Equiangular frames and generalizations of the Welch bound to dual pairs of frames

Abstract

The purpose of this paper is twofold. First, we determine the lower bound for the maximum coherence between a pair of dual frames in and state conditions under which the lower bound is attained. It is shown that the existence of frames and duals that attain the lower bound is related to the existence of equiangular tight frames (ETFs). Second, motivated by the scarcity of ETFs (which by default have dual ETFs), we examine the more general question of existence of equiangular frames that have equiangular duals. For the case where an equiangular dual cannot be found we provide conditions under which the number of angles among vectors in the canonical dual frame is small.