# On the Communication Performance of Airborne Distributed Coherent Radar
## 2 Materials and Methods
### Performance Analysis
#### 2.2.1 The PDF of \(\sqrt{\gamma_{c}}\)  
We first calculate the PDF of \(\sqrt{P_{r}^{i}\big{(}F_{1}^{i}\big{)}^{2}}\):  
\[ \begin{array}{rcl}f_{p(F_{1}^{i})^{2}}(r)\\ =&\int_{0}^{+\infty}\frac{1}{x}f_{p^{i}}(x)f_{(F_{1}^{i})^{2}}\big{(}\frac{x}{ x}\big{)}dx\\ =&\sum\limits_{g=1}^{G}\sum\limits_{k=0}^{+\infty}\int_{0}^{+\infty}\mathcal{A }(i,g,k)x^{k-b_{g}^{i}-\frac{1}{2}}r_{g}^{b_{g}^{i}-1}\exp\left(-\frac{x}{2 \big{(}\sigma_{r}^{i}\big{)}^{2}}-\frac{c_{r}^{i}}{x}\right)dx\end{array}  \tag{16}\]  
where  
\[\mathcal{A}(i,g,k)= \tag{17}\]  
Based on WA 203 (15) in [33], the PDF of \(\sqrt{P_{r}^{i}\big{(}F_{1}^{i}\big{)}^{2}}\) can be derived after some mathematical manipulation:  
\[f_{\sqrt{P_{r}^{i}\big{(}F_{1}^{i}\big{)}^{2}}}(x)=\sum\limits_{g=1}^{G}\sum \limits_{k=0}^{+\infty}\mathcal{B}(i,g,k)x^{b_{g}^{i}+k-\frac{1}{2}}K_{b_{g}^ {i}-k-\frac{1}{2}}\left(\frac{\sqrt{2c^{i}}}{\sigma_{r}^{i}}x\right), \tag{18}\]  
where \(K_{\beta}(x)\) represents the modified \(\beta\)-th order Bessel function of the second kind. There is:  
\[\mathcal{B}(i,g,k)=\frac{\big{(}\lambda_{r}^{i}\big{)}^{k}\exp\left(-\frac{ \lambda_{r}^{i}}{2}\right)a_{g}^{i}}{\Gamma\left(k+\frac{1}{2}\right)k!} \tag{19}\]  
Then we calculate the PDF of \(\sqrt{G_{i}^{i}G_{r}}\). Let \(E_{t}^{i}=\sqrt{G_{t}^{i}}\), \(E_{r}=\sqrt{G_{r}}\). Then \(E_{t}^{i}\) and \(E_{r}\) represent the transmitting antenna amplitude gain of the \(i\)-th radar UAV and the receiving antenna amplitude gain of the communication UAV respectively. Based on the modeling method of millimeter wave antenna in [34], there are:  
\[E_{t}^{i}=\left\{\begin{array}{ll}\left|\frac{\sin\left(\pi N_{t}^{i}\theta_ {t}^{i}\right)}{\pi N_{t}^{i}\theta_{t}^{i}}\right|&(\theta_{t}^{i}<\frac{1}{ N_{t}^{i}})\\ 0&\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  
where \(Q(\cdot)\) is the right tailed function with standard normal distribution, namely Q-function, and \(\delta(\cdot)\) is Dirac delta function. \(N_{s}\) is the number of discrete continuous antenna patterns mentioned in [19]. In addition:  
\[\mathcal{C}_{t}\big{(}i,i^{\prime}\big{)}=Q\Bigg{(}\frac{i^{\prime}+ N_{s}N_{t}\overline{\theta}_{t}^{i}}{N_{s}N_{t}\sigma_{\text{mt}}^{i}}\Bigg{)}-Q \Bigg{(}\frac{i^{\prime}+1+N_{s}N_{t}\overline{\theta}_{t}^{i}}{N_{s}N_{t} \sigma_{\text{mt}}^{i}}\Bigg{)}+Q\Bigg{(}\frac{i^{\prime}-N_{s}N_{t}\overline{ \theta}_{t}^{i}}{N_{s}N_{t}\sigma_{\text{mt}}^{i}}\Bigg{)}-Q\Bigg{(}\frac{i^{ \prime}+1-N_{s}N_{t}\overline{\theta}_{t}^{i}}{N_{s}N_{t}\sigma_{\text{mt}}^{ i}}\Bigg{)} \tag{23}\] \[\mathcal{C}_{\eta}(j^{\prime})=Q\Bigg{(}\frac{i^{\prime}+N_{s}N_{t }\overline{\theta}_{r}}{N_{s}N_{r}\sigma_{mr}}\Bigg{)}-Q\Bigg{(}\frac{i^{ \prime}+1+N_{s}N_{r}\overline{\theta}_{r}}{N_{s}N_{r}\sigma_{mr}}\Bigg{)}+Q \Bigg{(}\frac{i^{\prime}-N_{s}N_{t}\overline{\theta}_{r}}{N_{s}N_{r}\sigma_{mr }}\Bigg{)}-Q\Bigg{(}\frac{i^{\prime}+1-N_{s}N_{r}\overline{\theta}_{r}}{N_{s}N _{r}\sigma_{mr}}\Bigg{)}\] (24) \[\mathcal{D}\big{(}i,j^{\prime}\big{)}=\frac{1}{N_{t}N_{r}}\Bigg{|} \frac{\sin\left(\frac{\pi N_{t}j^{\prime}}{N_{t}N_{t}}\right)}{\frac{\pi N_{t }j^{\prime}}{N_{t}N_{t}}}\Bigg{|}\Bigg{|}\frac{\sin\left(\frac{\pi N_{t}j^{ \prime}}{N_{t}N_{r}}\right)}{\frac{\pi N_{t}j^{\prime}}{N_{t}N_{r}}}\Bigg{|} \tag{25}\]  
Then we calculate the PDF of \(\sqrt{P_{i}^{i}G_{i}^{\prime}G_{r}\big{(}F_{1}^{i}\big{)}^{2}}\). From (18) and (22) we can get:  
\[\begin{array}{ll}&f_{\sqrt{P_{i}^{i}G_{i}^{\prime}G_{r}\big{(}F_{1}^{i} \big{)}^{2}}}(r)\\ &=&\int_{0}^{+\infty}\frac{1}{2}\gamma  
\[ \begin{array}{ll}\mu_{Z}(n^{\prime})=&\sum\limits_{n_{1}=0}^{n^{\prime}}\sum \limits_{n_{2}=0}^{n_{1}}\cdots\sum\limits_{n_{N_{Under}-1}=0}^{n_{N_{Under}-2}} \left(\begin{array}{c}n^{\prime}\\ n_{1}\end{array}\right)\left(\begin{array}{c}n_{1}\\ n_{2}\end{array}\right)\cdots\left(\begin{array}{c}n_{N_{Under}-2}\\ n_{N_{Under}-1}\end{array}\right)\\ &\times\mu_{W^{1}}(n^{\prime}-n_{1})\mu_{W^{2}}(n_{1}-n_{2})\cdots\mu_{W^{N_{Under }-1}}(n_{N_{Under}-2}-n_{N_{Under}-1})\mu_{W^{N_{Under}}}(n_{N_{Under}-1})\end{array}