## 4 Analysis  
### CRLB Derivation  
The CRLB serves as a standard measure for assessing the accuracy of an unbiased estimator. Although the estimators for JLAS are biased, their biases asymptotically approach zeros if the noise and errors are not significant. Different from the CRLB, the Ziv-Zakai bound (ZZB) provides a more comprehensive lower bound on the MSE, tightly covering a wide range of noise [35; 36; 37]. However, the derivation and computation of the ZZB can be complex, making it less accessible for practical use in some situations. Thus, in this paper, we utilize the CRLB as a reference benchmark [38]. The CRLB of JLAS using sequential one-way TOAs was presented in [11]. We used for the comparison convenience. The expression of CRLB is  
\[\mathbf{CRLB}(\theta^{o})=\left(\frac{\partial\mathbf{m}^{o}{}^{T}}{\partial \theta^{o}}\mathbf{\partial}\frac{\partial\mathbf{m}^{o}}{\partial\theta^{o }{}^{T}}\right)^{-1}, \tag{66}\]  
where \(\mathbf{Q}=\text{Blkdiag}\{\mathbf{Q}_{t},\mathbf{Q}_{s}\}\), and \(\text{Blkdiag}\{\cdots\}\) is the block diagonal operator. The partial derivative of \(\mathbf{m}^{o}\) with respect to \(\theta^{o}\) is  
\[\frac{\partial\mathbf{m}^{o}}{\partial\theta^{o}{}^{T}}=\begin{bmatrix} \frac{\partial\mathbf{r}^{o}}{\partial\mathbf{\mu}^{oT}}&\frac{\partial \mathbf{r}^{o}}{\partial\mathbf{s}^{oT}}\\ \frac{\partial\mathbf{s}^{o}}{\partial\mathbf{\mu}^{oT}}&\frac{\partial \mathbf{s}^{o}}{\partial\mathbf{s}^{oT}}\end{bmatrix}=\begin{bmatrix}\mathbf{J} &\mathbf{S}\\ \mathbf{O}_{M\times(2N+2)}&\mathbf{I}_{MN}\end{bmatrix}, \tag{67}\]  
where  
\[\mathbf{m}^{o} =\left[\mathbf{r}^{T},\mathbf{s}^{oT}\right]^{T}, \tag{68}\] \[\theta^{o} =\left[\mathbf{\mu}^{oT},\mathbf{s}^{oT}\right]^{T}=\left[ \mathbf{p}^{oT},\mathbf{v}^{oT},\gamma^{o},\iota^{o},\mathbf{s}^{oT}\right], \tag{69}\]  
and \(\mathbf{s}^{o}=[\mathbf{s}_{1}^{oT},\ldots,\mathbf{s}_{M}^{oT}]^{T}\).  
The blocks in (67) are  
\[\mathbf{J}(i,\cdot)=\left[-\mathbf{I}_{i}^{oT},-\mathbf{I}_{i}^{oT}t_{i},1,t_ {i}\right], \tag{70}\]  
where  
\[\mathbf{I}_{i}^{o}=\frac{\mathbf{s}_{i}^{o}-\mathbf{p}^{o}-\mathbf{v}^{o}t_{i }}{\left\|\mathbf{s}_{i}^{o}-\mathbf{p}^{o}-\mathbf{v}^{o}t_{i}\right\|},i=1, \ldots,M, \tag{71}\]  
Figure 3: Example of divergence behavior: position error of each iteration.  
and  
\[\mathbf{S}=\begin{bmatrix}\mathbf{I}_{1}^{\mathrm{t}^{T}}&&\\ &\mathbf{I}_{2}^{\mathrm{t}^{T}}&&\\ &&\ddots&\\ &&&\mathbf{I}_{M}^{\mathrm{t}^{T}}\end{bmatrix}. \tag{72}\]  
In applying block inverse lemma ([PERSON], 1979) to (66), the upper-left block is the CRLB of \(\mathbf{\mu}^{o}\):  
\[\mathbf{CRLB}(\mathbf{\mu}^{o})=\bigg{(}\frac{\partial\mathbf{\tau}^{o}}{\partial\mathbf{ \mu}^{oT}}\mathbf{\mathcal{W}}\frac{\partial\mathbf{\tau}^{oT}}{\partial\mathbf{\mu}^{o}} \bigg{)}^{-1}. \tag{73}\]  
The CRLB above indicates the best achievable MSE performance of one-way sequential TOA-based localization. The optimization of the AN placement may improve the geometric dilution of precision and lower the CRLB ([PERSON], 1979), which can also realize better localization accuracy for the UN. The discussion of placement optimization is out of the scope of this paper, but it is valuable to be studied further in the future.