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Nonlinear and Nonideal Sampling Revisited

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Abstract

We revisit the problem of recovering a continuous-time signal lying within a known shift-invariant subspace from nonlinear and nonideal samples. Recently, an iterative algorithm for perfect recovery of such signals was proposed. This method requires operations which are not linear time-invariant (LTI), rendering it impractical due to its polynomial dependency on the data length. We describe an alternative iterative algorithm for recovering the signal, which involves only LTI operations. In the revised method, each iteration is much faster and implementation is simpler. Furthermore, the overall running time of our approach depends linearly on the number of samples.

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... So we may apply the established solvability and stability of the nonlinear functional equation (1.1) to study the stability of the nonlinear sampling procedure (1.7), and the Van-Cittert iteration method to reconstruct signals h ∈ V p (Φ) from its nonlinear sampling data ⟨F (h), Ψ⟩, see Theorems 7.1 and 7.7 for the theoretical results and Section 7.3 for the numerical simulations. The readers may refer to [14,16] for the study of the nonlinear sampling procedure (1.7) from an engineering viewpoint. ...
... As a stable nonlinear sampling procedure is one-to-one, from Theorem 7.1 we obtain the uniqueness of the nonlinear sampling procedure (1.7), which is established in [16] for p = 2 when µ < 1−δ(V 2 (Φ),V 2 (Ψ)) 1+δ(V 2 (Φ),V 2 (Ψ)) , a stronger assumption on F than (7.3) in Theorem 7.1. ...
... Hence the stability conclusion in Corollary 7.4 holds for p = 2 without the polynomial decay assumption on Φ. This is established for signals living in a Paley-Wiener space [32,41] or a shift-invariant space [16]. ...
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... That is, samples are equal to the values of the signal at a set of sampling points {t n } n∈Z . Due to physical limitation, e.g., the inertia of the acquisition devices [43], it is impossible to measure the values of a signal precisely at times t n [44]- [48]. In practice, only a local average of f (t) near t n can be obtained and used for signal reconstruction. ...
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... In [11], the authors generalized these developments to the challenging setting in which the linear part of the acquisition device does not necessarily match the prior on the signal (as opposed, e.g., to using a low-pass anti-aliasing filter when the original signal is known to be bandlimited). A simpler iterative algorithm, consisting of linear time-invariant (LTI) filtering operations, was recently developed in [16] for the same setting. ...
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... Clearly, the M Tlocal rate of innovation ρ MT of this type of signals is the same as that of the underlying SI function, and is thus given by (11). The recovery of nonlinearly distorted SI signals from noiseless samples was treated in [20] [23]. We are not aware of research works treating the noisy case. ...
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Fixed Point Theory Authorized licensed use limited to: Technion Israel School of Technology
  • A Granas
  • J Dugundji
A. Granas and J. Dugundji, Fixed Point Theory. New York: Springer-Verlag, 2003. Authorized licensed use limited to: Technion Israel School of Technology. Downloaded on April 17,2010 at 10:48:54 UTC from IEEE Xplore. Restrictions apply.