# An Interference Cancellation Scheme for High Reliability Based on MIMO Systems
## 3 Proposed Scheme  
The proposed two-path relay system is shown in Figure 2. All nodes have two antennas for simple representation. Also, DF method is used at R\({}_{1}\) and R\({}_{2}\) to relay the received symbol from S. In the proposed scheme, S transmits symbols by using CDD and R\({}_{1}\) and R\({}_{2}\) also use CDD after decoding and re-encoding the received symbols from S for obtaining spatial diversity gain. The CDD is spatial diversity scheme which is easily used in MIMO-OFDM systems because transmit symbols are cyclically delayed version for the original transmit symbol. Due to the cyclic shift, the frequency selectivity of channel is increased and the error distribution is changed. So, the error performance can be improved when the channel coding scheme is used because the error is distributed at the receiver[19]. In \(N_{t}\times N_{r}\) MIMO system where \(N_{t}\) and \(N_{r}\) are the number of transmit and receive antennas, the received symbol \(\mathbf{Y}=\left[ \begin{array}{cccc}y_{1}&y_{2}&\cdots&y_{N_{r}}\\ \end{array} \right]^{\text{T}}\) is as follows,\[\left[ \begin{array}{c}y_{1}\\ y_{2}\\ \vdots\\ y_{N_{r}}\end{array} \right]=\left[ \begin{array}{cccc}h_{11}&h_{12}&\cdots&h_{1N _{t}}\\ h_{21}&h_{11}&\cdots&h_{2N_{t}}\\ \vdots&\vdots&\ddots&\vdots\\ h_{N_{1}}&h_{N_{2}}&\cdots&h_{N_{t}N_{t}}\end{array} \right]\left[ \begin{array} []{c}x_{1}\\ x_{2}\\ \vdots\\ x_{N_{t}}\end{array} \right]+\left[ \begin{array}{c}z_{1}\\ z_{2}\\ \vdots\\ z_{N_{t}}\end{array} \right], \tag{12}\]  
where \(h_{ij}(i=1,2,\cdots,N_{r},j=1,2,\cdots,N_{t})\) is a channel coefficient from the \(j\)-th transmit antenna to the \(i\)-th receive antenna, \(x_{j}\) is transmit symbol from the \(j\)-th transmit antenna which is \(\delta_{j}\) cyclically delayed symbol for the original transmit symbol \(x\) and \(z_{i}\) is a zero-mean AWGN at the \(i\)-th receive antenna. In the CDD, the relationship for two different transmit symbols between \(x_{m}\) and \(x_{n}\) is as follows,  
\[x_{m}=e^{-j\frac{2\pi(i\delta_{m}-\delta_{m})}{N}}x_{n}, \tag{13}\]  
where \(N\) is fast Fourier transform (FFT) size. Substituting Equation (13) into (12) when it is assumed that the cyclic delay at the first antenna \(\delta_{1}\) is zero for simple representations, Equation (12) is rewritten as follows,  
\[\left[ \begin{array}{c}y_{1}\\ y_{2}\\ \vdots\\ y_{N_{r}}\end{array} \right]=\left[ \begin{array}{c}h_{\text{C},1}\\ h_{\text{C},2}\\ \vdots\\ h_{\text{C},N_{r}}\end{array} \right]x+\left[ \begin{array}{c}z_{1}\\ z_{2}\\ \vdots\\ z_{N_{r}}\end{array} \right], \tag{14}\]  
where \(h_{\text{C},j}=\sum\limits_{j=1}^{N_{t}}h_{ij}e^{-j\frac{2\pi kj}{N}}\) is composite channel due to property of Equation (13) and \(x\) is non-cyclic delayed original transmit symbol. Finally, transmit symbols are estimated by using maximum ratio combining (MRC) [20] in Equation (14) as follows,  
\[x=\underbrace{\sum_{i=1}^{N_{r}}h_{\text{C},i}^{H}y_{i}}_{\text{(a)}}\Bigg{/} \underbrace{\sum_{i=1}^{N_{r}}|h_{\text{C},i}|^{2}}_{\text{(b)}}. \tag{15}\]  
Table 2 shows configuration for the transmit symbols of the proposed scheme. It is assumed that \(n\) is an odd number. S transmits a symbol to R\({}_{1}\) in an odd time slot and to R\({}_{2}\) in an even time slot. For the transmit symbols in Table 2, the received symbols without noise at R\({}_{1}\) for an odd \(n\) except for 1 are as follows,  
Figure 2: The proposed two-path relay MIMO system. (**a**) Odd time slot; (**b**) Even time slot. MIMO: multiple input multiple output. \[\left[ \begin{array}{c}y_{\text{R}_{1},1}(n)\\ y_{\text{R}_{1},2}(n)\end{array} \right]=\underbrace{\left[ \begin{array}{cc}h_{ \text{SR}_{1},11}&h_{\text{SR}_{1},12}\\ h_{\text{SR}_{1},21}&h_{\text{SR}_{1},22}\end{array} \right]\left[ \begin{array}[ ]{c}x_{\delta_{1}}(n)\\ x_{\delta_{2}}(n)\end{array} \right]}_{\text{Desired}}+\underbrace{\left[ \begin{array}{cc}h_{\text{R}_{2}\text{R}_{1},11}&h_{\text{R}_{2}\text{R}_{1}, 12}\\ h_{\text{R}_{2}\text{R}_{1},21}&h_{\text{R}_{2}\text{R}_{1},22}\end{array}  \right]\left[