Factorization properties of optimum space-time processors in nonstationary environments

A structural analysis of space-time log-likelihood processors (LLP) that applies to arbitrary signal transmission models consisting of N sources, M sensors, a time-varying linear channel, and nonstationary Gaussian source signal and sensor noise processes is presented. The approach is based on representing the time-varying linear channel as a bounded linear operator L with closed range. By exploiting the properties of such operators and the specific structure of the array covariance function, it is shown that the classical M -dimensional integral equations defining the LLP can be transformed into equivalent N-dimensional integral equations. As a result, it is always possible to factor the LLP into a cascade of three specialized time-varying subprocessors, namely, a space-time whitening filter, an M-input N-output unitary beamformer (UB), and an N-input quadratic postprocessor (QPP). Both the UB and the QPP are given an interpretation and their most important features are indicated. To illustrate the theory, several examples of its application to signal models involving time-varying delays are given