Shannon Sampling Series Research Articles

On numerical realizations of Shannon’s sampling theorem

In this paper, we discuss some numerical realizations of Shannon’s sampling theorem. First we show the poor convergence of classical Shannon sampling sums by presenting sharp upper and lower bounds on the norm of the Shannon sampling operator. In addition, it is known that in the presence of noise in the samples of a bandlimited function, the convergence of Shannon sampling series may even break down completely. To overcome these drawbacks, one can use oversampling and regularization with a convenient window function. Such a window function can be chosen either in frequency domain or in time domain. We especially put emphasis on the comparison of these two approaches in terms of error decay rates. It turns out that the best numerical results are obtained by oversampling and regularization in time domain using a sinh\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sinh $$\\end{document}-type window function or a continuous Kaiser–Bessel window function, which results in an interpolating approximation with localized sampling. Several numerical experiments illustrate the theoretical results.

Approximation of Nonhomogeneous Random Field from Local Averages

In this article, we consider the extension of Shannon sampling series reconstruction theorem for nonhomogeneous random fields using local averages sampling, which helps improve certain earlier results. The upper bound of mean square truncation sampling approximation error is more precise, and we establish one approximation result in the almost sure sense.

On regularized Shannon sampling formulas with localized sampling

In this paper, we present new regularized Shannon sampling formulas which use localized sampling with special window functions, namely Gaussian, mathrm B-spline, and sinh -type window functions. In contrast to the classical Shannon sampling series, the regularized Shannon sampling formulas possess an exponential decay and are numerically robust in the presence of noise. Several numerical experiments illustrate the theoretical results.

Computable Time Concentration of Bandlimited Signals and Systems

Turing computability deals with the question of what is theoretically computable on a digital computer, and hence is relevant whenever digital hardware is used. In this paper we study different possibilities to define computable bandlimited signals and systems. We consider a definition that uses finite Shannon sampling series as approximating functions and another that employs computable continuous functions together with an effectively computable time concentration. We discuss the advantages and drawbacks of both definitions and analyze the connections and differences. In particular, we show that both definitions are equivalent for many practically relevant signal classes, e.g. for bandlimited signals with finite energy, but also that there are important differences, such as for the impulse responses of BIBO stable LTI systems.

Turing Meets Shannon: Computable Sampling Type Reconstruction With Error Control

The conversion of analog signals into digital signals and vice versa, performed by sampling and interpolation, respectively, is an essential operation in signal processing. When digital computers are used to compute the analog signals, it is important to effectively control the approximation error. In this paper we analyze the computability, i.e., the effective approximation of band limited signals in the Bernstein spaces B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">π</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sup> , 1 ≤ p <; ∞,andofthe corresponding discrete-time signals that are obtained by sampling. We show that for 1 <; p <; ∞, computability of the discrete-time signal implies computability of the continuous-time signal. For p = 1 this correspondence no longer holds. Further, we give a necessary and sufficient condition for computability and show that the Shannon sampling series provides a canonical approximation algorithm for p > 1. We discuss BIBO stable LTI systems and the time-domain concentration behavior of bandlimited signals as applications.

Sharp exponential bounds for the Gaussian regularized Whittaker–Kotelnikov–Shannon sampling series

Fast reconstruction of a bandlimited function from its finite oversampling data has been a fundamental problem in sampling theory. As the number of sample data increases to infinity, exponentially-decaying reconstruction errors can be achieved by many methods in the literature. In fact, it is generally conjectured that when the optimal method is used, the dominant term in the error of reconstructing a function bandlimited to [−δ,δ] (δ<π) from its data sampled at the integer points on [−n,n] is exp(−λ(π−δ)n). By far, the best estimate for the constant λ among regularization methods is 1∕2 and is achieved by the highly efficient Gaussian regularized Whittaker–Kotelnikov–Shannon sampling series. We prove in this paper that the exponential constant 1∕2 is optimal for this method. Moreover, the optimal variance of the Gaussian regularizer is provided.

Nonlinearly bandlimited signals

In this paper, we study the inverse scattering problem for a class of signals that have a compactly supported reflection coefficient. The problem boils down to the solution of the Gelfand–Levitan–Marchenko (GLM) integral equations with a kernel that is bandlimited. By adopting a sampling theory approach to the associated Hankel operators in the Bernstein spaces, a constructive proof of existence of a solution of the GLM equations is obtained under various restrictions on the nonlinear impulse response (NIR). The formalism developed in this article also lends itself well to numerical computations yielding algorithms that are shown to have algebraic rates of convergence. In particular, the use Whittaker–Kotelnikov–Shannon sampling series yields an algorithm that converges as whereas the use of Helms and Thomas (HT) version of the sampling expansion yields an algorithm that converges as for any m > 0 provided the regularity conditions are fulfilled. The complexity of the algorithms depend on the linear solver used. The use of conjugate-gradient (CG) method yields an algorithm of complexity per sample of the signal where N is the number of sampling basis functions used and is the number of CG iterations involved. The HT version of the sampling expansions facilitates the development of algorithms of complexity (per sample of the signal) by exploiting the special structure as well as the (approximate) sparsity of the matrices involved. The algorithms are numerically validated using Schwartz class functions as NIRs that are either bandlimited or effectively bandlimited. The results suggest that the HT variant of our algorithm is spectrally convergent for an input of the aforementioned class.

