# Precoded Generalized Spatial Modulation for Downlink MIMO Transmissions in Beyond 5G Networks
## 4 Numerical Results
### Link Level Simulations  
The first simulation results had the objective of evaluating the behavior of the iterative GSM receiver and of the overall proposed GSM MU-MIMO transmitter/receiver scheme. Figures 3 and 4 present the results of BER performance versus the signal-to-noise ratio (SNR) in dB of the proposed GSM MU-MIMO system with \(N_{tx}=255\), \(N_{rx}=10\), \(N_{u}=15\), \(N_{s}=17\) and \(N_{a}=2\), which corresponds to a spectral efficiency of 23 bpcu/user for 256-QAM and 27 bpcu/user for 1024-QAM. The expression \(n_{1}\times n_{2}\), mentioned in the legend of both figures, denotes that the receiver algorithm was ran with \(n_{1}\) restarts and \(n_{2}\) iterations. The type of polishing applied as well as the number of neighbors is also shown. Besides the expected improvement when using more iterations, it can be observed that by increasing the number of algorithm restarts, we can achieve a better system performance. When considering the \(1\times 500\) and \(10\times 50\) cases, which have the same total number of iterations, it is clear that the best results are achieved by the case with more restarts (\(10\times 50\)). Considering the scenarios where MMSE polishing is used and those where it is not, one can observe that those where polishing is applied have better performance. We also studied the impact of changing the number of neighbors on the performance of the algorithm and we concluded that the greater the number of neighbors, the better the performance will be (see the cases where P = 1, 4, 9 and 19). Moreover, the combination of the three proposed improvement strategies for the ADMM receiver lead to a better performance than the usage of the individual approaches. Globally, the proposed ADMM algorithm tends to lead to better results when compared to the case where the well-known OB-MMSE receiver (which we included as benchmark) is used [6]. Figure 3: Bit error rate (BER) performance of ADMM in a MU-MIMO scenario with \(N_{tx}=255\), \(N_{rx}=10\), \(N_{u}=15\), \(N_{s}=17\) and \(N_{a}=2\), 256-QAM. Our next goal is to provide a comparison between a conventional BD precoded MU-MIMO [9] and the proposed GSM MU-MIMO. Figure 5 shows the results for two different configurations. The first case concerns a comparison between the precoded GSM MU-MIMO with \(N_{tx}=160\), \(N_{rx}=6\), \(N_{u}=10\), \(N_{s}=16\), \(N_{a}=1\), 16-QAM and the conventional BD precoded MU-MIMO with \(N_{tx}=60\), \(N_{rx}=6\), \(N_{u}=10\), \(N_{s}=1\) and 256-QAM, both with a spectral efficiency of 8 bpcu/user. In the second case, we present a comparison between the precoder based on GSM MU-MIMO with \(N_{tx}=90\), \(N_{rx}=8\), \(N_{u}=10\), \(N_{s}=9\), \(N_{a}=3\), quadrature phase-shift keying (QPSK) and the precoder based on conventional MU-MIMO with \(N_{tx}=80\), \(N_{rx}=8\), \(N_{u}=10\), \(N_{s}=3\) and 16-QAM, both with a spectral efficiency of 12 bpcu/user. Regarding the GSM MU-MIMO scheme, results with the proposed receiver as well as other alternative ones are included, namely with a linear MMSE and with the sMMP (from [34]). In the case of sMMP, a lower number of child nodes (\(T=3\)) was adopted for 12 bpcu due to the very high computational complexity when operating with higher values in this scenario. In the results, it is clear that the proposed ADMM receiver achieves the best results when compared against sMMP and MMSE. In the case of MMSE, it simply cannot correctly detect the information (8 bps) or it has a high irreducible BLER (12 bps). This is due to the fact that from a receiver point of view, both scenarios correspond to underdetermined systems (\(N_{rx}<N_{s}\)), which a simple MMSE has a high difficulty to cope with. Through this figure, it can also be seen that the GSM MU-MIMO precoder with the proposed ADMM receiver achieves a better performance when compared to the conventional MU-MIMO precoder (which also uses the same receiver).