# The Role of Correlation in the Performance of Massive MIMO Systems
## 3 Received Signal Processing and Linear Precoder  
This study considers that each UE knows the average effective channel through \(\sqrt{\lambda}\mathbb{E}\big{[}\mathbf{h}_{u}^{\mathrm{H}}\mathbf{v}_{u}\big{]}\). To this end, the received signal is given as  
\[y_{u}=\underbrace{\sqrt{\lambda}\mathbb{E}[\mathbf{h}_{u}^{ \mathrm{H}}\mathbf{v}_{u}]x_{u}}_{\text{term1}}+\underbrace{\sqrt{\lambda} \big{(}\mathbf{h}_{u}^{\mathrm{H}}\mathbf{v}_{u}-\mathbb{E}[\mathbf{h}_{u}^{ \mathrm{H}}\mathbf{v}_{u}]\big{)}x_{u}}_{\text{term2}}\] \[+\underbrace{\sqrt{\lambda}\mathbf{h}_{u}^{\mathrm{H}}\sum_{l \
eq u}^{U}\mathbf{v}_{l}x_{l}}_{\text{term3}}+\underbrace{s_{u}}_{\text{term4}}, \tag{4}\]  
where the first term term1 in Equation (4) denotes the desired signal intended towards the \(u\)-th UE, while the second term, term2, and the third term, term3, represent the amount of interference due to the imperfect CSI knowledge and the residual inter-UE interference created by other UEs in the entire cell, respectively. The last term, term4, in Equation (4) denotes the additive white Gaussian noise, which is introduced at the UE side. It is worth noting that the interference and channel estimation error terms are neither Gaussian nor independent of the desired signal. Thus, the effective signal-to-interference-plus-noise ratio (SINR) is obtained by considering that both terms are complex Gaussian random variables and independent of the signal of interest, as considered in [34]. Therefore, the downlink SE at the \(u\)-th UE can be written as in Equation (5)  
\[\text{SE}_{u}=\bigg{(}1-\frac{T_{\text{tr}}}{T}\bigg{)}\text{log}_{2}(1+\text{ SINR}_{u})\quad\text{[bit/s/Hz]}, \tag{5}\]  
where \(\text{SINR}_{u}\) is the effective SINR associated with the \(u\)-th UE, which is given as  
\[\text{SINR}_{u}=\frac{\lambda\mid\mathbb{E}[\mathbf{h}_{u}^{\mathrm{H}}\mathbf{ v}_{u}]\mid^{2}}{\sigma_{\text{d}}^{2}+\lambda\sum_{l=1}^{U}\mathbb{E}[\mid \mathbf{h}_{u}^{\mathrm{H}}\mathbf{v}_{l}\mid^{2}]-\mathbb{E}[\mathbf{h}_{u}^ {\mathrm{H}}\mathbf{v}_{u}]\mid^{2}}. \tag{6}\]  
The expectation in Equation (6) is obtained by considering all sources of randomness. To this end, a Monte Carlo averaging process is used in order to compute the average SINR over random channel realizations. The channel vectors \(\mathbf{h}_{u}\), \(u=1,\ldots,U\), are modeled as an independent complex Gaussian random distribution with zero mean and the \(u\)-th UE's covariance matrix, \(\mathbf{G}_{u}\). \(\mathbf{G}_{u}\) at the base station side is given by \(\mathbf{G}_{U}=\mathbb{E}[\mathbf{h}_{u}\mathbf{h}_{u}^{\mathrm{H}}]\in \mathbb{C}^{M\times M}\). The Gaussian random distribution accounts for the random small-scale fading realization in each coherence block. The channel covariance matrix \(\mathbf{G}_{u}\) describes the macroscopic propagation characteristics. Clearly, the received effective signal in Equation (6) depends on the channel statistics, channel estimates, and the linear precoding technique that is used at the base station. For practical m-MIMO considerations, imperfect channel state information should be used at the base station. Hence, this study considers an imperfect CSI at the base station for precoding design. The channel estimation imperfection reduces the spectral efficiency due to the precoder mismatch with the actual channels. Therefore, it is particularly of interest to examine the system performance in terms of MSE and SE by considering robust precoding techniques. This study considers a fully digital precoding scheme with a frequency operation below 6 GHz bands. A well-known linear precoding technique is used, which is referred to as the RZF precoder.