Application of Hilbert Space Theory to Optimal and Adaptive Space-Time Processing.

The theory of optimal and adaptive space-time processing is developed from a fundamental and unified point of view. Because of its general nature and its geometric interpretations, an operator-theoretic approach in a Hilbert space is used. Based on measure theory in a Hilbert space, the likelihood-ratio receiver for optimal detection of a doubly -spread scatterer in the presence of nonstationary, anisotropic Gaussian interference is derived as a functional on the particular space. It is shown that the "white-noise" assumption is not a mathematical necessity for the detection problem to be nonsingular. However, by a slight extension of the theory, such an assmption can be easily included if it is convenient. The operator equations which define the individual elements in the optimal receiver structure are shown to be related to some common forms of signal processors. More specifically, using the concept of unitary equivalence of Hilbert spaces, the problems of frequency domain processing, beamsteering and space-time factorability, and processing of bandlimited signals are easily connected with the abstract theory. The link between the optimal detector and adaptive realizations is provided by an interpretation of the defining operator equations in the Hilbert space of random variables. It is shown that the inverse correlation operators, which are the basic elements of the likelihood-ratio processor, define orthogonal expansions of random variables, or "innovations" processes. Stochastic convergence of adaptive algorithms, in the mean-square sense, is based on a stochastic version of a fixed-point theorem. This is applied to Widrow's LMS algorithm and the garient lattice algorithm, and analytic expressions for convergence rate and misadjustment are obtained by detailed calculation of the appropriate operator -norms. The available theory is, however, sufficiently general for application to any recursive adaptive implementation.