# Deep-Unfolded Tikhonov-Regularized Conjugate Gradient Algorithm for MIMO Detection
## 2 Materials and Methods
### A Summary of MIMO Detection Methods  
MIMO detection methods are simply classified into two categories as linear and non-linear methods, where non-linear methods consist of non-iterative, iterative, and deep learning approaches. Additionally, deep learning-based methods encompass both data-driven and deep-unfolded methodologies also called as model-driven. Within this section, we will only discuss the detection methods that are in the focus of this study, CG and learned CG methods, which are classified as iterative and model-driven deep learning methods, respectively. Details of the other well-known methods such as ML, MMSE, Sphere Decoder, and DetNet can be found in [16; 34; 35; 36]. The conjugate gradient (CG) method, introduced by [PERSON] and [PERSON] in 1952 [37], is an iterative algorithm for solving systems of linear equations. CG as an iterative method is particularly useful in situations where direct methods are infeasible due to the large dimensions of the matrices involved. The CG method, summarized in Algorithm 1, provides an efficient and robust solution for signal detection in MIMO systems. ```
Inputs:\(y,H,\delta^{2}\) Output: Transmitted signal vector estimation \(\hat{s}\)
1: Initialization:

\(A=H^{H}H+\delta^{2}N_{N_{T}}\), \(b=H^{H}y\)

\(x_{0}=0\), \(r_{0}=b\), \(r_{0}=r_{0}\)

2:for\(i=0,\ldots,K\)do

3:\(\alpha_{i}=r_{i}^{T}r_{i}/p_{i}^{T}Ap_{i}\)

4:\(x_{i+1}=x_{i}+a_{i}p_{i}\)

5:\(r_{i+1}=r_{i}-a_{i}Ap_{i}\)

6:\(\beta_{i}=r_{i+1}^{T}r_{i+1}/r_{i}^{T}r_{i}\)

7:\(p_{i+1}=r_{i+1}-\beta_{i}p_{i}\)

8:end for

9:return\(\hat{s}=x_{i+1}\)
```  
**Algorithm 1**MIMO Detection with CG  
The CG method starts with the provided inputs, the channel matrix \(H\), the noise variance \(\delta^{2}\), and the received signal \(y\), and the algorithm initializes the relevant matrix and vectors. The search direction vector \(p_{i}\), the residual vector \(r_{i}\), and the solution vector \(x_{i}\) are set first. At each iteration, the solution vector is updated by calculating the step size \(\alpha\), which indicates how far to go in the current search direction. The new solution estimate is then reflected in the residual vector. The algorithm determines a coefficient, \(\beta\), which modifies the search direction to maintain efficiency in subsequent iterations. This iterative process continues until the desired number of iterations is reached or the solution is sufficiently accurate. Deep learning can also be applied to MIMO detection methods. Thus, employing the deep unfolding technique on iterative detection procedures provides faster convergence and higher generalization within modeled behaviors and requires less training data in iterative detection procedures. Incorporating deep learning techniques to adaptively find the optimal parameters during iterations ensures that the learned conjugate gradient method (LCG) [28] has higher performance over the traditional CG method. The LCG method, shown in Figure 2, alters the conjugate gradient algorithm by making the step size and search direction coefficient parameters, denoted by \(\alpha\) and \(\beta\), respectively, modifiable through training instead of mathematically calculating them on the relevant data sets. The method can now adaptively adjust its step sizes and directions, which enhances detection performance in MIMO systems and facilitates more effective convergence. The limitations of fixed-parameter approaches can be overcome by LCG through the utilization of data driven insights, thereby providing a robust foundation for the management of the complex and variable conditions of wireless communication channels. As explained in detail in [28], two types of LCG approaches are introduced, namely scalar, LCG-S, and vector, LCG-V. Scalar trainable parameters \(\alpha\) and \(\beta\) are used by LCG-S. By eliminating the requirement for matrix-vector multiplications and divisions, these scalar parameters simplify computations. Alternatively, vector trainable parameters \(\alpha^{i}\) and \(\beta^{i}\) are used by LCG-V.