# Joint Radar-Communication Optimization of Distributed Airborne Radar for AOA Localization
## 2 System Model
### Formal Representation of JRCO  
The state of the UAV swarm in time slot \(t\) could be represented by vector \(\mathbf{S}^{t}\):  
\[\mathbf{S}^{t}=\left[\left(\vec{\mathbf{r}}\right)^{T},\left(\mathbf{\omega}^{t} \right)^{T},\sigma_{\mathbf{R}}^{2}\right]^{T} \tag{23}\]  
The formal representation of JRCO at the \(t\)-th time slot:  
\[\begin{array}{rl}\min_{\vec{\mathbf{r}},\mathbf{\omega}^{t},\mathbf{C}_{\theta}^{t} }&\mathcal{J}\left(\vec{\mathbf{r}},\mathbf{\omega}^{t},\hat{\mathbf{h}},\mathbf{C}_{ \theta}^{t},\mathbf{C}_{\varphi}^{t},\mathbf{C}_{R}^{t}\right)\\ \text{s.t.}&d\left(\vec{\mathbf{r}}_{i}^{t},\vec{\mathbf{r}}_{i}^{t-1}\right)\leq d_{ \text{max}}(i=1,2,\cdots M)\\ &\frac{a_{r}}{\left(\sigma_{\theta,i}^{t}\right)^{2}}+a_{c}\big{(}\omega_{i}^{ t}\big{)}^{2}=E_{\text{max}}(i=1,2,\cdots M)\\ &\sigma_{\theta,i}^{t}=\sigma_{\varphi,i}^{t}(i=1,2,\cdots M)\end{array} \tag{24}\]  
where \(\mathcal{J}(\cdot)\) is the optimization objective function, which is the CRLB of the target location estimation for all UAVs in the swarm. This function could be calculated based on the trace of \(\mathbf{C}_{CRLB}\). \(d\left(\vec{\mathbf{r}}_{i},\vec{\mathbf{r}}_{i}^{t-1}\right)\), which refers to the Euclidean distance that the \(i\)-th UAV moves from the \(t\)-th time slot to the \(t+1\)-th time slot. Due to the limitations of the UAV payload, the maximum motion distance of each UAV in each time slot was limited and could not exceed \(d_{\max}\).  
It can be noted that the parameters \(\vec{\mathbf{r}}_{i}^{t}\), \(\mathbf{\omega}^{t}\), \(\mathbf{C}_{\theta}^{t}\), \(\mathbf{C}_{\varphi}^{t}\) in Equation (24) were based on information obtained from a global perspective (namely, this information was obtained by all the UAVs in the swarm). For a single UAV, it could only sense the information broadcasted by its neighboring UAVs. Therefore, directly adopting the optimization form in Equation (24) would be unrealistic.  
If \(\vec{\mathbf{r}}_{i}^{t}\) is used to represent the set of \(M_{i}\) UAV locations sensed by the current \(i\)-th UAV in the \(t\)-th time slot (including the current UAV itself), the elements in \(\vec{\mathbf{r}}_{i}^{t}\) could be represented as:  
\[\vec{\mathbf{r}}_{i}^{t}=\left[\left(\vec{\mathbf{r}}_{i,1}^{t}\right)^{T},\left(\vec {\mathbf{r}}_{i,2}^{t}\right)^{T},\cdots,\left(\vec{\mathbf{r}}_{i,M_{i}}^{t}\right)^{ T}\right]^{T} \tag{25}\]  
where \(\vec{\mathbf{r}}_{i,k}^{t}=\left[\vec{\mathbf{x}}_{i,k}^{t},\vec{\mathbf{y}}_{i,k}^{t} \right]^{T}\).  
From the perspective of the \(i\)-th UAV in the \(t\)-th time slot, the JRCO could be decomposed into:  
\[\begin{array}{cc}\min_{\vec{\mathbf{r}}_{i}^{t},\vec{\mathbf{r}}_{i}^{t},\vec{\mathbf{e }}_{\phi,i}^{t}}&\mathcal{J}_{i}\left(\vec{\mathbf{r}}_{i}^{t},\omega_{i}^{t},\hat {\mathbf{u}},\sigma_{\theta,i}^{t},\sigma_{\varphi,i}^{t},\sigma_{\theta,i}^{t}, \sigma_{\theta}^{2}\right)\\ \text{s.t.}&d\left(\vec{\mathbf{r}}_{i}^{t},\vec{\mathbf{r}}_{i}^{t-1}\right)\leq d_{ \max}\\ &\frac{d_{\tau}}{\left(\sigma_{\theta,i}^{t}\right)^{2}}+a_{c}\left(\omega_{i} ^{t}\right)^{2}=E_{\max}\\ &\sigma_{\theta,i}^{t}=\sigma_{\varphi,i}^{t}\end{array} \tag{26}\]  
where \(\mathcal{J}(\cdot)\) represents the objective function of the \(i\)-th UAV for optimization.  
It should be noted that the optimization problem shown in Equation (26) was executed simultaneously for all the UAVs in each time slot. All UAVs sensed the information and made decisions simultaneously.  
Section 4 demonstrates the numerical results that solved the optimization problem shown in Equation (26) through distributed algorithms that could ultimately achieve the sub-optimal solution of the optimization problem shown in Equation (24).