# Fourier Transform of a Covariance Function for Spectral Simulation

I am learning about generating Gaussian random fields by spectral simulation...

If I have a covariance function $C(h)$, then the spectral density is the Fourier transform of $C(h)$:

$S(\omega)=(1/N)\sum_{h=0}^{N-1}{C(h)[cos(2\pi\omega h/N)- i \sin(2\pi\omega h/N)]} \space\space\space\space\space [w=0,...,N-1]$

So, obviously there is are real and imaginary parts to the spectral density. Then, I take the square root of the spectral density to get the amplitude spectrum:

$|A(\omega)|=\sqrt{S(\omega)}$

So, there should also be real and imaginary parts of the amplitude spectrum too right?

I have been using a program called SPECSIM2, which I downloaded from here.

Using this program, (or another like ImageJ), I can plot the amplitude spectrum... Is that just taking the real component of the amplitude spectrum then? Is the imaginary component known as the phase spectrum?

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