## 3 System Model and Basic Assumptions
### A&F MIMO Signal Model with Pre-Equalization
#### 3.2.2 MMSE Pre-Equalization Scheme  
In order to maximize the equalization Signal-to-Noise Ratio (SNR), the MMSE [43] pre-equalizer can be used, which is given as  
\[\mathbf{M}_{\text{MMSE}}=(\mathbf{H}^{\text{H}}\mathbf{H}+\sigma^{2}_{n} \mathbf{I})^{-1}\mathbf{H}^{\text{H}}, \tag{12}\]  
where \(\mathbf{H}^{\text{H}}\) describes the hermitian transpose matrix of the channel and \(\sigma^{2}_{n}\) is the statistical information of the noise. The baseband processing by the A&F-MMSE protocol is given by Algorithm 2, which taking into account the received signal (2) and MMSE matrix (12), this algorithm performs the pre-equalization for SISO and MIMO strategies.  
```
Input :\(\mathbf{Y}=\mathbf{Y}^{1}_{\text{AF}}\), \(\hat{\mathbf{H}}=\hat{\mathbf{H}}^{1}\), \(\hat{\mathbf{N}}_{0}\) Output :\(\hat{\mathbf{Y}}^{1}_{\text{AF-MMSE}}=\hat{\mathbf{Y}}\) if\(\hat{\mathbf{N}}^{T}_{\text{T}}==1\)&&\(\mathbf{x}\) = 1 then \(\hat{\mathbf{Y}}=(\hat{\mathbf{H}}^{\text{H}}\hat{\mathbf{H}}+\hat{\mathbf{N}}_{0})^{-1} \hat{\mathbf{H}}^{\text{H}}\mathbf{Y}\); else for\(c\gets 1:N_{sub}\)do for\(s\gets 1:N_{sym}\)do \(\hat{\mathbf{Y}}^{(\cdot)}_{(c,s)}=\frac{(\hat{\mathbf{H}}^{\text{H}(\cdot)}_{(c,s)})^{-1}(\hat{\mathbf{N}}_{0})^{-1}}{\hat{\mathbf{I}}}\hat{\mathbf{H}}^{ \text{H}(\cdot;c)}_{(c,s)}\mathbf{Y}^{(\cdot)}_{(c,s)}\);
```  
**Algorithm 2**Implementation for MMSE Pre-Equalizer Scheme  
Considering the Algorithm 2 output, the pre-equalized signal is given by  
\[\hat{\mathbf{Y}}^{1}_{\text{AF-MMSE}}=\mathbf{X}+\hat{\mathbf{N}}^{1}_{\text{AF- MMSE}}, \tag{13}\]where \(\hat{\mathbf{N}}^{1}_{\text{AF-MMSE}}=(\hat{\mathbf{H}}^{1^{\text{H}}}\hat{\mathbf{ H}}^{1}+\sigma^{2}_{\text{N}^{1}_{\text{AF}}}\mathbf{I})^{-1}\mathbf{H}^{1^{ \text{H}}}\mathbf{N}^{1}_{\text{AF}}\in\mathbb{C}^{\text{N}^{\text{R}}_{\text{ R}}\times\text{T}}\) and \(\hat{\mathbf{H}}^{1}\in\mathbb{C}^{\text{N}^{\text{R}}_{\text{R}}\times\text{ N}^{\text{S}}_{\text{T}}}\) is the estimated channel frequency responses between eNodeB and A&F-MMSE strategy. Therefore, the re-transmitted signal from the A&F-MMSE protocol can be written as  
\[\mathbf{X}^{\text{MMSE}}_{\text{AF}}=\mathbf{M}_{\text{RT}}\mathbf{G}^{\text{ MMSE}}_{\text{AF}}\hat{\mathbf{\gamma}}^{1}_{\text{AF-MMSE}}, \tag{14}\]  
where \(\mathbf{G}^{\text{MMSE}}_{\text{AF}}\in\mathbb{C}^{\text{N}^{\text{R}}_{\text{ T}}\times\text{N}^{\text{R}}_{\text{R}}}\) denotes the diagonal matrix of the A&F-MMSE gain, \(\hat{\mathbf{\gamma}}^{1}_{\text{AF-MMSE}}\in\mathbb{C}^{\text{N}^{\text{R}}_ {\text{R}}\times\text{T}}\), and RF transmitter chains \(\mathbf{M}_{\text{RT}}\in\mathbb{C}^{\text{N}^{\text{R}}_{\text{T}}\times \text{N}^{\text{R}}_{\text{T}}}\). On the other hand, the received signal in the UE after re-transmission at the Relay Node can be expressed as  
\[\mathbf{Y}^{2}_{\text{AF-MMSE}}=\mathbf{H}^{2}\mathbf{X}^{\text{MMSE}}_{\text {AF}}+\mathbf{N}^{2}_{\text{AF-MMSE}} \tag{15}\]  
where \(\mathbf{N}^{2}_{\text{AF-MMSE}}\in\mathbb{C}^{\text{N}^{\text{R}}_{\text{R}} \times\text{T}}\) denotes the AWGN at the UE, \(N\sim(0,\sigma^{2}_{\text{N}^{2}_{\text{AF}}})\) and \(\sigma^{2}_{\text{N}^{2}_{\text{AF}}}\) represents the noise variance. \(\mathbf{H}^{2}\in\mathbb{C}^{\text{N}^{\text{R}}_{\text{T}}\times\text{N}^{ \text{D}}_{\text{R}}}\) is the channel matrix in the access link. Considering the data processing of the UE through the baseband \(\mathbf{G}_{\text{BB}}\) and RF receiver chains \(\mathbf{M}_{\text{DR}}\), the estimated received data at the destination can be obtained by  
\[\hat{\mathbf{S}} =\mathbf{G}^{\text{H}}_{\text{BB}}\mathbf{M}^{\text{H}}_{\text{ DR}}(\mathbf{H}^{2}\mathbf{X}^{\text{MMSE}}_{\text{AF}}+\mathbf{N}^{2}_{\text{AF- MMSE}}), \tag{16}\] \[=\mathbf{S}+\mathbf{N}_{\text{D}}\]  
where \(\mathbf{G}=\mathbf{G}_{\text{BB}}\mathbf{M}_{\text{DR}}\) and \(\mathbf{N}_{\text{D}}=\mathbf{G}^{\text{H}}(\mathbf{H}^{2}\mathbf{M}_{\text{ RT}}\mathbf{G}^{\text{MMSE}}_{\text{AF}}\hat{\mathbf{N}}^{1}_{\text{AF- MMSE}}+\mathbf{N}^{2}_{\text{AF-MMSE}})\).