## 4 Contribution
### Construction of SUUIL Model for Urban Environment Ultra-Low Altitude
#### 4.1.2 BER Analysis of SUUL Model  
According to the intermediate variable \(\gamma=\frac{sT_{s}}{N_{0}}\overline{\gamma}=\frac{T_{s}}{N_{0}}\overline{s}\), so:  
\[f_{\gamma}(\gamma)=\frac{N_{0}}{T_{s}}f_{S}(s)=\frac{N_{0}}{T_{s}}f_{S}\bigg{(} \frac{N_{0}}{T_{s}}\gamma\bigg{)}=\frac{\overline{s}}{\overline{\gamma}}f_{S }\bigg{(}\frac{\overline{s}}{\overline{\gamma}}\gamma\bigg{)} \tag{35}\]  
The probability density function of signal-to-noise ratio (SNR) is:  
\[f_{\gamma}(\gamma)=\frac{\overline{s}}{2\overline{\gamma}\lambda_{0}^{2} \sqrt{2\pi d_{0}}}\int_{0}^{\infty}\frac{1}{a}\exp\left(-\frac{\lambda_{0}^{2 }\left(\ln a-\mu\right)^{2}+d_{0}\left(\frac{\overline{s}}{\overline{\gamma}} \gamma+a^{2}\right)}{2d_{0}\lambda_{0}^{2}}\right)l_{0}\bigg{(}\frac{a}{ \lambda_{0}^{2}}\sqrt{\frac{\overline{s}}{\overline{\gamma}}\gamma}\bigg{)} \mathrm{d}a \tag{36}\]  
where \(\overline{\gamma}\) is the average signal-noise ratio of per symbol, \(\sigma_{X}^{2}\) is the shadow variance. According to the following equation [21] the expression of bit error rate (BER) can be obtained:  
\[P_{b}(E)=\int_{0}^{\infty}P_{b}(E;\gamma)f_{\gamma}(\gamma)\mathrm{d}\gamma \tag{37}\]  
For PSK modulation, the expression of BER is calculated as follows:  
\[\xi=\frac{\overline{s}}{2\lambda_{0}^{2}\gamma\sqrt{2\pi d_{0}}}\int_{0}^{ \infty}Q\bigg{(}\sqrt{2\gamma}\bigg{)}\int_{0}^{\infty}\frac{1}{a}\exp\left( -\frac{\lambda_{0}^{2}\overline{\  
However, Equation (42) cannot be implemented well in simulation. The model provided in [22] continues to be used as an approximation to derive the BER, taking into account computational complexity and other factors. This model discusses the same channel characteristics as the model in this article.  
The Nakagami model is proposed in [23], and the magnitude of the LOS component obeys the Nakagami distribution:  
\[f_{A}(a)=\frac{2m^{m}a^{2m-1}}{\mu_{R_{a}}{}^{m}\Gamma(m)}\exp\left(-\frac{ma^{2 }}{\mu_{R_{a}}}\right)\quad,\quad\quad\quad\quad a\geq 0 \tag{43}\]  
where \(\Gamma(\cdot)\) is the Gamma function and \(m\) is the parameter of the Nakagami distribution, \(m=\frac{\mu_{R_{a}}{}^{2}}{\
u_{M}\
u_{M}|R_{a}|}\geq 0\). In order to better model various LOS conditions in the channel, let the range of \(m\) be \([0,\infty)\). In a multipath fading environment [21] the distribution parameter of the traditional Nakagami model ranges from \([0.5,\infty)\). A suitable \(m\) value is chosen to simulate LOS conditions in urban areas. The model parameter \(\left(\mu_{R_{a}},m,\lambda_{0}^{2}\right)\) are associated with the measurement parameter \(\left(d,K_{r},\sigma_{X}^{2}\right)\) in order to better apply the approximate model. Derive the approximate probability density function of the variance of \(\Psi\):  
\[f_{\Psi}(\Psi)=\frac{1}{\Gamma(m)\xi}\left(P_{l}\frac{m}{\mu_{R_{a}}}\right)^{ m}\exp\left(-\frac{m}{\xi}\Psi-\frac{mP_{l}}{\mu_{R_{a}}\xi}\exp\left(-\frac{ \Psi}{\xi}\right)\right) \tag{44}\]  
The parameters are correlated with those in the LOO model and use the second-order moment [24] to get the expressions of the following parameters:  
\[d_{0}=\frac{1}{4}\Psi^{\prime}(m) \tag{45}\]  
\[\mu_{R_{a}}=m\exp(2\mu-\Psi(m)) \tag{46}\]  
where \(\Psi^{\prime}(\cdot)\) is the first partial derivative of \(\Psi(\cdot)\)[21]. \(\mu_{R_{a}}\) and \(m\) can be calculated according to Equations (46) and (45). The probability density function of the received signal envelope can be calculated by Equations (17) and (43) and conditional mathematical expectation:  
\[f_{R}(r)=\left(\frac{2\lambda_{0}^{2}m}{2\lambda_{0}^{2}m+\mu_{R_{a}}}\right)^ {m}\frac{r}{\lambda_{0}^{2}}\exp\left(-\frac{r^{2}}{2\lambda_{0}^{2}}\right)_ {1}F_{1}\left(m;1;\frac{\mu_{R_{a}}r^{2}}{2\lambda_{0}^{2}\left(2\lambda_{0}^ {2}m+\mu_{R_{a}}\right)}\right)\quad,\;\;r\geq 0 \tag{47}\]  
where \({}_{1}F_{1}(\quad;\quad;\quad)\) is the confluent hypergeometric function [21].