Matched Filter

The impulse response is the time-reversed conjugate of the signal you want to detect: h(n) = s*(N-1-n), where s is the signal, N is its length, and * denotes complex conjugate. This is called matched because the filter is perfectly matched to the expected signal pattern. The time reversal allows peak output to occur when the signal aligns with the filter.

Matched filtering is mathematically equivalent to computing the cross-correlation between the received signal and the reference signal. Correlation peaks where the signal best matches the template. Convolution uses the time-reversed filter, which is mathematically identical to correlation for the matched filter case. Most implementations use FFT-based convolution for efficiency.

For a matched filter applied to a known signal in white Gaussian noise, the output SNR equals 2E/N₀, where E is the signal energy and N₀ is the noise power spectral density. This SNR is independent of the signal waveform shape—it depends only on signal energy. Longer signals with more energy achieve higher SNR, all else equal.

When signal mismatch occurs, the matched filter output SNR degrades proportionally to how much the actual signal differs from the template. The SNR reduction factor is called the signal mismatch loss, quantified by the squared correlation between actual and expected signals. Even small mismatches can significantly degrade detection performance, making robust signal detection challenging in practice.