Covariance matrix as a diagonal matrix or not??????

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Sadiq Akbar
Sadiq Akbar on 24 May 2021
Commented: Sadiq Akbar on 25 May 2021
The literature says that when three signals are uncorrelated i.e., they have different frequencies, then their COVARIANCE MATRIX RxxUn is diagoanl i.e.,
w=[pi/4 pi/3 pi/2]'; % Un-correlated signals
N=5;
xx=2*exp(1j*(w*[1:N]));
RxxUn=xx*xx'
But when they are fully correlated i.e., their frequency is same, then their COVARIANCE MATRIX RxxCo is Non-diagoanl and singular i.e
w=[pi/4 pi/4 pi/4]'; % Correlated signals
N=5;
xx=2*exp(1j*(w*[1:N]));
RxxCo=xx*xx'
Likewise, when they are partially correlated i.e., some have same frequencies and remainig have different frequency, then their COVARIANCE MATRIX RxxPar is Non-diagoanl and Non-singular i.e.,
w=[pi/4 pi/4 pi/2]'; % Partially Correlated signals
N=5;
xx=2*exp(1j*(w*[1:N]));
RxxPar=xx*xx'
But when I run this code, It is not so. Why it is so?

Answers (1)

the cyclist
the cyclist on 24 May 2021
I am definitely not an expert in this, but my guess is that these statements might only be strictly true for signals of infinite time extent. You have chosen a very small N, and therefore only a short signal.
Perhaps if you make N larger, you'll see the behavior approach what you expect.
(Hopefully some with more knowledge than I have will weigh in, though.)
  3 Comments
Sadiq Akbar
Sadiq Akbar on 25 May 2021
Thank you very much dear the cyclist for your 2nd response. Ok I am goint to give it in the attachment here.

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