freqdom (version 2.0.5)

freqdom.product: Compute a matrix product of two frequency-domain operators

Description

For given frequency-domain operators F and G (freqdom) the function freqdom.kronecker computes their matrix product frequency-wise.

Usage

freqdom.product(F, G)

Value

Function returns a frequency domain object (freqdom) of dimensions $L\times p\times r$, where $L$ is the size of the evaluation grid. The elements correspond to $F_{\theta }\ast G_{\theta }$ defined above.

Arguments

F

frequency-domain filter of type freqdom, i.e. a set of linear operators $F_{\theta }\in {\mathbf{R}}^{p\times q}$ defined on a discreet grid defined $S\subset [-\pi ,\pi ]$.

G

frequency-domain filter of type freqdom, i.e. a set of linear operators $G_{\theta }\in {\mathbf{R}}^{q\times r}$ defined on a discreet grid defined $S\subset [-\pi ,\pi ]$.

Functions

• freqdom.product(): Frequency-wise matrix product of two frequency-domain operators

Details

Let $F=\{F_{\theta }:\theta \in S\}$, $G=\{G_{\theta }:\theta \in S\}$, where $S$ is a finite grid of frequencies in $[-\pi ,\pi ]$, $F_{\theta }\in {\mathbf{C}}^{p\times q}$ and $G_{\theta }\in {\mathbf{C}}^{q\times r}$.

We define $$H_{\theta }=F_{\theta }{G}_{\theta }$$ as a matrix product of $F_{\theta }$ and $G_{\theta }$, i.e. $H_{\theta }\in {\mathbf{R}}^{p\times r}$. Function freqdom.product returns $H=\{H_{\theta }:\theta \in S\}$.

Examples

n = 100
X = rar(n)
Y = rar(n)
SX = spectral.density(X)
SY = spectral.density(Y)
R = freqdom.product(SY,SX)