if $job = Nag_Convolution$ , the discrete convolution of $x$ and $y$ , defined by
$z_{k} = \sum_{j = 0}^{n - 1} x_{j} y_{k - j} = \sum_{j = 0}^{n - 1} x_{k - j} y_{j};$
if $job = Nag_Correlation$ , the discrete correlation of $x$ and $y$ defined by
$w_{k} = \sum_{j = 0}^{n - 1} x_{j} y_{k + j} .$

Here

x

and

y

are real vectors, assumed to be periodic, with period

n

, i.e.,

x_{j} = x_{j \pm n} = x_{j \pm 2 n} = \dots

;

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.

\hat{x}

\hat{y}

\hat{z}

and

\hat{w}

are the discrete Fourier transforms of these sequences, i.e.,

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

then

{\hat{z}}_{k} = \sqrt{n} . {\hat{x}}_{k} {\hat{y}}_{k}

and

{\hat{w}}_{k} = \sqrt{n} . {\bar{\hat{x}}}_{k} {\hat{y}}_{k}

(the bar denoting complex conjugate).

This function calls the same auxiliary functions as nag_sum_fft_realherm_1d (c06pac) to compute discrete Fourier transforms.

4

References

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

5

Arguments

1: $job$ – Nag_VectorOpInput

On entry: the computation to be performed.

$job = Nag_Convolution$: $z_{k} = \sum_{j = 0}^{n - 1} x_{j} y_{k - j}$ ;
$job = Nag_Correlation$: $w_{k} = \sum_{j = 0}^{n - 1} x_{j} y_{k + j}$ .

Constraint:

job = Nag_Convolution

Nag_Correlation

2: $x [n]$ – doubleInput/Output

On entry: the elements of one period of the vector

x

x [j]

must contain

x_{j}

, for

j = 0, 1, \dots, n - 1

On exit: the corresponding elements of the discrete convolution or correlation.

3: $y [n]$ – doubleInput/Output

On entry: the elements of one period of the vector

y

y [j]

must contain

y_{j}

, for

j = 0, 1, \dots, 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; if the components of the transform

z_{k}

are written as

a_{k} + i b_{k}

, then for

0 \leq k \leq n / 2

a_{k}

is contained in

y [k]

, and for

1 \leq k \leq n / 2 - 1

b_{k}

is contained in

y [n - k]

. (See also Section 2.1.2 in the c06 Chapter Introduction.)

4: $n$ – IntegerInput

On entry:

n

, the number of values in one period of the vectors x and y.

Constraint:

n > 1

5: $fail$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.

$〈value〉$ is an invalid value of job.
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

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

8

Parallelism and Performance

nag_sum_convcorr_real (c06fkc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

nag_sum_convcorr_real (c06fkc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9

Further Comments

The time taken is approximately proportional to

n \times \log (n)

, but also depends on the factorization of

n

. nag_sum_convcorr_real (c06fkc) is faster if the only prime factors of

n

are

2

3

5

; and fastest of all if

n

is a power of

2

10

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_sum_convcorr_real (c06fkc)). In realistic computations the number of data values would be much larger.

NAG Library Function Document

nag_sum_convcorr_real (c06fkc)

▸▿ 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

2

Specification

3

Description