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 $F_\theta * G_\theta$ 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

Run this code

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

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