Spatially distributed sampling and reconstruction

• Published in: Applied and computational harmonic analysis Vol. 47; no. 1; pp. 109 - 148
• Main Authors: Cheng, Cheng, Jiang, Yingchun, Sun, Qiyu
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.07.2019
• Subjects: Beurling dimension; Distributed algorithm; Distributed sampling and reconstruction system; Finite rate of innovation; Inverse-closed subalgebra; Localized matrix; Signal on a graph; Spatially distributed network; Signal on a graph; Distributed sampling and reconstruction system; Localized matrix; Distributed algorithm; Inverse-closed subalgebra; Spatially distributed network; Beurling dimension; Finite rate of innovation
• ISSN: 1063-5203; 1096-603X
• DOI: 10.1016/j.acha.2017.07.007

A spatially distributed network contains a large amount of agents with limited sensing, data processing, and communication capabilities. Recent technological advances have opened up possibilities to deploy spatially distributed networks for signal sampling and reconstruction. In this paper, we introduce a graph structure for a distributed sampling and reconstruction system by coupling agents in a spatially distributed network with innovative positions of signals. A fundamental problem in sampling theory is the robustness of signal reconstruction in the presence of sampling noises. For a distributed sampling and reconstruction system, the robustness could be reduced to the stability of its sensing matrix. In this paper, we split a distributed sampling and reconstruction system into a family of overlapping smaller subsystems, and we show that the stability of the sensing matrix holds if and only if its quasi-restrictions to those subsystems have uniform stability. This new stability criterion could be pivotal for the design of a robust distributed sampling and reconstruction system against supplement, replacement and impairment of agents, as we only need to check the uniform stability of affected subsystems. In this paper, we also propose an exponentially convergent distributed algorithm for signal reconstruction, that provides a suboptimal approximation to the original signal in the presence of bounded sampling noises.