# Analytical Performance Evaluation of Massive MIMO Techniques for SC-FDE Modulations
## 5 Theoretical BER Performance  
Each of the receivers presented so far aims to minimize the Mean Squared Error (MSE) which, in turn, decreases BER performance. This minimization takes place through precise selection of feedforward and feedback coefficients. For a QPSK constellation with Gray mapping rule, as considered in this paper, the BER can be calculated by:  
\[BER_{p}\simeq Q\left(\sqrt{\frac{1}{\theta_{p}}}\right), \tag{15}\]  
where \(Q(x)\) represents the Gaussian error function and \(\theta_{p}\) is given by:  
\[\theta_{p}=\frac{1}{N^{2}}\sum_{k=0}^{N-1}\Theta_{k,p}. \tag{16}\]  
The MSE, represented by \(\Theta_{k,p}\), can be calculated using:  
\[\Theta_{k,p}=\mathbb{E}\left[\left|\mathbf{\hat{X}}_{k,p}-X_{k,p}\right|^{2} \right]=\mathbb{E}\left[\left|\mathbf{F}_{k,p}^{T}\mathbf{Y}_{k}-\mathbf{B}_{ k,p}^{T}\overline{\mathbf{X}}_{k,p}-X_{k,p}\right|^{2}\right] \tag{17}\]  
As presented in [23] and [11], \(\overline{X}_{k,p}\simeq\rho_{p}\hat{X}_{k,p}\) where \(\hat{X}_{k,p}\approx\rho_{p}X_{k,p}+\Delta_{k,p}\) allowing the redefinition of \(\overline{X}_{k,p}\) to \(\overline{X}_{k,p}\approx\rho_{p}^{2}X_{k,p}+\rho_{p}\Delta_{k,p}\). In matrix format, \(\overline{\mathbf{X}}_{k,p}\) is given by:  
\[\overline{\mathbf{X}}_{k,p}\simeq\mathbf{P}^{2}\mathbf{S}_{k,p}+\mathbf{P} \mathbf{\Delta}_{k,p}, \tag{18}\]  
where \(\mathbf{\Delta}_{k,p}\) is a mean zero error vector for \(p^{th}\) MT and \(\mathbf{P}=diag(\rho_{1},\cdots,\rho_{p})\), with \(\rho\) defined by:  
\[\rho_{p}=\frac{\mathbb{E}\left[\hat{x}_{n,p}x_{n,p}^{*}\right]}{\mathbb{E} \left[\left|x_{n,p}\right|^{2}\right]}, \tag{19}\]  
Figure 2: Iterative Block Decision Feedback Equalization (IB-DFE) receiver combined with the Maximum Ratio Combining (MRC) structure. which, in turn, defines the correlation factor that supplies a blockwise reliability measure of the estimates employed in the feedback loop and associated to the \((i-1)th\) iteration [22; 24]. For a QPSK constellations, \(\rho_{p}=1-P_{e}\) and can be approximately given by:  
\[\rho_{p}\approx\frac{1}{2N}\sum_{n=0}^{N-1}\left(\left|\rho_{n,p}^{Re}\right|+ \left|\rho_{n,p}^{Im}\right|\right), \tag{20}\]  
with  
\[\rho_{n,p}^{Re^{(i)}}=\tanh\left(\frac{\left|L_{n,p}^{Re^{(i)}}\right|}{2} \right), \tag{21}\]  
and  
\[\rho_{n,p}^{Im^{(i)}}=\tanh\left(\frac{\left|L_{n,p}^{Im^{(i)}}\right|}{2} \right). \tag{22}\]  
Once the data and noise components have zero mean and are uncorrelated, Equation (17) which represents the MSE can be extended, resulting in Equation (23):  
\[ \begin{split}&\Theta=\mathbf{F}^{H}\mathbb{E}\left[\mathbf{Y}^{*} \mathbf{Y}^{T}\right]\mathbf{F}-\mathbf{F}^{H}\mathbb{E}\left[\mathbf{Y}^{*} \overline{\mathbf{X}}^{T}\right]\mathbf{B}-\mathbf{F}^{H}\mathbb{E}\left[ \mathbf{Y}^{*}X_{p}\right]\mathbf{B}-\mathbf{B}^{H}\mathbb{E}\left[\overline {\mathbf{X}}^{*}\mathbf{Y}^{T}\right]\mathbf{F}+\\ &\mathbf{B}^{H}\mathbb{E}\left[\overline{\mathbf{X}}^{*} \overline{\mathbf{X}}^{T}\right]\mathbf{B}+\mathbf{B}^{H}\mathbb{E}\left[ \overline{\mathbf{X}}^{*}X_{p}\right]-\mathbf{F}^{T}\mathbb{E}\left[\mathbf{Y }X_{p}^{*}\right]+\mathbf{B}^{T}\mathbb{E}\left[\overline{\mathbf{X}}X_{p}^{* }\right]+\mathbb{E}\left[X_{p}^{*}X_{p}\right],\end{split}  \tag{23}\]  
For simplicity, in Equation (23), the user and subcarrier dependency has been dropped, with the exception for the \(S_{p}\) factor.