# Gibbs Sampling Detection for Large MIMO and MTC Uplinks with Adaptive Modulation
## 7 Assessment of the Proposed GS for Large MIMO and AM-MIMO  
In order to verify the superiority of T-MGS over MGS and OGS, a comparison of the average BER performance against the number of iterations is first provided in Figure 9 for symmetric MIMO. The number of antennas was set to \(N=16\), a 16-QAM constellation was used, the number of iterations \(t_{\text{max}}\) was set to 1000, while the SNR was fixed to 20 dB. Additionally, the conventional GS method (\(\alpha^{2}=1\)) is shown, and the \"optimal\" TP parameter in the OGS method is computed according to [22] (Equation (23)). In all instances, the MMSE filter output was chosen as the initial candidate vector \(\mathbf{x}^{(0)}\). For comparison purposes, SD performance (which attains ML performance) is also depicted. As can be verified, different mixing ratios exhibit dissimilar behaviors. Firstly, it is confirmed that conventional GS has a rapid decay in BER in the first few iterations, but then becomes stalled in a local minima. Secondly, it is observed that the proposed method T-MGS outperforms both MGS and OGS by a non-negligible margin, thus showing the importance of choosing adequate mixing ratios. Nonetheless, the performance observed in Figure 9 tends to a BER that is still far from the optimal, which is achieved via SD, therefore proving the usefulness of executing MR, as suggested in [23; 27], and whose effect will be analyzed next. Figure 9: BER performance as a function of the number of iterations \(t\) for SNR = 20 dB for different TP options (not including MR). The simulations are for \(n=32\), with a 16-QAM constellation. For the assessment of BER as a function of the SNR, and taking into account the previous result, \(R=10\) restarts are performed, fulfilling \(t_{\max}=600\) iterations each, and the output coincides with the candidate vector yielding the lowest ML cost. For the first restart, the MMSE solution is used as the initial vector, and for the remaining ones, a randomly chosen candidate is picked from the lattice instead. The performance for these settings is depicted in Figure 10, which serves as a summary of the performance that the proposed T-MGS technique is able to achieve with a quite low complexity: in Figure 8, one only uses \(R=10\) restarts, rather than the \(R=10\) restarts in Figure 8, and the a BER and SNR are in the regions of practical interest. As one could have already seen in Figures 6 and 8, a first thing to be noted is the existence of a stalling effect in OGS, which proves that [22] (Equation (23)) does not necessary scale for constellations larger than 4-QAM. Further, it can be verified that, with the inclusion of the MR approach, T-MGS performs very closely to the SD, whilst achieving near-optimal diversity for the whole studied SNR range. Again, it is worth emphasizing that the MR instances are parallel-architecture friendly, therefore such hardware design model, increasingly available at the BS, can be exploited to increase speed. In order to further confirm the suitability of the proposed algorithm for a quite large symmetric MIMO system (\(M=N=32\)), a simulation using \(q_{\infty}=1/(10M)\), \(R=15\) and \(t_{max}=1000\) was performed. The result is depicted in Figure 11, and the previously obtained conclusions can also be applied here. Note that, further augmenting the number of antennas or the constellation size would lead to a drastic increase in complexity, even using the stopping criteria or the complexity reduction techniques suggested in Section 3.3. Nonetheless, the algorithm can be efficiently used to obtain an estimate of SD performance for any number of dimensions, provided the initial vector is close to the initial solution (for instance, feeding the solution of the D-ELR algorithm as the initial vector in T-MGS would lead to a lower number of average number of required iterations). Figure 10: Comparison summary of the BER performance as a function of the SNR of the proposed variant for large MIMO systems.