## 3 Joint Angle and Delay Estimation (JADE)
### GWO vs. Combining \(|\)Embedding Is (Is-Gwo\(|\)Gwoeis)
Therefore, the positions of the wolves to converge to local minima. **Step (3.b) [GWOEIS]:** To improve and speed up convergence, instead of a common convergence factor, each realization or individual in the pack is assigned one of its own that accounts both for the AoA and TD pseudo-CDFs calculated in Steps (4) and (7) of the IS technique (cf. Section 3.2) as follows:  
\[a_{j,t}^{(r)}=\frac{u_{j}^{(r)}}{t}, \tag{31}\]  
where \(u_{j}^{(r)}\sim\mathcal{U}\left([\epsilon_{j}^{(r)},\epsilon_{j}^{(r)}]\right)\), \(j\!=\!\{1,2,\cdots,2Q\}\) is the dimension index, and  
\[\epsilon_{j}^{(r)} = \begin{cases}\max\!\left\{\mathcal{J}_{\theta_{j}|_{7}}(\mathbf{ \Sigma}_{t}^{(r)})_{j},\mathcal{J}_{\theta_{j}|_{7}}(\tilde{\theta}_{j}^{0}) \right\}\!,j\!=\!\{1,\cdots,Q\}\\ \max\!\left\{\mathcal{J}_{\tau_{j}}(\mathbf{\Sigma}_{t}^{(r)})_{j},\mathcal{J}_{ \tau_{j}}(\tilde{\tau}_{j}^{0})\right\}\!,j\!=\!\{Q\!+\!1,\cdots,2Q\}\end{cases}  \tag{32}\] \[\epsilon_{j}^{(r)} = \begin{cases}\min\!\left\{\mathcal{J}_{\theta_{j}|_{7}}(\mathbf{ \Sigma}_{t}^{(r)})_{j},\mathcal{J}_{\theta_{j}|_{7}}(\tilde{\theta}_{j}^{0}) \right\}\!,j\!=\!\{1,\cdots,Q\}\\ \min\!\left\{\mathcal{J}_{\tau_{j}}(\mathbf{\Sigma}_{t}^{(r)})_{j},\mathcal{J}_{ \tau_{j}}(\tilde{\tau}_{j}^{0})\right\}\!,j\!=\!\{Q\!+\!1,\cdots,2Q\}\end{cases}  \tag{33}\]  
where \((\mathcal{S})_{i}\) denotes the \(i\)-th element of the set \(\mathcal{S}\). \(\tilde{\tau}_{q}^{0}\) and \(\tilde{\theta}_{q}^{0}\) are the initial TD and AoA IS estimates obtained in (20) and (25), respectively, and \(j^{\prime}=j-Q\). When using the linearly decreasing factor in Step (3.a), the positions of the wolves can converge to local minima. To mitigate this issue, we generate for each individual a specific convergence factor that decreases with the iteration index while accounting through the pseud-CDF values for the distance between the gray wolves and the prey. As long as a wolf is far away from the prey, the uniform variable will generate realizations closer to 1. Once this wolf gets closer to the prey, the realization becomes quasi-static since, the convergence factor is then mainly scaled by \(1/t\). **Step (4):** For each lead gray wolf \(*=\alpha\), \(\beta\), or \(\delta\), we generate two random values, \(b_{j}^{*}\) and \(d_{j}^{*}\), in \(\mathcal{U}([0,1])\) for the calculation of two update coefficients \(g_{j,t}^{*}\) (or \(g_{j,t}^{*,(r)}\) with respect to each realization or individual \(r\) in the pack), and \(c_{j}^{*}\) according to one of the following cases (a) or (b). **Step (4.a) [GWO or IS-GWO]:**  
\[g_{j,t}^{*}=2\ a_{t}\ b_{j}^{*}-a_{t},\ \ \ \text{and}\ \ \ c_{j}^{*}=2\ d_{j}^{*}. \tag{34}\]  
In other words, the update coefficients \(g_{j,t}^{*}\) and \(c_{j}^{*}\) are assigned random values in \([-a_{t},a_{t}]\) and \([0,2]\), respectively. **Step (4.b) [GWOEIS]:**  
\[g_{j,t}^{*,(r)}=2\ a_{j,t}^{(r)}\ b_{j}^{*}-a_{j,t}^{(r)},\ \ \ \text{and}\ \ \ c_{j}^{*}=2\ d_{j}^{*}. \tag{35}\]  
Here, \(g_{j,t}^{*,(r)}\) is assigned a random value in \([-a_{j,t}^{(r)},a_{j,t}^{(r)}]\). **Step (5):** Let \(\text{dist}(*,(r),j)=\left|c_{j}^{*}(\mathbf{\Sigma}_{t-1}^{*})_{j}-(\mathbf{\Sigma}_{t -1}^{(r)})\right|\) denote the distance between the lead wolf * and the \(r\)th individual in the pack (or search agent) across the \(j\)-th dimension. The lead wolves' positions are then updated with respect to each realization \(r\) in the pack through intermediate variables \(\mathbf{\chi}_{*}^{(r)}\) according to one of the following cases (a) or (b).