Math::FFT - Perl module to calculate Fast Fourier Transforms use Math::FFT; my $PI = 3.1415926539; my $N = 64; my ($series, $other_series); for (my $k=0; $k<$N; $k++) { $series->[$k] = sin(4*$k*$PI/$N) + cos(6*$k*$PI/$N); } my $fft = new Math::FFT($series); my $coeff = $fft->rdft(); my $spectrum = $fft->spctrm; my $original_data = $fft->invrdft($coeff); for (my $k=0; $k<$N; $k++) { $other_series->[$k] = sin(16*$k*$PI/$N) + cos(8*$k*$PI/$N); } my $other_fft = $fft->clone($other_series); my $other_coeff = $other_fft->rdft(); my $correlation = $fft->correl($other_fft); This module implements some algorithms for calculating Fast Fourier Transforms for one-dimensional data sets of size 2^n. The data, assumed to arise from a constant sampling rate, is represented by an array reference $data (as described in the methods below), which is then used to create a "Math::FFT" object as my $fft = new Math::FFT($data); The methods available include the following.

FFT METHODS
This calculates the complex discrete Fourier transform for a data set "x[j]". Here, $data is a reference to an array "data[0...2*n-1]" holding the data data[2*j] = Re(x[j]), data[2*j+1] = Im(x[j]), 0<=j<n An array reference $coeff is returned consisting of coeff[2*k] = Re(X[k]), coeff[2*k+1] = Im(X[k]), 0<=k<n where X[k] = sum_j=0^n-1 x[j]*exp(2*pi*i*j*k/n), 0<=k<n Calculates the inverse complex discrete Fourier transform on a data set "x[j]". If $coeff is not given, it will be set equal to an earlier call to "$fft->cdft()". $coeff is a reference to an array "coeff[0...2*n-1]" holding the data coeff[2*j] = Re(x[j]), coeff[2*j+1] = Im(x[j]), 0<=j<n An array reference $orig_data is returned consisting of orig_data[2*k] = Re(X[k]), orig_data[2*k+1] = Im(X[k]), 0<=k<n where, excluding the scale, X[k] = sum_j=0^n-1 x[j]*exp(-2*pi*i*j*k/n), 0<=k<n A scaling "$orig_data->[$i] *= 2.0/$n" is then done so that $orig_data coincides with the original $data. This calculates the real discrete Fourier transform for a data set "x[j]". On input, $data is a reference to an array "data[0...n-1]" holding the data. An array reference $coeff is returned consisting of coeff[2*k] = R[k], 0<=k<n/2 coeff[2*k+1] = I[k], 0<k<n/2 coeff[1] = R[n/2] where R[k] = sum_j=0^n-1 data[j]*cos(2*pi*j*k/n), 0<=k<=n/2 I[k] = sum_j=0^n-1 data[j]*sin(2*pi*j*k/n), 0<k<n/2 Calculates the inverse real discrete Fourier transform on a data set "coeff[j]". If $coeff is not given, it will be set equal to an earlier call to "$fft->rdft()". $coeff is a reference to an array "coeff[0...n-1]" holding the data coeff[2*j] = R[j], 0<=j<n/2 coeff[2*j+1] = I[j], 0<j<n/2 coeff[1] = R[n/2] An array reference $orig_data is returned where, excluding the scale, orig_data[k] = (R[0] + R[n/2]*cos(pi*k))/2 + sum_j=1^n/2-1 R[j]*cos(2*pi*j*k/n) + sum_j=1^n/2-1 I[j]*sin(2*pi*j*k/n), 0<=k<n A scaling "$orig_data->[$i] *= 2.0/$n" is then done so that $orig_data coincides with the original $data. Computes the discrete cosine tranform on a data set "data[0...n-1]" contained in an array reference $data. An array reference $coeff is returned consisting of coeff[k] = C[k], 0<=k<n where C[k] = sum_j=0^n-1 data[j]*cos(pi*(j+1/2)*k/n), 0<=k<n Computes the inverse discrete cosine tranform on a data set "coeff[0...n-1]" contained in an array reference $coeff. If $coeff is not given, it will be set equal to an earlier call to "$fft->ddct()". An array reference $orig_data is returned consisting of orig_data[k] = C[k], 0<=k<n where, excluding the scale, C[k] = sum_j=0^n-1 coeff[j]*cos(pi*j*(k+1/2)/n), 0<=k<n A scaling "$orig_data->[$i] *= 2.0/$n" is then done so that $orig_data coincides with the original $data. Computes the discrete sine transform of a data set "data[0...n-1]" contained in an array reference $data. An array reference $coeff is returned consisting of coeff[k] = S[k], 0<k<n coeff[0] = S[n] where S[k] = sum_j=0^n-1 data[j]*sin(pi*(j+1/2)*k/n), 0<k<=n Computes the inverse discrete sine transform of a data set "coeff[0...n-1]" contained in an array reference $coeff, arranged as coeff[j] = A[j], 0<j<n coeff[0] = A[n] If $coeff is not given, it will be set equal to an earlier call to "$fft->ddst()". An array reference $orig_data is returned consisting of orig_data[k] = S[k], 0<=k<n where, excluding a scale, S[k] = sum_j=1^n A[j]*sin(pi*j*(k+1/2)/n), 0<=k<n The scaling "$a->[$i] *= 2.0/$n" is then done so that $orig_data coincides with the original $data. Computes the real symmetric discrete Fourier transform of a data set "data[0...n]" contained in the array reference $data. An array reference $coeff is returned consisting of coeff[k] = C[k], 