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http://dx.doi.org/10.4134/BKMS.b200916

RANDOM SAMPLING AND RECONSTRUCTION OF SIGNALS WITH FINITE RATE OF INNOVATION  

Jiang, Yingchun (School of Mathematics and Computational Science Guilin University of Electronic Technology)
Zhao, Junjian (School of Mathematical Sciences TianGong University)
Publication Information
Bulletin of the Korean Mathematical Society / v.59, no.2, 2022 , pp. 285-301 More about this Journal
Abstract
In this paper, we mainly study the random sampling and reconstruction of signals living in the subspace Vp(𝚽, 𝚲) of Lp(ℝd), which is generated by a family of molecules 𝚽 located on a relatively separated subset 𝚲 ⊂ ℝd. The space Vp(𝚽, 𝚲) is used to model signals with finite rate of innovation, such as stream of pulses in GPS applications, cellular radio and ultra wide-band communication. The sampling set is independently and randomly drawn from a general probability distribution over ℝd. Under some proper conditions for the generators 𝚽 = {𝜙λ : λ ∈ 𝚲} and the probability density function 𝜌, we first approximate Vp(𝚽, 𝚲) by a finite dimensional subspace VpN (𝚽, 𝚲) on any bounded domains. Then, we prove that the random sampling stability holds with high probability for all signals in Vp(𝚽, 𝚲) whose energy concentrate on a cube when the sampling size is large enough. Finally, a reconstruction algorithm based on random samples is given for signals in VpN (𝚽, 𝚲).
Keywords
Random sampling; signals with finite rate of innovation; sampling stability; probability density function; reconstruction algorithm;
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