nag_fft_real (c06eac) (PDF version)
c06 Chapter Contents
c06 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_fft_real (c06eac)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_fft_real (c06eac) calculates the discrete Fourier transform of a sequence of n real data values.

2  Specification

#include <nag.h>
#include <nagc06.h>
void  nag_fft_real (Integer n, double x[], NagError *fail)

3  Description

Given a sequence of n real data values xj , for j=0,1,,n-1, nag_fft_real (c06eac) 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.
The function nag_multiple_hermitian_to_complex (c06gsc) may be used to convert a Hermitian sequence to the corresponding complex sequence.
To compute the inverse discrete Fourier transform defined by
w^k = 1n j=0 n-1 xj exp +i 2πjk n ,   for ​ k= 0, 1, , n-1 ,
this function should be followed by a call of nag_conjugate_hermitian (c06gbc) to form the complex conjugates of the z^k .
nag_fft_real (c06eac) uses the fast Fourier transform (FFT) algorithm (see Brigham (1974)). There are some restrictions on the value of n (see Section 5).

4  References

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

5  Arguments

1:     nIntegerInput
On entry: n, the number of data values.
Constraint: n>1 . 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.
2:     x[n]doubleInput/Output
On entry: x[j]  must contain x j , for j=0,1,,n-1.
On exit: the discrete Fourier transform stored in Hermitian form. If the components of the transform z^ k  are written as a k + ib k , then for 0 k n/2 , a k  is contained in x[k] , and for 1 k n-1/2 , b k  is contained in x[n-k] . Elements of the sequence which are not explicitly stored are given by a n-k = a k , b n-k = - b k , b o = 0  and, if n is even, b n/2 = 0 . (See also Section 9.)
3:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

At least one of the prime factors of n is greater than 19.
n has more than 20 prime factors.
On entry, n=value.
Constraint: n>1.

7  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).

8  Further Comments

The time taken is approximately proportional to nlogn , but also depends on the factorization of n. nag_fft_real (c06eac) is somewhat faster than average 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_fft_real (c06eac) is particularly slow if n has several unpaired prime factors, i.e., if the ‘square-free’ part of n has several factors.

9  Example

This example reads in a sequence of real data values and prints their discrete Fourier transform (as computed by nag_fft_real (c06eac)), after expanding it from Hermitian form into a full complex sequence. It then performs an inverse transform using nag_fft_hermitian (c06ebc) and nag_conjugate_hermitian (c06gbc), and prints the sequence so obtained alongside the original data values.

9.1  Program Text

Program Text (c06eace.c)

9.2  Program Data

Program Data (c06eace.d)

9.3  Program Results

Program Results (c06eace.r)

nag_fft_real (c06eac) (PDF version)
c06 Chapter Contents
c06 Chapter Introduction
NAG C Library Manual

© The Numerical Algorithms Group Ltd, Oxford, UK. 2012