Frames, group codes, and subgroups of (Z/pZ)×

The problem of designing low coherence matrices and low-correlation frames arises in a variety of fields, including compressed sensing, MIMO communications and quantum measurements. The challenge is that one must control the (n2) pairwise inner products of the columns of the matrix. In this paper, we follow the group code approach of David Slepian [1], which constructs frames using unitary group representations and which in general reduces the number of distinct inner products to n-1. When n is a prime p, we present a carefully chosen representation which reduces the number of distinct inner products further to n-1/m, where m is the number of rows in the matrix. The resulting frames have superior performance to many earlier frame constructions and, in some cases, yield frames with optimally low coherence. We further expand a connection between frames and difference sets noted first in [2] to find bounds on the coherence when n-1/m = 2 and 3.

[1]  Georgios B. Giannakis,et al.  Achieving the Welch bound with difference sets , 2005, IEEE Transactions on Information Theory.

[2]  Robert W. Heath,et al.  Designing structured tight frames via an alternating projection method , 2005, IEEE Transactions on Information Theory.

[3]  Robert W. Heath,et al.  Space-time signaling and frame theory , 2001, 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221).

[4]  T. Strohmer.  Approximation of Dual Gabor Frames, Window Decay, and Wireless Communications , 2000, math/0010244.

[5]  A. J. Scott.  Tight informationally complete quantum measurements , 2006, quant-ph/0604049.

[6]  Mátyás A. Sustik,et al.  On the existence of equiangular tight frames , 2007 .

[7]  Peter G. Casazza,et al.  Duality Principles in Frame Theory , 2004 .

[8]  Robert W. Heath,et al.  Linear dispersion codes for MIMO systems based on frame theory , 2002, IEEE Trans. Signal Process..

[9]  David S. Slepian,et al.  Group codes for the Gaussian channel (Abstr.) , 1968, IEEE Trans. Inf. Theory.

[10]  Hugh F. Jones.  Groups, representations, and physics , 1990 .

[11]  Joel A. Tropp,et al.  Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit , 2007, IEEE Transactions on Information Theory.

[12]  Michael Elad,et al.  Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[13]  S. Waldron,et al.  Tight Frames and Their Symmetries , 2004 .

[14]  J. Tropp,et al.  SIGNAL RECOVERY FROM PARTIAL INFORMATION VIA ORTHOGONAL MATCHING PURSUIT , 2005 .

[15]  Xiaoming Huo,et al.  Uncertainty principles and ideal atomic decomposition , 2001, IEEE Trans. Inf. Theory.

[16]  Yonina C. Eldar,et al.  Optimal tight frames and quantum measurement , 2002, IEEE Trans. Inf. Theory.

[17]  J. Seidel,et al.  Spherical codes and designs , 1977 .

[18]  Peter G. Casazza,et al.  Equal-Norm Tight Frames with Erasures , 2003, Adv. Comput. Math..

[19]  Emmanuel J. Candès,et al.  Decoding by linear programming , 2005, IEEE Transactions on Information Theory.

[20]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[21]  Thomas Strohmer,et al.  GRASSMANNIAN FRAMES WITH APPLICATIONS TO CODING AND COMMUNICATION , 2003, math/0301135.

[22]  Michael B. Wakin,et al.  Analysis of Orthogonal Matching Pursuit Using the Restricted Isometry Property , 2009, IEEE Transactions on Information Theory.

[23]  J. Kovacevic,et al.  Life Beyond Bases: The Advent of Frames (Part II) , 2007, IEEE Signal Processing Magazine.

[24]  D. Slepian.  Group codes for the Gaussian channel , 1968 .

[25]  Jelena Kovačević,et al.  Life Beyond Bases: The Advent of Frames , 2006 .

[26]  E. Candès.  The restricted isometry property and its implications for compressed sensing , 2008 .