# Bulletin of theKorean Mathematical SocietyBKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

## Article

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Bull. Korean Math. Soc. 2022; 59(2): 285-301

Published online March 31, 2022 https://doi.org/10.4134/BKMS.b200916

## Random sampling and reconstruction of signals with finite rate of innovation

Yingchun Jiang, Junjian Zhao

Guilin University of Electronic Technology; TianGong University

### Abstract

In this paper, we mainly study the random sampling and reconstruction of signals living in the subspace $V^p(\Phi,\Lambda)$ of $L^p(\mathbb{R}^d)$, which is generated by a family of molecules $\Phi$ located on a relatively separated subset $\Lambda\subset \mathbb{R}^d$. The space $V^p(\Phi,\Lambda)$ 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 $\mathbb{R}^d$. Under some proper conditions for the generators $\Phi=\{\phi_\lambda:\lambda\in \Lambda\}$ and the probability density function $\rho$, we first approximate $V^{p}(\Phi,\Lambda)$ by a finite dimensional subspace $V^{p}_N(\Phi,\Lambda)$ on any bounded domains. Then, we prove that the random sampling stability holds with high probability for all signals in $V^{p}(\Phi,\Lambda)$ 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 $V^{p}_N(\Phi,\Lambda)$.

Keywords: Random sampling, signals with finite rate of innovation, sampling stability, probability density function, reconstruction algorithm

MSC numbers: Primary 94A20, 42C40

Supported by: This work is supported by the National Natural Science Foundation of China (No. 11661024) and the Guangxi Natural Science Foundation (Nos. 2020GXNSFAA159076, 2019GXNSFFA245012), Natural Science Foundation of Tianjin City (No. 18JCYBJC16300), Guangxi Science and Technology Project (No. 2021AC06001), Guangxi Key Laboratory of Cryptography and Information Security (No. GCIS201925), Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation.