# Capacity Analysis of Lattice Reduction Aided Equalizers for Massive MIMO Systems
## 2 Related Work  
In [1], the authors investigated the performance of Seysen's Algorithmic (SA)-based LR-aided MMSE receiver. The performance is analyzed in terms of BER corresponding to a large scale MIMO system. The prime focus of this paper is to implement the SA algorithm using Very-large-scale integration (VLASI). In [21], the authors proposed element based lattice reduction (ELR) combined with K-best detector for the MIMO system. They demonstrated that the proposed system provides better BER performance than the conventional LLL methods with a reduction in complexity. Moreover, it also indicates the hardware architecture of the proposed algorithm. In [23], the authors proposed a MIMO detection technique with LR aided fixed-complexity tree search. As demonstrated, the intended receiver can achieve near ML performance for a massive MIMO system with higher-order modulation. This paper provides a detailed analysis of the complexity of the system and also BER comparison with 256 QAM modulation. The authors of [8] provided a detailed mathematical model for calculating the capacity of a MIMO system with complex LLL (CLLL) based LR-aided receivers. Furthermore, the performance is compared, keeping ZF as the basic building block for receiver systems. This paper also demonstrates the performance comparison of CLLL, dual CLLL, and dual ELR receiver systems. The authors of [32] studied the utility of complex KZ reduction (CKZ) algorithm for the MIMO system and figured out its advantage of reduced complexity with full diversity. The BER performance of the proposed CKZ algorithm along with ZF, MMSE, and SIC receivers are compared with the linear receiver in this paper. The most important outcome of this paper is that the CKZ-aided receiver outperforms the LLL-aided receiver and achieves full diversity. In [33], the author provided a detailed quantitative analysis of lattice decoding in terms of the proximity factor. They provided a comparative analysis of the primal and dual reduction corresponding to both LLL and KZ aided ZF and SIC receivers. This paper also includes mathematical expression and analytical results corresponding to the upper bounds of the proximity factor. In [34], the authors pointed out the shortcomings of the KZ and LLL algorithms and proposed boosted KZ and LLL algorithms. They also provided some quantitative analysis of ergodic rate and orthogonal deficiency of conventional LR algorithms along with the boosted ones. The complexity of those algorithms is analyzed as a function of the dimension of the massive MIMO system. The authors of [35] proposed a hybrid LLL algorithm for the MIMO system and showed that it has fast convergence about the conventional LLL algorithm. However, the proposed algorithm degrades system performance. In [36], the authors analyzed LR-aided receiver for the realistic MIMO channel and also emphasized the performance complexity tradeoff. They proposed an LR-aided sphere decoding MIMO detector for complexity reduction. Besides, the MIMO system performance is investigated based on BER performance concerning variation antenna configuration, channel correlation coefficient, and channel estimation error. The proposed LR-SD system can provide near-optimal performance. Furthermore, this study provides a detailed analysis of the system complexity. In [37], the authors proposed an LR-SIC receiver for the massive MIMO system with low-precision ADCs. The prime focus of this paper to select the proper receiver system to make the system more energy efficient. In [38], the authors investigated the performance of integer forcing (IF) receivers for the MIMO system. Moreover, to design IF the receiver, they used the LLL algorithm and proved that the IF receiver outperforms the MMSE and ZF receivers. Nevertheless, in our paper, we combine LLL and KZ reduction techniques with MMSE receivers and show that LLR-MMSE and KZ-MMSEreceivers provide much better performance than the IF receivers.