# Analysis and Evaluation of Performance Gains and Tradeoffs for Massive MIMO Systems
## 3 Massive MIMO Performance Gains
### Spatial Diversity Gain
#### 3.1.2 Uplink Side  
The \(M\times 1\) received vectors at the \(j\)th BS after applying the MRC detection at the BS is given by expression (7), then using expression (5) for channel estimation, so we have  
\[r_{j}=\left[\sqrt{p_{p}}\sum_{\ell=1}^{L}G_{j\ell}+V_{j1}\right]^{H}\left[ \sqrt{p_{u}}\sum_{\ell^{\prime}=1}^{L}G_{j\ell^{\prime}}b_{\ell^{\prime}}+z_{ j}\right]. \tag{23}\]  
Considering the properties of the correlation matrix for massive MIMO channel of the received signal vector \(r_{j}\), the diversity gain at the uplink side can be obtained in terms of correlation matrix where the correlation matrix is  
\[C_{ULL}=E\left[r_{j}r_{j}^{H}\right]. \tag{24}\]  
Then the diversity gain (\(G_{div-UL}\)) is determined under imperfect CSI in terms of diversity measurement (\(D_{div}\)) at the uplink channel side as  
\[G_{div-UL}=\mathrm{D}_{div}E\left(\left[r_{j}r_{j}^{H}\right]\right). \tag{25}\]  
For massive MIMO system, as \(M\) increases without limit, the effects of additive receiver noise and fast fading disappear, as well as intra-cell interference, then according to expressions (19) and (20) for channel orthogonality, the following is obtained  
Figure 3: Interference due to pilot contamination at the downlink channel.  
\[\begin{array}{rl}G_{div-UL}&=\mathrm{D}_{div}E\left(\left[r_{j}r_{j}^{H}\right] \right)\\ &=Mp_{p}p_{u}D_{jj}+Mp_{p}p_{u}\begin{array}{c}\sum\limits_{\ell=1}^{L}\\ \ell=1\\ \ell\
eq j\end{array}.\end{array} \tag{26}\]  
From expression (26), the first term represents the propagation channels between the users and BS in \(j\)th cell, while the second term represents propagation channels from terminals in other cells which use the same pilot sequences interfering with the transmission from the \(k\)th terminal in the \(j\)th cell to its own base station in \(j\)th cell as shown in Figure 4 below. Then, the final expression for diversity gain at uplink side is  
\[G_{div-UL}=D_{div}E\left(\left[r_{j}r_{j}^{H}\right]\right)=\frac{\left(trD_{ jj}D_{j\ell}\right)^{2}}{tr\left(D_{jj}^{2}D_{j\ell}^{2}\right)}. \tag{27}\]  
As a result, expressions (22) and (27) are given for diversity gain at downlink and uplink channels respectively, which are the same expression, but in fact they have different statistical characteristics, where at the downlink side the transmission from the base station in any cell to its \(K\) terminals suffering interference from transmissions of the base stations in other cells to their own \(K\) terminals. While at the uplink side, the transmission from the terminal in any cell to its base station is suffering the interferences from the transmissions of terminals in the other cells to their base stations. These interferences are caused due to the pilot contamination.  
At the downlink, BS uses the estimated channel to precode the transmit signals to users, while at the uplink side the signals transmitted from users can be decoded using the channel estimations. With massive MIMO systems, the number of base stations' antennas is unlimited, so that the number of cells that use the same band frequencies are increased which causing high time delay for the CSI between the uplink training sequence and downlink data transmission which increase the diversity gain loss [13].  
The total diversity gain is the product of the uplink and downlink diversity gains, it can be written as  
\[G_{div-TL}=G_{div-DL}\cdot G_{div-UL}. \tag{28}\]  
Figure 4: Interference due to the pilot contamination at uplink channel.  
This expression quantifies the degree of fluctuation in the received power signal and how much the uplink and the downlink side be affected by this fluctuation. The number of independent fading link offers by massive MIMO is equal to the number of transmit antennas and the number of received antennas, which is named the diversity order (\(\theta_{ord}\)). A massive MIMO channel with \(M\) antennas at the downlink side and \(K\) antennas at the receiving side has diversity order of  
\[\theta_{ord}=M\times K. \tag{29}\]