C06 Chapter Contents
C06 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentC06PKF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

C06PKF calculates the circular convolution or correlation of two complex vectors of period $n$.

## 2  Specification

 SUBROUTINE C06PKF ( JOB, X, Y, N, WORK, IFAIL)
 INTEGER JOB, N, IFAIL COMPLEX (KIND=nag_wp) X(N), Y(N), WORK(*)

## 3  Description

C06PKF computes:
• if ${\mathbf{JOB}}=1$, the discrete convolution of $x$ and $y$, defined by
 $zk = ∑ j=0 n-1 xj y k-j = ∑ j=0 n-1 x k-j yj ;$
• if ${\mathbf{JOB}}=2$, the discrete correlation of $x$ and $y$ defined by
 $wk = ∑ j=0 n-1 x-j y k+j .$
Here $x$ and $y$ are complex vectors, assumed to be periodic, with period $n$, i.e., ${x}_{j}={x}_{j±n}={x}_{j±2n}=\dots \text{}$; $z$ 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 $\stackrel{^}{x}$, $\stackrel{^}{y}$, $\stackrel{^}{z}$ and $\stackrel{^}{w}$ are the discrete Fourier transforms of these sequences, and $\stackrel{~}{x}$ is the inverse discrete Fourier transform of the sequence ${x}_{j}$, i.e.,
 $x^k = 1n ∑ j=0 n-1 xj × exp -i 2πjk n , etc.,$
and
 $x~k = 1n ∑ j= 0 n- 1 xj × exp i 2πjk n ,$
then ${\stackrel{^}{z}}_{k}=\sqrt{n}.{\stackrel{^}{x}}_{k}{\stackrel{^}{y}}_{k}$ and ${\stackrel{^}{w}}_{k}=\sqrt{n}.{\stackrel{-}{\stackrel{^}{x}}}_{k}{\stackrel{^}{y}}_{k}$ (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:
${\mathbf{JOB}}=1$
${z}_{k}=\sum _{j=0}^{n-1}{x}_{j}{y}_{k-j}$ (convolution);
${\mathbf{JOB}}=2$
${w}_{k}=\sum _{j=0}^{n-1}{\stackrel{-}{x}}_{j}{y}_{k+j}$ (correlation).
Constraint: ${\mathbf{JOB}}=1$ or $2$.
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 $\left(0:{\mathbf{N}}-1\right)$ in the subroutine from which C06PKF is called, then ${\mathbf{X}}\left(\mathit{j}\right)$ must contain ${x}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$.
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 $\left(0:{\mathbf{N}}-1\right)$ in the subroutine from which C06PKF is called, then ${\mathbf{Y}}\left(\mathit{j}\right)$ must contain ${y}_{\mathit{j}}$, for $\mathit{j}=0,1,\dots ,n-1$.
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: ${\mathbf{N}}\ge 1$.
5:     WORK($*$) – COMPLEX (KIND=nag_wp) arrayWorkspace
Note: the dimension of the array WORK must be at least $2×{\mathbf{N}}+15$.
The workspace requirements as documented for C06PKF may be an overestimate in some implementations.
On exit: the real part of ${\mathbf{WORK}}\left(1\right)$ 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 $0$, $-1\text{​ or ​}1$. 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 $-1\text{​ or ​}1$ 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 $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit: ${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6  Error Indicators and Warnings

If on entry ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
${\mathbf{IFAIL}}=1$
 On entry, ${\mathbf{N}}<1$.
${\mathbf{IFAIL}}=2$
 On entry, ${\mathbf{JOB}}\ne 1$ or $2$.
${\mathbf{IFAIL}}=3$
An unexpected error has occurred in an internal call. Check all subroutine calls and array dimensions. Seek expert help.
${\mathbf{IFAIL}}=4$
 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.

The time taken is approximately proportional to $n×\mathrm{log}n$, but also depends on the factorization of $n$. C06PKF is faster if the only prime factors of $n$ are $2$, $3$ or $5$; 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.

### 9.1  Program Text

Program Text (c06pkfe.f90)

### 9.2  Program Data

Program Data (c06pkfe.d)

### 9.3  Program Results

Program Results (c06pkfe.r)