# Analysis and Evaluation of Performance Gains and Tradeoffs for Massive MIMO Systems
## 1 Introduction
At downlink side, the M-base station antennas transmit \(M\)-orthogonal pilot sequences to \(K\)-users; each user will estimate the channel then feeds back its channel estimates to the base station. Therefore, the complexity scale for the channel estimation process needs at least \(M+K\) channel uses in the uplink and \(M\) channel uses in the downlink [7]. Three main performance gains can be achieved under massive MIMO system: array gain, spatial diversity gain and spatial multiplexing gain. An increase in the energy efficiency means that the received SNR is increased from the coherent combining of the signals toward where users are located. Massive MIMO can improve the coverage area and resistance to noise through the use of array gain. An improvement in the quality and reliability of a wireless link means that the use of multiple antennas at base station is providing a receiver with multiple copies of the transmitted signal, and as a result the probability of one of the copies is not experiencing a deep fade increases, so the massive MIMO can provide a robust wireless link through spatial diversity gain. A linear increasing in data rate means that with large number of antennas many tens of terminals can be served simultaneously within the same bandwidth using the same transmit power, so the overall spectral efficiency can be ten times higher than in conventional MIMO [8], i.e., the massive MIMO can achieve high spectral efficiency through spatial multiplexing gain. Massive MIMO can achieve these performance gains individually by proper signal processing and channel coding. All these gains depend on statistical properties of massive MIMO channel and tradeoffs among these performance gains will appear significantly. In massive MIMO system, the propagation losses due to coherent beamforming/combining are mitigated by the large number of antennas at the base station (i.e., a large array gain) [9]. With a large array gain, the transmitted power is reduced which leads to save terminals batteries on the uplink and reduces the power consumption by BS [10]. High spectral efficiencies are achieved per cell by scheduling many terminals for simultaneous transmission through exploiting the spatial multiplexing gain while the spectral efficiency per terminal might only be 1-4 bit/s/Hz [11]. Massive MIMO depends on reciprocity concept of time duplex division (TDD) system to create the precoding matrix for downlink transmission where the reverse channel is used to obtain the channel state information (CSI) of the forward channel, i.e., terminals send orthogonal pilot sequences in the uplink training. In multi-cell massive MIMO system, the same pilot sequences are re-used where the base station in one cell is polluted by users in other cells. This non-orthogonal nature is named pilot contamination [12], which causes CSI delay and channel estimation error between reverse channel estimation and forward channel data transmission which degrades the system performance gains significantly. CSI delay and channel estimation error cause diversity gain loss and array gain loss respectively [13; 14]. The authors of [8] derived expressions to quantify the amount of antenna gain, diversity gain and multiplexing gain for MIMO system single cell under perfect and imperfect CSI. In the present work, new expressions are derived to quantify the amount of array gain, spatial multiplexing gain and spatial diversity gain under perfect and imperfect CSI of multi-cell massive MIMO system in order to study the effect of interference from non-coherent channels in other cells. The authors of [10] derived the power scaling law in a single cell and multi-cell massive MIMO system, then they derived the lower bounds of achievable rates at the uplink side under perfect and imperfect CSI and they discussed the tradeoff between energy efficiency and spectral efficiency under the effect of pilot contamination.