# Joint Radar-Communication Optimization of Distributed Airborne Radar for AOA Localization
## 3 JRCO Algorithm
### Estimating Target Location  
Based on the measurement model given in Equation (8) and the AOA measurement values of neighboring \(M_{i}\) UAVs, this subsection designed relevant parameter estimation algorithms to estimate the target location. In each time slot, each UAV could only construct an objective function based on Equation (21) by estimating the target location. At present, most of the current research on optimizing the CRLB [36; 37; 38; 39] has assumed that the accurate target location is known. It is obviously unreasonable that the target location is known before the UAV swarm can obtain an accurate target location estimation. In the case of neighboring \(M_{i}\) UAVs forming a network together and, based on the geometric relationship between each UAV and the target, there was:  
\[\overline{\theta}_{i,k}^{t}=\arctan\left(\frac{y_{u}-\overline{y}_{k}^{t}}{x_{u} -\overline{x}_{k}^{t}}\right) \tag{27}\]  
Tan operations can be performed on both sides simultaneously and multiplied by \(\left(x_{u}-\overline{x}_{k}^{t}\right)\) to obtain:  
\[\left(x_{u}-\overline{x}_{k}^{t}\right)\sin\overline{\theta}_{i,k}^{t}-\left( y_{u}-\overline{y}_{k}^{t}\right)\cos\overline{\theta}_{i,k}^{t}=0 \tag{28}\]  
where \(k=1,2,\cdots,M_{i}\). After some mathematical manipulations, there were:  
\[\left(\overline{w}_{i,k}^{t}\right)^{T}\overline{h}-\left(\overline{w}_{i,k}^ {t}\right)^{T}\overline{r}_{i,k}^{t}=0 \tag{29}\]  
where:  
\[\overline{h}=\frac{1}{h}\left[\cos\overline{\theta}_{u},\sin\overline{\theta} _{u}\right]^{T} \tag{30}\]  
\[\overline{w}_{i,k}^{t}=\left[\sin\overline{\theta}_{i,k}^{t},-\cos\overline{ \theta}_{i,k}^{t}\right]^{T} \tag{31}\]  
\[\overline{\theta}_{u}=\arctan\left(\frac{y_{u}}{x_{u}}\right) \tag{32}\]  
\[\overline{h}=\sqrt{\left(x_{u}\right)^{2}+\left(y_{u}\right)^{2}} \tag{33}\]  
When the variance of AOA measurement errors for each UAV was relatively small, the following approximate relationship held:  
\[\sin\overline{\theta}_{i,k}^{t}\approx\sin\theta_{i,k}^{t}-n_{\theta,k}^{t} \cos\overline{\theta}_{i,k}^{t} \tag{34}\]  
\[\cos\overline{\theta}_{i,k}^{t}\approx\cos\theta_{i,k}^{t}+n_{\theta,k}^{t} \sin\overline{\theta}_{i,k}^{t} \tag{35}\]  
Thus, the following pseudo-linear equation could be obtained:  
\[\left(\mathbf{w}_{i,k}^{t}\right)^{T}\overline{h}-\left(\mathbf{w}_{i,k}^{t}\right)^{ T}\overline{r}_{i,k}^{t}\overline{h}\approx\overline{h}\overline{q}_{i,l}n_{ \theta,k}^{t}-\overline{h}\left(\overline{w}_{i,k}^{t}\right)^{T}\mathbf{n}_{i,k}^ {t} \tag{36}\]  
where:  
\[\left(x_{u}-\overline{x}_{k}^{t}\right)\sin\overline{\theta}_{i,k}^{t}-\left( y_{u}-\overline{y}_{k}^{t}\right)\cos\overline{\theta}_{i,k}^{t}=0 \tag{28}\]0  
\[\left(x_{u}-\overline{x}_{k}^{t}\right)\sin\overline{\theta}_{i,k}^{t}-\left( y_{u}-\overline{y}_{k}^{t}\right)\cos\overline{\theta}_{i,k}^{t}=0 \tag{28}\]1  
\[\left(x_{u}-\overline{x}_{k}^{t}\right)\sin\overline{\theta}_{i,k}^{t}-\left( y_{u}-\overline{y}_{k}^{t}\right)\cos\overline{\theta}_{i,k}^{t}=0 \tag{28}\]2  
Due to \(k=1,2,\cdots,M_{i}\), Equation (36) corresponded to \(M_{i}\) equations, namely:  
\[\left(x_{u}-\overline{x}_{k}^{t}\right)\sin\overline{\theta}_{i,k}^{t}-\left( y_{u}-\overline{y}_{k}^{t}\right)\cos\overline{\theta}_{i,k}^{t}=0 \tag{28}\]3  
where:  
\[\mathbf{H}_{\theta}=\left[ \begin{array}{cc}\left(\mathbf{w}_{l,1}^{t}\right)^{T }&-\left(\mathbf{w}_{l,1}^{t}\right)^{T}\overline{r}_{i,1}^{t}\\ \vdots&\vdots\\ \left(\mathbf{w}_{l,M_{i}}^{t}\right)^{T}&-\left(\mathbf{w}_{l,M_{i}}^{t}\right)^{T} \overline{r}_{i,M_{i}}^{t}\end{array} \right] \tag{41}\]\[\mathbf{S}_{\theta}=\overline{h}diag\big{\{}\left[\overline{u}_{1},\overline{u}_{2}, \cdots,\overline{u}_{M_{i}}\right]\big{\}} \tag{42}\]  
\[\mathbf{T}_{\theta}=