# Gibbs Sampling Detection for Large MIMO and MTC Uplinks with Adaptive Modulation
## 5 Proposed Triple-Mixed Gibbs Sampling
### Mixing Ratio Choice  
For demonstration purposes, consider \(n=32\), \(m_{k}=16\), for \(k=1,...,n\), and \(q_{1}=q_{\text{SNR}}=1/2\). The performance in terms of BER as a function of the number of iterations for different mixing ratios \(q_{\infty}\) is depicted in Figure 7, where two different SNR values were considered (18 and 19 dB). The first thing to be inferred, after the \(t_{\max}=800\) iterations, is that in both cases it is a non-zero mixing ratio (\(q_{\infty}\
eq 0\)) that minimizes the BER, hence proving the advantage of having a triple MGS algorithm. Additionally, the minimum value for different SNR values is attained for different values of \(q_{\infty}\), implying that the optimal value of \(q_{\infty}\) also has a dependence with the SNR. Figure 6: BER performance for optimized Gibbs sampling with multiple restarts (OGS-MR) detection algorithm for \(M=N=16\) and a 16-QAM constellation. Finding the optimal choice for the mixing ratios \(q\) is a non-trivial task; nevertheless, one is to find parameters that on average perform well for any SNR. The authors in [23] proposed the MGS method with \(q_{\infty}=1/n,q_{1}=1\), whereas [22] suggested the OGS with \(q_{\infty}=0,q_{\text{SNR}}=1\). With the T-MGS algorithm, one expects to experience fewer stalling occurrences in local minima than in MGS, as long as \(q_{\text{SNR}}>0\) (faster mixing time). This fact was verified through simulation and, as depicted in Figure 7, a choice of \(q_{\infty}=1/(10n),q_{1}=q_{\text{SNR}}=1/2\) proved to attain very satisfactory results, the reason why these values will be adopted henceforth in the paper for the T-MGS. Note that with this choice of parameters, one random walk is performed each 10 iterations on average, and whenever it is not carried out, there is a similar chance that \(\alpha^{2}=1\) or \(\alpha^{2}\propto\text{SNR}\) are chosen. A step-by-step description of the proposed algorithm including multiple restarts can be found in Algorithm 2. It is worth mentioning that the previously mentioned MGS-MR corresponds to doing \(q_{\infty}=1/n\) and \(q_{1}=1\), whilst OGS-MR coincides with setting \(q_{\infty}=q_{1}=0\) in Algorithm 2. The \"temperature\" parameters and corresponding mixing ratios are summarized in Table 1.  
Reconsidering the case where \(R=15\) and the stopping criteria as was done for Figures 3 and 6, the BER performance including the T-MGS detector is shown in Figure 8. The previously identified problems have been mitigated by T-MGS, and the curve in Figure 8 shows that optimal diversity is indeed obtained for the entirety of the studied SNR range. Moreover, for a reference BER value of \(10^{-3}\), a gain of 5 dB is attained by the proposed algorithm over the state-of-the-art LRA method (the green curve). Taking into account the previously mentioned complexity reduction techniques and the attained performance, T-MGS is certainly a suitable candidate for solving the detection problem in large MIMO systems. In addition, it is worth emphasizing that the algorithm is parallel architecture-friendly, which is nowadays becoming an increasingly important feature. Figure 7: BER performance as a function of the number of iterations for different mixing ratios \(q_{\infty}\) (with one restart only, and the MMSE solution was used as the initial vector). The simulations are for \(M=N=16\) (i.e., \(n=32\)) with a 16-QAM constellation.