NAG Library Routine Document

C06PKF

if $JOB = 1$ , 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 $JOB = 2$ , the discrete correlation of $x$ and $y$ defined by
$w_{k} = \sum_{j = 0}^{n - 1} {\bar{x}}_{j} y_{k + j} .$

Here

x

and

y

are complex 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, and

\tilde{x}

is the inverse discrete Fourier transform of the sequence

x_{j}

, 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.,

and

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

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).

4 References

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

5 Parameters

1: JOB – INTEGERInput

On entry: the computation to be performed:

$JOB = 1$: $z_{k} = \sum_{j = 0}^{n - 1} x_{j} y_{k - j}$ (convolution);
$JOB = 2$: $w_{k} = \sum_{j = 0}^{n - 1} {\bar{x}}_{j} y_{k + j}$ (correlation).

Constraint:

JOB = 1

2

2: X(N) – COMPLEX (KIND=nag_wp) arrayInput/Output

On entry: the elements of one period of the vector

x

. If X is declared with bounds

(0 : N - 1)

in the subroutine from which C06PKF is called, then

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) – COMPLEX (KIND=nag_wp) arrayInput/Output

On entry: the elements of one period of the vector

y

. If Y is declared with bounds

(0 : N - 1)

in the subroutine from which C06PKF is called, then

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.

4: N – INTEGERInput

On entry:

n

, the number of values in one period of the vectors X and Y. The total number of prime factors of N, counting repetitions, must not exceed

30

Constraint:

N \geq 1

5: WORK( $*$ ) – COMPLEX (KIND=nag_wp) arrayWorkspace

Note: the dimension of the array WORK must be at least

2 \times N + 15

The workspace requirements as documented for C06PKF may be an overestimate in some implementations.

On exit: the real part of

WORK (1)

contains the minimum workspace required for the current value of N with this implementation.

6: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,

N < 1

$IFAIL = 2$

On entry,

JOB \neq 1

2

$IFAIL = 3$: An unexpected error has occurred in an internal call. Check all subroutine calls and array dimensions. Seek expert help.

$IFAIL = 4$

On entry,

N has more than

30

prime factors.

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

n \times \log n

, but also depends on the factorization of

n

. C06PKF is faster if the only prime factors of

n

are

2

3

5

; and fastest of all if

n

is a power of

2

9 Example

This example reads in the elements of one period of two complex vectors

x

and

y

, and prints their discrete convolution and correlation (as computed by C06PKF). In realistic computations the number of data values would be much larger.

NAG Library Routine DocumentC06PKF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

C06PKF