# 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
The scheme is again based on a block Markov strategy where in each block, the two Txs send new independent data as well as common update information that will decoded at the Rx. More specifically, the scheme is illustrated in Figure 11 and each block consists of three layers, where the top-most layer is most difficult to decode and the lowest layer the easiest. Both Txs send the same lowest layer, which thus can be transmitted in a cooperative manner, while the upper two layers are independent across the two transmitters. The details of the three layers are as follows:  
* In the top layer, each Tx independently sends new data in each block. These data are decoded at the Rx only, following the backward decoding algorithm described later. * In the middle layer, each Tx independently sends new data in each block. These data are decoded at the other Tx at the end of the block and at the Rx following the backward decoding algorithm described later. * In the lowest layer, the two Txs cooperate and jointly resend the data sent by the two Txs in the middle layer of the previous block (recall that the medium layer data of the previous block has been decoded by the other Tx at the end of the previous block). These data are decoded at the Rx following the backward decoding algorithm described next. The receiver decodes all transmitted data using a backward decoding procedure, starting from the last block. Specifically, for each block it decodes the data in the top and lowest  
Figure 11: Block Markov strategy of [PERSON] multi-access scheme with generalized feedback. layer, while it already is informed of the data sent in the middle layer, because it has decoded it in the previous step. Each Tx \(k\) produces its state estimates \(\hat{S}_{k}^{n}\) by using an optimal symbolwise estimator based on its own inputs \(X_{k}^{n}\), its own observed generalized feedback signals \(Z_{k}^{n}\), and also the middle-layer codeword symbols \(U_{k^{\prime}}^{n}\) decoded from the other Tx \(k^{\prime}\
eq k\). It is clearly suboptimal for Tx \(k\) to estimate its state sequence \(S_{k}^{n}\) simply based on its inputs \(X_{k}^{n}\) and its feedback signals \(Z_{k}^{n}\), and an improved performance can be obtained by attempting to decode also the codewords transmitted by the other Tx \(k^{\prime}\). In [65], the sensing tasks and data communication tasks are thus still considered individually. A first joint approach was considered in [67; 68], where sensing information was introduced to the coding scheme to allow for collaborative sensing; in other words, to allow each Tx to exploit sensing information available at the other Tx. On a technical level, this was enabled by having each Tx \(k\) compress the signals \(U_{k^{\prime}}^{N}\), \(X_{k}^{N}\), and \(Z_{k}^{N}\) of a given block and send the compression information (described in bits) as additional information in the middle layer of the next codeword, see Figure 12. In this way, any of the two Txs can convey sensing information to the other Tx over the Tx-to-Tx path, because the information in the middle layer is decoded at the other Tx. The compression information is also decoded at the Rx, and used to improve decoding of the transmitted data. For the compression of the sensing information, the scheme in [67; 68] uses implicit binning, i.e., the Tx as well as the Rx use their information about the compressed sequences from their own inputs and observations to reconstruct the sensing information. This allows the scheme to occupy less rate in the middle layer codewords and thus improve communication efficiency. The other encoding and decoding steps are as in the scheme in [65] and described previously. Each Tx \(k\) now performs the sensing task by producing symbolwise estimates based on the triples \(U_{k^{\prime}}^{N}\), \(X_{k}^{N}\), and \(Z_{k}^{N}\) and of the compression information obtained from the other Tx \(k^{\prime}\). The above coding schemes establish the following inner bound to the capacity-distortion tradeoff region \(\mathcal{CD}\)[67; 68]. **Theorem 9**.