Turing Meets Shannon: Computable Sampling Type Reconstruction With Error Control

Abstract

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.