fourier.inverse {freqdom}

Coefficients of a discrete Fourier transform

Description

Computes Fourier coefficients of some functional represented by an object of class freqdom.

Usage

fourier.inverse(F, lags = 0)

Arguments

F

an object of class freqdom which is corresponding to a function with values in \mathbf{C}^{d_1\times d_2}. To guarantee accuracy of inversion it is important that F\$freq is a dense grid of frequencies in [-\pi,\pi].

lags

lags of the Fourier coefficients to be computed.

Details

Consider a function F \colon [-\pi,\pi]\to\mathbf{C}^{d_1\times d_2}. Its k-th Fourier coefficient is given as

\frac{1}{2\pi}\int_{-\pi}^\pi F(\omega) \exp(ik\omega)d\omega.

We represent the function F by an object of class freqdom and approximate the integral via

\frac{1}{|F\$freq|}\sum_{\omega\in {F\$freq}} F(\omega) \exp(i k\omega),

for k\in lags.

Value

An object of class timedom. The list has the following components:

• operators \quad an array. The k-th matrix in this array corresponds to the k-th Fourier coefficient.

• lags \quad the lags of the corresponding Fourier coefficients.

See Also

fourier.transform, freqdom

Examples

Y = rar(100)
grid = c(pi*(1:2000) / 1000 - pi) #a dense grid on -pi, pi
fourier.inverse(spectral.density(Y, q=2, freq=grid))

# compare this to
cov.structure(Y)