# Best Relay Selection Strategy in Cooperative Spectrum Sharing Framework with Mobile-Based End User
## Appendix A Derivation of \(F_{W_{ij}}(w)\)  
In general, at high vehicular speeds, the complex fading can be represented by a wide-sense stationary (WSS) complex random process (e.g., \(w(t)=u(t)v(t)\)), where \(u(t)\) and \(v(t)\) are independent random processes). Let the complex random process \(w(t)=r_{w}e^{\theta(t)}\), which represents the complex Nakgami-\(m\) fading process, be a WSS where \(r_{w}\) is the envelope of the Nakagami-\(m\) distribution with a second moment \(\Omega=E[{r_{w}}^{2}]\) and \(\theta(t)\) is uniformly distributed over \([0,2\pi]\). In addition, the PDF given below in Equation (A1) can be exploited [68]:  
\[f_{r_{w}}(w)=\frac{2m^{m}w^{2m-1}}{\Gamma(m)\Omega^{m}}e^{-\frac{w_{\Omega}^{ 2}m}{\Omega}}r_{w}>0,0.5\leq m<1\] (A1)  
Furthermore, let \(r_{u}\) be a random variable with Rayleigh distribution where \(E[{r_{u}}^{2}]=\Omega/\Omega\). If \(r_{w}\) and \(r_{u}\) satisfy the relationship shown in Equation (A2) below, then \(r_{v}\) has a standard Beta distribution with parameters \(m\) and \(m-1\)[61]:  
\[r_{w}=r_{u}\sqrt{r_{v}}\] (A2)  
where \(r_{v}\) is a non-negative random variable and independent of \(r_{u}\). It is worth noting that the random variable \(\theta(t)\) is independent of \(r_{w}\), \(r_{u}\), and \(r_{v}\). Thus, multiplying both sides of Equation (A2) by \(e^{\theta(t)}\) yields \(r_{w}e^{\theta(t)}=r_{u}\sqrt{r_{v}}e^{\theta(t)}\). Therefore, Equation (A2) can be rewritten as a product of two random processes as follows:  
\[w(t)=u(t)v(t)\] (A3)  
where \(u(t)\) is the zero mean complex Gaussian random process and the random process \(v(t)=\sqrt{r_{v}}\) is exponentially correlated. Hence, the CDF of the random variable \(W\) can be written as follows:  
\[F_{W}(w)=\int_{0}^{\infty}F_{U}\Big{(}\frac{w}{v}\Big{)}f_{V}(v)dy,0\leq v\leq 1\] (A4)  
On the other hand, the PDF of \({r_{v}}^{2}\) follows the standard Beta distribution [62], and it can be expressed as follows:  
\[f_{V}(v)=\frac{v^{m-1}(1-v)^{-m}}{\beta(m,1-m)},0\leq v\leq 1\] (A5)where \({r_{u}}^{2}\) has a Rayleigh distribution with PDF and CDF that can be written, respectively, as follows [69]:  
\[f_{U}(u)=\frac{1}{\tilde{\gamma}}e^{-\frac{u}{\tilde{\gamma}}},\tilde{\gamma}>0 \tag{10}\]  
and  
\[F_{U}(u)=1-e^{-\frac{u}{\tilde{\gamma}}} \tag{11}\]  
where \(\tilde{\gamma}\) is the average SNR, which is given as [69; 70]:  
\[\tilde{\gamma}=\frac{log_{2}M}{1-\frac{1}{N^{2}}\Big{[}N+2\sum_{i=1}^{N-1}(N-i )J_{0}(2\pi f_{c}(\frac{s}{c})T_{s}i)\Big{]}}+\frac{NT_{s}}{log_{2}M}\frac{1}{ \frac{E_{b}}{N_{0}}} \tag{12}\]  
where \(N\) is the number of the sub-carriers in the OFDM system with mobile nodes moving with one speed equals \(s\), \(c\) is the speed of light, \(f_{c}\) is the carrier frequency, \(T_{s}\) is the symbol time in M-ary quadrature amplitude modulation (QAM) modulation scheme, \(\frac{E_{b}}{N_{0}}\) is the SNR per bit, and \(J_{0}\) is the first-kind Bessel function. Substituting the PDF of \({r_{v}}^{2}\) (i.e., Equation (10)) in Equation (10) yields:  
\[F_{W}(w)=\int_{0}^{1}\Big{(}1-e^{-\frac{w}{\tilde{\gamma}\theta}}\Big{)}\left( \frac{v^{m-1}(1-v)^{-m}}{\beta(m,1-m)}\right)dv \tag{13}\]  
Interestingly, Equations (3.194.1) and (3.471.2) [63] can be used to obtain the CDF for Nakagami-\(m\) fading channels of the considered system model when \(m\leq 1\), as shown in Equation (6).