# Estimating the Performance of MIMO SC-FDE Systems Using SISO Measurements
## 3 Theoretical BER Performance  
As previously mentioned, coefficients \(F_{k}\) and \(B_{k}\) are chosen so as to minimise the MSE and BER. For a SISO system, these coefficients can be expressed by Equations (8) and (9), respectively.  
The BER performance for a QPSK constellation is given by:  
\[BER_{p}\simeq Q\left(\sqrt{\frac{1}{\theta}}\right), \tag{14}\]  
where \(Q(x)\) denotes the Gaussian error function and  
\[\theta=\frac{1}{N^{2}}\sum_{k=0}^{N-1}\Theta_{k}. \tag{15}\]  
From Equation (15), \(\Theta_{k}\) depicts the MSE which can be calculated by Equation (16):  
\[\Theta_{k}=\mathbb{E}\left[\left|\widehat{X}_{k}-X_{k}\right|^{2}\right]= \mathbb{E}\left[\left|F_{k}Y_{k}-B_{k}\overline{X}_{k}-X_{k}\right|^{2}\right] \tag{16}\]As \(\overline{X}_{k}\) can be given by:  
\[\overline{X}_{k}=\rho\hat{X}_{k}, \tag{17}\]  
where \(\hat{X}_{k}\) denotes \"hard decision\" and is expressed by:  
\[\hat{X}_{k}=\rho X_{k}+\Delta_{k}. \tag{18}\]  
Therefore, Equation (17) can be written as:  
\[\overline{X}_{k}=\rho^{2}X_{k}+\rho\Delta_{k}. \tag{19}\]  
Knowing the expression for \(\overline{X}_{k}\), Equation (16) can be expanded as:  
\[\Theta_{k}=\mathbb{E}\left[\left|F_{k}Y_{k}-B_{k}\rho^{2}X_{k}+\rho\Delta_{k}- X_{k}\right|^{2}\right] \tag{20}\]  
Since \(\mathbb{E}[\Delta_{k}]=0\), after some mathematical manipulation, Equation (20) can be rewritten as:  
\[\begin{split}\Theta_{k}=F_{k}^{*}\mathbb{E}\left[Y_{k}^{*}Y_{k} \right]F_{k}-F_{k}^{*}\mathbb{E}\left[Y_{k}^{*}X_{k}\right]\rho^{2}B_{k}^{*}- F_{k}^{*}\mathbb{E}\left[Y_{k}^{*}X_{k}\right]-B_{k}^{*}\rho^{2}\mathbb{E} \left[X_{k}^{*}Y_{k}\right]F_{k}+\\ B_{k}^{*}B_{k}\rho^{4}\mathbb{E}\left[X_{k}^{*}X_{k}\right]+B_{k}^ {*}\rho^{2}\mathbb{E}\left[X_{k}^{*}X_{k}\right]-F_{k}\mathbb{E}\left[X_{k}^{ *}Y_{k}\right]+B_{k}\rho^{2}\mathbb{E}\left[X_{k}^{*}X_{k}\right]+\mathbb{E} \left[X_{k}^{*}X_{k}\right]\end{split} \tag{21}\]  
Knowing that:  
\[\mathbb{E}\left[Y_{k}^{*}Y_{k}\right]=R_{Y}; \tag{22}\]  
\[\mathbb{E}\left[Y_{k}^{*}X_{k}\right]=R_{Y^{*}X} \tag{23}\]  
and  
\[\mathbb{E}\left[X_{p}X_{p}^{*}\right]=R_{X}. \tag{24}\]  
Equation (21) can be simplified to:  
\[\Theta_{k}=F_{k}^{*}R_{Y}F_{k}-2 Re\Big{\{}F_{k}^{*}\rho^{2}R_{Y^{*}X}B_{k} \Big{\}}-2 Re\Big{\{}F_{k}^{*}R_{Y^{*}X}\Big{\}}+B_{k}^{*}B_{k}\rho^{4}R_{X}-2 Re\Big{\{}B_{k}^{*}\rho^{2}R_{X}\Big{\}}+R_{X}. \tag{25}\]  
By using Equations (14) and (25) it is then possible to evaluate the BER performance of a SISO system. When considering a MIMO system, Equation (25) should be replaced by Equation (26). Equation (26) is fully derived and explained in [23], where the BER performance of receivers that can be used in MIMO systems are analysed and compared. Each element depends on both the user and the subcarrier and this increases the complexity, since, in MIMO systems, it is necessary to work with matrices. Therefore, in this expression, the elements \(\mathbf{F}\) and \(\mathbf{B}\) are matrices for each subcarrier. The matrix \(\mathbf{F}\) equalises the channel and the matrix \(\mathbf{B}\) is responsible for removing the ISI, using the previous iteration values, as explain in Section 2 for an SISO system. Just for simplicity, user and subcarrier dependency have been dropped out, with the exception for the \(X_{p}\) factor.  
\[\Theta=\mathbf{F}^{H}\mathbf{R}_{\mathbf{Y}}\mathbf{F}+\mathbf{B}^{H}\mathbf{ R}_{\overline{X},\overline{X}}\mathbf{B}-2 Re\Big{\{}\mathbf{F}^{H}\mathbf{R}_{ \mathbf{Y},X_{p}}\Big{\}}+2 Re\Big{\{}\mathbf{B}^{H}\mathbf{R}_{\overline{X},X _{p}}\Big{\}}-2 Re\Big{\{}\mathbf{B}^{H}\mathbf{R}_{\overline{X},\mathbf{Y}} \mathbf{F}\Big{\}}+\mathbf{R}_{\mathbf{X}}. \tag{26}\]