# Gibbs Sampling Detection for Large MIMO and MTC Uplinks with Adaptive Modulation
###### Abstract
However, finding efficient detection algorithms for such systems has been the focus of some literature; for example, in [20], the authors proposed SD-based methods for optimal decoding of AM-MIMO, even though it is not a scalable solution for large MIMO due to the rising complexity that SD entails. Advanced statistical methods, such as Markov chain Monte Carlo (MCMC), have already shown significant gains in signal processing for wireless communications. In particular, Gibbs sampling (GS) [21] (section 11-3), which is the particular MCMC technique considered in this paper, can attain ML-like performance, provided the algorithm is run for a sufficient long time [22]. Even so, the conventional GS technique suffers from the so-called \"stalling problem\", meaning that for high values of signal to noise ratio (SNR), the BER performance does not improve any further [23]. Gibbs sampling-based techniques have been considered for large symmetric MIMO in recent years [24; 25; 26; 27; 28; 29]. In this paper, capitalizing on these previous works that use multiple restarts [23; 27], we propose a variant of the GS method to perform detection in both large MIMO (with \(\frac{N}{M}>0.75\), as in [27]) and AM-MIMO systems in particular, making use of the findings in [22] regarding the adaptation of the so-called \"temperature\" parameter (TP). The core idea of the GS technique proposed, and corresponding to the main contribution of this paper, is that at each iteration of the Gibbs sampler, the alternative approach picks one of three possible TP according to a given probability distribution. The TP values and corresponding probability distribution are chosen such that the stalling problem is circumvented, whilst keeping convergence rates high. Numerical results confirm that the proposed variant outperforms both aforementioned works, and that by using a multiple restarts (MR) approach, quasi-optimal performance can be obtained without further increasing processing time. Following this introductory section, the paper proceeds with the definition of the MIMO channel model in Section 2, which makes use of the mapping of the MIMO communication problem in the complex-domain to one in the real-domain, which can be tackled by the GS method. Section 3 starts by explaining how the GS technique can be applied in solving the MIMO detection problem and shows its pitfalls when a plain GS is used without additional measures; the section then shows how multiple-restarts can be one important additional measure to alleviate those problems. Because the performance obtained via GS detection is highly dependent on the choice of the TP, Section 4 presents some ways of choosing the TP. In Section 5, the central proposal of the paper emerges by probabilistically combining different TP along the iterations of the GS detection process, taking into consideration the observations made in the previous sections. Besides symmetric large-MIMO systems with homogeneous constellations in the antennas, the proposed GS detection is directly extendable to the AM-MIMO systems that naturally arise in MTC scenarios, and that extension is presented in Section 6. In Section 7, we present numerical results that assess the performance of the proposal, and the paper ends by presenting the conclusions in the Section 8.