## 3 Channel Statistics
### Definition of SOFS Quantifiers  
The considered SOFS quantifiers are LCR, AFD, CD, and spatial auto-covariance. Mathematical relationship of SOFS quantifiers with multipath SFs is provided in [10]. The LCR, \(N_{R}\), measures the count of a signal envelope's crossings for a given threshold level \(R\) within a unit time interval; which can be represented for Nakagami-m distributed envelope as given below,  
\[N_{R}=\int_{0}^{\infty}\hat{r}f_{R}(R,\hat{r})d\hat{r}. \tag{6}\]  
AFD (\(\tilde{\tau}\)) represents the average time duration for which the signal envelope remains below the threshold level \(R\). AFD for Nakagami-m distributed envelope can be defined as below,  
\[\tilde{\tau}=\frac{1}{N_{R}}\int_{0}^{R}f_{R}(r)d\tau. \tag{7}\]  
In various scenarios of propagation environments, different distributions are adopted for modeling envelope \(R\) of the received signal. For LoS scenario, when a strong dominant component is present, the envelope follows Rician distribution. However, when there is no single dominant component present, the envelope follows Rayleigh distribution. Due to its suitability and flexibility, to model the fading channels, Nakagami-\(m\) distribution is widely used [28]. It has wide range ofapplicability and other mostly used distributions such as Rayleigh and Rician can easily be deduced as a special case. The PDF of Nakagami-_m_ is given as,  
\[f_{R}(r)=\frac{2m^{m_{T}(2m-1)}}{\Gamma(m)P_{\circ}^{m}}\exp\left(\frac{-m\tau^{ 2}}{P_{\circ}}\right), \tag{8}\]  
where \(m\) is the shape parameter which is greater than 0.5 and \(\Gamma(. )\) is the standard Gamma function. An expression for LCR and AFD can be derived for Nakagami-_m_ distribution by substituting (8) in (6) and (7), respectively. The derived equations for LCR is expressed as,  
\[N_{R}=\frac{\sqrt{2\ \pi}\ \Lambda_{\theta}m^{(m-1/2)}\rho^{(2m-1)}\text{ exp}(-m\rho^{2})\sqrt{1+\gamma_{\theta}\ \cos(2\ (\theta_{v}-\theta_{\text{MF}}))}}{\lambda\Gamma(m)}, \tag{9}\]  
where \(\lambda=c/f_{d}\), \(f_{d}=(v/c)\cos\theta_{v}\) is the maximum Doppler frequency, \(v\) is the velocity of receiver, \(\theta_{v}\) is the direction of receiver motion, \(c\) is the speed of light, and \(\rho=R^{2}/P_{\circ}\) is the normalized threshold level of observation. AFD is the measure of the duration a signal spends below a threshold level and relation of AFD for Nakagami-_m_ distribution is shown below,  
\[\tau=\frac{\lambda\hat{\gamma}(m,m\rho)\text{exp}(m\rho^{2})}{\sqrt{2\ \pi}\ \Lambda_{\theta}\sqrt{1+\gamma_{\theta}\ \cos(2\ (\theta_{v}-\theta_{\text{MF}}))}m^{(m-1/2)}\rho^{(2m-1)}}, \tag{10}\]  
where \(\hat{\gamma}(.,. )\) is the incomplete Gamma function. Another useful measure for characterization of SOFS is spatial auto-covariance, which determines the correlation between the signals received at two different spatial positions. The relationship of spatial auto-covariance with multipath SFs, for Nakagami-_m_ distribution is given in [29], is shown as follows,  
\[p(r,\theta_{v})\approx\exp\left[-\frac{2\ \pi^{2}\ m\Gamma^{2}(m)\ \Lambda_{\theta}^{2}\big{(}1+\gamma_{\theta}\ \cos(2\ (\theta_{v}-\theta_{\text{MF}}))\big{)}}{\left(m\Gamma^{2}(m)-\Gamma^{2} \left(m+\frac{1}{2}\right)\right)}\left(\frac{r}{\lambda}\right)^{2}\right]. \tag{11}\]  
It calculates the correlation between signals received at two different positions separated by a distance \(r\). The separation in space for which the channel response remains unchanged (i.e., more than 50% correlated) is termed as CD, denoted by \(D_{c}\). Studying CD is of high significance in terms of spatial diversity and receiver's design.