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arXiv 1503.00628cs.ITmath.FAmath.IT
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算子的采样与重构

Sampling and reconstruction of operators

Götz E. Pfander, David F. Walnut

摘要:

我们研究了恢复运营商与带限Kohn-Nirenberg符号的行动,这样的运营商的加权脉冲训练,一个程序,我们称之为运营商抽样。Kailath,后来Kozek和作者已经表明,运营商采样是可能的,如果运营商的符号是带限的一组面积小于1。在本文中,我们开发了显式重建公式的算子采样,推广重建公式的带限函数。我们给出的采样率的充分必要条件,取决于大小和几何的带限集。此外,我们表明,在温和的几何条件下,类的运营商bandlimited到一个未知的一组面积小于一半允许采样和重建。Heckel和Boelcskei独立地获得了一个类似的结果,考虑了未知的面积小于1的集合。具有带限符号的算子已经被用来模拟具有慢时变脉冲响应的双色散通信信道。本文的结果是植根于工作的贝洛和Kailath在20世纪60年代。

英文摘要:

We study the recovery of operators with bandlimited Kohn-Nirenberg symbol from the action of such operators on a weighted impulse train, a procedure we refer to as operator sampling. Kailath, and later Kozek and the authors have shown that operator sampling is possible if the symbol of the operator is bandlimited to a set with area less than one. In this paper we develop explicit reconstruction formulas for operator sampling that generalize reconstruction formulas for bandlimited functions. We give necessary and sufficient conditions on the sampling rate that depend on size and geometry of the bandlimiting set. Moreover, we show that under mild geometric conditions, classes of operators bandlimited to an unknown set of area less than one-half permit sampling and reconstruction. A similar result considering unknown sets of area less than one was independently achieved by Heckel and Boelcskei. Operators with bandlimited symbols have been used to model doubly dispersive communication channels with slowly-time-varying impulse response. The results in this paper are rooted in work by Bello and Kailath in the 1960s.

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