## 5 Performance with Hardware Impairments  
### Performance with Phase-Shifter Errors  
As shown in Figure 1, in RF beamforming, phase shifts are applied to each antenna element of the array in order to steer the beam towards a certain direction. In real implementations, the conformed beam is altered due to the phase errors introduced by the non-ideal phase shifters. In order to study the impact of this effect, the phase shifter error of the \(i\)-th phase shifter, denoted by \(\delta_{i}\), can be modeled as a uniformly-distributed random variable in the interval \([-\delta_{max},\delta_{max}]\)[27]:  
\[\delta_{i}\in[-\delta_{max},\delta_{max}],0\leq\delta_{max}\leq 180^{\circ}, \tag{6}\]  
where \(\delta_{max}\) is a parameter depending on the phase shifter implementation [14; 15] and the \(\delta_{i}\) variables are assumed independent among different phase shifters. Therefore, if we define \(\theta_{i}\) as the ideal phase shift to be applied by phase shifter \(i\), the actual phase shift applied by that phase shifter will be \(\theta_{i}^{\prime}=\theta_{i}+\delta_{i}\).  
This work focuses on phase shifter errors constant over the bandwidth of interest but variable with time. Note that phase shifter errors that are constant with time are mainly caused by manufacturing imperfections, and, thus, they could be estimated in the uplink and their effect could be compensated in the beamforming stage. Otherwise, phase shifter errors variable with time cause a random and unpredictable error that cannot be estimated and compensated. The reason for selecting the latter assumption is to emulate the effect of using different beams for the channel estimation and transmission phases due to the inclusion of different phase-shifter errors in each of them. This may happen in frequency division duplex (FDD) systems. As a result, when the phase-shifter error varies over time, the beam selected from the codebook during the scheduling phase can result in a suboptimum beam for the transmission phase. Note that this is a worst-case scenario for the evaluation of the impact of phase-shifter errors.  
Table 4 collects several performance results for an ABF scheme with \(P=4\) affected by phase shifter errors, for three different values of the \(\delta_{max}\) parameter in degrees: \(\delta_{max}=2^{\circ},6^{\circ}\) and \(10^{\circ}\). Note that, according to previous works [14; 15], the value of \(\delta_{max}=10^{\circ}\) is very unlikely in practice, and, thus, it is here included for a worst-case evaluation. In particular, average and 5 th percentile user throughput values have been obtained considering only the UEs in LoS, only the UEs in NLoS, or both types combined. Average cell throughput values are also included. In all cases, the performance loss with respect to the ideal case is also shown. The results reveal that the impact of phase shifter errors is very minor, and it only causes a maximum reduction of 3.5% in user throughput, particularly for LoS users with a pessimistic value of \(\delta_{max}=10^{\circ}\). Indeed, LoS UEs are more affected by this impairment  
Figure 9: Average cell throughput values for the ABF, MRT and EGT evaluated schemes.  
because they strongly depend on the accuracy of the direct beam. In addition, since the equivalent receive beams of LoS UEs are narrower (in meters) than those of NLoS UEs, generally located farther away from the BS, an error in the pointing direction is more critical when serving LoS UEs.