Random sampling and reconstruction of signals with finite rate of innovation

다국어 초록 (Multilingual 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 $V^{p}_N(\Phi,\Lambda)$.

참고문헌 (Reference)

1 M. Vetterli, "Sampling signals with finite rate of innovation" 50 (50): 1417-1428, 2002

2 G. A. Velasco, "Relevant sampling of the short-time Fourier transform of time-frequency localized functions"

3 R. F. Bass, "Relevant sampling of band-limited functions" 57 (57): 43-58, 2013

4 H. Fuhr, "Relevant sampling in finitely generated shift-invariant spaces" 240 : 1-15, 2019

5 Y. Li, "Reconstruction from convolution random sampling in local shift invariant spaces" 35 (35): 15-, 2019

6 Richard F. Bass, "Random sampling of bandlimited functions" Springer Science and Business Media LLC 177 (177): 1-28, 2010

7 Jianbin Yang, "Random sampling in shift invariant spaces" Elsevier BV 398 (398): 26-34, 2013

8 J. Yang, "Random sampling and reconstruction in multiply generated shift-invariant spaces" 17 (17): 323-347, 2019

9 Jianbin Yang, "Random sampling and approximation of signals with bounded derivatives" Springer Science and Business Media LLC 2019 (2019): 2019

10 Richard F. Bass, "Random Sampling of Multivariate Trigonometric Polynomials" Society for Industrial & Applied Mathematics (SIAM) 36 (36): 773-795, 2005

11 G. Bennett, "Probability inequalities for the sum of independent random variable" 57 (57): 33-45, 1962

12 STEVE SMALE, "ONLINE LEARNING WITH MARKOV SAMPLING" World Scientific Pub Co Pte Lt 07 (07): 87-113, 2009

13 Qiyu Sun, "Nonuniform Average Sampling and Reconstruction of Signals with Finite Rate of Innovation" Society for Industrial & Applied Mathematics (SIAM) 38 (38): 1389-1422, 2007

14 Yancheng Lu, "Non-uniform Random Sampling and Reconstruction in Signal Spaces with Finite Rate of Innovation" Springer Science and Business Media LLC 169 (169): 247-277, 2019

15 S. H. Chan, "Monte Carlo non-local means : random sampling for large-scale image filtering" 23 (23): 3711-3725, 2014

16 F. Cucker, "Learning theory: an approximation theory viewpoint, Cam bridge Monographs on Applied and Computational Mathematics, 24" Cambridge University Press 2007

17 Q. Sun, "Frames in spaces with finite rate of innovation" 28 (28): 301-329, 2008

18 Sunil Rudresh, "Finite-Rate-of-Innovation-Sampling-Based Super-Resolution Radar Imaging" Institute of Electrical and Electronics Engineers (IEEE) 65 (65): 5021-5033, 2017

19 Satish Mulleti, "Ellipse Fitting Using the Finite Rate of Innovation Sampling Principle" Institute of Electrical and Electronics Engineers (IEEE) 25 (25): 1451-1464, 2016

20 Y. C. Eldar, "Compressed sensing of analog signals in shift-invariant spaces" 57 (57): 2986-2997, 2009