# Multi-UAVs Communication-Aware Cooperative Target Tracking
## 3 Information Fusion Based on Extended Information Filtering
### Extend Information Filter and Information Fusion Phase
 \tag{36}\]  
where  
\[\Gamma_{ji}^{\rm c}=H_{j}{}^{T}[k](\Sigma_{ji}[k])^{-1}H_{j}[k]. \tag{37}\]  
From Equation (34) we can figure out that in this paper, the communication noise was treated as a kind of observation noise from others. Equation (34) can be written as Equation (36), which is different from the method proposed in Reference [11]. It makes UAVs reach a compromise between observation and communication, and track the target cooperatively. Similar expression can be written for \(\hat{y}_{i}[k]\). Then, we can use the determinant of \(Y_{i}[k]\) as the objective function for each UAV, which can be found in Section 4. Next, we give a mathematical proof for our method to explain how it works when we put communication noise into the objective function, and to prove how our proposed method finds a way to reach a compromise between observation and communication for each UAV. Firstly, we give the following definition:  
\[N_{i}\triangleq\textit{the set of UAV its neighbours}(without\ i)\]  
\[N_{i}^{+}\triangleq\textit{the set of UAV its neighbours}(including\ i)\]  
\[R_{i}=\left[ \begin{array}{c}\sigma_{r}^{2}\\ \sigma_{\varphi}^{2}\end{array} \right] \tag{38}\]  
\[\sum\
olimits_{ji}=\left[\begin{array}{c}c_{rj}^{2}\\ \sigma_{\varphi j}^{2}  
where \(c_{rj}=c_{\varphi j}=0\) when \(j=i\), which means there is no communication noise when an UAV communicates with itself. \[J_{i}\triangleq\sum\limits_{j\in N_{i}^{+}}\left[ \begin{array}{cc}\frac{\sin^ {2}\varphi_{j}}{\sigma_{r}^{2}+c_{rj}^{2}}+\frac{\cos^{2}\varphi_{j}}{r_{j}^{2} (\sigma_{\varphi}^{2}+c_{\varphi j}^{2})}&\frac{\cos\varphi_{j}\sin\varphi_{j}} {\sigma_{r}^{2}+c_{rj}^{2}}-\frac{\cos\varphi_{j}\sin\varphi_{j}}{r_{j}^{2}( \sigma_{\varphi}^{2}+c_{\varphi j}^{2})}\\ \frac{\cos\varphi_{j}\sin\varphi_{j}}{\sigma_{r}^{2}+c_{rj}^{2}}-\frac{\cos \varphi_{j}\sin\varphi_{j}}{r_{j}^{2}(\sigma_{\varphi}^{2}+c_{\varphi j}^{2}) }&\frac{\cos^{2}\varphi_{j}}{\sigma_{r}^{2}+c_{rj}^{2}}+\frac{\sin^{2} \varphi_{j}}{r_{j}^{2}(\sigma_{\varphi}^{2}+c_{\varphi j}^{2})}\end{array}  \right]. \tag{42}\]  
We defined det(\(J\)) as the determinant of \(J\). Then, from Equation (42) we get:  
\[\det(J_{i})=\det\left\{\sum\limits_{j\in N_{i}^{+}}\left[ \begin{array}{cc} \frac{\sin^{2}\varphi_{j}}{\sigma_{r}^{2}+c_{rj}^{2}}+\frac{\cos^{2}\varphi_{ j}}{r_{j}^{2}(\sigma_{\varphi}^{2}+c_{\varphi j}^{2})}&\frac{\cos\varphi_{j} \sin\varphi_{j}}{\sigma_{r}^{2}+c_{rj}^{2}}-\frac{\cos\varphi_{j}\sin\varphi_{ j}}{r_{j}^{2}(\sigma_{\varphi}^{2}+c_{\varphi j}^{2})}\\ \frac{\cos\varphi_{j}\sin\varphi_{j}}{\sigma_{r}^{2}+c_{rj}^{2}}-\frac{\cos \varphi_{j}\sin\varphi_{j}}{r_{j}^{2}(\sigma_{\varphi}^{2}+c_{\varphi j}^{2}) }&\frac{\cos^{2}\varphi_{j}}{\sigma_{r}^{2}+c_{rj}^{2}}+\frac{\sin^{2} \varphi_{j}}{r_{j}^{2}(\sigma_{\varphi}^{2}+c_{\varphi j}^{2})}\end{array}  \right]\right\} \tag{43}\]  
where  
\[B_{\varphi j}=\frac{1}{\sigma_{r}^{2}+c_{rj}^{2}}+\frac{1}{r_{j}^{2}(\sigma_{ \varphi}^{2}+c_{\varphi j}^{2})} \tag{44}\]  
Then, we get the following expression:  
\[\det(J_{i})= \frac{1}{4}\cdot\left(\sum\limits_{j\in N_{i}^{+}}\left(\frac{1} {\sigma_{r}^{2}+c_{rj}^{2}}+\frac{1}{r_{j}^{2}(\sigma_{\varphi}^{2}+c_{\varphi j }^{2})}\right)\right)^{2} \tag{45}\] \[-\left(\sum\limits_{j\in N_{i}^{+}}\left(\cos 2\varphi_{j} \cdot\left(\frac{1}{\sigma_{r}^{2}+c_{rj}^{2}}-\frac{1}{r_{j}^{2}(\sigma_{ \varphi}^{2}+c_{\varphi j}^{2})}\right)\right)\right)^{2}\] \[-\left(\sum\limits_{j\in N_{i}^{+}}\left(\sin 2\varphi_{j} \cdot\left(\frac{1}{\sigma_{r}^{2}+c_{rj}^{2}}-\frac{1}{r_{j}^{2}(\sigma_{ \varphi}^{2}+c_{\varphi j}^{2})}\right)\right)\right)^{2}\]  
We can see from Equation (45) that \(r_{j}=r_{\min}\) is a necessary condition for the maximum of \(\det(J_{i})\), and has an upper limit.