# Analysis and Evaluation of Performance Gains and Tradeoffs for Massive MIMO Systems
## 4 Performance Tradeoffs Analysis
### Array 2 Gain vs. Spatial Multiplexing Gain Tradeoff  
A system consists of a large number of antennas can improve both energy efficiency and spectral efficiency significantly. However, increasing the achievable rate is associated with increasing the power which includes the radiated power and the circuit power and hence decreasing the energy efficiency. Therefore, to maximize one gain the other must keep with limited region. Energy efficiency is defined as the achievable sum rate divided by the power expended measured in Joules per used channel, while the spectral efficiency is defined as the achievable sum rate per unit of bandwidth. In [36], the authors designed a macro-cellular large scale antenna system in order to maximize the total forward link energy efficiency, however the power consumed at \(\ell\)th cell is expressed as  
\[P_{consumed}=P_{RF\ell}\left(1-\frac{\tau}{T_{coh}}\right)/2+P_{cct\ell}, \tag{57}\]  
where \(P_{RF\ell}\) is the RF power of base station in \(\ell\)th cell and it is weighted by \((1-\frac{\tau}{T_{coh}})/2\) because downlink activities take half of the coherence interval. RF power can be defined in terms of total transmit power with RF efficiency (\(\epsilon_{RF}\)), \(R_{RF\ell}=\frac{1}{\epsilon_{RF}}p_{d\ell}\), where \(p_{d\ell}=\sum\limits_{k=1}^{K}p_{d\ell k}\) and \(P_{cct\ell}=Mp_{cct\ell}\) is the internal circuit power which is expended on massive MIMO system due to critical computations and other activities such as analog electronics and A/D and D/A conversions, \(P_{cct\ell}\) is internal circuit power of each antenna. To calculate the energy efficiency (\(\eta_{e,\ell}\)) at \(\ell\)th cell downlink side and using expressions (52) and (54) for channel capacity then we have  
\[\eta_{e,\ell}=\frac{\sum\limits_{k=1}^{K}R_{\ell k}}{P_{consumed}}, \tag{58}\]\[ \begin{split}\eta_{e,\ell}&=\frac{\left(1-\frac{\tau}{t_{coh}} \right)\sum\limits_{k=1}^{K}\log_{2}(1+Y_{\ell k})}{\frac{1}{t_{RF}}p_{d\ell} \left(1-\frac{\tau}{t_{coh}}\right)/2+P_{cct}}\\ \eta_{e,\ell}&=\frac{\left(1-\frac{\tau}{t_{coh}} \right)\sum\limits_{k=1}^{K}\log_{2}\left(1+\frac{Mf_{MRRTpp}g_{d\ell}\beta_{ d\ell}^{2}}{M_{MRRTpp}\frac{l}{j\
eq\ell}p_{d\ell}\beta_{j\ell}^{2}+\sigma_{n}^{2}} \right)}{\frac{1}{t_{RF}^{2}}p_{d\ell}\left(1-\frac{\tau}{t_{coh}}\right)/2+P_ {cct}}\end{split} . \tag{59}\]  
Using the expression of SINR in expression (52), then the spectral efficiency for \(\ell\)th cell (\(\eta_{\delta,\ell}\)) in downlink massive MIMO system is written as  
\[\eta_{s,\ell}=\left(1-\frac{\tau}{T_{coh}}\right)\sum\limits_{k=1}^{K}\log_{2 }\left(1+\frac{Mf_{MRRT}p_{p}p_{d\ell k}\beta_{d\ell}^{2}}{Mf_{MRRT}p_{p}\sum \limits_{j\
eq\ell}^{L}p_{djk}\beta_{j\ell}^{2}+\sigma_{n}^{2}}\right). \tag{60}\]  
Assuming all users are sharing the same constant bandwidth, so the bandwidth is ignored in this work. To study the tradeoff between energy efficiency and spectral efficiency, the expression mentioned in [16] is considered, where the authors discussed the tradeoff under two cases: perfect CSI and no CSI with single user and multi-user as follows  
\[\frac{d\eta_{e,\ell}}{d\eta_{s,\ell}}=\frac{\sum\limits_{k=1}^{K}\frac{\partial \eta_{e,\ell}}{\partial p_{d\ell}}dp_{d\ell k}}{\sum\limits_{k=1}^{K}\frac{ \partial\eta_{e,\ell}}{\partial p_{d\ell}}dp_{d\ell k}}. \tag{61}\]  
In the present work, the tradeoff in multi-cell massive MIMO system is discussed under imperfect CSI taking into account the effect of coherent interference, i.e., pilot contamination effect. Each performance gain needs different transmit power to reach the maximum value, so the partial derivative is taken with respect to transmit power for each user in the cell.