Fast Fourier Transforms (FFTs)¶
This chapter describes functions for performing Fast Fourier Transforms (FFTs). The library includes radix2 routines (for lengths which are a power of two) and mixedradix routines (which work for any length). For efficiency there are separate versions of the routines for real data and for complex data. The mixedradix routines are a reimplementation of the FFTPACK library of Paul Swarztrauber. Fortran code for FFTPACK is available on Netlib (FFTPACK also includes some routines for sine and cosine transforms but these are currently not available in GSL). For details and derivations of the underlying algorithms consult the document “GSL FFT Algorithms” (see References and Further Reading)
Mathematical Definitions¶
Fast Fourier Transforms are efficient algorithms for calculating the discrete Fourier transform (DFT),
The DFT usually arises as an approximation to the continuous Fourier transform when functions are sampled at discrete intervals in space or time. The naive evaluation of the discrete Fourier transform is a matrixvector multiplication . A general matrixvector multiplication takes operations for datapoints. Fast Fourier transform algorithms use a divideandconquer strategy to factorize the matrix into smaller submatrices, corresponding to the integer factors of the length . If can be factorized into a product of integers then the DFT can be computed in operations. For a radix2 FFT this gives an operation count of .
All the FFT functions offer three types of transform: forwards, inverse and backwards, based on the same mathematical definitions. The definition of the forward Fourier transform, , is,
and the definition of the inverse Fourier transform, , is,
The factor of makes this a true inverse. For example, a call
to gsl_fft_complex_forward()
followed by a call to
gsl_fft_complex_inverse()
should return the original data (within
numerical errors).
In general there are two possible choices for the sign of the exponential in the transform/ inversetransform pair. GSL follows the same convention as FFTPACK, using a negative exponential for the forward transform. The advantage of this convention is that the inverse transform recreates the original function with simple Fourier synthesis. Numerical Recipes uses the opposite convention, a positive exponential in the forward transform.
The backwards FFT is simply our terminology for an unscaled version of the inverse FFT,
When the overall scale of the result is unimportant it is often convenient to use the backwards FFT instead of the inverse to save unnecessary divisions.
Overview of complex data FFTs¶
The inputs and outputs for the complex FFT routines are packed arrays of floating point numbers. In a packed array the real and imaginary parts of each complex number are placed in alternate neighboring elements. For example, the following definition of a packed array of length 6:
double x[3*2];
gsl_complex_packed_array data = x;
can be used to hold an array of three complex numbers, z[3]
, in
the following way:
data[0] = Re(z[0])
data[1] = Im(z[0])
data[2] = Re(z[1])
data[3] = Im(z[1])
data[4] = Re(z[2])
data[5] = Im(z[2])
The array indices for the data have the same ordering as those in the definition of the DFT—i.e. there are no index transformations or permutations of the data.
A stride parameter allows the user to perform transforms on the
elements z[stride*i]
instead of z[i]
. A stride greater
than 1 can be used to take an inplace FFT of the column of a matrix. A
stride of 1 accesses the array without any additional spacing between
elements.
To perform an FFT on a vector argument, such as gsl_vector_complex * v
,
use the following definitions (or their equivalents) when calling
the functions described in this chapter:
gsl_complex_packed_array data = v>data;
size_t stride = v>stride;
size_t n = v>size;
For physical applications it is important to remember that the index appearing in the DFT does not correspond directly to a physical frequency. If the timestep of the DFT is then the frequencydomain includes both positive and negative frequencies, ranging from through 0 to . The positive frequencies are stored from the beginning of the array up to the middle, and the negative frequencies are stored backwards from the end of the array.
Here is a table which shows the layout of the array data
, and the
correspondence between the timedomain data , and the
frequencydomain data :
index z x = FFT(z)
0 z(t = 0) x(f = 0)
1 z(t = 1) x(f = 1/(n Delta))
2 z(t = 2) x(f = 2/(n Delta))
. ........ ..................
n/2 z(t = n/2) x(f = +1/(2 Delta),
1/(2 Delta))
. ........ ..................
n3 z(t = n3) x(f = 3/(n Delta))
n2 z(t = n2) x(f = 2/(n Delta))
n1 z(t = n1) x(f = 1/(n Delta))
When is even the location contains the most positive and negative frequencies (, ) which are equivalent. If is odd then general structure of the table above still applies, but does not appear.
Radix2 FFT routines for complex data¶
The radix2 algorithms described in this section are simple and compact, although not necessarily the most efficient. They use the CooleyTukey algorithm to compute inplace complex FFTs for lengths which are a power of 2—no additional storage is required. The corresponding selfsorting mixedradix routines offer better performance at the expense of requiring additional working space.
All the functions described in this section are declared in the header file gsl_fft_complex.h
.

