On optimum multiple-alternative detection of signals in noise

The problem of the optimum detection of signals in noise, when a possible signal can arise from more than one class of signal types, is considered from the point of view of decision theory. Specifically, optimization here consists of minimizing the average risk for preassigned costs appropriate to the possible correct and incorrect decisions, when only a single signal, of class k , may occur out of N + 1 mutually exclusive signal classes. The analysis is a generalization of the authors' earlier work on binary, or single alternative detection systems. The present treatment outlines the optimization procedure for additive signals and noise, indicates the general structure of the detector, and gives expressions for the probabilities of error and minimum average (or Bayes) risk. Some explicit results for normal statistics are included, and an example-- coherent detection of similar signals, differing only in amplitude-- illustrates the general approach.