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: