## 3 Fundamental Limits
### FIM of Synchronous Networks  
For a single target (\(N_{\rm t}=1\)) synchronous ISAL network applying the OWR method, another parameter vector that satisfies a mapping with the parameters \(\mathbf{\theta}=[\mathbf{p}_{\rm r}^{\rm T},\mathbf{p}_{\rm t}^{\rm T}]^{\rm T}\) is defined as  
\[\mathbf{\gamma}=[\tau_{\rm I,1}^{\rm sen},\cdots,\tau_{N_{\rm t}+N_{\rm a},N_{ \rm t}+N_{\rm a}}^{\rm sen},\tau_{\rm I,2}^{\rm ran},\cdots,\tau_{N_{\rm t}+N_ {\rm a},N_{\  
where \(\lambda_{m,n},m,n\in\{m,n|m\in\mathcal{N}_{r},n\in\mathcal{N}_{r}\cup\mathcal{N}_{a},m\
eq n\}\cup\{m,n|m\in\mathcal{N}_{a},n\in\mathcal{N}_{r}\}\) and \(\lambda_{m,tar,n},m,n\in\{m,n|m,n\in\mathcal{N}_{r}\cup\mathcal{N}_{a}\}\), respectively, represent the range information intensity (RII) of the localization and sensing links, which can be expressed as  
\[\lambda_{m,n}=\frac{8\pi^{2}B^{2}}{c^{2}}\frac{P_{m}G_{m}G_{n}c^{2}}{(4\pi)^{ 2}d_{m,n}^{2}N_{0}BLf_{c}^{2}}=\xi_{m,n}\frac{P_{m}}{d_{m,n}^{2}} \tag{14}\]  
\[\lambda_{m,tar,n}=\frac{8\pi^{2}B^{2}}{c^{2}}\frac{P_{m}G_{m}G_{n}c^{2}}{(4\pi) ^{2}d_{m,tar}^{2}d_{tar,n}^{2}N_{0}BLf_{c}^{2}}\frac{\sigma}{4\pi}=\xi_{m,tar,n }\frac{P_{m}}{d_{m,tar}^{2}d_{tar,n}^{2}}\frac{\sigma}{4\pi}, \tag{15}\]  
where \(m,tar\), and \(n\), respectively, represent the \(m\)th transmitting node, the target, and the \(n\)th receiving node; \(B\) is the signal bandwidth; \(c\) is the speed of light; \(P_{m}\) is the signal transmission power; \(G_{m}\) and \(G_{n}\), respectively, represent the antenna gain of the transmitter and receiver; \(N_{0}\) is the power spectral density of environmental thermal noise; \(L\) is the communication system loss; \(f_{c}\) is the carrier frequency of the signal; \(\sigma\) is the radar cross section; and \(\xi_{m,n}\) and \(\xi_{m,tar,n}\) are, respectively, the common constant coefficient in the RII of the localization link and the sensing link. By means of a bijective transformation [17], the FIM for the parameter vector \(\mathbf{\theta}\) can be expressed as  
\[\mathbf{I}_{\mathbf{\theta}}=\mathbf{J}^{\mathrm{T}}\mathbf{I}_{\mathbf{\gamma}} \mathbf{J}, \tag{16}\]  
where \(\mathbf{J}\) is the Jacobian matrix for the transmission from \(\mathbf{\theta}\) to \(\gamma\), which can be written as  
\[\mathbf{J}=\frac{\partial\mathbf{\gamma}}{\partial\mathbf{\theta}}=[\frac{\partial\bm {\gamma}}{\partial\mathbf{p}_{\mathrm{r}}},\frac{\partial\mathbf{\gamma}}{ \partial\mathbf{p}_{\mathrm{t}}}], \tag{17}\]  
where the specific expressions of \(\frac{\partial\mathbf{\gamma}_{\mathrm{r}}}{\partial\mathbf{p}_{\mathrm{r}}}\) and \(\frac{\partial\mathbf{\gamma}}{\partial\mathbf{p}_{\mathrm{r}}}\) are given in Appendix A.1. Based on the results above, the FIM expression for the single-slot synchronous ISAL networks can be derived, which can be expressed in the following form:  
\[\mathbf{I}_{\mathbf{\theta}}^{\mathrm{syn}}=\left[ \begin{array}{cc}\mathbf{J}_{ \mathrm{e}}^{\mathrm{R}}&\mathbf{S}_{\mathrm{R,T}}\\ \mathbf{S}_{\mathrm{R,T}}^{\mathrm{T}}&\mathbf{J}_{\mathrm{e}}^{\mathrm{T}} \end{array} \right]_{2(N_{\mathrm{r}}+N_{\mathrm{t}})\times 2(N_{\mathrm{r}}+N_{ \mathrm{t}})}, \tag{18}\]  
where \(\mathbf{J}_{\mathrm{e}}^{\mathrm{R}},\mathbf{J}_{\mathrm{e}}^{\mathrm{T}}\), and \(\mathbf{S}_{\mathrm{R,T}}\) are the submatrices in \(\mathbf{I}_{\mathbf{\theta}}^{\mathrm{syn}}\), with respect to the radar localization information, the target sensing information, and the spatial cooperation information between radar localization and target sensing. For the FIM of multi-slot synchronous ISAL networks, since we can obtain the velocity information of the moving active nodes, we can predict the positions of the moving active nodes in the next time slot, corresponding to the time cooperation information of radar positioning between time slots in the FIM. Considering the complexity of the multi-slot ISAL networks, we give the expression of the FIM of the dual-slot synchronous ISAL networks as follows [26]:  
\[\mathbf{I}_{\mathbf{\theta}}^{\mathrm{syn}(2)}=\left[