Gabor Time-Frequency Lattices and the Wexler-Raz Identity |
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Authors: | Ingrid Daubechies HJ Landau Zeph Landau |
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Institution: | (1) AT&T Bell Laboratories, Murray Hill, New Jersey 07974, USA;(2) Department of Mathematics, University of California, Berkeley, California 94720, USA |
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Abstract: | Gabor time-frequency lattices are sets of functions of the form
generated from a given function
by discrete translations in time and frequency. They are potential tools for the decomposition and handling of signals that,
like speech or music, seem over short intervals to have well-defined frequencies that, however, change with time. It was recently
observed that the behavior of a lattice
can be connected to that of a dual lattice
Here we establish this interesting relationship and study its properties. We then clarify the results by applying the theory
of von Neumann algebras. One outcome is a simple proof that for
to span
the lattice
must have at least unit density. Finally, we exploit the connection between the two lattices to construct expansions having
improved convergence and localization properties. |
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Keywords: | |
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