## 5 Performance with Hardware Impairments
### Performance with Combiner Losses  
In the analog beamforming scheme, the signals of the different RF chains are mixed together by means of a combiner to feed the antenna array. Non-ideal combiners are known to introduce a power loss in their outputs [28], which increases with the number of branches that the device has to combine. In this work, we denote with \(L\) the power loss of the basic combiner with two input branches, i.e., whose output power is given by the sum of the two input powers divided by \(L\). Then, we assume that a combiner for a generic number \(P\) of RF chains is implemented by means of a cascade of \(\log_{2}P\) two-branch combiners, leading to a total loss, expressed in logarithmic units:  
\[L^{(tot)}[dB]=\log_{2}P\cdot L[dB]. \tag{7}\]  
Note that, differently from the results with ideal hardware shown in Figure 3, where user throughput increases with the number of RF chains, by taking into account now the combiner power loss (7), there will be an optimum number of RF chains depending on the scenario. Indeed, when we increase the number of RF chains, on one side, we still increase the number of users that can be served by the BS (thus allowing higher spatial multiplexing), but, on the other side, we have a power loss that decreases the SINR measured at the UEs. Regarding the selection of \(L\) values, note that the case without any compensation of losses corresponds to \(L=3\) dB [16], where half of the input power is lost. However, in those cases where compensation of the losses is possible by adding power amplifiers after the combiner stages, realistic values for combiner losses range from 0 to 3 dB. In this sense, we selected \(L=1\) dB as a meaningful intermediate evaluation point. It is worth noting that compensation of losses involves an additional cost due to including extra per-antenna power amplifiers. Figure 10 shows the CDF of user throughput for ABF with different number of RF chains, \(P=4,~{}8\) and 16, and two values of the two-branch combiner loss parameter, \(L=1,~{}3\) dB. Results show that, for \(L=1\) dB, the best results are obtained with \(P=16\) RF chains. However, when \(L=3\) dB is considered, the scheme with \(P=8\) RF chains achieves the best performance. Therefore, the optimum number of RF chains depends strongly on the combiner losses. The results in Figure 10 are complemented by Table 5, which collects the average UE throughput, 5 th percentile UE throughput and average cell throughput for the same schemes. More specifically, the table contains the average values of each performance indicator in Gbps and also the performance loss with respect to having ideal combiners (\(L=0\) dB). It is again observed that, for \(L=1\) dB, \(P=16\) RF chains is the best option, despite the fact that the higher the number of RF chains, the higher the total losses due to realistic combiners (for instance, the loss in average UE throughput is around 5% for \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline  & \multicolumn{2}{c}{\(\delta_{max}=2^{\circ}\)} & \multicolumn{2}{c}{\(\delta_{max}=6^{\circ}\)} & \multicolumn{2}{c}{\(\delta_{max}=10^{\circ}\)} \\ \cline{2-7}  & **Thr.** & **Loss** & **Thr.** & **Loss** & **Thr.** & **Loss** \\ \cline{2-7}  & **[Gbps]** & **[\%]** & **[Gbps]** & **[\%]** & **[Gbps]** & **[\%]** \\ \hline
**Avg UE thr. LoS** & 1.871 & 0.12 & 1.847 & 1.41 & 1.807 & 3.55 \\
**Avg UE thr. NLoS** & 0.794 & \(-\)0.03 & 0.785 & 1.10 & 0.770 & 2.94 \\
**Avg UE thr.** & 1.518 & 0.12 & 1.500 & 1.34 & 1.467 & 3.51 \\
**5\%-ile UE thr. LoS** & 1.029 & 0.34 & 1.021 & 1.15 & 1.004 & 2.77 \\
**5\%-ile UE thr. NLoS** & 0.311 & \(-\)0.12 & 0.309 & 0.43 & 0.304 & 2.11 \\
**5\%-ile UE thr.** & 0.419 & 0.51 & 0.415 & 1.44 & 0.410 & 2.62 \\
**Avg cell thr.** & 25.01 & 0.12 & 24.7 & 1.37 & 24.2 & 3.46 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance results of the ABF scheme with \(P=4\) including phase shifter errors. \(P=4\) but around 9% for \(P=16\)).