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.
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References
Bastiaans, M.J. (1980). Gabor's expansion of a signal in Gaussian elementary signals,Proc. IEEE,68(1), 538–539, April.
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.
Boas, R.P. (1954).Entire Functions, Academic Press, New York.
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.
Bölcskei, H. and Hlawatsch, F. (1997). Discrete Zak transforms, polyphase transforms, and applications,IEEE Trans. Signal Processing,45(4), 851–866, April.
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.
Chen, C.T. (1984).Linear System Theory and Design, Oxford University Press, Oxford.
Daubechies, I. (1992).Ten Lectures on Wavelets, SIAM.
Feichtinger, H.G. and Strohmer, T., Eds. (1998).Gabor Analysis and Algorithms: Theory and Applications, Birkhäuser, Boston, MA.
Gel'fand, I.M. and Shilov, G.E. (1967).Generalized Functions, Vol. 3, Academic Press, New York.
Gohberg, I., Lancaster, P., and Rodman, L. (1982).Matrix Polynomials, Academic Press, New York.
Heil, C.E. and Walnut, D.F. (1989). Continuous and discrete wavelet transforms,SIAM Rev.,41(4), 628–666, December.
Jaffard, S. (1990). Propriétés des matrices bien localisées près de leur diagonale et quelques applications,Ann. Inst. Henri Poincaré,7(5), 461–476.
Janssen, A.J.E.M. (1982). Bargmann transform, Zak transform, and coherent states,J. Math. Phys.,23(5), 720–731.
Janssen, A.J.E.M. (1988). The Zak transform: A signal transform for sampled time-continuous signals,Philips J. Research,43(1), 23–69.
Janssen, A.J.E.M. (1994). Signal analytic proofs of two basic results on lattice expansions,Appl. Comp. Harmonic Anal.,1, 350–354.
Janssen, A.J.E.M. (1995). On rationally oversampled Weyl-Heisenberg frames,Signal Processing,47, 239–245.
Janssen, A.J.E.M. (1996). Some Weyl-Heisenberg frame bound calculations,Indag. Math.,7(2), 165–183.
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.
Kailath, T. (1980).Linear Systems, Prentice-Hall, Englewood Cliffs, NJ.
Landau, H.J. (1993). On the density of phase-space expansions,IEEE Trans. Inf. Theory,39, 1152–1156.
Del Prete, V. Rational oversampling for Gabor frames,J. Fourier Anal. Appl., submitted.
Strohmer, T. (1998). Rates of convergence for the approximation of dual shift-invariant systems ofl 2(Z),J. Fourier Anal. Appl., submitted.
Vaidyanathan, P.P. (1993).Multirate Systems and Filter Banks, Prentice-Hall, Englewood Cliffs, NJ.
Walnut, D.F. (1992). Continuity properties of the Gabor frame operator,J. Math. Anal. Appl.,165, 479–504.
Whittaker, E.T. and Watson, G.N. (1952).Modern Analysis, 4th ed. Cambridge University Press, Cambridge, MA.
Zibulski, M. and Zeevi, Y.Y. (1993). Oversampling in the Gabor scheme,IEEE. Trans. Signal Proc.,41(8), 2679–2687.
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.
Feichtinger, H.G. and Gröchenig, K. (1997). Gabor Frames and Time-Frequency Analysis of Distributions,J. Functional Anal.,146, 464–495.
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Bölcskei, H., Janssen, J.E.M. Gabor frames, unimodularity, and window decay. The Journal of Fourier Analysis and Applications 6, 255–276 (2000). https://doi.org/10.1007/BF02511155
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DOI: https://doi.org/10.1007/BF02511155