Coherence Optimization and Best Complex Antipodal Spherical Codes

TL;DR

This paper introduces a new method for finding complex antipodal spherical codes with minimal coherence, improving upon existing results and offering faster approximations for time-sensitive applications.

Contribution

It extends existing methods to efficiently find Best Complex Antipodal Spherical Codes, achieving lower coherence values than previous approaches.

Findings

Numerically improved coherence values over prior methods

Proposed faster approximation for coherence optimization

Enhanced applicability in wireless communication and compressed sensing

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.