int
gsl_fft_complex_radix2_forward
(gsl_complex_packed_array data, size_t stride, size_t n)¶ 
int
gsl_fft_complex_radix2_transform
(gsl_complex_packed_array data, size_t stride, size_t n, gsl_fft_direction sign)¶ 
int
gsl_fft_complex_radix2_backward
(gsl_complex_packed_array data, size_t stride, size_t n)¶ 
int
gsl_fft_complex_radix2_inverse
(gsl_complex_packed_array data, size_t stride, size_t n)¶ These functions compute forward, backward and inverse FFTs of length
n
with stridestride
, on the packed complex arraydata
using an inplace radix2 decimationintime algorithm. The length of the transformn
is restricted to powers of two. For thetransform
version of the function thesign
argument can be eitherforward
() orbackward
().The functions return a value of
GSL_SUCCESS
if no errors were detected, orGSL_EDOM
if the length of the datan
is not a power of two.

int
gsl_fft_complex_radix2_dif_forward
(gsl_complex_packed_array data, size_t stride, size_t n)¶ 
int
gsl_fft_complex_radix2_dif_transform
(gsl_complex_packed_array data, size_t stride, size_t n, gsl_fft_direction sign)¶ 
int
gsl_fft_complex_radix2_dif_backward
(gsl_complex_packed_array data, size_t stride, size_t n)¶ 
int
gsl_fft_complex_radix2_dif_inverse
(gsl_complex_packed_array data, size_t stride, size_t n)¶ These are decimationinfrequency versions of the radix2 FFT functions.
Here is an example program which computes the FFT of a short pulse in a sample of length 128. To make the resulting Fourier transform real the pulse is defined for equal positive and negative times (), where the negative times wrap around the end of the array.
#include <stdio.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_fft_complex.h>
#define REAL(z,i) ((z)[2*(i)])
#define IMAG(z,i) ((z)[2*(i)+1])
int
main (void)
{
int i; double data[2*128];
for (i = 0; i < 128; i++)
{
REAL(data,i) = 0.0; IMAG(data,i) = 0.0;
}
REAL(data,0) = 1.0;
for (i = 1; i <= 10; i++)
{
REAL(data,i) = REAL(data,128i) = 1.0;
}
for (i = 0; i < 128; i++)
{
printf ("%d %e %e\n", i,
REAL(data,i), IMAG(data,i));
}
printf ("\n\n");
gsl_fft_complex_radix2_forward (data, 1, 128);
for (i = 0; i < 128; i++)
{
printf ("%d %e %e\n", i,
REAL(data,i)/sqrt(128),
IMAG(data,i)/sqrt(128));
}
return 0;
}
Note that we have assumed that the program is using the default error
handler (which calls abort()
for any errors). If you are not using
a safe error handler you would need to check the return status of
gsl_fft_complex_radix2_forward()
.
The transformed data is rescaled by so that it fits on the same plot as the input. Only the real part is shown, by the choice of the input data the imaginary part is zero. Allowing for the wraparound of negative times at , and working in units of , the DFT approximates the continuum Fourier transform, giving a modulated sine function.
The output of the example program is plotted in Fig. 1.
Mixedradix FFT routines for complex data¶
This section describes mixedradix FFT algorithms for complex data. The mixedradix functions work for FFTs of any length. They are a reimplementation of Paul Swarztrauber’s Fortran FFTPACK library. The theory is explained in the review article “Selfsorting Mixedradix FFTs” by Clive Temperton. The routines here use the same indexing scheme and basic algorithms as FFTPACK.
The mixedradix algorithm is based on subtransform modules—highly optimized small length FFTs which are combined to create larger FFTs. There are efficient modules for factors of 2, 3, 4, 5, 6 and 7. The modules for the composite factors of 4 and 6 are faster than combining the modules for and .
For factors which are not implemented as modules there is a fallback to a general length module which uses Singleton’s method for efficiently computing a DFT. This module is , and slower than a dedicated module would be but works for any length . Of course, lengths which use the general length module will still be factorized as much as possible. For example, a length of 143 will be factorized into . Large prime factors are the worst case scenario, e.g. as found in , and should be avoided because their scaling will dominate the runtime (consult the document “GSL FFT Algorithms” included in the GSL distribution if you encounter this problem).
The mixedradix initialization function gsl_fft_complex_wavetable_alloc()
returns the list of factors chosen by the library for a given length
. It can be used to check how well the length has been
factorized, and estimate the runtime. To a first approximation the
runtime scales as , where the are the
factors of . For programs under user control you may wish to
issue a warning that the transform will be slow when the length is
poorly factorized. If you frequently encounter data lengths which
cannot be factorized using the existing smallprime modules consult
“GSL FFT Algorithms” for details on adding support for other
factors.
All the functions described in this section are declared in the header
file gsl_fft_complex.h
.

