# Power Spectral Densities and transfer functions

• Niles

#### 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.

Last edited:
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.

## 1. What is a power spectral density (PSD)?

A power spectral density is a measure of the distribution of power in a signal or system over a range of frequencies. It is often used in signal processing and engineering to analyze the frequency components of a signal or system.

## 2. How is PSD related to transfer functions?

Transfer functions are mathematical representations of the relationship between input and output signals in a system. The PSD of a system can be obtained by taking the magnitude squared of the transfer function, which gives the power at each frequency in the input signal.

## 3. What is the significance of the peaks in a PSD plot?

The peaks in a PSD plot represent the frequencies at which the signal or system has the most power. These peaks can provide insight into the behavior and characteristics of the system, such as resonant frequencies or dominant frequencies in the input signal.

## 4. How can PSD and transfer functions be used in practical applications?

PSD and transfer functions are commonly used in fields such as signal processing, control systems, and telecommunications. They can be used to analyze and design filters, identify system characteristics, and optimize system performance.

## 5. Can PSD and transfer functions be used for non-linear systems?

While PSD and transfer functions are most commonly used for linear systems, they can also be applied to non-linear systems by using linearization techniques. However, the accuracy of the results may be limited due to the non-linear nature of the system.