# Deep-Unfolded Tikhonov-Regularized Conjugate Gradient Algorithm for MIMO Detection
## 5 Simulation Results
The superior performance of DU-TCG for difficult conditions such as high-order modulation systems suggests that integrating a regularization term into a deep-unfolded method will help convergence stability and hence improve detection performance. Additionally, in the simulation results presented above, the BER values are limited to \(10^{-4}\), as in similar studies utilizing deep unfolding methods in the literature [45; 46]. Achieving lower BER values in the simulation environment, particularly for deep-unfolding-based methods, usually requires considerably longer simulation times and higher computational resources, which is impractical for our current computing environment. Nevertheless, we conducted an extensive simulation to achieve lower BER values, and the results are presented in Figure 8, which illustrates the BER performance of the LCG and DU-TCG techniques for 64-QAM and 256-QAM modulations in a 32 \(\times\) 64 MIMO system under Rayleigh fading conditions. The detection performance of higher-order modulation schemes, such as 64-QAM and 256-QAM, as shown in Figure 8, demonstrates that the proposed DU-TCG method retains its superior performance even with more complex modulation formats, which are widely used in sub-6 GHz 5G communication systems [45; 47]. In contrast, other deep unfolding techniques, such as in [19; 28; 29; 42], may show a degradation in detection performance as the modulation order increases. In Figure 9, we compare the detection performance of the proposed DU-TCG method with DetNet over a 32 \(\times\) 64 MIMO layout. For the 32 \(\times\) 64 configuration, DU-TCG's BER remains up to nine times lower than DetNet's, demonstrating its strong performance even at higher antenna counts. While the number of trainable parameters in DetNet is eight for each layer, \(\theta_{DetNet}=\{W_{1l},b_{1l},W_{2l},b_{2l},W_{3l},b_{1l},\delta_{1l},\delta _{2l}\}\), it is three for the proposed DU-TCG method, \(\theta_{DU-TCG}=\{\alpha_{\ell},\beta_{I},L_{l}\}\). DetNet has a large number of trainable parameters, which can increase computational complexity and resource requirements. These results highlight the  
Figure 7: BER performance for different modulation types: (**a**) BPSK; (**b**) 16-QAM. usefulness of DU-TCG, providing better performance without excessive computational cost. The NMSE performance of the proposed DU-TCG and LCG methods, shown in logarithmic scale, for scalar and vector parameterization, is illustrated in Figure 10 for a 32 \(\times\) 64 MIMO layout. Figure 10 is one of the most important pieces of evidence showing the superiority of using Tikhonov regularization as DU-TCG has lower NMSE values in both scalar and vector parameterization cases. As the simulation results show, in the case of scalar parameterization, the NMSE decreases with increasing SNR values for both DU-TCG and LCG methods, which indicates that the signal detection performance of the system increases. On the other hand, in the case of vector parameterization, the increase in SNR values for LCG does not affect the NMSE values after a point, while the NMSE value of the proposed  
Figure 8: BER performance for different higher-order modulation types. Figure 9: BER performance between DU-TCG and DetNet in a 32 \(\times\) 64 MIMO system. DU-TCG continues to decrease. This shows that the performance of the LCG in complex situations remains constant after a certain level and cannot learn effectively. In addition, the NMSE values for vector parameterization approaches are much smaller than those for scalar parametrization, indicating improved detection performance. Therefore, in case of vector parameterization, the effect of regularization on NMSE performance is higher, resulting in better BER performance. After discussing the effect of scalar or parameterization techniques on the model-driven approaches above, we will show the impact of the same techniques on BER values in Figure 11.  
Figure 11: BER performance with scalar and vector parameterization in 32 \(\times\) 64 Rayleigh channel.