# Exploring ISAC: Information-Theoretic Insights
## 5 Network ISAC with Sensing Distortion
### One-to-Many Communication (Broadcast Channels) with Tx Sensing  
#### 5.1.1 The Memoryless Model  
Consider the single-Tx two-Rx broadcast ISAC system, which is depicted in Figure 7.  
Extensions to multiple Rxs follow standard techniques. The setup is similar to the single-user setup in Section 3.1, however, communication is to two distinct Rxs, 1 and 2. Specifically, the Tx wishes to communicate the rate-\(R_{0}\) message \(W_{0}\) to both Rxs, the rate-\(R_{1}\) message \(W_{1}\) to Rx 1, and the rate-\(R_{2}\) message \(W_{2}\) to Rx 2. The Tx thus produces inputs of the form \(X_{i}=\phi_{i}(W_{0},W_{1},W_{2},Z_{1},\ldots,Z_{i-1})\), for \(i=1,\ldots,n\). The communication channel and the generalized feedback channel are governed by a state sequence \(S^{n}\) that is i.i.d. according to \(P_{S}\) and is jointly modeled by a stationary memoryless channel of transition probabilities \(P_{Y_{1}Y_{2}Z|XS}(y_{1},y_{2},z|x,s)\) determining the outputs \(\{Y_{1,i}\}\) at Rx 1, the outputs \(\{Y_{2,i}\}\) at Rx 2, and the generalized feedback signals \(\{Z_{i}\}\) at the Tx. Based on its observed channel outputs \(Y_{k,1}\),..., \(Y_{k,n}\), each Rx \(k\in\{1,2\}\) produces the guesses \(\hat{W}_{0,k}\) and \(\hat{W}_{k}\) of the messages \(W_{0}\) and \(W_{k}\) using appropriate decoding functions \((\hat{W}_{0,k},\hat{W}_{k})=g_{k}(Y_{k,1},\ldots,Y_{k,n})\), and the Tx estimates the state sequences as \((\hat{S}_{1},\ldots,\hat{S}_{n})=h(X_{1},\ldots,X_{n},Z_{1},\ldots,Z_{n})\). As  
Figure 7: State-dependent broadcast channel with generalized feedback and state estimator at the Tx.  
before, communication performance is measured in terms of decoding error probabilities and sensing performance in terms of expected average per-block distortion.  
Accordingly, we have the following achievability definition, for a given bounded and non-negative distortion function \(d(\cdot,\cdot)\).  
**Definition 3**.: _A rate-distortion tuple \((\mathsf{R}_{0},\mathsf{R}_{1},\mathsf{R}_{2},\mathsf{D})\) is achievable if there exists a sequence (in \(n\)) of encoding, decoding, and state estimation functions such that_  
\[rCl\lim_{n\to\infty}\Pr\bigl{(}\hat{W}_{k}\
eq W_{k}\quad\text{or} \quad\hat{W}_{0,k}\
eq W_{0}\bigr{)} = 0,\qquad k\in\{1,2\}, \tag{34a}\] \[\overline{\lim_{n\to\infty}}=\frac{1}{n}\sum_{i=1}^{n}\mathbb{E}[ d(S_{i},\hat{S}_{i})] \leq \mathsf{D} \tag{34b}\]  
_The closure of the set of all achievable rate-distortion tuples \((\mathsf{R}_{0},\mathsf{R}_{1},\mathsf{R}_{2},\mathsf{D})\) is called the capacity-distortion region \(\mathcal{CD}\)._  
**Remark 2**.: _The above model was considered in [63]. The authors of [64] considered a slightly different model where the state is composed of two components \(S=(S_{1},S_{2})\), where each \(S_{k}\) is revealed to the corresponding \(Rx\) and has to be estimated at the Tx up to a maximum allowed distortion \(D_{k}\). As already mentioned for the point-to-point setup, receiver state information is included in the above model as a special case, by including the state information as part of the channel outputs. Similarly, since the state alphabet can be arbitrary but finite, in the above model, \(S\) can also be a pair of finite states \((S_{1},S_{2})\). To be able to fully capture the setup and the results in [64] as special cases, it thus suffices to extend the above model from [63] to multiple distortion constraints, which can easily be done. The advantage of the model in [63] is that it is more general and allows the modeling of all kinds perfect or imperfect state information at the Rxs._