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 efficient 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.iv

We consider the problem of narrow-band signal detection in a passive sonar environment. The classical method employs a fast Fourier Transform (FFT) delay-sum beamformer in which the feature used in detection is the output of the FFT spectrum analyser in each frequency bin. This is compared to a locally estimated mean noise power to establish a likelihood ratio test (LRT). In this paper, we suggest to use the power spectral density (PSD) matrix of the spectrum analyser output as the feature for detection due to the additional cross-correlation information contained in such matrices. However, PSD matrices have structural constraints and describe a manifold in the signal space. Thus, instead of the widely used Euclidean distance (ED), we must use the Riemannian distance (RD) on the manifold for measuring the similarity between such features. Here, we develop methods for measuring the Fréchet mean of noise PSD matrices and optimum weighting matrices for measuring similarity of noise and signal PSD matrices. These are then used to develop a decision rule for the detection of narrow-band sonar signals using PSD matrices. The results yielded by the new detection method are very encouraging.

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