Gabor frames, unimodularity, and window decay

  • Helmut Bölcskei
  • J. E. M. Janssen
Article

Abstract

We study time-continuous Gabor frame generating window functions g satisfying decay properties in time and/or frequency with particular emphasis on rational time-frequency lattices. Specifically, we show under what conditions these decay properties of g are inherited by its minimal dual γ0 and by generalized duals γ. We consider compactly supported, exponentially decaying, and faster than exponentially decaying (i.e., decay like |g(t)|≤Ce−α|t| 1/α for some 1/2≤α<1) window functions. Particularly, we find that g and γ0 have better than exponential decay in both domains if and only if the associated Zibulski-Zeevi matrix is unimodular, i.e., its determinant is a constant. In the case of integer oversampling, unimodularity of the Zibulski-Zeevi matrix is equivalent to tightness of the underlying Gabor frame. For arbitrary oversampling, we furthermore consider tight Gabor frames canonically associated to window functions g satisfying certain decay properties. Here, we show under what conditions and to what extent the canonically associated tight frame inherits decay properties of g. Our proofs rely on the Zak transform, on the Zibulski-Zeevi representation of the Gabor frame operator, on a result by Jaffard, on a functional calculus for Gabor frame operators, on results from the theory of entire functions, and on the theory of polynomial matrices.

