Noise covariance matrix

Question

I am attempting to implement a generalized least squares estimator that uses a noise covariance matrix from measured data. The basic model is as follows

$$\hat{c}_{GLS} = \left(\bf{A}^T\bf{\Sigma}^{-1}\bf{A}\right)^{-1}\bf{A}^T\bf{\Sigma}^{-1}\bf{r}$$

where $\bf{A}$ is a matrix of my basis functions, $\bf{\Sigma}$ is the noise covariance matrix and $\bf{r}=[r_x r_y r_z]^T$ is a column vector of my vector measurements ordered as shown. Likewise the noise vector measurements are $\bf{n}=[n_x n_y n_z]^T$.

The statement is made in the paper that $\Sigma = E\left\{\bf{n}\bf{n}^T\right\}$, which results in $\Sigma$ being size $3N\times 3N$ which makes sense. However, when I do this I get horrible results and the structure of $\Sigma$ doesn't look right. I expect it to be diagonal dominant and then falling off.

• My question is what am I doing wrong?
• How do I build the noise covariance matrix correctly?

I have data for $\bf{n}=[n_x n_y n_z]^T$ It isn't just the $3\times 3$ covariance matrix I am use to dealing with.

Edit: added code

minutes = 5; NSamples = Fs*60*minutes; NOverlap = NSamples/2; Sigma = zeros(3*NSamples,3*NSamples); M=20;

for nn = 1:M

nStart = (nn-1)*NOverlap+1;
nStop  = nStart+NSamples-1;
nx = x1(nStart:nStop);
ny = y1(nStart:nStop);
nz = z1(nStart:nStop);
Rxx = xcov(nx );Rxx= (Rxx(NSamples:end));
Rxy = xcov(nx,ny );Rxy= (Rxy(NSamples:end));
Rxz = xcov(nx,nz );Rxz= (Rxz(NSamples:end));
Ryy = xcov(ny );Ryy= (Ryy(NSamples:end));
Rzz = xcov(nz );Rzz= (Rzz(NSamples:end));
Ryz = xcov(ny, nz );Ryz= (Ryz(NSamples:end));

N   = [toeplitz(Rxx) toeplitz(Rxy) toeplitz(Rxz);
       toeplitz(Rxy) toeplitz(Ryy) toeplitz(Ryz);
       toeplitz(Rxz) toeplitz(Ryz) toeplitz(Rzz)];
Sigma = N + Sigma; 

end Sigma = Sigma/M;