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NAG Toolbox: nag_sum_fft_hermitian_1d_rfmt (c06fb)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_sum_fft_hermitian_1d_rfmt (c06fb) calculates the discrete Fourier transform of a Hermitian sequence of n complex data values (using a work array for extra speed).


[x, ifail] = c06fb(x, 'n', n)
[x, ifail] = nag_sum_fft_hermitian_1d_rfmt(x, 'n', n)


Given a Hermitian sequence of n complex data values zj  (i.e., a sequence such that z0  is real and z n-j  is the complex conjugate of zj , for j=1,2,,n-1), nag_sum_fft_hermitian_1d_rfmt (c06fb) calculates their discrete Fourier transform defined by
x^k = 1n j=0 n-1 zj × exp -i 2πjk n ,   k= 0, 1, , n-1 .  
(Note the scale factor of 1n  in this definition.) The transformed values x^k  are purely real (see also the C06 Chapter Introduction).
To compute the inverse discrete Fourier transform defined by
y^k = 1n j=0 n-1 zj × exp +i 2πjk n ,  
this function should be preceded by forming the complex conjugates of the z^k ; that is, xk=-xk, for k=n/2+2,,n.
nag_sum_fft_hermitian_1d_rfmt (c06fb) 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
The sequence to be transformed stored in Hermitian form. If the data values zj are written as xj + i yj, and if x is declared with bounds 0:n-1 in the function from which nag_sum_fft_hermitian_1d_rfmt (c06fb) is called, then for 0 j n/2, xj is contained in xj, and for 1 j n-1 / 2 , yj is contained in xn-j. (See also Real transforms in the C06 Chapter Introduction and Example.)

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 components of the discrete Fourier transform x^k. If x is declared with bounds 0:n-1 in the function from which nag_sum_fft_hermitian_1d_rfmt (c06fb) is called, then x^k is stored in xk, for k=0,1,,n-1.
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_hermitian_1d_rfmt (c06fb) 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.


This example reads in a sequence of real data values which is assumed to be a Hermitian sequence of complex data values stored in Hermitian form. The input sequence is expanded into a full complex sequence and printed alongside the original sequence. The discrete Fourier transform (as computed by nag_sum_fft_hermitian_1d_rfmt (c06fb)) is printed out. It then performs an inverse transform using nag_sum_fft_real_1d_rfmt (c06fa) and conjugation, and prints the sequence so obtained alongside the original data values.
function c06fb_example

fprintf('c06fb example results\n\n');

% Hermitian sequence x, stored in Hermitian form.
n = 7;
x = [0.34907;  0.5489;   0.74776;   0.94459;
     1.1385;   1.3285;   1.5137];

% DFT of x
[xtrans, ifail] = c06fa(x);

% Display in full complex form
z = nag_herm2complex(xtrans);
disp('Discrete Fourier Transform of x:');

% Inverse DFT of xtrans
[xres] = nag_hermconj(xtrans);
[xres, ifail] = c06fb(xres);

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),xres(j));

function [z] = nag_herm2complex(x);
  n = int64(size(x,1));
  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));

function [xconj] = nag_hermconj(x);
  n = size(x,1);
  n2 = floor((n+4)/2);
  xconj = x;
  for j = n2:n
    xconj(j) = -x(j);
c06fb 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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