Random Processes: Filtering, Estimation, and Detection

9 Detection Theory: Discrete Observation

9.1 BASIC DETECTION PROBLEM

In this chapter decision rules for a specified performance measure will be formulated for detection problems that have a statistical framework. By a statistical framework it is meant that the underlying probability density functions for the observed vectors and the class or hypotheses a priori probabilities are known. Possible performance measures include a posteriori probability, probability of error, and cost per decision. We will see that likelihood ratios carry the information necessary to make optimum classifications and that other so-called sufficient statistics can be found for many problems.

Many problems involve just two signal classes, but most problems require classification of multiple signal classes such as recognition of handwritten numerals and recognition of vowel sounds. The observed vectors are rarely one dimensional, and in most cases they are finite dimensional. Problems that contain observations over a continuous interval are of infinite dimension.

Decision rules will be developed in this chapter for the two-class case with extensions to the K-class case for a single observation and a vector observation, while the continuous or infinite observation case will be presented in Chapter 10.

As will be shown, for the two-class case, the optimum decision rule, under rather wide assumptions and performance measures, is a likelihood ratio test. For the M-class problem, the optimum decision rule can ...