Downsampling of Bounded Bandlimited Signals and the Bandlimited Interpolation: Analytic Properties and Computability

Abstract

Downsampling and the computation of the bandlimited interpolation of discrete-time signals are two important concepts in signal processing. In this paper we analyze the downsampling operation regarding its impact on the existence and computability of the bounded bandlimited interpolation. We assume that the discrete-time signal is obtained by downsampling the samples of a bounded bandlimited signal that vanishes at infinity, and we study two problems. First, we investigate the existence of the bounded bandlimited interpolation for such discrete-time signals from a signal theoretic perspective and show that there exist signals for which the bounded bandlimited interpolation does not exist. Second, we analyze the algorithmic generation of the bounded bandlimited interpolation, using the concept of Turing computability. Turing computability models what is theoretically implementable on a digital computer. Interestingly, it turns out that even if the bounded bandlimited interpolation exists analytically, it is not always computable, which implies that there exists no algorithm on a digital computer that can always compute it. Computability is important in order that the approximation error be controlled. If a signal is not computable, we cannot ascertain whether the computed signal is sufficiently close to the true signal, i.e., we cannot verify every approximation accuracy.