## 4 Discussion  
In Section 2, we have established a formal framework of machine-readable languages for analog bandlimited signals \(\boldsymbol{f}\in\mathcal{C}\mathcal{B}(\star)\) (recall that we used the symbol '\(\star\)' as a placeholder that may be uniformly replaced by '\(1\)' or '\(\infty\)'). In particular, we have introduced the languages \(\mathfrak{X}_{\star}\) and \(\mathfrak{F}_{\star}\), which formalize discrete-time descriptions and continuous-time descriptions for the elements of \(\mathcal{C}\mathcal{B}(\star)\), respectively. In Section 3, we have applied the established framework to a standard engineering model for wireless communication networks. Particularly, the model is relevant in the context of digital twinning, an exemplary application of which we have discussed in Section 1. Then, we have shown that discrete-time descriptions are, despite being the quasi-standard in digital signal processing, an insufficient representative of the elements of \(\mathcal{C}\mathcal{B}(\star)\): In contrast to continuous-time descriptions, they do _not_ allow for computing the norm \(\|\cdot\|_{\star}\), which we substantiated to be a relevant property within the considered model for wireless communication networks. Finally, we have shown that any computation on the basis of discrete-time descriptions can be processed on the basis of continuous-time descriptions equally, i.e., continuous-time descriptions capture \"more\" information than discrete-time descriptions. Section 3 focuses on the \(\|\cdot\|_{\star}\)-norm as relevant property. Using the methods and techniques we established in Sections 2 and 3, we can extend the results to several other properties that can be considered relevant in particular applications. We will discuss several of them in the following. * Time concentration. For \(L\in\mathbb{R}_{\mu}\), denote by \(\operatorname{rct}(\cdot,L):\mathbb{R}\to\mathbb{R}\) the function that equals one for \(-L\leq t\leq L\), zero otherwise. For a signal \(\mathbf{f}\in\mathcal{CB}(\star)\), the function \(f_{L}:\mathbb{R}\to\mathbb{R},t\mapsto\operatorname{rct}(t,L)\mathbf{f}(t)\) is an element of \(\mathcal{L}^{\star}(\mathbb{R})\) (c.f. [12], Definition 2.1, p. 15). We refer to the mapping \[\Gamma:\mathcal{CB}(\star)\times\mathbb{R}_{\mu}\to\mathbb{R}_{\mu},(\mathbf{f},L) \mapsto\|f_{L}\|_{\star}\] as time concentration. It is computable as a mapping \(\Gamma:\mathfrak{F}_{\star}\times\mathfrak{R}\to\mathfrak{R}\), but _not_ as a mapping \(\Gamma:\mathfrak{F}_{\star}\times\mathfrak{R}\to\mathfrak{R}\). For details, we refer to [20];
* Time evaluation. We refer to the mapping \(\Phi:\mathcal{CB}(\star)\times\mathbb{R}_{\mu}\to\mathbb{C}_{\mu},(\mathbf{f},t) \mapsto\mathbf{f}(t)\) as time evaluation. It is computable as a mapping \(\Phi:\mathcal{\mathfrak{F}}_{1}\times\mathfrak{R}\to\mathfrak{C}\), \(\Phi:\mathfrak{F}_{1}\times\mathfrak{R}\to\mathfrak{C}\), and \(\Phi:\mathfrak{F}_{\infty}\times\mathfrak{R}\to\mathfrak{C}\), but _not_ as a mapping \(\Phi:\mathfrak{F}_{\infty}\times\mathfrak{R}\to\mathfrak{C}\);
* Time derivative. Denote the time derivative of \(\mathbf{f}\in\mathcal{CB}(\star)\) by \(\
icefrac{{d}}{{dt}}\mathbf{f}\) (among other things, computing \(\mathbf{f}\mapsto\
icefrac{{d}}{{dt}}\mathbf{f}\) is essential for estimating system dynamics by means of the mean-value theorem). We can directly compute the time derivative of elementary computable functions in \(\mathcal{CB}(\star)\). Further, we have \(\|\
icefrac{{d}}{{dt}}\mathbf{f}\|_{\star}\leq\pi\|\mathbf{f}\|_{\star}\) for all \(\mathbf{f}\in\mathcal{CB}(\star)\). Hence, \(\
icefrac{{d}}{{dt}}\) is computable as a mapping \(\
icefrac{{d}}{{dt}}:\mathfrak{F}_{\star}\to\mathfrak{F}_{\star}\). In contrast, \(\
icefrac{{d}}{{dt}}\) is _not_ computable as a mapping \(\
icefrac{{d}}{{dt}}:\mathfrak{F}_{\star}\to\mathfrak{F}_{\star}\). As indicated in Section 1, the upcoming 6G communication standard specifies large parts of the technological infrastructure required for applications of digital twinning.