## 3 System Model and Basic Assumptions
### A&F MIMO Signal Model with Pre-Equalization
#### 3.2.1 ZF Pre-Equalization Scheme  
The Zero-Forcing (ZF) [43] technique is the simplest equalizer algorithm, which nullifies the interference through use of the following matrix,  
\[\mathbf{M}_{\mathrm{ZF}}=(\mathbf{H}^{\mathrm{H}}\mathbf{H})^{-1}\mathbf{H}^{ \mathrm{H}}, \tag{7}\]  
where \((\cdot)^{\mathrm{H}}\) denotes the Hermitian transpose operation. Note that, the ZF pre-equalizer does not require the statistics of the noise. From Equation (2), Algorithm 1 describes the procedure of pre-equalization for SISO and MIMO techniques.  
```
Input :\(\mathbf{Y}=\mathbf{Y}_{\mathrm{AF}}^{1}\), \(\hat{\mathbf{H}}=\hat{\mathbf{H}}^{1}\) Output :\(\hat{\mathbf{Y}}_{\mathrm{AF}:\mathrm{ZF}}^{1}=\hat{\mathbf{Y}}\) if\(N_{\mathrm{T}}^{5}==1\)&& \(N_{\mathrm{R}}^{6}==1\)then \(\hat{\mathbf{Y}}=[\hat{\mathbf{H}}^{\mathrm{H}}\hat{\mathbf{H}}]^{-1}\hat{ \mathbf{H}}^{\mathrm{H}}\mathbf{Y}\); else for\(c\gets 1:N_{sub}\)do for\(s\gets 1:N_{sym}\)do \(\hat{\mathbf{Y}}_{(c,s)}^{()}=[\hat{\mathbf{H}}_{(c,s)}^{H(:)}\hat{\mathbf{H}}_ {(c,s)}^{(:)}]^{-1}\hat{\mathbf{H}}_{(c,s)}^{H(:)}\mathbf{Y}_{(c,s)}^{(:)}\);
```  
**Algorithm 1** Implementation for ZF Pre-Equalizer Scheme  
The received signal in the first-hop after the pre-equalization step, which is given by the output of the Algorithm 1 can be written as  
\[\hat{\mathbf{Y}}_{\mathrm{AF}:\mathrm{ZF}}^{1}=\mathbf{X}+\hat{\mathbf{N}}_{ \mathrm{AF}:\mathrm{ZF}}^{1}, \tag{8}\]  
Figure 3: Pre-Equalization Schemes.  
where \(\hat{\mathbf{N}}^{1}_{\text{AF-ZF}}\)=\((\hat{\mathbf{H}}^{1^{\text{H}}}\hat{\mathbf{H}}^{1})^{-1}\hat{\mathbf{H}}^{1^{ \text{H}}}\mathbf{N}^{1}_{\text{AF}}\in\mathbb{C}^{\mathbb{N}^{\mathbb{R}}_{ \text{R}}\times\mathbb{T}}\) and \(\hat{\mathbf{H}}^{1}\in\mathbb{C}^{\mathbb{N}^{\mathbb{R}}_{\text{R}}\times \mathbb{N}^{\mathbb{R}}_{\text{T}}}\) is the estimated channel matrix between source and A&F-ZF protocol. In A&F-ZF Relay Node, the baseband matrix \(\mathbf{R}_{\text{BB}}\) is given by the product of the diagonal matrix of the A&F-ZF gain (\(\mathbf{G}^{\text{ZF}}_{\text{AF}}\in\mathbb{C}^{\mathbb{N}^{\mathbb{R}}_{ \text{T}}\times\mathbb{N}^{\mathbb{R}}_{\text{R}}}\)) and ZF pre-equalization matrix (7) \(\in\mathbb{C}^{\mathbb{N}^{\mathbb{R}}_{\text{R}}\times\mathbb{N}^{\mathbb{R}}_ {\text{R}}}\). After that, the result signal is transmitted through the RF transmitter chains \(\mathbf{M}_{\text{RT}}\in\mathbb{C}^{\mathbb{N}^{\mathbb{R}}_{\text{T}}\times \mathbb{N}^{\mathbb{R}}_{\text{T}}}\). Taking into account the processing in the protocol, the re-transmitted signal from the A&F-ZF Relay Node can be expressed as  
\[\mathbf{X}^{\text{ZF}}_{\text{AF}}=\mathbf{M}_{\text{RT}}\mathbf{G}^{\text{ZF }}_{\text{AF}}\mathbf{S}^{1}_{\text{AF-ZF}}, \tag{9}\]  
On the other hand, in the second-hop, the signal received by the UE can be given by  
\[\mathbf{Y}^{2}_{\text{AF-ZF}}=\mathbf{H}^{2}\mathbf{X}^{\text{ZF}}_{\text{AF }}+\mathbf{N}^{2}_{\text{AF-ZF}}, \tag{10}\]  
where \(\mathbf{N}^{2}_{\text{AF-ZF}}\in\mathbb{C}^{\mathbb{N}^{\mathbb{R}}_{\text{ R}}\times\mathbb{T}}\) indicates the AWGN in the destination node, with \(N\sim(0,\sigma^{2}_{\text{N}^{2}_{\text{AF}}})\) and \(\sigma^{2}_{\text{N}^{2}_{\text{AF}}}\) denotes the noise variance. \(\mathbf{H}^{2}\in\mathbb{C}^{\mathbb{N}^{\mathbb{R}}\times\mathbb{N}^{\mathbb{ R}}_{\text{T}}}\) is the channel matrix in the access link. Finally at UE, the data symbol matrix \(\mathbf{S}\) can be estimated through the following expression  
\[\begin{split}\hat{\mathbf{S}}&=\mathbf{G}^{\text{H}}_ {\text{BB}}\mathbf{M}^{\text{H}}_{\text{DR}}(\mathbf{H}^{2}\mathbf{X}^{\text{ZF }}_{\text{AF}}+\mathbf{N}^{2}_{\text{AF-ZF}})\\ &=\mathbf{S}+\mathbf{N}_{\text{D}},\end{split} \tag{11}\]  
where \(\mathbf{G}=\mathbf{G}_{\text{BB}}\mathbf{M}_{\text{DR}}\) and \(\mathbf{N}_{\text{D}}=\mathbf{G}^{\text{H}}(\mathbf{H}^{2}\mathbf{M}_{\text{ RT}}\mathbf{G}^{\text{ZF}}_{\text{AF}}\hat{\mathbf{N}}^{1}_{\text{AF-ZF}}+\mathbf{N}^{2}_{ \text{AF-ZF}})\) is the received signal not desired.