Power Spectral Densities and transfer functions

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SUMMARY

The discussion centers on the relationship between Power Spectral Densities (PSD) and transfer functions in the context of a PID circuit affecting a free-running oscillator. It is established that the resulting PSD of the locked oscillator is indeed the product of the transfer function squared (H^2) and the original PSD (S). This relationship is confirmed through the principle that multiplication in the frequency domain corresponds to convolution in the time domain, reinforcing the logic behind the multiplication of these terms.

PREREQUISITES
  • Understanding of Power Spectral Densities (PSD)
  • Knowledge of PID control systems and their transfer functions
  • Familiarity with frequency domain analysis
  • Basic concepts of convolution and its relationship to multiplication in the frequency domain
NEXT STEPS
  • Research the mathematical foundations of Power Spectral Densities and their applications
  • Study the properties of PID controllers and their transfer functions
  • Explore the relationship between convolution and multiplication in signal processing
  • Read literature on cascaded systems in the frequency domain
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Engineers, control system designers, and signal processing professionals seeking to understand the effects of PID controllers on oscillator behavior and frequency domain analysis.

Niles
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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.
 
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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.
 

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