# Application Research of Multi-Mode Relay in Future Heterogeneous Networks
## 2 System Model
\[\mathbf{y_{a}}=\sqrt{p_{S}}\beta_{1}\mathbf{H}_{A}^{2}\mathbf{x_{A}}+\sqrt{p _{S}}\beta_{1}\mathbf{H}_{A}\mathbf{H}_{B}\mathbf{x_{B}}\]  
\[+\beta_{1}\mathbf{H}_{A}\sum\limits_{n=1}^{N}\sqrt{p_{I,n}}\mathbf{s_{n}} \mathbf{H}_{I,n}+\mathbf{N_{A}} \tag{6}\]  
\[\mathbf{y_{b}}=\sqrt{p_{S}}\beta_{2}\mathbf{H}_{B}^{2}\mathbf{x_{B}}+\sqrt{p _{S}}\beta_{2}\mathbf{H}_{A}\mathbf{H}_{B}\mathbf{x_{A}}\]  
\[+\beta_{2}\mathbf{H}_{B}\sum\limits_{n=1}^{N}\sqrt{p_{I,n}}\mathbf{s_{n}} \mathbf{H}_{I,n}+\mathbf{N_{B}} \tag{7}\]  
Here \(\mathbf{N_{A}}\) and \(\mathbf{N_{B}}\) are AWGN at two sources. It is assume that we have perfect channel estimation, therefore the resulting self-interference in Equations (6) and (7) can be eliminated accordingly; the received signals at sources A and B can be rewritten in Equations (8) and (9). \[\mathbf{y_{a}}=\sqrt{p_{S}}\beta_{1}\mathbf{\hat{H}}_{A}\mathbf{ \hat{H}}_{B}\mathbf{x_{B}}+\sqrt{p_{S}}\beta_{1}\left(\mathbf{\hat{H}}_{A} \mathbf{\hat{e}}_{B}+\mathbf{\hat{H}}_{B}\mathbf{e_{A}}+\mathbf{e_{A}} \mathbf{e_{B}}\right)\mathbf{x_{B}} \tag{8}\] \[+\ \sqrt{p_{S}}\beta_{1}\left(2\mathbf{\hat{H}}_{A}\mathbf{e_{A}} +\mathbf{e_{A}^{2}}\right)\mathbf{x_{A}}+\beta_{1}\left(\mathbf{\hat{H}}_{A} +\mathbf{e_{A}}\right)\sum\limits_{n=1}^{N}\sqrt{p_{I,n}}s_{n}\mathbf{H}_{I,n }+\mathbf{N_{A}}\]  
\[\mathbf{y_{b}}=\sqrt{p_{S}}\beta_{2}\mathbf{\hat{H}}_{A}\mathbf{ \hat{H}}_{B}\mathbf{x_{A}}+\sqrt{p_{S}}\beta_{2}\left(\mathbf{\hat{H}}_{A} \mathbf{e_{B}}+\mathbf{\hat{H}}_{B}\mathbf{e_{A}}+\mathbf{e_{A}}\mathbf{e_{B}} \right)\mathbf{x_{A}} \tag{9}\] \[+\ \sqrt{p_{S}}\beta_{2}\left(2\mathbf{\hat{H}}_{B}\mathbf{e_{B}} +\mathbf{e_{B}^{2}}\right)\mathbf{x_{B}}+\beta_{2}\left(\mathbf{\hat{H}}_{B} +\mathbf{e_{B}}\right)\sum\limits_{n=1}^{N}\sqrt{p_{I,n}}\mathbf{s_{n}} \mathbf{H}_{I,n}+\mathbf{N_{B}}\]Here the right item next to Equations (8) and (9) is the terms of signal, noise, and interference, respectively. Accordingly, two sources SINR of A and B can be calculated by expressions of Equations (10) and (11). \[f_{A}=\frac{p_{s}p_{r1}\left|\hat{\mathbf{H}}_{A}\right|^{2}\left|\hat{\mathbf{H }}_{B}\right|^{2}}{p_{s}\left(p_{r1}\sigma_{E}^{2}+1\right)\left(\left|\hat{ \mathbf{H}}_{A}\right|^{2}+\left|\hat{\mathbf{H}}_{B}\right|^{2}\right)+4p_{ s}p_{r1}\sigma_{E}^{2}\left|\hat{\mathbf{H}}_{A}\right|^{2}+\left(p_{r1} \left|\hat{\mathbf{H}}_{A}\right|^{2}+p_{r1}\sigma_{E}^{2}+1\right)\sum_{n=1}^ {N}p_{r1,n}\left|\hat{\mathbf{H}}_{r1,n}\right|^{2}+2p_{s}\sigma_{E}^{2}+2p_{s} p_{r1}\sigma_{E}^{4}} \tag{10}\]  
\[f_{B}=\frac{p_{s}p_{r2}\left|\hat{\mathbf{H}}_{A}\right|^{2}\left|\hat{ \mathbf{H}}_{B}\right|^{2}}{p_{s}\left(p_{r2}\sigma_{E}^{2}+1\right)\left( \left|\hat{\mathbf{H}}_{A}\right|^{2}+\left|\hat{\mathbf{H}}_{B}\right|^{2} \right)+4p_{s}p_{r2}\sigma_{E}^{2}\left|\hat{\mathbf{H}}_{B}\right|^{2}+\left( p_{r2}\left|\hat{\mathbf{H}}_{B}\right|^{2}+p_{r2}\sigma_{E}^{2}+1\right)\sum_{n=1}^ {N}p_{r1,n}\left|\hat{\mathbf{H}}_{r1,n}\right|^{2}+2p_{s}\sigma_{E}^{2}+2p_{s }p_{r2}\sigma_{E}^{4}} \tag{11}\]  
Let \(H_{a}=\left|\tilde{\mathbf{H}}_{A}\right|^{2}\), \(H_{b}=\left|\tilde{\mathbf{H}}_{B}\right|^{2}\), \(\mathbf{M}=\sum_{n=1}^{N}p_{r1,n}\left|h_{I,n}\right|^{2}\), Equations (10) and (11) can be rewritten as Equations (12) and (13)  
\[+\beta_{1}\mathbf{H}_{A}\sum\limits_{n=1}^{N}\sqrt{p_{I,n}}\mathbf{s_{n}} \mathbf{H}_{I,n}+\mathbf{N_{A}} \tag{6}\]0  
\[+\beta_{1}\mathbf{H}_{A}\sum\limits_{n=1}^{N}\sqrt{p_{I,n}}\mathbf{s_{n}} \mathbf{H}_{I,n}+\mathbf{N_{A}} \tag{6}\]1  
As shown in Formulas (12) and (13), the SINR is in direct proportion to the MMR power \(p_{ri}\) (\(i=1,2\)); the absolute value of channel coefficient \(\mathbf{H}_{A}\) and \(\mathbf{H}_{B}\) also has the influence on SINR.