## 3 Results
In the setup depicted in Figure 1, the choice of the pair \((\mathcal{E}_{t},\mathcal{D}_{t})\) will generally be based on the robot's digital twin, i.e., (assuming both the receiving end and the robot itself have access to the sequence \((\mathfrak{a}_{t})_{t\in\mathbb{N}}\)), it will be implemented through computable mappings \(\mathfrak{a}_{t}\mapsto\mathcal{E}_{t}\), \(\mathfrak{a}_{t}\mapsto\mathcal{D}_{t}\), involving an optimization of some kind. For example, we may aim to choose \(\mathbf{y}_{t,1},\dots,\mathbf{y}_{t,M}\) and \(y_{\text{de},t,1},\dots,y_{\text{de},t,N}\) such that  
\[m=D\big{(}\arg\min_{n\in\{1,\dots,N\}}\big{\|}\mathbf{w}+\left(x_{\text{ph},t}*y_{ t,m}\right)-y_{\text{de},t,n}\big{\|}_{\infty}\big{)}\]  
for all \(m\in\{1,\dots,M\}\) and all \(\mathbf{w}\in\ell_{0}^{\infty}\) that satisfy \(\|\mathbf{w}\|_{\infty}<\mathbf{e}_{\mathbf{w}}\) for some \(\mathbf{e}_{\mathbf{w}}>0\). Accordingly, upon implementing the communication system, we require \(y_{\text{de},t,1},\dots,y_{\text{de},t,N}\in\mathcal{C}\ell_{0}^{\infty}\) and \(\mathbf{y}_{t,1},\dots,\mathbf{y}_{t,M}\in\text{dom}(\mathcal{T}_{\infty})\cap \mathcal{C}\ell_{0}^{\infty}\). In theory, we can then entirely neglect the analog part of the real system, i.e., the transmission of the signals through the physical (analog) medium, in the design of our signal processing algorithms. Again, recall that both \(\mathfrak{X}_{\star}\) and \(\mathfrak{\tilde{X}}_{\star}\) are machine-readable languages for the set \(\mathcal{CB}(\star)\). In particular, per its definition, \(\mathfrak{X}_{\star}\) does _not_ include the entirety of \(\mathcal{C}\ell(\star)\), which appears unnecessary following the discussion above. In fact, there does not seem to be an obvious reason to consider the space \(\mathcal{CB}(\star)\) at all. Assuming we are able to prove that the implementation of our systems satisfies (4), we even can perform the optimization of \((\mathcal{E}_{t},\mathcal{D}_{t})\) within the discrete-time domain. Upon closer inspection, we find that despite the discussion above, there exists a variety of reasons why we cannot neglect the analog part of the communication system. We will discuss several of them in the following:  
1. As mentioned above, any real implementation of our system will involve a steps of digital-to-analog conversion of the transmission signal \(\mathbf{y}_{t,m}\). However, any real-world digital-to-analog converter will not be able to synthesize the signal \(\mathcal{T}_{\infty}\mathbf{y}_{t,m}\) perfectly. More realistically, we will be able to synthesize some signal \(\widehat{f}_{t,m}\), that exhibits distortion from effects such as quantization and imperfect filtering. Depending on the application, it may be necessary to compute the signal \(\widehat{f}_{t,m}\) in the first place, or at least compute the error \(\|\widehat{f}_{t,m}-\mathcal{T}_{\infty}\mathbf{y}_{t,m}\|_{\infty}\), in order to ensure the proper transmission of messages. 2. The generalized Shannon equivalence, which motivates the processing of signals in the digital domain, is applicable as longs as the entirety of the considered system is linear. In practical wireless communications systems, _non_-linear distortions are a common issue. In particular, the analog subsystems of both the transmitter and the receiving end have to operate within a certain dynamic range, which sets an upper limit to the \(\|\cdot\|_{\infty}\)-norm of the analog signals they can process properly. Accordingly, in order to avoid _non_-linear distortions, we need to be able to compute or at least upper-bound the values \(\|\widehat{f}_{t,m}\|_{\infty}\) and \[\big{\|}f_{\text{ph},t}*\widetilde{f}_{t,m}\big{\|}_{\infty}\leq\big{\|}f_{ \text{ph},t}\big{\|}_{1}\big{\|}\widehat{f}_{t,m}\big{\|}_{\infty}\leq\big{(} \big{\|}f_{t}\big{\|}_{1}+\epsilon_{\text{ph}}\big{)}\big{\|}\widehat{f}_{t,m} \big{\|}_{\infty}.\]The details of this issue are investigated in the context of bounded-input-bounded-output (BIBO) stability analysis and the peak-to-average power ratio (PAPR) problem. 3.