# Analysis and Evaluation of Performance Gains and Tradeoffs for Massive MIMO Systems
## 3 Massive MIMO Performance Gains
### Array Gain
#### 3.2.1 Array Gain under Perfect CSI  
With perfect CSI, the propagation channel \(G\) is known by the base station. At the downlink side, the received signal for \(k\) terminals in \(\ell\)th cell from their base station is given by expression (8), here it is assumed that each BS has its transmit power to their \(K\) terminals, which will be defined as [30]  
\[Q_{\ell}=\sum_{j=1}^{L}\sqrt{p_{dj}}G_{jt}^{T}O_{j}s_{j}+n_{d\ell}, \tag{31}\]  
Let \(U_{j}=\sqrt{p_{dj}}O_{j}s_{j}\) is the pre-coded transmit vector to \(K\) terminals in \(\ell\)th cell, where \(p_{dj}\) is the diagonal matrix and consists of the downlink transmit power for \(K\) terminals of BS, where \(P_{dj}\)=_diag_(\(p_{dj}\), \(p_{dj}\),..., \(p_{dKj}\)) and the propagation channel matrix \(G\) is being as in expression (2). Then the total transmit power at \(j\)th cell is  
\[P_{Tj}=E\left[\begin{array}{cc}1&1&U_{j}&1^{2}\end{array}\right]=tr(O_{j}^{ H}O_{j}P_{dj})=tr(P_{dj}), \tag{32}\]For massive MIMO system, as \(M\) increases without limit, the effects of additive receiver noise and fast fading disappear. With perfect CSI, BS knows the propagation channel matrix \(G\), the effect of the interference between users in the same cell is disappeared. However, still there is the interference among the users from other cells, and with perfect CSI this effect also can be eliminated supposing the system behaves as a single cell [31]. According to expressions (19) and (20), the channel vectors between the users and the BS become orthogonal, as \(M\) grows unlimited then the received signal power under perfect CSI is defined as  
\[\begin{array}{ll}P_{re}^{Massive\;MIMO}&=E\left[||\;Q_{\ell}||^{2}|G\right]=M \left(\sum\limits_{j=1}^{L}P_{dj}D_{j\ell}\right)\\ &=M\left(P_{d\ell}D_{\ell\ell}+\sum\limits_{j\
eq\ell}^{L}P_{dj}D_{j\ell} \right)\end{array}, \tag{33}\]  
where \(D_{j\ell}\) is a \(K\times K\) diagonal matrix whose diagonal elements are \([D_{j\ell}]_{kk}=\beta_{jk\ell}\). The first term in expression (33) represents the propagation channels between the users and the base station in \(\ell\)th cell, while the second term represents the propagation channels from the base stations in other cells to their users interfering with propagation channels between the users and the base station in \(\ell\)th cell.  
Finally, as \(M{\rightarrow}\infty\) (the case with massive MIMO) under perfect CSI, the effect of interference can be eliminated and the above expression will be  
\[P_{re}^{Massive\;MIMO}=M\,tr(P_{d\ell}D_{\ell\ell}). \tag{34}\]  
This expression is the received power in \(\ell\)th cell for massive MIMO regime. According to array gain definition, which is described in expression (30), the received power for MIMO system is given as  
\[P_{re}^{MIMO}=tr(P_{\overline{d}}\Lambda), \tag{35}\]  
Where \(P_{\overline{d}}\) is the diagonal matrix of the transmit power assigned to each data stream in MIMO and \(\Lambda\) is the diagonal matrix for the Eigen value of the propagation matrix \(H\) in MIMO system. Then the array gain can be given according to expression (30) with respect to massive MIMO under perfect CSI as  
\[G_{Array}=\frac{P_{re}^{Massive\;MIMO}}{P_{re}^{MIMO}}=\frac{M\,tr(P_{d\ell}D_{ \ell\ell})}{tr(P_{\overline{d}}\Lambda)}. \tag{36}\]  
Massive MIMO technology enhances the system performance through array gain by increasing the number of antennas at base station with an unlimited number of antennas \(M\).