## 3 Results
Analogous to the situation at the transmitter, we will not be able to measure the signal \(\mathbf{y}_{\text{re,}t}\) perfectly at the receiving end. Due to finite quantization accuracy and imperfect filtering, we obtain an approximate signal \(\widetilde{\mathbf{y}}_{\text{re,}t}\). Since the overall duration of sampling is finite, we can assume \(\widetilde{\mathbf{y}}_{\text{re,}t}\) and \(\mathbf{T}_{\infty}\widetilde{\mathbf{y}}_{\text{re,}t}=:\widetilde{f}_{\text{re,}t}\) are elementary computable. Choosing \(\mathbf{y}_{\text{de,}t,n}\in\text{dom}(\mathbf{T}_{\infty})\cap\mathcal{C}\ell_{0}^{ \infty},n\in\{1,\dots,N\}\), and denoting \(\mathbf{f}_{\text{de,}t,n}:=\mathbf{T}_{\infty}\mathbf{y}_{\text{de,}t,n},n\in\{1,\dots,N\}\), we generally have \(\|\widetilde{\mathbf{y}}_{\text{re,}t}-\mathbf{y}_{\text{de,}t,n}\|_{\infty}\
eq\| \widetilde{f}_{\text{re,}t}-\mathbf{f}_{\text{de,}t,n}\|_{\infty}\). Aside from computational convenience, there is no a priori reason to perform decoding based on \(\widetilde{\mathbf{y}}_{\text{re,}t}\) and \(\mathbf{y}_{\text{de,}t,1},\dots,\mathbf{y}_{\text{de,}t,N}\) rather than \(\widetilde{f}_{\text{re,}t}\) and \(\mathbf{f}_{\text{de,}t,1},\dots,\mathbf{f}_{\text{de,}t,N}\). In view of the mentioned limitations of real-world systems, this observation becomes even more relevant: since effects such as quantization are generally _non_-linear, information may actually be lost if the decoding is performed in the space \(\mathcal{C}\ell_{0}^{\infty}\) rather than \(\mathcal{C}\mathbf{B}_{0,\pi}^{\infty}\). Unless proven otherwise for a specific case, the same argument holds true when we consider decoding schemes other than region-based ones. 4. From a model-based perspective, taking imperfect sampling at the receiving end into account raises another issue when considering the entire system within \(\mathcal{C}\ell^{1}\). As indicated above, we may want to design the system with a specified margin-of-error, i.e., we want to guarantee that proper message transmission is possible as long as we have \(\|\mathbf{w}\|_{\infty}<\epsilon_{\mathbf{w}}\). If we instead require \(\|\mathbf{T}_{\infty}\mathbf{w}\|_{\infty}<\epsilon_{\mathbf{w}}\), we can provide the continuous-time-domain upper bound \[\big{\|}\widetilde{f}_{\text{re,}t}-\big{(}\mathbf{T}_{\infty}\mathbf{w}+(\mathbf{f}_{ \text{ph,}t}*\widetilde{f}_{t,m})\big{)}\big{\|}_{\infty}\leq\epsilon_{\mathbf{w} }+\epsilon_{\text{ph}}\big{\|}\widetilde{f}_{t,m}\big{\|}_{\infty}+\big{\|} \widetilde{f}_{\text{re,}t}-(\mathbf{f}_{t}*\widetilde{f}_{t,m})\big{\|}_{\infty}\] for the reconstruction error, which can then be transferred to the discrete-time domain. Requiring \(\|\mathbf{w}\|_{\infty}<\epsilon_{\mathbf{w}}\) alone is insufficient, since \(\mathbf{T}_{\infty}\mathbf{w}\) may be arbitrarily large in this case nevertheless. Thus, we can provide an estimate for the reconstruction error only if we consider the continuous-time domain of the system. Consequently, we consider signals \(\mathbf{f}_{\text{ph,}t}\in\mathcal{B}_{\pi}^{1}\), \(\mathbf{f}_{t}\in\mathcal{C}\mathbf{B}_{\pi}^{1}\), \(\widetilde{f}_{t,m},\mathbf{f}_{\text{de,}t,n}\in\mathcal{C}\mathbf{B}_{0,\pi}^{\infty}\), \(t\in\mathbb{N}\), \(m\in\{1,\dots,M\}\), \(n\in\{1,\dots,N\}\), and a communications system of the form \[d\Big{(}\widetilde{\varkappa}_{t},\big{(}\widetilde{\varkappa},\Lambda_{\Psi}, [\cdot]_{\Psi}\big{)}\Big{)}:=\big{\|}\widetilde{\varkappa}_{t}-\widetilde{ \varkappa}\big{\|}_{2},\qquad\widetilde{\varkappa}\in\widetilde{\mathbb{R}}^{2}\]0 (5) \[d\Big{(}\widetilde{\varkappa}_{t},\big{(}\widetilde{\varkappa},\Lambda_{\Psi}, [\cdot]_{\Psi}\big{)}\Big{)}:=\big{\|}\widetilde{\varkappa}_{t}-\widetilde{ \varkappa}\big{\|}_{2},\qquad\widetilde{\varkappa}\in\widetilde{\mathbb{R}}^{2}\]1 (6) in the following. Observe that we do not aim at actually implementing signal processing in the analog domain.