naka 7.1 uprqoun oi idiìthte
tou metasqhmatismoÔ z, en¸ ston P naka 7.2 uprqoun oi metasqhmatismo z kai oi
ant stoiqe perioqè sÔgklish gia ti plèon sunhjismène kai qr sime peript¸sei .
7.3 O MONOPLEUROS METASQHMATISMOS Z
Sthn pargrafo aut ja estisoume sta basik shme a tou monìpleurou metasqh-
matismoÔ z kai kur w se aut pou ton diaforopoioÔn apì to metasqhmatismì z. H
diafor metaxÔ twn dÔo metasqhmatism¸n entop zetai sta ìria tou ajro smato oris-
P1
moÔ X( )
z ()
n=0 x n z
n
Enìthta 7.3 O Monìpleuro Metasqhmatismì z 251
PINAKAS 7.10 Oi idiìthte tou MetasqhmatismoÔ z
Idiìthta S ma Metasqhmatismì z Perioq sÔgklish
x(n) X (z ) P =fz 2C :R+ <jz j<R g
x1 (n ) X1 (z ) P1 =fz 2C :R+
1 <jz j<R1 g
x2 (n ) X2 (z ) P2 =fz 2C :R+
2 <jz j<R2 g
Grammikìthta ax1 (n)+bx2 (n) aX1 (z )+bX2 (z ) Toulqiston P1 \P2
Qronik ol sjhsh x(n+n0 ) z n0 X (z ) P
Sunèlixh x1 (n)?x2 (n) X1 (z )X2 (z ) Toulqiston P1 \P2
sto ped o tou qrìnou
Klimkwsh sto ped o z
ol sjhsh suqnìthta
n x(n) X z() j jR+<jzj<j jR
Parag¸gish nx(n) z dXdn(z) R+ <jz j<R
sto ped o z
Suzug a x? (n) X ? (z ? ) P
Pn
k= 1 x(n) 1 z 1 X (z ) P1 \jz j>1
Ajro smato 1 Toulqiston
Katoptrismì ston x( n) X (z 1 ) 1 1
R <jz j< R+
xona tou qrìnou
Profan¸ an dÔo s mata diakritoÔ qrìnou e nai diaforetik gia n < kai sa 0
gia n 0 tìte èqoun ton dio MMz kai diaforetikì Mz. Gia aitiat s mata, x n ( )=0
0
gia n < , o Mz kai o MMz sump ptoun. Me lla lìgia o monìpleuro metasqh-
()
matismì z tou s mato x n taut zetai me ton amf pleuro metasqhmatismì z tou
()() ()()
s mato x n u n . Efìson to s ma x n u n e nai aitiatì s ma, h perioq sÔgk-
lish tou monìpleurou metasqhmatismoÔ z e nai pnta to exwterikì mèro kÔklou me
th mikrìterh akt na Rx pou perilambnei tou pìlou tou s mato .
Sqedìn ìle oi idiìthte tou metasqhmatismoÔ z isqÔoun kai gia to monìpleuro.
Epeid to ktw ìrio tou ajro smato orismoÔ tou monìpleurou metasqhmatismoÔ z
e nai to mhdèn o monìpleuro metasqhmatismì èqei thn idiìthta th dexi kai th
arister ol sjhsh oi opo e apoteloÔn th dÔnamh tou monìpleurou metasqhmatismoÔ
z. Oi idiìthte autè kai h idiìthta th sunèlixh parèqei sto monìpleuro metasqh-
matismì z th dunatìthta ep lush exis¸sewn diafor¸n, oi opo e èqoun mh mhdenikè
arqikè sunj ke .
(1) Idiìthta th dexi ol sjhsh - Kajustèrhsh
An x(n) Z! X (z ) me akt na sÔgklish R tìte
x
n0
X
x(n n0 ) Z! z n0 X (z) + z n0 x( i)z i gia kje n0 1 (7.3.1)
i=1
252 Metasqhmatismì z Keflaio 7
PINAKAS 7.11 Metasqhmatismo z merik¸n basik¸n shmtwn diakritoÔ qrìnou
S ma Metasqhmatismì z Perioq sÔgklish
1 Æ(n) 1 Gia kje z 6= 0
u(n) 1 1 = z 1
z jzj > 1
2 1 z
3 Æ(n m); m > 0 z m jzj 6= 0
an u(n) 1 = zza jzj > jaj
4 1 az 1
5 an 1 u(n 1) 1 jzj > jaj
z a
6 nan u(n) (1
az 1
az 1 )2 jzj > jaj
anu ( n 1) 1 = zza jzj < jaj
7 1 az 1
8 an 1 u( n) 1 jzj < jaj
z a
9 nan u( n 1) az 1
(1 az 1 )2 jzj < jaj
[ os( 0n)℄u(n) 1 ( os 0 )z 1 jzj > 1
10 1 (2 os 0 )z 1 +z 2
[sin( 0n)℄u(n) (sin 0 )z 1 jzj > 1
11 1 (2 os 0 )z 1 +z 2
[rn os( 0n)℄u(n) 1 (r os 0 )z 1 jzj > r
12 1 (2r os 0 )z 1 +r2 z 2
[rn sin( 0 n)℄u(n) (r sin 0 )z 1 jzj > r
13 1 (2r os 0 )z 1 +r2 z 2
Apìdeixh
H apìdeixh aporrèei mesa apì tou orismoÔ . Prgmati, èqoume
1
X 1
X
Z [x(n n0)℄ = x(n n0 )z n n n0 =i
= x(i)z i n0
n=0 i= n0
" #
X1 1
X
= z n0 x(i)z i + x(i)z i
i= n0 i=0
n0
X
= z n0 X (z) + z n0 x( i)z i
i=1
ParathroÔme ìti kat th dexi ol sjhsh nèa de gmata eisèrqontai sto disthma
[0 1)
; ja prèpei na lboun kai aut mèro stou upologismoÔ . Ta nèa de gmata
e nai ta x ( 1) ( 2)
;x ( )
; : : : ; x n0 .
Gia n0 =1 èqoumeZ [ ( 1)℄ =
xn z 1 z x X ( ) + ( 1)
.
Enìthta 7.3 O Monìpleuro Metasqhmatismì z 253
(2) Idiìthta th arister ol sjhsh - Pro ghsh
An x(n) Z! X (z ) me akt na sÔgklish R tìte
x
0 1
nX
z n0 Z [x(n + n0)℄ = X (z) x(i)z i gia kje n0 1 (7.3.2)
i=0
Apìdeixh
H apìdeixh aporrèei mesa apì tou orismoÔ . Prgmati, èqoume
1
X 1
X
Z [x(n + n0)℄ = x(n + n0 )z n n+n0 =i
= x(i)z i+n0
n=0 i=n0
" #
1
X 0 1
nX
= z n0 x(i)z i x(i)z i
i=0 i=0
" #
0 1
nX
= z n0 X (z ) x(i)z i
i=0
ParathroÔme ìti kat thn arister ol sjhsh kpoia apì ta uprqonta de gmata
br skontai ektì diast mato ; [0 1)
kai sunep¸ prèpei na afairejoÔn apì to suno-
(0) (1)
likì jroisma. Ta de gmata aut e nai ta x ; x ; : : : ; x n0 . ( 1)
Gia n0 èqoume z 1 x n
=1 Z [ ( + 1)℄ = X ( ) (0)
z x .
(3) Je¸rhma th Arqik Tim
An x(n) Z! X (z ) me akt na sÔgklish R tìte
x
x(0) = zlim
!1
X (z) (7.3.3)
(4) Je¸rhma th Telik Tim
An x(n) Z! X (z ) me akt na sÔgklish Rx tìte
lim x(n) = zlim
n!1 !1
(z 1)X (z) (7.3.4)
Parat rhsh: Ta jewr mata th arqik kai telik tim , ìpw kai ta ant s-
toiqa jewr mata tou metasqhmatismoÔ Laplace, ma parèqoun th dunatìthta upolo-
()
gismoÔ th asumptwtik tim th akolouj a x n ìtan e nai gnwstì o monìpleuro
metasqhmatismì z kai ètsi apofeÔgetai o upologismì tou x n apì ton ()
z . Ta X( )
jewr mata isqÔoun ìtan sth perioq sÔgklish tou z z X ( ) ( 1)X ( )
z perilambne-
tai o monadia o kÔklo ètsi ¸ste to s ma, ìpw ja doÔme, na e nai eustajè .
254 Metasqhmatismì z Keflaio 7
Pardeigma 7.3.1
Na upologiste o monìpleuro kai o amf pleuro metasqhmatismì z tou s mato
x(n) = an u(n) (7.3.5)
LÔsh Epeid x(n) = 0; n < 0, o monìpleuro kai o amf pleuro metasqhmatismì z
1
e nai soi
X (z ) = X (z ) =
1 az 1
= z z a an jzj > jaj (7.3.6)
Pardeigma 7.3.2
Na upologiste o monìpleuro kai o amf pleuro metasqhmatismì z tou s mato
y(n) = an+1 u(n + 1) (7.3.7)
LÔsh O monìpleuro metasqhmatismì z e nai
1
X
Y (z ) = y(n)z n
n=0
1
X
= an+1 z n
n=0
a
= 1 az 1
; me jz j > jaj (7.3.8)
O monìpleuro metasqhmatismì z mpore na upologiste kai w ex . Gnwr zoume
x(n) = an u (n) Z! X (z ) = 1=(1 az 1 ) me jz j > jaj. Me th bo jeia th idiìthta
th arister ol sjhsh èqoume
z 1 Z [x(n + 1)℄ = X (z) x(0)
= 1 1az 1 1
1
= 1 azaz 1 )
Z [y(n)℄ = 1 aaz 1
Gia ton amf pleuro metasqhmatismì z gnwr zoume ìti x(n) = anu(n) Z! X (z) =
1=(1 az 1 ) me jz j > jaj. Me th bo jeia th idiìthta th ol sjhsh èqoume
Z [x(n + 1)℄ = zX (z) = 1 zaz 1 )
z
Z [y(n)℄ =
1 az 1 ; jzj > jaj (7.3.9)
Enìthta 7.3 O Monìpleuro Metasqhmatismì z 255
(5) O monìpleuro metasqhmatismì z periodik¸n shmtwn
()
àstw to periodikì s ma x n me per odo N , dhlad , x n N x n . Tìte o ( + )= ( )
monìpleuro metasqhmatismì z X( ) ()
z , tou x n uprqei kai d netai apì th sqèsh
NX1
X (z) = 1 1z N x(n)z n me jzj > 1 (7.3.10)
n=0
Apìdeixh
1
X
X (z) = x(n)z n
n=0
NX1 2X
N 1 (k+1)
X N 1
= x(n)z n + x(n)z n + ::: + x(n)z n + :::
n=0 n=N n=kN
P(k+1)N 1 x(n)z
H antikatstash l = n kN sto jroisma n=kN
n d nei
(k+1)
X N 1 NX1
x(n)z n = x(l + kN )z (l+kn)
n=kN l=0
NX1
= z kN x(l)z l
l=0
ra,
NX1
X (z) = [1 + z N 2
+ z + : : : ℄ x(n)z
N n
n=0
NX1
= 1 1z N x(n)z n
n=0
Ston P naka 7.3 uprqoun oi idiìthte tou monìpleurou metasqhmatismoÔ z.
