10.6 GENERAL BINARY DETECTION WITH UNWANTED PARAMETERS

In this section, we consider the general binary detection of signals in an additive white Gaussian noise process with mean zero and power spectral density N ₀/2. However, the received waveform is not completely known in advance as in the previous section, where we assumed that the only uncertainties were due to additive white Gaussian noise. These signals, which are not completely known in advance, arise in many applications due to factors such as fading, random phase in an echo pulse, and so on. The unknown parameters of the signal are known as unwanted parameters.

Consider the general binary detection problem where the received signal under hypotheses H ₁ and H ₀ is given by

where ? ₁ and ? ₀ are the unknown random vectors. Note that if ? ₁ and ? ₀ are known, the signals s ₁( t, ? ₁) and s ₀( t, ? ₀) are deterministies, and thus they are completely specified.

The unknown parameter ? _j, j = 0,1, may be either random or nonrandom. In our case, we assume that ? _j, j = 0,1, is a random vector with a known a priori density function. That is, the joint density function of the components of ? _j, j = 0,1, is known. The approach to solve this problem is to obtain a set of K