A classification of the harmonic frames up to unitary equivalence

Abstract Up to unitary equivalence, there are a finite number of tight frames of n vectors for C d which can be obtained as the orbit of a single vector under the unitary action of an abelian group G (for nonabelian groups there may be uncountably many). These so called harmonic frames (or geometrically uniform tight frames) have recently been used in applications including signal processing (where G is the cyclic group). In an effort to find optimal harmonic frames for such applications, we seek a simple way to describe the unitary equivalence classes of harmonic frames. By using Pontryagin duality, we show that all harmonic frames of n vectors for C d can be constructed from d-element subsets of G ( | G | = n ). We then show that in most, but not all cases, unitary equivalence preserves the group structure, and thus can be described in a simple way. This considerably reduces the complexity of determining whether harmonic frames are unitarily equivalent. We then give extensive examples, and make some steps towards a classification of all harmonic frames obtained from a cyclic group.

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