Optimum Adaptive Reception for Binary Sequences

The a posteriori probability density function p(¿|X1, X2,...,Xk), where the Xi, i=1, 2, ..., K, represent Kvector-valued observations statistically related to the random vector ¿, appears in many applications of the methods of statistical inference to problems in pattern recognition and statistical communication theory. In this paper, it is shown that for equally likely binary sequences (M= 2) of anticorrelated patterns for signals observed in additive Gaussian noise, a device that computes p¿|X1, X2, XK) can be synthesized from a correlator, a simple instantaneous nonlinearity, and a multiplier. These results are used to derive some equally simple structures for various optimum nonsupervised estimators, pattern recognition machines, and signal detectors.