A classification of the harmonic frames up to unitary equivalence

Tuan Chien and Shayne Waldron


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 signal processing and quantum information theory (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.

Keywords: signal processing, information theory, finite abelian groups, character theory, roots of unity, Pontryagin duality, group characters, group frames, Gramian matrices, tight frames, harmonic frames, geometrically uniform frames, automorphism groups

Math Review Classification: Primary 11Z05, 20C15, 42C15, 94A12; Secondary 11R18, 20F28, 65T60, 94A11, 94A15

Length: 18 pages

Last Updated: 24 December 2009