Nested Variational Chain and Its Application in Massive MIMO Detection for HighOrder Constellations
Abstract
:1. Introduction
 Firstly, the basic idea of the nested variational chain is proposed, and an algorithm is then proposed to establish a general framework. By referring to ’general’, it means this framework is able to combine ’exclusive’ and ’inclusive’ KL divergences, or it degrades to either one as a special case.
 Secondly, providing several examples, we show that existing algorithms, such as MMSE, GTA, and EC, can be regarded as special cases of the variational chain.
 Finally, to provide an initial application of the variational chain into massive MIMO detection, a GTAembedded Expecatation Consistency (GTAEC) algorithm is proposed which proves to provide better detection performance, especially for highorder constellations. The complexity of GTAEC is analyzed as well along with comparisons.
2. Preliminary
2.1. Signal Model
2.2. MIMO Detection
3. Nested Variational Chain
3.1. A Generic Framework for Nested Variational Chain
Algorithm 1 An algorithm for nested variational chain 

3.2. MMSE, GTA and EC MIMO Detectors as Special Cases
4. Applications into MIMO HighOrder Detection
4.1. The GTAEC Algorithm
Algorithm 2 The GTAEC Algorithm 

4.2. Complexity Analysis
5. Numerical Results
5.1. Simulation Parameters
5.2. Performance Evaluation
 On one hand, both EC and GTAEC significantly outperform existing algorithms such as MMSE and GTA. In most scenarios, GTAEC can obviously outperform EC with either 16QAM, 64QAM, or 256QAM employed. The performance gain of GTAEC becomes larger when highorder constellation is employed. For example, both the 64QAM and 256QAM cases exhibit larger gain than the 16QAM case when employing 16 or 64 antennas. This indicates that GTAEC has superior performance gain and is especially suitable for highorder constellations. We believe that the performance gain comes from exploiting additonal relations (correlation) among symbols rather than treating them independently.
 On the other hand, as for the complexity issue, GTAEC with $four$ iterations may outperform or have comparable performance to EC with $six$ iterations, suggesting that GTAEC requires less complexity than EC by recalling that their computational burdens are dominated by the number of iterations needed. As a result, $four$ iterations are recommended for GTAEC according to the simulation results, and hence the complexity of GTAEC is approximately $four$ times more than MMSE, indicating that it is a practical method for massive MIMO systems.
6. Conclusions
Funding
Data Availability Statement
Conflicts of Interest
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Algorithm  Factor Substitution  Inner Approximation  Detection and Factor Updating  Total Complexity 

MMSE  $\mathcal{O}\left(N{M}^{2}\right)$  $\mathcal{O}\left(M\right)$  $\mathcal{O}\left(M\sqrt{A}\right)$  $\mathcal{O}\left(N{M}^{2}\right)$ 
GTA  $\mathcal{O}\left(N{M}^{2}\right)$  $\mathcal{O}\left({M}^{2}\right)$  $\mathcal{O}\left(MA\right)$  $\mathcal{O}\left(N{M}^{2}\right)$ 
EC  $\mathcal{O}\left(\left({N}_{iter}+1\right)N{M}^{2}\right)$  $\mathcal{O}\left(M\right)$  $\mathcal{O}\left({N}_{iter}M\sqrt{A}\right)$  $\mathcal{O}\left(\left({N}_{iter}+1\right)N{M}^{2}\right)$ 
GTAEC  $\mathcal{O}\left(\left({N}_{iter}+1\right)N{M}^{2}\right)$  $\mathcal{O}\left({M}^{2}\right)$  $\mathcal{O}\left({N}_{iter}MA\right)$  $\mathcal{O}\left(\left({N}_{iter}+1\right)N{M}^{2}\right)$ 
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Wang, Q. Nested Variational Chain and Its Application in Massive MIMO Detection for HighOrder Constellations. Entropy 2023, 25, 1621. https://doi.org/10.3390/e25121621
Wang Q. Nested Variational Chain and Its Application in Massive MIMO Detection for HighOrder Constellations. Entropy. 2023; 25(12):1621. https://doi.org/10.3390/e25121621
Chicago/Turabian StyleWang, Qiwei. 2023. "Nested Variational Chain and Its Application in Massive MIMO Detection for HighOrder Constellations" Entropy 25, no. 12: 1621. https://doi.org/10.3390/e25121621