An Optimal Convergence Rate for the Gaussian Regularized Shannon Sampling Series

We consider the reconstruction of a bandlimited function from its finite localized sample data. Truncating the classical Shannon sampling series results in an unsatisfactory convergence rate due to the slow decay of the sinc function. To overcome this drawback, a simple and highly effective method, called the Gaussian regularization of the Shannon series, was proposed in the engineering and has received remarkable attention. It works by multiplying the sinc function in the Shannon series with a regularized Gaussian function. Recently, it was proved that the upper error bound of this method can achieve a convergence rate of the order , where 0 < δ < π is the bandwidth and n the number of sample data. The convergence rate is by far the best convergence rate among all regularized methods for the Shannon sampling series. The main objective of this article is to present the theoretical justification and numerical verification that the convergence rate is optimal when 0 < δ < π/2 by estimating the lower error bound of the truncated Gaussian regularized Shannon sampling series.

Spaceability and strong divergence of the Shannon sampling series and applications

In this paper the structure of the set of functions for which the peak value of the Shannon sampling series is strongly divergent is analyzed. Strong divergence is closely linked to the non-existence of adaptive reconstruction methods. Signals in the Paley–Wiener space PWπ1 of bandlimited functions with absolutely integrable Fourier transform are considered, and it is shown that the set of strong divergence is spaceable and dense-lineable, i.e., that there exist an infinite dimensional closed subspace and an infinite dimensional dense subspace, such that we have strong divergence of the peak value of the Shannon sampling series for all functions from these sets, except the zero function. Further, it is proved that this result is not restricted to the Shannon sampling series, but rather holds for an entire class of reconstruction processes.

Convergence Analysis of the Gaussian Regularized Shannon Sampling Series

ABSTRACTIn this article, we consider the reconstruction of a bandlimited function from its finite localized sample data. Truncating the classical Shannon sampling series results in an unsatisfactory convergence rate due to the slow decay of the sinc function. To overcome this drawback, a simple and highly effective method called the Gaussian regularization of the Shannon series was proposed in engineering and has received remarkable attention. It works by multiplying the sinc function in the Shannon series with a regularized Gaussian function. L. Qian established the convergence rate of for this method, where δ<π is the bandwidth and n is the number of sample data. C. Micchelli et al. proposed a different regularized method and obtained the corresponding convergence rate of . This latter rate is by far the best among all regularized methods for the Shannon series. However, their regularized method involves the solving of a linear system and is implicit and more complicated. The main objective of this article is to show that the Gaussian regularization of the Shannon series can also achieve the same best convergence rate as that by C. Micchelli et al. We also show that the Gaussian regularization method can improve the convergence rate for the useful average sampling.

Recent Progress of Shannon Sampling Theory with Applications

Errors appear when the Shannon sampling series is applied to approximate a signal in practice since the measured sampled values are usually given by averages of a function or the sampled values with amplitude error and time-jitter error. In this paper we give a survey of recent progress of the study of error analysismulti-dimensional Whittaker-Shannon sampling expansion approximations with these measured values.

The Variation Detracting Property of Some Shannon Sampling Series and Their Derivatives

In this paper some generalized Shannon sampling operators which diminish the total variation of functions and their derivatives are considered. For that purpose will be used some related kernel functions of certain even bandlimited kernel functions.

Uniform Error Bounds for Reconstruct Functions from Weighted Bernstein Class

Errors appear when the Shannon sampling series is applied to reconstruct a signal in practice. This is because the sampled values may not be exact, or the sampling series may have to be truncated. In this paper, we study errors in truncated sampling series with localized sampling for band-limited functions from weighted Bernstein class. And we apply these results to some practical examples. DOI: http://dx.doi.org/10.11591/telkomnika.v11i8.3078

Error Analysis for Shannon Sampling Series Approximation with Measured Sampled Values

Errors appear when the Shannon sampling series is applied to reconstruct a signal in practice. In this paper, we study a general model that uses linear functional to cover several errors in one formula, and consider sampling series with measured sampled values for not band-limited signals but satisfying some decay condition. We obtain the uniform truncated error bound of Shannon series approximation for multivariate Besov class. The results show this kind of Shannon series can approximate a smooth signal well.

