From continuous to discrete Weyl-Heisenberg frames through sampling |
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Authors: | A. J. E. M. Janssen |
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Affiliation: | (1) Philips Research Laboratories, Endhoven, WY-81, 5656 AA Eindhoven, The Netherlands |
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Abstract: | In this article we consider the question when one can generate a Weyl- Heisenberg frame for l 2 (ℤ) with shift parameters N, M −1 (integer N, M) by sampling a Weyl-Heisenberg frame for L 2 (ℝ) with the same shift parameters at the integers. It is shown that this is possible when the window g ε L 2 (ℝ) generating the Weyl-Heisenberg frame satisfies an appropriate regularity condition at the integers. When, in addition, the Tolimieri-Orr condition A is satisfied, the minimum energy dual window o γ ε L 2 (ℝ) can be sampled as well, and the two sampled windows continue to be related by duality and minimality. The results of this article also provide a rigorous basis for the engineering practice of computing dual functions by writing the Wexler-Raz biorthogonality condition in the time-domain as a collection of decoupled linear systems involving samples of g and o γ as knowns and unknowns, respectively. We briefly indicate when and how one can generate a Weyl-Heisenberg frame for the space of K-periodic sequences, where K=LCM (N, M), by periodization of a Weyl-Heisenberg frame for ℓ 2 ℤ with shift parameters N, M −1 . |
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Keywords: | 41A58 42C15 94A12 |
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