Question about the Weiner-Khinchin theorem

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In summary, the conversation involved the speaker discussing their attempt to solve a problem involving decomposition coefficients and the use of the Wiener-Khinchin theorem or Fourier transform. They also mentioned finding success without using the relation for z(t) and asked for clarification on any assumptions made in their solution. They were then reminded of the purpose of the website they were on.
  • #1
mrMeister
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Homework Statement
The question is attached below.
Relevant Equations
correlation equations
Wiener-Khinchin theorem
My try:

I tried to take the expression for the decomposition coefficients and put it into the equation that I had to prove.
Then, I tried to work with the integral limits in order to get into Wiener-Khinchin theorem or maybe Fourier transform of delta function but I didn't see any success in those tries.

It would be nice to get a new point of view.

Thank you.

EDIT: I did solve this problem eventually, but I did not use the relation for z(t) - how is that?
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  • #2
Please shoe your work and also where you got stuck.
 
  • #3
Watch my comment - I have solved this question but without the use of the relation for z(t).
I wanted to ask if it is hidden somewhere throughout the solution.
 
  • #4
So you want someone to tell you if you assumed anything in a derivation that you refuse to share? I don’t think you understand how PF works.
 

Related to Question about the Weiner-Khinchin theorem

1. What is the Weiner-Khinchin theorem?

The Weiner-Khinchin theorem, also known as the Wiener-Khinchin-Kolmogorov theorem, is a fundamental theorem in the field of stochastic processes. It states that the power spectral density of a stationary random process can be obtained by taking the Fourier transform of the autocorrelation function of the process.

2. Who discovered the Weiner-Khinchin theorem?

The Weiner-Khinchin theorem was discovered independently by Norbert Wiener and Aleksandr Khinchin in the early 1920s. It was later expanded upon by Andrey Kolmogorov in the 1930s.

3. What is the importance of the Weiner-Khinchin theorem?

The Weiner-Khinchin theorem is important because it provides a mathematical relationship between the power spectral density and the autocorrelation function of a stationary random process. This relationship is used in many fields, such as signal processing, communication systems, and physics, to analyze and understand random processes.

4. Can the Weiner-Khinchin theorem be applied to non-stationary processes?

No, the Weiner-Khinchin theorem is only applicable to stationary random processes. A stationary process is one where the statistical properties, such as the mean and variance, do not change over time. Non-stationary processes do not have a constant autocorrelation function, so the theorem cannot be applied.

5. Are there any limitations to the Weiner-Khinchin theorem?

Yes, there are some limitations to the Weiner-Khinchin theorem. It assumes that the random process is wide-sense stationary, meaning that the mean and autocorrelation function are finite and well-behaved. It also assumes that the process is ergodic, meaning that the time average and ensemble average of the process are equal. These assumptions may not hold for all random processes, so the theorem may not be applicable in all cases.

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