gsl_fft_complex_wavetable *
gsl_fft_complex_wavetable_alloc
(size_t n)¶ This function prepares a trigonometric lookup table for a complex FFT of length
n
. The function returns a pointer to the newly allocatedgsl_fft_complex_wavetable
if no errors were detected, and a null pointer in the case of error. The lengthn
is factorized into a product of subtransforms, and the factors and their trigonometric coefficients are stored in the wavetable. The trigonometric coefficients are computed using direct calls tosin
andcos
, for accuracy. Recursion relations could be used to compute the lookup table faster, but if an application performs many FFTs of the same length then this computation is a oneoff overhead which does not affect the final throughput.The wavetable structure can be used repeatedly for any transform of the same length. The table is not modified by calls to any of the other FFT functions. The same wavetable can be used for both forward and backward (or inverse) transforms of a given length.

void
gsl_fft_complex_wavetable_free
(gsl_fft_complex_wavetable * wavetable)¶ This function frees the memory associated with the wavetable
wavetable
. The wavetable can be freed if no further FFTs of the same length will be needed.
These functions operate on a gsl_fft_complex_wavetable
structure
which contains internal parameters for the FFT. It is not necessary to
set any of the components directly but it can sometimes be useful to
examine them. For example, the chosen factorization of the FFT length
is given and can be used to provide an estimate of the runtime or
numerical error. The wavetable structure is declared in the header file
gsl_fft_complex.h
.

gsl_fft_complex_wavetable
¶ This is a structure that holds the factorization and trigonometric lookup tables for the mixed radix fft algorithm. It has the following components:
size_t n
This is the number of complex data points size_t nf
This is the number of factors that the length n
was decomposed into.size_t factor[64]
This is the array of factors. Only the first nf
elements are used.gsl_complex * trig
This is a pointer to a preallocated trigonometric lookup table of n
complex elements.gsl_complex * twiddle[64]
This is an array of pointers into trig
, giving the twiddle factors for each pass.

gsl_fft_complex_workspace
¶ The mixed radix algorithms require additional working space to hold the intermediate steps of the transform.

gsl_fft_complex_workspace *
gsl_fft_complex_workspace_alloc
(size_t n)¶ This function allocates a workspace for a complex transform of length
n
.