Pardeigma 7.3.3
Na upologiste o monìpleuro metasqhmatismì z tou periodikoÔ orjog¸niou kÔmato
me per odo N
x(t) = 1; kN n < kN + N1 k = 0; 1; 2; : : :
0; kN + N1 n < (k + 1)N N1 < N (7.3.11)
256 Metasqhmatismì z Keflaio 7
PINAKAS 7.12 Oi idiìthte tou monìpleurou metasqhmatismoÔ z
Idiìthta S ma Metasqhmatismì z Perioq sÔgklish
x(n) X (z) P =fz 2C :R<jz jg
x1 (n) X1 (z) P1 =fz 2C :R<jz jg
x2 (n) X2 (z) P2 =fz 2C :R<jz jg
Grammikìthta ax1 (n)+bx2 (n) aX1 (z )+bX2 (z ) P1 \P2
Dexi ol sjhsh
Kajustèrhsh
x(n n0 ); n0 1 [ P 0
z n0 X (z )+ ni=1 x( i)z i ℄ R<jz j
h Pn0 1 i
Arister ol sjhsh x(n+n0 ); n0 1 z n0 X (z) i=0 x(i)z
i R<jz j
Pro ghsh
Sunèlixh x1 (n)?x2 (n) X1 (z)X2 (z) P1 \P2
sto ped o tou qrìnou
Klimkwsh sto ped o z n x(n) X(z) j jR<jzj
ol sjhsh suqnìthta
Suzug a x? (n) X ? (z? ) R<jz j
PN 1
Periodikì s ma x(n+N )=x(n) 1 z N n=0 x(n)z
1 n jzj>1
Je¸rhma x(0) = limz!1 X (z )
arqik tim
Je¸rhma limn!1 x(n) = limz!1(z 1)X (z)
telik tim
LÔsh O metasqhmatismì z d netai apì thn ex swsh
1 1
X (z ) = 1 z 1 NX
z n
N
n=0
= 1 1z N
z N1
z 1 1
1 (7.3.12)
7.4 O ANTISTROFOS METASQHMATISMOS Z
Gnwr zoume
X (rej ) = F x(n) r n (7.4.1)
kai an jzj = r kai br sketai sthn perioq sÔgklish èqoume
x(n) r n =F 1 X (rej 1
) ) x(n) = 2
Z
X (rej )ej
rn nd
<2>
Enìthta 7.4 O Ant strofo Metasqhmatismì z 257
Me eisagwg th rn sto eswterikì tou oloklhr¸mato èqoume
x(n) =
1 Z
X rej
rej
n
d
2 <2>
(7.4.2)
Me allag metablht z = rej kai epeid r e nai stajer posìthta, prokÔptei ìti
dz = jrej d = jz d d = (1=j )z 1 dz . To olokl rwma or zetai pnw se
2
disthma tou . Gia th nèa metablht z to disthma autì antistoiqe se kampÔlh
gÔrw apì to kÔklo z j j=
r. àtsi èqoume
x(n) =
1 Z
1 dz
2j C X (z)z
n (7.4.3)
ìpou C e nai mia aristerìstrofh kleist kampÔlh olokl rwsh gÔrw apì thn arq
twn axìnwn, h opo a br sketai sto eswterikì th perioq sÔgklish tou metasqhma-
tismoÔ z , h de olokl rwsh g netai ant strofa apì th for twn deikt¸n tou rologioÔ.
7.4.1 Upologismì tou ant strofou metasqhmatismou z gia rhtè sunart sei
O apeuje a upologismì tou ant strofou metasqhmatismoÔ z mèsw tou oloklhr¸ma-
to th (7.4.3) e nai ep ponh diadikas a kai gi' autì sun jw akoloujoÔntai èmmesoi
trìpoi eÔresh tou ant strofou metasqhmatismoÔ z. An h morf th sunrthsh
()
X z e nai apl kai mpore eÔkola na ekfraste w jroisma epimèrou stoiqeiwd¸n
ìrwn, tìte me th qr sh gnwst¸n metasqhmatism¸n z kai twn idiot twn tou metasqh-
matismoÔ z mporoÔme ap' euje a na upolog soume ton ant strofo metasqhmatismì
()
z. An h sunrthsh X z e nai rht sunrthsh tìte thn anaptÔssoume se apl k-
lsmata kai upolog zoume ton ant strofo metasqhmatismì z, me th qr sh gnwst¸n
metasqhmatism¸n z se aut.
7.4.2 Upologismì me anptuxh se apl klsmata
Thn anptuxh mia sunrthsh se apl klsmata thn èqoume diapragmateuje sto
Parrthma B. ExeidikeÔonta thn teqnik sthn per ptwsh tou metasqhmatismoÔ z,
diakr noume dÔo peript¸sei
1. An o bajmì m, tou poluwnÔmou tou arijmht e nai mikrìtero tou bajmoÔ
n, tou poluwnÔmou tou paronomast h rht sunrthsh anaptÔssetai se apl
klsmata sÔmfwna me thn ex swsh
X (z ) = (z C11z1 ) + (z C12z1 )2 + + (z C1zr1)r
+ (z C21z2 ) + (z C31z3 ) + + (zC(nznr)1r ) (7.4.4)
258 Metasqhmatismì z Keflaio 7
=
ìpou z1 ; z2 ; : : : ; zl e nai oi l ()
n r pìloi th X z me pollaplìthte r; 1; : : : ; 1
ant stoiqa. Oi suntelestè C1;k prosdior zontai apì ti sqèsei
C1k =
1 dr k (z z1 )r X (z )
; k = 1; 2; : : : ; r
(r k)! dz r k
z =z1
(7.4.5)
Gia tou aploÔ pìlou oi suntelestè Ci1 prosdior zontai apì ti sqèsei
Ci1 = (z zi )X (z )jz=zi ; i = 2; 3; : : : ; n r (7.4.6)
()
àqonta analÔsei th sunrthsh X z se apl klsmata, mporoÔme sth sunè-
qeia sqetik eÔkola na upolog soume ton ant strofo metasqhmatismì z. Prg-
mati upolog zoume pr¸ta tou epimèrou metasqhmatismoÔ z twn apl¸n klas-
mtwn kai Ôstera ajro zoume ti prokÔptouse ekfrsei .
Orismèna apì ta pio sunhjismèna zeÔgh metasqhmatism¸n z parat jentai s-
ton P naka 7.2 Ta zeÔgh aut ma bohjoÔn ston upologismì tou antistrìfou
metasqhmatismoÔ z, ekfrzonta thn sunrthsh X z w grammikì sunduasmì ()
aploustèrwn sunart sewn.
2. An o bajmì m, tou poluwnÔmou tou arijmht e nai megalÔtero so tou ba-
jmoÔ n, tou poluwnÔmou tou paronomast tìte diairoÔme pr¸ta ta polu¸numa
kai katal goume se mia èkfrash pou èqei th morf
X (z ) = Bm n z m n + Bm n 1zm n 1 + : : : ; +B0 z 0 + X1 (z ) (7.4.7)
()
ìpou h sunrthsh X1 z èqei bajmì arijmht mikrìtero tou bajmoÔ tou parono-
mast , kai thn opo a anaptÔssoume se apl klsmata sÔmfwna me ta prohgoÔ-
mena.
àna sunhjismèno tèqnasma gia na apofÔgoume th dia resh poluwnÔmwn, an m =
( )= ( )
n, e nai na qrhsimopoi soume th sunrthsh X1 z X z =z auxnonta ètsi
to bajmì tou poluwnÔmou tou paronomast kat èna, kai na anaptÔxoume sth
()
sunèqeia se apl klsmata th sunrthsh X1 z ìpw fa netai sto Pardeigma
7.4.3.
Pardeigma 7.4.1
Na upologiste to s ma x(n) to opo o èqei metasqhmatismì z th sunrthsh
X (z ) =
3 5=6z 1 1
(1 1=4z 1)(1 1=3z 1) jzj > 3 (7.4.8)
LÔsh AnalÔoume se apl klsmata
C1 C2
X (z ) =
1 1=4z 1 + 1 1=3z 1
Enìthta 7.4 O Ant strofo Metasqhmatismì z 259
kai upolog zoume ti stajerè C1 kai C2 .
C1 = (1 1=4z 1)X (z)jz 1 =4 = 1 kai C2 = (1 1=3z 1)X (z)jz 1 =3 =2
àtsi o metasqhmatismì z apokt th morf
X (z ) =
1 2
1 1=4z 1 + 1 1=3z 1 = X1(z) + X2(z)
Me th bo jeia tou Parade gmato 7.1.2 èqoume
n
x1 (n) =
1 u(n) Z! X1(z) = 1 jzj >
1
4 1 1=4z 1 4
n
x2 (n) = 2
1 u(n) Z! X2(z) = 2 j zj >
1
3 1 1=3z 1 3
Telik
x(n) =
n
1 u(n) + 2
n
1 u(n) Z! X (z ) =
3 5=6z 1 1
4 3 (1 1=4z 1)(1 1=3z 1 jz j > 3
Pardeigma 7.4.2
Na upologiste to s ma x(n) to opo o èqei metasqhmatismì z th sunrthsh
X (z ) =
3 5=6z 1 1 < jzj < 1
(1 1=4z )(1 1=3z ) 4
1 1 3 (7.4.9)
LÔsh Apì to prohgoÔmeno pardeigma èqoume
X (z ) =
1 + 2
1 1=4z 1 1 1=3z 1 = X1(z) + X2(z)
lìgw th perioq sÔgklish kai me th bo jeia twn Paradeigmtwn 7.1.2 kai 7.1.5
èqoume
n
x1 (n) =
1 u(n) Z! X1 (z ) =
1 1
4 1 1=4z 1 jzj > 4
n
x2 (n) = 2 13 u( n 1) Z! X2(z) = 1 12=3z 1 jzj < 31
Telik n n
x(n) =
1 u(n) 2 31 u( n 1)
4
260 Metasqhmatismì z Keflaio 7
Pardeigma 7.4.3
Na upologiste to s ma x(n) to opo o èqei metasqhmatismì z th sunrthsh
z
X (z ) = 2
z +z 2
(7.4.10)
LÔsh AnalÔoume se apl klsmata th sunrthsh X (z )=z , kai èqoume
X (z )
z
= (z + 2)(1 z 1) = 13 z +1 2 + 13 z 1 1
àtsi prokÔptei
X (z ) =
1 z 1 z
3z+2 + 3z 1
Gia to GQA sÔsthma tou parade gmato den prosdior zetai h perioq sÔgklish th
sunrthsh metafor . Oi pijanè perioqè sÔgklish e nai oi trei , oi opo e eikon -
zontai sto Sq ma 7.7.
ℑm z ℑm z ℑm z
-2 0 1 ℜe z -2 0 1 ℜe z -2 0 1 ℜe z
( á) (â) ( ã)
Sq ma 7.7 Oi pijanè perioqè sÔgklish tou metasqhmatismoÔ z th akolouj a x(n) sto
Pardeigma 7.4.3.
1. Sth perioq fz 2 C : 0 < jz j < 1g upoqrewtik èqoume
z z
Z 1 = ( 2) ( n 1) kai Z 1
1 = u( 1)
nu n
z+2 z
sunep¸ , to s ma e nai
1 ( 2)nu( n 1) 1 u( n 1)
x(n) =
3 3
2. Sth perioq fz 2 C : 1 < jz j < 2g upoqrewtik èqoume
1 z 1 z
Z
z+2
= ( 2) u( n 1) kai Z z 1 = u(n)
n
sunep¸ , to s ma e nai
x(n) =
1 ( 2)nu( n 1) + 13 u(n)
3
Enìthta 7.4 O Ant strofo Metasqhmatismì z 261
3. Sth perioq fz 2 C : 2 < jz j < 1g upoqrewtik èqoume
z z
Z 1 = ( 2)nu(n) kai Z 1
z+2 z 1 = u(n)
sunep¸ , to s ma e nai
x(n) =
1 ( 2)nu(n) + 1 u(n)
3 3
7.4.3 Upologismì me anptuxh se dunamoseir
SÔmfwna me th mejodolog a aut anaptÔssoume th sunrthsh X z se dunamoseir ()
()
kai sth sunèqeia h akolouj a x n upolog zetai me antisto qish stou suntelestè
th dunamoseir . H anptuxh mia rht sunrthsh se dunamoseir epitugqnetai
sun jw me suneq dia resh. H mejodo aut den katal gei se m a analutik èkfrash
()
gia thn x n . E nai mia arijmhtik mèjodo me thn opo a mporoÔme na upolog zoume
()
èna stoiqe o th x n kje for.
