# Computationally Efficient Soft Detection Schemes for Coded Massive MIMO Systems +
## 2 Related Works
### MMSE-PIC Detection for MIMO System  
The MMSE-PIC detector in Figure 1 needs to estimate soft symbol values in order to incorporate soft information from the channel decoder. The first step is to estimate the expected soft-symbol value \(\widetilde{s}_{i}\) and the variance \(E_{i}\) of the transmitted symbol \(s_{i}\), by using the _a priori_ information, as follows [3; 14]:  
\[\widetilde{s}_{i}=\sum_{a\in\mathrm{O}}\frac{a}{2^{K}}\prod_{k=1}^{K}(1+ \widetilde{x}_{i,k}\zeta_{i,k}), \tag{2}\]  
\[E_{i}=\sum_{a\in\mathrm{O}}\frac{|a|^{2}}{2^{K}}\prod_{k=1}^{K}(1+\widetilde {x}_{i,k}\zeta_{i,k})-|\widetilde{s}_{i}|^{2}, \tag{3}\]  
where \(a\) is a constellation symbol of \(\mathrm{O}\), \(\widetilde{x}_{i,k}\) is set to be \(-1\) and \(1\) according to the \(k\)th bits of \(a\), and \(\zeta_{i,k}\) is considered as zero at the first stage, and in the following iterations, \(\zeta_{i,k}\) can be calculated as:  
\[\zeta_{i,k}=\tanh((L(x_{i,k})+L_{a}^{d}(x_{i,k}))/2). \tag{4}\]  
Then, we perform the parallel interference cancellation (PIC) process on the received symbol vector to cancel the interference between layers, and the interference-cancelled symbol vector for the \(i\)th layer, \(\hat{\mathbf{y}}_{i}\) is calculated by using the following equation:  
\[\hat{\mathbf{y}}_{i}=\mathbf{y}-\sum_{j\
eq i}\mathbf{h}_{i}\widetilde{s}_{j }=\mathbf{h}_{i}s_{i}+\widetilde{\mathbf{n}}, \tag{5}\]  
where \(\mathbf{h}_{i}\) denotes the \(i\)th column of \(\mathbf{H}\), and \(\widetilde{\mathbf{n}}=\sum_{j\
eq i}\mathbf{h}_{j}(s_{i}-\widetilde{s}_{i}) +\mathbf{n}\).  
Figure 1: Coded multiple-input multiple-output (MIMO) system with minimum mean square error parallel interference cancellation (MMSE-PIC)-based joint iterative detection and decoding (JIDD).  
The next step is the suppression of noise plus interference (NPI) term in (5) with the following equation:  
\[\hat{z}_{i}=\mathbf{w}_{i}^{H}\hat{\mathbf{y}}_{i}, \tag{6}\]  
where \(\mathbf{w}_{i}^{H}\) is the \(i\)th row of the MMSE filtering matrix, \(\mathbf{W}^{H}\), which is calculated as follows [16]:  
\[\mathbf{W}^{H}=(\mathbf{H}^{H}\mathbf{H}\boldsymbol{\Lambda}+\sigma^{2} \mathbf{I}_{M})^{-1}\mathbf{H}^{H}=\tilde{\mathbf{W}}^{-1}\mathbf{H}^{H}, \tag{7}\]  
where \(\boldsymbol{\Lambda}\) is a diagonal matrix whose diagonal element \(\lambda_{i,i}\) equals to \(E_{i}\) calculated in (3).  
For the estimation of SBI in the form of log-likelihood ratio (LLR) to the decoder, the channel-compensated value, \(z_{i}=\hat{z}_{i}/(\mathbf{w}_{i}^{H}\mathbf{h}_{i})\) needs to be used as follows [16], so that the SBI estimation is not subject to the channel gain.  
\[\begin{split} L(x_{i,k}|\mathbf{y},\mathbf{H})& \approx\min_{a\in O_{k}^{0}}\left(\rho_{i}\left|z_{i}-a\right|^{2}+ \sum_{k=1}^{K}\ln\left(1+e^{(-x_{i,k}L_{a}^{d}(x_{i,k}))}\right)\right)\\ &-\min_{a\in O_{k}^{1}}\left(\rho_{i}\left|z_{i}-a\right|^{2}+ \sum_{k=1}^{K}\ln\left(1+e^{(-x_{i,k}L_{a}^{d}(x_{i,k}))}\right)\right),\end{split} \tag{8}\]  
where \(a\in O_{k}^{0}\) and \(a\in O_{k}^{1}\) represent the constellation symbols with the \(k\)th bit of 0 and 1, respectively, and the PE-SINR for the \(i\)th transmitted symbol can be estimated as:  
\[\rho_{i}=\frac{\mu_{i}}{1-E_{i}\mu_{i}}, \tag{9}\]  
where  
\[\mu_{i}=\mathbf{w}_{i}^{H}\mathbf{h}_{i}. \tag{10}\]  
Estimation of SBI using (8) requires exhaustive search to find the minima. A complexity reduced scheme which eliminates the search process to find the minima can be used as follows [17]:  
\[L(x_{i,k}|\mathbf{y},\mathbf{H})\approx\omega_{i,k}\rho_{i}\left(\left|\oslash( z_{i},e_{k}^{0})-q_{k}^{0}\right|^{2}-\left|\oslash(z_{i},e_{k}^{1})-q_{k}^{1} \right|^{2}\right), \tag{11}\]  
where \(\omega_{i,k}\) is a sign function introduced by symbol mapping, and \(q_{k}^{b}\) is the unique symbol nearest to the mapped \(z_{i}\). In addition, \(\oslash(z_{i},e_{k}^{b})=\left|z_{i}\right|e^{ik_{i}^{b}}\), where \(e_{k}^{b},b\in\{0,1\}\) represents the phase of mapped symbol for the \(k\)th bit of \(b\).