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

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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
the cyclist
the cyclist on 25 May 2021
Would it be easy for you to post a screenshot of the formula from the text? (I realize that you have referenced 3 pages of the book, so you may not be able to include all the relevant information.)
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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