Multi-Channel Parametric Estimator Fast Block Matrix Inverses

The optimal (adaptive) linear combiner (beamformer) weights for a sensor array are expressed in terms of the inverse of the multi-channel (MC) covariance matrix. Rather than form an estimate of the covariance matrix directly from the available data and inverting it, an alternative direct estimate of the inverse may be obtained by forming parametric MC linear prediction estimates and then expressing the inverse in terms of these parametric MC estimates. The resulting parametric estimate of the inverse is typically more accurate than inverting the estimate of the covariance matrix. This paper reveals, for the first time, the structure of the the inverse of the covariance matrix for the MC version of the covariance least squares linear prediction algorithm. The inverse structure involves products of triangular block MC Toeplitz matrices, which leads to fast computational solutions for the optimal weights.