Pardeigma 7.4.4
Na upologiste h akolouj a x(n) h opo a èqei metasqhmatismì z th sunrthsh
X (z ) =
1 ; jz j > jaj
1 az 1
(7.4.11)
LÔsh H èkfrash aut mpore na anaptuqje se dunamoseir me suneqe diairèsei
1 1 az 1
1 +az1 1+az
1 +a2z 2 + : : :
+az1
az 1 +a2 z 2
+a22z 22 3 3
a z +a z
+a3z 3
ParathroÔme ìti to jroisma 1+ az
1 + a2 z 2 + a3z 3 + : : : sugkl nei an az 1 < 1,
dhlad , an jaj < jz j. àtsi èqoume to anptugma gia to metasqhmatismì z
X (z ) =
1 = 1 + az 1 + a2z 2 + a3z 3 + : : :
1 az 1
(7.4.12)
Sugkr nonta thn parapnw èkfrash me thn sqèsh orismoÔ tou metasqhmatismoÔ z
1
X
X (z ) = x(n) z n
n= 1
èqoume
: : : x( 2) = 0; x( 1) = 0; x(0) = 1; x(1) = a; x(2) = a2; x(3) = a3; : : :
262 Metasqhmatismì z Keflaio 7
dhlad x(n) = an u(n) apotèlesma to opo o anamèname lìgw tou Parade gmato 7.1.2.
An az 1 > 1, dhlad , an jaj > jz j tìte h anptuxh se seir tou metasqhmatismoÔ z
g netai me thn paraktw dia resh.
1 az 1 +1
1 +a 1z a 1z a 2z 2 a 3z 3 : : :
+a 1z
a 1z +a 2z 2
+a 2z2
a 2 z 2 +a 3z3
+a 3z3
Me parìmoio trìpo skèyh ìpw kai sthn prohgoÔmenh per ptwsh katal goume
: : : x( 3) = a 3 ; x(
2) = a 2; x( 1) = a 1; x(0) = 0; x(1) = 0; x(2) = 0; : : :
dhlad x(n) = anu( n 1) apotèlesma to opo o anamèname lìgw tou Parade gmato
7.1.6.
Pardeigma 7.4.5
Na upologiste h akolouj a x(n) h opo a èqei metasqhmatismì z th sunrthsh
X (z ) = log 1 + az 1 ; jz j > jaj (7.4.13)
LÔsh Gnwr zoume
1
X ( 1)n+1wn ; jwj < 1; (Anptugma se seir Taylor)
log(1 + w) = n
n=1
Me efarmog th parapnw sqèsh èqoume
1
X ( 1)n+1anz n
X (z ) =
n=1 n
Sugkr nonta thn parapnw èkfrash me thn sqèsh orismoÔ tou metasqhmatismoÔ z
èqoume
x(n) = ( 1)n+1 ann ; n1
0; n<0
Telik prokÔptei
x(n) =
( a)n u(n 1) (7.4.14)
n
7.5 EFARMOGES TOU METASQHMATISMOU Z
Sthn enìthta aut ja anaptÔxoume ti efarmogè twn metasqhmatism¸n z. Eidikìtera
ja susthmatikopoi soume th dunatìthta pou parèqei o monìpleuro metasqhmatismì
Enìthta 7.5 Efarmogè tou MetasqhmatismoÔ z 263
z gia thn ep lush exis¸sewn diafor¸n oi opo e èqoun mh mhdenikè arqikè sun-
j ke kai ja efarmìsoume th diadikas a aut sth melèth GQA susthmtwn diakritoÔ
qrìnou. Telei¸nonta , ja exetsoume th sqèsh pou uprqei metaxÔ th jèsh twn
pìlwn th sunrthsh metafor sto migadikì ep pedo me ti idiìthte th aitiìthta
kai th eustjeia enì GQA sust mato diakritoÔ qrìnou.
M a apì ti pio shmantikè idiìthte tou metasqhmatismoÔ z e nai aut th sunèli-
xh . Sto pardeigma pou akolouje prosdior zetai, qwr na katafÔgoume sto jroi-
sma th sunèlixh , h akolouj a exìdou enì GQA sust mato diakritoÔ qrìnou, an h
kroustik apìkrish kai h e sodì tou e nai akolouj e peperasmènh èktash .
Pardeigma 7.5.1
Na prosdioriste h akolouj a exìdou enì GQA sust mato diakritoÔ qrìnou, to opo o
èqei kroustik apìkrish h(n) = [ 1; 2; 3℄ ìtan diege retai apì thn akolouj a x(n) =
"
[ 3; 4; 5; 2℄.
"
LÔsh Oi metasqhmatismo z th kroustik apìkrish kai th eisìdou e nai
H (z ) = 1 + 2z 1 + 3z 2 kai X (z ) = 3 + 4z 1 + 5z 2 + 2z 3
H èxodo tou sust mato prosdior zetai apì to jroisma th sunèlixh y (n) = h(n) ?
x(n) kai lìgw th idiìthta th sunèlixh Y (z ) = H (z ) X (z ), ètsi èqoume
èqoume
Y (z ) = 3 + 10z 1 + 22z 2 + 24z 3 + 19z 4 + 6z 5
Me ant strofo metasqhmatismì z br skoume thn akolouj a exìdou.
y(n) = [ 3; 10; 22; 24; 19; 6℄
"
7.5.1 Sust mata ta opo a qarakthr zontai apì grammikè exis¸sei diafor¸n
me stajeroÔ suntelestè
Gia ta sust mata ta opo a qarakthr zontai apì grammikè exis¸sei diafor¸n me
stajeroÔ suntelestè , o metasqhmatismì z apotele isqurì ergale o gia ton pros-
diorismì th sunrthsh metafor sust mato th apìkrish suqnìthta th
kroustik apìkrish sust mato .
Genik, ìpw gnwr zoume, oi prxei oi opo e prèpei na g noun, sto ped o tou
qrìnou, apì èna GQA sÔsthma diakritoÔ qrìnou sta dedomèna eisìdou, ¸ste na
prokÔyei h akolouj a exìdou, perigrfontai apì mia grammik ex swsh diafor¸n
me stajeroÔ suntelestè . Me lla lìgia, gnwr zoume ìti h e sodo kai h èxodo enì
GQA sust mato diakritoÔ qrìnou ikanopoioÔn mia grammik ex swsh diafor¸n me
264 Metasqhmatismì z Keflaio 7
stajeroÔ suntelestè th morf
N
X M
X
ak y(n k) = bk x(n k) me a0 = 1 (7.5.1)
k=0 k=0
Efarmìzoume metasqhmatismì z kai sta dÔo mèlh th ex swsh . Jewr¸nta ti ar-
qikè sunj ke mhdenikè , lìgw twn idiot twn th grammikìthta kai th qronik
metatìpish pou èqei o metasqhmatismì z, èqoume thn ex swsh,
N
X M
X
ak z kY (z) = bk z k X (z ) (7.5.2)
k=0 k=0
Gnwr zoume
Y (z ) = H (z ) X (z ) (7.5.3)
()
ìpou H z e nai o metasqhmatismì z th kroustik apìkrish kai e nai h sunrthsh
metafor tou sust mato . àtsi h sunrthsh metafor sust mato to opo o qarak-
thr zetai apì thn ex swsh diafor¸n e nai
PM
Y (z ) k
H (z ) =
X (z )
= PNk=0 bk z k
(7.5.4)
k=0 ak z
ParathroÔme ìti h sunrthsh metafor enì GQA sust mato e nai rht sunrthsh.
H eustjeia kai h aitiatìthta tou sust mato ìpw ja doÔme prosdior zoun thn
akrib perioq sÔgklish .
Pardeigma 7.5.2 (SÔsthma diakritoÔ qrìnou pr¸th txh )
Na upologiste h sunrthsh metafor kai h kroustik apìkrish tou aitiatoÔ GQA
sust mato diakritoÔ qrìnou pr¸th txh , to opo o ìpw e nai gnwstì perigrfetai
apì thn ex swsh diafor¸n
y(n) ay(n 1) = bx(n); a kai b jetiko pragmatiko arijmo (7.5.5)
LÔsh Efarmìzoume metasqhmatismì z kai sta dÔo mèlh th ex swsh kai lìgw twn
idiot twn th grammikìthta kai th metatìpish tou metasqhmatismoÔ z, èqoume thn
ex swsh
Y (z ) az 1Y (z ) = bX (z )
Y (z ) 1 az 1 = X (z )
apì thn opo a br skoume th sunrthsh metafor
Y (z ) b
H (z ) = =1
X (z ) az 1
(7.5.6)
Enìthta 7.5 Efarmogè tou MetasqhmatismoÔ z 265
Upolog same thn algebrik èkfrash th H (z ) kai èqoume dÔo pijanè perioqè sÔgk-
lish , h mia e nai h jz j > a kai h llh h jz j < a. Epeid to sÔsthma e nai aitiatì h
perioq sÔgklish e nai jz j > a.
H kroustik apìkrish tou sust mato e nai
h(n) = Z 1 [H (z )℄ = b an u(n) (7.5.7)
Sto Sq ma 7.8 perigrfetai to sÔsthma pr¸th txh diakritoÔ qrìnou ìpw autì èqei
ulopoihje me th bo jeia mia monda kajustèrhsh enì de gmato , enì ajroistoÔ
kai dÔo pollaplasiast¸n kai h kroustik tou apìkrish, dhlad , h akolouj a exìdou
tou ìtan h e sodì tou e nai h kroustik akolouj a.
x(n)=ä(n) y(n)=h(n)
x(n) b y(n)
z -1
-2 0 2 4 n a -2 0 2 4 n
Sq ma 7.8 H kroustik apìkrish tou sust mato pr¸th txh diakritoÔ qrìnou.
Pardeigma 7.5.3
Na upologiste h sunrthsh metafor kai h kroustik apìkrish tou aitiatoÔ GQA
sust mato diakritoÔ qrìnou, to opo o perigrfetai apì thn ex swsh diafor¸n
y(n) a1 y(n 1) + a2y(n 2) = x(n) (7.5.8)
LÔsh Efarmìzoume metasqhmatismì z kai sta dÔo mèlh th ex swsh kai lìgw twn
idiot twn th grammikìthta kai th metatìpish tou metasqhmatismoÔ z, èqoume thn
ex swsh
Y (z ) + a1 z 1 Y (z ) + a2 z 2 = X (z )
Y (z ) 1 + a1 z 1 + a2 z 2 = X (z )
apì thn opo a br skoume th sunrthsh metafor
Y (z )
H (z ) = = 1 + a1z 11 + a2z 2
X (z )
(7.5.9)
Upolog same thn algebrik èkfrash th H (z ). Epeid to sÔsthma e nai aitiatì h
perioq sÔgklish e nai jz j > R.
An = 1; 2728 kai a2 = 0; 81 to sÔsthma èqei dÔo suzuge pìlou tou 0; 9ej 4 .
a1
H sunrthsh metafor H (z ) analÔetai se apl klsmata w
H (z ) =
0; 5(1 j ) + 0; 5(1 + j )
1p 0; 6364(1 + j )z 1 1 p0; 6364(1 j )z 1
= 22 e j 4 1 0; 91ej 4 z 1 + 22 ej 4 1 0; 9e1 j 4 z 1 (7.5.10)
266 Metasqhmatismì z Keflaio 7
kai h kroustik apìkrish tou sust mato e nai
p p
h(n) =
2 e j
4 0; 9e u(n) + 22 ej 4 0; 9e
j n
4 j 4 n u (n)
p2 h i
= 2(0; 9)n os (n 1) 4 u(n) (7.5.11)
Sto Sq ma 7.9 perigrfetai h perioq sÔgklish oi suzuge migadiko pìloi tou sust -
mato diakritoÔ qrìnou kai h kroustik tou apìkrish h opo a e nai m a fj nousa h-
mitonoeid akolouj a. To sÔsthma e nai eustajè .