void
gsl_fft_complex_workspace_free
(gsl_fft_complex_workspace * workspace)¶ This function frees the memory associated with the workspace
workspace
. The workspace can be freed if no further FFTs of the same length will be needed.
The following functions compute the transform,

int
gsl_fft_complex_forward
(gsl_complex_packed_array data, size_t stride, size_t n, const gsl_fft_complex_wavetable * wavetable, gsl_fft_complex_workspace * work)¶ 
int
gsl_fft_complex_transform
(gsl_complex_packed_array data, size_t stride, size_t n, const gsl_fft_complex_wavetable * wavetable, gsl_fft_complex_workspace * work, gsl_fft_direction sign)¶ 
int
gsl_fft_complex_backward
(gsl_complex_packed_array data, size_t stride, size_t n, const gsl_fft_complex_wavetable * wavetable, gsl_fft_complex_workspace * work)¶ 
int
gsl_fft_complex_inverse
(gsl_complex_packed_array data, size_t stride, size_t n, const gsl_fft_complex_wavetable * wavetable, gsl_fft_complex_workspace * work)¶ These functions compute forward, backward and inverse FFTs of length
n
with stridestride
, on the packed complex arraydata
, using a mixed radix decimationinfrequency algorithm. There is no restriction on the lengthn
. Efficient modules are provided for subtransforms of length 2, 3, 4, 5, 6 and 7. Any remaining factors are computed with a slow, , general module. The caller must supply awavetable
containing the trigonometric lookup tables and a workspacework
. For thetransform
version of the function thesign
argument can be eitherforward
() orbackward
().The functions return a value of
0
if no errors were detected. The followinggsl_errno
conditions are defined for these functions:GSL_EDOM
The length of the data n
is not a positive integer (i.e.n
is zero).GSL_EINVAL
The length of the data n
and the length used to compute the givenwavetable
do not match.
Here is an example program which computes the FFT of a short pulse in a sample of length 630 () using the mixedradix algorithm.
#include <stdio.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_fft_complex.h>
#define REAL(z,i) ((z)[2*(i)])
#define IMAG(z,i) ((z)[2*(i)+1])
int
main (void)
{
int i;
const int n = 630;
double data[2*n];
gsl_fft_complex_wavetable * wavetable;
gsl_fft_complex_workspace * workspace;
for (i = 0; i < n; i++)
{
REAL(data,i) = 0.0;
IMAG(data,i) = 0.0;
}
data[0] = 1.0;
for (i = 1; i <= 10; i++)
{
REAL(data,i) = REAL(data,ni) = 1.0;
}
for (i = 0; i < n; i++)
{
printf ("%d: %e %e\n", i, REAL(data,i),
IMAG(data,i));
}
printf ("\n");
wavetable = gsl_fft_complex_wavetable_alloc (n);
workspace = gsl_fft_complex_workspace_alloc (n);
for (i = 0; i < (int) wavetable>nf; i++)
{
printf ("# factor %d: %zu\n", i,
wavetable>factor[i]);
}
gsl_fft_complex_forward (data, 1, n,
wavetable, workspace);
for (i = 0; i < n; i++)
{
printf ("%d: %e %e\n", i, REAL(data,i),
IMAG(data,i));
}
gsl_fft_complex_wavetable_free (wavetable);
gsl_fft_complex_workspace_free (workspace);
return 0;
}
Note that we have assumed that the program is using the default
gsl
error handler (which calls abort()
for any errors). If
you are not using a safe error handler you would need to check the
return status of all the gsl
routines.
Overview of real data FFTs¶
The functions for real data are similar to those for complex data. However, there is an important difference between forward and inverse transforms. The Fourier transform of a real sequence is not real. It is a complex sequence with a special symmetry:
A sequence with this symmetry is called conjugatecomplex or
halfcomplex. This different structure requires different
storage layouts for the forward transform (from real to halfcomplex)
and inverse transform (from halfcomplex back to real). As a
consequence the routines are divided into two sets: functions in
gsl_fft_real
which operate on real sequences and functions in
gsl_fft_halfcomplex
which operate on halfcomplex sequences.
Functions in gsl_fft_real
compute the frequency coefficients of a
real sequence. The halfcomplex coefficients of a real sequence
are given by Fourier analysis,
Functions in gsl_fft_halfcomplex
compute inverse or backwards
transforms. They reconstruct real sequences by Fourier synthesis from
their halfcomplex frequency coefficients, ,
The symmetry of the halfcomplex sequence implies that only half of the
complex numbers in the output need to be stored. The remaining half can
be reconstructed using the halfcomplex symmetry condition. This works
for all lengths, even and odd—when the length is even the middle value
where is also real. Thus only n
real numbers are
required to store the halfcomplex sequence, and the transform of a real
sequence can be stored in the same size array as the original data.
The precise storage arrangements depend on the algorithm, and are different for radix2 and mixedradix routines. The radix2 function operates inplace, which constrains the locations where each element can be stored. The restriction forces real and imaginary parts to be stored far apart. The mixedradix algorithm does not have this restriction, and it stores the real and imaginary parts of a given term in neighboring locations (which is desirable for better locality of memory accesses).
Radix2 FFT routines for real data¶
This section describes radix2 FFT algorithms for real data. They use the CooleyTukey algorithm to compute inplace FFTs for lengths which are a power of 2.
The radix2 FFT functions for real data are declared in the header files
gsl_fft_real.h

