Subsampling in Information Theory and Data Processing a Dissertation Submitted to the Department of Electrical Engineering and the Committee on Graduate Studies of Stanford University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

An ubiquitous challenge in modern data and signal acquisition arises from the evergrowing size of the object under study. Hardware and power limitations often preclude sampling with the desired rate and precision, which motivates the exploitation of signal and/or channel structures in order to enable reduced-rate sampling while preserving information integrity. This thesis is devoted to understanding the fundamental interplay between the underlying signal structures and the data acquisition paradigms, as well as developing efficient and provably effective algorithms for data reconstruction. The main contributions of this thesis are as follows. • We investigate the effect of sub-Nyquist sampling upon the capacity of a continuous-time channel. We start by deriving the sub-Nyquist sampled channel capacity under periodic sampling systems that subsume three canonical sampling structures, and then characterize the fundamental upper limit on the capacity achievable by general time-preserving sub-Nyquist sampling methods. Our findings indicate that the optimal sampling structures extract out the set of frequencies that exhibits the highest signal-to-noise ratio and is aliassuppressing. In addition, we illuminate an intriguing connection between sampled channels and MIMO channels, as well as a new connection between sampled capacity and MMSE. • We study the universal sub-Nyquist design when the sampler is designed to operate independent of instantaneous channel realizations, under a sparse multiband channel model. We evaluate the sampler design based on the capacity loss due to channel-independent sub-Nyquist sampling, and characterize the minimax