On the Arithmetic Complexity of the Bandwidth of Bandlimited Signals

TL;DR

This paper investigates the computability and arithmetic complexity of the bandwidth of bandlimited signals, revealing fundamental limitations in computing exact or approximate bandwidths using Turing machines.

Contribution

It characterizes the arithmetic complexity of signal bandwidths and demonstrates the impossibility of computing non-trivial bounds for them.

Findings

Existence of computable signals with non-computable bandwidths

Impossibility of computing non-trivial bounds for bandwidths

Connection between bandwidth computation and oracle machine theory

Abstract

The bandwidth of a signal is an important physical property that is of relevance in many signal- and information-theoretic applications. In this paper we study questions related to the computability of the bandwidth of computable bandlimited signals. To this end we employ the concept of Turing computability, which exactly describes what is theoretically feasible and can be computed on a digital computer. Recently, it has been shown that there exist computable bandlimited signals with finite energy, the actual bandwidth of which is not a computable number, and hence cannot be computed on a digital computer. In this work, we consider the most general class of band-limited signals, together with different computable representations thereof. Among other things, our analysis includes a characterization of the arithmetic complexity of the bandwidth of such signals and yields a negative answer to the question of whether it is at least possible to compute non-trivial upper or lower bounds for the bandwidth of a bandlimited signal. Furthermore, we relate the problem of bandwidth computation to the theory of oracle machines. In particular, we consider halting and totality oracles, which belong to the most frequently investigated oracle machines in the theory of computation.