## 4 Simulation Results and Complexity Analysis  
### Simulation Results and Discussion  
In this section, we follow the multi-user uplink M-MIMO environment illustrated in Section 2 to perform numerical simulations with OFDM and UFMC waveforms in order to evaluate and verify the performance of our proposed CHSOR receiver. The number of subcarriers of OFDM is set to be equal to the amount of data (512). In addition, it is necessary to add a CP with a length of 128, i.e., a quarter of the number of subcarriers, and 52 pilot data in one symbol. For UFMC, we set the number of subcarriers to 1024 and the total data volume to 512. This is divided into \(B=16\) sub-bands, each with \(M=32\) data. With these settings, 256 zeros must be padded at the head and the tail of the symbol to account for the difference between the number of subcarriers and the amount of data. For the filter type, we adopt a Chebyshev FIR filter [57,58] with a length of 43 and a side attenuation of 40 dB. To assess the impact of antenna configuration on performance, we set the number of transmit antennas \(N_{T}\) to 16 and the number of receive antennas \(N_{R}\) to 64, 128, 192, and 256. To facilitate further analysis, we used the parameter \(\beta\) as the ratio of the receiving to transmitting antennas, or antenna ratio, denoted as \(\beta=\frac{N_{R}}{N_{T}}\)[50,59]. In terms of channels, we established two flat Rayleigh fading channels with additive white Gaussian noise (AWGN). The maximum signal-to-noise ratio (SNR) for these channels was set at 50 dB and least-squares (LS) estimation was used to obtain the channel state information (CSI). For clarity, the above experimental parameters are tabulated in Table 2.  
The following experiments were applied to the OFDM and UFMC waveforms. Our proposed CHSOR scheme is compared with nine detectors, including SOR [17], MSOR [19], AOR [18], SSOR [20], SAOR [21], CSOR [22], Chebyshev-MAOR [27], Chebyshev-RI [26], and MMSE, with the MMSE scheme used as a benchmark. The MATLAB version R2022a mathematical software tool was used to simulate the numerical results and graphics. We executed 500,000 Monte Carlo rounds for each graph and used Microsoft Excel for the calculations and statistical tables. To observe which of UFMC and OFDM is more suitable for B5G and future communication environments, we used BER performance and power spectral density (PSD) indices to make an intuitive comparison. Figure 7 shows the BER vs. SNR for both waveforms detected using MMSE when \(N_{R}\times N_{T}=64\times 16\). It can be seen that UFMC is better than OFDM; when the SNR is 35 dB, the BER of OFDM and UFMC are \(1.599\times 10^{-3}\) and \(4.946\times 10^{-5}\), respectively, which is a 96.907% improvement. On the other hand, the PSD comparison in Figure 8 shows that the power sidelobe level of the OFDM waveform is larger than that of the UFMC waveform, indicating higher out-of-band emission and a tendency towards ACI. Therefore, we observe that the UFMC waveform has higher spectral efficiency and can better suppress the occurrence of ACI. In summary, the UFMC waveform is superior to the OFDM waveform regarding both BER performance and PSD, and is a promising B5G candidate waveform in future applications. For BER performance, Figures 9-12 show the BER vs. SNR curves of each method under different antenna configurations; Figures 9a-12a show the results for OFDM, while Figures 9b-12b show the results for UFMC. It is worth noting that the BER of the Gauss-Seidel (GS) and Conjugate-Gradient (CG) methods in Figures 9 and 10 is relatively poor