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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_sum_fft_complex_1d_multi_col (c06ps)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_sum_fft_complex_1d_multi_col (c06ps) computes the discrete Fourier transforms of m sequences, stored as columns of an array, each containing n complex data values.

Syntax

[x, ifail] = c06ps(direct, n, m, x)
[x, ifail] = nag_sum_fft_complex_1d_multi_col(direct, n, m, x)

Description

Given m sequences of n complex data values zjp , for j=0,1,,n-1 and p=1,2,,m, nag_sum_fft_complex_1d_multi_col (c06ps) simultaneously calculates the (forward or backward) discrete Fourier transforms of all the sequences defined by
z^kp=1n j=0 n-1zjp×exp±i2πjkn ,  k=0,1,,n-1​ and ​p=1,2,,m.  
(Note the scale factor 1n  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_multi_col (c06ps) with direct='F' followed by a call with direct='B' will restore the original data.
The function uses a variant of the fast Fourier transform (FFT) algorithm (see Brigham (1974)) known as the Stockham self-sorting algorithm, which is described in Temperton (1983). Special code is provided for the factors 2, 3 and 5.

References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Self-sorting mixed-radix fast Fourier transforms J. Comput. Phys. 52 1–23

Parameters

Compulsory Input Parameters

1:     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: direct='F' or 'B'.
2:     n int64int32nag_int scalar
n, the number of complex values in each sequence.
Constraint: n1.
3:     m int64int32nag_int scalar
m, the number of sequences to be transformed.
Constraint: m1.
4:     x n×m – complex array
The complex data values zjp stored in xp-1×n+j+1, for j=0,1,,n-1 and p=1,2,,m.

Optional Input Parameters

None.

Output Parameters

1:     x n×m – complex array
Stores the complex transforms.
2:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
   ifail=1
On entry,m<1.
   ifail=2
On entry,n<1.
   ifail=3
On entry,direct'F' or 'B'.
   ifail=5
An unexpected error has occurred in an internal call. Check all function calls and array dimensions. Seek expert help.
   ifail=-99
An unexpected error has been triggered by this routine. Please contact NAG.
   ifail=-399
Your licence key may have expired or may not have been installed correctly.
   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 by nag_sum_fft_complex_1d_multi_col (c06ps) is approximately proportional to nm logn, but also depends on the factors of n. nag_sum_fft_complex_1d_multi_col (c06ps) is fastest if the only prime factors of n are 2, 3 and 5, and is particularly slow if n is a large prime, or has large prime factors.

Example

This example reads in sequences of complex data values and prints their discrete Fourier transforms (as computed by nag_sum_fft_complex_1d_multi_col (c06ps) with direct='F'). Inverse transforms are then calculated using nag_sum_fft_complex_1d_multi_col (c06ps) with direct='B' and printed out, showing that the original sequences are restored.
function c06ps_example


fprintf('c06ps example results\n\n');

m = int64(3);
n = int64(6);
zr = [0.3854  0.6772  0.1138  0.6751  0.6362  0.1424;
      0.9172  0.0644  0.6037  0.6430  0.0428  0.4815;
      0.1156  0.0685  0.2060  0.8630  0.6967  0.2792];
zi = [0.5417  0.2983  0.1181  0.7255  0.8638  0.8723; 
      0.9089  0.3118  0.3465  0.6198  0.2668  0.1614; 
      0.6214  0.8681  0.7060  0.8652  0.9190  0.3355];
z = transpose(zr + i*zi);

title = 'Original sequences:';
[ifail] = x04da('General','Non-unit', z, title);

% transform
direct = 'F';
[zt, ifail] = c06ps(direct, n, m, z);
disp(' ');
title = 'Discrete Fourier Transforms:';
[ifail] = x04da('General','Non-unit', zt, title);

% Restore by back-transform
direct = 'B';
[zr, ifail] = c06ps(direct, n, m, zt);
disp(' ');
title = 'Original data as restored by inverse transform';
[ifail] = x04da('General','Non-unit', zr, title);


c06ps example results

 Original sequences:
          1       2       3
 1   0.3854  0.9172  0.1156
     0.5417  0.9089  0.6214

 2   0.6772  0.0644  0.0685
     0.2983  0.3118  0.8681

 3   0.1138  0.6037  0.2060
     0.1181  0.3465  0.7060

 4   0.6751  0.6430  0.8630
     0.7255  0.6198  0.8652

 5   0.6362  0.0428  0.6967
     0.8638  0.2668  0.9190

 6   0.1424  0.4815  0.2792
     0.8723  0.1614  0.3355
 
 Discrete Fourier Transforms:
             1          2          3
 1      1.0737     1.1237     0.9100
        1.3961     1.0677     1.7617

 2     -0.5706     0.1728    -0.3054
       -0.0409     0.0386     0.0624

 3      0.1733     0.4185     0.4079
       -0.2958     0.7481    -0.0695

 4     -0.1467     0.1530    -0.0785
       -0.1521     0.1752     0.0725

 5      0.0518     0.3686    -0.1193
        0.4517     0.0565     0.1285

 6      0.3625     0.0101    -0.5314
       -0.0321     0.1403    -0.4335
 
 Original data as restored by inverse transform
          1       2       3
 1   0.3854  0.9172  0.1156
     0.5417  0.9089  0.6214

 2   0.6772  0.0644  0.0685
     0.2983  0.3118  0.8681

 3   0.1138  0.6037  0.2060
     0.1181  0.3465  0.7060

 4   0.6751  0.6430  0.8630
     0.7255  0.6198  0.8652

 5   0.6362  0.0428  0.6967
     0.8638  0.2668  0.9190

 6   0.1424  0.4815  0.2792
     0.8723  0.1614  0.3355

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