In compressed sensing theory, small dictionary coherence is desirable in both obtaining and recovering sparse signal representations in a redundant system. It has been known that equiangular tight frames are optimal redundant systems with minimal coherence in R n .On the other hand, maximal robustness of redundant system is also desirable in applications such as signal transmission. However, equiangularity does not generate maximal robustness. In the paper, it is proved that one class of equiangular tight frames, the absolutely equiangular tight frames, have both minimal coherence and maximal robustness. It also is proved that the absolute equiangularity simultaneously assumes many good mathematical properties such as tightness, unique existence, minimal coherence, maximal robustness and self-offset.A frame Φ = {ϕ i } m i=1 for R n is said to be absolutely equiangular, if there exists µ ∈ R such that ϕ i , ϕ j = µ, for all 1 ≤ i = j ≤ m. Obviously, an absolutely equiangular frame has minimal coherence, since it is already equiangular. A frame Φ = {ϕ i } m i=1 in R n is said to be maximally robust, if any n elements of Φ are linearly independent. Here are some related results:be an absolutely equiangular tight frame for R n . Then, any n elements of Φ are linearly independent.Theorem 2. Let c be a representation of a signal x ∈ R n in a maximally robustthen c is the unique sparsest representation of x. And, it can be exactly obtained by both orthogonal greedy algorithm and basis pursuit.in R n is absolutely equiangular, then it is also tight and self-offset, i.e., m i=1 ϕ i = 0.(2)In R n , the absolutely equiangular tight frames can be constructed either by unitary decomposition of Gram matrices or from given Hadamard matrices.
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