# A Survey on Massive MIMO Systems in Presence of Channel and Hardware Impairments
## 5 Hardware Impairments
### Carrier Frequency Offset (CFO)  
Using inexpensive local oscillators increases the probability of oscillator instability and causes carrier frequency offset, which are due to the mismatch between the oscillator of the transmitter and the receiver. The effect of CFO is modeled as a phase rotation in the received time domain signal or a frequency shift in the spectrum of the received frequency domain signal [41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51]. In massive MIMO systems, since we have multiple users we also have multiple CFOs to estimate and compensate. In centralized massive MIMO systems, since the BS antennas use a common oscillator, the number of CFOs that have to be estimated is equal to the number of users. However, in the distributed massive MIMO system each BS antenna has a separate oscillator which increases the number of CFOs to be estimated. So, the problem of estimating CFO is more severe in distributed massive MIMO systems [41]. Previous CFO estimation techniques which were proposed for conventional systems are not suitable for massive MIMO systems, since their complexity increases significantly with the number of BS antennas [42]. Therefore, studying the effect of CFO on the performance of massive MIMO systems and proposing low complexity algorithms for CFO estimation and compensation in these systems is vital. In [41] the authors have investigated the effect of CFO on the sum-rate of centralized and distributed massive MIMO system in the uplink. In this paper, it is assumed that the channel is flat fading and perfect CSI is available. The results indicate that even with perfect CSI, if we do not have a good CFO estimation then the sum-rate of the system will degrade significantly, specially in the distributed case. The simulation based studies in [43] also confirm that in OFDM massive MIMO systems if the time and frequency offset are not estimated well, the performance of the system degrades significantly. In [42; 44] the effect of residual CFO after CFO compensation on the sum-rate of massive MIMO systems in the uplink has been studied. In [42] the studies were done on flat fading channels with MF and ZF detectors and in [44] the studies were done on frequency selective channels with MF detector. It was concluded that with residual CFO, by increasing the number of antennas at the BS we can achieve an array gain similar to the no CFO case. In [45] the authors derived the sum-rate of OFDM and SC-FDP massive MIMO systems in the downlink and in presence of CFO. The sum-rates were analytically derived for MF and ZF precoders for both systems. The results show that OFDM massive MIMO system is more sensitive to CFO and its sum-rate is bounded even when the number of BS antennas tends to infinity while that of SC-FDP increases unlimitedly with the number of BS antennas. This is due to the fact that the interference caused by the CFO in the OFDM massive MIMO does not vanish as \(M\) increases. In [46], the effect of channel knowledge at the user side on the sum-rate of SC-FDP massive MIMO system affected by CFO has been investigated. The sum-rates were derived for three different channel knowledge scenarios. In the first and the second scenario, the users have knowledge of the expectation of the total effective channel including the effect of CFO in time and frequency domain, respectively. In the third scenario the users only know the expectation of the effective channel without considering the effect of CFO. It was shown that in the first scenario, the effect of CFO can be completely compensated and the sum-rate is identical to the sum-rate of a fully synchronized system. It is also showed that the sum-rate in the second scenario is higher than the sum-rate in the third scenario when the number of BS antennas is finite. But, when the number of BS antennas tends to infinity the sum-rate in the second scenario is bounded and in the third scenario it tends to infinity. In summary, it can be concluded that CFO can have considerable effect on the performance of massive MIMO systems.