Uncertainty Principles for Time-Frequency Representations

We present a machine to produce new uncertainty principles from old ones. Traditionally an uncertainty principle is an inequality involving both a functionfand its Fourier transformf.To generate new uncertainty principles, we interpret the pair(f,f)as a time-frequency representation, replace it by a different time-frequency representation, and formulate a corresponding inequality. We discuss a few recent uncertainty principles for the short-time Fourier transform and the Wigner distribution that can be obtained in this way and suggest further problems.

[1]  G. Hardy.  A Theorem Concerning Fourier Transforms , 1933 .

[2]  Harry Dym,et al.  Fourier series and integrals , 1972 .

[3]  A. Berthier,et al.  On support properties of Lp-functions and their Fourier transforms , 1977 .

[4]  H. Feichtinger.  On a new Segal algebra , 1981 .

[5]  M. Cowling,et al.  Bandwidth Versus Time Concentration: The Heisenberg–Pauli–Weyl Inequality , 1984 .

[6]  M. Benedicks.  On Fourier transforms of functions supported on sets of finite Lebesgue measure , 1985 .

[7]  G. Folland.  Harmonic analysis in phase space , 1989 .

[8]  D. Donoho,et al.  Uncertainty principles and signal recovery , 1989 .

[9]  L. Cohen,et al.  Time-frequency distributions-a review , 1989, Proc. IEEE.

[10]  E. Lieb.  Integral bounds for radar ambiguity functions and Wigner distributions , 1990 .

[11]  L. Hörmander.  A uniqueness theorem of Beurling for Fourier transform pairs , 1991 .

[12]  F. Hlawatsch,et al.  Linear and quadratic time-frequency signal representations , 1992, IEEE Signal Processing Magazine.

[13]  J. Benedetto,et al.  Fourier transform inequalities with measure weights , 1992 .

[14]  V. Havin.  The Uncertainty Principle in Harmonic Analysis , 1994 .

[15]  K. Gröchenig.  An uncertainty principle related to the Poisson summation formula , 1996 .

[16]  G. Folland,et al.  The uncertainty principle: A mathematical survey , 1997 .

[17]  S. K. Ray,et al.  Uncertainty principles like Hardy's theorem on some Lie groups , 1998, Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics.

[18]  Philippe Jaming,et al.  Principe d'incertitude qualitatif et reconstruction de phase pour la transformée de Wigner , 1998 .

[19]  H. Feichtinger,et al.  A Banach space of test functions for Gabor analysis , 1998 .

[20]  A. Janssen.  Proof of a conjecture on the supports of Wigner distributions , 1998 .

[21]  Karlheinz Gröchenig,et al.  Foundations of Time-Frequency Analysis , 2000, Applied and numerical harmonic analysis.

[22]  K. Gröchenig,et al.  Hardy's Theorem and the Short‐Time Fourier Transform of Schwartz Functions , 2001 .

[23]  Joseph D. Lakey,et al.  Embeddings and Uncertainty Principles for Generalized Modulation Spaces , 2001 .

[24]  P. Jaming,et al.  Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms , 2001, math/0102111.

[25]  K. Gröchenig,et al.  Uncertainty principles as embeddings of modulation spaces , 2002 .