Vector sets with optimal coherence according to the Welch bound cannot exist for all pairs of dimension and cardinality. If such an optimal vector set exists, it is an equiangular tight frame and represents the solution to a Grassmannian line packing problem. Best Complex Antipodal Spherical Codes (BCASCs) are the best vector sets with respect to the coherence. By extending methods used to find best spherical codes in the real-valued Euclidean space, the proposed approach aims to find BCASCs, and thereby, a complex-valued vector set with minimal coherence. There are many applications demanding vector sets with low coherence. Examples are not limited to several techniques in wireless communication or to the field of compressed sensing. Within this contribution, existing analytical and numerical approaches for coherence optimization of complex-valued vector spaces are summarized and compared to the proposed approach. The numerically obtained coherence values improve previously reported results. The drawback of increased computational effort is addressed and a faster approximation is proposed which may be an alternative for time critical cases.

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In sparse recovery, the unique sparsest solution to an under-determined system of linear equations is of main interest. This scheme is commonly proposed to be applied to signal acquisition. In most cases, the signals are not sparse themselves, and therefore, they need to be sparsely represented with the help of a so-called dictionary being specific to the corresponding signal family. The dictionaries cannot be used for optimization of the resulting under-determined system because they are fixed by the given signal family. However, the measurement matrix is available for optimization and can be adapted to the dictionary. Multiple properties of the resulting linear system have been proposed which can be used as objective functions for optimization. This paper discusses two of them which are both related to the coherence of vectors. One property aims for having incoherent measurements, while the other aims for insuring the successful reconstruction. In the following, the influences of both criteria are compared with different reconstruction approaches.

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