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Coherence Optimization and Best Complex Antipodal Spherical Codes
Vector sets with optimal coherence according to the Welch bound cannot exist
for all pairs of dimension and cardinality. If such an optimal vector set
exists, it is an equiangular tight frame and represents the solution to a
Grassmannian line packing problem. Best Complex Antipodal Spherical Codes
(BCASCs) are the best vector sets with respect to the coherence. By extending
methods used to find best spherical codes in the real-valued Euclidean space,
the proposed approach aims to find BCASCs, and thereby, a complex-valued vector
set with minimal coherence. There are many applications demanding vector sets
with low coherence. Examples are not limited to several techniques in wireless
communication or to the field of compressed sensing. Within this contribution,
existing analytical and numerical approaches for coherence optimization of
complex-valued vector spaces are summarized and compared to the proposed
approach. The numerically obtained coherence values improve previously reported
results. The drawback of increased computational effort is addressed and a
faster approximation is proposed which may be an alternative for time critical
cases
Harmonic equiangular tight frames comprised of regular simplices
An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a
Euclidean space whose coherence achieves equality in the Welch bound, and thus
yields an optimal packing in a projective space. A regular simplex is a simple
type of ETF in which the number of vectors is one more than the dimension of
the underlying space. More sophisticated examples include harmonic ETFs which
equate to difference sets in finite abelian groups. Recently, it was shown that
some harmonic ETFs are comprised of regular simplices. In this paper, we
continue the investigation into these special harmonic ETFs. We begin by
characterizing when the subspaces that are spanned by the ETF's regular
simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of
optimal packing in a Grassmannian space. We shall see that every difference set
that produces an EITFF in this way also yields a complex circulant conference
matrix. Next, we consider a subclass of these difference sets that can be
factored in terms of a smaller difference set and a relative difference set. It
turns out that these relative difference sets lend themselves to a second,
related and yet distinct, construction of complex circulant conference
matrices. Finally, we provide explicit infinite families of ETFs to which this
theory applies
Geometry of the Welch Bounds
A geometric perspective involving Grammian and frame operators is used to
derive the entire family of Welch bounds. This perspective unifies a number of
observations that have been made regarding tightness of the bounds and their
connections to symmetric k-tensors, tight frames, homogeneous polynomials, and
t-designs. In particular. a connection has been drawn between sampling of
homogeneous polynomials and frames of symmetric k-tensors. It is also shown
that tightness of the bounds requires tight frames. The lack of tight frames in
symmetric k-tensors in many cases, however, leads to consideration of sets that
come as close as possible to attaining the bounds. The geometric derivation is
then extended in the setting of generalized or continuous frames. The Welch
bounds for finite sets and countably infinite sets become special cases of this
general setting.Comment: changes from previous version include: correction of typos,
additional references added, new Example 3.
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