The Math of Sonar and Radar – The Matched Filter

Sonar and radar send a known signal (e.g., a chirp) and listen for echoes. The matched filter, derived from the Cauchy–Schwarz inequality, maximises signal‑to‑noise ratio. It correlates the received signal with a time‑reversed copy of the transmitted signal. The Fourier transform converts this into a simple product. This technique is used in GPS, ultrasound imaging, and seismology (locating earthquakes). The mathematics is the same as the autocorrelation function, which measures self‑similarity. It’s the unsung hero of remote sensing.