Article

# 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]. ...
Article
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... The metaheuristic optimization based on the differential evolution (DE) strategy restricts this drawback. It optimizes iteratively a candidate solution with regard to a given objective function (7). However, DE optimization does not guarantee that optimal solution is suitable in the case when the objective function is not differentiable corrupted by noise and even not continuous [18], [19]. ...
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Compressive sensing is a processing approach aiming to reduce the data stream from the observed object with the inherent sparsity using the optimal signal models. The compression of the sparse input signal in time or in the transform domain is performed in the transmitter by the Analog to Information Converter (AIC). The recovery of the compressed signal using optimization based on the differential evolution algorithm is presented in the article as an alternative to the faster pseudoinverse algorithm. Pseudoinverse algorithm results in an unambiguous solution associated with lower compression efficiency. The selection of the mathematically appropriate signal model affects significantly the compression efficiency. On the other hand, the signal model influences the complexity of the algorithm in the receiving block. The suitability of both recovery methods is studied on examples of the signal compression from the passive infrared (PIR) motion sensors or the ECG bioelectric signals.
... 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. ...
Preprint
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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. ...
Article
We address the problem of recovering signals from samples taken at their rate of innovation. Our only assumption is that the sampling system is such that the parameters defining the signal can be stably determined from the samples, a condition that lies at the heart of every sampling theorem. Consequently, our analysis subsumes previously studied nonlinear acquisition devices and nonlinear signal classes. In particular, we do not restrict attention to memoryless nonlinear distortions or to union-of-subspace models. This allows treatment of various finite-rate-of-innovation (FRI) signals that were not previously studied, including, for example, continuous phase modulation transmissions. Our strategy relies on minimizing the error between the measured samples and those corresponding to our signal estimate. This least-squares (LS) objective is generally non-convex and might possess many local minima. Nevertheless, we prove that under the stability hypothesis, any optimization method designed to trap a stationary point of the LS criterion necessarily converges to the true solution. We demonstrate our approach in the context of recovering pulse streams in settings that were not previously treated. Furthermore, in situations for which other algorithms are applicable, we show that our method is often preferable in terms of noise robustness.
... 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. ...
Article
In this paper, we consider the problem of estimating finite rate of innovation (FRI) signals from noisy measurements, and specifically analyze the interaction between FRI techniques and the underlying sampling methods. We first obtain a fundamental limit on the estimation accuracy attainable regardless of the sampling method. Next, we provide a bound on the performance achievable using any specific sampling approach. Essential differences between the noisy and noise-free cases arise from this analysis. In particular, we identify settings in which noise-free recovery techniques deteriorate substantially under slight noise levels, thus quantifying the numerical instability inherent in such methods. This instability, which is only present in some families of FRI signals, is shown to be related to a specific type of structure, which can be characterized by viewing the signal model as a union of subspaces. Finally, we develop a methodology for choosing the optimal sampling kernels based on a generalization of the Karhunen--Lo\`eve transform. The results are illustrated for several types of time-delay estimation problems. Comment: 23 pages, 4 figures. Submitted to IEEE Trans. Information Theory
Conference Paper
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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.