# Computationally Efficient Soft Detection Schemes for Coded Massive MIMO Systems +
## 3 Proposed Method
### Soft-Symbol Value Estimation  
Direct estimation of (2) and (3) requires exponentially increasing number of computations by the number of bits in a symbol. If we use symmetric property of QAM constellations, then we can highly reduce the complexity. For this, we first decompose the complex value \(\widetilde{s}_{i}\) into real and imaginary parts as follows:  
\[\widetilde{s}_{i}=\Re(\widetilde{s}_{i})+j\Im(\widetilde{s}_{i}), \tag{18}\]  
where \(\Re(x)\) and \(\Im(x)\) are the real and imaginary parts of the complex number \(x\). With the above decomposition, computations in (2) is now performed with real number operations, and \(\Re(\widetilde{s}_{i})\) can be derived as follows. \[ \begin{split}\Re(\widetilde{s}_{i})&=\sum_{a\in \mathbb{O}}\frac{\Re(a)}{2^{K}}\prod_{k=1}^{K}(1+\widetilde{x}_{i,k}\xi_{i,k}) \\ &=\sum_{\Re(a)\in\mathbb{O}^{K}}\sum_{\Im(a)\in O^{I}}\frac{\Re(a )}{2^{K}}\prod_{k\in\pi^{K}}(1+\widetilde{x}_{i,k}\xi_{i,k})\prod_{k\in\pi^{I} }(1+\widetilde{x}_{i,k}\xi_{i,k})\\ &=\sum_{\Re(a)\in\mathbb{O}^{K}}\frac{\Re(a)}{2^{K}}\prod_{k\in \pi^{K}}(1+\widetilde{x}_{i,k}\xi_{i,k})\sum_{\Im(a)\in O^{I}}\prod_{k\in\pi^{ I}}(1+\widetilde{x}_{i,k}\xi_{i,k})\\ &=\sum_{\Re(a)\in\mathbb{O}^{K}}\frac{\Re(a)}{2^{K}}\prod_{k\in \pi^{K}}(1+\widetilde{x}_{i,k}\xi_{i,k})2^{K/2}\\ &=\sum_{\Re(a)\in\mathbb{O}^{K}}\frac{\Re(a)}{2^{K/2}}\prod_{k \in\pi^{K}}(1+\widetilde{x}_{i,k}\xi_{i,k}),\end{split}  \tag{19}\]  
where \(O^{R}=\{a|a\) is a symbol projected on to the real axis], \(\kappa^{R}\) represents a set of bit indexes of the symbols which are projected to real axis. As shown in (19), the number of computations in \(\prod_{k=1}^{K}(1+\widetilde{x}_{i,k}\xi_{i,k})\) of (2) is reduced from \(2^{K}\) to \(2^{K/2}\). We derive \(\Im(\widetilde{s}_{i})\) with the same way as follows:  
\[ \begin{split}\Im(\widetilde{s}_{i})&=\sum_{a\in \mathbb{O}}\frac{\Im(a)}{2^{K}}\prod_{k=1}^{K}(1+\widetilde{x}_{i,k}\xi_{i,k}) \\ &=\sum_{\Im(a)\in O^{I}}\frac{\Im(a)}{2^{K/2}}\prod_{k\in\pi^{I} }(1+\widetilde{x}_{i,k}\xi_{i,k}),\end{split}  \tag{20}\]  
where \(O^{I}=\{a|a\) is a symbol projected on to the imaginary axis], \(\kappa^{I}\) represent the bit indexes of the symbols which are projected to imaginary axis. Now we prove that \(E_{i}\) can be decomposed into \(E_{i}^{R}\) and \(E_{i}^{I}\) which are the variances estimated for \(\Re(s_{i})\) and \(\Im(s_{i})\), respectively, as follows:  
\[ \begin{split} E_{i}&=\sum_{a\in\mathbb{O}}\frac{|a| ^{2}}{2^{K}}\prod_{k=1}^{K}(1+\widetilde{x}_{i,k}\zeta_{i,k})-|\widetilde{s}_{ i}|^{2}\\ &=\sum_{\Re(a)\in O^{K}}\sum_{\Im(a)\in O^{I}}\frac{\Re(a)^{2}}{ 2^{K}}\prod_{k\in\pi^{K}}(1+\widetilde{x}_{i,k}\zeta_{i,k})\prod_{k\in\pi^{I}} (1+\widetilde{x}_{i,k}\zeta_{i,k})-|\widetilde{s}_{i}|^{2}\\ &=\sum_{\Re(a)\in O^{K}}\sum_{\Im(a)\in O^{I}}\frac{\Re(a)^{2}}{ 2^{K}}\prod_{k\in\pi^{K}}(1+\widetilde{x}_{i,k}\xi_{i,k})\prod_{k\in\pi^{I}}( 1+\widetilde{x}_{i,k}\zeta_{i,k})\\ &\quad+\sum_{\Re(a)\in O^{K}}\sum_{\Im(a)\in O^{I}}\frac{\Im(a)^{2 }}{2^{K}}\prod_{k\in\pi^{K}}(1+\widetilde{x}_{i,k}\zeta_{i,k})\prod_{k\in\pi^{ I}}(1+\widetilde{x}_{i,k}\zeta_{i,k})-|\widetilde{s}_{i}|^{2}\\ &=\sum_{\Re(a)\in O^{K}}\frac{\Re(a)^{2}}{2^{K/2}}\prod_{k\in\pi^{ K}}(1+\widetilde{x}_{i,k}\xi_{i,k})-\Re(\widetilde{s}_{i})^{2}+\sum_{\Im(a)\in O^{I}} \frac{\Im(a)^{2}}{2^{K/2}}\prod_{k\in\pi^{I}}(1+\widetilde{x}_{i,k}\xi_{i,k})- \Im(\widetilde{s}_{i})^{2}\\ &=E_{i}^{R}+E_{i}^{I},\end{split}  \tag{21}\]where  
\[ \begin{split} E_{i}^{R}&=\sum_{\Re(a)\in O^{R}}\frac{\Re(a )^{2}}{2^{K/2}}\prod_{k\in\kappa^{R}}(1+\widetilde{x}_{i,k}\zeta_{i,k})-\Re( \widetilde{s}_{i})^{2},\\ E_{i}^{I}&=\sum_{\Im(a)\in O^{I}}\frac{\Im(a)^{2}}{2 ^{K/2}}\prod_{k\in\kappa^{I}}(1+\widetilde{x}_{i,k}\zeta_{i,k})-\Im(\widetilde{ s}_{i})^{2}.\end{split}  \tag{22}\]  
With the above decomposition, we further reduce the computational complexity of (22) by expanding \(E_{i}^{R}\) as follows:  
\[