int
gsl_fft_real_radix2_transform
(double data[], size_t stride, size_t n)¶ This function computes an inplace radix2 FFT of length
n
and stridestride
on the real arraydata
. The output is a halfcomplex sequence, which is stored inplace. The arrangement of the halfcomplex terms uses the following scheme: for the real part of the th term is stored in location , and the corresponding imaginary part is stored in location . Terms with can be reconstructed using the symmetry . The terms for and are both purely real, and count as a special case. Their real parts are stored in locations and respectively, while their imaginary parts which are zero are not stored.The following table shows the correspondence between the output
data
and the equivalent results obtained by considering the input data as a complex sequence with zero imaginary part (assumingstride
= 1}):complex[0].real = data[0] complex[0].imag = 0 complex[1].real = data[1] complex[1].imag = data[n1] ............... ................ complex[k].real = data[k] complex[k].imag = data[nk] ............... ................ complex[n/2].real = data[n/2] complex[n/2].imag = 0 ............... ................ complex[k'].real = data[k] k' = n  k complex[k'].imag = data[nk] ............... ................ complex[n1].real = data[1] complex[n1].imag = data[n1]
Note that the output data can be converted into the full complex sequence using the function
gsl_fft_halfcomplex_radix2_unpack()
described below.
The radix2 FFT functions for halfcomplex data are declared in the
header file gsl_fft_halfcomplex.h
.

int
gsl_fft_halfcomplex_radix2_inverse
(double data[], size_t stride, size_t n)¶ 
int
gsl_fft_halfcomplex_radix2_backward
(double data[], size_t stride, size_t n)¶ These functions compute the inverse or backwards inplace radix2 FFT of length
n
and stridestride
on the halfcomplex sequencedata
stored according the output scheme used bygsl_fft_real_radix2()
. The result is a real array stored in natural order.

