Rudolf L. Stens

Sampling theory for not necessarily band-limited functions: a historical overview

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 Vallee 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.

Prediction by Samples From the Past With Error Estimates Covering Discontinuous Signals

There are several reasons why the classical sampling theorem is rather impractical for real life signal processing. First, the sinc-kernel is not very suitable for fast and efficient computation; it decays much too slowly. Second, in practice only a finite number N of sampled values are available, so that the representation of a signal f by the finite sum would entail a truncation error which decreases rather slowly for N¿ ¿, due to the first drawback. Third, band-limitation is a definite restriction, due to the nonconformity of band and time-limited signals. Further, the samples needed extend from the entire past to the full future, relative to some time t = t0. This paper presents an approach to overcome these difficulties. The sinc-function is replaced by certain simple linear combinations of shifted B-splines, only a finite number of samples from the past need be available. This deterministic approach can be used to process arbitrary, not necessarily bandlimited nor differentiable signals, and even not necessarily continuous signals. Best possible error estimates in terms of an Lp-average modulus of smoothness are presented. Several typical examples exhibiting the various problems involved are worked out in detail.

Linear Prediction by Samples from the Past

If a signal f is band-limited to [—πW, πW] for some W > 0, then f can be completely reconstructed for all values of t ∈ R from its sampled values f (k/W), k ∈ Z, taken just at the nodes k/W, equally spaced apart on the whole R, in terms of

Multiplex signal transmission and the development of sampling techniques: the work of Herbert Raabe in contrast to that of Claude Shannon

\begin{array}{*{20}{c}} {f(t) = \sum\limits_{{k = - \infty }}^{\infty } {f\left( {\frac{k}{W}} \right)\frac{{\sin \pi (Wt - k)}}{{\pi (Wt - k)}}} } & {(t \in R).} \\ \end{array}

Function Spaces, Approximation Theory, and Their Applications

(5.1.1)

An Introduction to Sampling Analysis

The paper is concerned with Shannon sampling reconstruction formulae of derivatives of bandlimited signals as well as of derivatives of their Hilbert transform, and their application to Boas-type formulae for higher order derivatives. The essential aim is to extend these results to non-bandlimited signals. Basic is the fact that by these extensions aliasing error terms must now be added to the bandlimited reconstruction formulae. These errors will be estimated in terms of the distance functional just introduced by the authors for the extensions of basic relations valid for bandlimited functions to larger function spaces. This approach can be regarded as a mathematical foundation of aliasing error analysis of many applications.

Approximation of continuous and discontinuous functions by generalized sampling series

This article discusses the interplay between multiplex signal transmission in telegraphy and telephony, and sampling methods. It emphasizes the works of Herbert Raabe (1909–2004) and Claude Shannon (1916–2001) and the context in which they occurred. Attention is given to the role that the exceptional research atmosphere in Berlin during the 1920s and early 1930s played in the development of some of the ideas underlying these works, first in Germany and then in the USA, as some of the protagonists moved there. Raabes thesis, published in 1939, describes and analyses a time-division multiplex system for telephony. In order to build his working prototype, Raabe had to develop the theoretical tools he needed and achieved a thorough understanding of sampling, including sampling with pulses of finite duration and sampling of low-pass and band-pass signals. His condition for reconstruction was known as ‘Raabes condition’ in the German literature of the time. On the other hand, Shannons works of 1948 and 1949 contain the classical sampling theorem, but go much further and lay down the abstract theoretical framework that underlies much of the modern digital communications. It is interesting to compare Raabes very practical approach with Shannons abstract work: Raabe independently developed his methods to the degree he needed, but his main purpose was to build a working prototype. Shannon, on the other hand, approached sampling independently of practical constraints, as part of information theory – which became tremendously influential.

Convergence in Variation and Rates of Approximation for Bernstein-Type Polynomials and Singular Convolution Integrals

The aim of the paper is to extend some results concerning univariate generalized sampling approximation to the multivariate frame. We give estimates of the approximation error of the multivariate generalized sampling series for not necessarily continuous functions in \(L^{p}(\mathbb{R}^{n})\)-norm, using the averaged modulus of smoothness of Sendov and Popov type. Finally, we study some concrete examples of sampling operators and give applications to image processing dealing, in particular, with biomedical images.