We consider the problem of narrow-band signal detection in a passive sonar environment. The collected signals are passed to a fast Fourier Transform (FFT) delay-sum beamformer. In classical signal detection, the output of the FFT spectrum analyser in each frequency bin is the signal power spectrum which is used as the signal feature for detection. The observed signal power is compared to a locally estimated mean noise power and a log likelihood ratio test (LLRT) can then be established. In this thesis, we propose the use of 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 are structurally constrained and therefore form a manifold in the signal space. Thus, to find the distance between two matrices, the measurement must be carried out using Riemannian distance (RD) along the tangent of the manifold, instead of using the common Euclidean distance (ED). In this thesis, we develop methods for measuring the Frechet mean of noise PSD matrices using the RD and weighted RD. Further, we develop an optimum weighting matrix for use in signal detection by RD so as to further enhance the detection performance. These concepts and properties 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. / Thesis / Master of Applied Science (MASc)