Wiener filter to estimate constant signal

Question

I want to compare the performance of a Wiener Filter and the Kalman filer to estimate the value of a constant $d$ using mesurements corrupted by a white noise. That is, my measurements are of the form $$x(n) = d + v(n)$$ where $v(n)$ have a normal distribution with mean $0$ and known variance $\sigma^2$.

Using the Kalman filter, I could put it on State Space form $$d(n+1) = d(n)$$ $$x(n) = d(n) + v(n)$$ and solve the problem. But I am having difficults to set the problem so I can solve with the Wiener filter. My desided signal is the constant signal $x$. My filter input is the measurements $x$. But I have doubts if I am doing something wrong, because my estimated signal is not a really good estimative of the desired response. I used the following matlab code:

function test()
    n = 0:511;
    d = 10 * ones(1,512);
    v = 0.5*randn(1,512);
    x = d + v;
    w = WienerFIRFilter(x, d, 12);
    y = filter(w', 1, x);

    plot(x)
    hold on
    plot(y, 'r')
end

The function WienerFIRFilter is defined as following

function w=WienerFIRFilter(u,d,M) 
    aux = xcorr(d,u,'biased'); 
    p = aux(1,(length(aux)+1)/2:((length(aux)+1)/2)+M-1); 
    [U, R] = corrmtx(u,M-1);
    w=inv(R)*p'; 
end

Am I doing something wrong?

Answer 1

You can use this code

 function w=WienerFIRFilter(input,desired,M) 
     auto_corr=xcorr(input,M,'unbiased');
     r=auto_corr(M+1:end); % positive lags only
     R=toeplitz(r); %correlation matrix
     cross_corr=xcorr(input,desired,M,'unbiased');
     p=(cross_corr(M+1:end));
     w=inv(R)*p'
 end