On the Behavior of the Threshold Operator for Bandlimited Functions

One interesting question is how the good local approximation behavior of the Shannon sampling series for the Paley–Wiener space $\mathcal {PW}_{\pi}^{1}$ is affected if the samples are disturbed by the non-linear threshold operator. This operator, which is important in many applications, sets all samples whose absolute value is smaller than some threshold to zero. In this paper we analyze a generalization of this problem, in which not the Shannon sampling series is disturbed by the threshold operator but a more general system approximation process, were a stable linear time-invariant system is involved. We completely characterize the stable linear time-invariant systems that, for some functions in $\mathcal {PW}_{\pi}^{1}$ , lead to a diverging approximation process as the threshold is decreased to zero. Further, we show that if there exists one such function then the set of functions for which divergence occurs is in fact a residual set. We study the pointwise behavior as well as the behavior of the L ∞-norm of the approximation process. It is known that oversampling does not lead to stable approximation processes in the presence of thresholding. An interesting open problem is the characterization of the systems that can be stably approximated with oversampling.

Uniform Bounds of Aliasing and Truncated Errors in Sampling Series of Functions from Anisotropic Besov Class

Errors appear when the Shannon sampling series is applied to approximate a signal in real life. This is because a signal may not be bandlimited, the sampling series may have to be truncated, and the sampled values may not be exact and may have to be quantized. In this paper, we truncate the multidimensional Shannon sampling series via localized sampling and obtain the uniform bounds of aliasing and truncation errors for functions from anisotropic Besov class without any decay assumption. The bounds are optimal up to a logarithmic factor. Moreover, we derive the corresponding results for the case that the sampled values are given by a linear functional and its integer translations. Finally we give some applications.

On Some Truncated Shannon Sampling Series

On Some Truncated Shannon Sampling Series

Unboundedness of thresholding and quantization for bandlimited signals

It is well-known that the Shannon sampling series is locally uniformly convergent for all signals in the Paley–Wiener space PWπ1. An interesting question is how this good local approximation behavior is affected if the samples are disturbed by the non-linear threshold operator. This operator, which sets to zero all samples with absolute value smaller than some threshold, arises in the modeling of many applications, e.g., in wireless sensor networks. Moreover, it constitutes an essential part of a large class of quantizers, and consequently is important for all digital signal processing applications that involve conversion between analog and digital domains. In this paper, the approximation behavior of the Shannon sampling series that only uses the samples with absolute value larger than or equal to some threshold is analyzed. It is shown that there exists a signal in PWπ1 such that the local approximation error increases unboundedly as the threshold tends to zero. Moreover, for a fixed threshold, the local approximation error can grow arbitrarily large on the set of signals whose norm is bounded by one. With this, we generalize results of Butzer et al. that were given in the paper “On quantization, truncation and jitter errors in the sampling theorem and its generalizations,” Signal Processing (2) 1980 [1]. We conclude the paper with a discussion about the differences in the reconstruction behavior between the sampling series which is truncated in the domain of the sampled signal, i.e., time-domain truncation, and the sampling series which is truncated in the range of the sampled signal.

Approximation of Wide-Sense Stationary Stochastic Processes by Shannon Sampling Series

In this paper, the convergence behavior of the symmetric and the nonsymmetric Shannon sampling series is analyzed for bandlimited continuous-time wide-sense stationary stochastic processes that have absolutely continuous spectral measure. It is shown that the nonsymmetric sampling series converges in the mean-square sense uniformly on compact subsets of the real axis if and only if the power spectral density of the process fulfills a certain integrability condition. Moreover, if this condition is not fulfilled, then the pointwise mean-square approximation error of the nonsymmetric sampling series and the supremum of the mean-square approximation error over the real axis of the symmetric sampling series both diverge. This shows that there is a significant difference between the convergence behavior of the symmetric and the nonsymmetric sampling series.

There Exists No Globally Uniformly Convergent Reconstruction for the Paley–Wiener Space ${{\cal PW}}_{\pi}^{1}$ of Bandlimited Functions Sampled at Nyquist Rate

The bandlimited interpolation of signals and the convergence behavior of the Shannon sampling series are discussed in order to show that it is desirable to have a uniformly convergent reconstruction, for as large a space of signals as possible. In this paper general sampling series are analyzed for the frequently utilized Paley-Wiener space <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PW</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pi</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> , which is the largest space in the scale of Paley-Wiener spaces. The analysis is done not only for the Shannon sampling series, but for a whole class of axiomatically defined reconstruction processes. It is shown that for this very general class, which contains all common sampling series including the Shannon sampling series, a uniformly convergent reconstruction is not possible for the space <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">PW</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pi</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> . Moreover, a universal signal is identified that causes the divergence behavior for all sampling series. Finally, a lower and an upper bound are derived and used to describe the asymptotic behavior of the peak value of the finite sampling series.