.. index::

   single: FFT

   single: Fast Fourier Transforms, see FFT

   single: Fourier Transforms, see FFT

   single: Discrete Fourier Transforms, see FFT

   single: DFTs, see FFT

******************************

Fast Fourier Transforms (FFTs)

******************************

.. include:: include.rst

This chapter describes functions for performing Fast Fourier Transforms

(FFTs).  The library includes radix-2 routines (for lengths which are a

power of two) and mixed-radix routines (which work for any length).  For

efficiency there are separate versions of the routines for real data and

for complex data.  The mixed-radix 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 :ref:`References and Further Reading <fft-references>`)

.. index:: FFT mathematical definition

Mathematical Definitions

========================

Fast Fourier Transforms are efficient algorithms for

calculating the discrete Fourier transform (DFT),

.. math:: x_j = \sum_{k=0}^{n-1} z_k \exp(-2 \pi i j k / n) 

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

matrix-vector multiplication :math:`W\vec{z}`.

A general matrix-vector multiplication takes

:math:`O(n^2)` operations for :math:`n` data-points.  Fast Fourier

transform algorithms use a divide-and-conquer strategy to factorize the

matrix :math:`W` into smaller sub-matrices, corresponding to the integer

factors of the length :math:`n`.  If :math:`n` can be factorized into a

product of integers :math:`f_1 f_2 \ldots f_m`

then the DFT can be computed in :math:`O(n \sum f_i)`

operations.  For a radix-2 FFT this gives an operation count of

:math:`O(n \log_2 n)`.

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*,

:math:`x = \hbox{FFT}(z)`, is,

.. math:: x_j = \sum_{k=0}^{n-1} z_k \exp(-2 \pi i j k / n) 

and the definition of the *inverse Fourier transform*,

:math:`x = \hbox{IFFT}(z)`, is,

.. math:: z_j = {1 \over n} \sum_{k=0}^{n-1} x_k \exp(2 \pi i j k / n).

The factor of :math:`1/n` makes this a true inverse.  For example, a call

to :func:`gsl_fft_complex_forward` followed by a call to

:func:`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/ inverse-transform 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,

.. math:: z^{backwards}_j = \sum_{k=0}^{n-1} x_k \exp(2 \pi i j k / n)

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.

.. index::

   single: FFT, complex data

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, :code:`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 :code:`z[stride*i]` instead of :code:`z[i]`.  A stride greater

than 1 can be used to take an in-place 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 :code:`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 time-step of the DFT is :math:`\Delta` then the

frequency-domain includes both positive and negative frequencies,

ranging from :math:`-1/(2\Delta)` through 0 to :math:`+1/(2\Delta)`.  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:`data`, and the

correspondence between the time-domain data :math:`z`, and the

frequency-domain data :math:`x`::

  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))

  .        ........        ..................

  n-3      z(t = n-3)      x(f = -3/(n Delta))

  n-2      z(t = n-2)      x(f = -2/(n Delta))

  n-1      z(t = n-1)      x(f = -1/(n Delta))

When :math:`n` is even the location :math:`n/2` contains the most positive

and negative frequencies (:math:`+1/(2 \Delta)`, :math:`-1/(2 \Delta)`)

which are equivalent.  If :math:`n` is odd then general structure of the

table above still applies, but :math:`n/2` does not appear.

.. index::

   single: FFT of complex data, radix-2 algorithm

   single: Radix-2 FFT, complex data

Radix-2 FFT routines for complex data

=====================================

The radix-2 algorithms described in this section are simple and compact,

although not necessarily the most efficient.  They use the Cooley-Tukey

algorithm to compute in-place complex FFTs for lengths which are a power

of 2---no additional storage is required.  The corresponding

self-sorting mixed-radix 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 :file:`gsl_fft_complex.h`.

.. function:: 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

   :data:`n` with stride :data:`stride`, on the packed complex array :data:`data`

   using an in-place radix-2 decimation-in-time algorithm.  The length of

   the transform :data:`n` is restricted to powers of two.  For the

   :code:`transform` version of the function the :data:`sign` argument can be

   either :code:`forward` (:math:`-1`) or :code:`backward` (:math:`+1`).

   The functions return a value of :macro:`GSL_SUCCESS` if no errors were

   detected, or :macro:`GSL_EDOM` if the length of the data :data:`n` is not a

   power of two.

