luozq@mcmaster.ca, and wongkm@mcmaster.ca Communicated by P. P. VaidyanathanReceived May 18, 2000; revised May 16, 2001 In this paper we characterize all totally interpolating biorthogonal finite impulse response (FIR) multifilter banks of multiplicity two, and provide a design framework for corresponding compactly supported multiwavelet systems with high approximation order. In these systems, each component of the analysis and synthesis portions possesses the interpolating property. The design framework is based on scalar filter banks, and examples with approximation order two and three are provided. We show that our multiwavelet systems preserve almost all of the desirable properties of the generalized interpolating scalar wavelet systems, including the dyadic-rational nature of the filter coefficients, equality of the flatness degree of the low-pass filters and the approximation order of the corresponding functions, and equality between the uniform samples of a signal and its projection coefficients for a given scale. This last property allows us to avoid the cumbersome prefiltering associated with standard multiwavelet systems. We also show that there are no symmetric totally interpolating biorthogonal multifilter banks of multiplicity two. Finally, we point out that our design framework incorporates a simple relationship between the multiscaling functions and multiwavelets that substantially simplifies the implementation of the system. 

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.