Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Geometric Properties of Grassmannian Frames for R2 and R3 | Hindawi Publishing Corporation EURASIP Journal on Applied Signal Processing Volume 2006 Article lD49850 Pages 1-17 DOI ASP 2006 49850 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 V 1 c Rd Rd is d-dimensional Euclidean space is characterized by the property that its span is Rd see 1 . The norm xh ofx e 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 . Definition 1. Let N d and let XN xk N 1 be a subset of Rd with each xk II 1. The maximum correlation of XN M Xn is defined as ACfiXf max I x Xi . 1 t d k i 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