ℑm z
Ìïíáäéáßïò
êýêëïò
h(n)
j 4ð
0,9 e
0 1 ℜe z 0 n
j 4ð
0,9 e
Sq ma 7.9 H perioq sÔgklish , oi suzuge migadiko pìloi, to mhdenikì me pollaplìthta
2 kai h kroustik apìkrish tou sust mato diakritoÔ qrìnou sto Pardeigma 7.5.3.
An a1 = 1; 5556 kai a2 = 1; 21 to sÔsthma èqei dÔo suzuge pìlou tou 1; 1ej 4 .
H H (z ) analÔetai se apl klsmata w
H (z ) = 1 0;07778(1
; 5(1 j )
+ + 0; 5(1 + j )
j )z 1 1 0; 7778(1 j )z 1
p p
= 2 e 1 1; 1ej 4 z 1 + 22 ej 4 1 1; 1e1 j 4 z
2 j
4
1
1 (7.5.12)
kai h kroustik apìkrish tou sust mato e nai
p p
h(n) =
2 e j
4 1; 1e u(n) + 22 ej 4 1; 1e
j n
4 j 4 n u (n)
p2 h i
= 2(1; 1)n os (n 1) 4 u(n) (7.5.13)
Sto Sq ma 7.10 perigrfetai h perioq sÔgklish oi suzuge migadiko pìloi, to mh-
denikì me pollaplìthta 2 tou sust mato diakritoÔ qrìnou kai h kroustik tou apìkr-
ish h opo a e nai m a aÔxousa hmitonoeid akolouj a. To sÔsthma e nai mh eustajè .
Pardeigma 7.5.4
àstw to aitiatì sÔsthma tou opo ou h e sodo kai h èxodo ikanopoioÔn th grammik
ex swsh diafor¸n
y(n)
1 y(n 1) = x(n) + 1 x(n 1)
2 3 (7.5.14)
Enìthta 7.5 Efarmogè tou MetasqhmatismoÔ z 267
ℑm z
Ìïíáäéáßïò
êýêëïò
j 4ð
1,1 e h(n)
0 1 ℜe z 0 n
j 4ð
1,1 e
Sq ma 7.10 H perioq sÔgklish , oi suzuge migadiko pìloi kai h kroustik apìkrish tou
sust mato diakritoÔ qrìnou sto Pardeigma 7.5.3.
Na upologiste h kroustik apìkrish tou sust mato .
LÔsh Efarmìzoume metasqhmatismì z kai sta dÔo mèlh th ex swsh kai lìgw twn
idiot twn th grammikìthta kai th metatìpish tou metasqhmatismoÔ z, èqoume thn
ex swsh
Y (z )
1 z 1 Y (z ) = 1
X (z ) + z 1 )
2 3
Y (z ) 1
1 z 1 =
1
X (z ) 1 + z
1 )
2 3
apì thn opo a br skoume thn sunrthsh metafor
Y (z ) 1
H (z ) =
X (z )
= 11 + 11==32zz 1 (7.5.15)
Upolog same thn algebrik èkfrash th H (z ) kai èqoume dÔo pijanè perioqè sÔgk-
lish , h mia e nai h jz j > 1=2 kai h llh h jz j < 1=2. Epeid to sÔsthma e nai aitiatì
h perioq sÔgklish e nai jz j > 1=2.
Epeid o bajmì tou poluwnÔmou tou arijmht e nai so me to bajmì tou poluwnÔmou
tou paronomast prèpei na g nei dia resh prin thn anlush se apl klsmata. àtsi
èqoume
H (z ) =
2+5 1
3 3 1 1=2z 1 (7.5.16)
H kroustik apìkrish tou sust mato e nai
h(n) = Z 1 [H (z )℄ =
2 Æ(n) + 5 1 n u(n)
3 3 2 (7.5.17)
Oi idiìthte th dexi kai th arister ol sjhsh se sunduasmì me thn idiìthta th
sunèlixh d noun ax a sto monìpleuro metasqhmatismì z, giat ma epitrèpoun na lÔ-
noume diaforikè exis¸sei me arqikè sunj ke , kai na upolog zoume thn èxodo GQA
susthmtwn, ta opo a den br skontai arqik se hrem a, an gnwr zoume thn sunrthsh
268 Metasqhmatismì z Keflaio 7
()
metafor tou sust mato H z kai to metasqhmatismì z tou s mato eisìdou X (z ).
Efarmìzoume ta parapnw sto pardeigma pou akolouje .
Pardeigma 7.5.5
D netai to aitiatì GQA sÔsthma diakritoÔ qrìnou tou opo ou h e sodo kai h èxodo
sundèontai apì thn ex swsh diafor¸n
y(n) 0; 5y(n 1) = x(n) (7.5.18)
Na breje h sunrthsh metafor kai h kroustik apìkrish tou sust mato .
LÔsh Efarmìzoume metasqhmatismì z kai sta dÔo mèrh th ex swsh diafor¸n kai
èqoume
Y (z ) 0; 5z 1Y (z ) = X (z ) (7.5.19)
Apì thn opo a br skoume th sunrthsh metafor tou sust mato
H (z ) =
1
1 0; 5z 1 (7.5.20)
me perioq sÔgklish jz j > 0; 5 afoÔ to sÔsthma e nai aitiatì. H kroustik apìkrish
tou sust mato upolog zetai me ant strofo metasqhmatismì z
h(n) = (0; 5)nu(n) (7.5.21)
An den èqoume arqikè sunj ke tìte h èxodo tou sust mato prosdior zetai me th
bo jeia tou jewr mato th sunèlixh gnwr zonta th sunrthsh metafor kai to
metasqhmatismì z tou s mato eisìdou.
Pardeigma 7.5.6
D netai to aitiatì GQA sÔsthma diakritoÔ qrìnou tou opo ou h e sodo kai h èxodo
sundèontai apì thn ex swsh diafor¸n
y(n) 0; 5y(n 1) = x(n) (7.5.22)
Na upologiste h èxodo tou sust mato an to s ma eisìdou e nai x(n) = u(n) kai to
sÔsthma br sketai se hrem a.
LÔsh Sto prohgoÔmeno pardeigma èqei upologiste h sunrthsh metafor tou
sut mato H (z ) = 1 0;15z 1 jz j > 0; 5. O metasqhmatismì z th
me ped o sÔgklish
eisìdou e nai X (z ) =
1 1 me ped o sÔgklish jz j > 1. O metasqhma-
akolouj a
1 z
tismì z th akolouj a exìdou e nai
Y (z ) = H (z )X (z ) =
1 1
1 0; 5z 1 1 z 1 (7.5.23)
me perioq sÔgklish thn tom twn dÔo epimèrou perioq¸n sugkl sh , dhlad , jz j > 1.
AnalÔoume ton Y (z ) se apl klsmata kai èqoume
Y (z ) =
1 2
1 0; 5z 1 + 1 z 1 (7.5.24)
Enìthta 7.5 Efarmogè tou MetasqhmatismoÔ z 269
kai me ant strofo metasqhmatismì z br sketai h akolouj a exìdou tou sust mato .
y(n) = (0; 5)nu(n) + 2u(n) (7.5.25)
y(n)
0 2 4 6 8 10 12 14 n
ìåôáâáôéêÞ Sq ma 7.11 H akolouj a exìdou tou
ìüìéìç
êáôÜóôáóç êáôÜóôáóç Probl mato 7.5.6.
Sto Sq ma 7.11 fa netai h èxodo tou sust mato . ParathroÔme ìti n!1 ;
n lim (0 5)
()=0
nn , epomènw h èxodo tou sust mato gia n >> e nai h n un. 0 ()=2()
H katstash aut qarakthr zetai w mìnimh katstash (steady-state response). To
disthma tim¸n sto opo o o ìro (0 5) ( )
; n u n den e nai per pou so me mhdèn qarak-
thr zetai w metabatik katstash (transient response). H metabatik katstash ek-
te netai sto disthma 0 7
n kai h mìnimh katstash gia ti timè tou n pou e nai
megalÔtere se apì 8.
An èqoume arqikè sunj ke tìte sthn ex swsh diafor¸n lìgw th idiìthta th
arister ol sjhsh tou metasqhmatismoÔ z sumperilambnoume ti arqikè sunj ke .
Pardeigma 7.5.7
D netai to GQA sÔsthma diakritoÔ qrìnou tou opo ou h e sodo kai h èxodo sundèontai
apì thn ex swsh diafor¸n
y(n) 0; 5y(n 1) = x(n) (7.5.26)
Na upologiste h èxodo tou sust mato an to s ma eisìdou e nai x(n) = u(n) me
arqik sunj kh y( 1) = 1
LÔsh Efarmìzoume monìpleuro metasqhmatismì z kai sta dÔo mèrh th (7.5.26) kai
èqoume
Y (z ) 0; 5 z 1 Y (z ) + y( 1) = X (z ) (7.5.27)
Y (z ) 0; 5z 1Y (z ) 0; 5 = X (z ) (7.5.28)
LÔnonta thn (7.5.27) w pro Y (z ) èqoume
Y (z ) = 1 01; 5z 1 X (z ) + 1 00;; 55z 1
= H (z)X (z) + 1 00;; 55z 1
= Yo (z ) + Yi (z ) (7.5.29)
270 Metasqhmatismì z Keflaio 7
ìpou
Yo (z ) = H (z )X (z ) = 1 01; 5z 1 1 1z 1
= 1 01; 5z 1 + 1 2z 1 (7.5.30)
e nai o metasqhmatismì z th exìdou tou sust mato gia mhdenikè arqikè sunj ke .
O ant strofo metasqhmatismì z d nei
yo (n) = (0; 5)nu(n) + 2u(n) (7.5.31)
kai e nai gnwst w apìkrish mhdenik katstash (zero stage response)
kai
Yi (z ) = 0; 5
1
1 0; 5z 1 (7.5.32)
e nai o metasqhmatismì z th exìdou tou sust mato o opo o proèrqetai apì ti ar-
qikè sunj ke tou sust mato . H suneisfor tou ìrou sthn èxodo tou sust mato
br sketai me ant strofo metasqhmatismì z kai e nai
yi (n) = 0; 5(0; 5)nu(n) (7.5.33)
kai e nai gnwst w apìkrish mhdenik eisìdou (zero input response).
H èxodo tou sust mato ja e nai
y(n) = yo(n) + yi (n) = (0; 5)nu(n) + 2u(n) + 0; 5(0; 5)nu(n)
= [ 0; 5(0; 5)n + 2℄ u(n) (7.5.34)
7.5.2 Melèth GQA sust mato me th bo jeia metasqhmatismoÔ z
. Apì thn (7.5.4) parathroÔme ìti h sunrthsh metafor sust mato e nai rht
sunrthsh. Upenjum zetai ìti oi r ze tou arijmht onomzontai mhdenik th H z ()
kai oi r ze tou paronomast pìloi th H z . ()
Apì thn perioq sÔgklish kai th jèsh twn pìlwn kai twn mhdenik¸n mporoÔme
na exgoume sumpersmata gia thn eustjeia kai thn aitiatìthta tou sust mato ,
prgmati
àna GQA sÔsthma diakritoÔ qrìnou e nai aitiatì an h n gia n < . Sthn( )=0 0
per ptwsh aut h perioq sÔgklish tou metasqhmatismoÔ z th kroustik
apìkrish , e nai to exwterikì enì kÔklou me akt na R+ sh me to mètro tou
pìlou pou èqei mègisto mètro. Me lla lìgia gia na e nai èna sÔsthma diakritoÔ
qrìnou aitiatì prèpei h perioq sÔgklish na e nai to exwterikì kÔklou me thn
mikrìterh akt na pou perièqei tou pìlou .
