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.
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