Minimum Bayes Risk Adaptive Linear Equalizers

This paper introduces Bayes risk (expected loss) as a criterion for linear equalization. Since the probability of error is equal to the Bayes risk (BR) for a particular binary loss function, this work is a natural generalization of previous works on minimum probability of error (PE) equalizers. Adaptive equalization algorithms are developed that minimize the BR. Like the minimum PE equalizers, the BR algorithms have low computational complexity which is comparable to that of the LMS algorithm. The advantage of the BR criterion is that the loss function can be specified in a manner that accelerates adaptive equalizer convergence relative to the minimum PE adaptive algorithm as illustrated in simulation examples. Besides introducing a new criterion, this paper provides another independent contribution to the field of PE minimizing equalization. While most prior works focus on
M
-ary QAM type modulations with rectangular decision regions, this paper uses upper bounds on the probabilities of certain events to yield tractable mathematics that apply to two-dimensional constellations with arbitrarily shaped decision regions. The resulting adaptive algorithm use the full information available in the phase of the error signal, whereas previous algorithms use a quantized version of this error phase.