Exploitation of Geometry in Signal Processing and Sensing

Abstract

Developments in hardware and massive sensors or transducers have quickly shifted many disciplines and applications from a scarcity of data to a deluge of data, presenting new challenges in signal processing of high-dimensional and large-scale datasets. One key insight is that the signals of interest often possess certain geometry, such that the information hidden in the signals usually exhibits a much lower dimension than that of the signals themselves. New hardware and sensor developments also offer new degrees of freedom in signal acquisition through design of pilot or probing sequences that are tuned to the underlying geometry of systems, which is defined by the choice of sensing sequences that optimizes the objective function of the system.

The first half of this thesis presents algorithms take advantage of geometry in the form of sparsity and low rank representations to minimize complexity and increase system capability. Performance is evaluated both in theory and in experiments. Chapter 2 discusses compressive sensing and sparse signal processing using Orthogonal Matching Pursuit, with applications to asynchronous multi-user detection in random access channels and diagnostic grade wireless ECG transmission and monitoring. Chapter 3 analyzes the sensitivity of compressive sensing to basis mismatch when the sparsity basis of the signal realized by the physics is differed from the one assumed in compressive sensing, and examines its implications for spectrum analysis and beamforming. Chapter 4 presents the Parallel Estimation and Tracking via Recursive Least Squares (PETRELS) algorithm for online estimation and tracking of a low-dimensional linear subspace from highly incomplete streaming data.

The second half of this thesis presents deterministic sensing sequences for active sensing and wireless communications. Chapter 5 presents coordination schemes of the transmission of a pair of Golay complementary waveforms to suppress the range sidelobes in a desired Doppler interval using the Prouhet-Thue-Morse sequence and its generalizations. Chapter 6 presents a family of minimum mean squared error (MMSE) optimal training sequences for channel state estimation in multi-user Multiple Input Multiple Output (MIMO)-Orthogonal Frequency Division Multiplexing (OFDM) systems.