int
gsl_fft_halfcomplex_radix2_unpack
(const double halfcomplex_coefficient[], gsl_complex_packed_array complex_coefficient, size_t stride, size_t n)¶ This function converts
halfcomplex_coefficient
, an array of halfcomplex coefficients as returned bygsl_fft_real_radix2_transform()
, into an ordinary complex array,complex_coefficient
. It fills in the complex array using the symmetry to reconstruct the redundant elements. The algorithm for the conversion is:complex_coefficient[0].real = halfcomplex_coefficient[0]; complex_coefficient[0].imag = 0.0; for (i = 1; i < n  i; i++) { double hc_real = halfcomplex_coefficient[i*stride]; double hc_imag = halfcomplex_coefficient[(ni)*stride]; complex_coefficient[i*stride].real = hc_real; complex_coefficient[i*stride].imag = hc_imag; complex_coefficient[(n  i)*stride].real = hc_real; complex_coefficient[(n  i)*stride].imag = hc_imag; } if (i == n  i) { complex_coefficient[i*stride].real = halfcomplex_coefficient[(n  1)*stride]; complex_coefficient[i*stride].imag = 0.0; }
Mixedradix FFT routines for real data¶
This section describes mixedradix FFT algorithms for real data. The mixedradix functions work for FFTs of any length. They are a reimplementation of the realFFT routines in the Fortran FFTPACK library by Paul Swarztrauber. The theory behind the algorithm is explained in the article “Fast MixedRadix Real Fourier Transforms” by Clive Temperton. The routines here use the same indexing scheme and basic algorithms as FFTPACK.
The functions use the FFTPACK storage convention for halfcomplex sequences. In this convention the halfcomplex transform of a real sequence is stored with frequencies in increasing order, starting at zero, with the real and imaginary parts of each frequency in neighboring locations. When a value is known to be real the imaginary part is not stored. The imaginary part of the zerofrequency component is never stored. It is known to be zero (since the zero frequency component is simply the sum of the input data (all real)). For a sequence of even length the imaginary part of the frequency is not stored either, since the symmetry implies that this is purely real too.
The storage scheme is best shown by some examples. The table below
shows the output for an oddlength sequence, . The two columns
give the correspondence between the 5 values in the halfcomplex
sequence returned by gsl_fft_real_transform()
, halfcomplex[]
and the
values complex[]
that would be returned if the same real input
sequence were passed to gsl_fft_complex_backward()
as a complex
sequence (with imaginary parts set to 0
):
complex[0].real = halfcomplex[0]
complex[0].imag = 0
complex[1].real = halfcomplex[1]
complex[1].imag = halfcomplex[2]
complex[2].real = halfcomplex[3]
complex[2].imag = halfcomplex[4]
complex[3].real = halfcomplex[3]
complex[3].imag = halfcomplex[4]
complex[4].real = halfcomplex[1]
complex[4].imag = halfcomplex[2]
The upper elements of the complex
array, complex[3]
and
complex[4]
are filled in using the symmetry condition. The
imaginary part of the zerofrequency term complex[0].imag
is
known to be zero by the symmetry.
The next table shows the output for an evenlength sequence, . In the even case there are two values which are purely real:
complex[0].real = halfcomplex[0]
complex[0].imag = 0
complex[1].real = halfcomplex[1]
complex[1].imag = halfcomplex[2]
complex[2].real = halfcomplex[3]
complex[2].imag = halfcomplex[4]
complex[3].real = halfcomplex[5]
complex[3].imag = 0
complex[4].real = halfcomplex[3]
complex[4].imag = halfcomplex[4]
complex[5].real = halfcomplex[1]
complex[5].imag = halfcomplex[2]
The upper elements of the complex
array, complex[4]
and
complex[5]
are filled in using the symmetry condition. Both
complex[0].imag
and complex[3].imag
are known to be zero.
All these functions are declared in the header files
gsl_fft_real.h
and gsl_fft_halfcomplex.h
.

gsl_fft_real_wavetable
¶ 
gsl_fft_halfcomplex_wavetable
¶ These data structures contain lookup tables for an FFT of a fixed size.

gsl_fft_real_wavetable *
gsl_fft_real_wavetable_alloc
(size_t n)¶ 
gsl_fft_halfcomplex_wavetable *
gsl_fft_halfcomplex_wavetable_alloc
(size_t n)¶ These functions prepare trigonometric lookup tables for an FFT of size real elements. The functions return a pointer to the newly allocated struct if no errors were detected, and a null pointer in the case of error. The length
n
is factorized into a product of subtransforms, and the factors and their trigonometric coefficients are stored in the wavetable. The trigonometric coefficients are computed using direct calls tosin
andcos
, for accuracy. Recursion relations could be used to compute the lookup table faster, but if an application performs many FFTs of the same length then computing the wavetable is a oneoff overhead which does not affect the final throughput.The wavetable structure can be used repeatedly for any transform of the same length. The table is not modified by calls to any of the other FFT functions. The appropriate type of wavetable must be used for forward real or inverse halfcomplex transforms.

