# Analysis and Evaluation of Performance Gains and Tradeoffs for Massive MIMO Systems
## 4 Performance Tradeoffs Analysis
### Spatial Diversity Gain vs. Spatial Multiplexing Gain Tradeoff  
Two main performance metrics are associated with any communication system: the data transmission rate and the outage probability (\(P_{out}\)). The data transmission rate (\(R_{rate}\)) is defined as the data rate forwarded per unit bandwidth, while the information outage probability (\(P_{out}\)) is defined as the probability that an instantaneous channel capacity is less than the transmitted encoding rate. (\(P_{out}\)) is a function of a signal to interference plus noise ratio (Y1) and data rate (\(R_{rate}\)). With a fixed data rate an increase in (Y1) will result in decreasing (\(P_{out}\)), while with a fixed (\(P_{out}\)) an increase in (Y1) will lead to increase the data rate. Many researches had discussed the tradeoff between diversity gain and multiplexing gain, the most important research was performed by [PERSON] and [PERSON] [15], where they worked with point-to-point MIMO system in rich scattering Rayleigh fading channel. They derived the optimal tradeoff curve and evaluated the performance of MIMO channel assuming perfect CSI. The same authors extended their work with MU-MIMO and provided a complete picture on the tradeoff between the three types of gains (i.e., diversity gain, multiplexing gain and multiple-access gain) [39]. A ray tracking propagation model is utilized to look up the tradeoff between diversity and multiplexing gain in massive MIMO as in [40]. Another discussion is for the tradeoff between diversity and multiplexing gain under imperfect CSI and no CSI in point to point MIMO system as in [41].With multi-cell massive MIMO system under imperfect CSI, the average diversity order (\(\theta_{avg}\)) and average multiplexing gain (\(G_{M\Omega X}^{avg. }\)) can be given according to the following definitions:  
**Definition 1:**_A block with \(T_{coh}\) be the length (time-bandwidth product) of the coherence interval, \(\tau\) is the number of symbols used for pilots (\(\tau\geq K\)) and transmission rate (\(R_{rate}\)) can achieve average multiplexing gain if_  
\[G_{M\Omega X}^{avg.}=\left(\frac{T_{coh}}{T_{coh}-\tau}\right)\lim_{Y\to \infty}\frac{E\left(R_{rate}|\hat{G}\right)}{\log Y}. \tag{71}\]  
**Definition 2:**_A block with \(T_{coh}\) be the length (time-bandwidth product) of the coherence interval, \(\tau\) be the number of symbols used for pilots (\(\tau\geq K\)) and package outage probability can achieve average diversity order \(\theta_{avg}\) if_  
\[\theta_{avg}=-\lim_{Y\to\infty}\frac{\log E\left(P_{out}|\hat{G}\right)}{\log Y}. \tag{72}\]  
_Assuming the average packet outage probability equals the average packet error probability when random coding is applied. The outage probability is calculated according to the following definition._  
**Definition 3:**_Mutual information is the characterization of channel capacity between the input and the output of the channel in term of input distributed and channel realization, the channel capacity equation with unity power constraint is given as_  
\[I(x:y|H)=\text{logdet}(1+\frac{\text{SNR}}{m}\text{HH}^{H}), \tag{73}\]  
_where \(y=Hx+n\) is a basic expression for MIMO system and \(I\) (\(x:y|H\)) is the mutual information between \(X\) and \(Y\).