Xiaohui Luo

A Multivariate Non-Gaussian Distribution Theory with Applications To Array Detection

May 21, 1998
3:00 PM
422 Snell Building

Abstract

A multivariate non-Gaussian distribution theory for complex random matrices is developed in this dissertation and a variety of adaptive array detection problems in interference modeled by the complex matrix distributions are addressed.

We develop two separate classes of complex non-Gaussian matrix distributions by weakening the necessary and sufficient conditions for multivariate complex Gaussian matrix distributions. The resulting two classes of distributions are (i) the family of unitary invariant distributions and (ii) the family of complex random matrices with independent spherically invariant columns. It is to be noted that the multivariate complex Gaussian matrix distribution is the only common member of both distribution famailes. We analytically characterize these matrix distributions and prove a number of useful theorems including a result on the statistical representation of the random matrices. The circular symmetry property of each element of the random matrices is also proved. Some examples of matrix distributions from both classes are provided. Wherever applicable results for the corresponding two classes of distributions for real random matrices are also included in the dissertation.

Next, we consider the problem of signal detection in interference modeled by each of the two non-Gaussian distribution families introduced in this dissertation. The class of problems considered here is primarily motivated by array detection applications. Our problem formulation is fairly general and includes subspace signal models and multiple observations resulting from transmitter/receiver diversity. The usual assumption that the receiver has no prior knowledge of the interference covariance matrix is made and group invariant tests for signal detection in this context are developed. A number of novel results with regard to obtaining a maximum likelihood (ML) estimate of the interference covariance matrix for both non-Gaussian distribution families are developed. For the unitary invariant distribution family, a quasi-generalized likelihood ratio test (QGLRT) is derived and we show that the size of a given invariant test is identical for every member of this distribution family. Analytical expressions for the probability of detection for an entire class of invariant tests are derived for some examples from the first distribution family. For the second distribution family, necessary and sufficient conditions for the existence of a ML covariance estimate are obtained and a novel recursive algorithm that guarantees convergence to a ML solution is derived. Examples illustrating the detection performance in the second class of interference distributions are included.

Thesis Committee:
Prof. R. Raghavan (advisor)
Prof. H. Lev-Ari
Prof. S. Gutmann (Mathematics Dept.)