# Best Relay Selection Strategy in Cooperative Spectrum Sharing Framework with Mobile-Based End User
## 2 System Model  
The cognitive underlay cooperative system considered in this work includes one stationary PU \(P\) and one SU. Specifically, the SU consists of one stationary source \(S\), which acts as a cognitive base station; \(N\) stationary AF relays \(R_{k}\) (\(k=1,2,\ldots,N\)); and one mobile destination \(D\), which acts as an end user that may move at high vehicular speeds. The system model is illustrated in Figure 1. The SU \(S\) and SU \(D\) have no direct link due to deep fading. The scenario here is similar to a downlink scenario operating in a half-duplex scheme where the base station needs to transmit the data to its users. In addition, the time division multiple access (TDMA) technique is exploited, as well as a transmission scheme in both hops. In particular, two hops are deployed in this model where the SU \(S\) broadcasts the data to \(N\) SU relays with interference power constraints \(I\) at the PU receiver. After that,the best selected relay amplifies this data using a variable gain \(G_{R_{k}}\), and then resends it toward the SU \(D\). \(G_{R_{k}}^{2}\) can be written as follows:  
\[G_{R_{k}}^{2}=\frac{P_{R_{k}}}{d_{R_{k}D}{}^{-\beta}\Big{(}P_{s}d_{SR_{k}}{}^{- \beta}|h_{SR_{k}}|^{2}+\sigma^{2}\Big{)}}=\frac{\bar{I}d_{2}|h_{SP}|^{2}}{|h_{R_ {k}P}|^{2}\big{(}\bar{I}d_{1}|h_{SR_{k}}|^{2}+|h_{SP}|^{2}\big{)}} \tag{1}\]  
where \(P_{s}=Id_{SP}{}^{\beta}/|h_{SP}|^{2}\) is the power of the transmitted signal at \(S\), \(P_{R_{k}}=Id_{R_{k}P}{}^{\beta}/|h_{R_{k}P}|^{2}\) is the transmitted signal power at \(R_{k}\), \(d_{1}=(d_{SP}/d_{SR_{k}}){}^{\beta}\), \(d_{2}=(d_{R_{k}P}/d_{R_{k}D}){}^{\beta}\), \(d_{ij}\) is the distance between node \(i\) and \(j\), \(h_{ij}\) represents the channel coefficient of the link \(i\)\(\rightarrow\)\(j\) where \(i\in\{S,R_{k}\}\) and \(j\in\{R_{k},P,D\}\), \(\beta\in[2,6]\) is the exponent of the path loss, and \(\sigma^{2}\) is the variance of the additive white Gaussian noise (AWGN) term at \(R_{k}\). Assuming the power \(P_{s}\) is used to transmit the signal \(x\) from \(S\) to \(R_{k}\), then the received signal \(y_{R_{k}}\) at the SU \(R_{k}\) can be represented as:  
\[y_{R_{k}}=\sqrt{P_{s}d_{SR_{k}}{}^{-\beta}}h_{SR_{k}}x+n_{1k} \tag{2}\]  
where \(n_{1k}\) represents the zero mean AWGN with variance \(\sigma^{2}\) at the \(k\)-th link between \(S\) and \(R_{k}\). The signal is amplified then by \(R_{k}\) with a suitable gain. Thereafter, it is forwarded to \(D\) with a power \(P_{R_{k}}\). Therefore, the received power \(y_{D}\) at the SU \(D\) can be written as:  
\[y_{D}=G_{R_{k}}h_{R_{k}D}\sqrt{d_{R_{k}D}{}^{-\beta}}y_{R_{k}}+n_{2k} \tag{3}\]  
where \(n_{2k}\) represents the zero mean AWGN with variance \(\sigma^{2}\) at the \(k\)-th link between \(R_{k}\) and \(D\). However, the links between the stationary nodes (SU \(S\), SU \(R_{k}\), and PU \(P\)) are assumed to experience Rayleigh distribution. Specifically, the links \(S\)\(\rightarrow\)\(R_{k}\), \(S\)\(\rightarrow\)\(P\), and \(R_{k}\)\(\rightarrow\)\(P\) are modelled by Rayleigh distribution. Therefore, \(\{|h_{SR_{k}}|^{2},|h_{SP}|^{2},|h_{R_{k}P}|^{2}\}\) have exponential distribution, with mean values \(1/\lambda_{SR_{k}},1/\lambda_{SP}\), and \(1/\lambda_{R_{k}P}\), respectively. Therefore, the cumulative distribution function (CDF) of these links can be expressed as follows. \[F_{X_{ij}}(x)=1-e^{-\lambda_{ij}x} \tag{4}\]  
Figure 1: Cognitive relay network with mobile end user. In addition, the probability density function (PDF) of these links can be expressed as follows:  
\[f_{X_{ij}}(x)=\lambda_{ij}e^{-\lambda_{ij}x} \tag{5}\]  
where \(X_{ij}\in\{|h_{SR_{k}}|^{2},|h_{SP}|^{2},|h_{R_{k}P}|^{2}\}\) and \(\lambda_{ij}\in\{\lambda_{SR_{k}},\lambda_{SP},\lambda_{R_{k}P}\}\). As mentioned previously, the SU \(D\) is considered to be mobile and, thus, all links connected to this node are assumed to be i.n.i.d. Nakagami-\(m\) channels with fading parameter \(m\leq 1\)[61; 62].