Generalized rectification of cross spectral matrices for arrays of arbitrary geometry

In high-resolution methods applied to uniform linear arrays (ULA), the preprocessing that consists of forcing the estimated cross spectral matrix (CSM) to be Toeplitz by averaging its elements along its diagonals is known to increase the resolving power drastically. That is why it is always done in practice. However, this approach is limited to linear arrays because of the required Toeplitz structure for the CSM. This paper generalizes this technique to arrays of arbitrary geometry; the developed method is referred to as rectification. It proceeds by searching first for a vector subspace of Hermitian matrices that contains the manifold generated by the CSMs when the angle of arrival (AOA) varies. This preliminary step is performed only once for a given array geometry. Next, rectification of estimated CSMs is achieved by projecting them onto this subspace, resulting in denoising and increased resolving power of source localization methods at a very low computational cost. As a byproduct, the storage requirements for the CSMs are greatly reduced.