0<=k<=n where C[k] = sum_j=0^n data[j]*cos(pi*j*k/n), 0<=k<=n Computes the inverse real symmetric discrete Fourier transform of a data set "coeff[0...n]" contained in the array reference $coeff. If $coeff is not given, it will be set equal to an earlier call to "$fft->dfct()". An array reference $orig_data is returned consisting of orig_data[k] = C[k], 0<=k<=n where, excluding the scale, C[k] = sum_j=0^n coeff[j]*cos(pi*j*k/n), 0<=k<=n A scaling "$coeff->[0] *= 0.5", "$coeff->[$n] *= 0.5", and "$orig_data->[$i] *= 2.0/$n" is then done so that $orig_data coincides with the original $data. Computes the real anti-symmetric discrete Fourier transform of a data set "data[0...n-1]" contained in the array reference $data. An array reference $coeff is returned consisting of coeff[k] = C[k], 0<k<n where C[k] = sum_j=0^n data[j]*sin(pi*j*k/n), 0<k<n ("coeff[0]" is used for a work area) Computes the inverse real anti-symmetric discrete Fourier transform of a data set "coeff[0...n-1]" contained in the array reference $coeff. If $coeff is not given, it will be set equal to an earlier call to "$fft->dfst()". An array reference $orig_data is returned consisting of orig_data[k] = C[k], 0<k<n where, excluding the scale, C[k] = sum_j=0^n coeff[j]*sin(pi*j*k/n), 0<k<n A scaling "$orig_data->[$i] *= 2.0/$n" is then done so that $orig_data coincides with the original $data.

CLONING
The algorithm used in the transforms makes use of arrays for a work area and for a cos/sin lookup table dependent only on the size of the data set. These arrays are initialized when the "Math::FFT" object is created and then are populated when a transform method is first invoked. After this, they persist for the lifetime of the object. This aspect is exploited in a "cloning" method; if a "Math::FFT" object is created for a data set $data1 of size "N": $fft1 = new Math::FFT($data1); then a new "Math::FFT" object can be created for a second data set $data2 of the same size "N" by $fft2 = $fft1->clone($data2); The $fft2 object will copy the reuseable work area and lookup table calculated from $fft1.

APPLICATIONS
This module includes some common applications - correlation, convolution and deconvolution, and power spectrum - that arise with real data sets. The conventions used here follow that of Numerical Recipes in C, by Press, Teukolsky, Vetterling, and Flannery, in which further details of the algorithms are given. Note in particular the treatment of end effects by zero padding, which is assumed to be done by the user, if required. The correlation between two functions is defined as / Corr(t) = | ds g(s+t) h(s) / This may be calculated, for two array references $data1 and $data2 of the same size $n, as either $fft1 = new Math::FFT($data1); $fft2 = new Math::FFT($data2); $corr = $fft1->correl($fft2); or as $fft1 = new Math::FFT($data1); $corr = $fft1->correl($data2); The array reference $corr is returned in wrap-around order - correlations at increasingly positive lags are in "$corr->[0]" (zero lag) on up to "$corr->[$n/2-1]", while correlations at increasingly negative lags are in "$corr->[$n-1]" on down to "$corr->[$n/2]". The sign convention used is such that if $data1 lags $data2 (that is, is shifted to the right), then $corr will show a peak at positive lags. The convolution of two functions is defined as / Convlv(t) = | ds g(s) h(t-s) / This is similar to calculating the correlation between the two functions, but typically the functions here have a quite different physical interpretation - one is a signal which persists indefinitely in time, and the other is a response function of limited duration. The convolution may be calculated, for two array references $data and $respn, as $fft = new Math::FFT($data); $convlv = $fft->convlv($respn); with the returned $convlv being an array reference. The method assumes that the response function $respn has an odd number of elements $m less than or equal to the number of elements $n of $data. $respn is assumed to be stored in wrap-around order - the first half contains the response at positive times, while the second half, counting down from "$respn->[$m-1]", contains the response at negative times. Deconvolution undoes the effects of convoluting a signal with a known response function. In other words, in the relation / Convlv(t) = | ds g(s) h(t-s) / deconvolution reconstructs the original signal, given the convolution and the response function. The method is implemented, for two array references $data and $respn, as $fft = new Math::FFT($data); $deconvlv = $fft->deconvlv($respn); As a result, if the convolution of a