# Analysis and Evaluation of Performance Gains and Tradeoffs for Massive MIMO Systems
## 3 Massive MIMO Performance Gains
### Spatial Multiplexing Gain
#### 3.3.2 Multiplexing Gain under Imperfect CSI  
In order to use the advantages that massive MU-MIMO can offer, accurate channel state information (CSI) is required at the BS. The impact of up-link training is significantly appearing in multi-cell system due to the same band of frequencies, which is shared by a multiplicity of cells. If each cell is serving the maximum number of terminals, then the pilot signals received by a base station are contaminated by pilots transmitted by terminals in other cells. Therefore, the user in the \(j\)th cell for example receives signal from its base station and from other base stations. If \(q_{jk}\) is the received signal for \(k\)th user, then  
\[q_{jk}=\sqrt{p_{dj}\beta_{jkj}}h_{jkj}o_{j}s_{jk}+\sum\limits_{\ell\
eq j} \sqrt{p_{d\ell}\beta_{k\ell j}}\,h_{\ell kj}o_{\ell}s_{\ell k}+n_{j}. \tag{48}\]  
The first term in the above expression represents the desired received signal while the second term reflects the effect of inter-cell interference. Pilot contamination affects system performance parameters such as achievable sum rate or spectral efficiency, channel capacity, received signal to interference plus noise and so on. In [34], the authors proposed a new detection method and derived a formula for signal to interference plus noise ratio (SINR) and spectral efficiency (SE), then calculated optimal number of terminals to achieve better performance is in term of achievable rate (\(R\)). Many researches discuss how to mitigate the effect of polluted signals by modifying either the precoding methods which use at the forward channel or the discovery methods which is used at the reverse channel [35]. Now, let consider the received signal vector which is described previously in expression (31), and substitute expressions (5) and (9) in expression (31), the following will be obtained,  
\[Q_{\ell}=f_{MRT}\sum_{j=1}^{L}\sqrt{p_{dj}}Q_{j\ell}^{T}\left[\sqrt{p_{p}}\sum _{\ell^{\prime}=1}^{L}G_{j\ell^{\prime}}+V_{j1}\right]^{*}s_{j}+n_{d\ell}. \tag{49}\]  
Since the number of base station's antennas increases unlimitedly, the effect of intra-cell interference will vanish, while the effect of inter-cell interference still affects system performance. Using expressions (19) and (20) which is the key property of large MIMO system, then expression (49) will become  
\[Q_{\ell}=Mf_{MRT}\sqrt{p_{p}}\sqrt{p_{d\ell}}D_{\ell\ell}s_{\ell}+Mf_{MRT} \sqrt{p_{p}}\sum_{j\
eq\ell}^{L}\sqrt{p_{dj}}D_{j\ell}s_{j}+\overline{n}_{d \ell}, \tag{50}\]  
where \(\overline{n}_{d\ell}\) is uncorrelated received AWGN noise with variance \(\sigma^{2}{}_{n}\). The received signal for \(k\)th user in \(\ell\)th cell is given by  
\[Q_{\ell k}=Mf_{MRT}\sqrt{p_{p}}\sqrt{p_{d\ell}}\beta_{\ell k\ell}s_{\ell k}+ Mf_{MRT}\sqrt{p_{p}}\sum_{j\
eq\ell}^{L}\sqrt{p_{djk}}\beta_{jk\ell}s_{jk}+ \overline{n}_{d\ell}. \tag{51}\]  
The first term is for the \(k\)th user in \(\ell\)th cell receives the transmitted data from its BS in cell, while the second term is for the effect of pilot contamination on the received data to \(k\)th cell user in \(\ell\)th cell, which interferes with the sending of data from other BSs to their users in other cells. As a result, the received SINR (Y\({}_{\ell k}\)) for \(k\)th user in \(\ell\)th cell is given by  
\[\mathrm{Y}_{\ell k}=\frac{Mf_{MRT}p_{p}p_{d\ell k}\beta_{\ell k\ell}^{2}}{Mf_ {MRT}p_{p}\sum_{j\
eq\ell}^{L}p_{djk}\beta_{jk\ell}^{2}+\sigma_{n}^{2}}. \tag{52}\]  
Let \(T_{coh}\) be the length of coherence interval and \(\tau\) be the number of symbols used for pilots, the achievable rate of the \(k\)th user within the multi-cell system and the effect of interference due to pilot contamination is given by  
\[R_{\ell k}=\left(\frac{T_{coh}-\tau}{T_{coh}}\right)\log_{2}(1+\mathrm{Y}_{ \ell k}). \tag{53}\]  
The term (\(T_{coh}-\tau/T_{coh}\))represents the pilot overhead, which is the ratio of the time needed in sending data to the coherence interval.