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Homework Help: Wiener Filter, Correlation Matrices

  1. Feb 21, 2010 #1
    I'm trying to understand the Wiener Filter, and I have a few questions.

    1. How can there be such a thing as a correlation matrix of 1 vector. I read here:

    R_yy = E[ y(k) * y^T(k) ]

    where y(k) is a vector, and y_T(x) is the same vector transposed. I thought correlation represents the degree of correspondence between 2 variables, so how can we say there is a correlation between 2 equivalent vectors?

    2. They present a matrix R_yy as the correlation matrix (mentioned above). Then they say vector "r_yy" is the first column of R_yy. If that is so, what are the other columns of R_yy?

    3. The book first presents the Wiener filter formulation by saying that in minimizing the MSE criterion, we can find "h_w" (the wiener filter vector). They go on to another form of the Wiener filter by saying:

    h_w = h_1 - inv(R_yy)*r_vv

    where h_1 was defined as [1 0 0 0 ...]^T and r_vv, I suppose is the correlation (once again I dont know how one can have correlation between 2 equivalent vectors).

    Thank you for any help.
  2. jcsd
  3. Feb 24, 2010 #2
    Any ideas?
  4. Feb 24, 2010 #3
    ryy is the auto-correlation, witch you can think of as the correlation between the signal and shifted versions of itself. The relationship posted by you is actually ryy(0).
    ryy(n)= E{y(k)*yT(k-n)}
    That matrix is a symmetric toeplitz matrix. Indeed the first column is ryy, the second is [ryy(1) ryy(0) ryy(1) ryy(2) ... ryy(N-1)]T and so on the last one is ryy reversed.
    I don't quite understand what you mean by your notation here.
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