Statistical Characterization of the Optimal Detector for a Signal with Time-Varying Phase Based on the Edgeworth Series

This paper focuses on approximating the false alarm and detection probabilities of the optimal non-coherent detector for a signal, which contains a constant amplitude and unknown phase, corrupted by Gaussian noise. Several closed-form approximations of these probabilities are obtained using different truncations of the Edgeworth series and the Central Limit Theorem (CLT). The accuracy of the different approximations is contrasted to the performance of the optimal non-coherent detector revealing that the best approximation corresponds to the Edgeworth expansion using the longest series, which offers a great precision. The CLT approximation is not accurate enough to predict the performance of the optimal detector. The closed-form expression based on the Edgeworth series allows us to set a detection threshold for a false alarm probability value and obtain the detection probability of the detector with extreme accuracy.