Applications of state-variable techniques in detection theory

The central issue in detection theory is that of detecting known or random signals in the presence of random noise. The detailed problem description depends on the physical situation of interest. In most of the original work on detection theory the random processes were modeled as Gaussian processes and characterized by their covariance function. In many cases the solution for the optimum detector is expressed in terms of an integral equation which is difficult to solve. In this paper we demonstrate how to formulate and solve detection theory problems using state-variable techniques. These techniques enable us to obtain complete solutions to almost all problems of interest. In addition, they offer new insights into the problems. We first formulate our basic state-variable model for real and complex random processes. We then study five applications of state-variable techniques. The first application is in the solution of linear homogeneous Fredholm equations. This problem arises whenever we want to find the eigenvalues and eigenfunction of a random process. The second application is the detection of a slowly fluctuating point target in the presence of colored noise. The third application is the detection of Gaussian processes in Gaussian noise. The fourth application is detection of Doppler spread targets and communication over Doppler spread channels. In the final application, we introduce distributed state variables in order to study the detection of doubly spread targets. The goal of the paper is to demonstrate the importance of state-variable techniques in a wide variety of detection theory problems.