# An Interference Cancellation Scheme for High Reliability Based on MIMO Systems
## 2 Conventional Two-Path Relay System Model  
The conventional two-path relay system is shown in Figure 1. S denotes a source, R\({}_{1}\) and R\({}_{2}\) denotes half-duplex relays and D denotes a destination. \(h_{\text{SR}_{1}}\) and \(h_{\text{SR}_{2}}\) are channel coefficients from S to R\({}_{1}\) and S to R\({}_{2}\), respectively. \(h_{\text{R}_{1}\text{D}}\) and \(h_{\text{R}_{2}\text{D}}\) are channel coefficients from R\({}_{1}\) to D and R\({}_{2}\) to D, respectively. \(h_{\text{R}_{1}\text{R}_{2}}\) and \(h_{\text{R}_{2}\text{R}_{1}}\) are inter-relay channel coefficients from R\({}_{1}\) to R\({}_{2}\) and R\({}_{2}\) to R\({}_{1}\), respectively. The relays forward received signals from S by using the AF scheme. In an odd time slot, S transmits symbol to R\({}_{1}\) and the other relay R\({}_{2}\) relays the symbol which is received from S at previous time slot. Likewise, in an even time slot, S transmits symbol to R\({}_{2}\) and the other relay R\({}_{1}\) relays the symbol which is received from S. In the two-path relay system, since S and one of the relay nodes transmit symbol simultaneously in a specific time slot, IRI occur at relay nodes. Table 1 shows configuration for transmission symbols of conventional scheme. According to transmit symbol for each time slot in Table 1, the received symbol at R\({}_{1}\) for time slot \(n\) is as follows,  
\[y_{\text{R}_{1}}(n)=h_{\text{SR}_{1}}x_{\text{S}}(n)+h_{\text{R}_{1}\text{R}_{ 2}}x_{\text{R}_{2}}(n)+z_{\text{R}_{1}}(n), \tag{1}\]  
where \(x_{\text{S}}\) is transmit symbol from S, \(x_{\text{R}_{2}}\) is transmit symbol from R\({}_{2}\), \(z_{\text{R}_{1}}\) is zero-mean additive white Gaussian noise (AWGN) at R\({}_{1}\) and \(n\) is assumed to be odd number. In Equation (1), the second term of the right-hand side is interference. In the next time slot, \(\rm R_{1}\) relays the received symbol to \(\rm D\) using the AF and the received symbol at \(\rm D\) is as follows,  
\[ \begin{array}{ll}y_{\rm D}(n+1)&=\beta_{1}y_{\rm R_{1}}(n)h_{\rm R_{1}D}+z_{\rm D }(n+1),\\ &=\beta_{1}\big{(}h_{\rm SR_{1}}x_{\rm s}(n)+h_{\rm R_{1}R_{2}}x_{\rm R_{2}}(n) \big{)}+\widetilde{z}_{\rm D}(n+1),\end{array}  \tag{2}\]  
where \(z_{\rm D}\) is zero-mean AWGN at \(\rm D\), \(\widetilde{z}_{\rm D}=z_{D}+\beta_{1}z_{R_{1}}\) is new zero-mean AWGN and \(\beta_{1}\) is an amplifying factor as follows,  
\[\beta_{1}=\sqrt{\frac{P_{\rm R_{1}}}{\big{|}h_{\rm SR_{1}}\big{|}^{2}P_{\rm S} +\big{|}h_{\rm R_{1}R_{2}}\big{|}^{2}P_{\rm R_{2}}+\sigma^{2}}}, \tag{3}\]  
where \(P_{\rm S}\), \(P_{\rm R_{1}}\) and \(P_{\rm R_{2}}\) are transmit power at \(\rm S\), \(\rm R_{1}\) and \(\rm R_{2}\), respectively, and \(\sigma^{2}\) is noise power. Substituting Equation (1) into (2), the received symbol at \(\rm D\) is as follows,  
\[y_{\rm D}(n+1)=\beta_{1}x_{\rm S}(n)h_{\rm SR_{1}}h_{\rm R_{1}D}+\beta_{1}x_{ \rm R_{2}}(n)h_{\rm R_{1}R_{2}}h_{\rm R_{1}D}+\beta_{1}h_{\rm R_{1}D}z_{\rm R_ {1}}(n)+z_{\rm D}(n+1). \tag{4}\]  
Because \(\rm R_{1}\) relays the symbol to \(\rm D\) at time slot \(n\), \(y_{\rm D}(n)\) is given by,  
\[y_{\rm D}(n)=x_{\rm R_{2}}(n)h_{\rm R_{2}D}+z_{\rm D}(n), \tag{5}\]  
or Equation (5) is written as follows, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Time Slot** & \(\rm S\) & \(\rm R_{1}\) & \(\rm R_{2}\) \\ \hline \multirow{2}{*}{\(t=1\)} & \(x_{\rm S}(1)\) & - & - \\  & (\(\rm S\to R_{1}\)) & & \\ \hline \multirow{2}{*}{\(t=2\)} & \(x_{\rm S}(2)\) & \(\beta_{1}y_{\rm R_{1}}(1)\) & - \\  & (\(\rm S\to R_{2}\)) & (\(\rm R_{1}\to D\), \(\rm R_{1}\to R_{2}\)) & \\ \hline \multirow{2}{*}{\(t=3\)} & \(x_{\rm S}(3)\) & - & \(\beta_{2}y_{\rm R_{2}}(2)\) \\  & (\(\rm S\to R_{1}\)) & - & (\(\rm R_{2}\to D\), \(\rm R_{2}\to R_{1}\)) \\ \hline \multirow{2}{*}{\(\vdots\)} & \(\vdots\) & \(\vdots\) & \(\vdots\) \\ \hline \multirow{2}{*}{\(t=n\) (Odd \(n\))} & \(x_{\rm S}(n)\) & - & \(\beta_{n-1}y_{\rm R_{2}}(n-1)\) \\  & (\(\rm S\to R_{1}\)) & - & (\(\rm R_{2}\to D\), \(\rm R_{2}\to R_{1}\)) \\ \hline \end{tabular}
\end{table}