Closed form analysis of the normalized matched filter with a test case for detection of underwater acoustic signals

In this paper, closed-form expressions for the performance of the normalized matched filter (NMF) detector are developed specifically for the case of large time-bandwidth product, N. As a test case, the task of detecting underwater acoustic signals is considered. While the matched filter is the most common detector used, the NMF detector is used in cases where the ambient noise is fast time varying and is hard to estimate. While the performance of the NMF has been studied, no closed-form expressions are given for the detection and false alarm probabilities, and the accuracy of the available approximations greatly deteriorates with N. As a result, evaluating the detection threshold from the receiver operating characteristic requires significant, and sometimes untraceable, numerical calculations. This is specifically important for underwater acoustic signals, where due to the low signal-to-noise ratio, N is very large. The analysis performed in this paper solves this problem. The analysis is based on the probability distribution of the NMF to give an exact closed-form (tabulized) expression for the false alarm probability, and a relatively accurate approximation for the probability of detection, both for the large N case. These approximations are found accurate in numerical simulations. Results from an experiment conducted in the Mediterranean sea at the depth of roughly 1000 m validate the analysis.