.. function:: 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 decimation-in-frequency versions of the radix-2 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 (:math:`-10 \dots 10`),

where the negative times wrap around the end of the array.

.. include:: examples/fft.c

   :code:

Note that we have assumed that the program is using the default error

handler (which calls :func:`abort` for any errors).  If you are not using

a safe error handler you would need to check the return status of

:func:`gsl_fft_complex_radix2_forward`.

The transformed data is rescaled by :math:`1/\sqrt n` 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

wrap-around of negative times at :math:`t=128`, and working in units of

:math:`k/n`, the DFT approximates the continuum Fourier transform, giving

a modulated sine function.

.. math:: \int_{-a}^{+a} e^{-2 \pi i k x} dx = {\sin(2\pi k a) \over\pi k}

The output of the example program is plotted in :numref:`fig_fft-complex-radix2`.

.. _fig_fft-complex-radix2:

.. figure:: /images/fft-complex-radix2.png

   :scale: 60%

   A pulse and its discrete Fourier transform, output from

   the example program.

.. index::

   single: FFT of complex data, mixed-radix algorithm

   single: Mixed-radix FFT, complex data

Mixed-radix FFT routines for complex data

=========================================

This section describes mixed-radix FFT algorithms for complex data.  The

mixed-radix 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 "Self-sorting

Mixed-radix FFTs" by Clive Temperton.  The routines here use the same

indexing scheme and basic algorithms as |fftpack|.

The mixed-radix algorithm is based on sub-transform 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 :math:`2*2` and :math:`2*3`.

For factors which are not implemented as modules there is a fall-back to

a general length-:math:`n` module which uses Singleton's method for

efficiently computing a DFT. This module is :math:`O(n^2)`, and slower

than a dedicated module would be but works for any length :math:`n`.  Of

course, lengths which use the general length-:math:`n` module will still

be factorized as much as possible.  For example, a length of 143 will be

factorized into :math:`11*13`.  Large prime factors are the worst case

scenario, e.g. as found in :math:`n=2*3*99991`, and should be avoided

because their :math:`O(n^2)` scaling will dominate the run-time (consult

the document "GSL FFT Algorithms" included in the GSL distribution

if you encounter this problem).

The mixed-radix initialization function :func:`gsl_fft_complex_wavetable_alloc`

returns the list of factors chosen by the library for a given length

:math:`n`.  It can be used to check how well the length has been

factorized, and estimate the run-time.  To a first approximation the

run-time scales as :math:`n \sum f_i`, where the :math:`f_i` are the

factors of :math:`n`.  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 small-prime modules consult

"GSL FFT Algorithms" for details on adding support for other

factors.

.. First, the space for the trigonometric lookup tables and scratch area is

.. allocated by a call to one of the :code:`alloc` functions.  We

.. call the combination of factorization, scratch space and trigonometric

.. lookup arrays a *wavetable*.  It contains the sine and cosine

.. waveforms for the all the frequencies that will be used in the FFT.

.. The wavetable is initialized by a call to the corresponding :code:`init`

.. function.  It factorizes the data length, using the implemented

.. subtransforms as preferred factors wherever possible.  The trigonometric

.. lookup table for the chosen factorization is also computed.

.. An FFT is computed by a call to one of the :code:`forward`,

.. :code:`backward` or :code:`inverse` functions, with the data, length and

.. wavetable as arguments.

All the functions described in this section are declared in the header

file :file:`gsl_fft_complex.h`.

.. function:: 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 :data:`n`. The function returns a pointer to the newly allocated

   :type:`gsl_fft_complex_wavetable` if no errors were detected, and a null

   pointer in the case of error.  The length :data:`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 to :code:`sin` and :code:`cos`, 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 one-off 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.

.. function:: void gsl_fft_complex_wavetable_free (gsl_fft_complex_wavetable * wavetable)

   This function frees the memory associated with the wavetable

   :data:`wavetable`.  The wavetable can be freed if no further FFTs of the

   same length will be needed.

These functions operate on a :type:`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 run-time or

numerical error. The wavetable structure is declared in the header file

:file:`gsl_fft_complex.h`.

.. type:: 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:

   ================================= ==============================================================================================

   :code:`size_t n`                  This is the number of complex data points

   :code:`size_t nf`                 This is the number of factors that the length :code:`n` was decomposed into.

