pdl::fft(3) [redhat man page]
FFT(3) User Contributed Perl Documentation FFT(3) 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. Each routine works `inplace'. 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,$kernel); ALTERNATIVE FFT PACKAGES
Various other modules - such as PDL::FFTW and PDL::Slatec - contain FFT routines. However, unlike PDL::FFT, these modules are optional, and so may not be installed. FUNCTIONS
fft Signature: ([o,nc]real(n); [o,nc]imag(n)) Complex FFT of the "real" and "imag" arrays [inplace] ifft Signature: ([o,nc]real(n); [o,nc]imag(n)) Complex Inverse FFT of the "real" and "imag" arrays [inplace] realfft() One-dimensional FFT of real function [inplace]. The real part of the transform ends up in the first half of the array and the imaginary part of the transform ends up in the second half of the array. realfft($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 $kernel = kernctr($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. 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. kernctr() `centre' a kernel (auxiliary routine to fftconvolve) $kernel = kernctr($image,$smallk); fftconvolve($image,$kernel); kernctr centres a small kernel to emulate the behaviour of the direct convolution routines. convmath Signature: ([o,nc]a(m); [o,nc]b(m)) Internal routine doing maths for convolution cmul Signature: (ar(); ai(); br(); bi(); [o]cr(); [o]ci()) Complex multiplication cdiv Signature: (ar(); ai(); br(); bi(); [o]cr(); [o]ci()) Complex division BUGS
Where the source is marked `FIX', could re-implement using phase-shift factors on the transforms and some real-space bookkeeping, to save some temporary space and redundant transforms. AUTHOR
This file copyright (C) 1997, 1998 R.J.R. Williams (rjrw@ast.leeds.ac.uk), Karl Glazebrook (kgb@aaoepp.aao.gov.au), Tuomas J. Lukka, (lukka@husc.harvard.edu). All rights reserved. There is no warranty. You are allowed to redistribute this software / documentation under certain conditions. For details, see the file COPYING in the PDL distribution. If this file is separated from the PDL distribution, the copyright notice should be included in the file. perl v5.8.0 2003-01-29 FFT(3)
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Complex(3pm) User Contributed Perl Documentation Complex(3pm) NAME
PDL::Complex - handle complex numbers SYNOPSIS
use PDL; use PDL::Complex; DESCRIPTION
This module features a growing number of functions manipulating complex numbers. These are usually represented as a pair "[ real imag ]" or "[ angle phase ]". If not explicitly mentioned, the functions can work inplace (not yet implemented!!!) and require rectangular form. While there is a procedural interface available ("$a/$b*$c <=> Cmul (Cdiv $a, $b), $c)"), you can also opt to cast your pdl's into the "PDL::Complex" datatype, which works just like your normal piddles, but with all the normal perl operators overloaded. The latter means that "sin($a) + $b/$c" will be evaluated using the normal rules of complex numbers, while other pdl functions (like "max") just treat the piddle as a real-valued piddle with a lowest dimension of size 2, so "max" will return the maximum of all real and imaginary parts, not the "highest" (for some definition) TIPS, TRICKS &; CAVEATS o "i" is a constant exported by this module, which represents "-1**0.5", i.e. the imaginary unit. it can be used to quickly and conviniently write complex constants like this: "4+3*i". o Use "r2C(real-values)" to convert from real to complex, as in "$r = Cpow $cplx, r2C 2". The overloaded operators automatically do that for you, all the other functions, do not. So "Croots 1, 5" will return all the fifths roots of 1+1*i (due to threading). o use "cplx(real-valued-piddle)" to cast from normal piddles into the complex datatype. Use "real(complex-valued-piddle)" to cast back. This requires a copy, though. o This module has received some testing by Vanuxem Gregory (g.vanuxem at wanadoo dot fr). Please report any other errors you come across! EXAMPLE WALK-THROUGH The complex constant five is equal to "pdl(1,0)": pdl> p $x = r2C 5 5 +0i Now calculate the three roots of of five: pdl> p $r = Croots $x, 3 [1.70998 +0i -0.854988 +1.48088i -0.854988 -1.48088i] Check that these really are the roots of unity: pdl> p $r ** 3 [5 +0i 5 -1.22465e-15i 5 -7.65714e-15i] Duh! Could be better. Now try by multiplying $r three times with itself: pdl> p $r*$r*$r [5 +0i 5 -4.72647e-15i 5 -7.53694e-15i] Well... maybe "Cpow" (which is used by the "**" operator) isn't as bad as I thought. Now multiply by "i" and negate, which is just a very expensive way of swapping real and imaginary parts. pdl> p -($r*i) [0 -1.70998i 1.48088 +0.854988i -1.48088 +0.854988i] Now plot the magnitude of (part of) the complex sine. First generate the coefficients: pdl> $sin = i * zeroes(50)->xlinvals(2,4) + zeroes(50)->xlinvals(0,7) Now plot the imaginary part, the real part and the magnitude of the sine into the same diagram: pdl> line im sin $sin; hold pdl> line re sin $sin pdl> line abs sin $sin Sorry, but I didn't yet try to reproduce the diagram in this text. Just run the commands yourself, making sure that you have loaded "PDL::Complex" (and "PDL::Graphics::PGPLOT"). FUNCTIONS
