In engineering, many signals are mixture of diverse, physical phenomena. Various mathematical transforms are good at describing different physical phenomena. Therefore, redundant systems might do better job at describing signals than orthonormal bases. Besides, redundant signal representations are more robust to noise, quantization and losses than orthonormal decompositions. This dissertation is dedicated to resolve two closely related mathematical problems regarding redundant systems. Firstly, a signal may have more than one representations in a redundant system. In many applications, the sparsest signal representation is desirable. Sufficient conditions are studied for a signal representation to be the uniquely sparsest. Also, it is proved that if a signal representation is sparse enough, it can be obtained by both orthogonal greedy algorithm and basis pursuit exactly. The conditions are given in terms of dictionary coherence and certainty respectively. Secondly, the construction of two classes of optimal redundant systems are discussed. Based on the research into equiangular tight frames, the concept of absolutely equiangular tight frame is introduced here. It is discovered that the absolute equiangularity simultaneously implies many good properties such as unique existence, minimal coherence, maximal robustness, tightness and self-offset . More important, an efficient, fast construction of absolutely equiangular tight frames using Hadamard matrices is proposed. As another class of optimal redundant systems, the construction of maximally robust tight frames is also studied.
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