Conventional techniques for characterizing clutter depend on covariance-based statistical modeling. This presents a disadvantage to cognitive radar/sonar since optimizing waveform design becomes highly nonconvex. Modeling the clutter and target responses via random transfer functions known as channel matrices simplifies this waveform optimization problem. The goal of this paper is to explore the optimal receive architectures for target detection that emerge when these channel matrices are modeled as deterministic, and then as random using a Ricean channel model. A likelihood ratio test (LRT) is derived yielding the well-known coherent matched filter, and an average LRT (ALRT) test is derived using Bayesian integration. The detection performance of these receivers is assessed and compared via standard analyses yielding receiver operating characteristic (ROC) curves. It is shown that the optimal ALRT is not strictly a linear function of the data.