# A Survey on Massive MIMO Systems in Presence of Channel and Hardware Impairments
## 6 Time Varying Channels
### Time Varying Frequency Selective Channels  
In this subsection we consider multi-path channels were the channel coefficients change within a block due to Doppler. We name these channels doubly dispersive since the channel has time dispersion due to its multi-path nature and frequency dispersion due to its time variation. In [83], a single-carrier massive MIMO system in doubly dispersive channel has been considered. It is assumed that this system is operating in millimeter wave band and TDD mode. An adaptive channel estimation technique based on reduced rank Kalman filtering has been proposed to cope with channel variation effects. To reduce the channel estimation complexity a pre-beamformer, which considers the channel variation in time, was proposed. In massive MIMO systems, pilots are used for channel estimation. Since the channel changes in time, another pilot block has to be sent again after a while. When the channel time variations are high, the time distance between two consecutive pilot blocks reduces which leads to low spectral efficiency. In [84], the authors investigated the possibility of increasing the time distance between pilot blocks with the aid of channel prediction. Therefore, a channel prediction technique based on Kalman filter and autoregressive moving average (ARMA) channel time variation model was proposed. Simulation analysis shows that for high Doppler frequencies, the use of channel prediction increases the rate compared to the case where we have no channel prediction. However, for low Doppler frequencies, channel prediction improves the performance only when a long time has passed from the last channel estimation. The downlink performance of SC-FDP and OFDM massive MIMO systems in doubly dispersive channels were analyzed in [85]. The sum-rates of both systems were derived analytically in doubly dispersive channels, considering the effect of channel aging and channel estimation error due to Doppler drifts. The analytical and simulation results indicate that SC-FDP is more robust to channel double dispersiveness and channel aging effects than OFDM. In fact, when we have no channel estimation errors due to Doppler drifts, the sum-rate of SC-FDP massive MIMO, unlike OFDM massive MIMO, increases unlimitedly as the number of BS antennas tends to infinity. However, when we have channel estimation errors the sum-rate of both systems saturates when the number of BS antennas gets large with SC-FDP having a slightly higher sum-rate. Figure 9 shows the sum-rate in four scenarios versus the number of BS antennas. In this figure, scenarios I, II, and III refer to the case where we have double dispersive channel with perfect CSI, double dispersive channel with channel aging, double dispersive channel with channel aging, and channel estimation error. In scenario III.A the channel estimation error is modeled with additive noise and in scenario III.B the effect of Doppler shift on channel estimation has been considered. A summary of channel aging and channel time variation effects in massive MIMO systems is presented in Table 2.  
Figure 9: Sum-rate versus the number of BS antennas.