## 1 Introduction  
As technology rapidly develops and information updates become swift, people's demand for the internet is increasing daily. This means that a large amount of data needs to be transmitted, presenting a significant challenge for wireless communication systems. Single-carrier modulation has long been unable to handle the needs of modern communications due to its susceptibility to inter-symbol interference (ISI), high bit error rate (BER), and large bandwidth requirements. In contrast, multi-carrier modulation effectively overcomes the limitations of single-carrier modulation by decomposing high-rate data streams into multiple low-rate streams for parallel transmission. Among such schemes, orthogonal frequency-division multiplexing (OFDM) introduces the concept of sub-carrier orthogonality and is the most famous technology [1; 2; 3]. It significantly reduces inter-carrier interference (ICI), further improves spectrum utilization efficiency, and is especially suitable for multi-path propagation and high-speed data transmission scenarios [4]. Consequently, OFDM has become a key technology in mobile communications. With the ongoing development of the internet of things (IoT) [5; 6; 7], application fields are expanding to include smart homes, smart manufacturing, and telemedicine, leading to a significant increase in connected devices. Simultaneously, these devices' connection and information exchange will inevitably bring about increasing data transmission requirements. To meet various applications of the IoT as proposed by international mobile telecommunications (IMT) [8; 9], there are three important usage scenarios for fifth-generation (5G) devices, namely, enhanced mobile broadband (eMBB), ultra-reliable and low latency communications (URLLC), and massive machine-type communications (mMTC). Thus, achieving higher data transmission rates and lower latency is crucial. In light of these concerns, OFDM is no longer sufficient due to defects such as high side lobe loss and adjacent channel interference (ACI) [10]. Hence, the priority should be to find a multi-carrier waveform that can overcome the shortcomings of OFDM and make contact with the application scenarios of B5G. Universal filtered multi-carrier (UFMC) is a feasible candidate waveform that combines the benefits of OFDM and filter bank multi-carrier (FBMC), which can effectively suppress out-of-band (OOB) transmissions and reduce the impact of ACI. In addition to better coping with the three major application scenarios of B5G, massive multiple input-multiple output (M-MIMO) technology [11; 12], which mounts many antennas at the transmitter or receiver to promote space diversity, has become indispensable in modern wireless communications. It is implemented through beamforming and coherent superposition [11]. M-MIMO can improve system data throughput without increasing transmission power or bandwidth to efficiently achieve diversity gain, array gain, and capacity gain. Therefore, applying M-MIMO for UFMC waveforms can better meet high data rates and link reliability requirements. Unfortunately, even though M-MIMO brings many benefits, it is still applied to traditional linear detectors such as zero-forcing (ZF) and minimum mean square error (MMSE) detectors [13], which results in excessive computational complexity. Because the calculation process of these methods involves matrix inversion with complexity of \(\mathcal{O}(N^{3})\), the computational burden increases dramatically, especially when the number of antennas increases; therefore, to eliminate the harm caused by matrix inversion due to traditional linear detectors, many researchers are currently developing linear detectors based on iterative algorithms such as the Jacobi method [14], Gauss-Seidel (GS) method [15], conjugate gradient (CG) method [16], successive over-relaxation (SOR) method [17], and accelerated over-relaxation (AOR) [18]. Through iterative operations, these detectors have successfully reduced complexity from the cubed signal dimension to the squared dimension.