Introduction

In this chapter we finally address the detection problem in continuous time. The setup is described in Section 26.2. The key result of this chapter is that—even though in this setup the observation consists of a stochastic process (i.e., a continuum of random variables)—the problem can be reduced without loss of optimality to a finite-dimensional problem where the observation consists of a random vector. Before stating this result precisely in Section 26.4, we shall take a detour in Section 26.3 to discuss the definition of sufficient statistics when the observation consists of a continuous-time SP. The proof of the main result is delayed until Section 26.8. In Section 26.5 we analyze the conditional law of the sufficient statistic vector under each of the hypotheses. This analysis enables us in Section 26.6 to derive an optimal guessing rule and in Section 26.7 to analyze its performance. Section 26.9 addresses the front-end filter, which is a critical element of any practical implementation of the decision rule. Extensions to passband detection are then described in Section 26.10, followed by some examples in Section 26.11. Section 26.12 treats the problem of detection in “colored” noise, and the chapter concludes with a discussion of the detection problem for mean signals that are not bandlimited.