Random Sampling and Reconstruction of Sparse Time- and Band-Limited Signals. (23rd May 2022)

• Record Type: Journal Article
• Authors: Li, Song-Hua; Liu, Zhilong; Xian, Jun
• Is Part Of: Numerical functional analysis and optimization. Volume 43:Number 5(2022)
• Page Start: 613
• Page End: 629
• Publication Date: 2022-05-23
• Subjects: Reconstruction algorithm -- random sampling -- band-limited functions; 65D15 -- 94A20 -- 62D05 -- 46N10; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2022.2051184
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: The random sampling and reconstruction algorithm of sparse time- and band-limited signals, which stem from bandlimited functions, is studied in this paper. For time- and band-limited signals, considering the projection on a finite-dimensional solution space, the reconstruction algorithm can therefore be described by a matrix. Especially, we study the case of the finite-dimensional spaces with regard to time- and band-limited signals. First, as the cardinality of the sample value { f ( x j ) : j ∈ J } is not less than the dimension of solution space, we prove that the coefficient matrix has full rank. Second, as sample value { f ( x j ) : j ∈ J } is insufficient, especially spare time- and band-limited signals, based on the Restricted Isometry Property (RIP) condition and the concentration inequality of a probability distribution for random sampling, we present a ℓ 1 − minimization reconstruction from few randomly chosen samples on [ − T, T ] . Finally, numerical experiments show that with high probability the time- and band-limited signals can be approximately recovered provided τ -sparsity of non-vanishing coefficients ak .