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A necessary and sufficient condition for dual Weyl-Heisenberg frames to be compactly supported

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Abstract

In this note we consider continuous-time Weyl-Heisenberg (Gabor) frame expansions with rational oversampling. We present a necessary and sufficient condition on a compactly supported function g(t) generating a Weyl-Heisenberg frame for L2 (ℝ) for its minimal dual (Wexler-Razdual) γ0 (t) to be compactly supported. We furthermore provide a necessary and sufficient condition for a band-limited function g(t) generating a Weyl-Heisenberg frame for L2 (ℝ) to have a band-limited minimal dual γ0 (t). As a consequence of these conditions, we show that in the cases of integer oversampling and critical sampling a compactly supported (band-limited) g(t) has a compactly supported (band-limited) minimal dual γ0(t) if and only if the Weyl-Heisenberg frame operator is a multiplication operator in the time (frequency) domain. Our proofs rely on the Zak transform, on the Zibulski-Zeevi representation of the Weyl-Heisenberg frame operator, and on the theory of polynomial matrices.

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References

  1. Bastiaans, M.J. (1980). Gabor's expansion of a signal in Gaussian elementary signals,Proc. IEEE,68(1), 538–539, April.

    Google Scholar 

  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. Bölcskei, H. and Hlawatsch, F. (1997). Discrete Zak transforms, polyphase transforms, and applications,IEEE Trans. Signal Processing,45(4), 851–866, April.

    Google Scholar 

  4. Daubechies, I. (1992).Ten Lectures on Wavelets, SIAM.

  5. Daubechies, I., Landau, H.J., and Landau, Z. (1995). Gabor time-frequency lattices and the Wexler-Raz identity,J. Fourier Anal. Appl.,1(4), 437–478.

    Google Scholar 

  6. Feichtinger, H.G. and Strohmer, T., Eds., (1998).Gabor Analysis and Algorithms: Theory and Applications, Birkhäuser, Boston, MA.

    Google Scholar 

  7. Gohberg, I., Lancaster, P., and Rodman, L. (1982).Matrix Polynomials, Academic Press.

  8. Heil, C.E. and Walnut, D.F. (1989). Continuous and discrete wavelet transforms,SIAM Rev.,31(4), 628–666, December.

    Google Scholar 

  9. Janssen, A.J.E.M. (1981). Gabor representation of generalized functions,J. Math. Anal. Appl.,83, 377–394.

    Google Scholar 

  10. Janssen, A.J.E.M. (1988). The Zak transform: A signal transform for sampled time-continuous signals,Philips J. Res.,43(1), 23–69.

    Google Scholar 

  11. Janssen, A.J.E.M. (1994). Signal analytic proofs of two basic results on lattice expansions,Applied and Computational Harmonic Analysis,1, 350–354.

    Google Scholar 

  12. Janssen, A.J.E.M. (1995). Duality and biorthogonality for Weyl-Heisenberg frames,J. Fourier Anal. Appl.,1(4), 403–436.

    Google Scholar 

  13. Janssen, A.J.E.M. (1995). On rationally oversampled Weyl-Heisenberg frames,Signal Processing,47, 239–245.

    Google Scholar 

  14. 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 

  15. Landau, H.J. (1993). On the density of phase-space expansions,IEEE Trans. Inf. Theory,39, 1152–1156.

    Google Scholar 

  16. Ron, A. and Shen, Z. (1995). Frames and stable bases for shift-invariant subspaces of L2(Rd),Can. J. Math.,47(5), 1051–1094.

    Google Scholar 

  17. Walnut, D.F. (1992). Continuity properties of the Gabor frame operator,J. Math. Anal. Appl.,165, 479–504.

    Google Scholar 

  18. Wexler, J. and Raz, S. (1990). Discrete Gabor expansions,Signal Processing,21, 207–220.

    Google Scholar 

  19. Zibulski, M. and Y.Y. Zeevi. (1993). Oversampling in the Gabor scheme,IEEE Trans. Signal Processing,41(8), 2679–2687, August.

    Google Scholar 

  20. Zibulski, M. and Zeevi, Y.Y. (1997). Analysis of multiwindow Gabor-type schemes by frame methods,Applied and Computational Harmonic Analysis,4(2), 188–221, April.

    Google Scholar 

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Authors and Affiliations

  1. Information Systems Laboratory, Dept. of Electrical Engineering, Stanford University Packard 223, 300 Serra Mall, 94305-9510, Stanford, CA

    Helmut Bölcskei

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  1. Helmut Bölcskei
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Communicated by Akram Aldroubi

on leave from Department of Communications, Vienna University of Technology

This work was supported in part by FWF grants P10531-ÖPH, P12228-TEC, and J1629-TEC.

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Bölcskei, H. A necessary and sufficient condition for dual Weyl-Heisenberg frames to be compactly supported. The Journal of Fourier Analysis and Applications 5, 409–419 (1999). https://doi.org/10.1007/BF01261635

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  • DOI: https://doi.org/10.1007/BF01261635

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