Non-uniform Random Sampling and Reconstruction in Signal Spaces with Finite Rate of Innovation
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Non-uniform Random Sampling and Reconstruction in Signal Spaces with Finite Rate of Innovation Yancheng Lu1 · Jun Xian1,2
Received: 2 April 2019 / Accepted: 26 October 2019 © Springer Nature B.V. 2019
Abstract We consider non-uniform random sampling in a signal space with finite rate of innovation V 2 (Λ, Φ) ⊂ L2 (Rd ) generated by a series of functions Φ = (φλ )λ∈Λ . A sub2 (Λ, Φ) of V 2 (Λ, Φ) is consisting of functions concentrates at least 1 − δ of the set VR,δ whole energy in a cube with side lengths R. Under mild assumptions on the generators and the probability distribution, we show that for R sufficiently large, taking O(R d log(R d )) 2 (Λ, Φ) many samples with such the non-uniform distribution yields a sampling set for VR,δ with high probability. We impose compact support on the generators as an additional constraint for obtaining a reconstruction algorithm from non-uniform random sampling with high probability. Keywords Random sampling · Non-uniform sampling · Spaces with finite rate of innovation · Non-uniform distribution · Reconstruction algorithm Mathematics Subject Classification (2000) 94A20 · 42C15 · 60E15 · 62M30
1 Introduction The space of signals with finite degree of freedom per unit of time is called the space with finite rate of innovation (FRI) [2, 5, 12, 15, 17, 19, 23]. The concept is introduced by Vetterli et al. [23] at first. The FRI model is ubiquitous and has wide scientific applications such as radar imaging [15], compression of electrocardiogram signals [2], curve fitting [12]. The model of signals with finite rate of innovation can cover following cases: (i) band-limited signals [3], (ii) signals in some shift-invariant spaces [1, 7–9, 24, 25], (iii) non-uniform
B J. Xian
[email protected] Y. Lu [email protected]
1
Department of Mathematics, Sun Yat-sen University, 510275 Guangzhou, China
2
Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, 510275 Guangzhou, China
Y. Lu, J. Xian
splines [16, 20, 21], (iv) stream of pulses k ak p(t − tk ) where p is some pulse signal shape, applied in GPS applications and cellular radio, (v) sum of some of the signals above [19], and so on. In this paper, we consider the signal spaces of finite rate of innovation V 2 (Λ, Φ) ⊂ 2 L (Rd ), defined as the closed linear span of a tuple of generators {φλ }λ∈Λ ⊂ L2 (Rd ), which is introduced in [17, 19]: V 2 (Λ, Φ) = c(λ)φλ : (cλ )λ∈Λ ∈ 2 (Λ) , λ∈Λ
where Λ is a relatively-seperated subset of Rd . Realistically we can learn about f ∈ V 2 (Λ, Φ) only if the samples are taken in the set where most of the L2 -norm is localized. So motivated by [3, 8], we focus on the sampling problem on the following subset of V 2 (Λ, Φ): 2 f (x)2 dx ≥ (1 − δ) f (x)2 dx , VR,δ (Λ, Φ) := f ∈ V 2 (Λ, Φ) : Rn
CR
2 where CR = [−R/2, R/2]n and 0 < δ < 1. Thus VR,δ (Λ, Φ) represents the subset consisting of those functions whose energy is largely concentrated on CR . Usually, in order to guarantee the success of reconstruction, we demand the s
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