# Deterministic Propagation Modeling for Intelligent Vehicle Communication in Smart Cities
## 3 Simulation Results
### Multipath Metrics
#### 3.3.3 Doppler Spread (B\({}_{\rm D}\)) and Doppler Shift (fd)  
Doppler Spread (B\({}_{\rm D}\)) and coherence time (T\({}_{\rm C}\)) describe the time-varying nature of the channel, caused by either relative motion between the mobile and base station or by the movement of objects in the channel, in the small-scale region. B\({}_{\rm D}\) is defined as the range of frequencies (spectral broadening) over which the received Doppler Spectrum is essentially non-zero (if the baseband signal bandwidth is much greater than B\({}_{\rm D}\), the effects of Doppler spread are negligible at the receiver--slow fading channel). Doppler Shift (fd) is the Doppler spectrum of the received signal in the range fc - fd to fc + fd, where fc is the transmitted frequency [34].  
Figure 12 shows the effect of the relative velocity on the maximum Doppler Shift, which is when Cos\(\theta\) = 1 (\(\theta\) is the angle between the direction of motion of the mobile and direction of arrival of scattered waves). Figure 12a,b depicts Doppler Shift with respect to TX1 of three different cars PA, PB and PC (see Figure 10a) which are traveling at constant velocities (v) of 40 km/h and 60 km/h respectively.  
The relative motion between cars and TX, results in random frequency modulation due to different Doppler Shifts on each of the multipath components. If the Doppler Shift is given by [34]:  
\[\rm{fd}=\ \frac{1}{2\pi}\cdot\frac{\Delta\omega}{\Delta t}=\frac{v}{\lambda} \cdot Cos\theta \tag{5}\]  
the theoretical fd\({}_{\rm max}\) (40 km/h) = 218.52 Hz and f\({}_{\rm{dmax}}\) (60 km/h) = 327.78 Hz, when Cos\(\theta\) = 1.  
When PA is traveling at a constant velocity of 40 km/h, the simulated fd\({}_{\rm max}\) were reported between \(-\)207,821 and 213.33 Hz, while for 60 km/h the simulated fd\({}_{\rm max}\) were reported between \(-\)311.72 and 319.99 Hz. The reported fd positive values is an indicator that PA is moving directly toward the TX and the apparent received frequency is defined by: f = (fc + fd) Hz, meanwhile negative fd values means that PA is moving directly away the TX and the apparent received frequency is f = (fc - fd) Hz. According to this definition and simulation results, the received signal spectrum will have components in the range f = (5.9 \(\times\) 10\({}^{9}\) - 311.72 to 5.9 \(\times\) 10\({}^{9}\) + 319.99) Hz. Additional simulation shows that the signal spectrum components for PB and PC (traveling at constant velocity of 60 km/h) are in the range of f\({}_{\rm PB}\) = (5.9 \(\times\) 10\({}^{9}\) - 297.05 to 5.9 \(\times\) 10\({}^{9}\) + 299.42) Hz and f\({}_{\rm PC}\) = (5.9 \(\times\) 10\({}^{9}\) - 148.80 to 5.9 \(\times\) 10\({}^{9}\) + 192.65) Hz respectively.  
Figure 13a,b show the TX1-B\({}_{\rm D}\) for velocities of 40 km/h and 60 km/h respectively. Higher B\({}_{\rm D}\) values are observed in the vicinity of the TX1 and higher B\({}_{\rm D}\) values are depicted for velocity of 60 km/h. Figure 13a clearly illustrates the positive and negative range of received frequencies when the PA is traveling at 40 km/h. Factors such as the relative velocity and direction of motion respect to TX, have an impact on the apparent received frequency which impact B\({}_{\rm D}\), T\({}_{\rm C}\) and fd values, as aforementioned.  
Figure 12: TX1-Doppler Shift (fd): (**a**) car velocity of 40 km/h, (**b**) car velocity of 60 km/h.