freqdom: Create an object corresponding to a frequency domain functional

Description

Creates an object of class freqdom. This object corresponds to a functional with domain $[-\pi,\pi]$ and some complex vector space as codomain.

Usage

freqdom(F, freq)

Value

Returns an object of class freqdom. An object of class freqdom is a list containing the following components:

operators $\quad$ the array F as given in the argument.
freq $\quad$ the vector freq as given in the argument.

Arguments

F: a vector, a matrix or an array. For vectors $F[k], 1\leq k\leq K$ are complex numbers. For matrices $F[k,]$ are complex vectors. For arrays the elements $F[,,k]$, are complex valued $(d_1\times d_2)$ matrices (all of same dimension).
freq: a vector of dimension $K$ containing frequencies in $[-\pi,\pi]$.

Details

This class is used to describe a frequency domain functional (like a spectral density matrix, a discrete Fourier transform, an impulse response function, etc.) on selected frequencies. Formally we consider a collection $[F_1,\ldots,F_K]$ of complex-valued matrices $F_k$, all of which have the same dimension $d_1\times d_2$. Moreover, we consider frequencies $\{\omega_1,\ldots, \omega_K\}\subset[-\pi,\pi]$. The object this function creates corresponds to the mapping $f: \mathrm{freq}\to \mathbf{C}^{d_1\times d_2}$, where $\omega_k\mapsto F_k$.

Consider, for example, the discrete Fourier transform of a vector time series $X_1,\ldots, X_T$:. It is defined as $$ D_T(\omega)=\frac{1}{\sqrt{T}}\sum_{t=1}^T X_t e^{-it\omega},\quad \omega\in[-\pi,\pi]. $$ We may choose $\omega_k=2\pi k/K-\pi$ and $F_k=D_T(\omega_k)$. Then, the object freqdom creates, is corresponding to the function which associates $\omega_k$ and $D_T(\omega_k)$.

Examples

Run this code

i = complex(imaginary=1)
OP = array(0, c(2, 2, 3))
OP[,,1] = diag(2) * exp(i)/2
OP[,,2] = diag(2)
OP[,,3] = diag(2) * exp(-i)/2
freq = c(-pi/3, 0, pi/3)
A = freqdom(OP, freq)