Performance convergence of the adaptive matched filter

Reed et al. (1974) examined the output SNR of an adaptive beamformer, or coherent adaptive matched filter (Chen and Reed 1991, Robey et al. 1992), conditioned on the sample covariance matrix. This output SNR varies with different realizations of the sample covariance. When normalized by the known-covariance SNR, it is a beta random variable. Its closeness to unity is an indicator of how close the performance of the AMF approaches that of the non-adaptive matched filter. In this paper we examine the SNR, not conditioned on the sample covariance, as a figure-of-merit for detection performance. We find an additional criterion for performance convergence that is not obtained from the conditional SNR analysis. For our analysis, we make use of our method of "stochastic representations," first presented in Kraut et al. (1997), by which a whole class of adaptive detectors can be expressed as simple functions of statistically independent, scalar random variables.