# Analysis and Evaluation of Performance Gains and Tradeoffs for Massive MIMO Systems
## 3 Massive MIMO Performance Gains  
In this work, the performance of massive MIMO system will be measured under two main considerations: perfect and imperfect channel state information (CSI). After that, we will discuss how to measure the massive MIMO channel performance in terms of spatial multiplexing, spatial diversity and array gains.  
Perfect channel state information means the base station has perfect knowledge about the propagation channel matrix \(G\). The base station can use any linear detector to perform well i.e., (MRC, ZF... ) as (\(M>>K\)). Theoretically, the singular value decomposition (SVD) of the channel matrix \(G\) would be taken in precoding and detecting process with removing the two unitary matrices, hence one data stream per singular value can be transmitted without creating any interference, i.e., the effect of non-orthogonal channel vectors from other cells. Channel capacity could be achieved in massive MIMO without pilot overhead effect on time coherence interval [25].  
Practically, the channel matrix \(G\) has to be estimated at BS during the uplink training phase. If \(T_{coh}\) is the length of the coherence interval and \(\tau\) be the number of symbols used for pilot sequences, all users concurrently transmit reciprocal orthogonal pilot sequences of length \(\tau\) symbols. The channel state information (CSI) at the base station is a core component when trying to maximize network throughput. The only factor that remains limiting performance in massive MIMO is the inter-cell interference since when it is associated with the finite time existing to send pilot sequences, it makes the appreciated CSI at one BS \"contaminated\" by the CSI of users in adjacent cells, which is called pilot contamination effect. This phenomenon results from unavoidable re-use of reverse link pilot sequences by terminals in different cells. Clearly, in multi-cell massive MIMO system, as \(M\) tends to infinity, the downlink SINR ( Y\({}_{\ell k}\)) of \(k\)th user at \(\ell\)th cell is proportional according to the following expression  
\[\mathrm{Y}_{\ell k}\propto\frac{p_{dk\ell}\beta_{\ell k\ell}^{2}}{\sum\limits _{\begin{array}{c}j=1\\ j\
eq\ell\end{array}}^{L}p_{dkj}\beta_{jk\ell}^{2}}, \tag{10}\]  
where \(p_{dk}\) is the downlink transmit power delivered by BS to \(k\)th user in its cell. \(\beta_{\ell\ell}\) and \(\beta_{jk\ell}\) are the direct and the cross gains respectively, which represent the effect of pilot contamination [12].  
However, the mobility of users is closely related to the quality of the channel state information (CSI), which can be acquired at the BSs, high mobility implies low-quality of CSI. With multi-cell massive MIMO, base stations transmit the forward data symbols during the coherent time to their terminals depending on the uplink training sequences for channel estimations, but actually imperfect  
Figure 2: Explanation of inter-cell interference due to pilot contamination at downlink transmission.  
channel information may be obtained because of imperfect uplink training sequences channel estimation, delay in the acquisition protocols and user mobility. Because the coherence time of the MIMO channel inversely proportional with high user mobility, large channel imperfections will be as terminal mobility increases [19]. The channel model is suggested to represent the imperfect CSI by modeling the fading channel as  
\[h_{\ell,k}^{m}=\sqrt{X_{\ell,k}^{m}}\left[\sqrt{\left(1-\left(\tau_{\ell,k}^{m} \right)^{2}\right)}h_{\ell,k}^{m}+\tau_{\ell}^{m}h_{\ell,k}^{\sim m}\right], \tag{11}\]  
where \(h_{\ell,k}^{m}\) is the channel between the BS at \(m\)th cell and \(k\)th terminal in \(\ell\)th cell, \(h_{\ell,k}^{\sim m}\) is the channel estimation error. In this formula, the accuracy of channel request can be setting between the terminals in \(\ell\)th cell and BS in cell \(m\)th by setting \(\tau_{\ell}^{m}\in[0,1]\), which is the training sequences time interval and a small number on \(\tau\) means low mobility and vice versa [26].