Enìthta 7.5 Efarmogè tou MetasqhmatismoÔ z 271
àna GQA sÔsthma diakritoÔ qrìnou e nai eustajè an gia fragmènh e sodo,
j ( )j
x n < M1 , kai h èxodo e nai fragmènh. Prgmati
1
X 1
X 1
X
jy(n)j = jh(k)x(n k)j M1 jh(k)j < 1 ) jh(k)j < 1
k= 1 k= 1 k= 1
ParathroÔme ìti an to sÔsthma e nai eustajè h kroustik apìkrish e nai
apolÔtw fragmènh kai ètsi uprqei o metasqhmatismì Fourier th . ra gia na
e nai to sÔsthma eustajè prèpei h perioq sÔgklish tou H z na perièqei to ()
monadia o kÔklo, oÔtw ¸ste na sugkl nei o metasqhmatismì Fourier tou h n . ()
Gia na e nai eustajè kai aitiatì prèpei na isqÔoun kai oi dÔo parapnw sun-
j ke ìloi oi pìloi prèpei na br skontai sto eswterikì tou monadia ou kÔklou.
()
Genikìtera h jèsh twn pìlwn th H z sto ep pedo z prosdior zei th sumperifor
th kroustik apìkrish tou sust mato .
AnaptÔssonta th sunrthsh metafor se apl klsmata kai upojètonta ìti
èqoume aploÔ pìlou 1 ; 2 ; : : : ; ; N èqoume
z z z
H (z ) = C1 + C2 z + : : : + CN z (7.5.35)
z 1 2 N
Apì thn opo a br skoume thn kroustik apìkrish tou sust mato
h(n) = Z 1 [H (z )℄ = (C1 n1 + C2 n2 + : : : CN nN ) u(n) (7.5.36)
An èqoume pragmatikì pìlo , tìte h kroustik apìkrish èqei ti akìlouje
idiìthte
a) 0<1 limn!1(n) = 0
h suneisfor tou ìrou sth h(n) e nai m a fj nousa ekjetik akolouj a
b) =1 n = 1 gia ìle ti timè tou n
h suneisfor tou ìrou sth h(n) e nai h bhmatik akolouj a
g) < 1 limn!1(n) = 1
h suneisfor tou ìrou sth h(n) e nai m a aÔxousa ekjetik akolouj a
d) 1
<< 0 limn!1(n) = 0 kai to n enallssei prìshmo
h suneisfor tou ìrou sth h(n) e nai m a fj nousa ekjetik akolouj a me ìrou
pou enallsoun prìshmo
e) = 1 n = 1;1; n = 2k
n = 1k + 1
h suneisfor tou ìrou sth h(n) e nai h bhmatik akolouj a me ìrou pou enal-
lssoun prìshmo
st) < 1 limn!1 jn j = 1 kai to n enallssei prìshmo
h suneisfor tou ìrou sth h(n) e nai m a aÔxousa ekjetik akolouj a me ìrou
pou enallssoun prìshmo
272 Metasqhmatismì z Keflaio 7
An to polu¸numo tou paronomast èqei dÔo migadikè suzuge r ze kai ? h
kroustik apìkrish tou sust mato e nai
h(n) = [C + C ? (? )n ℄ u(n)
upojètonta C = jC jej kai = jjej èqoume
h(n) = jC jjjn ej (n +) + jC jjjn e j (n +) u(n)
= 2C jjjn os(n + )u(n)
O ìro os(n + ) e nai fragmèno apì to 1. H sÔgklish mh th kroustik
jj
apìkrish tou sust mato ja prosdior zetai apì ton ìro n . An < h jj 1
kroustik apìkrish apotele fj nousa hmitonoeid seir (blèpe Pardeigma
7.5.3). Sth per ptwsh aut to sÔsthma e nai eustajè . Ant jeta an > h jj 1
kroustik apìkrish apotele aÔxousa hmitonoeid seir kai to sÔsthma e nai
astajè . An j j=1 h kroustik apìkrish tou sust mato e nai hminonoeid
seir me stajerì plto .
Sto Sq ma 7.12 paristnetai h sumperifor th kroustik apìkroush enì
aitiatoÔ sust mato diakritoÔ qrìnou, ìpw aut prosdior zetai apì th jèsh twn
pìlwn tou sto migadikì ep pedo.
ℑm z
Ìïíáäéáßïò
êýêëïò
ℜe z
Sq ma 7.12 H sumperifor th kroustik tou apìkrish enì sust mato diakritoÔ qrìnou
anloga me th jèsh twn pìlwn th sunrthsh metafor tou sto migadikì ep pedo z.
Pardeigma 7.5.8
JewroÔme to sÔsthma diakritoÔ qrìnou, me e sodo x(n) kai èxodo y(n), to opo o peri-
grfetai apì thn ex swsh
3y(n) 7y(n 1) + 2y(n 2) = 3x(n 2) (7.5.37)
Enìthta 7.5 Efarmogè tou MetasqhmatismoÔ z 273
Na upologiste h kroustik apìkrish tou sust mato gia na e nai to sÔsthma a) aitiatì
kai b) eustajè . Mpore na e nai to sÔsthma suqrìnw aitiatì kai eustajè ;
LÔsh Efarmìzonta metasqhmatismì z kai sta dÔo mèlh th ex swsh èqoume
Z y(n)
7 2
3 y(n 1) + 3 y(n 2) = Z [x(n 2)℄
Y (z ) 1
7z 1 + 2z 2 = z 2X (z )
3 3
kai h sunrthsh metafor tou sust mato e nai
Y (z ) 2
H (z ) =
X (z )
= 1 7 zz 1 + 2 z 2
3 3
1
= z2 7 z + 2
3 3
= z 1 1(z 2)
3
AnaptÔssonta thn H (z ) se apl klsmata èqoume
H (z ) = z
C1
+ zC 22
1
3
= 3 1 +3 1
5 z 13 5 z 2 (7.5.38)
(a) Gia na e nai to sÔsthma aitiatì prèpei h perioq sÔgklish na e nai jz j > 2. àtsi h
kroustik apìkrish tou aitiatoÔ sust mato e nai
3 n
1 1
h(n) =
5 3 u(n 1) + 35 (2)n 1 u(n 1) (7.5.39)
ìpou qrhsimopoi jhke to zeugri Mz 5 tou P naka 7.2.
(b) Gia na e nai to sÔsthma eustajè prèpei h perioq sÔgklish na perièqei to mona-
1 < jz j < 2.
dia o kÔklo dhlad na e nai
3 àtsi h kroustik apìkrish tou eustajoÔ
sust mato e nai
3 n
1 1
h(n) =
5 3 u(n 1) 35 (2)n 1u( n) (7.5.40)
ìpou qrhsimopoi jhkan ta zeugria Mz 5 kai 8 tou P naka 7.2.
Parathr sei
H parapnw ex swsh diafor¸n den mpore na perigrfei sÔsthma pou na e nai
sugqrìnw eustajè kai aitiatì.
Gia thn per ptwsh aitiatoÔ sust mato n!1 h n . lim ( )=1
274 Metasqhmatismì z Keflaio 7
SÔnoyh Kefala ou
Sto Keflaio autì or same to metasqhmatismì z kai to monìpleuro metasqhma-
tismì z, parousisthkan oi idiìthtè tou kai upolog same tou metasqhmatismoÔ z
orismènwn basik¸n shmtwn diakritoÔ qrìnou, ta opo a sunantme sth melèth gram-
mik¸n susthmtwn. Sth sunèqeia prosdior same ton ant strofo metasqhmatismì z.
E dame ìti an h morf tou metasqhmatismoÔ z e nai apl , tìte mporoÔme na up-
olog soume ton ant strofo metasqhmatismì z me th bo jeia tou P naka 7.2. An o
metasqhmatismì z den èqei apl morf all e nai rht sunrthsh, tìte analÔoume
th sunrthsh se apl klsmata kai me th bo jeia twn idiot twn tou metasqhma-
tismoÔ z kai tou Pinka 7.2 upolog zoume eÔkola to s ma qwr na katafÔgoume
sthn ex swsh antistrof .
Ep sh sto Keflaio autì anaptÔxame ti efarmogè tou metasqhmatismoÔ z.
Eidikìtera exetsame th dunatìthta pou èqei o monìpleuro metasqhmatismì z na
epilÔei grammikè exis¸sei diafor¸n me stajeroÔ suntelestè oi opo e den èqoun
mhdenikè arqikè sunj ke . H dunatìthta aut ofe letai sti idiìthte tou monì-
pleurou metasqhmatismoÔ z pou anafèrontai sth dexi kai arister ol sjhsh. Sth
sunèqeia parousisthkan oi efarmogè tou metasqhmatismoÔ z se ìti afor th melèth
GQA susthmtwn diakritoÔ qrìnou. Prosdior same th sunrthsh metafor tou
sust mato apì thn ex swsh diafor¸n pou sqet zei thn èxodo kai thn e sodo tou
sust mato , upojètonta ìti oi arqikè sunj ke e nai mhdenikè . Ep sh me th bo jeia
th ex swsh diafor¸n, prosdior same to monìpleuro metasqhmatismì z th exìdou
tou sust mato , to opo o mpore na mh br sketai se katstash hrem a kai antistrè-
fonta to monìpleuro metasqhmatismì z prosdior same thn èxodo tou sust mato .
Tèlo parousisame ta sumpersmata pou exgoume apì thn perioq sÔgklish kai
th jèsh twn pìlwn th sunrthsh metafor tou sust mato sto migadikì ep pedo
kai ta opo a aforoÔn sthn eustjeia kai thn aitiìthta tou sust mato diakritoÔ
sust mato kaj¸ kai sth sumperifor th kroustik apìkrish tou sust mato .
7.6 PROBLHMATA
7.1 D netai to GQA sÔsthma to opo o qarakthr zetai apì thn ex swsh diafor¸n
y(n) = x(n) + x(n 2)
()
Na prosdioriste o sunrthsh metafor tou sust mato H z . Me th bo jeia
()
th H z na upologiste h apìkrish suqnìthta tou sust mato H ( )
. Na
j ( )j
g noun oi grafikè parastsei tou mètrou H kai th fsh arg ( )
H se
sunrthsh me thn .
Enìthta 7.6 Probl mata 275
7.2 D netai to eustajè kai aitiatì sÔsthma, to opo o perigrfetai apì thn ex swsh
diafor¸n
y(n) + y(n
1 1) = x(n)
2
1. Na upologiste h apìkrish suqnìthta tou sust mato
2. Na upologiste h apìkrish tou sust mato an to s ma eisìdou e nai
x(n) = Æ(n)
1 Æ(n 1)
2
7.3 Aitiatì sÔsthma diakritoÔ qrìnou èqei sunrthsh metafor
z+1
H (z ) = 2
z 0; 9z + 0; 81
1. Na sqediaste to ped o sÔgklish oi pìloi kai ta mhdenik tou sust mato .
2. Na upologiste h apìkrish suqnìthta tou sust mato .
3. Na prosdioriste h ex swsh diafor¸n, me stajeroÔ suntelestè , h opo a
qarakthr zei to sÔsthma.
7.4 D netai sÔsthma diakritoÔ qrìnou to opo o èqei sunrthsh metafor
1 z 2
H (z ) = jzj > 0; 9
1 0; 81z 2 ;
1. Na upologiste h kroustik apìkrish tou sust mato .
2. Na prosdioriste h èxodo tou sust mato an h e sodì tou e nai h sunrthsh
monadia ou b mato .
7.5 D netai GQA sÔsthma to opo o èqei kroustik apìkrish
n
h(n) =
1 u(n)
3
Na upologiste h èxodo tou sust mato an h e sodì tou e nai to s ma
n
x(n) =
1 u(n)
2
276 Metasqhmatismì z Keflaio 7
7.6 Na upologiste h aitiat lÔsh th ex swsh
y(n)
3 y(n 1) + 1 y(n 2) = x(n); n0
2 2
ìtan n
x(n) =
1 u(n)
4
an y( 1) = 4 kai y( 2) = 10
7.7 àna grammikì qronik anallo wto sÔsthma èqei kroustik apìkrish
n
h(n) = 2 1 u(n)
2
1. Na breje h èxodo tou sust mato ìtan to s ma eisìdou e nai x(n) =
1 n u n . Oi arqikè sunj ke tou sust mato e nai y
() ( 1) = 4 kai
4
y( 2) = 10 .
