# Improved Likelihood Probability in MIMO Systems Using One-Bit ADCs
## 4 Proposed Method
### Proposed Receiver  
The proposed receiver shown in Figure 2 is described in this subsection. Because the optimization problem in (15) improves the expected likelihood probability, the likelihood probability is updated for every data block \(d\). Initially, the proposed receiver obtains the channel estimates during the pilot transmission. The pilot-based likelihood probability \(p_{r,k}^{pi}\) is computed using (8). During data transmission, the receiver detects the data symbol based on the ML criterion expressed in (20). Using the estimated likelihood probability, the detection method can be expressed as follows:  
\[\hat{k}[n] =\operatorname*{argmax}_{k\in\mathcal{K}}\hat{\mathbb{P}}(\mathbf{q} [n]|\mathbf{s}[n]=\mathbf{s}_{k})\] \[=\operatorname*{argmax}_{k\in\mathcal{K}}\prod_{r:\hat{\mu}:[n]=1 }\hat{\rho}_{r,k}\prod_{r:\hat{\mu}:[n]=-1}(1-\hat{\rho}_{r,k}). \tag{20}\]  
The detected data symbol is used to update the data-based likelihood probability \(p_{r,k}^{da}\) based on (13). In addition, the estimated likelihood probability \(\hat{\rho}_{r,k}\) is used in the log-likelihood ratio (LLR) calculation, which is expressed as  
\[ \begin{split}\text{LLR}(b_{ij}[n])&=\log\frac{ \mathbb{P}(\mathbf{q}[n]|b_{i,j}[n]=1)}{\mathbb{P}(\mathbf{q}[n]|b_{i,j}[n]=0)} =\log\frac{\sum\limits_{\mathbf{s}_{k}\in\mathcal{X}_{ij}^{1}}\mathbb{P}( \mathbf{q}[n]|\mathbf{s}[n]=\mathbf{s}_{k})}{\sum\limits_{\mathbf{s}_{k}\in \mathcal{X}_{ij}^{0}}\mathbb{P}(\mathbf{q}[n]|\mathbf{s}[n]=\mathbf{s}_{k})} \\ &\overset{(a)}{=}\log\frac{\sum\limits_{\mathbf{s}_{k}\in \mathcal{X}_{ij}^{1}}\hat{\mathbb{P}}(\mathbf{q}[n]|\mathbf{s}[n]=\mathbf{s}_ {k})}{\sum\limits_{\mathbf{s}_{k}\in\mathcal{X}_{ij}^{0}}\hat{\mathbb{P}}( \mathbf{q}[n]|\mathbf{s}[n]=\mathbf{s}_{k})}=\log\frac{\sum\limits_{\mathbf{s} _{k}\in\mathcal{X}_{ij}^{1}}\left(\prod\limits_{r:\hat{\mu}:[n]=1}\hat{\rho}_{ r,k}\prod\limits_{r:\hat{\mu}:[n]=-1}(1-\hat{\rho}_{r,k})\right)}{ \sum\limits_{\mathbf{s}_{k}\in\mathcal{X}_{ij}^{0}}\left(\prod\limits_{r: \hat{\mu}:[n]=1}\hat{\rho}_{r,k}\prod\limits_{r:\hat{\mu}:[n]=-1}(1-\hat{\rho} _{r,k})\right)},\end{split}  \tag{21}\]  
where estimated likelihood probability in (20) is applied to true likelihood probability \((a)\) because the true likelihood probability is unknown at the receiver. \(b_{i,j}[n]\in\{0,1\}\) is the \(j\)-th bit of the \(i\)-th stream and \(\mathcal{X}_{i,j}^{b}\) is the set of symbols \(\mathbf{s}_{k}\) for which \(b_{i,j}=b\in\{0,1\}\). This LLR value is used to obtain the decoded data \(\hat{b}_{i,j}\) from the channel decoder. After the data block ends, the proposed receiver updates the estimated likelihood probability \(\hat{\rho}_{r,k}\) using the pilot- and data-based likelihood probabilities. The trained weights \((\alpha^{*},\beta^{*})\) in memory are used based on the data block number \(d\). The estimated likelihood probability was used for the subsequent data block. The sequential algorithm for the proposed receiver is presented in Algorithm 1. Figure 2: Proposed receiver with the linear combination of pilot- and data-based likelihood probabilities. ```
1 Pilot transmission
2 Compute initial channel estimate \(\hat{\mathbf{H}}=\left[\hat{\mathbf{h}}_{1},\cdots,\hat{\mathbf{h}}_{N_{r}}\right]\). 3 Obtain pilot-based likelihood probability \(p_{r,k}^{pi}\), \(r\in\mathcal{R},k\in\mathcal{K}\) using (8). 4 Set initial estimated likelihood probability \(\hat{p}_{r,k}=p_{r,k}^{pi},r\in\mathcal{R},k\in\mathcal{K}\). 5 Data transmission
6 for\(d=1\) to \(D\)do
7 for\(n=1\) to \(N_{d}\)do
8 Detect data symbol based on ML criterion (20). 9 Calculate data-based likelihood probability \(p_{r,k}^{da}\) using (13). 10 Compute LLR based on (21). 11 Obtain decoded data \(\hat{b}_{i,j}\) from channel decoder. 12
13 end for
14 Obtain trained weights \((\alpha,\beta)\) from memory obtained using (19). 15 Update estimated likelihood probability \(\hat{p}_{r,k}=\alpha p_{r,k}^{pi}+\beta p_{r,k}^{da}\) from (14).