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

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?

Answer 1

[the cyclist]

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.)

[Sadiq Akbar]

Thank you very much dear the cyclist for your kind response. May be you are rigth but I think there is no such restrictions given in the book "Array Signal Processing" section 2.3.1 page 33-35. So lets wait for some other one as well.

[the cyclist]

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]

Thank you very much dear the cyclist for your 2nd response. Ok I am goint to give it in the attachment here.