Matrix conjugate gradients for the generation of high-resolution spectrograms

When only a limited number of snapshots is available for estimating the spatial correlation matrix, a low-rank solution of the MVDR equations, obtained via a small number of iterations of conjugate gradients (CG), can yield a higher SINK than the full-rank MVDR beamformer. The primary issue addressed in this paper is whether the unity gain constraint in the look direction should be enforced a-priori via the use of a blocking matrix, constituting steering dependent conjugate gradients (SD-CG), or effected a-posteriori through simple scaling of the beamforming vector, constituting steering independent conjugate gradients (SI-CG). It is proven that the two methods yield exactly the same low-rank beamformer. This is an important result since the construction, and application to the data, of a blocking matrix for each "look" direction represents a very substantial computational burden. A simplified expression for the power estimate obtained with the SI-CG beamformer is also presented. Simulations verify the efficacy of the theory.