Suppose a known pulse $s(t)$ is transmitted through additive white noise and the receiver must decide whether that pulse is present.

A matched filter is chosen so its impulse response is a time-reversed conjugate of the expected signal:

$$h(t)=k,s^*(T-t).$$

Sampling the filter output at time $T$ produces a value proportional to the inner product between the received waveform and the expected signal.

The same operation can therefore be implemented conceptually as a correlator:

$$z=\int_0^T r(t)s^*(t),dt.$$

Among linear filters observed at the chosen decision time, the matched filter maximizes output SNR for white noise.

Detection then compares one or more correlation outputs with decision thresholds or with each other. The receiver does not try to reproduce every instantaneous detail of the noisy waveform; it extracts the components that are most informative for distinguishing the candidate transmitted signals.