void
gsl_fft_real_wavetable_free
(gsl_fft_real_wavetable * wavetable)¶ 
void
gsl_fft_halfcomplex_wavetable_free
(gsl_fft_halfcomplex_wavetable * wavetable)¶ These functions free the memory associated with the wavetable
wavetable
. The wavetable can be freed if no further FFTs of the same length will be needed.
The mixed radix algorithms require additional working space to hold the intermediate steps of the transform,

gsl_fft_real_workspace
¶ This workspace contains parameters needed to compute a real FFT.

gsl_fft_real_workspace *
gsl_fft_real_workspace_alloc
(size_t n)¶ This function allocates a workspace for a real transform of length
n
. The same workspace can be used for both forward real and inverse halfcomplex transforms.

void
gsl_fft_real_workspace_free
(gsl_fft_real_workspace * workspace)¶ This function frees the memory associated with the workspace
workspace
. The workspace can be freed if no further FFTs of the same length will be needed.
The following functions compute the transforms of real and halfcomplex data,

int
gsl_fft_real_transform
(double data[], size_t stride, size_t n, const gsl_fft_real_wavetable * wavetable, gsl_fft_real_workspace * work)¶ 
int
gsl_fft_halfcomplex_transform
(double data[], size_t stride, size_t n, const gsl_fft_halfcomplex_wavetable * wavetable, gsl_fft_real_workspace * work)¶ These functions compute the FFT of
data
, a real or halfcomplex array of lengthn
, using a mixed radix decimationinfrequency algorithm. Forgsl_fft_real_transform()
data
is an array of timeordered real data. Forgsl_fft_halfcomplex_transform()
data
contains Fourier coefficients in the halfcomplex ordering described above. There is no restriction on the lengthn
. Efficient modules are provided for subtransforms of length 2, 3, 4 and 5. Any remaining factors are computed with a slow, , generaln module. The caller must supply awavetable
containing trigonometric lookup tables and a workspacework
.

int
gsl_fft_real_unpack
(const double real_coefficient[], gsl_complex_packed_array complex_coefficient, size_t stride, size_t n)¶ This function converts a single real array,
real_coefficient
into an equivalent complex array,complex_coefficient
, (with imaginary part set to zero), suitable forgsl_fft_complex
routines. The algorithm for the conversion is simply:for (i = 0; i < n; i++) { complex_coefficient[i*stride].real = real_coefficient[i*stride]; complex_coefficient[i*stride].imag = 0.0; }

