Optimal Correlators for Detection and Estimation in Optical Receivers

Motivated by modern applications of light detection and ranging (LIDAR), we study the model of an optical receiver based on an avalanche photo-diode (APD), followed by electronic circuitry for detection of reflected optical signals and estimation of their delay. This model is complicated as it consists of three types of noise: thermal noise, shot noise, and multiplicative noise (excess noise) that stems from the random gain of the APD. Consequently, the derivation of the optimal likelihood ratio test (LRT) for signal detection is non-trivial. We consider instead simple detectors, that are based on correlating the received signal with a given deterministic waveform, and our purpose is to characterize the waveform that best trades off between the false-alarm (FA) error exponent and the missed-detection (MD) error exponent. We also study the problem of estimating the delay by maximizing the correlation between the received signal and a time-shifted waveform, as a function of this time shift. We characterize the optimal correlator waveform that minimizes the mean square error (MSE) for SNR. The optimal correlator waveforms for detection and for estimation turn out to be different, but their limiting behavior is the same: when the thermal noise is dominant, the optimal correlator waveform becomes proportional to the clean signal, but when the thermal noise is negligible compared to the other noises, then it becomes a logarithmic function of the clean signal, as expected.