From matched filters to martingales
T. Kailath · 2002
Summary form only given. The famous matched filter solution for maximizing the output SNR of a known signal in additive white noise was independently discovered by several investigators in the mid-forties. Since then it has appeared as a key component of optimal detectors in a variety of scenarios. Its first appearance was perhaps in D.O. North's 1943 RCA report, which is remarkable for the facility with which the author exploits sophisticated (for the times) mathematical analysis to obtain useful physical results and insights; among other items, the Rice distribution is introduced and used in a routine way. Since then, the concept has evolved and grown in a fascinating way, which is outlined, chiefly through the early work of V.A. Kotelnikov (1947) and of P.M. Woodward, and its notable extensions to multipath problems through the estimator-correlator ideas of P. Price and P.E. Green. We describe how the effort to extend these results to non-Gaussian signals led back in a fascinating way to the original likelihood ratio formulas. Martingale theory and the need for attention to the definition of stochastic integrals arose in a natural way in the course of this development and later enabled, among other things, the development of close parallels between detection problems for signals with additive Gaussian noise and with "multiplicative" Poisson-type noise. As interest in such areas declined in the information theory community, the methods began to appear in finance theory, where they were soon also regarded as the natural tool. Moreover the growing emphasis on soft-decision rules in the new turbo coding schemes may renew interest in general likelihood ratio formulas.