## 1 Introduction
However, while iterative-based detectors benefit from lower complexity, this comes at the expense of some BER performance. More importantly, the initial estimate significantlyimpacts the conjugate direction sequence produced by the CG method, which has a greater impact than the SOR-based method, thereby affecting the overall convergence speed. Given this, [PERSON] et al. proposed a modified SOR (MSOR) [19] that adjusts the parameters in SOR to reduce its complexity; however, the BER performance also declined. Moreover, [PERSON] et al. proposed a symmetric successive over-relaxation (SSOR) method [20] based on SOR. Their method combines forward and backward scanning SOR to optimize performance; however, the complexity is greatly increased. In [21], [PERSON] et al. further proposed a symmetric accelerated over-relaxation (SAOR) method, using AOR to replace SOR and to achieve better BER performance but also making greater sacrifices in complexity. Therefore, [PERSON] et al. [22] and our previous work in [23] both proposed two-stage methods, respectively, Chebyshev successive over-relaxation (CSOR) and Chebyshev accelerated over-relaxation (CAOR). In this light, combining the Chebyshev acceleration method with the SOR and AOR methods can achieve rapid convergence while still reducing complexity. In comparison, [PERSON] et al. proposed an iterative block decision-feedback equalizer (IB-DFE) [24] based on an iterative architecture with nested inner and outer loops. Their approach combines the equalizer with traditional iterative methods such as GS, SOR, etc., to reduce complexity. However, it requires many iterations to converge, which leads to high complexity. Moreover, the authors only simulated 32 antennas and low-order QAM, which fails to cover the actual requirements of M-MIMO in 5G networks. In another work the same authors proposed AC-AORNet [25], which optimizes the parameters of the iterative algorithm through a deep unfolding network and incorporates Chebyshev acceleration to expedite convergence speed and improve BER performance. However, its training phase incurs considerable computational cost and time, resulting in increased overall complexity. To improve the initial state of the Chebyshev acceleration, they proposed related methods, namely, Chebyshev-RI [26] and Chebyshev-MAOR [27]. The former uses the stair matrix as the basis of the initial vector and provides a favorable input state for Chebyshev acceleration through the Richardson (RI) method in order to improve convergence speed and complexity performance. However, Chebyshev-RI suffers from significant performance degradation under high-order modulation, making it unsuitable for application in 5G systems. On the other hand, Chebyshev-MAOR [27] combines the modified AOR with Chebyshev acceleration to bring it closer to the MMSE detector's performance. Nevertheless, this approach exhibits sensitivity to a low receive-to-transmit antenna ratio, and more iterations are required to meet performance expectations. On the other hand, the new randomized iterative detection algorithm (NRIDA) [28] proposed by [PERSON] et al. has more efficient detection capabilities than the original randomized iterative detection algorithm (RIDA). It introduces a conditional sampling mechanism that selectively updates symbols in each iteration through a sampling distribution, effectively reducing unnecessary calculations. However, this method still has a significant performance gap with the MMSE detector. In addition, it was only simulated at 16-QAM, which is insufficient for B5G applications that need to support high-order modulation. In another study, [PERSON] et al. proposed an iterative parallel interference cancellation approach based on the lattice reduction-aided (IPIC-LRA) method [29], introducing the LRA approach into parallel interference cancellation (PIC) to improve the accuracy of the initial estimate. Then, the iterative architecture makes corrections step-by-step to achieve better detection results.