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

Arguments

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,

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,

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 argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 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

c06pkf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

c06pkf 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 routine. 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

. 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

10

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 Document

c06pkf (convcorr_complex)

▸▿ 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

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