Modern statistical approaches to reception in communication theory

Abstract

When reception in the theory of communication is recognized as a problem in statistical inference, system design and system analysis appear as the counterparts of designing and evaluating statistical tests. This paper discusses the optimum properties of designs based on statistical decision theory from the risk point of view, and from that of information theory. Connections between risk and information loss are established, which result in a unified theory of system design. This includes Minimax methods capable in principle of handling all degrees of a priori knowledge of signal and noise statistics, new methods for comparing actual and ideal systems for the same purpose, and new interpretations of previously used formulations as special cases of the more general theory. Both detection and extraction of signals in noise are considered, the former as a problem of testing statistical hypotheses and the latter as one of estimating parameters. Formulation of the general reception problem as a decision operation is followed by a summary of statistical decision theory from the risk point of view, with some examples of Bayes and Minimax tests and optimum classes of decision rules. Applications to detection show the optimum nature of likelihood ratio receivers as a class, and indicate methods for defining the minimum detectable signal and for comparing system performance. As an illustration, curves of Bayes and Minimax risk are given for detection of a pulsed carrier in noise. Applications to extraction show the nature of optimum extraction and the roles of the mean square error and maximum likelihood criteria from the more general point of view of risk theory. Conditions under which information loss is an extremum in detection and extraction are established, and information loss itself as a criterion of performance is compared with that of the risk measure.