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

In freqdom: Frequency Domain Based Analysis: Dynamic PCA

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)

Arguments

F	frequency-domain filter of type freqdom, i.e. a set of linear operators F_θ \in \mathbf{R}^{p \times q} defined on a discreet grid defined S \subset [-π,π].
G	frequency-domain filter of type freqdom, i.e. a set of linear operators G_θ \in \mathbf{R}^{q \times r} defined on a discreet grid defined S \subset [-π,π].

Details

Let F = \{ F_θ : θ \in S \}, G = \{ G_θ : θ \in S \}, where S is a finite grid of frequencies in [-π,π], F_θ \in \mathbf{C}^{p \times q} and G_θ \in \mathbf{C}^{q \times r}.

We define

H_θ = F_θ G_θ

as a matrix product of F_θ and G_θ, i.e. H_θ \in \mathbf{R}^{p\times r}. Function freqdom.product returns H = \{ H_θ : θ \in S \}.

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_θ * G_θ defined above.

Functions

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

Examples

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

freqdom documentation built on Oct. 4, 2022, 5:05 p.m.