cplx real-valued-pdl Cast a real-valued piddle to the complex datatype. The first dimension of the piddle must be of size 2. After this the usual (complex) arithmetic operators are applied to this pdl, rather than the normal elementwise pdl operators. Dataflow to the complex parent works. Use "sever" on the result if you don't want this. complex real-valued-pdl Cast a real-valued piddle to the complex datatype without dataflow and inplace. Achieved by merely reblessing a piddle. The first dimension of the piddle must be of size 2. real cplx-valued-pdl Cast a complex valued pdl back to the "normal" pdl datatype. Afterwards the normal elementwise pdl operators are used in operations. Dataflow to the real parent works. Use "sever" on the result if you don't want this. r2C Signature: (r(); [o]c(m=2)) convert real to complex, assuming an imaginary part of zero r2C 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. i2C Signature: (r(); [o]c(m=2)) convert imaginary to complex, assuming a real part of zero i2C 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. Cr2p Signature: (r(m=2); float+ [o]p(m=2)) convert complex numbers in rectangular form to polar (mod,arg) form. Works inplace Cr2p 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. Cp2r Signature: (r(m=2); [o]p(m=2)) convert complex numbers in polar (mod,arg) form to rectangular form. Works inplace Cp2r 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: (a(m=2); b(m=2); [o]c(m=2)) 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. Cprodover Signature: (a(m=2,n); [o]c(m=2)) Project via product to N-1 dimension Cprodover 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. Cscale Signature: (a(m=2); b(); [o]c(m=2)) mixed complex/real multiplication Cscale 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: (a(m=2); b(m=2); [o]c(m=2)) 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. Ccmp Signature: (a(m=2); b(m=2); [o]c()) Complex comparison oeprator (spaceship). It orders by real first, then by imaginary. Hm, but it is mathematical nonsense! Complex numbers cannot be ordered. Ccmp 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. Cconj Signature: (a(m=2); [o]c(m=2)) complex conjugation. Works inplace Cconj 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. Cabs Signature: (a(m=2); [o]c()) complex "abs()" (also known as modulus) Cabs 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. Cabs2 Signature: (a(m=2); [o]c()) complex squared "abs()" (also known squared modulus) Cabs2 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. Carg Signature: (a(m=2); [o]c()) complex argument function ("angle") Carg 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. Csin Signature: (a(m=2); [o]c(m=2)) sin (a) = 1/(2*i) * (exp (a*i) - exp (-a*i)). Works inplace Csin 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. Ccos Signature: (a(m=2); [o]c(m=2)) cos (a) = 1/2 * (exp (a*i) + exp (-a*i)). Works inplace Ccos 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. Ctan a [not inplace] tan (a) = -i * (exp (a*i) - exp (-a*i)) / (exp (a*i) + exp (-a*i)) Cexp Signature: (a(m=2); [o]c(m=2)) exp (a) = exp (real (a)) * (cos (imag (a)) + i * sin (imag (a))). Works inplace Cexp 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. Clog Signature: (a(m=2); [o]c(m=2)) log (a) = log (cabs (a)) + i * carg (a). Works inplace Clog 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. Cpow Signature: (a(m=2); b(m=2); [o]c(m=2)) complex "pow()" ("**"-operator) Cpow 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. Csqrt Signature: (a(m=2); [o]c(m=2)) Works inplace Csqrt 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. Casin Signature: (a(m=2); [o]c(m=2)) Works inplace Casin 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. Cacos Signature: (a(m=2); [o]c(m=2)) Works inplace Cacos 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. Catan cplx [not inplace] Return the complex "atan()". Csinh Signature: (a(m=2); [o]c(m=2)) sinh (a) = (exp (a) - exp (-a)) / 2. Works inplace Csinh 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. Ccosh Signature: (a(m=2); [o]c(m=2)) cosh (a) = (exp (a) + exp (-a)) / 2. Works inplace Ccosh 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. Ctanh Signature: (a(m=2); [o]c(m=2)) Works inplace Ctanh 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. Casinh Signature: (a(m=2); [o]c(m=2)) Works inplace Casinh 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. Cacosh Signature: (a(m=2); [o]c(m=2)) Cacosh 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. Catanh Signature: (a(m=2); [o]c(m=2)) Works inplace Catanh 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. Cproj Signature: (a(m=2); [o]c(m=2)) compute the projection of a complex number to the riemann sphere. Works inplace Cproj 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. Croots Signature: (a(m=2); [o]c(m=2,n); int n => n) Compute the "n" roots of "a". "n" must be a positive integer. The result will always be a complex type! Croots 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. re cplx, im cplx Return the real or imaginary part of the complex number(s) given. These are slicing operators, so data flow works. The real and imaginary parts are returned as piddles (ref eq PDL). rCpolynomial Signature: (coeffs(n); x(c=2,m); [o]out(c=2,m)) evaluate the polynomial with (real) coefficients "coeffs" at the (complex) position(s) "x". "coeffs[0]" is the constant term. rCpolynomial 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. AUTHOR
Copyright (C) 2000 Marc Lehmann <pcg@goof.com>. All rights reserved. There is no warranty. You are allowed to redistribute this software / documentation as described in the file COPYING in the PDL distribution. SEE ALSO
perl(1), PDL. perl v5.14.2 2012-05-30 Complex(3pm)