mathematical explanation of diagonal loading

Diagonal loading is a technique used in signal processing to improve the performance of linear filters. It involves adding a small positive constant to the diagonal elements of the correlation matrix before computing its inverse.

In mathematical terms, suppose we have a correlation matrix R, which is a symmetric matrix of size N x N. The diagonal loading technique involves adding a positive constant, denoted by λ, to the diagonal elements of R, resulting in a modified correlation matrix Rλ:

Rλ = R + λ*I

where I is the identity matrix of size N x N.

The purpose of diagonal loading is to ensure that the inverse of Rλ exists and is well-conditioned, even if R is ill-conditioned. This is because adding λ*I to R effectively increases the diagonal elements of the matrix, which can help to stabilize the inverse computation and improve the numerical accuracy.

In practice, the value of λ is often chosen empirically, based on the specific application and the characteristics of the input data. Too small a value of λ may not have a noticeable effect on the filter performance, while too large a value may lead to over-regularization and reduced filter performance.

Overall, diagonal loading is a useful technique for improving the stability and performance of linear filters in various signal processing applications.