## 3 An Overview of Existing Works
### Correlation between the Actual and the Corresponding Estimate Channels  
The considered system model is referred to as Model A in Table 2 and illustrated in Figure 2a, in which the direct link between the source and the destination is assumed to be unavailable. The end-to-end SNR at the destination with the support of the relay \(R_{k}\) is given in Equation (6). In [17; 18; 19], the PRS and the BRS, and the \(l^{th}\)-PRS and the \(l^{th}\)-BRS schemes are considered, respectively. More specifically, a single relay is selected based on outdated channels, which is  
\[b=\operatorname*{arg\,max}_{k=1,\cdots,K}\left\{\min\left\{\hat{\gamma}_{SR_{k }},\hat{\gamma}_{R_{k}D}\right\}\right\}, \tag{10}\]  
for the BRS (\(l^{th}\)-BRS) scheme. In addition,  
\[p=\operatorname*{arg\,max}_{k=1,\ldots,K}\left[\hat{\gamma}_{SR_{k}}\right] \tag{11}\]  
for PRS (\(l^{th}\)-PRS) scheme, where \(\hat{\gamma}_{ij}=\frac{p}{N_{0}}\left|\hat{h}_{ij}\right|^{2}\) denotes the (outdated) estimate of \(\gamma_{ij}\). We should note that the estimate end-to-end SNR at the destination with the support of the selected relay is different from what the destination is supposed to receive. The authors then derive the system performances, i.e., outage probability and symbol error probability, under Rayleigh fading and show that, when CSI is outdated, the system DO is reduced to one. They also show that a 3% reduction in correlation causes up to an order of magnitude increase in the system outage probability. They then suggest that a high feedback rate may be required in practice in order to attain the full benefits of the BRS scheme. Moreover, it is shown that, depending on the correlation value, the PRS scheme may outperform the BRS scheme and vice versa. However, it is worth pointing out that these works are limited to the case of equally correlation coefficients, i.e., \(\left\{\hat{\rho}_{SR_{k}}\right\}_{k=1}^{K}=\hat{\rho}_{1}\) and \(\left\{\hat{\rho}_{R_{k}D}\right\}_{k=1}^{K}=\hat{\rho}_{2}\). The systems with different correlation coefficients have not been investigated yet. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Model** & **Reference** & \(\left\{\hat{\rho}_{SR_{k}}\right\}_{k=1}^{K}\) & \(\left\{\hat{\rho}_{R_{k}D}\right\}_{k=1}^{K}\) & \(\left\{\hat{\rho}_{SR_{k}}^{SR_{k}}\right\}_{k=1}^{K}\) & \(\left\{\hat{\rho}_{R_{k}D}\right\}_{k=1}^{K}\) & \(\left\{\hat{\rho}_{R_{k}}^{R_{k}}\right\}_{k=1}^{K}\) & \(\left\{\hat{\rho}_{SD}^{SR_{k}}\right\}_{k=1}^{K}\) & \(\left\{\hat{\rho}_{SD}^{SR_{k}D}\right\}_{k=1}^{K}\) \\ \hline A & [17; 18; 19] & \(\hat{\rho}_{1}\) & \(\hat{\rho}_{2}\) & 0 & 0 & 0 & 0 & 0 \\ B & [20] & \(1\) & \(1\) & 0 & 0 & \(\rho_{R}\) & \(\rho_{SD}^{SR}\) & \(\rho_{SD}^{RD}\) \\ C & [21] & \(1\) & \(1\) & \(\rho_{SR}\) & \(\rho_{RD}\) & 0 & 0 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: A summary of channel correlation forms which have been investigated. Correlations between Source-Relay and Relay-Destination Channels, between Source-Destination and Source-Relay Channels, and between Source-Destination and Relay-Destination Channels  
The investigated system is illustrated in Figure 2b and Model B in Table 2, where only a single relay case is considered. The direct source-destination link is also taken into account. The total end-to-end SNR at the destination is given in Equation (5) by dropping the index \(k\). The authors of [20] consider the impact of each channel correlation on the system performance separately. The moment generating function of \(\gamma_{D}^{T}\), \(M_{\gamma_{D}^{T}}(s)\) is first derived in closed-form under conditions of Nakagami-\(m\) fading. Consequently, the bit error rate of the differential binary phase shift keying is obtained as \(M_{\gamma_{D}^{T}}(1)\). They show that all three of the correlation forms degrade the system performance. Moreover, they also point out that the effect of correlation is often noticeable when the correlation coefficient is larger than 0.5. However, this work is limited for single relay systems only.