NAME¶
PDL::FFT - FFTs for PDL
DESCRIPTION¶
FFTs for PDL. These work for arrays of any dimension, although ones with small
prime factors are likely to be the quickest.
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 FFT of the "real" and "imag" arrays [inplace].
fft($real,$imag);
ifft()¶
Complex inverse FFT of the "real" and "imag" arrays
[inplace].
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 (inplace)
fftnd($real,$imag);
ifftnd()¶
N-dimensional inverse FFT
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.