A series technique for the optimum detection of stochastic signals in noise

A procedure for the optimum detection of stochastic signals in noise is discussed. The optimum test function is expanded in a point-wise convergent series for which a bound on the convergence properties can be obtained. Knowledge of this bound permits the substitution of a truncated version of the series for the optimum test function. This leads to a test procedure that uses a variable number of terms of the series for each decision and also gives the same decision as the optimum detector. For detection of stochastic signals in Gaussian noise, an expansion is obtained in terms of the eigenfunctions associated with the Gaussian probability density function, which leads to optimum decisions with a moderate number of terms of the series. It is also well suited for adaptive detection in which the distribution function of the stochastic signal is unknown--the coefficients of the expansion factor into two terms, one dependent only on the noise distribution and the other dependent on the distribution of the stochastic signal. Computer results for Gaussian noise are given. For this case, the test procedure can be viewed as a sequence of linear, quadratic, etc., detectors that, when a basic inequality is met, terminates with an optimum decision.