naginterfaces.library.sum.convcorr_​complex

naginterfaces.library.sum.convcorr_complex(job, x, y)

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

For full information please refer to the NAG Library document for c06pk

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/c06/c06pkf.html

Parameters

jobint

The computation to be performed:

$job=1$

$z_k={\sum }_{j=0}^{n-1}{x}_j{y}_{k-j}$ (convolution);

$job=2$

$w_k={\sum }_{j=0}^{n-1}{\overline{x}}_j{y}_{k+j}$ (correlation).

xcomplex, array-like, shape $\left(n\right)$

The elements of one period of the vector $x$. If $x$ is declared with bounds $(0:n-1)$ in the function from which convcorr_complex is called, $x\left[\text{j}\right]$ must contain $x_{\text{j}}$, for $\text{j}=0,1,\dots ,n-1$.

ycomplex, array-like, shape $\left(n\right)$

The elements of one period of the vector $y$. If $y$ is declared with bounds $(0:n-1)$ in the function from which convcorr_complex is called, $y[\text{j}-1]$ must contain $y_{\text{j}}$, for $\text{j}=0,1,\dots ,n-1$.

Returns

xcomplex, ndarray, shape $\left(n\right)$

The corresponding elements of the discrete convolution or correlation.

ycomplex, ndarray, shape $\left(n\right)$

The discrete Fourier transform of the convolution or correlation returned in the array $x$.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 1$.

(errno $2$)

On entry, argument $job$ had an illegal value.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

convcorr_complex computes:

if $job=1$, the discrete convolution of $x$ and $y$, defined by

$$z_k=\sum _{j=0}^{n-1}{x}_j{y}_{k-j}=\sum _{j=0}^{n-1}{x}_{k-j}{y}_j\text{;}$$

if $job=2$, the discrete correlation of $x$ and $y$ defined by

$$w_k=\sum _{j=0}^{n-1}{\overline{x}}_j{y}_{k+j}\text{.}$$

Here $x$ and $y$ are complex vectors, assumed to be periodic, with period $n$, i.e., $x_j=x_{j\pm n}=x_{j\pm 2n}=\cdots$; $z$ 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 $\hat{x}$, $\hat{y}$, $\hat{z}$ 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.,

$${\hat{x}}_k=\frac{1}{\sqrt{n}}\sum _{j=0}^{n-1}{x}_j\times exp\left(-i\frac{2\pi jk}{n}\right)\text{, etc.,}$$

and

$${\tilde{x}}_k=\frac{1}{\sqrt{n}}\sum _{j=0}^{n-1}{x}_j\times exp\left(i\frac{2\pi jk}{n}\right)\text{,}$$

then ${\hat{z}}_k=\sqrt{n}.{\hat{x}}_k{\hat{y}}_k$ and ${\hat{w}}_k=\sqrt{n}.{\overline{\hat{x}}}_k{\hat{y}}_k$ (the bar denoting complex conjugate).

References

Brigham, E O, 1974, The Fast Fourier Transform, Prentice–Hall