# Analysis and Evaluation of Performance Gains and Tradeoffs for Massive MIMO Systems
## 2 System Description
### Uplink Communication Phase  
At the beginning of each coherence interval \(T_{coh}\), the terminals transmit pilot sequences with length \(\tau\). Each \(m\)th antenna at base station in \(j\)th cell will receive a vector \(y_{jm}\) with length \(\tau\) from \(k\)th user in \(\ell\)th cell as shown in Figure 1, where each BS received pilot sequence from its cell and other cells due to limited number of orthogonal pilot sequences that leads to problem of pilot contamination, then the received signal is defined by  
\[y_{jm}=\sum_{\ell=1}^{L}\sum_{k=1}^{K}\sqrt{p_{u}\tau\beta_{jk\ell}}h_{jk\ell }\psi_{\ell k}+w_{jm}, \tag{3}\]  
where \(w_{jm}\) is the additive white Gaussian noise (AWGN) whose elements are i.i.d. with zero mean and variance \(\sigma^{2}_{w}\), so the noise vector at \(j\)th cell is defined by \(W_{j}\) which is a \(M\times\tau\) vector and \(W_{j}=[w_{j1},\,w_{j2},\,\ldots,\,w_{jm}]\). \(Y_{j}\) is the received signal vector with dimension \(M\times\tau\), so the received vector is defined by \(Y_{j}=[y_{j1},\,y_{j2},\,\ldots,\,y_{jm}]\), while the terms \(\{(\tau)^{1/2}\psi_{j\ell}\}\) is the training sequence vector transmitted by \(k\)th user in \(\ell\)th cell where \(\psi_{\ell}=[\psi_{\ell 1},\,\psi_{\ell 2},\,\ldots,\,\psi_{\ell k}]\) with \(\tau\times K\). Then the received signal at BS is given by  
\[Y_{j}=\sqrt{p_{u}\tau}\sum_{\ell=1}^{L}\psi_{\ell}G_{j\ell}+W_{j}. \tag{4}\]  
After receiving the signal, the BS estimates the propagation channel matrix \(G\) using linear estimation theory such as MMSE estimator [23]. Each base station transmits a vector of message-bearing  
Figure 1: Uplink pilot training sequences. symbols through a precoding matrix which depends on estimation channel matrix \(\hat{G}_{jj}\), then linearly precoded these symbols. After that, the BS will move the precoded signal vector to all users. The estimated channel matrix for \(M\times K\) propagation matrix \(G_{jj}\) between the \(M\) base station antennas of \(j\)th cell and the \(K\) terminals in the same \(j\)th cell is defined by \(\hat{G}_{jj}\) which is obtained according to the following expression:  
\[\hat{G}_{jj}=\sqrt{p_{p}}\sum_{\ell=1}^{L}G_{j\ell}+V_{j}, \tag{5}\]  
where \(V_{j}\) is the noise matrix with \(M\times K\) elements which are AWGN with zero mean mutually uncorrelated, and uncorrelated with the propagation matrices, \(P_{p}\) is the pilot signal to noise ratio, and since \(M\) goes to infinity, the effects of noise will be vanished, as in the present work which it is defined as the average pilot transmitted power. At the uplink communication phase, \(K\) users in each cell independently send data streams to their related base station. The base station at \(j\)th cell will receive \(M\times 1\) vectors including transmissions from all of the terminals in the \(L\) cells. Then the received signal vector at base station is defined by  
\[\overline{r}_{j}=\sqrt{p_{p}}\sum_{\ell=1}^{L}G_{j\ell}b_{\ell}+z_{j}, \tag{6}\]  
where \(b_{\ell}\) is the \(K\times 1\)vector of message-bearing symbols from the terminals of the \(\ell\)th cell, \(z_{j}\) is a vector of received noise whose components are AWGN with zero-mean, mutually uncorrelated and uncorrelated with the propagation matrices. The message-bearing signals are transmitted by the terminals and they are independent and distributed with zero-mean, unit-variance and complex Gaussian variables. The base station uses its estimated channel to perform linear detecting process such as MRC, ZF and MMSE detectors. By using the linear detector, the received signal is divided into streams by multiplying it with the linear detector matrix, which depends on the pseudo inverse of the propagation channel \(G\). With maximum ratio combining (MRC) detector, the base station processes its received signal by multiplying it with the conjugate-transpose of the estimated channel.