From continuous to discrete Weyl-Heisenberg frames through sampling |
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Authors: | A J E M Janssen |
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Institution: | (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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