# Why does diagonal loading of a covariance matrix make an adaptive beamformer more robust in the case of a perturbed array?

It has been shown that 'diagonal loading' a covariance matrix derived for an adaptive beamformer can improve robustness of the beamformer when the antenna array is perturbed, albeit at the expense of background noise power.

The question is why does this work? Is the diagonal loading stopping the adaptive filter from reaching a local minima? Or is there some other mechanism at work here?

## 2 Answers

I had worked on an array processing problem where I had used diagonal loading of the measurement covariance matrix. But I had used diagonal loading as a solution to what I thought was a numerical issue with my eigen values being too small. Since I had to invert the covariance matrix with small eigen values, I had numerical issues with the result. Diagonal loading of the covariance matrix helped me get rid of the issue as the eigen values were now bounded by the diagonal loading value.

I am not sure, but one place where diagonal loading helps is in numerical stability of inverting the covariance matrix which is most likely encountered in adaptive beamforming.

Morgan and Elliot showed that an array's sensitivity to perturbation is directly related to the white noise gain of the array,

E. N. Gilbert and S. P. Morgan. Optimum design of directive antenna arrays subject to random variations. Bell System Technical Journal, 34(3):637

They showed that the most robust array has equal magnitude weight coefficients.

There are more than a few reasons to diagonally load a covariance matrix, like in Dominant Mode Rejection, but for the case of a perturbed array, on page the sensitivity increases as $|| \mathbf{w} ||^2$ increases.

If you refer to page 505 of :

H.L. Van Trees. Detection, Estimation, and Modulation Theory, vol 4: Optimum Array Processing. Wiley, 2004

It is shown that adding a quadratic inequality constraint, $\mathbf{w^H} \mathbf{w} \le T$ (i.e. bounding the coefficients), MVDR beamformer will have the form:

$$\mathbf{w^H} =\frac{\mathbf{v^H} (\mathbf{S_x} + \lambda \mathbf{I})^{-1} }{\mathbf{v^H} (\mathbf{S_x} + \lambda \mathbf{I})^{-1} \mathbf{v} }$$

So, diagonal loading has the effect of reducing the spread in amplitude of the weights. A large of spread of magnitude weights is also a characteristic of having interferers in the main beam, and a quadratic constraint "protects the main response pattern"