# Impact of Weather Factors on Unmanned Aerial Vehicles' Wireless Communications
## 2 Methodology
### Mathematical Models
#### 2.1.6 Effect of Mobility and Multipath Propagation on BER  
In this section, we examine the effects of mobility and time-variant characteristics of the wireless channel, which negatively impact the BER performance of GCS-to-UAV wireless communications. Accordingly, we develop a mathematical model to compute effective BER after modifying the computed SNR based on a static channel model. For this modeling, we assume OFDM based multi-carrier wireless communication between the GCS and the UAV, since it is a widely adopted modulation scheme due to its superior spectral efficiency. Both cellular and Wi-Fi communication networks, utilized for UAV command and control, employ this modulation scheme. The complex envelope of the transmitted OFDM symbol \(s(t)\), based on \(N\) sub-carriers, is given by [26] as follows:  
\[s(t)=\left(\sum_{m=0}^{N-1}a_{m}e^{\beta{\pi}t}\right)e^{i\theta(t)} \tag{12}\]  
where \(a_{m}\) is the data symbol that modulates the \(m^{th}\) sub-carrier at frequency \(m/T\) over the symbol period \(T\), and \(\theta(t)\) is a time varying phase resulting from the carrier offset between the transmitter and the receiver. Based on different scenarios, a multitude of UAV channel models have been studied [17; 24; 27; 28; 29]. A general approach of channel modeling is to conduct a channel impulse response-based data collection campaign for a given region of interest, followed by statistical analysis and curve fitting to derive an empirical model. Figure 5 illustrates a time-variant generic wireless channel through which the transmitted symbol \(s(t)\) propagates to arrive at the receiver. The wireless channel includes different types of moving and stationary objects, with varying configurations and concentrations. The transmitted signal \(s(t)\) is reflected, refracted, and scattered by these objects. As a result, the signal propagation takes multiple paths to reach the receiver. In this illustration, only four paths are shown. However, based on the objects' configurations and concentrations in the channel, many paths may exist. Figure 5: Illustration of mobility and multipath propagation in UAV wireless channel. Each path \(p_{k}\), causes unique amplitude \(a_{p_{k}}\), Doppler shift \(f_{d_{p_{k}}}\), and propagation delay \(\tau_{p_{k}}\) variations to the transmitted symbol. The received symbol y(t) is the aggregation of all symbol instances arriving via different paths, plus the all-white Gaussian noise (AWGN) \(N_{0}\) as applicable to the operating bandwidth B. Figure 6 shows the mathematical formulations for the amplitude, Doppler shift, and propagation delay variations applicable to \(L\) propagation paths [30]. The received symbol y(t) can then be given by  
\[y(t)=\left(\sum_{k=0}^{L-1}s(t).a_{p_{k}}e^{ja_{k}}e^{ja_{k}}e^{j2\pi f_{d_{p_{k }}}t}.\delta\left(t-\tau_{p_{k}}\right)\right)+N_{0} \tag{13}\]  
where corresponding to path \(p_{k}\), \(a_{p_{k}}e^{ja\tau_{p_{k}}}\), \(e^{2\pi f_{d_{p_{k}}}t}\) and \(\tau_{p_{k}}\), represent the amplitude, Doppler shift, and path delay applied to the symbol passing through it, and \(N_{0}\) is the temperature and bandwidth dependent AWGN noise. The phase variations caused by clock phase noise, Doppler shift, and propagation delays result in loss of OFDM sub-carrier orthogonality. Consequently, the OFDM sub-carriers interfere with one another. This inter sub-carrier interference (ICI) manifests as near-Gaussian noise [30], resulting in SNR degradation. As SNR degrades, link performance deteriorates due to an increase in BER. The studies described in [26; 31] investigated \(y(t)\) due to the response of the multipath propagation channel on \(s(t)\). The time-variant channel conditions degrade SNR due to ICI resulted noise. The SNR degradation is analyzed for two scenarios: Doppler shift, and clock phase noise.