PDF's, confidence regions, and relevant statistics for a class of sample covariance-based array processors

In this paper, we add to the many results on sample covariance matrix (SCM) dependent array processors by i) weakening the traditional assumption of Gaussian data and ii) providing for a class of array processors additional performance measures that are of value in practice. The data matrix is assumed drawn from a class of multivariate elliptically contoured (MEC) distributions. The performance measures include the exact probability density functions (pdf's), confidence regions, and moments of the weight vector (matrix), beam response, and beamformer output of certain SCM-based (SCB) array processors. The array processors considered include the SCB: i) maximum-likelihood (ML) signal vector estimator ii) linearly constrained minimum variance beamformer (LCMV) iii) minimum variance distortionless response beamformer (MVDR) iv) generalized sidelobe canceller (GSC) implementation of the LCMV beamformer. It is shown that the exact joint pdf's for the weight vectors/matrices of the aforementioned SCB array processors are a linear transformation from a complex multivariate extension of the standardized t-distribution. The SCB beam responses are generalized t-distributed, and the pdf's of the SCB beamformer outputs are given by Kummer's function. All but the beamformer outputs are shown to be completely invariant statistics over the class of MEC's considered.