Wiener Filter, Correlation Matrices

[the_dialogue]

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 don't know how one can have correlation between 2 equivalent vectors).



Thank you for any help.

 

[the_dialogue]

Any ideas?

 

[emanuel_hr]

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?

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)*y^T(k-n)}

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?

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.

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 don't know how one can have correlation between 2 equivalent vectors).

I don't quite understand what you mean by your notation here.