freqdom {freqdom} | R Documentation |
Creates an object of class freqdom
. This object corresponds to a functional with domain [-\pi,\pi]
and some complex vector space as codomain.
freqdom(F, freq)
F |
a vector, a matrix or an array. For vectors |
freq |
a vector of dimension |
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)
.
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.
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)