An elucidating proof that a matched filter is optimum

Abstract

A matched filter is an essential element in digital communications receivers. It is known that it is optimal, in the sense of producing sufficient statistics of maximum signal-to-noise ratio; a proof is presented in many undergraduate textbooks. Recent research in mathematics education suggests that the purpose of a proof is not merely to convince students that a theorem is true by pure logic; rather, a proof should be explanatory, with semantic content that elucidates the relationship between the concepts involved. In this paper, a time-domain proof that the matched filter is optimal is presented, and it is argued that it has larger explanatory value than alternative proofs.

Data synchronization for gas emission measurements from dairy cattle: A matched filter approach

“…Assuming the noise covariance matrix R is known and the filter, characterized by the impulse response h(t), is linear and time-invariant, the MF will be based on maximization of the output signal-to-noise ratio (SNR) (Turin 1960, Robey et al 1992, Jiang et al, 2012, Bazdresch 2018, Ren et al 2022:…”

“…where h H indicates the conjugate transpose of h. It follows from Eq.2 that the only type of linear filter that maximizes SNR is the following MF (Turin 1960, Haykin and Moher 2007, Bazdresch 2018, Zubeldia, et al 2021:…”