## 2 Skywave Ionosphere Channel for Hypersonic Vehicles  
During the OTH propagation of hypersonic vehicles, multipath fading is caused by ionospheric reflections, as shown in Figure 1.  
Based on the above propagation conditions, let the baseband signal be \(s(t)\), and the carrier modulation (transmission signal) of the baseband signal is denoted as \(s(t)e^{j2\pi\#t}\), where \(f\) is the carrier frequency, and \(e^{j2\pi\#t}\) has an exponential complex number derivation process, derived from [PERSON]'s formula. Therefore, the received signal be written as  
\[y(t)=\sum_{1}^{N(t)}h_{k}(t)s(t-\tau_{k}(t))e^{j2\pi(f+f_{dk}(t))(t-\tau_{k}(t ))} \tag{1}\]  
where \(N\) (\(t\)) is the number of multipaths at time \(t\); \(f_{dk}\) (\(t\)) and \(\tau_{k}\) (\(t\)) represent the \(k\)-th path's Doppler frequency and delay, respectively; and \(h_{k}\) (\(t\)) represents ionospheric fading for the \(k\)-th path. According to Equation (1), the parameters that affect a skywave ionosphere channel are \(f_{dk}\) (\(t\)), \(\tau_{k}\) (\(t\)), and \(h_{k}\). In fact, time-varying characteristics of the communication channel, including Doppler effects, delays, and attenuation, significantly impact system performance. Doppler effects cause frequency shifts and broadening, leading to difficulties in signal demodulation and increased error rates. Delays result in multipath interference and inter-symbol interference, reducing signal quality. Attenuation leads to a decreased signal-to-noise ratio (SNR) and signal fading, affecting communication reliability. Therefore, analyzing the dynamic changes in channels and designing and optimizing communication systems with targeted approaches is the foundation for achieving a high-quality and stable signal transmission. The physical environment of the ionosphere in channel propagation belongs to the category of plasma. According to classical theory, plasma can be considered a special conductive dispersion loss medium. In what follows, the propagation theory of the ionosphere in skywave is used to derive the multipath fading and Doppler calculation of the skywave ionosphere channel for hypersonic vehicles. The skywave ionosphere channel fading \(h_{k}\) consists of two components:  
\[h_{k}(t)=l_{ak}(t)\cdot a_{iok}(t) \tag{2}\]  
where \(l_{ak}\) (\(t\)) includes free-space attenuation, ground reflection loss, and other additional losses, which are detailed in the ITU recommendations for predicting skywave transmission loss and field strength distribution (ITU-R.P series) [39; 40; 41]; they are not discussed in this section. Aside from the aforementioned losses, the absorption loss of the ionosphere \(a_{iok}\) (\(t\)) is the focus in this section. Before radio waves enter the ionosphere, neutral molecules and ions, along with electrons, undergo diffuse and irregular thermal motion. When radio waves enter the ionized gas, free electrons experience harmonic motion under the influence of the incident wave's electric field. In general, moving electrons also collide with other particles, transferring the kinetic energy gained from the radio waves to neutral molecules and ions, resulting in  
Figure 1: Scenarios for skywave OTH propagation of hypersonic vehicles. thermal energy loss, known as an ionospheric absorption loss of radio waves. There are two types of ionospheric absorption losses: an offset absorption loss and a non-offset absorption loss. Offset absorption is generally less than 1 dB and can be ignored [42]. Therefore, this section primarily focuses on the impact of non-offset absorption on electromagnetic waves.