## 4 Performance with Practical Limitations  
### Performance with Outdated Channel State Information  
A very usual ideal assumption for simulation is the perfect knowledge of the instantaneous (per-subframe) channel at every BS, which allows the BS precoder to be adapted to the actual channel. However, practical systems must often deal with the problem of having imperfect CSI. In frequency division duplex (FDD) systems, the downlink channel is estimated at the UE and fed back to the BS. These estimation and feedback stages can, in practice, impair the CSI with quantization noise, due to the usually limited resources on the feedback channel. In addition, the presence of noise in the channel during the transmission of training sequences for CSI estimation contributes to an imperfect channel estimation. The latter effect is also present in the channel estimation stage of time division duplex (TDD) systems. Another important practical limitation in both FDD and TDD systems is the unavoidable delay between the instant the channel is estimated and the instant the channel is used in designing the precoder for data transmission, which, in turn, can further degrade the system performance. In this work, we will focus on the effect of outdated CSI over the performance of ABF with \(P=4\), DP NFS and DP FS. To this end, it is here assumed that each BS receives an update of the channel coefficients only every certain period \(T\), higher than the subframe duration \(T_{s}=1\) ms. It is worth noting that we consider \(T\) values that cause changes only in the small scale fading components of the channel, i.e., the large scale fading components like path loss, shadowing and path angles of arrival/departure remain constant during \(T\). For a better illustration of the small scale channel variation during the period \(T\), the results will be presented as a function of the product \(f_{D}\Delta T\), where \(f_{D}=vf/c\) is the maximum Doppler shift at carrier frequency \(f\) with user speed equal to \(v\), and \(\Delta T=T-T_{s}\) is the CSI delay with respect to the subframe duration. Figure 6 shows the CDF of user throughput for the ABF and DP FS schemes, considering different values of the channel update period, in particular \(T=1,~{}50,~{}150\) and \(500\) ms, which lead to \(f_{D}\Delta T=0\), \(3.8\), \(11.6\) and \(38.8\), respectively. Note that \(f_{D}\Delta T=0\) corresponds to the ideal case, where the channel is supposed to be updated every subframe. It can be first observed that the ABF scheme is more robust to outdated CSI, as it presents a nearly negligible performance degradation with \(f_{D}\Delta T\). Such degradation of ABF is also much lower than for the DP FS approach, which results in a reduced performance gap between ABF and DP as \(T\) increases. The lower robustness to outdated CSI of DP is well justified by the fact that this scheme has a strong dependence on the small scale channel fading, whereas the ABF mainly relies on the large scale components, since it typically tends to point the beamformer towards the angle of departure of the strongest path. In fact, while in a practical mmW system it is expected that the channel is tracked every few milliseconds or few tens of milliseconds, the higher values of \(T\) considered here, e.g., \(T=150,500\) ms, are useful to understand the robustness of the different schemes when there is basically no knowledge of the small scale fading, and, thus, beamformers are designed mainly based on the large scale fading components of the channel. In such a scenario, ABF, by simply selecting the beam that points toward the direction that maximizes the