On the Need of Analog Signals and Systems for Digital-Twin Representations Holger Boche1 Yannik N. Böck2 Ullrich J. Mönich3 Frank H. P. Fitzek4

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On the Need of Analog Signals and Systems for

Digital-Twin Representations

Holger Boche1, Yannik N. Böck2, Ullrich J. Mönich3, Frank H. P. Fitzek4

Abstract

We consider the task of converting diﬀerent digital descriptions of analog ban-

dlimited signals and systems into each other, with a rigorous application of

mathematical computability theory. Albeit very fundamental, the problem ap-

pears in the scope of digital twinning, an emerging concept in the ﬁeld of digital

processing of analog information that is regularly mentioned as one of the key en-

ablers for next-generation cyber-physical systems and their areas of application.

In this context, we prove that essential quantities such as the peak-to-average

power ratio and the bounded-input/bounded-output norm, which determine the

behavior of the real-world analog system, cannot generally be determined from

the system’s digital twin, depending on which of the above-mentioned descrip-

tions is chosen. As a main result, we characterize the algorithmic strength of

Shannon’s sampling type representation as digital twin implementation and also

introduce a new digital twin implementation of analog signals and systems. We

show there exist two digital descriptions, both of which uniquely characterize

a certain analog system, such that one description can be algorithmically con-

verted into the other, but not vice versa.

Keywords: Bandlimited signal, digital twin, PAPR problem, BIBO stability,

compiler.

1Holger Boche is with the Technische Universität München, Lehrstuhl für Theoretische

Informationstechnik, 80290 Munich, Germany, and the Munich Center for Quantum Science

and Technology (MCQST), Schellingstr. 4, 80799 Munich, Germany. e-mail: boche@tum.de.

2Yannik N. Böck is with the Technische Universität München, Lehrstuhl für Theoretische

Informationstechnik, 80290 Munich, Germany. e-mail: yannik.boeck@tum.de.

3Ullrich J. Mönich is with the Technische Universität München, Lehrstuhl für Theoretische

Informationstechnik, 80290 Munich, Germany. e-mail: moenich@tum.de.

4F. H. P. Fitzek is with the Deutsche Telekom Chair of Communication Networks, Technical

University of Dresden, 01187 Dresden, Germany, and the Cluster of Excellence “Centre for

Tactile Internet with Human-in-the-Loop” (CeTI. email: frank.ﬁtzek@tu-dresden.de

Preprint submitted to Journal of L

X Templates October 11, 2022

arXiv:2210.04313v1 [cs.IT] 9 Oct 2022

1. Introduction

Bandlimited signals are essential to state-of-the-art information processing,

especially at the border between analog and digital systems. In the physical

world, be it in signal processing, control, communication or measurement tech-

nology, information is usually carried by analog, continuous-time signals. In

contrast, in the digital world, where most of the data processing takes place,

information is processed in discrete-time computational cycles. According to

Shannon’s sampling theorem, a sampling series can be used to uniquely recover

a bandlimited continuous-time signal from a discrete-time sequence of samples,

provided that the signal’s energy is ﬁnite and the samples are taken at least at

Nyquist rate [1].

Since the publication of Shannon’s seminal article, the development and in-

vestigation of sampling-type representations of analog signals and systems has

been an active ﬁeld of research. In order to meet the progressive requirements

of applied engineering, the relevant theory has been advanced into various di-

rections. Among others, this includes extensions to generalized function spaces

[2, 3, 4], modiﬁed sampling sequences [5], sampling-type representations of op-

erators [6, 7], and non-deterministic frameworks [8, 6]. The collected results

form the foundation of modern digital processing of analog information.

One of the most recent concepts in the ﬁeld of digital processing of analog

information, which is regularly referred to as one of the key enablers for next-

generation cyber-physical systems [9], is known as digital twinning. Originally

associated primarily with Industry 4.0 [10], the concept is now also attracting

great interest in many other areas of modern technology. For recent examples

from networking or medicine technologies, see [11, 12]. With the introduction of

the metaverse, real worlds will be transferred to virtual space. Here, information

processing will be even more dependent on human multi-modalities (human

senses) and its interaction with the digital domain, c.f. [13]. In order to make

human senses experienceable, the information will have to be processed in real

time. This raises the question of whether this processing can be done in the

digital world at all, or whether analog approaches might be the solution.

