# 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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No, the amplitude spectrum is a nonnegative function (as you can see in the paper too), which does not have an imaginary part. One can represent a complex number, such as $S(\omega)$, as $re^{i\phi}$ with $r\ge 0$ and $\phi\in [0,2\pi)$. Here $r$ is the amplitude and $\phi$ is the phase. The amplitude spectrum is the plot of the absolute value of Fourier transform, while the phase spectrum is the plot of the phase (argument). The prevailing idea is that the amplitude spectrum is the useful part, and one would like to recover the signal from it as much as possible.