Coherence Optimization and Best Complex Antipodal Spherical Codes

Abstract

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.

A Coherence-Based Algorithm for Optimizing Rank-1 Grassmannian Codebooks

“…The value of this approach is shown by the fact that the BCASC algorithm achieves lower coherence values for rank-1 codebooks than previous approaches [5].…”

“…For unit-length vectors, the denominator in (1) is 1 and may be suppressed. The lower bound on the achievable coherence is given by [5] …”

“…In order to obtain a smooth approximation to the max operator a series of sub-problems given by [4], [5] …”

“…Minimising the maximum coherence of a codebook is equivalent to maximising the minimum distance between the lines represented by the codebook [5]. The approach in [4] uses optimisation algorithms which are able to enforce the orthonormality constraint in higherrank codebooks.…”

“…By considering rank-1 codebooks only, the BCASC algorithm is able to use a simpler optimisation technique which maximises the minimum Euclidean distance between codewords [5]. The Euclidean distance between the codewords alone does not lie on the Grassmannian manifold since it measures distances between codewords and not the lines spanned by the codewords.…”