# Joint Radar-Communication Optimization of Distributed Airborne Radar for AOA Localization
## 3 JRCO Algorithm
### Composite Variance Calculation of AOA Measurement  
At present, all input variables for constructing the objective function of \(i\)-th UAV in the \(t\)-th time slot could be obtained, namely \(\bar{\mathbf{r}}_{i}^{t}\), \(\hat{\mathbf{u}}_{i}\), and \(\bar{\mathbf{\mathcal{C}}}_{\theta,i}^{t}\). According to Equation (21), the objective function could be constructed at this time. However, \(\bar{\mathbf{\mathcal{C}}}_{\theta,i}^{t}\) did not include the impact of the UAV location error on AOA measurement at this time.  
This section considers the impact of the UAV location error on the AOA measurement based on the covariance matrix \(\bar{\mathbf{\mathcal{C}}}_{\theta,i}^{t}\). For a local networking system composed of \(M_{i}\) UAVs around the \(i\)-th UAV, the errors in the \(x\), \(y\) and \(z\) coordinates of each UAV could be defined as \(\Delta x_{i,k}^{t}\), \(\Delta y_{i,k}^{t}\) and \(\Delta z_{i,k}^{t}\) respectively, and the AOA measurement error was \(n_{\theta,i,k}^{t}\). Thus, they were all Gaussian random variables with a zero mean, i.e., \(\Delta x_{i,k}^{t}\sim\mathcal{N}(0,\sigma_{1}^{t})\), \(\Delta y_{i,k}^{t}\sim\mathcal{N}\big{(}0,\sigma_{1}^{t}\big{)}\) and \(n_{\theta,i,k}^{t}\sim\mathcal{N}\bigg{(}0,\left(\sigma_{\theta,i}^{t}\right) ^{2}\bigg{)}\). For the theoretical basis of Gaussian random variables for the above random variables, please refer to the literature [40].  
The AOA estimation error caused by the location error of the \(k\)-th UAV in the local networking system of \(i\)-th UAV could be defined as \(n_{s,i,k}^{t}\). At this point, the azimuth AOA measurement error could be determined by \(n_{\theta^{\prime},i,k}^{t}=n_{\theta,i,k}^{t}+n_{s,i,k}^{t}\), and the elevation AOA measurement error could be determined by \(n_{\theta^{\prime},i,k}^{t}=n_{\theta,i,k}^{t}+n_{s,i,k}^{t}\). Based on the geometric relationship, there were:  
\[n_{\theta,i,k}^{t}=\arctan\left(\frac{y_{u}-y_{i,k}^{t}-\Delta y_{i,k}^{t}}{x _{u}-x_{i,k}^{t}-\Delta x_{i,k}^{t}}\right) \tag{53}\]  
\[n_{\theta,i,k}^{t}=\arctan\left(\frac{z_{u}-z_{i,k}^{t}-\Delta z_{i,k}^{t}}{ \sqrt{\left(x_{u}-x_{i,k}^{t}-\Delta x_{i,k}^{t}\right)^{2}+\left(y_{u}-y_{i, k}^{t}-\Delta y_{i,k}^{t}\right)^{2}}}\right) \tag{54}\]  
The derivation of PDFs for \(n_{\varphi,i,k}^{t}\) and \(n_{\theta,i,k}^{t}\) was not an easy process, but numerical simulations could verify that \(n_{\varphi,i,k}^{t}\) and \(n_{\theta,i,k}^{t}\) approximately followed Gaussian distributions. The PDF based on sample fitting and Gaussian distribution fitting was verified under the condition of \(x_{u}=50\), \(y_{u}=50\), \(z_{u}=50\), \(\sigma_{1}=1\) (shown in Figure 3). The locations of the sensor are all \(x_{i,k}^{t}=0\), \(y_{i,k}^{t}=0\) in Figure 3. Based on the properties of the Gaussian distribution, it could be concluded that both \(n_{\theta^{\prime},i,k}^{t}\) and \(n_{\theta^{\prime},i,k}^{t}\) follow the Gaussian distribution at this time. Their variances were defined as \(\left(\sigma_{\theta^{\prime},i,k}^{t}\right)\) and \(\left(\sigma_{\theta^{\prime},i,k}^{t}\right)^{2}\), and their corresponding Covariance matrix was defined as \(\bar{\mathbf{\mathcal{C}}}_{\theta^{\prime},i}^{t}\) and \(\bar{\mathbf{\mathcal{C}}}_{\theta^{\prime},i}^{t}\), respectively.  
Figure 3: Approximated PDF for \(n_{\varphi,i,k}^{t}\).