# Comparative Analysis of Data Detection Techniques for 5G Massive MIMO Systems
## 3 Proposed Methods  
The equalization matrix (\(\mathbf{A}\)) is diagonally dominant and utilization of the diagonal matrix (\(\mathbf{D}\)) leads to a good convergence. In this paper, we exploit the diagonally dominant property of \(\mathbf{A}\) in initializing the AMP-based detector instead of a zero vector. In other words, \(\hat{\mathbf{x}}_{(0)}\) is computed based on \(\mathbf{D}\). Then, low complexity iterative methods are utilized to estimate the signal \(\hat{\mathbf{x}}_{(1)}\). Figure 2 shows the general block diagram of proposed detector. Details and mathematical description are shown below:  
* Step 1: Compute the initial solution \(\hat{\mathbf{x}}_{(0)}\) as \[\hat{\mathbf{x}}_{(0)}=\mathbf{D}^{-1}\mathbf{y}_{MF}.\] (9)
* Step 2: Apply the SOR, GS, or JA methods where \(n=0\), as
* If SOR: \[\hat{\mathbf{x}}_{(1)}=(\mathbf{D}-\frac{1}{\omega}\mathbf{L})^{-1}(\mathbf{y }_{MF}+((1-\frac{1}{\omega})\mathbf{D}+\frac{1}{\omega}\mathbf{U})\hat{ \mathbf{x}}_{(0)}).\] (10)
* If GS: \[\hat{\mathbf{x}}_{(1)}=(\mathbf{D}-\mathbf{L})^{-1}(\mathbf{y}_{MF}+\mathbf{ U}\hat{\mathbf{x}}_{(0)}).\] (11)
* If JA: \[\hat{\mathbf{x}}_{(1)}=\mathbf{D}^{-1}(\hat{\mathbf{x}}_{MF}+(\mathbf{D}- \mathbf{A})\hat{\mathbf{x}}_{(0)}).\] (12)
* Step 3: Apply the AMP algorithm as shown in (8), where \(n=1\).  
Algorithms 1-3 show the proposed iterative detectors based on the AMP algorithm, the SOR, the GS, and JA methods. Each of them contains the initialization, iterations, and the final estimation. In addition, the main difference between the three of them is the initialization part.  
```
Input:\(\mathbf{y},\mathbf{H},\sigma^{2},n,\omega\) Output: Estimated signal \(\mathbf{\hat{x}}\) Initialization: \(\mathbf{A}=\mathbf{H}^{H}\mathbf{H}+\sigma^{2}\mathbf{I}_{K}\) \(\mathbf{D}=diag(\mathbf{A})\), \(\mathbf{U}=-triu(\mathbf{A})\), \(\mathbf{L}=-tril(\mathbf{A})\) \(\mathbf{y}_{MF}=\mathbf{H}^{H}\mathbf{y}\)  Initial estimations: \(\mathbf{z}_{(0)}=[0\cdots 0]\) \(\mathbf{\hat{x}}_{(0)}=\mathbf{D}^{-1}\mathbf{y}_{MF}\) \(\mathbf{\hat{x}}_{(1)}=\left(\mathbf{D}-\frac{1}{\omega}\mathbf{L}\right)^{- 1}\left(\mathbf{y}_{MF}+\left(\left(1-\frac{1}{\omega}\right)\mathbf{D}+\frac {1}{\omega}\mathbf{U}\right)\mathbf{\hat{x}}_{(0)}\right)\) Iteration:  for j = 2 : 1 : \(n\)  Apply the AMP algorithm using (6)-(8)  end Return\(\mathbf{\hat{x}}\).
```  
**Algorithm 1**Initialize the AMP using the SOR method  
```
Input:\(\mathbf{y},\mathbf{H},\sigma^{2},n,\omega\) Output: Estimated signal \(\mathbf{\hat{x}}\) Initialization: \(\mathbf{A}=\mathbf{H}^{H}\mathbf{H}+\sigma^{2}\mathbf{I}_{K}\) \(\mathbf{D}=diag(\mathbf{A})\), \(\mathbf{U}=-triu(\mathbf{A})\), \(\mathbf{L}=-tril(\mathbf{A})\) \(\mathbf{y}_{MF}=\mathbf{H}^{H}\mathbf{y}\)  Initial estimations: \(\mathbf{z}_{(0)}=[0\cdots 0]\) \(\mathbf{\hat{x}}_{(0)}=\mathbf{D}^{-1}\mathbf{y}_{MF}\) \(\mathbf{\hat{x}}_{(1)}=\left(\mathbf{D}-\mathbf{L}\right)^{-1}\left(\mathbf{y }_{MF}+\mathbf{U}\mathbf{\hat{x}}_{(0)}\right)\) Iteration:  for j = 2 : 1 : \(n\)  Apply the AMP algorithm using (6)-(8)  end Return\(\mathbf{\hat{x}}\).
```  
**Algorithm 2**Initialize the AMP detector using the GS method  
Figure 2: Block diagram of proposed detector.