Power Spectral Densities and transfer functions

[Niles]

Hi

I know the PSD S of a free-running oscillator, describing its frequency fluctuations. Now, say that I want to consider the effect a standard PID circuit (with transfer function H) has on the oscillator.

I read in a paper that the resulting PSD of the locked oscillator becomes the product between H^2 and S. Is this really the case? It doesn't seem very logical to me that we are allowed to simply multiply these.

I'd be happy if someone can confirm this and possibly (shortly!) explain why. As an alternative, I'd be happy to get a reference to an article/book that describes it.

Thanks in advance.

 

[meBigGuy]

Think of two filters in the frequency domain. How do you determine the cascaded response? Multiplication in the frequency domain is convolution in the time domain. H^2 because S is energy.