Robust detection: A geometric approach

The detection of signals distorted by systems/channels with unknown and possibly time-varying characteristics has been a problem of critical importance in many signal processing applications. A major problem in this area is the lack of robustness of detectors to fluctuations in the system/channel parameters. In this thesis, we propose a framework for the design of robust and simple detection algorithms using wavelet and multiresolution analysis tools. Our approach is based on a geometric interpretation of the detection problem.; We first present a geometric framework for representing the signal set {dollar}{lcub}cal S{rcub}{dollar} (which is the set of all possible noise free distorted signals) by a representation subspace {dollar}{lcub}cal G{rcub}{dollar}. We use the energy of the orthogonal projection of the received signal on the representation subspace {dollar}{lcub}cal G{rcub}{dollar} as the test statistic. The representation subspace {dollar}{lcub}cal G{rcub}{dollar} is designed to be close to the signal set {dollar}{lcub}cal S{rcub}{dollar} in some sense and to be such that the orthogonal projection on this representation subspace {dollar}{lcub}cal G{rcub}{dollar} is easy to compute.; We discuss how to apply the geometric framework to robust multipath signal detection and synthetic aperture radar (SAR)/inverse synthetic aperture radar (ISAR) image focusing. We use multiresolution subspaces as the representation subspaces because of their time-frequency localization properties and their computational efficiency. We propose three different similarity measures--the gap metric, the deflection, and the modified deflection--to measure the closeness between the signal set {dollar}{lcub}cal S{rcub}{dollar} and the representation subspace {dollar}{lcub}cal G{rcub}{dollar}. We present theorems that provide explicit formulas for the gap metric, the deflection, and the modified deflection. We describe algorithms that design the multiresolution subspaces to minimize these similarity measures. These algorithms utilize the parameterization of the compactly supported orthonormal scaling functions. We present simulation results that show significant improvements in performance over alternative detectors.