# Deep-Unfolded Tikhonov-Regularized Conjugate Gradient Algorithm for MIMO Detection
## 6 Discussion  
In this study, unlike traditional techniques, a method was proposed that integrates the iterative conjugate gradient method with Tikhonov regularization. The regularization matrix, step size, and search direction coefficients were used as trainable parameters in a deep unfolding approach. Extensive simulation results that demonstrate the superiority of the proposed method over existing methods were presented.  
The results of this study clearly demonstrate that the proposed method, DU-TCG, is an improved detection strategy for MIMO systems, outperforming both state-of-the-art deep learning techniques such as DetNet and LCG, as well as traditional approaches such as MMSE and CG. Combining the conjugate gradient method with the Tikhonov regularization approach in a deep learning framework successfully reduces the degrading effects of channel conditions and higher-order modulation.  
The scalability of DU-TCG is demonstrated by its consistent performance in both large (32 \(\times\) 128) and small (10 \(\times\) 10) MIMO systems, providing broad applicability to MIMO configurations of varying size and complexity. Additionally, the better performance of the DU-TCG is also shown under various channel conditions such as Kronecker, TDL-A, and TDL-E channel models, which are widely used in advanced communication systems. This research also reveals that DU-TCG surpasses DetNet in a 32 \(\times\) 64 MIMO layout when system complexity increases. Simulation results also show that the proposed method provides up to 4 dB SNR gain compared to CG with considerably less iterations. In addition, DU-TCG stays ahead of CG and LCG as the number of layers increase, even surpassing LCG's high-layer performance with fewer layers. Ultimately, as compared to the vector parameterization of LCG, the NMSE and BER performances of DU-TCG are superior. For both scalar and vector parameterization, DU-TCG decreased the NMSE values.  
This work combines Tikhonov regularization and the CG technique with deep unfolding, which significantly improves MIMO signal detection under various MIMO layouts, modulation orders, and channel conditions. The proposed method inserts a regularization term to the system, which improves the stability and generality of the solution. In particular, when there are noisy or imperfect data conditions, this regularization helps the CG converge more consistently. Thus, the approach efficiently handles the complexity present in high-dimensional MIMO systems while iteratively improving its signal estimates.  
Consequently, this study highlights the advantages of the proposed DU-TCG method for MIMO detection over different scenarios. As a future work, combining Tikhonov regularization with other iterative detection algorithms for MIMO systems may be considered. Additionally, reducing computational complexity of model-driven approaches is also another challenge that needs to be discussed in the area of deep-unfolded algorithms.  
Conceptualization, S.N.K.; methodology, S.N.K. and A.K.; software, S.N.K.; validation, S.N.K. and A.K.; formal analysis, S.N.K. and A.K.; investigation, S.N.K. and A.K.; resources, S.N.K. and A.K.; data curation, S.N.K.; writing--original draft preparation, S.N.K. and A.K.; writing--review and editing, A.K.; visualization, S.N.K.; supervision, A.K. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Data Availability StatementData set available on request from the authors.  
The authors declare no conflicts of interest.