## 3 Fundamental Limits
### FIM of Asynchronous Networks  
For a single target \((N_{\text{t}}=1)\) asynchronous ISAL network applying the OWR method, another parameter vector that satisfies a mapping with the parameters \(\mathbf{\theta}=[\mathbf{p}_{\text{r}}^{\text{T}},\mathbf{r}^{\text{T}},\mathbf{p }_{\text{t}}^{\text{T}}]^{\text{T}}\) is defined as  
\[\mathbf{\zeta}=[v_{1,1}^{\text{sen}},\cdots,v_{N_{\text{t}}+N_{\text{u}},N_{ \text{t}}+N_{\text{u}}}^{\text{sen}},v_{1,2}^{\text{ran}},\cdots,v_{N_{\text{ t}}+N_{\text{u}},N_{\text{t}}}^{\text{ran}}]^{\text{T}}, \tag{21}\]  
where \(v_{m,n}^{\text{sen}}=\frac{d_{m,n}+d_{tar,n}}{\tau_{m,n}}+\tau_{m,n},m,n\in\{m, n|m,n\in\mathcal{N}_{\text{r}}\cup\mathcal{N}_{\text{a}}\}\),\(v_{m,n}^{\text{ran}}=\frac{d_{m,n}}{c}+\tau_{m,n},m,n\in\{m,n|m\in\mathcal{N}_{\text{r}},n\in\mathcal{N}_{\text{r}},m\
eq n\}\cup\{m,n|m\in\mathcal{N}_{\text{a}},n\in \mathcal{N}_{\text{r}}\}\). Similar to the synchronous ISAL networks, we can obtain the FIM of \(\mathbf{\zeta}\) expressed as (13), and the FIM of \(\mathbf{\theta}\) can also be obtained from (16). The Jacobian matrix for the transformation from \(\mathbf{\theta}\) to \(\mathbf{\zeta}\) in the asynchronous ISAL network can be expressed as  
\[\mathbf{J}=\frac{\partial\mathbf{\zeta}}{\partial\mathbf{\theta}}=[\frac{\partial\bm {\zeta}}{\partial\mathbf{\rho}_{\text{r}}},\frac{\partial\mathbf{\zeta}}{\partial\bm {\tau}},\frac{\partial\mathbf{\zeta}}{\partial\mathbf{\rho}_{\text{t}}}], \tag{22}\]  
where the specific expressions of \(\frac{\partial\mathbf{\zeta}}{\partial\mathbf{\rho}_{\text{r}}},\frac{\partial\mathbf{ \zeta}}{\partial\mathbf{\tau}}\), and \(\frac{\partial\mathbf{\zeta}}{\partial\mathbf{\rho}_{\text{t}}}\) are given in Appendix A.2. Based on the results above, the FIM expression of the single-slot asynchronous ISAL networks can be derived, which can be expressed in the following form:  
\[\mathbf{I}_{\mathbf{\theta}}^{\text{asyn}}=\left[ \begin{array}{cccc}\mathbf{J}_ {\text{e}}^{\text{R}}&\mathbf{S}_{\text{R},\text{r}}&\mathbf{S}_{\text{R}, \text{T}}\\ \mathbf{S}_{\text{R},\text{T}}^{\text{T}}&\mathbf{J}_{\text{e}}^{\text{T}}& \mathbf{S}_{\text{T},\text{T}}\\ \mathbf{S}_{\text{R},\text{T}}^{\text{T}}&\mathbf{S}_{\text{T},\text{r}}^{ \text{T}}&\mathbf{J}_{\text{e}}^{\text{T}}\end{array} \right]_{[2(N_{\text{t} }+N_{\text{t}})+|\mathcal{N}_{\text{t}}|]\times[2(N_{\text{t}}+N_{\text{t}}) +|\mathcal{N}_{\text{t}}|]}, \tag{23}\]  
where \(\mathbf{J}_{\text{e}}^{\text{T}}\), \(\mathbf{S}_{\text{R},\text{T}}\), and \(\mathbf{S}_{\text{T},\text{r}}\) are the submatrices in \(\mathbf{I}_{\mathbf{\theta}}^{\text{asyn}}\) with respect to the clock offset information, the cooperation information between radar localization and clock offset, and the cooperation information between target sensing and clock offset. According to the principle of RLM-OWR, for the FIM of multi-slot asynchronous ISAL networks, in addition to introducing radar positioning temporal cooperation information between time slots through Doppler measurement, we can also obtain the relative clock drift rate through RLM-OWR, thereby further introducing temporal cooperation information for estimating clock offsets between time slots. Considering the complexity of the multi-slot ISAL networks, here we give the expression of the FIM of the dual-slot asynchronous ISAL networks after applying RLM-OWR, which is shown as (24)  
\[\mathbf{I}_{\mathbf{\theta}}^{\text{asyn}(2)}=\left[\begin{array}{cccc}\mathbf{J }_{\text{e}}^{\text{R}(1)}+\mathbf{T}_{\text{Loc}}^{(1)}&\mathbf{S}_{\text{R}, \tau}^{(1)}&\mathbf{S}_{\text{R},\text{T}}^{(1)}&\mathbf{T}_{\text{Loc}}^{(1) }&\mathbf{0}&\mathbf{0}\\ \mathbf{S}_{\text{R},\text{T}}^{(1)\text{T}}&\mathbf{J}_{\text{e}}^{\text{T}(1) }+\mathbf{T}_{\text{T}}^{(1)}&\mathbf{S}_{\text{T},\text{T}}^{(1)}&\mathbf{0  
where submatrix \(\mathbf{T}_{\tau}\) contains the temporal cooperation information on the clock offset, which is expressed as  
\[\mathbf{T}_{\tau}=\left[