## 4 System Model
### Fading Model
#### 4.3.2 Doppler Effect  
The Doppler effect is also a fading effect: when the receiver or an object inside the propagation channel is moving, it creates the Doppler effect. The consequence is a carrier frequency shift. The maximum frequency variation is the Doppler band Bd. Everyone has listened to the sound of an ambulance when it approaches and when it leaves: this is the Doppler effect. If the receiver is in motion, the propagation delay of the electromagnetic wave will therefore increase or decrease. The coherence time (Tc) of the propagation channel is the time over which the channel is not varying. This delay is equal to \(\frac{1}{(2.B_{d})}\): if Tc is higher than the symbol duration (Ts), it means that the propagation channel is time invariant, and then, the Doppler effect is low. On the opposite, if Tc is lower than Ts, the  
Figure 12: BER as a function of signal to noise ratio.  
propagation delay is variant over time, and therefore, the fading will be fast. The frequency shift (\(f_{d}\)) due to the Doppler effect is:  
\[\text{fd}=\frac{v_{r}}{\lambda}\times\cos(\theta)=\text{Fc}\times\frac{v_{r}}{ \text{v}_{\text{p}}}\times\cos(\theta) \tag{7}\]  
where \(v\) is the receiver velocity, \(\lambda\) is the wavelength of the electromagnetics wave, \(\text{v}_{\text{p}}\) is the propagation velocity, \(\theta\) is the angle of the electromagnetic wave when it arrives at the receiver, and Fc is the carrier frequency. Therefore, the maximum Doppler frequency shift is \(\text{Fc}\times\frac{v_{r}}{\text{v}_{\text{p}}}\), and the carrier frequency will be in the interval:  
\[[\text{Fc}-\text{vr}/\lambda\ ;\text{Fc}+\text{vr}/\lambda\ ]. \tag{8}\]  
The locomotive and the Hi-Rail pickup are most of the time in motion. It is legitimate to think that there will be a Doppler effect that might degrade the quality of the transmission. If we consider the speed (vr) of the Hi-Rail pickup relative to the train, it is as if the Hi-Rail pickup, therefore the receiver, was moving with a relative vr speed. In the study, if we consider a relative speed vr maximum of 5 km/h, the carrier frequency will vary then in the interval [915 MHz-4.2 Hz; 915 MHz + 4.2 Hz]. Therefore, the frequency variation \(\Delta\)F is equal to 4.2 Hz or 0.00459 ppm. The XTend modules used in the receiver tolerate a frequency shift of 40 ppm: the Doppler effect will have no impact on this railway telecommunications system. The conclusion is the same if we calculate the coherence time Tc: Tc = 1/(2.Bd); then, 60 ms. For a data rate at 125 kbps, we have calculated earlier that the symbol duration value was 8 \(\upmu\)s: therefore, the coherence time is much longer than the symbol duration. Even if we think of the case when the locomotive is moving and the Hi-Rail pickup is stopped: the relative speed is higher, and this is the worst case. vr equals 35 km/h (these are the rules of Transport Canada on this type of railroad), and then, the frequency variation interval is [915 MHz \(-\) 29.68 Hz; 915 MHz + 29.68 Hz] which gives a Doppler band to 59.36 Hz and then a coherence time of 8.42 ms, which is much higher than 8 \(\upmu\)s (duration time). Even in this last case, the Doppler effect has no negative impact.  
In conclusion, the Doppler effect due to the Hi-Rail pickup relative velocity does not affect the railway telecommunications system.  
Now, let us consider the influence of the geographical and atmospheric criteria specific to the implementation area.