2. Poio e nai to s ma sthn èxodo tou sust mato sth mìnimh katstash;
7.8 D netai èna GQA sÔsthma diakritoÔ qrìnou me kroustik apìkrish h(0) = 1,
h(1) = 2 kai h (2) = 1
1. Na upolog sete th apìkrish suqnìthta H( ) tou sust mato kai
2. na knete th grafik parastsh tou mètrou th apìkrish suqnìthta tou
sust mato se sunrthsh me th suqnìthta.
7.9 Na breje to s ma diakritoÔ qrìnou, tou opo ou o metasqhmatismì z èqei peri-
oq sÔgklish , tou pìlou kai ta mhdenik pou eikon zontai sto Sq ma 7.13.
Im
1 1 1 1 Re
2 2 Sq ma 7.13 Oi pìloi, ta mhdenik kai h perioq
sÔgklish sto Prìblhma 7.9.
7.10 D netai to aitiatì GQA sÔsthma diakritoÔ qrìnou me sunrthsh metafor
1 kz 1
H (z ) = 4
1 + 3z
k 1
Enìthta 7.6 Probl mata 277
1. Gia poie timè tou k to sÔsthma e nai eustajè ;
n
2. Gia k= 1 kai s ma eisìdou x(n) = 32 , poia e nai h èxodo tou sust -
mato ;
7.11 Gia èna grammikì qronik anallo wto sÔsthma d nontai
1. An to s ma eisìdou e nai to s ma x (n) = ( 2)n, tìte h èxodo tou sust -
mato e nai to s ma y n ( )=0 kai
2. an to s ma eisìdou e nai to s ma x(n) = 12 n u(n) tìte h èxodo tou
sust mato e nai to s ma y n () = Æ(n) + 41 n u(n), ìpou stajer
posìthta.
1. Na breje h stajer .
2. Na upologiste h sunrthsh metafor tou sust mato , kai
3. na prosdioriste h èxodo tou sust mato ìtan h e sodì tou e nai to s ma
( )=1
xn .
7.12 D netai to aitiatì GQA sÔsthma diakritoÔ qrìnou tou opo ou h e sodo x(n) kai
()
h èxodo y n ikanopoioÔn thn ex swsh diafor¸n
k
y(n) + y(n
3 1) = x(n) k4 x(n 1)
To opo o br sketai se hrem a.
1. Na breje h sunrthsh metafor tou sust mato kai na sqedisete to
digramma twn pìlwn kai mhdenik¸n kai na sqedisete th perioq sÔg-
klis th .
2. Gia poie timè th paramètrou k to sÔsthma e nai eustajè ;
3. Na prosdioriste h èxodo tou sust mato an k = 1 kai to s ma eisìdou
e nai n
x(n) =
2
3
7.13 àna aitiatì sÔsthma diakritoÔ qrìnou qarakthr zetai apì thn ex swsh diafor¸n
1
y(n) = y(n 2) + x(n) x(n 2)
4
ìpou x(n) e nai to s ma eisìdou kai y(n) to s ma exìdou. Na upologistoÔn
1. H sunrthsh metafor tou sust mato .
278 Metasqhmatismì z Keflaio 7
2. Na g nei to digramma twn pìlwn kai mhdenik¸n kai h perioq sÔgklish
th sunrthsh metafor tou sust mato
3. H kroustik apìkrish tou sust mato .
4. H èxodo tou sust mato an x(n) = u(n) ìpou u(n) e nai h monadia a
bhmatik akolouj a.
To sÔsthma br sketai se hrem a.
7.14 D netai sÔsthma diakritoÔ qrìnou to opo o èqei sunrthsh metafor
1 z 2
H (z ) = jzj > 0; 9
1 0; 81z 2 ;
1. Na upologiste h kroustik apìkrish tou sust mato .
2. Na prosdioriste h èxodo tou sust mato an h e sodì tou e nai h sunrthsh
monadia ou b mato .
7.15 An x(n) = an na upologistoÔn oi monìpleuroi metasqhmatismo z
1. X [x(n 2)℄
2. X [x(n + 2)℄
7.16 Na breje h kroustik apìkrish thlepikoinwniakoÔ kanalioÔ sto opo o parousi-
zontai dÔo diadìsei , dhlad , perigrfetai apì thn ex swsh diafor¸n,
y(n) = x(n) + ax(n 1)
Gia poie timè th paramètrou a to ant stofo sÔsthma e nai aitiatì kai eusta-
jè ;
Bibliograf a
7.1 S. Jeodwr dh , K. Mpermper dh , L. Kof dh , “Eisagwg sth Jewr a Shmtwn
kai Susthmtwn”, Tupwj tw - Gi¸rgo Dardanì , Aj na 2003.
7.2 N. Kaloupts dh , “S mata Sust mata kai Algìrijmoi”, D aulo , Aj na, 1994.
7.3 J. G. Proakis, D. G. Manolakis, “Introduction to Digital Signal Processing”,
MacMillan Publishing Company, 1994.
7.4 A. V. Oppenheim, R. W. Schafer, “Digital Signal Processing”, Prentice - Hall Inc.,
N. Y., 1975.
ÐÁÑÁÑÔÇÌÁ Á
ÌÅÑÉÊÁ ÂÁÓÉÊÁ ÓÔÏÉ×ÅÉÁ ÃÉÁ
ÔÏÕÓ ÌÉÃÁÄÉÊÏÕÓ ÁÑÉÈÌÏÕÓ
Sto parrthma autì anafèrontai oi trìpoi parstash enì migadikoÔ arijmoÔ
sto migadikì ep pedo kai or zontai merikè basikè ènnoie , ìpw mètro, fsh, prag-
matikì mèro , fantastikì mèro migadikoÔ arijmoÔ kai suzug migadikì arijmì .
A.1 PARASTASH MIGADIKOU ARIJMOU STO MIGADIKO EPIPEDO
Se kartesianè suntetagmène h morf enì migadikoÔ arijmoÔ z d netai apì thn ex sw-
sh
z = x + jy (A.1.1)
ìpou j= p 1 kai x kai y e nai pragmatiko arijmo , oi opo oi ant stoiqa onomzontai
pragmatikì mèro kai fantastikì mèro , tou migadikoÔ arijmoÔ. Sun jw
tm ma
sumbol zoume x = <e[z ℄ kai y = =m[z ℄. O migadikì arijmì z mpore ep sh na
parastaje se polikè suntetagmène apì thn ex swsh
z = r ej (A.1.2)
ìpou r > 0e nai to mètro tou migadikoÔ arijmoÔ z , to mètro sumbol zetai kai me
jj
z kai e nai h gwn a h fsh tou migadikoÔ arijmoÔ z ( = argz =\)
z.
Sto Sq ma A.1 uprqei h parstash enì migadikoÔ arijmoÔ sto migadikì ep pedo.
H sqèsh metaxÔ twn dÔo aut¸n ekfrsewn twn migadik¸n arijm¸n aporrèei apì th
sqèsh tou Euler
ej = os + j sin (A.1.3)
apì thn sqed ash tou z sto migadikì ep pedo, Sq ma A.1. ParathroÔme ìti
x=r os kai y = r sin (A.1.4)
280 Merik Basik Stoiqe a gia tou MigadikoÔ ArijmoÔ . Parrthma A
ℑm z
y Á
r
è Sq ma A.1 Parstash enì migadikoÔ arijmoÔ
0 x ℜe z sto migadikì ep pedo.
ìpou
p
r = x2 + y 2 ! !
y
= sin 1 p = os 1 p x = tan 1 y (A.1.5)
x + y2
2 x2 + y 2 x
Me th bo jeia th sqèsh tou Euler èqoume
os = 12 ej + e j
ej = os + j sin
sin = 21j ej e j
e j = os j sin (A.1.6)
A.2 SUZUGHS MIGADIKOS ARIJMOS - IDIOTHTES
An z = + = ej e nai èna
x jy r migadikì arijmì tìte o suzug migadikì tou z
pou sumbol zetai w z ? , d netai apì th sqèsh
z ? = x jy = r e j (A.2.1)
Gia dÔo suzuge migadikoÔ arijmoÔ èqoume ti idiìthte
1. An èna migadikì arijmì e nai so me to suzug tou migadikì arijmì, tìte to
fantastikì tou mèro e nai so me mhdèn, dhlad o arijmì e nai pragmatikì ,
prgmati
z = z ? ) x + jy = x jy ) y = 0
2. To tetrgwno tou mètrou migadikoÔ arijmoÔ e nai so me to ginìmeno tou mi-
gadikoÔ arijmoÔ ep to suzug tou migadikì arijmì, prgmati
z z ? = r ej r e j = r2
3. To pragmatikì mèro enì migadikoÔ arijmoÔ e nai so me to hmijroisma tou
migadikoÔ arijmoÔ kai tou suzug tou migadikoÔ arijmoÔ, prgmati
z + z?
z + z ? = x + jy + x jy = 2x kai <e[z ℄ =
2
Enìthta A.2 Suzug migadikì arijmì - Idiìthte 281
4. To fantastikì mèro enì migadikoÔ arijmoÔ e nai so me thn hmidiafor tou
suzugoÔ migadikoÔ arijmoÔ tou apì to migadikì arijmì diairoÔmenh me j , prg-
mati
z z?
z z ? = x + jy x + jy = 2jy kai =m[z ℄ =
2j
Pardeigma A.1
Na ekfraste o migadikoÔ arijmì z = 4+3
2 j
j se polik morf kai na parastaje sto
migadikì ep pedo.
LÔsh Pollaplasizoume arijmht kai paronomast me to suzug tou paronomast ,
ètsi èqoume
z=
4 + 3j = (4 + 3j )(2 + j ) = 8 + 4j + 6j 3 = 5 + 10j = 1 + 2j
2 j (2 j )(2 + j ) 4+1 5
Sto Sq ma A.2 blèpoume th parstash tou migadikoÔ arijmoÔ sto migadikì ep pedo.
ℑm z
y=2 Á
Sq ma A.2 H grafik parstash tou migadikoÔ
0 x=1 ℜe z arijmoÔ sto Pardeigma A.1.
Pardeigma A.2
p
Na ekfraste o migadikì arijmì z= 3 + j se polik morf kai na parastaje sto
migadikì ep pedo.
LÔsh Me th bo jeia twn sqèsewn (A.1.5) upolog zetai to mètro tou migadikoÔ arijmoÔ
p
r= 3+1=2
kai h fsh tou
p
tan = p13 = 33 ) = 6
ètsi h polik morf tou migadikoÔ arijmoÔ e nai
z = 2 ej 6
Sto Sq ma A.3 uprqei h parstash tou migadikoÔ arijmoÔ sto migadikì ep pedo
282 Merik Basik Stoiqe a gia tou MigadikoÔ ArijmoÔ . Parrthma A
ℑm z
Á
r=2
è= 6ð
Sq ma A.3 H grafik parstash tou migadikoÔ
0 ℜe z arijmoÔ sto Pardeigma A.2.
A.3 PROBLHMATA
A.1 Na ekfraste kje èna apì tou migadikoÔ se kartesian morf kai na paras-
taje sto migadikì ep pedo sto opo o na fa netai to pragmatikì kai to fan-
tastikì tm ma kje arijmoÔ.
p 6 ej 4 3
z1 = 2 ej 4 z2 = 3 ej4 +2 ej5 z3 =
1 j z4 = 2j ej 94
A.2 Na ekfraste kje èna apì tou paraktw migadikoÔ arijmoÔ se polik mor-
f kai na parastajoÔn sto migadikì ep pedo ìpou na fa netai to mètro kai h
fsh kje arijmoÔ.
p p p
z1 = 3 z2 = (1 j )4 z3 =
3 + j3
3+j
2
z4 =
3 +pj
1+j 3
ÐÁÑÁÑÔÇÌÁ B
ÁÍÁÐÔÕÎÇ ÑÇÔÇÓ ÓÕÍÁÑÔÇÓÇÓ
ÓÅ ÁÐËÁ ÊËÁÓÌÁÔÁ
O basikì skopì tou parart mato e nai na parousisei ton trìpo anlush
mia rht sunrthsh , dhlad , mia sunrthsh h opo a mpore na ekfraste w
lìgo dÔo poluwnÔmwn th metablht , se jroisma apl¸n klasmtwn.