Math Subject Classifications

42C15 94A12 

Key words and Phrases

Gabor frame Zak transform entire function polynomial matrix unimodularity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Bastiaans, M.J. (1980). Gabor's expansion of a signal in Gaussian elementary signals,Proc. IEEE,68(1), 538–539, April.CrossRefGoogle Scholar
  2. [2]
    Benedetto, J.J. and Walnut, D.F. (1994). Gabor frames for L2 and related spaces. In Benedetto, J.J. and Frazier, M.W., Eds.,Wavelets: Mathematics and Applications, CRC Press, Boca Raton, FL, 97–162.Google Scholar
  3. [3]
    Boas, R.P. (1954).Entire Functions, Academic Press, New York.MATHGoogle Scholar
  4. [4]
    Bölcskei, H. (1999). A necessary and sufficient condition for dual Weyl—Heisenberg frames to be compactly supported,J. Fourier Anal. Appl.,5(5), 409–419.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    Bölcskei, H. and Hlawatsch, F. (1997). Discrete Zak transforms, polyphase transforms, and applications,IEEE Trans. Signal Processing,45(4), 851–866, April.CrossRefGoogle Scholar
  6. [6]
    Casazza, P.G., Christensen, O., and Janssen, A.J.E.M. (1999). Weyl-Heisenberg frames, translation invariant systems and the Walnut representation, submitted toJ. Funct. Anal. Google Scholar
  7. [7]
    Chen, C.T. (1984).Linear System Theory and Design, Oxford University Press, Oxford.Google Scholar
  8. [8]
    Daubechies, I. (1992).Ten Lectures on Wavelets, SIAM.Google Scholar
  9. [9]
    Feichtinger, H.G. and Strohmer, T., Eds. (1998).Gabor Analysis and Algorithms: Theory and Applications, Birkhäuser, Boston, MA.MATHGoogle Scholar
  10. [10]
    Gel'fand, I.M. and Shilov, G.E. (1967).Generalized Functions, Vol. 3, Academic Press, New York.MATHGoogle Scholar
  11. [11]
    Gohberg, I., Lancaster, P., and Rodman, L. (1982).Matrix Polynomials, Academic Press, New York.MATHGoogle Scholar
  12. [12]
    Heil, C.E. and Walnut, D.F. (1989). Continuous and discrete wavelet transforms,SIAM Rev.,41(4), 628–666, December.MathSciNetCrossRefGoogle Scholar
  13. [13]
    Jaffard, S. (1990). Propriétés des matrices bien localisées près de leur diagonale et quelques applications,Ann. Inst. Henri Poincaré,7(5), 461–476.MATHMathSciNetGoogle Scholar
  14. [14]
    Janssen, A.J.E.M. (1982). Bargmann transform, Zak transform, and coherent states,J. Math. Phys.,23(5), 720–731.MATHMathSciNetCrossRefGoogle Scholar
  15. [15]
    Janssen, A.J.E.M. (1988). The Zak transform: A signal transform for sampled time-continuous signals,Philips J. Research,43(1), 23–69.MATHMathSciNetGoogle Scholar
  16. [16]
    Janssen, A.J.E.M. (1994). Signal analytic proofs of two basic results on lattice expansions,Appl. Comp. Harmonic Anal.,1, 350–354.MATHMathSciNetCrossRefGoogle Scholar
  17. [17]
    Janssen, A.J.E.M. (1995). On rationally oversampled Weyl-Heisenberg frames,Signal Processing,47, 239–245.MATHMathSciNetCrossRefGoogle Scholar
  18. [18]
    Janssen, A.J.E.M. (1996). Some Weyl-Heisenberg frame bound calculations,Indag. Math.,7(2), 165–183.MATHMathSciNetCrossRefGoogle Scholar
  19. [19]
    Janssen, A.J.E.M. (1998). The duality condition for Weyl-Heisenberg frames. In Feichtinger, H.G. and Strohmer, T., Eds.,Gabor Analysis and Algorithms: Theory and Applications, Birkhäuser, Boston, MA, 33–84.Google Scholar
  20. [20]
    Kailath, T. (1980).Linear Systems, Prentice-Hall, Englewood Cliffs, NJ.MATHGoogle Scholar
  21. [21]
    Landau, H.J. (1993). On the density of phase-space expansions,IEEE Trans. Inf. Theory,39, 1152–1156.MATHCrossRefGoogle Scholar
  22. [22]
    Del Prete, V. Rational oversampling for Gabor frames,J. Fourier Anal. Appl., submitted.Google Scholar
  23. [23]
    Strohmer, T. (1998). Rates of convergence for the approximation of dual shift-invariant systems ofl 2(Z),J. Fourier Anal. Appl., submitted.Google Scholar
  24. [24]
    Vaidyanathan, P.P. (1993).Multirate Systems and Filter Banks, Prentice-Hall, Englewood Cliffs, NJ.MATHGoogle Scholar
  25. [25]
    Walnut, D.F. (1992). Continuity properties of the Gabor frame operator,J. Math. Anal. Appl.,165, 479–504.MATHMathSciNetCrossRefGoogle Scholar
  26. [26]
    Whittaker, E.T. and Watson, G.N. (1952).Modern Analysis, 4th ed. Cambridge University Press, Cambridge, MA.Google Scholar
  27. [27]
    Zibulski, M. and Zeevi, Y.Y. (1993). Oversampling in the Gabor scheme,IEEE. Trans. Signal Proc.,41(8), 2679–2687.MATHCrossRefGoogle Scholar
  28. [28]
    Zibulski, M. and Zeevi, Y.Y. (1997). Analysis of multiwindow Gabor-type schemes by frame methods.Appl. Comp. Harmonic Anal.,4(2), 188–221, April.MATHMathSciNetCrossRefGoogle Scholar
  29. [29]
    Feichtinger, H.G. and Gröchenig, K. (1997). Gabor Frames and Time-Frequency Analysis of Distributions,J. Functional Anal.,146, 464–495.MATHCrossRefGoogle Scholar

Copyright information

© Birkhäuser 2000

Authors and Affiliations

  • Helmut Bölcskei
    • 1
    • 2
  • J. E. M. Janssen
    • 3
  1. 1.Information Systems LaboratoryStanford UniversityStanford
  2. 2.Dept. of CommunicationsVienna University of TechnologyViennaAustria
  3. 3.Philips Research Laboratories EindhovenAA EindhovenThe Netherlands

Personalised recommendations