# The Role of Correlation in the Performance of Massive MIMO Systems
## 4 Imperfect Channel Estimation Analysis
#### Closed-Form Analysis for the Mean Square Error of the Channel Estimation  
In this subsection, we provide a closed-form analysis of the MSE of the channel estimate. Let the eigenvalue decomposition (EVD) of the channel covariance matrices be  
\[\mathbf{G}_{u}=\mathbf{U}_{u}\mathbf{\Lambda}_{u}\mathbf{U}_{u}^{\text{H}}, \tag{17}\]  
where \(\mathbf{U}_{u}=[\mathbf{u}_{u,1},\ldots,\mathbf{u}_{u,M}]\in\mathbb{C}^{M \times M}\) is a unitary matrix of the eigenvectors and \(\mathbf{\Lambda}_{u}\) is the eigenvalues of \(\mathbf{G}_{u}\) ordered as \(\lambda_{u,1}\geq\lambda_{u,2}\geq\cdots\geq\lambda_{u,M}\). Using the EVD of the covariance of the channel and channel estimate, the MSE \(\text{C}_{\text{an}}\) between the actual channel and the channel estimate is obtained as  
\[\text{C}_{\text{an}}=\sum_{u=1}^{U}\text{trace}\left(\mathbf{U}_{u}\left(\mathbf{ \Lambda}_{u}-\rho_{\text{tr}}T_{\text{tr}}\mathbf{\Lambda}_{u}(\rho_{\text{tr}}T_{ \text{tr}}\mathbf{\Lambda}_{u}\mathbf{I})^{-1}\mathbf{\Lambda}_{u}\right)\mathbf{U}_{u}^ {\text{H}}\right) \tag{18}\]  
```
1:for every user \(u\)do
2: Calculate user distance from the BS \"\(D_{u}\)\"
3:if\(D_{u}<200\)then
4: Generate training sequence using DFT (Equation: (8))
5: Calculate the channel covariance matrix (Equations: (21) for ULA or (25) for URA)
6: Apply MMSE filter
7: Estimate the user channel at the BS (Equation: (10))
8: Compute the EVD (Equation: (17))
9: Calculate the MSE (Equation: (16) for simulation and Equation: (20) for analytical)
10: Precode the data to the users using RZF precoding (Equation: (7))
11: Calculate the spectral efficiency (Equation: (5))
12:endif
13:endif
14:endfor
```  
**Algorithm 1**  
Noting that \(\text{trace}(ABC)=\text{trace}(CAB)\)[38], hence, the overall MSE across the UEs can be further simplified into the expressions given by Equations (19) and (20), respectively.  
\[\text{C}_{\text{an}}=\sum_{u=1}^{U}\text{trace}\Big{(}\mathbf{\Lambda}_{u}\big{(}T _{\text{tr}}\rho_{\text{tr}}\mathbf{\Lambda}_{u}+\mathbf{I}\big{)}^{-1}\Big{)} \tag{19}\]  
\[\text{C}_{\text{an}}=\sum_{u=1}^{U}\sum_{m=1}^{M}\frac{\lambda_{u,m}}{1+T_{\text {tr}}\rho_{\text{tr}}\lambda_{u,m}} \tag{20}\]  
The normalized MSE (NMSE) per symbol is obtained by dividing Equation (20) into the total number of base station antennas \(M\). The expression in Equation (20) can be applied for any channel covariance matrices. The expression in Equation (20) reveals that as the SNR increases, the MSE performance is improved where the estimated channel tends to be exactly the same as the actual channel. It is also explained that increasing the pilot length would enhance the MSE performance as well. Importantly, increasing the level of correlations helps in reducing the estimation error variance (the uncertainty in the channel directions) and, hence, improving the MSE performance. This is because the power in the channel will be concentrated in the strong eigendirections only and, thus, making the channel easy to estimate. Overall, the formulation in Equation (20) for the MSE is analytically convenient and, more importantly, it is straightforward to use to reproduce the numerical results. In the following section, we introduce a realistic physical channel model that is used in the performance evaluation of the TDD m-MIMO systems under consideration.