## 3 Tapped Delay Line Model
### A2G TDL Model
#### 3.2.2 Amplitude Fading  
To accurately understand and model the power variations characteristics of individual channel taps, the amplitude distribution of taps are evaluated. Depending on the feature of radio propagation channels, different modeling approaches are used to represent the statistical fading behavior of the envelope amplitude [32]. To mitigate the fast fading effect, CIRs are usually averaged over a certain measurement duration. In this work, we use the window length of 100 wavelengths to average the CIRs according to the speed of UAV during the measurement.  
Assuming that the distance between transmitter and receiver is sufficiently large, all received signals belong to the same channel tap are plane waves without a dominant component. As a result, the amplitude of the channel tap generally follows the Rayleigh distribution, and the gain of the tap can be modeled as a zero mean Gaussian random variable [33]. The probability density function (PDF) of Rayleigh distribution is given as  
\[p_{f}(|\alpha|,\sigma_{l})=\frac{|\alpha|}{\sigma_{l}^{2}}e^{-\frac{|\alpha|^{ 2}}{2\sigma_{l}^{2}}}, \tag{7}\]  
where \(\sigma_{l}\) is the standard deviation. \(|\alpha|\) is the normalized magnitude, which is given by  
\[\alpha(n\Delta t,(m-1)\Delta\tau)=\frac{h^{*}(n\Delta t,(m-1)\Delta\tau)}{ \sqrt{P((m-1)\Delta\tau)}}\,. \tag{8}\]  
When a dominant path or LoS component is present, the amplitude distribution of the taps can be modeled by Rice distribution; the PDF of the Rice distribution is [34]  
\[\begin{split} p_{f}(|\alpha|,K_{r},\Omega_{r})&= \frac{1+K_{r}}{\Omega_{r}}I_{0}\Bigg{(}2\sqrt{\frac{K_{r}(K_{r}+1)|\alpha|}{ \Omega_{r}}}\Bigg{)}\\ &\exp\Bigg{(}-\frac{K_{r}\overline{\Omega_{r}}-(K_{r}+1)|\alpha| }{\overline{\Omega_{r}}}\Bigg{)},\end{split} \tag{9}\]  
where \(K_{r}\) denotes the K-factor corresponding to the power ratio between the LoS component and other MPCs. \(I_{0}(\cdot)\) is the modified Bessel function, and \(\Omega_{r}\) is the sum of the power of there two kinds of path. The stronger the LoS component is, the less likely the deep fading occurs. The K-factor reflects the power ratio between the dominant path and the scattering path in multipath propagation, which directly affects signal quality, the design of equalization and detection algorithms, and the choice of modulation and coding schemes, as well as linking budget and power control strategies. The design of UAV communication  
Figure 4: An example of normalized APDP.  
system needs to be adjusted according to the K-factor in the actual environment to ensure system stability and efficiency. The value of the Rice K-factor is calculated using the method of the moment [35].  
Other typical distributions used to describe the amplitude are Gaussian distribution, Weibull distribution [36], Nakagami distribution, which are given as [32]:  
* Gaussian distribution: \[p_{f}\big{(}|\alpha|,\mu_{g},\sigma_{g}\big{)}=\frac{1}{\sigma_{g}\sqrt{2\pi}} \exp\left(-\frac{(|\alpha|-\mu_{g})^{2}}{2\sigma_{g}^{2}}\right),\] (10) where \(\mu_{g}\) and \(\sigma_{g}\) denote the mean and standard deviation of the distribution, respectively.
* Weibull distribution: \[p_{f}(|\alpha|,\beta_{w},\Omega_{w})=\frac{\beta_{w}}{\Omega_{w}}|\alpha|^{ \beta_{w}-1}\exp\left(-\frac{|\alpha|^{\beta_{w}}}{\Omega_{w}}\right),\] (11) where \(\Omega_{w}\) and \(\beta_{w}\) denote the shape parameter and scale parameter, respectively.
* Nakagami distribution: \[p_{f}(|\alpha|,\varphi,\Omega)=\frac{2q^{\varphi}}{\Gamma(\varphi)\Omega^{ \varphi}}|\alpha|^{2\varphi-1}\exp(-\frac{\varphi}{\Omega}|\alpha|^{2}),\] (12) where \(\varphi\) and \(\Omega\) denote the shape and spread parameter, respectively. \(\Gamma(\varphi)\) is the Gamma function evaluated at \(\varphi\).