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NAG Toolbox: nag_sum_fft_realherm_1d_multi_col (c06pq)

 Contents

    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example

Purpose

nag_sum_fft_realherm_1d_multi_col (c06pq) computes the discrete Fourier transforms of m sequences, each containing n real data values or a Hermitian complex sequence stored column-wise in a complex storage format.

Syntax

[x, ifail] = c06pq(direct, n, m, x)
[x, ifail] = nag_sum_fft_realherm_1d_multi_col(direct, n, m, x)

Description

Given m sequences of n real data values xjp , for j=0,1,,n-1 and p= 1,2,,m, nag_sum_fft_realherm_1d_multi_col (c06pq) simultaneously calculates the Fourier transforms of all the sequences defined by
z^kp = 1n j=0 n-1 xjp × exp -i 2πjk n ,   k0,1,,n-1 ​ and ​ p=1,2,,m .  
The transformed values z^ k p  are complex, but for each value of p the z^ k p  form a Hermitian sequence (i.e., z^ n-k p  is the complex conjugate of z^ k p ), so they are completely determined by mn  real numbers (since z^0p  is real, as is z^ n/2 p  for n even).
Alternatively, given m Hermitian sequences of n complex data values zjp , this function simultaneously calculates their inverse (backward) discrete Fourier transforms defined by
x^kp = 1n j=0 n-1 zjp × exp i 2πjk n ,   k=0,1,,n-1 ​ and ​ p=1,2,,m .  
The transformed values x^kp  are real.
(Note the scale factor 1n  in the above definition.)
A call of nag_sum_fft_realherm_1d_multi_col (c06pq) 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 coding is provided for the factors 2, 3, 4 and 5.

References

Brigham E O (1974) The Fast Fourier Transform Prentice–Hall
Temperton C (1983) Fast mixed-radix real Fourier transforms J. Comput. Phys. 52 340–350

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 real or 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+2×m – double array
The data must be stored in x as if in a two-dimensional array of dimension 0:n+1,1:m; each of the m sequences is stored in a column of the array. In other words, if the data values of the pth sequence to be transformed are denoted by xjp, for j=0,1,,n-1, then:
  • if direct='F', xp-1×n+2+j must contain xjp, for j=0,1,,n-1 and p=1,2,,m;
  • if direct='B', xp-1 × n+2 + 2×k and xp-1 × n+2 + 2×k+1 must contain the real and imaginary parts respectively of z^kp, for k=0,1,,n/2 and p=1,2,,m. (Note that for the sequence z^kp to be Hermitian, the imaginary part of z^0p, and of z^ n/2 p  for n even, must be zero.)

Optional Input Parameters

None.

Output Parameters

1:     x n+2×m – double array
  • if direct='F' and x is declared with bounds 0:n+1,1:m then x2×kp and x2×k+1p will contain the real and imaginary parts respectively of z^ k p , for k=0,1,,n/2 and p=1,2,,m;
  • if direct='B' and x is declared with bounds 0:n+1,1:m then xjp will contain xjp, for j=0,1,,n-1 and p=1,2,,m.
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=4
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_realherm_1d_multi_col (c06pq) is approximately proportional to nm logn, but also depends on the factors of n. nag_sum_fft_realherm_1d_multi_col (c06pq) 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 real data values and prints their discrete Fourier transforms (as computed by nag_sum_fft_realherm_1d_multi_col (c06pq) with direct='F'), after expanding them from complex Hermitian form into a full complex sequences.
Inverse transforms are then calculated by calling nag_sum_fft_realherm_1d_multi_col (c06pq) with direct='B' showing that the original sequences are restored.
function c06pq_example


fprintf('c06pq example results\n\n');

% 3 real sequences stored as rows
m = int64(3);
n = int64(6);
x = [0.3854   0.6772   0.1138   0.6751   0.6362  0.1424  0  0;
     0.5417   0.2983   0.1181   0.7255   0.8638  0.8723  0  0;
     0.9172   0.0644   0.6037   0.6430   0.0428  0.4815  0  0];
x = transpose(x);
disp('Original data values:');
disp(x(1:n,:));

% Transform to get Hermitian sequences
direct = 'F';
[xt, ifail] = c06pq(direct, n, m, x);

zt = xt(1:2:n+1,:) + i*xt(2:2:n+2,:);
title = 'Discrete Fourier transforms in complex Hermitian format:';
[ifail] = x04da('General','Non-unit', zt, title);

for j = 1:m
  zt(1:n,j) = nag_herm2complex(n,xt(:,j));
end
title = 'Discrete Fourier transforms in full complex format:';
disp(' ');
[ifail] = x04da('General','Non-unit', zt, title);

% Restore data by back transform
direct = 'B';
[xr, ifail] = c06pq(direct, n, m, xt);
disp(' ');
disp('Original data as restored by inverse transform:');
disp(xr(1:n,:));



function [z] = nag_herm2complex(n,x);
  z(1) = complex(x(1));
  for j = 2:floor(double(n)/2) + 1
    z(j) = x(2*j-1) + i*x(2*j);
    z(n-j+2) = x(2*j-1) - i*x(2*j);
  end
c06pq example results

Original data values:
    0.3854    0.5417    0.9172
    0.6772    0.2983    0.0644
    0.1138    0.1181    0.6037
    0.6751    0.7255    0.6430
    0.6362    0.8638    0.0428
    0.1424    0.8723    0.4815

 Discrete Fourier transforms in complex Hermitian format:
             1          2          3
 1      1.0737     1.3961     1.1237
        0.0000     0.0000     0.0000

 2     -0.1041    -0.0365     0.0914
       -0.0044     0.4666    -0.0508

 3      0.1126     0.0780     0.3936
       -0.3738    -0.0607     0.3458

 4     -0.1467    -0.1521     0.1530
        0.0000     0.0000     0.0000
 
 Discrete Fourier transforms in full complex format:
             1          2          3
 1      1.0737     1.3961     1.1237
        0.0000     0.0000     0.0000

 2     -0.1041    -0.0365     0.0914
       -0.0044     0.4666    -0.0508

 3      0.1126     0.0780     0.3936
       -0.3738    -0.0607     0.3458

 4     -0.1467    -0.1521     0.1530
        0.0000     0.0000     0.0000

 5      0.1126     0.0780     0.3936
        0.3738     0.0607    -0.3458

 6     -0.1041    -0.0365     0.0914
        0.0044    -0.4666     0.0508
 
Original data as restored by inverse transform:
    0.3854    0.5417    0.9172
    0.6772    0.2983    0.0644
    0.1138    0.1181    0.6037
    0.6751    0.7255    0.6430
    0.6362    0.8638    0.0428
    0.1424    0.8723    0.4815


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