Remarks on the statistical behaviour of orthogonal beamforming

Orthogonal beamforming is the name given to certain high-resolution methods for estimating the spectra of a wave field received by an array of sensors. These methods use the eigenvalues and eigenvectors of the spectral matrix of the sensor outputs. The problem is to predict the behaviour of such methods when only an estimate of the matrix is known. The sensor outputs may consist of sensor noise, ambient noise and noise from a finite set of discrete sources. The properties of the eigensystem of the spectral matrix in the case of weak ambient noise motivate the methods of orthogonal beamforming, for example Pisarenko´s nonlinear peak estimates and the projection estimates of Owsley and Liggett. If the spectral matrix is estimated by one of the classical methods, some asymptotic distributional properties of the matrix estimate and its eigensystem are well known. They can be used to determine asymptotic expressions, for example for the first and second moments of the peak estimators, and to approximate the distributions. The parameters, however, cannot be calculated in applications since the eigensystem of the exact spectral matrix is required. Therefore we recently developed bounds for the deviation of the peak estimates which only use limited knowledge about the matrix. We applied some results on perturbations of Hermitian operators. The asymptotic behaviour of the bounds for the projection estimator is investigated, and possibilities for their estimation are indicated. Finally, we report on extensive simulations with random matrices to evaluate the new bounds. As a result, we found that the projection estimator behaves stably and that there are tight bounds if the eigenvalues of interest are sufficiently separated from the rest.