   :code:`size_t factor[64]`         This is the array of factors.  Only the first :code:`nf` elements are used. 

   :code:`gsl_complex * trig`        This is a pointer to a preallocated trigonometric lookup table of :code:`n` complex elements.

   :code:`gsl_complex * twiddle[64]` This is an array of pointers into :code:`trig`, giving the twiddle factors for each pass.

   ================================= ==============================================================================================

.. (FIXME: factor[64] is a fixed length array and therefore probably in

.. violation of the GNU Coding Standards).

.. type:: gsl_fft_complex_workspace

   The mixed radix algorithms require additional working space to hold

   the intermediate steps of the transform.

.. function:: gsl_fft_complex_workspace * gsl_fft_complex_workspace_alloc (size_t n)

   This function allocates a workspace for a complex transform of length

   :data:`n`.

.. function:: void gsl_fft_complex_workspace_free (gsl_fft_complex_workspace * workspace)

   This function frees the memory associated with the workspace

   :data:`workspace`. The workspace can be freed if no further FFTs of the

   same length will be needed.

The following functions compute the transform,

.. function:: 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

   :data:`n` with stride :data:`stride`, on the packed complex array

   :data:`data`, using a mixed radix decimation-in-frequency algorithm.

   There is no restriction on the length :data:`n`.  Efficient modules are

   provided for subtransforms of length 2, 3, 4, 5, 6 and 7.  Any remaining

   factors are computed with a slow, :math:`O(n^2)`, general-:math:`n`

   module. The caller must supply a :data:`wavetable` containing the

   trigonometric lookup tables and a workspace :data:`work`.  For the

   :code:`transform` version of the function the :data:`sign` argument can be

   either :code:`forward` (:math:`-1`) or :code:`backward` (:math:`+1`).

   The functions return a value of :code:`0` if no errors were detected. The

   following :data:`gsl_errno` conditions are defined for these functions:

   =================================== =========================================================================================================

   :macro:`GSL_EDOM`                   The length of the data :data:`n` is not a positive integer (i.e. :data:`n` is zero).

   :macro:`GSL_EINVAL`                 The length of the data :data:`n` and the length used to compute the given :data:`wavetable` do not match.

   =================================== =========================================================================================================

Here is an example program which computes the FFT of a short pulse in a

sample of length 630 (:math:`=2*3*3*5*7`) using the mixed-radix

algorithm.

.. include:: examples/fftmr.c

   :code:

Note that we have assumed that the program is using the default

:code:`gsl` error handler (which calls :func:`abort` for any errors).  If

you are not using a safe error handler you would need to check the

return status of all the :code:`gsl` routines.

.. index:: FFT of real data

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:

.. math:: z_k = z_{n-k}^*

A sequence with this symmetry is called *conjugate-complex* or

*half-complex*.  This different structure requires different

storage layouts for the forward transform (from real to half-complex)

and inverse transform (from half-complex back to real).  As a

consequence the routines are divided into two sets: functions in

:code:`gsl_fft_real` which operate on real sequences and functions in

:code:`gsl_fft_halfcomplex` which operate on half-complex sequences.

Functions in :code:`gsl_fft_real` compute the frequency coefficients of a

real sequence.  The half-complex coefficients :math:`c` of a real sequence

:math:`x` are given by Fourier analysis,

.. math:: c_k = \sum_{j=0}^{n-1} x_j \exp(-2 \pi i j k /n)

Functions in :code:`gsl_fft_halfcomplex` compute inverse or backwards

transforms.  They reconstruct real sequences by Fourier synthesis from

their half-complex frequency coefficients, :math:`c`,

.. math:: x_j = {1 \over n} \sum_{k=0}^{n-1} c_k \exp(2 \pi i j k /n)

The symmetry of the half-complex sequence implies that only half of the

complex numbers in the output need to be stored.  The remaining half can

be reconstructed using the half-complex symmetry condition. This works

for all lengths, even and odd---when the length is even the middle value

where :math:`k=n/2` is also real.  Thus only :data:`n` real numbers are

required to store the half-complex 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 radix-2 and mixed-radix routines.  The radix-2 function

operates in-place, which constrains the locations where each element can

be stored.  The restriction forces real and imaginary parts to be stored

far apart.  The mixed-radix 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).

