Geometric Properties of Grassmannian Frames for and
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Geometric Properties of Grassmannian Frames for R2 and R3 John J. Benedetto and Joseph D. Kolesar Department of Mathematics, University of Maryland, College Park, MD 20742, USA Received 16 September 2004; Revised 19 January 2005; Accepted 21 January 2005 Grassmannian frames are frames satisfying a min-max correlation criterion. We translate a geometrically intuitive approach for two- and three-dimensional Euclidean space (R2 and R3 ) into a new analytic method which is used to classify many Grassmannian frames in this setting. The method and associated algorithm decrease the maximum frame correlation, and hence give rise to the construction of specific examples of Grassmannian frames. Many of the results are known by other techniques, and even more generally, so that this paper can be viewed as tutorial. However, our analytic method is presented with the goal of developing it to address unresovled problems in d-dimensional Hilbert spaces which serve as a setting for spherical codes, erasure channel modeling, and other aspects of communications theory. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1.
INTRODUCTION
A finite frame {xk }Nk=1 ⊆ Rd , Rd is d-dimensional Euclidean space, is characterized by the property that its span is Rd , see [1]. The norm x of x ∈ Rd is the usual Euclidean distance. Given a finite frame for Rd with N elements, we would like to measure the correlation between frame elements and in particular to decide when the correlation is small. We consider the following metric which is similar to an ∞ norm [2]. XdN
{x k }N k=1
Definition 1. Let N ≥ d and let = be a subset of Rd with each xk = 1. The maximum correlation of XdN , M∞ (XdN ), is defined as
M∞ XdN = max xk , xl .
(1)
k=l
Note that because we consider the absolute value of the inner product rather than just the inner product, if the angle between a pair of vectors is closer to 90◦ , then the pair is less correlated, while if the angle is closer to 0◦ or 180◦ , then the pair is more correlated. Thus, we are measuring the smaller angle between the lines (one-dimensional subspaces) spanned by these vectors. We could instead consider an 1 -, 2 -, or p -type norm to measure correlation, that is,
M p XdN
=
xk , xl p
1/ p
,
(2)
k=l
or even weighted versions of (2). (See [3] for a discussion of the case p = 1, 2.) Fix d and N with N ≥ d. Our goal is to construct N-element unit-norm frames, XdN , with smallest maximum
correlation, M∞ (XdN ), that is, unit-norm frames that are maximally spread apart. To this end, we make the following definition. Definition 2. Let N ≥ d. A sequence UdN = {uk }Nk=1 ⊆ Rd of unit-norm vectors is an (N, d)-Grassmannian frame if it is a frame and if
M∞ UdN = inf M∞ XdN
,
(3)
where the infimum is taken over all unit-norm, N-element frames for Rd . A compactness argument shows that Grassmannian frames exist (see Appendix A), but constructing Grassmannian frames is challenging [4–6]. As is described in [2], the conce
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