# Joint Radar-Communication Optimization of Distributed Airborne Radar for AOA Localization
## 3 JRCO Algorithm
### Estimating Target Location
\begin{bmatrix}-\overline{h}\Big{(}\mathbf{w}_{i,1}^{t}\Big{)} ^{T}&\mathbf{0}_{2}^{T}&\cdots&\mathbf{0}_{2}^{T}\\ \mathbf{0}_{2}^{T}&-\overline{h}\Big{(}\mathbf{w}_{i,2}^{t}\Big{)}^{T}&\cdots& \mathbf{0}_{2}^{T}\\ \vdots&\vdots&\ddots&\vdots\\ \mathbf{0}_{2}^{T}&\mathbf{0}_{2}^{T}&\cdots&-\overline{h}\Big{(}\mathbf{w}_{i,M _{i}}^{t}\Big{)}^{T}\end{bmatrix}  \tag{43}\]  
The above process was repeated and we obtained:  
\[\mathbf{H}_{\varphi}\overline{\mathbf{x}}=\mathbf{S}_{\varphi}\overline{\mathbf{ n}}_{\varphi,i}^{t}+\mathbf{T}_{\varphi}\mathbf{n}_{l,k}^{t} \tag{44}\]  
where:  
\[\mathbf{H}_{\varphi}= \begin{bmatrix}\Big{(}z_{i,1}^{t}\Big{)}^{T}&-\Big{(}z_ {i,1}^{t}\Big{)}^{\overline{T}}\overline{\mathbf{r}}_{i,1}^{t}\\ \vdots&\vdots\\ \Big{(}z_{i,M_{i}}^{t}\Big{)}^{T}&-\Big{(}z_{i,M_{i}}^{t}\Big{)}^{\overline{T }}\overline{\mathbf{r}}_{i,M_{i}}^{t}\end{bmatrix}  \tag{45}\]  
\[\mathbf{S}_{\varphi}=\overline{h}diag\big{\{}\left[\overline{v}_{1}, \overline{v}_{2},\cdots,\overline{v}_{M_{i}}\right]\big{\}} \tag{46}\]  
\[\overline{v}_{k}=\big{(}x_{u}-\overline{x}_{k}^{t}\big{)}\cos\overline{\theta }_{i,k}^{t}\cos\overline{\varphi}_{i,k}^{t}+\big{(}y_{u}-\overline{y}_{k}^{t} \big{)}\sin\overline{\theta}_{i,k}^{t}\cos\overline{\varphi}_{i,k}^{t}+\big{(} z_{u}-\overline{z}_{k}^{t}\big{)}\sin\overline{\varphi}_{i,k}^{t} \tag{48}\]  
\[\overline{z}_{i,k}^{t}=\left[\cos\overline{\theta}_{i,k}^{t}\sin\overline{ \varphi}_{i,k}^{t}\sin\overline{\theta}_{i,k}^{t}\sin\overline{\varphi}_{i,k} ^{t},-\cos\overline{\varphi}_{i,k}^{t}\right]^{T} \tag{49}\]  
\[z_{i,k}^{t}=\left[\cos\theta_{i,k}^{t}\sin\varphi_{i,k}^{t}\sin\theta_{i,k}^{ t}\sin\varphi_{i,k}^{t},-\cos\varphi_{i,k}^{t}\right]^{T} \tag{50}\]  
Defining \(\mathbf{H}=\left[(\mathbf{H}_{\theta})^{T},\big{(}\mathbf{H}_{\varphi}\big{)} ^{T}\right]^{T}\), \(\mathbf{S}=blkdiag\big{\{}\mathbf{S}_{\theta},\mathbf{S}_{\varphi}\big{\}}\) and \(\mathbf{T}=\left[(\mathbf{T}_{\theta})^{T},\big{(}\mathbf{T}_{\varphi}\big{)} ^{T}\right]^{T}\). \(\overline{\mathbf{x}}=\left[\overline{h}^{T},\overline{h}\right]^{T}\) represented the vector of the location parameter, and Equations (40) and (44) could be solved using methods such as weighted least squares (WLS). If \(\lambda=\mathbf{S}\bigg{[}\left(\overline{\mathbf{n}}_{\theta,i}^{t}\right)^{T}, \left(\widetilde{\mathbf{n}}_{\varphi,i}^{t}\right)^{T}\bigg{]}^{T}+\mathbf{T} \mathbf{n}_{l,k}^{t}\), then after some mathematical manipulation, the covariance matrix of \(\lambda\) could be obtained as:  
\[E\Big{[}\lambda\lambda^{T}\Big{]}=\mathbf{S}\widetilde{\mathbf{C}}_{\theta,i}^{t} \mathbf{S}^{T}+\mathbf{TC}_{l}^{t}\mathbf{T}^{T}\triangleq\mathbf{U}^{-1} \tag{51}\]  
After some further mathematical manipulation, the cost function of the WLS method could be obtained:  
\[\mathcal{C}=\mathbf{x}^{T}\mathbf{H}\mathbf{U}\mathbf{H}^{T}\mathbf{x} \tag{52}\]  
where \(\mathbf{x}\) was the estimated value of \(\overline{\mathbf{x}}\). The cost function could be minimized to obtain \(\mathbf{x}\). The solving process of WLS need not be elaborated here.