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NAG Toolbox: nag_sum_withdraw_fft_real_1d_nowork (c06ea)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_sum_fft_real_1d_nowork (c06ea) calculates the discrete Fourier transform of a sequence of n real data values. (No extra workspace required.)
Note: this function is scheduled to be withdrawn, please see c06ea in Advice on Replacement Calls for Withdrawn/Superseded Routines..


[x, ifail] = c06ea(x, 'n', n)
[x, ifail] = nag_sum_withdraw_fft_real_1d_nowork(x, 'n', n)


Given a sequence of n real data values xj , for j=0,1,,n-1, nag_sum_fft_real_1d_nowork (c06ea) calculates their discrete Fourier transform defined by
z^k = 1n j=0 n-1 xj × exp -i 2πjk n ,   k= 0, 1, , n-1 .  
(Note the scale factor of 1n  in this definition.) The transformed values z^k  are complex, but they form a Hermitian sequence (i.e., z^ n-k  is the complex conjugate of z^k ), so they are completely determined by n real numbers (see also the C06 Chapter Introduction).
To compute the inverse discrete Fourier transform defined by
w^k = 1n j=0 n-1 xj × exp +i 2πjk n ,  
this function should be followed by a call of nag_sum_conjugate_hermitian_rfmt (c06gb) to form the complex conjugates of the z^k .
nag_sum_fft_real_1d_nowork (c06ea) uses the fast Fourier transform (FFT) algorithm (see Brigham (1974)). There are some restrictions on the value of n (see Arguments).


Brigham E O (1974) The Fast Fourier Transform Prentice–Hall


Compulsory Input Parameters

1:     xn – double array
xj+1 must contain xj, for j=0,1,,n-1.

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the dimension of the array x.
n, the number of data values. The largest prime factor of n must not exceed 19, and the total number of prime factors of n, counting repetitions, must not exceed 20.
Constraint: n>1.

Output Parameters

1:     xn – double array
The discrete Fourier transform stored in Hermitian form. If the components of the transform z^k are written as ak + i bk, and if x is declared with bounds 0:n-1 in the function from which nag_sum_fft_real_1d_nowork (c06ea) is called, then for 0 k n/2, ak is contained in xk, and for 1 k n-1 / 2 , bk is contained in xn-k. (See also Real transforms in the C06 Chapter Introduction and Example.)
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:
At least one of the prime factors of n is greater than 19.
n has more than 20 prime factors.
On entry,n1.
An unexpected error has occurred in an internal call. Check all function calls and array dimensions. Seek expert help.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


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 × logn, but also depends on the factorization of n. nag_sum_fft_real_1d_nowork (c06ea) 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.
On the other hand, nag_sum_fft_real_1d_nowork (c06ea) is particularly slow if n has several unpaired prime factors, i.e., if the ‘square-free’ part of n has several factors. For such values of n, nag_sum_fft_real_1d_rfmt (c06fa) (which requires additional double workspace) is considerably faster.


This example reads in a sequence of real data values and prints their discrete Fourier transform (as computed by nag_sum_fft_real_1d_nowork (c06ea)), after expanding it from Hermitian form into a full complex sequence. It then performs an inverse transform using nag_sum_conjugate_hermitian_rfmt (c06gb) followed by nag_sum_fft_hermitian_1d_nowork (c06eb), and prints the sequence so obtained alongside the original data values.
function c06ea_example

fprintf('c06ea example results\n\n');

% real data
n = 7;
x = [0.34907  0.54890  0.74776  0.94459  1.13850  1.32850  1.51370];

% transform
[xt, ifail] = c06ea(x);

% get result in form useful for printing.
zt = nag_herm2complex(xt);
disp('Discrete Fourier Transform of x:');

% restore by conjugating and backtransforming
xt(floor(n/2)+2:n) = -xt(floor(n/2)+2:n);
[xr, ifail] = c06eb(xt);

fprintf('Original sequence as restored by inverse transform\n\n');
fprintf('       Original   Restored\n');
for j = 1:n
  fprintf('%3d   %7.4f    %7.4f\n',j, x(j),xr(j));

function [z] = nag_herm2complex(x);
  n = size(x,2);
  z(1) = complex(x(1));
  for j = 2:floor((n-1)/2) + 1
    z(j) = x(j) + i*x(n-j+2);
    z(n-j+2) = x(j) - i*x(n-j+2);
  if (mod(n,2)==0)
    z(n/2+1) = complex(x(n/2+1));
c06ea example results

Discrete Fourier Transform of x:
   2.4836 + 0.0000i
  -0.2660 + 0.5309i
  -0.2577 + 0.2030i
  -0.2564 + 0.0581i
  -0.2564 - 0.0581i
  -0.2577 - 0.2030i
  -0.2660 - 0.5309i

Original sequence as restored by inverse transform

       Original   Restored
  1    0.3491     0.3491
  2    0.5489     0.5489
  3    0.7478     0.7478
  4    0.9446     0.9446
  5    1.1385     1.1385
  6    1.3285     1.3285
  7    1.5137     1.5137

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Chapter Contents
Chapter Introduction
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