# Accurate Performance Analysis of Coded Large-Scale Multiuser MIMO Systems with MMSE Receivers
## 5 Numerical Results and Simulations  
In this section, numerical results and simulations are performed to validate the accuracy of the theoretical analysis. We consider a practical scenario where a BS equipped with \(R\) receiving antennas is located at the center of a hexagonal cell with a radius of 1 km. We assume that \(K=5,10\) users are distributed independently and uniformly at random in the cell and the distance between each user and the base station is greater than \(r_{h}=100\) meters, that is, for the \(k\)-th user, \(r_{k}>r_{h}\). The largescale fading parameter \(\zeta_{k}\) associated with geometric attenuation and shadow fading can be estimated by \(\zeta_{k}=\tau_{k}/(r_{k}/r_{h})^{v}\), where \(\tau_{k}>0\) denotes the shadow fading and obeys \(10\log_{10}(\tau_{k})\sim\mathcal{N}(0,\sigma_{sf}^{2})\). \(v\) represents the pathloss exponent. In the simulation, we choose \(\sigma_{sf}^{2}=8\) dB and \(v=3.8\). Furthermore, the gray labeled 16-QAM constellation is considered and the statistical model of the channel is assumed to be perfectly known at the receiver. In each of the pictures, the left-hand side is the case of the conventional multiuser MIMO, and the right-hand side corresponds to the large-scale multiuser MIMO scenes. We can see that, in all cases, the theoretical curves of our approximate approaches are well fit to the simulations. We first provide results for the PDF of SINR based on linear MMSE detection for multiuser MIMO systems in Figure 1, where SNR = 10 dB. Obviously, for a fixed number of users \(K\), when the number of receiving antennas of the BS increases, the mean value of the SINR also increases, and it is more capable of providing a larger value of SINR. Therefore, the performance of the system tends to be better. Figure 2 presents the SEP performance of 16 QAM, and 64 QAM signaling for an uncoded Large-ScaleMIMO system with an MMSE receiver. It should be explained that, based on the derived SINR distribution, the corresponding SEP expression can be obtained by using the method of [16]. The black curves correspond to the system used in [16], where only Rayleigh fading channels were considered. By comparison, it can be seen that large-scale fading has a greater impact on the performance of the system. However, which must be considered in a practical multi-user scenario. In Figure 3, we plot the upper-bound on BER of of 16-QAM signaling scheme, for the encoded MMSE-MIMO systems, where the half rate \((133,171)_{8}\) convolutional code was used and the free distance is \(d_{f}=10\). We can see that, in the high-SNR region, our theoretical curves perfectly tend to the experimental curves. In the low SNR region, the anti-interference ability of the system under consideration is relatively weak, and the desired signal is severely affected by the interference. Therefore, the value of PEP is relatively large, and the BER upper-bound obtained based on PEP is relatively loose. It is not difficult to observe that, when the number of users \(K\) and the number of receiving antennas \(R\) of the BS have been given, the BER performance deteriorates as the modulation order increases. From another perspective, QAM,16 QAM and 64 QAM require a higher SNR than binary phase shift keying (BPSK) for a given BER. Furthermore, the system performance is deteriorated as the difference between the number of transmit and receive antennas decreases. This is because, when SNR grows larger, the encoded MMSE-MIMO receiver realizes an asymptotic diversity of \(d_{free}(R-K+1)\). Since asymptotic diversity (33) requires SNR to be infinite, the simulations in the actual SNR region deviate from the theoretical result (33). In addition, since the ZF algorithm ignores the effects of noise, the MMSE exhibits better BER performance than the ZF algorithm over the entire SNR region. However, as the number of antennas continues to increase, the performance of ZF and MMSE algorithms is similar. Figure 1: PDF of the SINR based on linear MMSE detection for multiuser MIMO systems at SNR = 10 dB.