freqdom (version 2.0.2)

fourier.inverse: 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)

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.

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.

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)