c06ff:: Summation of Series (NAG Toolbox)

nag_sum_fft_complex_multid_1d_sep (c06ff) computes the discrete Fourier transform of one variable (the

l

th say) in a multivariate sequence of complex data values

z_{j_{1} j_{2} \dots j_{m}}

, for

j1 = 0, 1, \dots, n_{1} - 1

and

j2 = 0, 1, \dots, n_{2} - 1

, and so on. Thus the individual dimensions are

n_{1}, n_{2}, \dots, n_{m}

, and the total number of data values is

n = n_{1} \times n_{2} \times \dots \times n_{m}

The function computes

n / n_{l}

one-dimensional transforms defined by

{\hat{z}}_{j_{1} \dots k_{l} \dots j_{m}} = \frac{1}{\sqrt{n_{l}}} \sum_{j_{l} = 0}^{n_{l} - 1} z_{j_{1} \dots j_{l} \dots j_{m}} \times \exp (- \frac{2 π i j_{l} k_{l}}{n_{l}}),

where

k_{l} = 0, 1, \dots, n_{l} - 1

To compute the inverse discrete Fourier transforms, defined with

\exp (+ \frac{2 π i j_{l} k_{l}}{n_{l}})

in the above formula instead of

\exp (- \frac{2 π i j_{l} k_{l}}{n_{l}})

, this function should be preceded and followed by the complex conjugation of the data values and the transform (by negating the imaginary parts stored in

y

The data values must be supplied in a pair of one-dimensional arrays (real and imaginary parts separately), in accordance with the Fortran convention for storing multidimensional data (i.e., with the first subscript

j_{1}

varying most rapidly).

This function calls nag_sum_fft_complex_1d_sep (c06fc) to perform one-dimensional discrete Fourier transforms by the fast Fourier transform (FFT) algorithm in Brigham (1974), and hence there are some restrictions on the values of

n_{l}

(see Arguments).

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

The time taken is approximately proportional to

n \times \log n_{l}

, but also depends on the factorization of

n_{l}

. nag_sum_fft_complex_multid_1d_sep (c06ff) is faster if the only prime factors of

n_{l}

are

2

3

5

; and fastest of all if

n_{l}

is a power of

2

Example

This example reads in a bivariate sequence of complex data values and prints the discrete Fourier transform of the second variable. It then performs an inverse transform and prints the sequence so obtained, which may be compared with the original data values.

function c06ff_example


fprintf('c06ff example results\n\n');

x = [ 1.000     0.999     0.987     0.936     0.802;
      0.994     0.989     0.963     0.891     0.731;
      0.903     0.885     0.823     0.694     0.467];
y = [ 0.000    -0.040    -0.159    -0.352    -0.597;
     -0.111    -0.151    -0.268    -0.454    -0.682
     -0.430    -0.466    -0.568    -0.720    -0.884];
nd = int64(size(x));
l  = int64(2);

% transform, then inverse transform to restore data
[xt, yt, ifail] = c06ff(l, nd, x, y);
[xr, yr, ifail] = c06ff(l, nd, xt, -yt);

% Display as complex matrices
z = x + i*y;
zt = reshape(xt+i*yt,nd);
zr = reshape(xr-i*yr,nd);

matrix = 'general';
diag = ' ';
usefrm = 'Above';
format = 'F9.3';
labrow = 'None';
labcol = 'None';
ncols  = int64(80);
indent = int64(0);

title  = 'Original data:';
[ifail] = x04db(...
    matrix, diag, z, usefrm, format, title, labrow, labcol, ncols, indent);
disp(' ');
title = 'Discrete Fourier transform of variable 2:';
[ifail] = x04db(...
    matrix, diag, zt, usefrm, format, title, labrow, labcol, ncols, indent);
disp(' ');
title = 'Original sequence as restored by inverse transform:';
[ifail] = x04db(...
    matrix, diag, zr, usefrm, format, title, labrow, labcol, ncols, indent);

c06ff example results

 Original data:
      1.000    0.999    0.987    0.936    0.802
      0.000   -0.040   -0.159   -0.352   -0.597

      0.994    0.989    0.963    0.891    0.731
     -0.111   -0.151   -0.268   -0.454   -0.682

      0.903    0.885    0.823    0.694    0.467
     -0.430   -0.466   -0.568   -0.720   -0.884
 
 Discrete Fourier transform of variable 2:
      2.113    0.288    0.126   -0.003   -0.287
     -0.513   -0.000    0.130    0.190    0.194

      2.043    0.286    0.139    0.018   -0.263
     -0.745   -0.032    0.115    0.189    0.225

      1.687    0.260    0.170    0.079   -0.176
     -1.372   -0.125    0.063    0.173    0.299
 
 Original sequence as restored by inverse transform:
      1.000    0.999    0.987    0.936    0.802
      0.000   -0.040   -0.159   -0.352   -0.597

      0.994    0.989    0.963    0.891    0.731
     -0.111   -0.151   -0.268   -0.454   -0.682

      0.903    0.885    0.823    0.694    0.467
     -0.430   -0.466   -0.568   -0.720   -0.884

NAG Toolbox: nag_sum_fft_complex_multid_1d_sep (c06ff)

▸▿ Contents

Purpose

Syntax

Description