# Capacity Analysis of Lattice Reduction Aided Equalizers for Massive MIMO Systems
## 1 Introduction  
The continuous demand for high speed and reliable communication systems puts the MIMO systems the most demanding research topic [1]. As a further enhancement of MIMO technology, massive MIMO can increase the capacity in many folded and also enhance the system energy efficiency [2, 3]. However, all the benefits come with the burden of challenges. In the MIMO system, the design of low complex signal processing techniques is the main challenge of concern. Therefore, it is critical to address the challenge of developing low complex signal processing techniques for increasing the capacity of the massive MIMO system. In this paper, the authors discuss this critical problem and examine the low complex LR-aided receivers for the channel capacity enhancement. Several studies state that maximum-likelihood (ML) is the optimal receiver for a MIMO system. Nevertheless, it is practically unimplementable as its complexity increases exponentially with the number of antennas used in a MIMO system [4, 5, 6]. The linear receivers such as matched filter (MF),zero-forcing (ZF), and minimum mean squared error (MMSE) equalizers gain lots of attention because of their low signal processing complexity [7]. However, these receivers have limitations such as the cost and poor bit error rate (BER) performance [1; 8]. This results in the development of nonlinear receivers, such as successive interference cancellation (SIC) [9], decision feedback equalizer (DFE) [10; 11; 12; 13], probabilistic data association (PDA)-based MIMO receiver, and semidefinite programming relaxation (SDPR)-based MIMO detector [14; 15; 16; 17]. These receiver systems can minimize the performance gap in comparison with the ML receiver but with a significant increase in the system complexity. The sphere detector (SD) receiver [18] is introduced, and it can provide near-optimal performance to enhance the performance of the MIMO system. The basic approach of this SD algorithm is concentrated on a tree searching algorithm. More advanced receiver systems such as fixed-complexity SD (FSD) [19] and the K-best detector [20] have been developed, but the system complexity remains very high considering the massive MIMO system [21]. Lattice reduction (LR) [21] is a powerful and efficient algorithm to improve system performance with reduced complexity [22]. The basic approach of the LR-aided receiver system is to convert the lattice basis formed by the channel matrix into a near orthogonal lattice basis. The existence of this short and orthogonal basis leads to a tremendous improvement in the MIMO system performance [1; 23]. Extensive research work is ongoing, and different variants of LR algorithms have been proposed, such as Lenstra-Lenstra-Lovasz algorithm (LLL) algorithm, [PERSON]'s algorithm, and [PERSON]'s algorithm [1; 24; 25]. In the context of the MIMO detection, LLL algorithm is widely used because of its polynomial time complexity [21]. Furthermore, numerous research works are motivated to develop low complexity LR algorithms [20; 22; 26; 27; 28]. In [8; 22], the complex-valued LLL algorithm is proposed to reduce the size of the channel matrix by half in comparison to that of a real-valued channel matrix. The authors of [8] investigated the improvement of the capacity due to LR-aided receivers. They utilized LLL algorithm-based lattice reduction-aided ZF receiver for their analysis. Apart from the LLL algorithm, researchers are continuously working on the KZ algorithm and Minkowski reduction algorithm [21; 29; 30]. The KZ algorithm is more efficient in terms of generating more orthogonal basis vector than the LLL algorithm. Therefore, KZ reduced basis is considered to be the best reduced basis for the decoding purpose in a MIMO system. Consequently, it results in significant improvement in the MIMO system. As in [31; 32], the KZ algorithm, and particularly complex KZ algorithm, is able to provide full diversity with reduced complexity.