  Many emerging applications involve sparse signals, and their processing is a
subject of active research. We desire a large class of sensing matrices which
allow the user to discern important properties of the measured sparse signal.
Of particular interest are matrices with the restricted isometry property
(RIP). RIP matrices are known to enable efficient and stable reconstruction of
sufficiently sparse signals, but the deterministic construction of such
matrices has proven very difficult. In this thesis, we discuss this matrix
design problem in the context of a growing field of study known as frame
theory. In the first two chapters, we build large families of equiangular tight
frames and full spark frames, and we discuss their relationship to RIP matrices
as well as their utility in other aspects of sparse signal processing. In
Chapter 3, we pave the road to deterministic RIP matrices, evaluating various
techniques to demonstrate RIP, and making interesting connections with graph
theory and number theory. We conclude in Chapter 4 with a coherence-based
alternative to RIP, which provides near-optimal probabilistic guarantees for
various aspects of sparse signal processing while at the same time admitting a
whole host of deterministic constructions.
