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Gabor Time-Frequency Lattices and the Wexler-Raz Identity

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Gabor time-frequency lattices are sets of functions of the form \(g_{m \alpha , n \beta} (t) =e^{-2 \pi i \alpha m t}g(t-n \beta)\) generated from a given function \(g(t)\) 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 \((m \alpha , n \beta )\) can be connected to that of a dual lattice \((m/ \beta , n /\alpha ).\) 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 \(g_{m \alpha , n \beta}\) to span \(L^2,\) the lattice \((m \alpha , n \beta )\) 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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Daubechies, I., Landau, H. & Landau, Z. Gabor Time-Frequency Lattices and the Wexler-Raz Identity. J Fourier Anal Appl 1, 437–478 (1994).

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