The Equivalence of Optimum Transducers and Sufficient and Most Efficient Statistics

Abstract

The problem of the design of transducers for the purpose of restoring the fidelity of a signal (which has been corrupted by noise), while preserving information in the sense of Shannon is shown to be a particular though unusual one of statistical inference and to be amendable to the methods of mathematical statistics. Adopting the view that the inference of the approximate form of a function s(t) (the signal) from the available mixture of the signal and the noise time series n(t) is the natural limiting case of inference from discrete series of random variables, it is seen that the transducer under discussion is the physical analog of a ``statistic''—in the sense of Fisher—and that the properties of the optimum transducer are equivalent to the statistical properties of sufficiency (preservation of information) and efficiency (maximization of fidelity). The case of a Gaussian signal to which Gaussian noise has been added is considered in detail. Using the probability density functional of the Gaussian time series, the maximum likelihood estimate (which in this case is efficient, hence sufficient) of the signal is derived, and it's physical analog is identified as the smoothing filter of Wiener with infinite lag.