# Adaptive Threshold-Aided K-Best Sphere Decoding for Large MIMO Systems
## 1 Introduction  
Large multiple-input multiple-output (MIMO) systems have received enormous attention from researchers in the field of wireless communication for their high spectral and power efficiency [1]. However, the promised benefits of large MIMO are expensive in terms of computational complexity at the receiver compared to the conventional MIMO systems [1, 2]. In conventional MIMO systems, to simplify the exhaustive search of the optimal maximum likelihood (ML) receiver, a sphere decoder (SD) can be employed, which only searches for the ML solution inside a sphere to reduce computational complexity. Furthermore, low-complexity sub-optimal algorithms including parallel interference cancellation (PIC) [3] and successive interference cancellation (SIC) [4] can also be considered. However, to approach the channel capacity, it is required to employ a near-optimal receiver such as the SD. Even though the SD performs close to the ML receiver, its complexity grows exponentially with the number of transmit antennas, which results in excessively large complexity in large MIMO systems [2, 5]. Among variants of SD, K-best SD (KSD) begins its search from the first layer, and \(K\) candidate paths that are associated with the smallest path metrics at each layer are preserved for the subsequent layers until the iteration process reaches the leaf layer [6, 7]. Then, the one with the smallest path metric is chosenas the hard decision output of the KSD algorithm. The KSD requires fixed detection complexity [7] and can be implemented in a parallel fashion, which makes it suitable for hardware implementation [8]. However, to approach the bit-error rate (BER) performance of the original SD, the KSD needs to retain a large number of nodes in each layer, as presented in [9], which leads to a large computational complexity. This paper presents an adaptive threshold-aided K-best SD (AKSD) algorithm. Unlike the traditional KSD, where the number of survival nodes is constant, in AKSD, the number of retained nodes dynamically changes in each layer of the tree. Specifically, to reduce the complexity while maintaining the near-optimal BER performance, we keep the nodes whose path metric is smaller than the \(K\)th node's metric plus threshold \(\Delta_{i}\). At each layer, the threshold is updated dynamically based on the signal-to-noise ratio (SNR) and the index of the tree-search layer. Furthermore, the ratio between the first and second smallest path metrics is also employed to determine the threshold. Intuitively, if the first minimum path metric is far less than the second, the path associated with the first is very likely to be an ML solution. Therefore, there is no need to keep a large number of nodes. In contrast, if this ratio is small, which decreases the probability that the path with the minimum path metric is the ML solution, then we need to keep a large number of nodes. The main contributions of this study can be summarized as follows:  
1. We propose the AKSD algorithm, in which the adaptive threshold controls the number of visited nodes in each layer of the tree. In this algorithm, the threshold for retaining the most promising nodes at each layer is adaptively determined based on the ratio between the first and second minimum path metrics at each layer, the SNR, and the layer index. 2. To evaluate the performance of the proposed AKSD algorithm, we have performed simulations for large MIMO systems. The simulation results show that, compared to the conventional KSD algorithm, the AKSD algorithm requires up to 71% less computational complexity. The rest of this paper is organized as follows. Section 2 describes the MIMO system model and the conventional SD algorithm. Section 3 explains the traditional KSD and proposed AKSD algorithm. The simulation results and discussions with respect to the BER performance and computational complexity are presented in Section 4. Finally, the conclusions of the study are described in Section 5.