

 
Manual Reference Pages  PDL::FFT (3)
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NAME
PDL::FFT  FFTs for PDL
CONTENTS
DESCRIPTION
!!!!!!!!!!!!!!!!!!!!!!!!!!WARNING!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
As of PDL2.006_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 inplace and do not recognize
the inplace 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 doubleprecision piddle. If you pass in a float, the
singleprecision FFT will be performed.
FREQUENCIES
For evensized 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/21)/ND, 1/2D, (N/21)/ND, ..., 1/ND.
which can easily be obtained (taking the Nyquist frequency to be
positive) using
$kx = $real>xlinvals(($N/21)/$N/$D,1/2/$D)>rotate(($N/2 1));
For oddsized input arrays the Nyquist frequency is not directly
acessible, and the frequencies are
0, 1/ND, 2/ND, ..., (N/20.5)/ND, (N/20.5)/ND, ..., 1/ND.
which can easily be obtained using
$kx = $real>xlinvals(($N/20.5)/$N/$D,($N/20.5)/$N/$D)>rotate(($N1)/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 1D FFT of the real and imag arrays [inplace].
Signature: ([o,nc]real(n); [o,nc]imag(n))
fft($real,$imag);
Complex inverse 1D FFT of the real and imag arrays [inplace].
Signature: ([o,nc]real(n); [o,nc]imag(n))
ifft($real,$imag);
Onedimensional 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($real);
realifft()
Inverse of onedimensional realfft routine [inplace].
realifft($real);
fftnd()
Ndimensional FFT over all pdl dims of input (inplace)
fftnd($real,$imag);
ifftnd()
Ndimensional inverse FFT over all pdl dims of input (inplace)
ifftnd($real,$imag);
fftconvolve()
Ndimensional 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 speedoptimized 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 crossover 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 badvalue 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 badvalue 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 badvalue flag of all output piddles if the flag is set for any of the input piddles.
BUGS
Where the source is marked ‘FIX’, could reimplement using phaseshift
factors on the transforms and some realspace 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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