Likelihood Detection with a Multiple-Markov Process

The problem of signal detection in the presence of noise has received considerable attention in the literature. The problem is approached historically by minimization of a signal-to-noise ratio. This formulation utilizes the properties of the process under consideration in terms of the frequency content as expressed by the correlation function and/or the power spectral density. The integral equations that evolve have, as their solution, the impulse response or the Fourier transform of the optimum processing device. With this approach, the statistics of the process do not enter the mechanism of solution explicity. An alternative approach to the problem is to consider the statistics expressed in terms of the distribution function, explicitly. In general, a process with a frequency distribution that is other than white will yield a likelihood ratio in a form that is complicated algebraically. The linear prediction possible with processes that exhibit mth-order Markovian properties allows a solution, however, and this solution embodies both the statistical and the correlation properties explicitly. For the special case of Gaussian statistics, the equivalence with other techniques is demonstrated.