freqdom {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)

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).

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.

See Also

fourier.transform

Examples

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)