Cross-correlation of white noise process with its conjugate

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Discussion Overview

The discussion revolves around the properties of a white Gaussian noise (WGN) process, specifically focusing on the expected value of the product of samples of the process and its conjugate. Participants explore whether the expression E[w[n1] w*[n2]] equals zero, considering both complex and real white noise processes.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant states that for a WGN process, E[w[n1] w[n2]] = 0, and questions whether E[w[n1] w*[n2]] would also be zero.
  • Another participant suggests that if the discussion pertains to complex white noise, where the real and imaginary parts are uncorrelated, then the expected value should also be zero.
  • A different viewpoint raises a question about the implications of reversing samples in the time domain and its effect on the conjugate in the frequency domain.
  • A later reply summarizes that for a complex random process, E[w[n1] w*[n2]] = 0, and for a real random process, E[w[n1] w[n2]] = 0, indicating that the ensemble average of the auto-correlation function at times n1 and n2 is generally zero.

Areas of Agreement / Disagreement

Participants express differing views regarding the implications of the expected value for complex versus real white noise processes. While some assert that the expected value is zero in both cases, the discussion does not reach a consensus on the broader implications of time reversal and its effects.

Contextual Notes

There are unresolved assumptions regarding the definitions of the white noise processes being discussed, particularly the distinction between complex and real processes. The implications of time reversal on the expected values are also not fully explored.

nitisha
If w[n] are samples of the white gaussian noise process, I know that
E[w[n1] w[n2]] = 0 for a WGN process.

what would the following expression lead to:

E[w[n1] w*[n2]] = ?

Would it also be zero?

Thanks a lot!
 
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nitisha said:
If w[n] are samples of the white gaussian noise process, I know that
E[w[n1] w[n2]] = 0 for a WGN process.

what would the following expression lead to:

E[w[n1] w*[n2]] = ?

Would it also be zero?

Thanks a lot!
Welcome to the PF. :smile:

What do you think the answer is and why? Also, is this for homework?
 
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Are you talking about complex white noise where the real and imaginary parts are uncorrelated white noise processes? If so, the expected value should be 0.
 
If you reverse the samples in the time domain, you get the conjugate, = negative phase, in the frequency domain.

Why does it matter if a Gaussian white noise is played forwards or backwards in time?
 
Thanks all! Got the answer.

If white noise is a complex random process, we say that E[w[n1] w*[n2]] = 0;
If it is a real random process, we say that E[w[n1] w[n2]] = 0

Generally speaking, the ensemble average of the auto-correlation function at times n1 and n2 of a WGN process is 0.
 

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