()
àstw h rht sunrthsh f x h opo a èqei th morf
f (x) =
N (x)
= bamxxn ++ abm
m
1 xm 1 + : : : + b1 x + b0
D(x) n n 1 xn 1 + : : : + a1 x + a0
Ja exetsoume pr¸ta thn per ptwsh sthn opo a o bajmì tou arijmht , m, e nai
( )
mikrìtero tou bajmoÔ tou paronomast , n m < n kai sth sunèqeia ja exetsoume
thn per ptwsh pou o bajmì tou arijmht e nai megalÔtero so tou bajmoÔ tou
paronomast .
B.1 O BAJMOS TOU N (x) EINAI MIKROTEROS TOU BAJMOU TOU
D(x).
()
ätan o bajmì tou polu¸numou tou arijmht N x , e nai mikrìtero tou bajmoÔ tou
()
poluwnÔmou tou paronomast D x , dhlad e nai m < n, analÔoume ton paronomas-
t se ginìmeno paragìntwn
n
Y
D(x) = (x i ) (B.1.1)
i=1
ìpou 1 ; 2 ; : : : ; n oi r ze tou D(x). Anloga me th fÔsh twn riz¸n diakr noume
ti peript¸sei :
B.1.1 R ze diakekrimène kai pragmatikè
A jewr soume ìti o bajmì tou paronomast e nai 2, opìte h sunrthsh f (x) grfe-
tai diadoqik
b x + b0 b1 x + b0
f (x) = 2 1
x + a1 x + a0
= (x 1 )(x 2 )
= x C11 + x C22 (B.1.2)
284 Anptuxh rht sunrthsh se apl klsmata. Parrthma B
To prìblhma e nai na upolog soume ti stajerè C1 kai C2 . Knonta apaloif
paronomast¸n èqoume diadoqik:
b1 x + b0 = C1 (x 2 ) + C2 (x 1 ) = (C1 + C2 )x (C12 + C21 )
apì thn opo a èqoume to sÔsthma twn dÔo exis¸sewn
C1 + C2 = b1
C1 2 + C2 1 = b0
H lÔsh tou sust mato d nei ti timè twn stajer¸n C1 kai C2
b1 1 + b0
C1 = 1 2
b1 2 + b0
C2 = 2 1
(B.1.3)
An kai o trìpo autì isqÔei pnta, uprqei mia pio eÔkolh mèjodo . An jèloume na
upolog soume th stajer C1 pollaplasizoume thn (B.1.2) me x 1 kai èqoume
x 1
(x 1 )f (x) = C1 + C2
x 2
(B.1.4)
AfoÔ oi r ze 1 kai 2 e nai diakritè , o deÔtero ìro tou dexioÔ mèlou th (B.1.4)
e nai so me mhdèn gia x = 1 , ètsi èqoume
b +b
C1 = (x 1 )f (x)jx=1 = 1 1 0 (B.1.5)
1 2
ìmoia br skoume kai
b +b
C2 = (x 2 )f (x)jx=2 = 1 2 0 (B.1.6)
2 1
Oi dÔo auto trìpoi genikeÔontai, an o bajmì tou paronomast e nai n, kai èqoume
C1 C2
f (x) = +
x 1 x 2
+ : : : + x Cnn (B.1.7)
kai oi stajerè upolog zontai apì ton tÔpo
Ck = (x k )f (x)jx=k ; k = 1; 2; : : : ; n (B.1.8)
Enìthta B.1 O bajmì tou N (x) e nai mikrìtero tou bajmoÔ tou D(x) . 285
B.1.2 R ze pollaplè kai pragmatikè
A upojèsoume ìti o paronomast èqei m a dipl pragmatik r za, thn 1 , kai m a
()
apl pragmatik r za, thn 2 , tìte h sunrthsh f x grfetai:
b x2 + b1 x + b0
f (x) = 2
(x 1)2 (x 2 ) (B.1.9)
Sthn per ptwsh aut anazhtoÔme èna anptugma th morf :
C11
f (x) =
(x + C12 + C21
1 ) (x 1 )2 (x 2 )
(B.1.10)
Gia na upolog soume tou suntelestè C11 , C12 kai C21 mporoÔme kai ed¸ na knoume
apaloif paronomast¸n na exis¸soume tou suntelestè twn omobjmiwn ìrwn kai
na lÔsoume to sÔsthma. Uprqei ìmw kai giaut thn per ptwsh èna aploÔstero
trìpo .
Pollaplasizoume thn (B.1.10) me x 1 2 kai èqoume:
( )
C (x 1 )2
(x 1 )2 f (x) = C11 (x 1 ) + C12 + 21
(x 2 ) (B.1.11)
Apì thn (B.1.11) upolog zoume thn C12 me th sqèsh:
b 2 + b + b
C12 = (x 1 )2 f (x) x=1 = 2 1 1 1 0 (B.1.12)
1 2
Gia na upolog soume to C11 diafor zoume thn (B.1.11) w pro x kai èqoume:
d
(
x 1 )2 f (x) = C11 + C21
2(x 1)(x 2) (x 1 )2
dx
(x 2)2
2
= C11 + C21 2((xx 21)) 2((xx 21))2 (B.1.13)
o teleuta o ìro th (B.1.13) e nai so me mhdèn gia x = 1 ètsi èqoume:
d
C11 =
dx
(x 1 )2 f (x)
x=1
2
= 2b2 1 + b1 b2(1+ b11)+2 b0 (B.1.14)
1 2 1 2
O suntelest C21 upolog zetai apì th gnwst sqèsh:
C21 = (x 2 )f (x)jx=2
2
= b2(2+2 b112)+2 b0 (B.1.15)
286 Anptuxh rht sunrthsh se apl klsmata. Parrthma B
Genik, an to polu¸numo tou paronomast èqei th r za 1 me pollaplìthta r kai n r
aplè r ze 2 ; 3 ; : : : ; n r+1 tìte o paronomast analÔetai
r+1
nY
D(x) = (x 1 )r (x i ) (B.1.16)
i=2
ètsi sunrthsh f (x) analÔetai se apl klsmata w
C11 C12 C1r C21 C(n r)1
f (x) = + + + + + +
(x 1) (x 1 ) 2 (x 1) (x 2 )
r (x n r ) (B.1.17)
kai oi suntelestè C1i ; i = 1; 2; : : : ; r upolog zontai apì thn:
C1i =
1 dr i [(x 1)r f (x)℄
(r i)! dxr i x=i
(B.1.18)
oi upìloipe stajerè Cki ; 4 = 2; 3; : : : ; n r upolog zontai me thn (B.1.8).
B.1.3 Ìparxh migadik¸n riz¸n
An to polu¸numo D(x) èqei èna zeÔgo suzug¸n migadik¸n riz¸n 1 = + j! kai
2 = 1 =
? ()
j!, tìte h sunrthsh f x analÔetai w :
C1 C2
f (x) = +
x 1 x ?1
+ (x C3 )r + + x Cn (B.1.19)
3 n
äloi oi suntelestè upolog zontai apì thn sqèsh:
Ck = (x k )f (x)jx=k ; k = 1; 2; : : : ; n (B.1.20)
shmei¸netai ìti oi suntelestè C1 kai C2 e nai suzuge migadiko (C2
C1? . Se = )
per ptwsh pou oi r ze emfan zontai me kpoia pollaplìthta, akolouje tai h pro-
hgoÔmenh mejodolog a.
B.2 O BAJMOS TOU N (x) EINAI MEGALUTEROS H ISOS TOU BA-
JMOU TOU D(x)
()
An o bajmì tou polu¸numou tou arijmht N x , e nai megalÔtero so tou bajmoÔ
()
tou polu¸numou tou paronomast D x , dhlad , m
n, knoume th dia resh kai h
()
sunrthsh f x grfetai:
N (x) g(x)
f (x) = = ( x) +
D(x) D(x)
(B.2.1)
Epeid o bajmì tou poluwnÔmou g (x) e nai mikrìtero apì to bajmì tou D (x), o ìro
g(x)
D(x) sthn (B.2.1) analÔetai se apl klsmata ìpw se kpoia apì ti peript¸sei
pou perigryame.
ÐÁÑÁÑÔÇÌÁ Ã
×ÑÇÓÉÌÏÉ ÌÁÈÇÌÁÔÉÊÏÉ ÔÕÐÏÉ
O basikì skopì tou parart mato e nai na parousisei qr sime sqèsei apì
ta majhmatik.
G.1 Trigwnometr a.
Gia to orjog¸nio tr gwno tou sq mato isqÔoun oi sqèsei
sin = yr (G.1.1)
os = xr (G.1.2)
r
tan = xy = sin
y
os (G.1.3) è
x
2 2
sin + os = 1 (G.1.4) To orjog¸nio tr gwno.
os 2 = sin (G.1.5)
sin 2 = os (G.1.6)
Sth sunèqeia parousizontai trigonwmetrikè tautìthte
sin( ) = sin os os sin (G.1.7)
os( ) = os os sin sin (G.1.8)
2sin sin = os( ) os( + ') (G.1.9)
2 os os = os( ) + os( + ') (G.1.10)
2sin os = sin( ) + sin( + ') (G.1.11)
os2 = 12 (1 + os 2) (G.1.12)
288 Qr simoi Majhmatiko TÔpoi. Parrthma G
sin2 = 21 (1 os 2) (G.1.13)
os(2) = os2 sin2
= 2 os2 1
= 1 2sin2 (G.1.14)
4 os3 = 3 os + os(3) (G.1.15)
4sin3 = 3sin sin(3) (G.1.16)
a os b sin = A os( + ) (G.1.17)
ìpou p
A = a2 + b2
= tan 1 (b=a)
a = A os
b = A sin
G.2 Aìrista oloklhr¸mata.
G.2.1 Rht¸n alebrik¸n sunart sewn
Z
xn dx =
1 xn+1; n 6= 1
n+1
(G.2.1)
Z
dx
= 1 ln ja + bxj
a + bx b
(G.2.2)
n+1
(a + bx)n dx = (ab+(nbx+)1) ; n > 0
Z
(G.2.3)
Z
dx 1
(a + bx) (n 1)b(a + bx)n 1 ; n > 1
n = (G.2.4)
Z
dx
= 1 tan 1 bx
a +b x
2 2 2 (G.2.5)
ab a
= 1 ln(a2 + x2)
Z
xdx
a2 + x2 2
(G.2.6)
x2 dx
Z x
=x a tan 1
a2 + x2
(G.2.7)
a
Enìthta G.2 Aìrista oloklhr¸mata 289
G.2.2 Trigwnometrik¸n sunart sewn
os( x) dx = 1 sin( x)
Z
(G.2.8)
os( x) dx = 1 [ os( x) + x sin( x)℄
Z
x 2 (G.2.9)
Z
x2 os x dx = 2x os x + (x2 2)sin x (G.2.10)
sin( x) dx = 1 os( x)
Z
(G.2.11)
Z
x sin( x) dx = 2 [sin( x) + x
1 os( x)℄ (G.2.12)
Z
x2 sin x dx = 2x sin x (x2 2) os x (G.2.13)
G.2.3 Ekjetik¸n sunart sewn
Z
eax dx = eax
1 (G.2.14)
a
x 1
Z
xe dx = e
ax ax
a a2
(G.2.15)
Z
eax
eax sin( x) dx = 2 2 [a sin( x) os( x)℄
a +
(G.2.16)
Z
eax
eax os( x) dx = 2 2 [a os( x) + sin( x)℄
a +
(G.2.17)
G.2.4 Orismèna oloklhr¸mata
Z 1
x=22 dx p
e = 2; >0 (G.2.18)
1
Z 1 p
x2 e x=22 dx = 3 2; >0 (G.2.19)
1
Z 1 p b2
e a2 x+ bx dx = a
e 4a2 ; a > 0 (G.2.20)
1
Z 1 Z 1 sin x
sin (x) dx = x
dx =
2 (G.2.21)
0 Z
0
1
sin 2 (x) dx = 2 (G.2.22)
0
290 Qr simoi Majhmatiko TÔpoi. Parrthma G
G.3 Gewmetrikè seirè
N
X N (N + 1)
n=
n=1 2 (G.3.1)
XN
N (N + 1)(2N + 1)
n2 =
n=1
6 (G.3.2)
XN
N 2 (N + 1)2
n3 =
n=1
4 (G.3.3)
(
NX1 1 xN ; x 6= 1
xn = 1 x
N; x=1
(G.3.4)
n=0
m (
X
xn =
xk xm+1 ;
1 x x 6= 1
m k + 1; x = 1
(G.3.5)
n=k
1
X 1
xn = ; jxj < 1
n=0
1 x
(G.3.6)
1
X xk
xn = ; jxj < 1
n=k
1 x
(G.3.7)
1
X x
nxn =
n=k
(1 x)2 ; jxj < 1 (G.3.8)
XN
ej (+n ) =
sin[(N + 1) =2℄ ej[+(N =2)℄
n=0 sin( =2) (G.3.9)
x2 x 3 1 xn
X
ex = 1 + x +
2! 3! + =
+
n=0
n!
