This dissertation addresses the problem of deciding which of two random signals is being received after observing a finite number of observations of that signal. The methods presented are varied, each addressing a different aspect of the problem.

First, we consider discriminators that involve a test statistic formed as a sum of one-step memory nonlinearities. The optimal one-step memory nonlinearity is obtained as the solution of an integral equation so as to maximize a generalized signal- to-noise ratio. The optimal one-step memory nonlinearity for weak-signal detection is also discussed. We obtain a partial "robustness" result, in which a guaranteed level of performance is proved. Our simulation results show that optimal one-step memory discriminators yield a significant improvement over optimal memoryless discriminators.

Second, we consider likelihood ratio test (LRT) discriminators based on non-Gaussian multivariate probability density functions (pdfs). We propose that multivariate pdfs be constructed from the X2 distribution with a memoryless transformation to shape the univariate marginal pdf to any specified univariate pdf. With the X2 random variables constructed from underlying Gaussian autoregressive processes, we show that the multivariate X2 pdfs may be numerically computed. As an alternative to multivariate pdfs constructed as transformations of the multivariate Gaussian pdf, we show that the transformed X2 multivariate pdfs can yield much better LRT discriminators in some cases.

Third, we consider LRT discriminators based on multivariate pdfs constructed from specified univariate pdfs via a hidden Markov model (HMM). These are not the usual nonstationary HMMs ; HMMs we consider are stationary with fixed univariate marginals. Estimation methods for identifying the parameters of the HMMs are discussed in detail. Parsimonious models with a small number of parameters are also discussed. Simulation results show that the models can be used to construct good LRT discriminators.

Finally, we consider the problem of designing a worst-case signal for the signal discrimination problem. The problem is formulated as a game that pits a signal designer against a discriminator designer. The signal designer tries to deceive the discriminator by replicating a signal of interest to the discriminator designer. Power constraints on the spurious signal signal prevent it from exactly replicating the statistics of the genuine signal. The problem is solved for a lower bound on time-averaged power of the spurious signal and a lower bound on expected power of the spurious signal.