# Analysis and Evaluation of Performance Gains and Tradeoffs for Massive MIMO Systems
## 3 Massive MIMO Performance Gains
### Spatial Diversity Gain
#### 3.1.1 Downlink Side  
Let \(X\) be defined as the effective channel matrix at the transmit side, which includes the propagation channel matrix \(G_{j\!j}\) and the precoding matrix \(O_{j}\). Here \(G_{j\ell}\) is the propagation channel matrix between the base stations in other cells and the \(k\)th terminals in \(\ell\)th cell where they are served by their BS, i.e., pilot contamination effects. Then, from Equation (8) the effective channel matrix can be written as  
\[X=\sqrt{p_{d}}\sum_{j=1}^{L}G_{j\ell}^{T}O_{j}. \tag{16}\]  
Now, using expressions (9), expression (16) can be rewritten as below  
\[X=\sqrt{p_{d}}\sum_{j=1}^{L}G_{j\ell}^{T}f_{MRT}\left[\sqrt{p_{d}}\sum_{\ell^{ \prime}=1}^{L}G_{j\ell^{\prime}}+V_{j1}\right]^{*}. \tag{17}\]  
Diversity gain (\(G_{div-DL}\)) measured under imperfect CSI in terms of diversity measurement (\(D_{div}\)) at the downlink side is given by  
\[G_{div-DL}=\mathrm{D}_{div}\left(E[X^{H}X]\right)=\frac{\left(tr\,C_{DL}\right) ^{2}}{tr(C_{DL}^{2})}. \tag{18}\]  
As \(M\) grows unlimited, only the products of identical quantities remain significant, i.e., the propagation matrices which appear in both of the bracketed expressions, and if the BS has a large number of antennas and the channels are i.i.d fast flat Rayleigh fading channels, then the channel vectors between the users and the BS become orthogonal. Massive MIMO channel vectors are regulated by the law of orthogonality, which is clarified in Theorem 1 below.  
**Theorem 1:**_Let \(W=[w_{1},w_{2},\ldots,w_{M}]^{T}\) and \(A=[a_{1},a_{2},\ldots,a_{M}]^{T}\) are vectors whose elements are independent, identically distributed (i.i.d) random variables with zero mean and variance \(E[\,|\,w_{i}\,|^{2}]=\sigma^{2}w\) and \(E[\,|\,a_{i}\,|^{2}]=\sigma^{2}a\) for i=1, 2, \(\ldots\), \(M\) respectively. As a result from the law of large numbers, then the following will be obtained:_  
\[\frac{1}{M}W^{H}W\rightarrow\sigma_{w}^{2}\text{ and }\frac{1}{M}W^{H}A \to 0\text{ as }M\rightarrow\infty\]  
Now, if the conditions of the above theorem are satisfied for such as any vectors \(W\) and \(A\) are being any two distinct columns of \(G\) propagation matrix, then  
\[\frac{1}{M}G_{j\ell 1}^{H}G_{j\ell 2}=D_{j\ell 1}^{1/2}D_{j\ell 2}^{1/2}\left( \frac{H_{j\ell 1}^{H}H_{j\ell 2}}{M}\right)\approx D, \tag{19}\]  
where,  
\[\frac{1}{M}H_{j\ell 1}^{H}H_{j\ell 2}=I_{K}\delta_{\ell 1\ell 2}\text{ \ as \ }M \rightarrow\infty\text{,} \tag{20}\]  
where \(I_{K}\) is \(K\times K\) identity matrix. Therefore, favorable propagation is attained with i.i.d random variables with zero mean and (\(M>>K\)). Hence, the diversity gain at downlink side can be written as  
\[\begin{array}{rl}G_{div-DL}&=\mathrm{D}_{div}\left(E[X^{H}X]\right)\\ &=Mf_{MRT}p_{p}p_{d}D_{\ell\ell}+Mf_{MRT}p_{p}p_{d}\begin{array}{l}L\\ \sum\\ j=1\\ j\
eq\ell\end{array}\text{.}\end{array} \tag{21}\]  
From expression (21), the first term represents the propagation channels between the users and BS in \(\ell\)th cell, while the second term represents propagation channels from base stations in other cellsdirected to their own \(K\) terminals interfering with the propagation channels from the base station in the \(\ell\)th cell to its \(K\) terminals. This interference due to the effect of pilot contamination is shown in Figure 3 below. With keeping the originality for \(M\) to grow with unlimited number, the final expression for diversity gain at downlink side is  
\[G_{div-DL}=\mathrm{D}_{div}\left(E[X^{H}X]\right)=\frac{\left(trD_{\ell\ell}D_{ j\ell}\right)^{2}}{tr\left(D_{\ell\ell}^{2}D_{j\ell}^{2}\right)}. \tag{22}\]