NAME

PDL::FFT - FFTs for PDL

DESCRIPTION

!!!!!!!!!!!!!!!!!!!!!!!!!!WARNING!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! As of PDL-2.006_04, the direction of the FFT/IFFT has been reversed to match the usage in the FFTW library and the convention in use generally. !!!!!!!!!!!!!!!!!!!!!!!!!!WARNING!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

FFTs for PDL. These work for arrays of any dimension, although ones with small prime factors are likely to be the quickest. The forward FFT is unnormalized while the inverse FFT is normalized so that the IFFT of the FFT returns the original values.

For historical reasons, these routines work in-place and do not recognize the in-place flag. That should be fixed.

SYNOPSIS

        use PDL::FFT qw/:Func/;
        fft($real, $imag);
        ifft($real, $imag);
        realfft($real);
        realifft($real);
        fftnd($real,$imag);
        ifftnd($real,$imag);
        $kernel = kernctr($image,$smallk);
        fftconvolve($image,$kernel);

DATA TYPES

The underlying C library upon which this module is based performs FFTs on both single precision and double precision floating point piddles. Performing FFTs on integer data types is not reliable. Consider the following FFT on piddles of type 'double':

        $r = pdl(0,1,0,1);
        $i = zeroes($r);
        fft($r,$i);
        print $r,$i;
        [2 0 -2 0] [0 0 0 0]

But if $r and $i are unsigned short integers (ushorts):

        $r = pdl(ushort,0,1,0,1);
        $i = zeroes($r);
        fft($r,$i);
        print $r,$i;
        [2 0 65534 0] [0 0 0 0]

This used to occur because PDL::PP converts the ushort piddles to floats or doubles, performs the FFT on them, and then converts them back to ushort, causing the overflow where the amplitude of the frequency should be -2.

Therefore, if you pass in a piddle of integer datatype (byte, short, ushort, long) to any of the routines in PDL::FFT, your data will be promoted to a double-precision piddle. If you pass in a float, the single-precision FFT will be performed.

FREQUENCIES

For even-sized input arrays, the frequencies are packed like normal for FFTs (where N is the size of the array and D is the physical step size between elements):

 0, 1/ND, 2/ND, ..., (N/2-1)/ND, 1/2D, -(N/2-1)/ND, ..., -1/ND.

which can easily be obtained (taking the Nyquist frequency to be positive) using

"$kx = $real->xlinvals(-($N/2-1)/$N/$D,1/2/$D)->rotate(-($N/2 -1));"

For odd-sized input arrays the Nyquist frequency is not directly acessible, and the frequencies are

 0, 1/ND, 2/ND, ..., (N/2-0.5)/ND, -(N/2-0.5)/ND, ..., -1/ND.

which can easily be obtained using

"$kx = $real->xlinvals(-($N/2-0.5)/$N/$D,($N/2-0.5)/$N/$D)->rotate(-($N-1)/2);"

ALTERNATIVE FFT PACKAGES

Various other modules - such as PDL::FFTW and PDL::Slatec - contain FFT routines. However, unlike PDL::FFT, these modules are optional, and so may not be installed.

FUNCTIONS

  Signature: ([o,nc]real(n); [o,nc]imag(n))

  Signature: ([o,nc]real(n); [o,nc]imag(n))

        realfft($real);

        realifft($real);

        fftnd($real,$imag);

        ifftnd($real,$imag);

        $kernel = kernctr($image,$smallk);
        fftconvolve($image,$kernel);

  Signature: ([o,nc]a(m); [o,nc]b(m))

  Signature: (ar(); ai(); br(); bi(); [o]cr(); [o]ci())

  Signature: (ar(); ai(); br(); bi(); [o]cr(); [o]ci())

BUGS

Where the source is marked `FIX', could re-implement using phase-shift factors on the transforms and some real-space bookkeeping, to save some temporary space and redundant transforms.

AUTHOR

This file copyright (C) 1997, 1998 R.J.R. Williams (rjrw@ast.leeds.ac.uk), Karl Glazebrook (kgb@aaoepp.aao.gov.au), Tuomas J. Lukka, (lukka@husc.harvard.edu). All rights reserved. There is no warranty. You are allowed to redistribute this software / documentation under certain conditions. For details, see the file COPYING in the PDL distribution. If this file is separated from the PDL distribution, the copyright notice should be included in the file.