data set $data with a response function $respn is calculated as $fft1 = new Math::FFT($data); $convlv = $fft1->convlv($respn); then the deconvolution $fft2 = new Math::FFT($convlv); $deconvlv = $fft2->deconvlv($respn); will give an array reference $deconvlv containing the same elements as the original data $data. If the FFT of a real function of "N" elements is calculated, the "N/2+1" elements of the power spectrum are defined, in terms of the (complex) Fourier coefficients "C[k]", as P[0] = |C[0]|^2 / N^2 P[k] = 2 |C[k]|^2 / N^2 (k = 1, 2 ,..., N/2-1) P[N/2] = |C[N/2]|^2 / N^2 Often for these purposes the data is partitioned into "K" segments, each containing "2M" elements. The power spectrum for each segment is calculated, and the net power spectrum is the average of all of these segmented spectra. Partitioning may be done in one of two ways: non-overlapping and overlapping. Non-overlapping is useful when the data set is gathered in real time, where the number of data points can be varied at will. Overlapping is useful where there is a fixed number of data points. In non-overlapping, the first <2M> elements constitute segment 1, the next "2M" elements are segment 2, and so on up to segment "K", for a total of "2KM" sampled points. In overlapping, the first and second "M" elements are segment 1, the second and third "M" elements are segment 2, and so on, for a total of "(K+1)M" sampled points. A problem that may arise in this procedure is leakage: the power spectrum calculated for one bin contains contributions from nearby bins. To lessen this effect data windowing is often used: multiply the original data "d[j]" by a window function "w[j]", where j = 0, 1, ..., N-1. Some popular choices of such functions are | j - N/2 | w[j] = 1 - | ------- | ... Bartlett | N/2 | / j - N/2 \ 2 w[j] = 1 - | ------- | ... Welch \ N/2 / 1 / \ w[j] = --- |1 - cos(2 pi j / N) | ... Hamm 2 \ / The "spctrm" method, used as $fft = Math::FFT->new($data); $spectrum = $fft->spctrm([$key => $value, ...]); returns an array reference $spectrum representing the power spectrum for a data set represented by an array reference $data. The options available are This specifies the window function; if not given, no such function is used. Accepted values (see above) are "bartlett", "welch", "hamm", and "\&my_window", where "my_window" is a user specified subroutine which must be of the form, for example, sub my_window { my ($j, $n) = @_; return 1 - abs(2*($j-$n/2)/$n); } which implements the Bartlett window. This specifies whether overlapping should be done; if true (1), overlapping will be used, whereas if false (0), or not specified, no overlapping is used. This specifies that the data will be partitioned into "n" segments. If not specified, no segmentation will be done. This specifies that "2m" data points will be used for each segment, and must be a power of 2. The power spectrum returned will consist of "m+1" elements.

STATISTICAL FUNCTIONS
For convenience, a number of common statistical functions are included for analyzing real data. After creating the object as my $fft = new Math::FFT($data); for a data set represented by the array reference $data of size "N", these methods may be called as follows. This returns the mean 1/N * sum_j=0^N-1 data[j] If an array reference $data is not given, the data set used in creating $fft will be used. This returns the standard deviation sqrt{ 1/(N-1) * sum_j=0^N-1 (data[j] - mean)**2 } If an array reference $data is not given, the data set used in creating $fft will be used. This returns the minimum and maximum values of the data set. If an array reference $data is not given, the data set used in creating $fft will be used. This returns the median of a data set. The median is defined, for the sorted data set, as either the middle element, if the number of elements is odd, or as the interpolated value of the the two values on either side of the middle, if the number of elements is even. If an array reference $data is not given, the data set used in creating $fft will be used. Please report any to Randy Kobes <randy@theoryx5.uwinnipeg.ca> Math::Pari and PDL The algorithm used in this module to calculate the Fourier transforms is based on the C routine of fft4g.c available at http://momonga.t.u-tokyo.ac.jp/~ooura/fft.html, which is copyrighted 1996-99 by Takuya OOURA. The file arrays.c included here to handle passing arrays to and from C comes from the PGPLOT module of Karl Glazebrook <kgb@aaoepp.aao.gov.au>. The perl code of Math::FFT is copyright 2000 by Randy Kobes, and is distributed under the same terms as Perl itself.