Autocorrelation Matrix from a vector

2012 9 26
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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?

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Wayne King
Wayne King 2012-9-26
编辑：Wayne King 2012-9-26

1 个投票

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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xplore29
xplore29 2012-9-27

0 个投票

Thank you.
In case of cross correlation between y(nx1) and x(mx1) (n>m) vectors
rxy is a nxn matrix and following code should work
[xc,lags] = xcorr(y,x,n-1); r = xc(n:end); rxy = toeplitz(r,conj(r)); % the conj() of course here is not needed
but the true X-correlation matrix is T (nxm) matrix
T = rxy(:,1:m);
Is this correct?

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