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

The rich information in the power spectral density (PSD) matrix of a received signal can be extracted in different ways for various purposes in signal processing. Care, however, must be taken when distances between PSD matrices are considered because these matrices form a Riemannian manifold in the signal space. In this paper, two Riemannian distances (RD) are introduced for the measurement of distances between PSD matrices on the manifold. The principle by which the geodesics on the manifold can be lifted to a Euclidean subspace isometric with the tangent space of the manifold is also explained and emphasized. This leads to the concept that any optimization involving the RD on the manifold can be equivalently performed in this Euclidean subspace. The application of this principle is illustrated by the development of iterative algorithms to find the Riemannian means and Riemannian medians according to the two RD. These distance measures are then applied to the detection of narrow-band sonar signals. Exploration of the translation of measure reference as well as the application of optimum weighting show substantial improvement in detection performance over classical detection method.