Probability density function afterring

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SUMMARY

The discussion focuses on the transformation of a probability density function (PDF) after a random variable undergoes convolution with a filter, specifically a low-pass filter. The user inquires about the analytical methods to determine how the mean and variance of a Gaussian random variable change post-filtering. It is established that applying a low-pass filter to a Gaussian signal results in another Gaussian signal, adhering to the Central Limit Theorem, but the specifics of how the PDF characteristics are altered require further exploration.

PREREQUISITES
  • Understanding of convolution in signal processing
  • Familiarity with probability density functions (PDFs)
  • Knowledge of Gaussian distributions and their properties
  • Basic concepts of filtering, particularly low-pass filters
NEXT STEPS
  • Research the analytical methods for convolution of probability distributions
  • Study the effects of low-pass filtering on Gaussian signals
  • Explore the Central Limit Theorem and its implications on signal processing
  • Learn about the transformation of statistical moments (mean and variance) through filtering
USEFUL FOR

Mathematicians, statisticians, signal processing engineers, and anyone interested in the effects of filtering on probability distributions.

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Probability density function after filtering

Hello,

I am trying to find how a random variable will transform once gone through
a filter.

For example, I have a random sequence x(t), going through a filter h(t). Thus,
y(t) = x(t)*h(t) ; % '*' is convolution.

Now I want to find out how the PDF of y(t). Is there a certain analytical method to find this?
Suppose x(t) is Gaussian with a certain mean and variance. How will this mean and variance will be chaned when this signal goes through a low-pass filter. (Going through a low pass filter will give a Gaussian signal due to Central Limit theorem, but how will the PDF characteristics change).

Any advice or references are greatly appreciated.

Thank you.
 
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