# Computationally Efficient Soft Detection Schemes for Coded Massive MIMO Systems +
## 3 Proposed Method
### Algorithm for the Proposed Methods and Complexity Comparisons  
With the proposed techniques, diagonal elements of the Gram matrix are approximately represented by a universal constant value, and the estimation of soft-symbol values are performed with  
\begin{table}
\begin{tabular}{c c c c} \hline \(K\) & \(A\) & \(\widetilde{s}_{i}\) & \(E_{i}\) \\ \hline
4 & \(1/\sqrt{10}\) & \(A(-2\zeta_{i,1}+\zeta_{i,1}\zeta_{i,3})+jA(-2\zeta_{i,2}+\zeta_{i,2}\zeta_{i,A})\) & \(A^{2}(10-4\zeta_{i,3  
much lower complexity. Eventually, it results in that the computational complexity of matrix inversion is greatly reduced to a linear order, and the estimation of the PE-SINR is also greatly simplified. The complete process of the proposed method is summarized in Algorithm 1, where \(\eta\) and \(l\) denote the number of iterations between the MMSE-PIC detector and decoder, and the number of iterations inside the detector, respectively. ```
Initialize: \(\mathbf{L}_{a}^{d}=0\), \(\mathbf{L}=0\) for\(1\leq\eta\leq\eta_{max}\)do for\(1\leq l\leq l_{max}\)do  Calculate \(\zeta_{i,k}\) with (4), \(\widetilde{s}_{i}\) and \(E_{i}\) with Table 1 \(\widetilde{w}_{i,i}=\frac{N}{K}E_{i}+\sigma^{2}\) \(\hat{z}_{i}=\mathbf{h}_{i}^{H}\hat{\mathbf{y}}_{i}/\widetilde{w}_{i,i}\) \(\mu_{i}=\frac{N}{\frac{N}{K}E_{i}+\sigma^{2}}\), and \(\rho_{i}=\frac{N}{M\sigma^{2}}\) \(L(x_{i,k}|\mathbf{y},\mathbf{H})\) estimation with (11) endfor  Iterative decoding, output \(\mathbf{L}_{a}^{d}\) endfor
```  
**Algorithm 1** Pseudo-code for the proposed complexity reduced MMSE-PIC-based JIDD algorithm  
The computational complexity of the proposed method is compared with two conventional schemes in terms of the number of multiplications per iteration, and the results are shown in Table 2. We compare the computational complexity of major computations during the MMSE-PIC process, which are matrix-inversion processes as well as calculating \(\mathbf{G}\), \(\mu_{i}\), \(\widetilde{s}_{i}\), and \(E_{i}\) as listed in Table 2. As shown in the table, the computational complexity of matrix-inversion is reduced to a linear order in the proposed scheme, benefiting from the diagonal approximation. The complexity of the Gram-matrix estimation in the proposed scheme is reduced to \(O(1)\) due to the further approximation with a universal constant value. Given \(\mathbf{W}^{-1}\) and \(\mathbf{G}\), the computational complexities of estimating \(\mu_{i}\) in the conventional schemes are \(O(M^{2})\), while that of the proposed scheme is reduced to \(O(M)\). Especially, we compare the complexity of calculating \(\widetilde{s}_{i}\) and \(E_{i}\) in more details. Table 3 compares the number of multiplications required in the conventional schemes using (2) and (3), with the proposed compact methods using (19)-(24). To estimate \(\widetilde{s}_{i}\) with (2), we need \(K-1\) multiplications in \(\prod_{k=1}^{K}\) for each constellation symbol \(a\), in addition to three multiplications with \(\Re(a)\), \(\Im(a)\), and \(1/2^{K}\), respectively, and thus in total we need \((K-1+3)2^{K}\) multiplications. Similarly, to estimate \(E_{i}\) in (3), we need \(K-1\) multiplications in \(\prod_{k=1}^{K}\) for each constellation symbol \(a\), in addition to two multiplications with \(1/2^{K}\) and \(|a|^{2}\), respectively, and then two multiplication for \(|\widetilde{s}_{i}|^{2}\). Therefore, we need \(((K-1+2)2^{K}+2)\) multiplications in total. On the other hand, the proposed method reduces the number of multiplications from \(O(K)\) to \(O(K/2)\), due to symbol projections to real and imaginary axes, respectively. Therefore, we need \(2(K/2-1+2)2^{K/2}\) multiplications to estimate \(\widetilde{s}_{i}\), and \((2(K/2-2+2)2^{K/2-1}+2)\) multiplications to estimate \(E_{i}\). We note that if we use the expanded results in Table 1, then the complexity is less than 1% of the direct estimation of (2) and (3).