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Equal-Norm Tight Frames with Erasures

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Abstract

Equal-norm tight frames have been shown to be useful for robust data transmission. The losses in the network are modeled as erasures of transmitted frame coefficients. We give the first systematic study of the general class of equal-norm tight frames and their properties. We search for efficient constructions of such frames. We show that the only equal-norm tight frames with the group structure and one or two generators are the generalized harmonic frames. Finally, we give a complete classification of frames in terms of their robustness to erasures.

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References

  1. N.I. Akhiezer and I.M. Glazman, Theory of Linear Operators in Hilbert Spaces, Vol. 1 (Frederick Ungar, New York, 1966).

    Google Scholar 

  2. J.J. Benedetto and D. Colella, Wavelet analysis of spectogram seizure chips, in: Proc. of SPIE Conf. on Wavelet Application in Signal and Image Processing, San Diego, CA, July 1995, pp. 512–521.

  3. J.J. Benedetto and G.E. Pfander, Wavelet periodicity detection algorithms, in: Proc. of SPIE Conf. on Wavelet Application in Signal and Image Processing, San Diego, CA, July 1998, pp. 48–55.

  4. H. Bolcskei and Y. Eldar, Geometrically uniform frames, Preprint (2001).

  5. H. Bolcskei, F. Hlawatsch and H.G. Feichtinger, Frame-theoretic analysis of oversampled filter banks, IEEE Trans. Signal Process. 46(12) (1998) 3256–3269.

    Google Scholar 

  6. P.G. Casazza, The art of frame theory, Taiwanese J. Math. 4(2) (2000) 129–202.

    Google Scholar 

  7. P.G. Casazza, Modern tools forWeyl-Heisenberg (Gabor) frame theory, Adv. in Imag. Electron. Phys. 115 (2000) 1–127.

    Google Scholar 

  8. Z. Cvetkovi´c and M. Vetterli, Oversampled filter banks, Signal Process. 46(5) (1998) 1245–1255.

    Google Scholar 

  9. I. Daubechies, The wavelet transform, time-frequency localization and signal anaysis, IEEE Trans. Inform. Theory 36(5) (1990) 961–1005.

    Google Scholar 

  10. I. Daubechies, Ten Lectures on Wavelets (SIAM, Philadelphia, PA, 1992).

    Google Scholar 

  11. R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952) 341–366.

    Google Scholar 

  12. Y. Eldar and G.D. Forney, Jr., Optimal tight frames and quantum measurement, Preprint (2001).

  13. G.J. Foschini and M.J. Cans, On the limits of wireless communications in a fading environment when using multiple antennas, Wireless Personal Comm. 6(3) (1998) 311–335.

    Google Scholar 

  14. V.K Goyal, Single and Multiple Description Transform Coding with Bases and Frames (SIAM, Philadelphia, PA, 2002).

    Google Scholar 

  15. V.K Goyal and J. Kovačević, Optimal multiple description transform coding of Gaussian vectors, in: Proc. of Data Compr. Conf., Snowbird, UT, March 1998, pp. 388–397.

  16. V.K Goyal, J. Kovačević and J.A. Kelner, Quantized frame expansions with erasures, J. Appl. Comput. Harmon. Anal. 10(3) (2001) 203–233.

    Google Scholar 

  17. V.K Goyal, J. Kovačević and M. Vetterli, Multiple description transform coding: Robustness to erasures using tight frame expansions, in: Proc. of IEEE Internat. Symp. on Inform. Theory, Cambridge, MA, August 1998.

  18. V.K Goyal, J. Kovačević and M. Vetterli, Quantized frame expansions as source-channel codes for erasure channels, in: Proc. of Data Compr. Conf., Snowbird, UT, March 1999, pp. 388–397.

  19. V.K Goyal, M. Vetterli and N.T. Thao, Quantized overcomplete expansions in ℝN: Analysis, synthesis, and algorithms, IEEE Trans. Inform. Theory 44(1) (1998) 16–31.

    Google Scholar 

  20. D. Han and D.R. Larson, Frames, Bases and Group Representations, Memoirs of American Mathematical Society, Vol. 697 (Amer. Math. Soc., Providence, RI, 2000).

    Google Scholar 

  21. B. Hassibi, B. Hochwald, A. Shokrollahi and W. Sweldens, Representation theory far high-rate multiple-antenna code design, IEEE Trans. Inform. Theory 47(6) (2001) 2355–2367.

    Google Scholar 

  22. C. Heil and D. Walnut, Continuous and discrete wavelet transforms, SIAM Rev. 31 (1989) 628–666.

    Google Scholar 

  23. B. Hochwald, T. Marzetta, T. Richardson, W. Sweldens and R. Urbanke, Systematic design of unitary space-time constellations, IEEE Trans. Inform. Theory (2000) submitted.

  24. J.R. Holub, Pre-frame operators, Besselian frames and near-Riesz bases in Hilbert spaces, Proc. Amer. Math. Soc. 122(3) (1994) 779–785.

    Google Scholar 

  25. E. Šoljanin, Frames and quantum information theory, Preprint (2000).

  26. T. Strohmer, Modern Sampling Theory: Mathematics and Applications, chapter Finite and infinitedimensional models for oversampled filter banks (Birkhäuser, Boston, 2000) pp. 297–320.

    Google Scholar 

  27. I.E. Telatar, Capacity of multiantenna Gaussian channels, European Trans. Telecomm. 10(6) (1999) 585–595.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri-Columbia, Columbia, MO, 65211, USA

    Peter G. Casazza

  2. Mathematics of Communication Research, Bell Labs, Lucent Technologies, 600 Mountain Avenue, Murray Hill, NJ, 07974, USA

    Jelena Kovačević

Authors
  1. Peter G. Casazza
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  2. Jelena Kovačević
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Casazza, P.G., Kovačević, J. Equal-Norm Tight Frames with Erasures. Advances in Computational Mathematics 18, 387–430 (2003). https://doi.org/10.1023/A:1021349819855

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