int
gsl_fft_halfcomplex_unpack
(const double halfcomplex_coefficient[], gsl_complex_packed_array complex_coefficient, size_t stride, size_t n)¶ This function converts
halfcomplex_coefficient
, an array of halfcomplex coefficients as returned bygsl_fft_real_transform()
, into an ordinary complex array,complex_coefficient
. It fills in the complex array using the symmetry to reconstruct the redundant elements. The algorithm for the conversion is:complex_coefficient[0].real = halfcomplex_coefficient[0]; complex_coefficient[0].imag = 0.0; for (i = 1; i < n  i; i++) { double hc_real = halfcomplex_coefficient[(2 * i  1)*stride]; double hc_imag = halfcomplex_coefficient[(2 * i)*stride]; complex_coefficient[i*stride].real = hc_real; complex_coefficient[i*stride].imag = hc_imag; complex_coefficient[(n  i)*stride].real = hc_real; complex_coefficient[(n  i)*stride].imag = hc_imag; } if (i == n  i) { complex_coefficient[i*stride].real = halfcomplex_coefficient[(n  1)*stride]; complex_coefficient[i*stride].imag = 0.0; }
Here is an example program using gsl_fft_real_transform()
and
gsl_fft_halfcomplex_inverse()
. It generates a real signal in the
shape of a square pulse. The pulse is Fourier transformed to frequency
space, and all but the lowest ten frequency components are removed from
the array of Fourier coefficients returned by
gsl_fft_real_transform()
.
The remaining Fourier coefficients are transformed back to the timedomain, to give a filtered version of the square pulse. Since Fourier coefficients are stored using the halfcomplex symmetry both positive and negative frequencies are removed and the final filtered signal is also real.
#include <stdio.h>
#include <math.h>
#include <gsl/gsl_errno.h>
#include <gsl/gsl_fft_real.h>
#include <gsl/gsl_fft_halfcomplex.h>
int
main (void)
{
int i, n = 100;
double data[n];
gsl_fft_real_wavetable * real;
gsl_fft_halfcomplex_wavetable * hc;
gsl_fft_real_workspace * work;
for (i = 0; i < n; i++)
{
data[i] = 0.0;
}
for (i = n / 3; i < 2 * n / 3; i++)
{
data[i] = 1.0;
}
for (i = 0; i < n; i++)
{
printf ("%d: %e\n", i, data[i]);
}
printf ("\n");
work = gsl_fft_real_workspace_alloc (n);
real = gsl_fft_real_wavetable_alloc (n);
gsl_fft_real_transform (data, 1, n,
real, work);
gsl_fft_real_wavetable_free (real);
for (i = 11; i < n; i++)
{
data[i] = 0;
}
hc = gsl_fft_halfcomplex_wavetable_alloc (n);
gsl_fft_halfcomplex_inverse (data, 1, n,
hc, work);
gsl_fft_halfcomplex_wavetable_free (hc);
for (i = 0; i < n; i++)
{
printf ("%d: %e\n", i, data[i]);
}
gsl_fft_real_workspace_free (work);
return 0;
}
The program output is shown in Fig. 2.
References and Further Reading¶
A good starting point for learning more about the FFT is the following review article,
 P. Duhamel and M. Vetterli. Fast Fourier transforms: A tutorial review and a state of the art. Signal Processing, 19:259–299, 1990.
To find out about the algorithms used in the GSL routines you may want
to consult the document “GSL FFT Algorithms” (it is included
in GSL, as doc/fftalgorithms.tex
). This has general information
on FFTs and explicit derivations of the implementation for each
routine. There are also references to the relevant literature. For
convenience some of the more important references are reproduced below.
There are several introductory books on the FFT with example programs, such as “The Fast Fourier Transform” by Brigham and “DFT/FFT and Convolution Algorithms” by Burrus and Parks,
 Oran Brigham. “The Fast Fourier Transform”. Prentice Hall, 1974.
 C. S. Burrus and T. W. Parks. “DFT/FFT and Convolution Algorithms”, Wiley, 1984.
Both these introductory books cover the radix2 FFT in some detail. The mixedradix algorithm at the heart of the FFTPACK routines is reviewed in Clive Temperton’s paper,
 Clive Temperton, Selfsorting mixedradix fast Fourier transforms, Journal of Computational Physics, 52(1):1–23, 1983.
The derivation of FFTs for realvalued data is explained in the following two articles,
 Henrik V. Sorenson, Douglas L. Jones, Michael T. Heideman, and C. Sidney Burrus. Realvalued fast Fourier transform algorithms. “IEEE Transactions on Acoustics, Speech, and Signal Processing”, ASSP35(6):849–863, 1987.
 Clive Temperton. Fast mixedradix real Fourier transforms. “Journal of Computational Physics”, 52:340–350, 1983.
In 1979 the IEEE published a compendium of carefullyreviewed Fortran FFT programs in “Programs for Digital Signal Processing”. It is a useful reference for implementations of many different FFT algorithms,
 Digital Signal Processing Committee and IEEE Acoustics, Speech, and Signal Processing Committee, editors. Programs for Digital Signal Processing. IEEE Press, 1979.
For largescale FFT work we recommend the use of the dedicated FFTW library by Frigo and Johnson. The FFTW library is selfoptimizing—it automatically tunes itself for each hardware platform in order to achieve maximum performance. It is available under the GNU GPL.
 FFTW Website, http://www.fftw.org/
The source code for FFTPACK is available from http://www.netlib.org/fftpack/