A Physical Interpretation of Tight Frames

We characterize the existence of finite tight frames whose frame elements are of predetermined length. In particular, we derive a “fundamental inequality” which completely characterizes those sequences which arise as the lengths of a tight frame’s elements. Furthermore, using concepts from classical physics, we show that this characterization has an intuitive physical interpretation.

[1]  I. Daubechies,et al.  PAINLESS NONORTHOGONAL EXPANSIONS , 1986 .

[2]  Vivek K Goyal.  Quantized Overcomplete Expansions : Analysis , Synthesis and Algorithms , 1995 .

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

[4]  Vivek K Goyal,et al.  Quantized Frame Expansions with Erasures , 2001 .

[5]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[6]  Shayne Waldron,et al.  Generalized Welch bound equality sequences are tight fram , 2003, IEEE Trans. Inf. Theory.

[7]  V. Paulsen,et al.  Optimal frames for erasures , 2004 .

[8]  A. Horn.  Doubly Stochastic Matrices and the Diagonal of a Rotation Matrix , 1954 .

[9]  Nate Strawn,et al.  Manifold structure of spaces of spherical tight frames , 2003, math/0307367.

[10]  Thomas L. Marzetta,et al.  Systematic design of unitary space-time constellations , 2000, IEEE Trans. Inf. Theory.

[11]  Georg Zimmermann,et al.  Normalized Tight Frames in Finite Dimensions , 2001 .

[12]  Helmut Bölcskei,et al.  Frame-theoretic analysis of oversampled filter banks , 1998, IEEE Trans. Signal Process..

[13]  Keri Kornelson,et al.  Ellipsoidal tight frames and projection decompositions of operators , 2003 .

[14]  Vivek K. Goyal,et al.  Filter bank frame expansions with erasures , 2002, IEEE Trans. Inf. Theory.

[15]  Zoran Cvetkovic.  Resilience properties of redundant expansions under additive noise and quantization , 2003, IEEE Trans. Inf. Theory.

[16]  Michael Unser,et al.  Texture classification and segmentation using wavelet frames , 1995, IEEE Trans. Image Process..

[17]  T. Strohmer.  Finite-and Infinite-Dimensional Models for Oversampled Filter Banks , 2001 .

[18]  Michael Unser,et al.  Wavelet Applications in Signal and Image Processing II: 27-29 July 1994, San Diego, California , 1994 .

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

[20]  John J. Benedetto,et al.  Periodic Wavelet Transforms and Periodicity Detection , 2002, SIAM J. Appl. Math..

[21]  Kurt Jetter,et al.  Recent Progress in Multivariate Approximation , 2001 .

[22]  John J. Benedetto,et al.  Wavelet analysis of spectrogram seizure chirps , 1995, Optics + Photonics.

[23]  I. Daubechies.  Ten Lectures on Wavelets , 1992 .

[24]  R. Duffin,et al.  A class of nonharmonic Fourier series , 1952 .

[25]  John J. Benedetto,et al.  Wavelet periodicity detection algorithms , 1998, Optics & Photonics.

[26]  Venkat Anantharam,et al.  Optimal sequences and sum capacity of synchronous CDMA systems , 1999, IEEE Trans. Inf. Theory.

[27]  Martin Vetterli,et al.  Oversampled filter banks , 1998, IEEE Trans. Signal Process..

[28]  John J. Benedetto,et al.  Finite Normalized Tight Frames , 2003, Adv. Comput. Math..

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

[30]  Dennis Gabor,et al.  Theory of communication , 1946 .