# Exploring ISAC: Information-Theoretic Insights
## 3 Mono-Static ISAC with Sensing Distortion
### The Capacity-Distortion-Cost Tradeoff  
The goal of information-theoretic studies is to identify the optimal performance that can be achieved by choosing the best system implementation under given modeling assumptions. In the present case, we consider the memoryless model introduced in the previous section, Section 3.1, and wish to determine the performance of the optimal data encoders and decoders, and state estimators. We will limit the study to encoders/decoders that have arbitrarily small error probabilities when the blocklength \(n\) grows without bounds. A rate-distortion-cost tuple \((\mathsf{R},\mathsf{D},\mathsf{B})\) is said achievable if there exists a sequence (in \(n\)) of encoding, decoding, and estimation functions \((\phi_{1},\ldots,\phi_{n},g,h)\) that simultaneously satisfy  
\[rCl\lim_{n\to\infty}\Pr[\hat{W}\
eq W] = 0, \tag{4a}\] \[\overline{\lim}_{n\to\infty}\;\Delta^{(n)} \leq \mathsf{D},\] (4b) \[\overline{\lim}_{n\to\infty}\;\frac{1}{n}\sum_{i=1}^{n}\mathbb{E }[b(X_{i})] \leq \mathsf{B}. \tag{4c}\]  
The capacity-distortion-cost tradeoff \(\mathsf{C}(\mathsf{D},\mathsf{B})\) is the largest rate \(\mathsf{R}\) such that the rate-distortion-cost triple \((\mathsf{R},\mathsf{D},\mathsf{B})\) is achievable. The main result of [PERSON] et al. in [56], see Theorem 1 below, provides an exact characterization of \(\mathsf{C}(\mathsf{D},\mathsf{B})\). It assumes that the receiver has perfect state information at the receiver. Here, we give the more general result that also includes the case without receiver state information. Note that this model is more general because any kind (even imperfect) of receiver-side information can be provided to the receiver as part of the output \(Y\), in which case the original results of [56] are recovered. For a more detailed discussions on this, see [63]. A first step to obtain this result is to describe the optimal estimator, which in the present memoryless model is pleasingly simple, because it operates on a symbol-by-symbol basis. That means, estimate \(\hat{S}_{i}\) of the \(i\)-th state symbol \(S_{i}\) is solely based on the \(i\)-th input \(X_{i}\) and feedback signal \(Z_{i}\). **Lemma 1** ([56]). : _Define the function_  
\[r\mathsf{Cl}\hat{s}^{*}(x,z) := \arg\min_{s^{\prime}\in\mathcal{S}}\sum_{s\in\mathcal{S}}P_{S|XZ }(s|x,z)d(s,s^{\prime}), \tag{5}\]  
_where ties can be broken arbitrarily, and_  
\[P_{S|XZ}(s|x,z)=\frac{P_{S}(s)P_{Z|SX}(z|s,x)}{\sum_{s\in\mathcal{S}}P_{S}( \bar{s})P_{Z|SX}(z|\bar{s},x)}. \tag{6}\]  
_Irrespective of the choice of encoding and decoding functions, distortion \(\Delta^{(n)}\) in (4b) is minimized by the estimator_  
\[h^{*}(x^{n},z^{n}):=(\hat{s}^{*}(x_{1},z_{1}),\hat{s}^{*}(x_{2},z_{2}),\ldots, \hat{s}^{*}(x_{n},z_{n})). \tag{7}\]  
_Note that the function \(\hat{s}^{*}(\cdot,\cdot)\) only depends on the SDMC channel law \(P_{YZ|SX}\) and the state distribution \(P_{S}\)._  
To be used later, define the following sets of input distributions:  
\[\mathcal{P}_{\mathsf{B}} =\bigg{\{}P_{X}\,\bigg{|}\,\sum_{x\in\mathcal{X}}P_{X}(x)b(x)\leq \mathsf{B}\bigg{\}}, \tag{8a}\] \[\mathcal{P}_{\mathsf{D}} =\bigg{\{}P_{X}\,\bigg{|}\,\sum_{x\in\mathcal{X}}P_{X}(x)\mathbb{ E}[d(S,\hat{s}^{*}(X,Z))|X=x]\leq\mathsf{D}\bigg{\}}. \tag{8b}\]  
Then, the minimum distortion for a given cost \(\mathsf{B}\) is given by  
\[\mathsf{D}_{\min}(\mathsf{B}):=\min_{P_{X}\in P_{\mathsf{B}}}\sum_{x\in \mathcal{X}}P_{X}(x)\mathbb{E}[d(S,\hat{s}^{*}(X,Z))|X=x]. \tag{9}\]  
The main result in [56; 63] is the following theorem:  
**Theorem 1**. : _The capacity-distortion-cost tradeoff of an SDMC \(P_{YZ|SX}\) with state distribution \(P_{S}\) is:_  
\[\mathsf{C}(\mathsf{D},\mathsf{B})=\max_{P_{X}\in(P_{\mathsf{B}}\cap P_{ \mathsf{D}})}I(X;Y),\qquad\mathsf{D}\geq\mathsf{D}_{\min}\ \mathsf{B}\geq \mathsf{0}. \tag{10}\]  
**Remark 1**.