Provided by: pdl_2.007-2build1_amd64 
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
fft()
Complex 1-D FFT of the "real" and "imag" arrays [inplace].
Signature: ([o,nc]real(n); [o,nc]imag(n))
fft($real,$imag);
ifft()
Complex inverse 1-D FFT of the "real" and "imag" arrays [inplace].
Signature: ([o,nc]real(n); [o,nc]imag(n))
ifft($real,$imag);
realfft()
One-dimensional FFT of real function [inplace].
The real part of the transform ends up in the first half of the array and the imaginary part of the
transform ends up in the second half of the array.
realfft($real);
realifft()
Inverse of one-dimensional realfft routine [inplace].
realifft($real);
fftnd()
N-dimensional FFT over all pdl dims of input (inplace)
fftnd($real,$imag);
ifftnd()
N-dimensional inverse FFT over all pdl dims of input (inplace)
ifftnd($real,$imag);
fftconvolve()
N-dimensional convolution with periodic boundaries (FFT method)
$kernel = kernctr($image,$smallk);
fftconvolve($image,$kernel);
fftconvolve works inplace, and returns an error array in kernel as an accuracy check -- all the values in
it should be negligible.
See also PDL::ImageND::convolveND, which performs speed-optimized convolution with a variety of boundary
conditions.
The sizes of the image and the kernel must be the same. kernctr centres a small kernel to emulate the
behaviour of the direct convolution routines.
The speed cross-over between using straight convolution (PDL::Image2D::conv2d()) and these fft routines
is for kernel sizes roughly 7x7.
convmath
Signature: ([o,nc]a(m); [o,nc]b(m))
Internal routine doing maths for convolution
convmath does not process bad values. It will set the bad-value flag of all output piddles if the flag
is set for any of the input piddles.
cmul
Signature: (ar(); ai(); br(); bi(); [o]cr(); [o]ci())
Complex multiplication
cmul does not process bad values. It will set the bad-value flag of all output piddles if the flag is
set for any of the input piddles.
cdiv
Signature: (ar(); ai(); br(); bi(); [o]cr(); [o]ci())
Complex division
cdiv does not process bad values. It will set the bad-value flag of all output piddles if the flag is
set for any of the input piddles.
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.
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