## 4 Performance with Practical Limitations
### Performance with PAPC  
In practical multi-antenna systems, each element of the antenna array may be powered by its own amplifier and, thus, is limited by the linearity of that amplifier. For that reason, precoding schemes should apply a more restrictive assumption than the sum power constraint, that is, a per-antenna power constraint. In order to study the effect of limiting the maximum available power per antenna in the DP NFS and DP FS schemes, we consider, in this section, a second digital precoding alternative that takes into account this practical limitation: the Equal-Gain Transmission (EGT) scheme with per-antenna power constraints [19]. EGT ensures the allocation of the same amount of power per  
Figure 6: CDF of user throughput for the ABF with \(P=4\) and digital precoding frequency selective (DP FS) schemes considering different values of \(f_{D}\Delta T\).  
Figure 7: Average cell throughput for the ABF with \(P=4\) and DP evaluated schemes considering different values of \(f_{D}\Delta T\).  
element of the antenna array. Moreover, its implementation is simple and has shown a bounded performance degradation in comparison to the MRT precoder when the number of antenna elements in the array is sufficiently high [26]. If the channel vector between UE \(k\) and its serving BS in FB \(b\) is expressed as:  
\[\mathbf{h}_{k,b}=\left[|h_{1}|e^{i\theta_{1}},|h_{2}|e^{i\theta_{2}},\ldots,|h_{ N_{t}}|e^{i\theta_{N_{t}}}\right]^{T}, \tag{4}\]  
then, the corresponding EGT precoding vector for UE \(k\) in FB \(b\), \(\mathbf{w}_{EGT_{k,b}}\), is  
\[\mathbf{w}_{EGT_{k,b}}=\frac{1}{\sqrt{N_{t}}}\left[e^{-j\theta_{1}},e^{-j \theta_{2}},\ldots,e^{-j\theta_{N_{t}}}\right]^{T}. \tag{5}\]  
As before, equal power allocation among users is considered. Note that PAPC is implicitly satisfied by the ABF scheme considered in this work, and, thus, the selection of EGT for DP allows a fair comparison from a power allocation perspective.  
Figure 8 shows the CDF of user throughput for the two digital precoding alternatives, MRT and EGT with PAPC, both of them configured according to the two scheduling variants under study (DP NFS and DP FS). The performance results of ABF with \(P=4\) and \(P=16\) RF chains with ideal assumptions are also included in the figure for the sake of comparison. In general, the EGT precoder worsens the throughput performance of both DP variants, although not very significantly. A meaningful result is that MRT DP NFS achieves very similar performance to EGT DP FS. Therefore, the performance loss due to the PAPC constraint can be, in this case, compensated by a more elaborated scheduler (per FB instead of per subframe). Moreover, the power per antenna element limitation brings the performance of digital precoders a bit closer to the one of ABF with \(P=16\) RF chains, i.e., with the same multiplexing capacity, although there is still a substantial performance gap for NLoS UEs (lower part of the CDF curve). Finally, the average cell throughput values for the ABF, MRT and EGT evaluated schemes are included in Figure 9. From this figure, we can confirm that the EGT-PAPC precoder causes approximately a 4% performance loss over both DP variants, a percentage much lower than the one caused by outdated CSI over DP.  
Figure 8: CDF of user throughput for the ABF, Maximum Ratio Transmission (MRT) and Equal-Gain Transmission (EGT) evaluated schemes.