# A classification of the harmonic frames up to unitary equivalence

## Tuan Chien and Shayne Waldron

## 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
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

## Availability: