# Joint Radar-Communication Optimization of Distributed Airborne Radar for AOA Localization
## 2 System Model
### The Model of AOA Measurement  
For the \(i\)-th UAV, when it sensed the information broadcasted by the neighboring \(M_{i}\) UAVs (specific information will be introduced in Section 3.3, \(M_{i}\) UAVs including the \(i\)-th UAV itself), the \(i\)-th UAV could fuse the information broadcasted by \(M_{i}\) UAVs and obtain an estimation of the target location. It should be noted that the focus of this article was not on target localization algorithms. However, Section 3 used the CRLB of the estimated target location as the objective function to achieve the joint optimization of each UAV location and electromagnetic domain power. The calculation of CRLB required the target location to be known; therefore, it was necessary to estimate the target location before executing the joint optimization algorithm. The AOA measurement based on multiple UAVs was the foundation of target location estimation; therefore, it is necessary to model the measurement process of the AOA first. The specific target location estimation method is provided in Section 3.4. We first provided a measurement model for azimuth AOA. The azimuth AOA measurement model for the \(i\)-th UAV in the \(t\)-th time slot was:  
\[\theta_{i}^{t}=\overline{\theta}_{i}^{t}+\bar{n}_{\theta,i}^{t} \tag{8}\]  
where \(\theta_{i}^{t}\) is the azimuth AOA measurement vector of the neighboring UAVs sensed by the \(i\)-th UAV in the \(t\)-th time slot. \(\overline{\theta}_{i}^{t}\) is the mean vector of the azimuth AOA measured by UAVs around the \(i\)-th UAV in the \(t\)-th time slot (the \(i\)-th UAV cannot sense this value). The measurement error carried by the \(i\)-th UAV in the \(t\)-th time slot was \(\bar{n}_{\theta,i}^{t}\). The elements of \(\theta_{i}^{t}\) and \(\overline{\theta}_{i}^{t}\) could be represented as:  
\[\theta_{i}^{t}=\left[\theta_{i,1}^{t},\theta_{i,2}^{t},\cdots,\theta_{i,M_{i }}^{t}\right]^{T} \tag{9}\]  
\[\overline{\theta}_{i}^{t}=\left[\overline{\theta}_{i,1}^{t},\overline{\theta }_{i,2}^{t},\cdots,\overline{\theta}_{i,M_{i}}^{t}\right]^{T} \tag{10}\]  
where \(\theta_{i,k}^{t}\) represents the measured AOA of the neighboring \(j\)-th UAV sensed by the \(i\)-th UAV in the \(t\)-th time slot, and \(k=1,2,\cdots,M_{i}\). \(\overline{\theta}_{i,k}^{t}\) represents the mean azimuth AOA of the \(j\)-th UAV sensed by the \(i\)-th UAV in the \(t\)-th time slot. Based on geometric relationships, \(\overline{\theta}_{i,k}^{t}\) could be represented as:  
\[\overline{\theta}_{i,k}^{t}=\arctan\left(\frac{y_{u}-y_{i,k}^{t}}{x_{u}-x_{i, k}^{t}}\right) \tag{11}\]  
where \(x_{i,k}^{t}\) and \(y_{i,k}^{t}\) represent the \(x\) and \(y\) coordinates of the surrounding \(k\)-th UAV sensed by \(i\)-th UAV in the \(t\)-th time slot, respectively. Each element in \(\bar{n}_{\theta,i}^{t}\) is an independent zero-mean Gaussian random variable, while the elements in \(\bar{n}_{\theta,i}^{t}\) could be represented as:  
\[\bar{n}_{\theta,i}^{t}=\left[n_{\theta,i,1}^{t},n_{\theta,i,2}^{t},\cdots,n_{ \theta,i,M_{i}}^{t}\right]^{T} \tag{12}\]where \(n_{\theta,j,k}^{t}\) represents the AOA measurement error of the neighboring \(k\)-th UAV sensed by the \(i\)-th UAV in the \(t\)-th time slot. The variance of \(n_{\theta,i,k}^{t}\) was \(\left(\sigma_{\theta,i,k}^{t}\right)^{2}\). \(\left(\sigma_{\theta,i,k}^{t}\right)^{2}\), which represented the azimuth AOA measurement accuracy of the UAV system. The covariance matrix of \(n_{\theta,i}^{t}\) was:  
\[\tilde{\mathbf{C}}_{\theta,i}^{t}=diag\bigg{\{}\left(\sigma_{\theta,i,1}^{t} \right)^{2},\left(\sigma_{\theta,i,2}^{t}\right)^{2},\cdots,\left(\sigma_{ \theta,i,M_{i}}^{t}\right)^{2}\bigg{\}} \tag{13}\]  
where \(\tilde{\mathbf{C}}_{\theta,i}^{t}\) represents a diagonal matrix with the dimension \(M_{i}\times M_{i}\). Equations (12) and (13) describe the azimuth AOA measurement error of the neighboring UAVs from the perspective of the \(i\)-th UAV. At this point, it was also necessary to describe the azimuth AOA measurement error of all UAVs.