# Adaptive Relay Selection Scheme by Using Compound Channel
## 4 The Proposed Scheme
### Relay Selection with Outage  
\(S\) receives feedback for the CSI before calculating the outage probability. Then, \(S\) calculates the outage probability of \(\mathbf{H_{k}}\). The outage probability is calculated as follows,  
\[O_{k}=\Pr[C_{\mathbf{H_{k}}}<B_{\mathbf{x}}], \tag{7}\]  
where \(C_{\mathbf{H_{k}}}\) is channel capacity of \(\mathbf{H_{k}}\). \(B_{\mathbf{x}}\) denotes the total number of bits in \(\mathbf{x}\). If the transmitted data exceeds the channel capacity, outage occurs. If \(C_{\mathbf{H_{k}}}\) is greater than \(B_{\mathbf{x}}\), outage is not occurred. Relay satisfying (6) is defined as \(R_{q}\). In other words, \(R_{q}\) is a set of qualified relays with non-outage. When \(N_{p}\) is the number of relays using on the system, \(N_{p}\) is compared with the number of \(R_{q}\). In this paper explain that the case where \(N_{p}\) is 2. The proposed scheme classifies three cases according to \(Q\). \(Q\) is the number of \(R_{q}\) and \(R_{q}\) is a relay that outage does not occur among \(K\) relays. The relay is selected differently in each case. The case 1 considers \(Q<2\). If the \(Q=0\), throughput of all links through relays cannot be guaranteed due to the outage and \(Q=1\), only one relay can guarantee the reliability of data. Therefore, regardless of multiple relays, the relay that link between \(S\) and \(R_{k}\) has the best quality is selected. The number of the combination for case 1 is one. If \(Q\) is two, the case is defined as case 2. In case 2, all \(R_{q}\) are selected. Therefore, the number of the combination for case 2 is two. Figure 2: Flow chart of the proposed scheme. where \(i\) and \(j\) denote each relay index of all \(R_{q}\) unlike fixed \(i\) and \(j\) in case 2. The Frobenius norm for \(\bar{\mathbf{H}}_{c3}\) is as follows,  
\[\Gamma\left(\mathbf{P}_{w},\bar{\mathbf{H}}_{i,u},\bar{\mathbf{H}}_{j,v}\right) =\text{Tr}\left\{\left(\bar{\mathbf{H}}_{c3}\right)\left(\bar{\mathbf{H}}_{c3} \right)^{H}\right\}. \tag{17}\]  
Case 2 and 3 calculate each \(\Gamma\) based on (15) or (17). The selected antennas of case 2 and 3 are as follows,  
\[\mathbf{P}_{w}^{opt},\bar{\mathbf{H}}_{i,u}^{opt},\bar{\mathbf{H}}_{j,v}^{opt }=\underset{\mathbf{P}_{w},\bar{\mathbf{H}}_{i,u},\bar{\mathbf{H}}_{j,v}}{ \arg\max}\;\Gamma\left(\mathbf{P}_{w},\bar{\mathbf{H}}_{i,u},\bar{\mathbf{H}} _{j,v}\right), \tag{18}\]  
where \(\mathbf{P}_{w}^{opt}\) is optimal permutation matrix for \(\bar{\mathbf{H}}_{i,u}^{opt}\) and \(\bar{\mathbf{H}}_{j,v}^{opt}\). \(\bar{\mathbf{H}}_{i,u}^{opt}\) and \(\bar{\mathbf{H}}_{j,v}^{opt}\) are optimal antenna combination matrix of each \(R_{q}\). Both cases select \(\bar{\mathbf{H}}_{i,u}^{opt}\) and \(\bar{\mathbf{H}}_{j,v}^{opt}\) with maximum \(\Gamma\). In cases 2 and 3, the compound channel is as follows,  
\[\bar{\mathbf{H}}_{c}^{opt}=\bar{\mathbf{H}}_{i,u}^{opt}+\mathbf{P}_{w}^{opt} \bar{\mathbf{H}}_{j,v}^{opt}. \tag{19}\]  
The total combination number of case 2 is as follows,  
\[T_{2}=T_{1}^{2}N_{m}!. \tag{20}\]  
Furthermore, the total number of the combination for case 3 is as follows,  
\[T_{3}=T_{2}N_{c}. \tag{21}\]  
The proposed scheme is to select the optimal relay considering the number of relays with an outage occurrence. Furthermore, the proposed scheme considers the compound channel for selection of relay and antenna. The proposed scheme achieves performance enhancement through the optimal compound channel and diversity from each case. Equations (13), (20) and (21) show the diversity order of each case. In the proposed scheme, the relay can be selected adaptively according to the system. The proposed scheme transmits same \(\mathbf{x}\) to \(D_{u}\). Multi-user detection scheme requires more channel feedback information than single-user detection scheme. The proposed scheme requires only a calculated Frobenius norm of channel quality indicator (CQI). However, the precoding scheme requires a CQI, a precoding matrix indicator (PMI) and rank indicator (RI). The system using the proposed scheme can be utilized in two cases.