Random Sampling and Reconstruction of Sparse Time- and Band-Limited Signals. (23rd May 2022) - Li, Song-Hua; Liu, Zhilong; Xian, Jun
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 .