Time-Frequency Localization via the Weyl Correspondence

Abstract

Public reporting burden for this collection of information is estimated to average I houri per resOpnse. including the time for reviewing Instructions. searching existing data sources,. gathering and maintaining the data needed, and completing and reviewing the collection of information Send comments rearding this burden estimate or any other aspect of this Approved for public release; distribution unlimited A ABSTRACT (Maximum 200 word)A technique of producing signals whose energy is concentrated in a given region of the time-frequency plane is examined. The degree to which a particular signal is concentrated is measured by integrating the Wigner distribution over the given region.This procedure was put forward by Flandrin, and has been used for timevarying filtering in the recent work of Hlawatsch, Kozek, and Krattenthaler. In this paper, the associated operator is studied.Estimates for the eigenvalue decay and the smoothness and decay of the eigenfunctions are established. MITREApproved for public release;Bedford, Massachusetts distribution unlimited. ABSTRACTA technique of producing signals whose energy is concentrated in a given region of the time-frequency plane is examined. The degree to which a particular signal is concentrated is measured by integrating the Wigner distribution over the given region. This procedure was put forward by Flandrin, and has been used for time-varying filtering in the recent work of Hlawatsch, Kozek, and Krattenthaler. In this paper, the associated operator is studied. Estimates for the eigenvalue decay and the smoothness and decay of the eigenfunctions are established.iii

The chirplet transform: physical considerations

“…Others have: done work to further generalize energy concentration to arbitrarily shaped regions of the TF plane [55] rather than just parallelograms. It would therefore be possible to use these results to define more general parameterizable transforms based on families of multiple analysis primitives acting collectively in the TF plane.…”

Localization operators associated with the windowed Opdam-Cherednik transform on modulation spaces

“…The role of these operators to localize a signal simultaneously in time and frequency domains, this can be seen as the uncertainty principle. The localization operators were introduced and studied by Daubechies [9][10][11], Ramanathan and Topiwala [30], and extensively investigated in [16,33,35]. This class of operators occurs in various branches of pure and applied mathematics and has been studied by many authors.…”

“…This class of operators occurs in various branches of pure and applied mathematics and has been studied by many authors. Localization operators are recognized as an important new mathematical tool and have found many applications to time-frequency analysis, quantum mechanics, the theory of differential equations, and signal processing (see [7,18,22,23,30,35]). They are also known as anti-Wick operators, wave packets, Toeplitz operators, or Gabor multipliers (see [5,8,16,23]).…”

Localization of multidimensional Wigner distributions

“…The localization problem, i.e., estimating the integral of the Wigner distribution over a subregion of the phase space, and the closely related problem of the optimal simultaneous concentration of ψ and its Fourier transform ψ, have received much attention in the literature both in quantum mechanics, mathematical time-frequency analysis, and signal processing (see e.g. [1,2,3,4,5,6,9,10,12,13,11], and the references within). Bounds on the L p norms were found in [7].…”