NAG Library Function Document

nag_fft_hermitian (c06ebc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_fft_hermitian (c06ebc) calculates the discrete Fourier transform of a Hermitian sequence of

n

complex data values. (No extra workspace required.)

2 Specification

#include <nag.h>

#include <nagc06.h>

void	nag_fft_hermitian (double x[], Integer n, NagError *fail)

3 Description

Given a Hermitian sequence of

n

complex data values

z_{j}

(i.e., a sequence such that

z_{0}

is real and

z_{n - j}

is the complex conjugate of

z_{j}

, for

j = 1, 2, \dots, n - 1

), nag_fft_hermitian (c06ebc) calculates their discrete Fourier transform defined by

{\hat{x}}_{k} = \frac{1}{\sqrt{n}} \sum_{j = 0}^{n - 1} z_{j} \times \exp (- i \frac{2 π j k}{n}), k = 0, 1, \dots, n - 1 .

(Note the scale factor of

\frac{1}{\sqrt{n}}

in this definition.) The transformed values

{\hat{x}}_{k}

are purely real (see also the c06 Chapter Introduction).

To compute the inverse discrete Fourier transform defined by

{\hat{y}}_{k} = \frac{1}{\sqrt{n}} \sum_{j = 0}^{n - 1} z_{j} \times \exp (+ i \frac{2 π j k}{n}),

this function should be preceded by a call of nag_conjugate_hermitian (c06gbc) to form the complex conjugates of the

z_{j}

nag_fft_hermitian (c06ebc) 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: $x [n]$ – doubleInput/Output: On entry: the sequence to be transformed stored in Hermitian form. If the data values $z_{j}$ are written as $x_{j} + i y_{j}$ , and if x is declared with bounds $(0 : n - 1)$ in the function from which nag_fft_hermitian (c06ebc) is called, then for $0 \leq j \leq n / 2$ , $x_{j}$ is contained in $x [j - 1]$ , and for $1 \leq j \leq (n - 1) / 2$ , $y_{j}$ is contained in $x [n - j]$ . (See also Section 2.1.2 in the c06 Chapter Introduction and Section 10.)

On exit: the components of the discrete Fourier transform ${\hat{x}}_{k}$ . If x is declared with bounds $(0 : n - 1)$ in the function from which nag_fft_hermitian (c06ebc) is called, then ${\hat{x}}_{k}$ is stored in $x [k]$ , for $k = 0, 1, \dots, n - 1$ .
2: $n$ – IntegerInput: On entry: $n$ , 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: $n > 1$ .
3: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.2.1.2 in the Essential Introduction for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT: On entry, $n = 〈value〉$ .
Constraint: $n > 1$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

An unexpected error has been triggered by this function. Please contact NAG.
See Section 3.6.6 in the Essential Introduction for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 3.6.5 in the Essential Introduction for further information.
NE_PRIME_FACTOR: At least one of the prime factors of n is greater than $19$ . $n = 〈value〉$ .
NE_TOO_MANY_FACTORS: n has more than $20$ prime factors. $n = 〈value〉$ .

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 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken is approximately proportional to

n \times \log (n)

, but also depends on the factorization of

n

. nag_fft_hermitian (c06ebc) is faster if the only prime factors of

n

are

2

3

5

; and fastest of all if

n

is a power of

2

On the other hand, nag_fft_hermitian (c06ebc) is particularly slow if

n

has several unpaired prime factors, i.e., if the ‘square-free’ part of

n

has several factors.

10 Example

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_fft_hermitian (c06ebc)) is printed out. It then performs an inverse transform using nag_fft_real (c06eac) and nag_conjugate_hermitian (c06gbc), and prints the sequence so obtained alongside the original data values.

NAG Library Function Documentnag_fft_hermitian (c06ebc)