# Gibbs Sampling Detection for Large MIMO and MTC Uplinks with Adaptive Modulation
## 4 Optimization of the Temperature Parameter  
Performance of GS based methods are highly dependent on the chosen TP. If \(a^{2}\) is set to equal the noise variance (that is \(a^{2}=1\)), as in the conventional GS, a fast convergence to a local minimum is guaranteed but an error floor in the high SNR regime is to be expected. This is related with the existence of several local minima where the algorithm may be stuck for a very long time, a situation that becomes increasingly preponderant when either the \(m_{k}\)-arity or \(n\) are large. On the other hand, if \(a^{2}\) is chosen large, convergence rates of the underlying Markov chain increase (faster mixing time), but the likelihood of visiting the optimal solution in the steady state decreases, hence resulting in a trade-off situation in the choice of the temperature parameter. Figure 3: BER performance for both conventional Gibbs sampling and mixed Gibbs sampling with multiple restarts detection algorithms for \(M=N=16\) and 16-QAM constellation. In order to overcome this problem, the authors in [23] proposed a mixed GS (MGS) where at each iteration there is a small chance that a random walk is performed (which coincides with setting \(\alpha^{2}=\infty\) in (6)). However, this method proves to mix slowly to the steady state, hence requiring a considerable amount of restarts and iterations in order to perform reliable detection. On the other hand, ref. [22] proposed a method comprising of a fixed, optimized TP \(\alpha_{\text{opt. }}^{2}\), proportional to SNR, as follows:  
\[\alpha_{\text{opt.}}^{2}=\frac{\text{SNR}}{\log(n)}+\sqrt{\left(\frac{\text{ SNR}}{\log(n)}\right)^{2}-2\frac{\text{SNR}}{\log(n)}}, \tag{17}\]  
only valid for \(\text{SNR}>2\log n\), as one is interested in. This corresponds to the largest possible TP, such that, once in the stationary distribution, there is polynomially small and not exponentially small chance of finding the optimal vector. The method using the fixed TP as in (17) will henceforth be denoted as optimized GS (OGS). Notwithstanding the breakthrough developments, the results in [22] were only obtained for a \(\{-1,1\}\) constellation set, which is rather limited. It is interesting to note that the result in (17) suggests that \(\alpha_{\text{opt. }}\) scales with \(\mathcal{O}(\sqrt{\text{SNR}})\), for fixed \(n\), which means that faster mixing times are required in the high SNR regime. In order to demonstrate that MCMC methods employing Gibbs sampling converge to the optimal solution and that the parameter \(\alpha\) plays a crucial role in that convergence, a study of BER evolution in terms of the number of iterations will next be entailed. For that purpose, the SNR was fixed, and a set of values for \(\alpha^{2}\) was chosen, including \(\alpha^{2}=1\) and \(\alpha^{2}=\alpha_{\text{opt. }}^{2}\) Using the MMSE filter output as the initial vector, Figures 4 and 5 show the change in BER with an increasing number of iterations. For comparison purposes, conventional detectors are also depicted. The first thing to be noticed in Figure 4 is the fast convergence of the GS method to the ML performance when \(\alpha^{2}=``\text{optimal}\"\) (blue line). Moreover, it is interesting to observe the rapid BER decay in the first few iterations for when \(\alpha^{2}=1\) and then the stalling effect, as had been previously verified. Higher choices of \(\alpha^{2}\) (namely, \(\alpha^{2}=9\) and \(\alpha^{2}=18\)) proved not to improve the MMSE initial solution any further. Let one denote the detector using \(\alpha^{2}=``\text{optimal}\"\), mixing ratio \(q_{\infty}=0\) and multiple restarts, as optimized Gibbs sampling with multiple restarts (OGS-MR). The results corresponding to OGS-MR are depicted in Figure 6. For comparison purposes, the remaining used parameters are identical to the ones used to obtain Figure 3. As can be seen and as expected, the problem in the high SNR regime was partially mitigated, confirming that choosing an \(\alpha\) different from 1 is indeed beneficent.