Mean and Median of PSD Matrices on a Riemannian Manifold: Application to Detection of Narrowband Sonar Signals

Abstract

We investigate the employment of power spectral density (PSD) matrix, which is constructed by the received signals in a multi-sensor system and contains additional cross-correlation information, as a feature in signal processing. Since the PSD matrices are structurally constrained, they form a manifold in signal space. The commonly used Euclidean distance (ED) to measure the distance between two such matrices are not informative or accurate. Riemannian distances (RD), which measure distances along the surface of the manifold, should be employed to give more meaningful measurements. Furthermore, the principle that the geodesics on the manifold can be lifted to an isometric Euclidean space is emphasized since any processing involving the optimization of the geodesics can be lifted to the isometric Euclidean space and be carried out in terms of the equivalent Euclidean metric. Application of this principle is illustrated by having e cient algorithms locating the mean and median of the PSD matrices on the manifold developed. These concepts are then applied to the detection of narrow-band sonar signals from which the decision rule is set up by translating the measure reference. In order to further enhance the detecton performance, an algorithm is developed for obtaining the optimum weighting matrix which can better classify the signal from noise. The experimental results show that the performance by the PSD matrices being the detection feature is very encouraging.