.. index::

   single: FFT of real data, radix-2 algorithm

   single: Radix-2 FFT for real data

Radix-2 FFT routines for real data

==================================

This section describes radix-2 FFT algorithms for real data.  They use

the Cooley-Tukey algorithm to compute in-place FFTs for lengths which

are a power of 2. 

The radix-2 FFT functions for real data are declared in the header files

:file:`gsl_fft_real.h` 

.. function:: int gsl_fft_real_radix2_transform (double data[], size_t stride, size_t n)

   This function computes an in-place radix-2 FFT of length :data:`n` and

   stride :data:`stride` on the real array :data:`data`.  The output is a

   half-complex sequence, which is stored in-place.  The arrangement of the

   half-complex terms uses the following scheme: for :math:`k < n/2` the

   real part of the :math:`k`-th term is stored in location :math:`k`, and

   the corresponding imaginary part is stored in location :math:`n-k`.  Terms

   with :math:`k > n/2` can be reconstructed using the symmetry 

   :math:`z_k = z^*_{n-k}`.

   The terms for :math:`k=0` and :math:`k=n/2` are both purely

   real, and count as a special case.  Their real parts are stored in

   locations :math:`0` and :math:`n/2` respectively, while their imaginary

   parts which are zero are not stored.

   The following table shows the correspondence between the output

   :data:`data` and the equivalent results obtained by considering the input

   data as a complex sequence with zero imaginary part (assuming :data:`stride` = 1})::

      complex[0].real    =    data[0] 

      complex[0].imag    =    0 

      complex[1].real    =    data[1] 

      complex[1].imag    =    data[n-1]

      ...............         ................

      complex[k].real    =    data[k]

      complex[k].imag    =    data[n-k] 

      ...............         ................

      complex[n/2].real  =    data[n/2]

      complex[n/2].imag  =    0

      ...............         ................

      complex[k'].real   =    data[k]        k' = n - k

      complex[k'].imag   =   -data[n-k] 

      ...............         ................

      complex[n-1].real  =    data[1]

      complex[n-1].imag  =   -data[n-1]

   Note that the output data can be converted into the full complex

   sequence using the function :func:`gsl_fft_halfcomplex_radix2_unpack`

   described below.

The radix-2 FFT functions for halfcomplex data are declared in the

header file :file:`gsl_fft_halfcomplex.h`.

.. function:: 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 in-place radix-2 FFT of

   length :data:`n` and stride :data:`stride` on the half-complex sequence

   :data:`data` stored according the output scheme used by

   :func:`gsl_fft_real_radix2`.  The result is a real array stored in natural

   order.

.. function:: 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 :data:`halfcomplex_coefficient`, an array of

   half-complex coefficients as returned by :func:`gsl_fft_real_radix2_transform`, into an ordinary complex array, :data:`complex_coefficient`.  It fills in the

   complex array using the symmetry :math:`z_k = z_{n-k}^*`

   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[(n-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;

        }

.. index::

   single: FFT of real data, mixed-radix algorithm

   single: Mixed-radix FFT, real data

Mixed-radix FFT routines for real data

======================================

This section describes mixed-radix FFT algorithms for real data.  The

mixed-radix functions work for FFTs of any length.  They are a

reimplementation of the real-FFT routines in the Fortran |fftpack| library

by Paul Swarztrauber.  The theory behind the algorithm is explained in

the article "Fast Mixed-Radix 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 half-complex

sequences.  In this convention the half-complex 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 zero-frequency 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 :math:`n/2` is not stored

either, since the symmetry :math:`z_k = z_{n-k}^*`

implies that this is purely real too.

The storage scheme is best shown by some examples.  The table below

shows the output for an odd-length sequence, :math:`n=5`.  The two columns

give the correspondence between the 5 values in the half-complex

sequence returned by :func:`gsl_fft_real_transform`, :data:`halfcomplex[]` and the

values :data:`complex[]` that would be returned if the same real input

sequence were passed to :func:`gsl_fft_complex_backward` as a complex

sequence (with imaginary parts set to :code:`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 :data:`complex` array, :code:`complex[3]` and

:code:`complex[4]` are filled in using the symmetry condition.  The

imaginary part of the zero-frequency term :code:`complex[0].imag` is

known to be zero by the symmetry.

