# Exploring ISAC: Information-Theoretic Insights
## 5 Network ISAC with Sensing Distortion
### Multi-Access ISAC: Collaborative Sensing and Suboptimality of Symbolwise Estimators
#### 5.2.2 Results  
Determining the set of all achievable rates for the MAC with feedback is even open for only data communication, without the sensing task. Only a non-computable multi-letter expression is known in the general case [96]. Exceptions are the Gaussian MAC with perfect feedback [97] and a class of semi-deterministic MACs with one- or two-sided perfect feedback [98]. Various coding schemes [99; 100; 95; 99; 101] have been proposed, as well as an outer bound on the feedback capacity based on the dependence balance bound [102]. A recurrent theme in the presented coding schemes is that the feedback links to the two Txs allow to build up cooperation between the Txs. In fact, the feedback links establish a communication path from one Tx to the other, and the two Txs can thus (either implicitly as in [97] or explicitly as in [99; 100; 95; 101]) align future channel inputs through cooperation, which amplifies the signals compared to the noise and allows for a better decoding performance at the Rx. As we shall see, the same idea is also key for proposing good MAC ISAC schemes. There is a second fundamental idea that is required to achieve good ISAC MAC schemes, as we shall see in the following. It is inspired from and closely related to the works on multi-access communication over state-dependent channels where the Txs both have state information, see for example [103; 104; 105; 106; 107]. We first present infeasibility results for the ISAC multi-access problem. A first outer bound on the capacity distortion region \(\mathcal{CD}\) was established in [65] and then improved in [66]. The outer bound in [66] is:  
**Theorem 8** (Outer Bound). : _The capacity-distortion region of the ISAC MAC \(\mathcal{CD}\) is included in the set of all tuples \((R_{1},R_{2},D_{1},D_{2})\) that for some pmf \(P_{\text{QQ}_{2}}P_{\text{X}_{1}\text{X}_{2}|\text{Q}\text{Q}_{2}}\) satisfy:_  
\[R_{1} \leq I(X_{1};YZ_{1}Z_{2}|X_{2}Q_{2}), \tag{41}\] \[R_{2} \leq I(X_{2};YZ_{1}Z_{2}|X_{1}QQ_{2}),\] (42) \[R_{1}+R_{2} \leq I(X_{1}X_{2};YZ_{1}Z_{2}|QQ_{Z}),\] (43) \[R_{1}+R_{2} \leq I(X_{1}X_{2};Y), \tag{44}\]  
_with the dependence balance constraint:_  
\[I(X_{1};X_{2}|QQ_{2})\leq I(X_{1};X_{2}|Z_{1}Z_{2}QQ_{Z}), \tag{45}\]and the sensing constraints:_  
\[D_{k} \geq\mathsf{E}d(S_{k},s^{*}(Z_{1},Z_{2},X_{1},X_{2})),\quad k\in\{1,2\}, \tag{46}\] \[f_{k,R-D}(D_{k}) \leq I(S_{k}X_{k^{\prime}};Z_{k}|X_{k}Q),\quad k^{\prime},k\in\{1, 2\},\;k^{\prime}\
eq k,\] (47) \[f_{k,R-D}(D_{k}) \leq I(S_{k};Z_{1}Z_{2}|X_{1}X_{2}Q),\quad k\in\{1,2\}, \tag{48}\]  
_where \(f_{k,R-D}(D_{k})\) is the standard rate-distortion function of source \(S_{k}\)._  
_It suffices to consider \(Q\) and \(Q_{Z}\) whose alphabets \(Q\) and \(Q_{Z}\) have cardinalities satisfying \(|\mathcal{Q}|\cdot|\mathcal{Q}_{Z}|\leq|\mathcal{X}_{1}|\cdot|\mathcal{X}_{2}|+3\)._  
The outer bound is obtained by combining standard information-theoretic bounding steps with the following three key ideas: (1) providing Tx \(k\) also with Tx \(k^{\prime}\) inputs and outputs \((X_{k^{\prime}}^{n},Z_{k^{\prime}}^{n})\), for \(k,k^{\prime}\in\{1,2\}\) and \(k\
eq k^{\prime}\), during the sensing task can only improve sensing performance and leads to constraint (46); (2) applying dependence balance considerations a la [PERSON] and [PERSON] [102] based on the pair of generalized feedback outputs \((Z_{1},Z_{2})\) yields a valid constraint, see (45); and (3) the sensing distortion at Tx \(k\) cannot be smaller than the minimum sensing distortion in a joint source channel coding problem where the source \(S_{k}^{n}\) is transmitted from Tx \(k^{\prime}\) to \(k\), see constraints (47) and (48). The former two key ideas were already exploited to derive the outer bound in [65]. Idea (3) was proposed in [66] and allows a strictly improved bound to be obtained. A first coding scheme (and thus achievability result) for the ISAC MAC was proposed in [65] based on [PERSON] scheme for multi-access communication with feedback in [98].