## 3 Proposed Solutions
### Robust Iterative Solution
In applying the SPA, the outgoing message from the factor node \(h_{i}\) to the variable node \(\mathbf{\mu}\) is the integration of the product of \(h_{i}\) and all incoming messages to \(h_{i}\) over the variable \(\xi_{i}\):  
\[\varpi_{h_{i}\to\mathbf{\mu}}(\mathbf{\mu})=\int h_{i}(\xi_{i},\mathbf{\mu}) \varpi_{\xi_{i}\to h_{i}}(\xi_{i})d\xi_{i}. \tag{48}\]  
Expanding \(\overline{r}_{i}(\mathbf{\mu})\) in \(h_{i}(\xi_{i},\mathbf{\mu})\) by the first-order Taylor series at the estimate \(\tilde{\mathbf{\mu}}\) results in  
\[\mathcal{J}(\mathbf{\mu})=\exp\Big{\{}(\mathbf{r}-\mathbf{\bar{r}}(\mathbf{\mu}))^{T} \mathbf{\mathcal{W}}(\mathbf{r}-\mathbf{\bar{r}}(\mathbf{\mu}))\Big{\}}, \tag{40}\]0  
where  
\[\mathcal{J}(\mathbf{\mu})=\exp\Big{\{}(\mathbf{r}-\mathbf{\bar{r}}(\mathbf{\mu}))^{T} \mathbf{\mathcal{W}}(\mathbf{r}-\mathbf{\bar{r}}(\mathbf{\mu}))\Big{\}}, \tag{40}\]1  
and  
\[\mathcal{J}(\mathbf{\mu})=\exp\Big{\{}(\mathbf{r}-\mathbf{\bar{r}}(\mathbf{\mu}))^{T} \mathbf{\mathcal{W}}(\mathbf{r}-\mathbf{\bar{r}}(\mathbf{\mu}))\Big{\}}, \tag{40}\]2  
where  
\[\mathcal{J}(\mathbf{\mu})=\exp\Big{\{}(\mathbf{r}-\mathbf{\bar{r}}(\mathbf{\mu}))^{T} \mathbf{\mathcal{W}}(\mathbf{r}-\mathbf{\bar{r}}(\mathbf{\mu}))\Big{\}}, \tag{40}\]3  
Substituting (49) into (43) approximates \(h_{i}(\xi_{i},\mathbf{\mu})\) to  
\[\mathcal{J}(\mathbf{\mu})=\exp\Big{\{}(\mathbf{r}-\mathbf{\bar{r}}(\mathbf{\mu}))^{T} \mathbf{\mathcal{W}}(\mathbf{r}-\mathbf{\bar{r}}(\mathbf{\mu}))\Big{\}}, \tag{40}\]4  
where \(b_{i}=\overline{r}_{i}(\tilde{\mathbf{\mu}})-\mathbf{a}_{i}^{\mathrm{T}}\tilde{ \mathbf{\mu}}\). In considering that \(\varpi_{\xi_{i}\to h_{i}}(\xi_{i})\) is Gaussian as well, it is concluded from (53) that \(\varpi_{h_{i}\to\mathbf{\mu}}(\mathbf{\mu})\) in (48) obeys  
\[\mathcal{J}(\mathbf{\mu})=\exp\Big{\{}(\mathbf{r}-\mathbf{\bar{r}}(\mathbf{\mu}))^{T} \mathbf{\mathcal{W}}(\mathbf{r}-\mathbf{\bar{r}}(\mathbf{\mu}))\Big{\}}, \tag{40}\]5  
where  
\[\mathcal{J}(\mathbf{\mu})=\exp\Big{\{}(\mathbf{r}-\mathbf{\bar{r}}(\mathbf{\mu}))^{T} \mathbf{\mathcal{W}}(\mathbf{r}-\mathbf{\bar{r}}(\mathbf{\mu}))\Big{\}}, \tag{40}\]6 \[\mathcal{J}(\mathbf{\mu})=\exp\Big{\{}(\mathbf{r}-\mathbf{\bar{r}}(\mathbf{\mu}))^{T} \mathbf{\mathcal{W}}(\mathbf{r}-\mathbf{\bar{r}}(\mathbf{\mu}))\Big{\}}, \tag{40}\]7  
Therefore, the marginal (45) is also Gaussian due to (54), where \(\mathcal{B}(\mathbf{\mu})\sim\mathcal{N}(\mathbf{\mu};\tilde{\mathbf{\mu}},\mathbf{\mathbf{X}}_{ \mathbf{\mu}}^{-1})\), and  
\[\mathcal{J}(\mathbf{\mu})=\exp\Big{\{}(\mathbf{r}-\mathbf{\bar{r}}(\mathbf{\mu}))^{T} \mathbf{\mathcal{W}}(\mathbf{r}-\mathbf{\bar{r}}(\mathbf{\mu}))\Big{\}}, \tag{40}\]8  
\[\mathcal{J}(\mathbf{\mu})=\exp\Big{\{}(\mathbf{r}-\mathbf{\bar{r}}(\mathbf{\mu}))^{T} \mathbf{\mathcal{W}}(\mathbf{r}-\mathbf{\bar{r}}(\mathbf{\mu}))\Big{\}}, \tag{40}\]9 Factor graph of sequential TOA JLAS. In denoting  
\[\mathbf{\mathcal{W}}^{-1}=\mathbf{Q}_{\varepsilon}+\left(\frac{\partial\mathbf{r} ^{o}}{\partial\mathbf{s}^{oT}}\mathbf{Q}_{s}\frac{\partial\mathbf{r}^{T}}{ \partial\mathbf{s}^{o}}\right). \tag{41}\]0  
the refined estimation of \(\mu^{o}\) is given by  
\[\mathbf{\mathcal{W}}^{-1}=\mathbf{Q}_{\varepsilon}+\left(\frac{\partial\mathbf{r} ^{o}}{\partial\mathbf{s}^{oT}}\mathbf{Q}_{s}\frac{\partial\mathbf{r}^{T}}{ \partial\mathbf{s}^{o}}\right). \tag{41}\]1  
Starting the iteration following (59) can lead to an inaccurate estimate if the initialization is poor, resulting in worse values for \(\mathbf{X}_{\mathbf{\mu}}\) and \(\mathbf{w}_{\mu}\), which can cause divergence. Through retaining some information from previous iterations in \(\mathbf{X}_{\mu}\) and \(\mathbf{w}_{\mu}\), this divergence may be mitigated, potentially allowing the iteration to converge asymptotically.