# Exploring ISAC: Information-Theoretic Insights
## 5 Network ISAC with Sensing Distortion
### One-to-Many Communication (Broadcast Channels) with Tx Sensing
#### 5.1.2 Results  
The optimal estimator is again given by Lemma 1. That means, the optimal estimator is (irrespective of the choice of the encoding and decoding functions)  
\[h_{k}^{*}(x^{n},z^{n})=(\hat{s}_{k}^{*}(x_{1},z_{1}),\hat{s}_{k}^{*}(x_{2},z_{ 2}),\ldots,\hat{s}_{k}^{*}(x_{n},z_{n})). \tag{35}\]  
where \(\hat{s}^{*}(x,z)=\arg\min_{s^{\prime}\in\mathcal{S}}\sum_{s\in S}P_{S|XZ}(s|x, z)d(s,s^{\prime})\) was defined in (5) and the conditional probability distribution \(P_{S|XZ}(s|x,z)=\frac{P_{S}(s)P_{Z|XZ}(s|x,x)}{\sum_{i\in S}P_{S}(s)P_{Z|XZ}( s|x,x)}\) again only depends on the channel and the state distribution. Identification of this optimal estimator immediately allows a reduction of the problem of characterizing the capacity-distortion tradeoff region \(\mathcal{CD}\) of the broadcast channel (BC) to the problem of identifying the set of communication rates that are achievable under a given constraint on the statistics of the channel input symbols. In this sense, we again notice a decoupling of the sensing problem and the communication problem for the BC as for the point-to-point channel. The communication problem needs to be solved under a constraint on the channel input statistics, but otherwise the sensing part does not interfere. The pure communication problem over a memoryless BC with feedback is still open, and only inner and outer bounds are known for general channels. Notable exceptions are the classes of physically degraded BCs [94] (where feedback does not increase capacity) and other classes of BCs with states [7]. For these classes, with the help of the optimal estimator in (35), one can immediately characterize the capacity-distortion region \(\mathcal{CD}\), see [64]. For all other classes, the optimal estimator can be combined with the proposed coding schemes for BCs with feedback [2; 3; 4; 5; 6] the known infeasibility proofs (converses) to obtain inner and outer bounds on \(\mathcal{CD}\) for general ISAC BCs. The following inner and outer bounds on \(\mathcal{CD}\) were reported in [63], see also [64]. **Theorem 7** (Outer Bound). : _If \((R_{0},R_{1},R_{2},D)\) lies in \(\mathcal{CD}\), then there exist pmfs \(P_{X},P_{U_{1}|X},P_{U_{2}|X}\) such that the random tuple \((U_{k},X,S,Y_{1},Y_{2},Z)\sim P_{U_{k}|X}P_{X}P_{S}P_{Y_{1}Y_{2}Z|SX}\) satisfies the rate constraints_  
\[rCIR_{0}+R_{k} \leq I(U_{k};Y_{k}),\quad k=1,2, \tag{36a}\] \[R_{0}+R_{1}+R_{2} \leq I(X;Y_{1},Y_{2}), \tag{36b}\]  
_and the average distortion constraints_  
\[\mathbb{E}[d(S,\hat{s}^{*}(X,Z))]\leq D. \tag{37}\]  
**Proposition 1** (Inner Bound). : _The capacity-distortion region \(\mathcal{CD}\) includes all tuples \((R_{0},R_{1},R_{2},D)\) that for some choice of the auxiliaries \((U_{0},U_{1},U_{2},X,S,Y_{1},Y_{2},Z,V_{0},V_{1},V_{2})\sim P_{U_{0}U_{1}U_{ 2}X}P_{S}P_{Y_{1}Y_{2}}\)\({}_{Z|SX}P_{Y_{0}V_{1}V_{2}|U_{0}U_{1}U_{2}Z}\) satisfy inequalities (38) above and the distortion constraint (37)._  
\[R_{0}+R_{1} \leq I(U_{0},U_{1};Y_{1},V_{1})-I(U_{0},U_{1},U_{2},Z;V_{0},V_{1} \mid Y_{1}) \tag{38a}\] \[R_{0}+R_{2} \leq I(U_{0},U_{2};Y_{2},V_{2})-I(U_{0},U_{1},U_{2},Z;V_{0},V_{2} \mid Y_{2})\] (38b) \[R_{0}+R_{1}+R_{2} \leq I(U_{1};Y_{1},V_{1}|U_{0})+I(U_{2};Y_{2},V_{2}\mid U_{0})+\min_{k \in\{1,2\}}I(U_{0};Y_{k},V_{k})-I(U_{1};U_{2}\mid U_{0})\] \[-I(U_{0},U_{1},U_{2},Z;V_{1}\mid V_{0},Y_{1})-I(U_{0},U_{1},U_{2},Z;V_{2}|V_{0},Y_{2})\] \[-\max_{k\in\{1,2\}}I(U_{0},U_{1},U_{2},Z;V_{0}\mid Y_{k})\] (38c) \[2R_{0}+R_{1}+R_{2} \leq I(U_{0},U_{1};Y_{1},V_{1})+I(U_{0},U_{2};Y_{2},V_{2})-I(U_{1};U_{ 2}\mid U_{0})\] \[rCIR_{0}+R_{k} \leq I(U_{k};Y_{k}),\quad k=1,2, \tag{36a}\]0  
The outer bound is obtained by considering a genie-aided system where Rx 2 observes not only the \(Y_{2}\) outputs but also the \(Y_{1}\) outputs, and by using the optimal estimator in (35). The inner bound is obtained by again combining this optimal estimator with the scheme in [5] for broadcast communication with generalized feedback.