# The Role of Correlation in the Performance of Massive MIMO Systems
## 4 Imperfect Channel Estimation Analysis  
In practice, channel estimation is essential for the receiver combining process and downlink data precoding. This allows the use of a simplified detection in the UL and an interference suppression in precoding. In addition, the system performance relies on the effective received signal, which depends heavily on the channel estimation. Therefore, this section presents the channel estimation process by using the practical Bayesian channel estimator. The Bayesian channel estimator uses the minimum mean square error (MMSE) filter to estimate the UL channel. The MMSE estimator makes use of noise and channel statistics to improve the accuracy of channel estimation [35; 36]. In this study, we focus on the TDD transmission mode over which the channels in the uplink and downlink transmissions are assumed to be reciprocal within the coherence block \(T\). Typically, the channel statistics remain constant over a specific coherence block. Each coherence transmission block consists of a number of time and frequency resources as depicted in Figure 1.  
The base station can estimate the downlink channel by using predefined training signals that can be sent by the UEs in the uplink direction. The base station uses multiple antennas to make an accurate decision for the channel estimation by using the channel statistical distributions of the UEs. The MSE that represents the quality of the channel estimates can be minimized using the MMSE estimator. The MMSE estimator uses the received pilot signal and the known pilot book \(\mathbf{P}^{\mathrm{H}}\) with mutually orthogonal sequences, which satisfy the constraint \(\mathbf{P}^{\mathrm{H}}\mathbf{P}=T_{\mathrm{tr}}\mathbf{I}\). The finite length of the coherence blocks requires the sequence length to be as minimal as possible to reduce the system overhead since  
Figure 1: The resource block in time and frequency. The channel is assumed to be frequency flat over a specified period, which corresponds to the coherence time \(T\). The pilot symbols \(T_{\mathrm{tr}}\) are used in each coherence block for the uplink channel estimation in the TDD system and the rest of the resources are used for uplink and downlink data transmissions. longer training length comes at the cost of having a shorter period for data transmission. A rule-of-thumb is to assume the pilot length required in the uplink training is equal to the number of serving UEs in the network. The same set of orthogonal sequences is reused across the entire cell. We use a discrete Fourier transform (DFT) to generate an efficient set of mutually orthogonal sequences. An example of generating training sequences using a discrete Fourier transform (DFT) matrix is given in Equation (8):  
\[\mathbf{P}= \begin{bmatrix}1&1&1&\cdots&1\\ 1&\omega_{\text{T}_{\text{tr}}}&\omega_{\text{T}_{\text{tr}}}^{2}&\cdots& \omega_{\text{T}_{\text{tr}}}^{(\text{T}_{\text{tr}}-1)}\\ \vdots&\vdots&\ddots&\ddots&\vdots\\ 1&\omega_{\text{T}_{\text{tr}}}^{(\text{T}_{\text{tr}}-1)}&\omega_{\text{T}_{ \text{tr}}}^{2(\text{T}_{\text{tr}}-1)}&\ddots&\omega_{\text{T}_{\text{tr}}}^{( \text{T}_{\text{tr}}-1)(\text{T}_{\text{tr}}-1)}\end{bmatrix} , \tag{8}\]  
where \(\omega_{\text{T}_{\text{tr}}}=e^{-\beta\pi/T_{\text{tr}}}\) is a \(T_{\text{tr}}\)-th primitive root of 1. Orthogonal sequences of length \(T_{\text{tr}}\) are transmitted by the UEs, which allow the base station to estimate the uplink channel. A flowchart of the transmission technique is provided in Figure 2.  
As discussed earlier, the uplink channel estimate is used for the downlink precoding and there is no need to estimate the downlink channel. To this end, the received training signal, \(\mathbf{y}_{u}\in\mathbb{C}^{\text{T}u}\), at the base station is given by  
\[\mathbf{y}_{u}^{\text{tr}}=\mathbf{h}_{u}+\frac{1}{\sqrt{T_{\text{tr}}\rho_{ \text{tr}}}}\mathbf{s}_{u}^{\text{tr}}, \tag{9}\]  
where \(\mathbf{s}_{u}^{\text{tr}}\) is the receiver noise that exhibits a complex Gaussian distribution with \(\mathcal{CN}\big{(}\mathbf{0},\mathbf{I}\big{)}\).