correlation

Correlation measures how similar two signals are as you slide one past the other — its peak tells you both 'how alike' and 'by what time offset' they best line up. Picture overlaying two heartbeat traces and shifting one until the spikes snap together: the lag where they match best is the correlation peak. Cross-correlation compares two different signals; autocorrelation compares a signal with a shifted copy of itself, exposing hidden periodicities like the pitch of a voice.

This is the engine of detection and timing. A GPS receiver finds a satellite by correlating the noisy antenna signal against the known code it expects; the sharp correlation peak reveals the exact delay, and from delay comes distance. Radar, sonar, OFDM frame synchronization, and DNA sequence alignment all lean on the same idea. The optimal detector for a known waveform buried in white noise — the matched filter — is literally a correlation against that waveform.