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NAG Toolbox: nag_sum_fft_complex_1d (c06pc)
Purpose
nag_sum_fft_complex_1d (c06pc) calculates the discrete Fourier transform of a sequence of $n$ complex data values (using complex data type).
Syntax
Description
Given a sequence of
$n$ complex data values
${z}_{\mathit{j}}$, for
$\mathit{j}=0,1,\dots ,n1$,
nag_sum_fft_complex_1d (c06pc) calculates their (
forward or
backward) discrete Fourier transform (DFT) defined by
(Note the scale factor of
$\frac{1}{\sqrt{n}}$ in this definition.) The minus sign is taken in the argument of the exponential within the summation when the forward transform is required, and the plus sign is taken when the backward transform is required.
A call of nag_sum_fft_complex_1d (c06pc) with ${\mathbf{direct}}=\text{'F'}$ followed by a call with ${\mathbf{direct}}=\text{'B'}$ will restore the original data.
nag_sum_fft_complex_1d (c06pc) uses a variant of the fast Fourier transform (FFT) algorithm (see
Brigham (1974)) known as the Stockham selfsorting algorithm, which is described in
Temperton (1983). If
$n$ is a large prime number or if
$n$ contains large prime factors, then the Fourier transform is performed using Bluestein's algorithm (see
Bluestein (1968)), which expresses the DFT as a convolution that in turn can be efficiently computed using FFTs of highly composite sizes.
References
Bluestein L I (1968) A linear filtering approach to the computation of the discrete Fourier transform Northeast Electronics Research and Engineering Meeting Record 10 218–219
Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Selfsorting mixedradix fast Fourier transforms J. Comput. Phys. 52 1–23
Parameters
Compulsory Input Parameters
 1:
$\mathrm{direct}$ – string (length ≥ 1)

If the forward transform as defined in
Description is to be computed, then
direct must be set equal to 'F'.
If the backward transform is to be computed then
direct must be set equal to 'B'.
Constraint:
${\mathbf{direct}}=\text{'F'}$ or $\text{'B'}$.
 2:
$\mathrm{x}\left({\mathbf{n}}\right)$ – complex array

If
x is declared with bounds
$\left(0:{\mathbf{n}}1\right)$ in the function from which
nag_sum_fft_complex_1d (c06pc) is called, then
${\mathbf{x}}\left(\mathit{j}\right)$ must contain
${z}_{\mathit{j}}$, for
$\mathit{j}=0,1,\dots ,n1$.
Optional Input Parameters
 1:
$\mathrm{n}$ – int64int32nag_int scalar

Default:
the dimension of the array
x.
$n$, the number of data values.
Constraint:
${\mathbf{n}}\ge 1$.
Output Parameters
 1:
$\mathrm{x}\left({\mathbf{n}}\right)$ – complex array

The components of the discrete Fourier transform.
If
x is declared with bounds
$\left(0:{\mathbf{n}}1\right)$ in the function from which
nag_sum_fft_complex_1d (c06pc) is called, then
${\hat{z}}_{k}$ is contained in
${\mathbf{x}}\left(k\right)$, for
$0\le k\le n1$.
 2:
$\mathrm{ifail}$ – int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see
Error Indicators and Warnings).
Error Indicators and Warnings
Errors or warnings detected by the function:
 ${\mathbf{ifail}}=1$

On entry,  ${\mathbf{n}}<1$. 
 ${\mathbf{ifail}}=2$

On entry,  ${\mathbf{direct}}\ne \text{'F'}$ or $\text{'B'}$. 
 ${\mathbf{ifail}}=4$

An unexpected error has occurred in an internal call. Check all function calls and array dimensions. Seek expert help.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
Accuracy
Some indication of accuracy can be obtained by performing a subsequent inverse transform and comparing the results with the original sequence (in exact arithmetic they would be identical).
Further Comments
The time taken is approximately proportional to $n\times \mathrm{log}\left(n\right)$, but also depends on the factorization of $n$. nag_sum_fft_complex_1d (c06pc) is faster if the only prime factors of $n$ are $2$, $3$ or $5$; and fastest of all if $n$ is a power of $2$.
When the Bluestein’s FFT algorithm is in use, an additional complex workspace of size approximately $8n$ is allocated.
Example
This example reads in a sequence of complex data values and prints their discrete Fourier transform (as computed by nag_sum_fft_complex_1d (c06pc) with ${\mathbf{direct}}=\text{'F'}$). It then performs an inverse transform using nag_sum_fft_complex_1d (c06pc) with ${\mathbf{direct}}=\text{'B'}$, and prints the sequence so obtained alongside the original data values.
Open in the MATLAB editor:
c06pc_example
function c06pc_example
fprintf('c06pc example results\n\n');
direct = 'F';
x = [ 0.34907  0.37168i;
0.54890  0.35669i;
0.74776  0.31175i;
0.94459  0.23702i;
1.13850  0.13274i;
1.32850 + 0.00074i;
1.51370 + 0.16298i];
[xt, ifail] = c06pc(direct, x);
disp('Discrete Fourier Transform of x:');
disp(xt);
direct = 'B';
[xr, ifail] = c06pc(direct, xt);
fprintf('Original sequence as restored by inverse transform\n\n');
fprintf(' Original Restored\n');
z = [x xr];
disp(z);
c06pc example results
Discrete Fourier Transform of x:
2.4836  0.4710i
0.5518 + 0.4968i
0.3671 + 0.0976i
0.2877  0.0586i
0.2251  0.1748i
0.1483  0.3084i
0.0198  0.5650i
Original sequence as restored by inverse transform
Original Restored
0.3491  0.3717i 0.3491  0.3717i
0.5489  0.3567i 0.5489  0.3567i
0.7478  0.3118i 0.7478  0.3118i
0.9446  0.2370i 0.9446  0.2370i
1.1385  0.1327i 1.1385  0.1327i
1.3285 + 0.0007i 1.3285 + 0.0007i
1.5137 + 0.1630i 1.5137 + 0.1630i
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