Two-channel signal processing in canonical coordinates

Abstract

Canonical coordinates provide an elegant framework for analyzing and solving many two-channel problems in signal processing, communications, radar and sonar, and sensor fusion. This dissertation addresses some of the existing issues in canonical correlation analysis of two-channel data, establishes a direct connection between canonical coordinates and certain two-channel signal processing problems, and exploits canonical correlation analysis to solve some real two-channel signal processing problems. More specifically, in this dissertation, connections between two-channel constrained least squares (CLS) problems and various canonical coordinate systems are established. It is shown that under certain sets of constraints a two-channel CLS problem will produce one of the important canonical coordinate systems, namely canonical coordinates, half-canonical coordinates, or programmable canonical correlation analysis (PCCA) coordinates. Further, a unified framework for building reduced-rank Wiener filters is developed. It is demonstrated that, depending on the objective of reduced-rank estimation, either canonical coordinates or half-canonical coordinates are optimal for building the reduced-rank Wiener filter. Simple algorithms, called alternating power methods, are also developed that allow for both recursive and real-time computation of canonical coordinates, half-canonical coordinates, and reduced-rank Wiener filters. The developed algorithms may be viewed as two-step decompositions of the standard power method, as they solve a coupled generalized eigenvalue problem through power iterations. In addition, a network structure, with lateral connections that implement a deflation process, is developed for recursive extraction of canonical coordinates. This dissertation also addresses the empirical canonical coordinate decompositions of two-channel data, where the channel covariances are estimated from a limited number of data samples and are not necessarily full-rank. It clarifies how the number of samples (sample support) drawn from two-channel data, and the ranks of the data matrices, affect the algebraic and geometric properties of empirical canonical correlations and coordinates. It is shown that empirical canonical correlations are maximal invariants that measure the cosines of the principal angles between the row spaces of the data matrices for the two data channels. When the sample support is smaller than the sum of the ranks of the two data matrices, some of the empirical canonical correlations become one, regardless of the two-channel model that generates the samples. In such cases, the empirical canonical correlations may not be used as estimates of correlation between random variables. This has interesting implications for canonical correlation analysis of nonlinear functions of two-channel data, where the aim is to capture coherence between the two channels by estimating correlation between their high-order attributes. This will be possible only if the sample support is greater than the sum of the ranks of the nonlinearly mapped data matrices. In these cases, however, the so-called kernel formulations of canonical correlation analysis are computationally disadvantageous with respect to the direct formulations. Finally, canonical correlation analysis is employed to develop a multi-aspect feature extraction method for underwater target classification. The developed feature extraction method exploits the linear dependence or coherence between two consecutive sonar returns. This is accomplished by extracting the dominant canonical correlations between the two sonar returns and using them as features for classifying mine-like objects from non-mine-like objects. The experimental results on a wideband acoustic backscattered data set, which contains sonar returns from several mine-like and non-mine-like objects in two different environmental conditions, show that canonical correlation features can offer good discrimination between mine-like and non-mine-like objects. Further, the results show that in a fixed bottom condition, canonical correlation features do not vary with changes in aspect angle.