Sampling Theory for not Necessarily Band-Limited Functions: A Historical Overview

Abstract

Abstract. Shannon's sampling theorem is one of the most powerful results in signal analysis. The aim of this overview is to show that one of its roots is a basic paper of de la Vall6e Poussin of 1908. The historical development of sampling theory from 1908 to the present, especially the matter dealing with not necessarily band-limited functions (which includes the duration-limited case actually studied in 1908), is sketched. Emphasis is put on the study of error estimates, as well as on the delicate point-wise behavior of sampling sums at discontinuity points of the signal to be reconstructed.

Nonuniform Interpolation of Noisy Signals Using Support Vector Machines

“…Given , the interpolation problem consists of finding an approximating function that fits the observations as follows: (1) where ; is the bandwidth of the interpolating units (and in general has to be determined from some a priori knowledge or search strategy); and represents the noise. The previous continuous time series model, after nonuniform sampling, is expressed as the following discrete time model: (2) An optimal bandlimited interpolation algorithm, in the least squares (LS) sense, was first proposed by Yen [11]. The problem can be expressed as the minimization of the quadratic loss function, given by (3) which, in matrix notation, consists of minimizing (4) where is the vector of model coefficients, , and is a square matrix whose elements are (5) It can be seen that the solution vector is (6) This is a critically determined problem, as we have as many free parameters as observations, and in the presence of noise this yields an ill-posed problem [12].…”

“…In this section, we propose to use several SVM approaches for estimating efficiently coefficients in signal model (2). In the SVM framework for digital signal processing [21], the optimality criterion is a regularized and constrained version of the regularized LS criterion.…”

“…The first one consists of using signal model (2) as the primal problem in the SVM formulation. The second one consists of considering a generic nonlinear SVM regression, obtaining the SVM dual and solution equations for a generic Mercer's kernel, and finally introducing the sinc kernel as a Mercer's kernel for obtaining signal model (2) in the dual solution.…”

“…Specifically, a finite set of samples from a continuous-time function can be seen as a duration-limited discrete-time signal in practice, and then it cannot be bandlimited. In this case, the reconstruction of a signal from its samples depends on the a priori information that we have, and the classical sinc kernel has been replaced by more general kernels that are not necessarily bandlimited [2], [3]. This issue has been mostly studied in problems of uniformly sampled time series.…”

A formula for the error of finite sinc-interpolation over a finite interval

“…Approximating a function f from a sample at equidistant abscissae is a classical problem in engineering that gives rise to interesting mathematics and recurrently leads to the publication of survey papers and books such as [4,5,8,9,15]. One approach to Shannon's sampling theory takes advantage of the Lagrange property…”