(G.3.10)
ÅõñåôÞñéï
jroisma th sunèlixh , 51 Grafikì prosdiorismì th Jemeli¸dh analogik
Ajroist , 32 sunèlixh , 47 per odo , 11
Aitiatì s ma, 4 suqnìthta, 11
Aitiatì sÔsthma, 37 Decibel, 125 kuklik suqnìthta, 11
Aitiokratikì s ma, 7 Deigmatolhy a, 2 Jemeli¸dh yhfiak kuklik
Akolouj a eterosusqètish , 249 per odo deigmatolhy a , 2 suqnìthta, 14
Akolouj a susqètish , 249 Dexiìpleurh akolouj a, 244 Je¸rhma Parseval, 99
Amf pleurh akolouj a, 242 Dexiìpleurh s ma, 209 Je¸rhma th sunèlixh tou ML,
Amf pleuro s ma, 207
Diagrmmata Bode, 125 211
Amf pleuro metasqhmatismì
Diakritì Fsma, 79, 74 Je¸rhma th sunèlixh tou MF, 97
Diakritì metasqhmatismì
Laplace, 200
Amf pleuro metasqhmatismì z, Fourier, 171 Idanikì katwperatì f ltro, 128
Diamìrfwsh, 95, 135
234 Idanikì f ltro basik z¸nh , 128
Diamorfwt , 61
Anklash, 8 Idiìthte susthmtwn, 37
Distash dianusmatikoÔ q¸rou, 64
Analogoyhfiakì metatropèa , 33 Idiìthte th sunèlixh , 45
Diat rhsh th suqnìthta , 55, 57
Analutik sunrthsh, 204 antimetajetik , 45
Diaforik ex swsh
Anptugma Fourier, 69 epimeristik , 46
deÔterh txh , 35
Antistrof sust mato , 35 proseteristik , 46
me stajeroÔ suntelestè , 34
Ant strofo diakritì tautotik , 46
pr¸th txh , 34
metasqhmatismì Fourier, 171 Idiìthte tou ML, 209, 217
Diaforist , 41
Apìkrish isqÔo , 113 Idiìthte tou MF, 151, 176
Duðsmì , 105
Apìkrish mhdenik eisìdou, 270 Idiìthte tou Mz, 245, 251
Dunamikì sÔsthma, 39
Apìkrish mhdenik eisìdou, 270 IsqÔ s mato , 7
Apìkrish mhdenik katstash ,
Euler sqèsh, 279
270 Kajustèrhsh, 249
E sodo sust mato , 32
Apìkrish monadia ou de gmato , 51 Katstash hrem a , 37
Ekjetik seir Fourier, 69
Apìkrish pltou , 55 Kbntish, 2
Enèrgeia s mato , 6
Apìkrish suqnìthta sust mato , Kentrikì lobì , 115
Energeiakì s ma, 6
55, 57
Ex swsh anlush , 69, ,139, 145
Krit rio Nyquist, 168
Apìkrish fsh , 55 Kroustik akolouj a, 26
Ex swsh sÔnjesh , 70, 139, 145
Aristerìplerh akolouj a, 244 Kroustik apìkrish sust mato ,
àxodo sust mato , 32
Aristerìpleuro s ma, 208 44, 51
Eswterikì ginìmeno
Armonik susqetizìmena ekjetik Kroustik sunrthsh, 19
dianusmtwn, 65
s mata Kuklik anklash akolouj a , 173
shmtwn, 66
diakritoÔ qrìnou, 137 Kuklik ol sjhsh akolouj a , 174
Eukle dio q¸ro shmtwn, 66
suneqoÔ qrìnou, 67, 68 Eustajè sÔsthma, 40 Kuklik sunèlixh akolouji¸n, 175
Armonik sunist¸sa fsmato , 70 Efarmogè twn MF, 119 Kuklik suqnìthta, 4
rtio s ma, 5 Efarmogè twn ML, 219 Kuklik suqnìthta -3 dB, 127
Efarmogè twn Mz, 262 Kwdikopo hsh, 3
L2
Perittì s ma, 5
Bajmwtì pollaplasiast , 32 Z¸nh -mètro
apokop , 128
Grammik sunèlixh, 51, 177 dièleush , 128 Meikt sÔndesh susthmtwn, 35
Grammikì sÔsthma, 37 metbash , 130 Metabatik katstash, 269
Grammikì fsma, 83 Mèsh tim s mato , 70
292 Euret rio
Mèsh qronik sunrthsh deigmatolhy a , 164 analutik , 204
autosusqètish , 113 jemeli¸dh , 3 autosusqètish , 99, 111, 250
Metasqhmatismì Fourier, 55, 57, Perioq sÔgklish , 200, 251 deigmatolhy a , 93
88, 145, 171 Perittì s ma, 5 kl sh , 25
Metasqhmatismì Laplace, 54, 200, Pollaplasiast , 32 monadia ou b mato , 18
Metasqhmatismì z, 57, 234 Pìlo sunrthsh , 204 pros mou, 25
Mètro Poluwnumikì ekjetikì s ma, 205 Sunrthsh dèlta kroustik
dianÔsmato , 65 Pragmatikì ekjetikì s ma, 11 Dirac, 19
migadikoÔ arijmoÔ, 279 Pragmatikì mèro migadikoÔ idiìthta ol sjhsh , 20
s mato , 66 arijmoÔ, 279 Sunrthsh metafor sust mato ,
Mhdenikì sunrthsh , 204 54, 50, 57
M ko dianÔsmato , 65 Rht sunrthsh, 121, 204 Sundèsei susthmtwn, 35
Migadikì ekjetikì s ma me anatrofodìthsh me
diakritoÔ qrìnou, 14 Seir Fourier diakritoÔ qrìnou, andrash, 35
suneqoÔ qrìnou, 10 139, meikt , 35
Monadia a bhmatik akolouj a, 26 Seiriak sÔndesh susthmtwn, 35 parllhlh, 35
Monadia o b ma diakritoÔ qrìnou, S ma, 1 seiriak , 35
26 aitiatì, 4 Sunèlixh, 44, 51
Monadia o de gma, 26 aitiokratikì, 7 Suneqè fsma, 145
Monadia o kÔklo , 235 analogikì, 2 Suneq sunist¸sa fsmato , 70
Mìnimh katstash, 269 peirh dirkeia , 5 Sunj ke Dirichlet, 74
Monodistato s ma, 1 apl suqnìthta , 12 Suntelestè Fourier, 70, 139
Monìpleuro Metasqhmatismì rtio, 5 Suntelest
Laplace, 216 diakritoÔ qrìnou, 2 susqètish , 249
Monìpleuro Metasqhmatismì z, didistato, 1 autosusqètish , 250
234 enèrgeia , 7 SÔsthma, 32
ekjetik txh , 208 aitiatì, 37
Nomoteleiakì s ma, 7 isqÔo , 7 aitiokratikì, 32
monodistato, 1 analogikì, 32
Olokl rwma th sunèlixh , 44 nomoteleiakì, 7 antistrèyimo, 38
Oloklhrwt , 61 periodikì, 3 grammikì, 37
Orjog¸nia perittì, 5 diamìrfwsh , 61
dianÔsmata, 65 peperasmènh dirkeia , 5 deÔterh txh , 35, 122
s mata, 66 peperasmèno, 5 diakritoÔ qrìnou, 28
Orjog¸nio sÔnolo shmtwn, 137 poludistato, 1 diafìrish , 41
Orjog¸nio palmì , 24 stoqastikì,7 eustajè , 40
Orjokanonik bsh shmtwn, 66 suneqoÔ qrìnou, 2 kajustèrhsh , 32
Orjokanonikì sÔnolo dianusmtwn, tuqa o, 7 me mn mh, 39
65 uyhl¸n suqnot twn, 72 mèsh tim , 38, 45, 62
fj nwn hmitonoeidè , 13 m a eisìdou mia exìdou, 32
Palmì qamhl¸n suqnot twn, 72 olokl rwsh , 61
orjog¸nio , 24 yhfiakì, 3 pl rou anìrjwsh , 60
trigwnikì ,24 Shme o -3dB, 114 poll¸n eisìdwn mia exìdou, 32
Palmokwdik diamìrfwsh, 3 Shme o anwmal a sunrthsh , 204 poll¸n eisìdwn poll¸n exìdwn,
ParajÔrwsh, 174 Stajer apìsbesh sust mato , 34 32
Parllhlh sÔndesh susthmtwn, Stajer elathr ou, 34 pr¸th txh , 34, 121, 188
31 Stajer sunist¸sa fsmato , 70 stoqastikì, 32
Parembol , 154 Perittì s ma, 5 suneqoÔ qrìnou, 32
Peperasmèno s ma, 5 Stajer qrìnou, 126 qronik anallo wto, 39
Peribllousa, 13 Stoqastikì s ma, 7 qwr mn mh, 34
Periodikì s ma, 3 Stoqastikì sÔsthma, 32 ubridikì, 33
Periodik sunèlixh, 157 Suzug migadikì arijmì , 280 Suqnìthta -3 dB,
Per odo , 3 Sunrthsh Suqnìthta Nyquist, 168
Euret rio 293
Suqnìthta apokop , 128 Fsh migadikoÔ arijmoÔ, 279 zwnodiabatì, 130
Suqnìthta deigmatolhy a , 164 Fsma s mato , 70 zwnofraktikì, 130
Sqèsh metaxÔ ML kai MF, 201 diakritì, 88 katwperatì, 128
Sqèsh tou Euler, 279 suneqè , 88, 145 uyiperatì, 130
Fsma suqnot twn, 104 apìrriyh suqnìthta , 130
Txh sust mato , 34
Fasmatik mhdenik, 90
Tautìthta Parseval, 77 Fasmatikè grammè , 70, 139 Qronik anallo wto sÔsthma, 39
Tmhmatik omalè sunart sei , 74
Fasmatik puknìthta enèrgeia , Qronik
Trigwnikì palmì , 24
100 diastol , 8
Trigwnometrik seir Fourier, 70 Fasmatik puknìthta isqÔo , 113 sustol , 9
Tuqa o s ma, 7
Fasmatik puknìthta pltou , 88 Qronik metatìpish, 9
Fèrousa suqnìthta, 96, 135 Qronik stajer, 126
Ubridikì sÔsthma, 33
Fj nonta hmitonoeid s mata, 13 Q¸ro shmtwn, 66
Ìparxh MF, 72
FEFE eustajè sÔsthma, 40
F ltro, 128 Yhfiakì s ma, 2
Fainìmeno Gibbs, 82, 90
bajuperatì, 130 Yhfiak kuklik suqnìthta, 14
Fantastikì mèro migadikoÔ
basik z¸nh , 128, 130 Yhfioanalogikì metatropèa , 33
arijmoÔ, 279