nag_convolution_real (c06ekc) (PDF version)
c06 Chapter Contents
c06 Chapter Introduction
NAG C Library Manual

NAG Library Function Document

nag_convolution_real (c06ekc)

+ Contents

    1  Purpose
    7  Accuracy

1  Purpose

nag_convolution_real (c06ekc) calculates the circular convolution or correlation of two real vectors of period n.

2  Specification

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

3  Description

nag_convolution_real (c06ekc) computes:
Here x and y are real vectors, assumed to be periodic, with period n, i.e., xj = xj±n = xj±2n = ; z and w are then also periodic with period n.
Note:  this usage of the terms ‘convolution’ and ‘correlation’ is taken from Brigham (1974). The term ‘convolution’ is sometimes used to denote both these computations.
If x^ , y^ , z^  and w^  are the discrete Fourier transforms of these sequences, i.e.,
x^k = 1n j=0 n-1 xj exp -i 2πjk n , etc.,
then z^k = n x^k y^k  and w^k = n x^- k y^k  (the bar denoting complex conjugate).
This function calls the same auxiliary functions as nag_fft_real (c06eac) and nag_fft_hermitian (c06ebc) to compute discrete Fourier transforms, and there are some restrictions on the value of n.

4  References

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

5  Arguments

1:     operationNag_VectorOpInput
On entry: the computation to be performed.
z k = j=0 n-1 x j y k-j .
w k = j=0 n-1 x j y k+j .
Constraint: operation=Nag_Convolution or Nag_Correlation.
2:     nIntegerInput
On entry: n, the number of values, in one period of the vectors x and y.
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.
3:     x[n]doubleInput/Output
On entry: the elements of one period of the vector x. x[j]  must contain x j , for j=0,1,,n - 1.
On exit: the corresponding elements of the discrete convolution or correlation.
4:     y[n]doubleInput/Output
On entry: the elements of one period of the vector y. y[j]  must contain y j , for j=0,1,,n - 1.
On exit: the discrete Fourier transform of the convolution or correlation returned in the array x; the transform is stored in Hermitian form, exactly as described in the document nag_fft_real (c06eac).
5:     failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

On entry, argument operation had an illegal value.
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

The results should be accurate to within a small multiple of the machine precision.

8  Further Comments

The time taken is approximately proportional to nlogn , but also depends on the factorization of n. nag_convolution_real (c06ekc) 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_convolution_real (c06ekc) 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 the elements of one period of two real vectors x and y and prints their discrete convolution and correlation (as computed by nag_convolution_real (c06ekc)). In realistic computations the number of data values would be much larger.

9.1  Program Text

Program Text (c06ekce.c)

9.2  Program Data

Program Data (c06ekce.d)

9.3  Program Results

Program Results (c06ekce.r)

nag_convolution_real (c06ekc) (PDF version)
c06 Chapter Contents
c06 Chapter Introduction
NAG C Library Manual

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