## 4 System Model
### Fading Model  
#### 4.3.1 Fading Due to Time Spread  
The reality of the propagation medium is a little more complex: the signal transmitted by the locomotive undergoes multiple effects as it travels to the Hi-Rail pickup. It undergoes reflections, diffraction, and scattering. This phenomenon is called multipath fading [15]. Reflection must be considered when the size of the objects on which the wave is reflected is greater than the signal's wavelength, which is 33.33 cm for our carrier frequency at 915 MHz. Alert signs on the railway track have large dimensions compared to 33.33 cm: therefore, the reflections of the wave must be considered. When small objects of sizes less than the wavelength are present, the transmitted signal may be subject to scattering. Figure 11: BER as a function of signal to noise ratio. When the signal encounters obstacles of small dimensions compared to its wavelength, a phenomenon is observed corresponding to the superposition of many random diffractions. Then, the incident wave is processed statistically. In our study, the rail convoy moves through a forest with \"spruce\"-type conifers. The dimension of thorns and other leaves present on deciduous trees is less than 33.33 cm: therefore, the phenomenon of scattering must be considered. Likewise, the edges of certain track signs will create a diffraction phenomenon on the transmitted wave. The three phenomena (reflection, diffusion, diffraction) will create different signals arriving at the receiver with a time delay and different amplitudes. The resulting signal at the receiver is the sum of these signals. We can imagine in the worst case that this sum taking account of multipath delays would create a destructive result, thus condemning the reception of the signal transmitted by the locomotive. This multipath fading will degrade the transmission of information. There are two predominant statistical models to highlight this phenomenon:  
* Rician model [16];
* Rayleigh model [17]. The Rician model involves multiple paths but also the main path called line of sight. In our study, there is no direct path between the locomotive and the Hi-Rail pickup due to the evolution of the railway in a wooden environment with small hills. Therefore, to model the phenomenon of multipath fading, the statistical model used is Rayleigh. As a reminder, the Rayleigh probability density is:  
\[fr=e^{-r^{2}/2.\alpha^{2}} \tag{5}\]  
where the effective value (RMS) of the distribution is \(\sqrt{2\alpha}\). Then, Gaussian white noise representing the noise at the receiving antenna is added to obtain the complete model of the telecommunication channel. In MATLAB [18], the Rayleigh communication channel between the locomotive and the Hi-Rail pickup is inspired by [PERSON] [19]. The received signal is modeled as follows:  
\[x(k)=\sum_{i=1}^{N}path(k,i) \tag{6}\]  
where  
\(N\): number of paths;  
Contribution of the ith multipath signal:  
\[path(k,i)=Ai.x(k-di)\]  
_Ai_: Gain of the ith path;  
_di_: delay of the ith sample. This Rayleigh model is a digital finite impulse response (FIR) filter. The additive noise is a Gaussian white noise (AWGN channel in MATLAB) [20; 21], and the Rayleigh channel is the \"SISO channel\" block: this last block is configured for four paths for the signal transmitted with delays (\(\tau\)) of respectively 0 \(\upmu\)s, 8 \(\upmu\)s, 12 \(\upmu\)s, and 16 \(\upmu\)s, which is usually used for outdoor telecommunication systems. These values correspond to distances traveled for the incident wave of 0 km, 2.4 km, 3.6 km, and 4.8 km, respectively. First, the FSK transmitter is set to 125 kbps. It is followed by the additive white noise channel and Rayleigh [22] channel. Then, the FSK demodulation block intervenes, and a comparison is made between the random binary sequence transmitted and the one received at the end of the telecommunication channel modelized. The MATLAB script produced allows us to obtain the curve of the BER as a function of the signal-to-noise ratio (EbNo) (Figure 12).