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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 section References and Further Reading)
Fast Fourier Transforms are efficient algorithms for calculating the discrete Fourier transform (DFT),
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 W\vec{z}. A general matrix-vector multiplication takes O(n^2) operations for n data-points. Fast Fourier transform algorithms use a divide-and-conquer strategy to factorize the matrix W into smaller sub-matrices, corresponding to the integer factors of the length n. If n can be factorized into a product of integers f_1 f_2 ... f_m then the DFT can be computed in O(n \sum f_i) operations. For a radix-2 FFT this gives an operation count of 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, x = FFT(z), is,
x_j = \sum_{k=0}^{n-1} z_k \exp(-2\pi i j k / n)
and the definition of the inverse Fourier transform, x = IFFT(z), is,
z_j = {1 \over n} \sum_{k=0}^{n-1} x_k \exp(2\pi i j k / n).
The factor of 1/n 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/ 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,
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.
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 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 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 \Delta then the frequency-domain includes both positive and negative frequencies, ranging from -1/(2\Delta) through 0 to +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, and the correspondence between the time-domain data z, and the frequency-domain data 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 n is even the location n/2 contains the most positive and negative frequencies (+1/(2 \Delta), -1/(2 \Delta)) which are equivalent. If n is odd then general structure of the table above still applies, but n/2 does not appear.
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 `gsl_fft_complex.h'.
These functions compute forward, backward and inverse FFTs of length
n with stride stride, on the packed complex array data
using an in-place radix-2 decimation-in-time algorithm. The length of
the transform n is restricted to powers of two. For the
transform
version of the function the sign argument can be
either forward
(-1) or backward
(+1).
The functions return a value of GSL_SUCCESS
if no errors were
detected, or GSL_EDOM
if the length of the data n is not a
power of two.
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 (-10 ... 10), 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,128-i) = 1.0; } for (i = 0; i < 128; i++) { printf ("%d %e %e\n", i, REAL(data,i), IMAG(data,i)); } printf ("\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 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 t=128, and working in units of k/n, the DFT approximates the continuum Fourier transform, giving a modulated sine function.
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 2*2 and 2*3.
For factors which are not implemented as modules there is a fall-back to a general length-n module which uses Singleton's method for efficiently computing a DFT. This module is O(n^2), and slower than a dedicated module would be but works for any length n. Of course, lengths which use the general length-n module will still be factorized as much as possible. For example, a length of 143 will be factorized into 11*13. Large prime factors are the worst case scenario, e.g. as found in n=2*3*99991, and should be avoided because their 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 gsl_fft_complex_wavetable_alloc
returns the list of factors chosen by the library for a given length
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 n \sum f_i, where the f_i are the
factors of 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.
All the functions described in this section are declared in the header file `gsl_fft_complex.h'.
gsl_fft_complex_wavetable
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 to sin
and 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.
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 run-time or
numerical error. The wavetable structure is declared in the header file
`gsl_fft_complex.h'.
size_t n
size_t nf
n
was decomposed into.
size_t factor[64]
nf
elements are
used.
gsl_complex * trig
n
complex elements.
gsl_complex * twiddle[64]
trig
, giving the twiddle
factors for each pass.
The mixed radix algorithms require additional working space to hold the intermediate steps of the transform.
The following functions compute the transform,
These functions compute forward, backward and inverse FFTs of length
n with stride stride, on the packed complex array
data, using a mixed radix decimation-in-frequency algorithm.
There is no restriction on the length n. Efficient modules are
provided for subtransforms of length 2, 3, 4, 5, 6 and 7. Any remaining
factors are computed with a slow, O(n^2), general-n
module. The caller must supply a wavetable containing the
trigonometric lookup tables and a workspace work. For the
transform
version of the function the sign argument can be
either forward
(-1) or backward
(+1).
The functions return a value of 0
if no errors were detected. The
following gsl_errno
conditions are defined for these functions:
GSL_EDOM
GSL_EINVAL
Here is an example program which computes the FFT of a short pulse in a sample of length 630 (=2*3*3*5*7) using the mixed-radix 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,n-i) = 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 < wavetable->nf; i++) { printf ("# factor %d: %d\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.
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:
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
gsl_fft_real
which operate on real sequences and functions in
gsl_fft_halfcomplex
which operate on half-complex sequences.
Functions in gsl_fft_real
compute the frequency coefficients of a
real sequence. The half-complex coefficients c of a real sequence
x are given by Fourier analysis,
c_k = \sum_{j=0}^{n-1} x_j \exp(-2 \pi i j k /n)
Functions in gsl_fft_halfcomplex
compute inverse or backwards
transforms. They reconstruct real sequences by Fourier synthesis from
their half-complex frequency coefficients, c,
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 k=n/2 is also real. Thus only 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).
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 `gsl_fft_real.h'
This function computes an in-place radix-2 FFT of length n and stride stride on the real array 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 k < n/2 the real part of the k-th term is stored in location k, and the corresponding imaginary part is stored in location n-k. Terms with k > n/2 can be reconstructed using the symmetry z_k = z^*_{n-k}. The terms for k=0 and k=n/2 are both purely real, and count as a special case. Their real parts are stored in locations 0 and n/2 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 (assuming 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 gsl_fft_halfcomplex_radix2_unpack
described below.
The radix-2 FFT functions for halfcomplex data are declared in the header file `gsl_fft_halfcomplex.h'.
These functions compute the inverse or backwards in-place radix-2 FFT of
length n and stride stride on the half-complex sequence
data stored according the output scheme used by
gsl_fft_real_radix2
. The result is a real array stored in natural
order.
gsl_fft_real_radix2_transform
, into an ordinary complex array, complex_coefficient. It fills in the
complex array using the symmetry
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; }
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 n/2 is not stored either, since the symmetry 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, n=5. The two columns
give the correspondence between the 5 values in the half-complex
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 zero-frequency term complex[0].imag
is
known to be zero by the symmetry.
The next table shows the output for an even-length sequence, 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 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'.
sin
and 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.
The mixed radix algorithms require additional working space to hold the intermediate steps of the transform,
The following functions compute the transforms of real and half-complex data,
gsl_fft_real_transform
data is an array of
time-ordered real data. For gsl_fft_halfcomplex_transform
data contains Fourier coefficients in the half-complex ordering
described above. There is no restriction on the length n.
Efficient modules are provided for subtransforms of length 2, 3, 4 and
5. Any remaining factors are computed with a slow, O(n^2),
general-n module. The caller must supply a wavetable containing
trigonometric lookup tables and a workspace work.
This function converts a single real array, real_coefficient into
an equivalent complex array, complex_coefficient, (with imaginary
part set to zero), suitable for 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; }
This function converts halfcomplex_coefficient, an array of
half-complex coefficients as returned by gsl_fft_real_transform
, into an
ordinary complex array, complex_coefficient. It fills in the
complex array using the symmetry
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 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 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 <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; }
A good starting point for learning more about the FFT is the review article Fast Fourier Transforms: A Tutorial Review and A State of the Art by Duhamel and Vetterli,
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,
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,
The derivation of FFTs for real-valued data is explained in the following two articles,
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,
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.
The source code for FFTPACK is available from Netlib,
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