## 3 Channel Statistics
### Characterization of SOFS for Outdoor and Indoor Radio Propagation Channels  
The analysis is further extended to study the SOFS of mmWave band channels at 28 GHz and 38 GHz. The impact of change in the maximum Doppler shift, received signal's envelope's threshold level and wavelength on the LCR, AFD, spatial auto-covariance, and CD are studied in this section. The Nakagami model is used to approximate the PDF of the power of a Rician fading signal as well as Rayleigh fading signal. For the scenario, where the dominant LoS component is present, the Nakagami shape parameter m is taken as half of ratio of LoS power and NLoS power. In case where dominant component is absent, Rayleigh fading occurs, shape parameter m is taken as one. For Case 1, when the measurement setup is outdoor urban environment and measurements are taken at 38 GHz, the normalized LCR and AFD are plotted in Figure 4a,b, respectively for different values of maximum Doppler frequency. Impact of changing the separation distance between the Tx and Rx on SOFS is studied. It is observed that LCR is increasing with increase in the Doppler frequency, but the change of separation distance has minor impact on LCR. For the AFD case at 38 GHz, it has converse behavior with respect to LCR. AFD is decreasing with increase in maximum Doppler frequency. Moreover, change in separation distance of Tx and Rx has clear impact on AFD which is observed as increasing while distance is kept higher than 250 m. Similarly, for the same Case 1, when the measurement setup is outdoor urban environment and measurements are taken at 38 GHz, the spatial auto-covariance and CD are plotted in Figure 5a,b, respectively for different values of wavelength and distance. Impact of changing the separation distance between the Tx and Rx on these SOFS is studied. It is observed that auto-covariance is decreasing from 1 to 0 when \(\frac{r}{\lambda}\) is increased from 0 to 0.5 and then it remains constant at 0 for further increase. However, the change of separation distance has minor impact on auto-covariance. For the CD case at 38 GHz, it has converse behavior with respect to auto-covariance. CD increases linearly with increase in \(\lambda\). Moreover, change in separation distance of Tx and Rx has clear impact on CD which is observed as increasing while distance is changed from 150 m to 250 m.  
Figure 4: Level-crossing-rate (LCR) and average fade duration (AFD) plotted against different values of maximum Doppler shift in (**a**,**b**), respectively; for different scenarios of outdoor mmWave (case 1; 38 GHz) radio propagation environments. (\(m=1\), \(\rho=2\)). Figure 5: Spatial auto-covariance and coherence distance (CD) for different scenarios of outdoor mmWave (case 1; 38 GHz) radio propagation environments in (**a**,**b**), respectively. (\(m=1\), \(f_{d}=1\) Hz). For Case 2 (a), when the measurement setup is indoor corridor environment and measurements are taken at 28 GHz, the normalized LCR and AFD are plotted in Figure 6a,b, respectively for different values of maximum Doppler frequency. Impact of changing the separation distance between the Tx and Rx on SOFS is studied. It is observed that LCR is increasing while we increase the Doppler frequency as well as the change of separation distance has significant impact on LCR. When the distance between Tx and Rx inside corridor is changed from 3 m to 6 m, LCR is observed to increase. For the indoor case at 28 GHz, AFD has converse behavior with respect to LCR. AFD is decreasing with increase in maximum Doppler frequency. Moreover, change in separation distance of Tx and Rx has clear impact on AFD which is observed as decreasing while distance between Tx and Rx inside corridor is changed from 3 m to 6 m. Similarly, for the same Case 2 (a), when the measurement setup is indoor corridor environment and measurements are taken at 28 GHz, the spatial auto-covariance and CD are plotted in Figure 7a,b, respectively for different values of wavelength and distance.