Detection performance of the parametric correlator

We consider the parametric correlator with complex inputs and derive the expressions for the probability of false alarm and probability of detection in terms of the finite integration time and, for the probability of detection, in terms of the SNR. From these expressions performance results are presented. Furthermore, asymptotic expressions for the minimum detectable signal are obtained that allow evaluation of the detection performance of the parametric correlator and of magnitude squared coherence. We conclude this analysis by comparing the parametric correlator with the detection performance of magnitude squared coherence (background noise levels unknown) along with the performance of the optimal energy detector.