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Joint Workshop on Coding and Communication (JWCC'07), Dürnstein, Austria, October 15–16, 2007.

The matched filter done right

Abstract

One of the key results of Digital Communications can be paraphrased very roughly as follows: “in guessing which of two deterministic signals is being observed in white Gaussian noise, the inner products between the observed waveform and each of the signals form a sufficient statistic. Consequently, it is optimal to base one's decision on these two inner products.” It is surprising that this basic result is never formulated as a theorem in any of the textbooks on the subject. This may be because of the difficulties in defining white Gaussian noise, in defining sufficient statistics for waveform observations, and in relating sufficiency to optimal detection.

In this talk I shall describe a number of approaches to formulating the above statement as a theorem and point out some of their short-comings. I will finally describe my proposed approach, formulate the theorem, and prove it from first principles.

The proposed approach does not rely on the Itô Calculus, on Brownian Motion, or on generalized stochastic processes. It does not introduce non-physical infinite-power noise processes. Moreover, it is suitable for treating colored noise.

The video is 35 minutes long.