Question
I have observations of a noisy channel and i form a vector y of length n from those observations. I need to compute the autocorrelation matrix from this vector y which should be just the outer product y*y' i-e E[y*y']=y*y'. This matrix is a rank=1 nxn matrix. Is this correct?
Expert Answer
John Michell
PhD Expert
Answered Aug 17, 2026
No, that is not correct. The way you are doing it is not giving the autocorrelation matrix. For one thing your matrix is not going to be Toeplitz.
For example:
x = randn(10,1);
rxx = x*x';
Note the above, rxx, is not Toeplitz.
But
[xc,lags] = xcorr(x,x,9,'biased'); r = xc(10:end); rxx = toeplitz(r,conj(r)); % the conj() of course here is not needed
Now, rxx is Toeplitz and not that the autocorrelation matrix has full column rank.
rank(rxx)
Or you could do:
X = fft(x,2^nextpow2(2*size(x,1)-1));
R = ifft(abs(X).^2);
m = length(x);
R = R./m; % Biased autocorrelation estimate
rxx = toeplitz(R(1:length(x)),conj(R(1:length(x))));
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