# Radio Channel Scattering in a 28 GHz Small Cell at a Bus Stop: Characterization and Modelling
## 3 Results
### Doppler Spectrum  
To analyze the time-varying characteristics of the radio channel, we have calculated the Doppler spectrum corresponding to the time variation of the strongest components present in the complex impulse response. First, we identified the delay, \(\tau_{i}\), of the main contributions in the PDPs. Then, we took each of the 0.1058 s time records of the time-varying impulse response corresponding to those delays with h(t, \(\tau_{i}\)). We divided each record into 11 sections of 2048 samples, lasting 9.33 ms each, and calculated the correlation function, and the associated Doppler spectrum, for each section. In Figure 12, we plot the results for the eleven sections of the main component, the one with a relative delay of \(\tau_{i}=0\) ns, coming from the same record of a car.  
As can be seen, the DS shapes resemble those in the literature for F2F links in the presence of moving scatterers. However, the Doppler spectra change with time. When the car is far away from the sounder (solid thick blue line), the DS is quite narrow. This is clearly due to the filtering effect of the directive antenna pattern already described in [36; 38]. However, as the car approaches the sounder, the Doppler spectrum is broadened, reaching a maximum broadening as shown in the dashed, thick red line. As there are significant DS changes with time, the channel cannot be considered stationary. This is reasonable because the incidence and observation angles change as the vehicle moves, and this will change the measured field scattered by the vehicle. A non-stationary model approximation would be needed. Any model would require a spread factor that varies with time as the vehicle moves.  
In the literature, there are several DS models of F2F links in presence of moving scatterers. Closed-form expressions are given in [32] for an environment where the transmitter and receiver are in a fixed position and surrounded by a large number of moving scatterers. The emitted wave is supposed  
Figure 12: Time-varying Doppler spectrum for the main component (\(\tau_{i}=0\) ns). Each drawn line corresponds to a different time interval.  
to reach the receiver after a single bounce on the moving scatterer. Isotropic scattering is also assumed. Under these hypotheses, it is shown that for a constant velocity, \(v\), of the scatterers, the DS will be  
\[DS(f)=\frac{\sigma^{2}}{\pi^{2}f_{d}}\;K\left(\sqrt{1-\left(\frac{f}{2\;f_{d}} \right)^{2}}\right) \tag{2}\]  
where \(\sigma^{2}\) is the average received power, \(f_{d}\) is the maximum Doppler frequency, being \(f_{d}=v/\lambda\), \(\lambda\) is the wavelength and \(K\) denotes the complete elliptic integral of the first kind. However, if \(v\) is a random variable with exponential distribution and mean value \(v_{m}\), then  
\[DS(f)=\frac{\sigma^{2}}{\pi^{2}f_{m}}\;K_{0}\left(\frac{|f|}{2\;f_{m}}\right) \tag{3}\]  
where \(f_{m}=v_{m}/\lambda\) and \(K_{0}\) represents the zeroth-order modified Bessel function of the second kind.  
The case of an F2F link with a single moving scatterer is presented in [34]. If the scatterer has a constant scattering cross-section and is moving in one direction, then  
\[DS(f)=\frac{2}{\sqrt{1-\left(\cos(\beta)+\frac{f}{f_{d}}\right)^{2}}} \tag{4}\]  
where \(\beta\) is related to the direction of movement. If, however, it is moving in many directions, then  
\[DS(f)=\frac{1}{\pi^{2}}\;K\left(\sqrt{1-\left(\frac{f  
A complete scattering model for this channel should account for clustering of the contributions with different delays, as well as the Doppler spread for each delay component. The Doppler spread model for each component could be the same and even have the same parameter values, but should account for the non-stationary nature of the channel by varying the \(v_{m}\) value in (7).