The expectations towards digital twin technologies are ambitious. In medical

research, for example, the idea of implementing digital twins of humans is con-

sidered, that can be employed for medical purposes such as virtual surgery. For

this kind of technologies, requirements regarding trustworthiness are clearly of

special relevance. In general, the number of every-day technologies that poten-

tially aﬀect sensitive human goods, like ﬁnancial resources, private information

or health, can be expected to rise signiﬁcantly with the increasing establishment

of digital twinning. The need to follow strict speciﬁcations on privacy, integrity,

reliability, safety and alike with regards to these technologies, is manifest. Con-

siderations of this kind are essential in view of future robotic systems, medicine

applications and 6G communication technologies, see e.g. [14]. We will discuss

this topic in detail in Section 10.

Although an unambiguous and widely accepted deﬁnition of the term digital

twin has not yet been established, the approach usually exhibits the following

abstract characteristics:

1) The starting point is an abstract set of arbitrary entities from the physical

world, usually in the context of some engineering problem. The abstract

set is often deﬁned implicitly by the problem statement. For example, it

may consist of all conﬁgurations and properties of a network of interacting

autonomous vehicles, or of all possible conﬁgurations and properties of

the individual parts of a combustion engine. Depending on the speciﬁc

application, there is a number of object related properties that we want

to predict. For example, in the case of combustion engines, this may, be

the expected fuel consumption in a certain operating state.

2) The objects in the abstract set are assigned a description in some language

that is readable by digital machines, see below. For example, the individ-

ual parts of the above-mentioned combustion engine may be characterized

by means of the ﬁnite element method within some computer-aided-design

(CAD) software. The machine-readable description is the object’s digital

representation, i.e., a digital twin. In real-time systems, the object’s digi-

tal twin is sequentially updated to match its real-world counterpart. This

is usually implemented by means of (physical) measurements and analog

to digital conversion.

3) The digital twin of the object is used as input for an algorithm, which in

turn is supposed to predict one of the object-related properties. In real-

time systems, the output of the algorithm can be used to control the real

system.

Schematically, the concept of a digital twin described above is shown in Figure 1.

Physical (analog) plane

Virtual (digital) plane

Property

prediction

—

Predictive

control

—

Decision

making

Digital

computing platform

Figure 1: Schematic representation of the digital twin approach according to the formalization

given in Section 1.

In the scope of this article, the term “language” as used above refers to a ﬁxed

method for characterizing abstract objects that is accessible to Turing machines,

e.g., the concept of discrete- and continuous-time descriptions of computable

bandlimited signals introduced in Section 5. It is not to be confused with a

formal language according to the strict mathematical deﬁnition. However, it

is noteworthy that, since the theory of Turing machines can be equivalently

formalized by the theory of formal languages, it is (in principle) possible to

formalize our framework in a manner such that it does coincide with formal

languages in the mathematical sense. In this context, the problem of converting

diﬀerent digital descriptions of analog objects into each other may be regarded

as a compiler problem.

Depending on the individual application, the implementation of a digital

twin can be arbitrarily complex. For example, the source code of a CAD ap-

plication can be thousands of lines long, and the data describing a particular

object can be several gigabytes in size. Digital twins of humans in healthcare

and robotics can be expected to be even more complex than that. Thus, the

question of the "proper" way to describe a real world object arises: if a certain

property about the real system should be predicted, which characteristics does

the language describing the system has to satisfy? The problem is visualized in

Figure 2. The answer to this question highly depends on the speciﬁc applica-

Physical (analog) plane

Language A Language B

A or B?

Figure 2: Digital twins of the same abstract object in diﬀerent machine-readable languages.

Even if both twins uniquely characterize the real object, not all information about it may be

algorithmically accessible in both languages. The choice of which language to use is thus a

creative engineering task and depends strongly on the speciﬁc application.

tion and the properties to be predicted. Often, the choice of language is also a

matter of feasibility and convenience.

In this article, we consider the standard digital signal processing description

of bandlimited signals and systems in connection with digital twin technology.

Albeit digital twins are mostly associated to complex systems like networks

of autonomous vehicles or combustion engines, the digital processing of ban-

dlimited signals satisﬁes all above criteria, as will be discussed in Section 2.

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OntheNeedofAnalogSignalsandSystemsforDigital-TwinRepresentationsHolgerBoche1,YannikN.Böck2,UllrichJ.Mönich3,FrankH.P.Fitzek4AbstractWeconsiderthetaskofconvertingdierentdigitaldescriptionsofanalogban-dlimitedsignalsandsystemsintoeachother,witharigorousapplicationofmathematicalcomputabilitytheory.Alb...

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On the Need of Analog Signals and Systems for Digital-Twin Representations Holger Boche1 Yannik N. Böck2 Ullrich J. Mönich3 Frank H. P. Fitzek4

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