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

摘要

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-theorthic 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 assumption 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. Stochastic convergence of adaptive algorithms, in the mean-square sense, is based on a stochastic version of a fixed-point theorem.