# On the Communication Performance of Airborne Distributed Coherent Radar
## 2 Materials and Methods
### System Model
#### 2.1.2 Statistical Model of \(P_{r}^{i}\) and \(P_{c}\)  
Due to the limitation of UAV load, the power amplifier output of each UAV may be unstable. Therefore, in this subsection, we will give the transmitting power model of radar UAV and communication UAV. Since power is closely related to amplitude, we define the amplitude factor corresponding to the \(i\)-th radar UAV as \(A_{r}^{i}\), and the amplitude factor corresponding to the communication UAV as \(A_{c}\). So obviously there is \(P_{r}^{i}=\left(A_{r}^{i}\right)^{2}\), \(P_{c}=\left(A_{c}\right)^{2}\). If \(A_{r}^{i}\) follows the Gaussian distribution with mean \(\mu_{r}^{i}\) and variance \(\left(\sigma_{r}^{i}\right)^{2}\). \(A_{c}\) follows the Gaussian distribution with mean \(\mu_{c}\) and variance \(\sigma_{c}^{2}\). i.e., \(A_{r}\sim\mathcal{N}\left(\mu_{r}^{i}\left(\sigma_{r}^{i}\right)^{2}\right)\)and \(A_{c}\sim\mathcal{N}\big{(}\mu_{c},\sigma_{c}^{2}\big{)}\). We define \(B_{r}^{i}=A_{r}^{i}/\sigma_{r}^{i}\) and \(B_{c}=A_{c}/\sigma_{c}\). Then \(\big{(}B_{r}^{i}\big{)}^{2}\) follows the noncentral chi-squared distribution with scale parameter \(\lambda_{r}^{i}=\left(\frac{\mu_{c}^{i}}{\sigma_{r}^{i}}\right)^{2}\) and degree of freedom 1. The PDF of \(\big{(}B_{r}^{i}\big{)}^{2}\) is:  
\[f_{(B_{r}^{i})^{2}}(x)=\frac{1}{2}\exp\left[-\frac{x+\lambda_{r}^{i}}{2} \right]\left(\frac{x}{\lambda_{r}^{i}}\right)^{-\frac{1}{2}}I_{-\frac{1}{2}} \left(\sqrt{\lambda_{r}^{i}x}\right) \tag{5}\]  
where \(I_{\beta}(x)\) represents the modified \(\beta\)-th order Bessel function of the first kind. Since the expression in (5) contains exponential function, power function and Bessel function, it is not conducive to subsequent derivation in this paper. According to [28], we will apply another approximate expression for the non-central chi square distribution:  
\[f_{(B_{r}^{i})^{2}}(x)=\sum_{k=0}^{+\infty}\frac{\left(\lambda_{r}^{i}\right)^ {k}\exp\left(-\frac{\lambda_{r}^{i}}{2}\right)}{2^{2k+\frac{1}{2}}\Gamma \left(k+\frac{1}{2}\right)k!}\exp\left(-\frac{x}{2}\right)x^{k-\frac{1}{2}}, \tag{6}\]  
where \(\Gamma(\cdot)\) is the gamma function. The PDF of \(P_{r}^{i}\) can be obtained from (6)  
\[f_{P_{r}^{i}}(x)=\sum_{k=0}^{+\infty}\frac{\left(\lambda_{r}^{i}\right)^{k} \exp\left(-\frac{\lambda_{r}^{i}}{2}\right)}{2^{2k+\frac{1}{2}}\sigma_{c}^{2l +1}\Gamma\left(k+\frac{1}{2}\right)k!}\exp\left(-\frac{x}{2\left(\sigma_{r}^{ i}\right)^{2}}\right)x^{k-\frac{1}{2}} \tag{7}\]  
Similarly, the PDF of \(P_{c}\) can be obtained:  
\[f_{P_{c}}(x)=\sum_{l=0}^{+\infty}\frac{\lambda_{c}^{l}\exp\left(-\frac{ \lambda_{r}}{2}\right)}{2^{2l+\frac{1}{2}}\sigma_{c}^{2l+1}\Gamma\left(l+ \frac{1}{2}\right)l!}\exp\left(-\frac{x}{2\sigma_{c}^{2}}\right)x^{l-\frac{1}{ 2}} \tag{8}\]  
It should be noted that the PDFs represented by (7) and (8) involve the summation of functions to infinite times, which is not feasible for the numerical simulation in Section 3. However, when the summation order is high, the higher-order components of PDF impose little effect on the PDF value, so the subsequent simulation only takes a finite number of summations to approximate the real distribution.