THE MATCHED FILTER — a dart

THE MATCHED FILTER correlate against the template — the peak marks the hit

To find a known shape buried in noise, correlate the incoming signal against a copy of that shape — slide the template along and multiply-add at each position. Where they align, the products reinforce and the correlation peaks; elsewhere they cancel. The matched filter is provably the optimal detector for a known signal in white noise: it maximizes the signal-to-noise ratio at the moment of alignment. It is how radar finds an echo, how GPS locks a ranging code, how a barcode scanner finds the pattern. Detection as a search for a peak.

THE TECHNIQUE correlate signal × template at each lag; the max lag is the hit

The demo correlates a signal against a 4-sample template and finds the lag of maximum correlation — the alignment position: live demo

HISTORY & CREDIT North 1943 · matched filter

“A signal too weak to see is lost.” — correlate it against its known shape and the peak lifts it out of the noise. cited

the correlation · c[lag] = Σᵢ sig[lag+i]·t[i] — slide and multiply-add.
the peak · the lag of maximum c marks the alignment — optimal SNR in white noise.
1943 · D. O. North — the matched filter; radar, GPS, comms.

A known shape pulled from noise by its own echo — detection as the search for a peak. correlation

RECOMMEND FOR I-13 the peak lag, on the compiler

On the canonical compiler, correlating a signal against its template peaks at lag 3 — exactly where the template was embedded:

$ i13 run cr_matchedfilter.i13 # correlation over lags RUN OK · 777 step(s) · peak stack 10 · call depth 10 peak_lag = 3 -- maximum correlation lag found_at_3 = 1 -- the template sits at offset 3

Recommend as the batch’s close — a NULL, and a fitting end. The matched filter is correlation — a dot product at each lag — and finding the peak is a search over a pinned computed quantity; its optimality (max SNR) is a theorem (B39), and the correlation itself is a linear operation, not a carried channel. NULL. It closes THE CARRIER where the batch began: the phase channel was the real sixth-axis bid, and the panel’s ruling on it — new axis, or a broadening of generative-dual — is the batch’s verdict. [ 0.m | <-~-~-~-~-( . ||