The next table shows the output for an even-length sequence, :math:`n=6`.

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 :data:`complex` array, :code:`complex[4]` and

:code:`complex[5]` are filled in using the symmetry condition.  Both

:code:`complex[0].imag` and :code:`complex[3].imag` are known to be zero.

All these functions are declared in the header files

:file:`gsl_fft_real.h` and :file:`gsl_fft_halfcomplex.h`.

.. type:: gsl_fft_real_wavetable

          gsl_fft_halfcomplex_wavetable

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

.. function:: 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

   :math:`n` 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 :data:`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 to :code:`sin` and :code:`cos`, 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 one-off 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 half-complex transforms.

.. function:: 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

   :data:`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,

.. type:: gsl_fft_real_workspace

   This workspace contains parameters needed to compute a real FFT.

.. function:: gsl_fft_real_workspace * gsl_fft_real_workspace_alloc (size_t n)

   This function allocates a workspace for a real transform of length

   :data:`n`.  The same workspace can be used for both forward real and inverse

   halfcomplex transforms.

.. function:: void gsl_fft_real_workspace_free (gsl_fft_real_workspace * workspace)

   This function frees the memory associated with the workspace

   :data:`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 half-complex

data,

.. function:: 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:`data`, a real or half-complex

   array of length :data:`n`, using a mixed radix decimation-in-frequency

   algorithm.  For :func:`gsl_fft_real_transform` :data:`data` is an array of

   time-ordered real data.  For :func:`gsl_fft_halfcomplex_transform`

   :data:`data` contains Fourier coefficients in the half-complex ordering

   described above.  There is no restriction on the length :data:`n`.

   Efficient modules are provided for subtransforms of length 2, 3, 4 and

   5.  Any remaining factors are computed with a slow, :math:`O(n^2)`,

   general-n module.  The caller must supply a :data:`wavetable` containing

   trigonometric lookup tables and a workspace :data:`work`. 

.. function:: 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, :data:`real_coefficient` into

   an equivalent complex array, :data:`complex_coefficient`, (with imaginary

   part set to zero), suitable for :code:`gsl_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;

        }

.. function:: int gsl_fft_halfcomplex_unpack (const double halfcomplex_coefficient[], gsl_complex_packed_array complex_coefficient, size_t stride, size_t n)

   This function converts :data:`halfcomplex_coefficient`, an array of

   half-complex coefficients as returned by :func:`gsl_fft_real_transform`, into an

   ordinary complex array, :data:`complex_coefficient`.  It fills in the

   complex array using the symmetry :math:`z_k = z_{n-k}^*`

   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 :func:`gsl_fft_real_transform` and

:func:`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

:func:`gsl_fft_real_transform`.

The remaining Fourier coefficients are transformed back to the

time-domain, to give a filtered version of the square pulse.  Since

Fourier coefficients are stored using the half-complex symmetry both

positive and negative frequencies are removed and the final filtered

signal is also real.

.. include:: examples/fftreal.c

   :code:

The program output is shown in :numref:`fig_fft-real-mixedradix`.

.. _fig_fft-real-mixedradix:

.. figure:: /images/fft-real-mixedradix.png

   :scale: 100%

   Low-pass filtered version of a real pulse, output from the example program.

.. _fft-references:

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 :file:`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,

* E. 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 radix-2 FFT in some detail.

The mixed-radix algorithm at the heart of the |fftpack| routines is

reviewed in Clive Temperton's paper,

* Clive Temperton, Self-sorting mixed-radix fast Fourier transforms,

  Journal of Computational Physics, 52(1):1--23, 1983.

The derivation of FFTs for real-valued data is explained in the

following two articles,

* Henrik V. Sorenson, Douglas L. Jones, Michael T. Heideman, and C. Sidney

  Burrus.  Real-valued fast Fourier transform algorithms.

  "IEEE Transactions on Acoustics, Speech, and Signal Processing",

  ASSP-35(6):849--863, 1987.

* Clive Temperton.  Fast mixed-radix real Fourier transforms.

  "Journal of Computational Physics", 52:340--350, 1983.

In 1979 the IEEE published a compendium of carefully-reviewed 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 large-scale FFT work we recommend the use of the dedicated FFTW library

by Frigo and Johnson.  The FFTW library is self-optimizing---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/