# Gibbs Sampling Detection for Large MIMO and MTC Uplinks with Adaptive Modulation
###### Abstract  
Wireless networks beyond 5G will mostly be serving myriads of sensors and other machine-type communications (MTC), with each device having different requirements in respect to latency, error rate, energy consumption, spectral efficiency or other specifications. Multiple-input multiple-output (MIMO) systems remain a central technology towards 6G, and in cases where massive antenna arrays or cell-free networks are not possible to deploy and only moderately large antenna arrays are allowed, the detection problem at the base-station cannot rely on zero-forcing or matched filters and more complex detection schemes have to be used. The main challenge is to find low complexity, hardware feasible methods that are able to attain near optimal performance. Randomized algorithms based on Gibbs sampling (GS) were proven to perform very close to the optimal detection, even for moderately large antenna arrays, while yielding an acceptable number of operations. However, their performance is highly dependent on the chosen \"temperature\" parameter (TP). In this paper, we propose and study an optimized variant of the GS method, denoted by triple mixed GS, and where three distinct values for the TP are considered. The method exhibits faster convergence rates than the existing ones in the literature, hence requiring fewer iterations to achieve a target bit error rate. The proposed detector is suitable for symmetric large MIMO systems, however the proposed fixed complexity detector is highly suitable to spectrally efficient adaptively modulated MIMO (AM-MIMO) systems where different types of devices upload information at different bit rates or have different requirements regarding spectral efficiency. The proposed receiver is shown to attain quasi-optimal performance in both scenarios. Gibbs sampling; large MIMO detection; adaptively modulated MIMO; MTC 20  
number of receiving antenna elements in a cell-free network). The conditions for such types of operation may not always exist, for example in heavily loaded cells (with a number of terminals approaching the number of antennas at the BS), or when the dimensions of the antenna arrays are physically constrained, particularly if using low-frequency carriers. In the case of an uplink MIMO channel, when the number of (single-antenna) users \(M\) is far less than the number of BS antennas \(N\), linear processing techniques, such as zero-forcing (ZF) or minimum mean-square error (MMSE), attain near-optimal performance [9], and detectors based on Neumann series expansions can further reduce that complexity [10]. However, when \(M\to N\), i.e., as the ratio of the number of antenna elements on both sides of a MIMO system gets closer to 1, linear detection is well-known to severely degrade [11; 12]. It is therefore important to use detection methods that perform close to maximum likelihood (ML), i.e., exhaustive search, while yielding low complexity. Despite being optimal, ML and sphere decoding detection (SD) methods are prohibitive in large dimensions due to their exponential complexity [13]. Other possible alternatives are lattice-reduction-aided (LRA) methods [14], punctured SD-based techniques [15], or trellis-based detection approximating the lattice problem by one with a trellis representation [16], all of which hold performance-complexity trade-offs that have been proven to be acceptable for symmetric large MIMO systems. To address the need to serving devises with widely different requirements (in terms of bit rate, error probability, and spectral efficiency), adaptively modulated MIMO (AM-MIMO) systems, where each user has the possibility of adjusting the utilized modulation scheme according to its channel conditions and rate requirements, are of practical interest [17; 18]. The great advantage of these schemes over conventional fixed-MIMO are their high spectral efficiencies, whilst keeping bit error rates (BER) close to a given target [19].