# Regression of Likelihood Probability for Time-Varying MIMO Systems with One-Bit ADCs
## 4 Proposed Solution
### Proposed Receiver  
This subsection describes the proposed receiver structure. The receiver initiates by estimating the channel during pilot transmission. The initial probability \(\hat{p}_{r,k}[0]\) can then be obtained using the channel estimate. During data transmission, the receiver calculates the probability proposed in (12). The calculated likelihood probability is used in the log-likelihood ratio (LLR) calculation, given by  
\[\begin{split} LLR(b_{i,j}[n])=\log\frac{\mathbb{P}(\mathbf{q}[n]|b _{i,j}[n]=1)}{\mathbb{P}(\mathbf{q}[n]|b_{i,j}[n]=0)}=\log\frac{\sum\limits_{ \mathbf{s}_{k}\in\mathcal{X}_{i,j}^{1}}\mathbb{P}(\mathbf{q}[n]|\mathbf{s}[n ]=\mathbf{s}_{k})}{\sum\limits_{\mathbf{s}_{k}\in\mathcal{X}_{i,j}^{0}} \mathbb{P}(\mathbf{q}[n]|\mathbf{s}[n]=\mathbf{s}_{k})}\\ \stackrel{{(a)}}{{=}}\log\frac{\sum\limits_{ \mathbf{s}_{k}\in\mathcal{X}_{i,j}^{1}}\hat{\mathbb{P}}(\mathbf{q}[n]|\mathbf{s }[n]=\mathbf{s}_{k})}{\sum\limits_{\mathbf{s}_{k}\in\mathcal{X}_{i,j}^{0}} \widehat{\mathbb{P}}(\mathbf{q}[n]|\mathbf{s}[n]=\mathbf{s}_{k})}=\log\frac{ \sum\limits_{\mathbf{s}_{k}\in\mathcal{X}_{i,j}^{1}}\left(\prod\limits_{r:q _{r}[n]=1}\hat{p}_{r,k}^{\text{pro}}\prod\limits_{r:q_{r}[n]=-1}(1-\hat{p}_{r, k}^{\text{pro}})\right)}{\sum\limits_{\mathbf{s}_{k}\in\mathcal{X}_{i,j}^{0}} \left(\prod\limits_{r:q_{r}[n]=1}\hat{p}_{r,k}^{\text{pro}}\prod\limits_{r:q_{ r}[n]=-1}(1-\hat{p}_{r,k}^{\text{pro}})\right)},\end{split} \tag{24}\]  
where the proposed likelihood probability in (17) is applied to the true likelihood probability \((a)\). \(b_{i,j}[n]\in\{0,1\}\) 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.  
At the end of the data block, the receiver computes the channel-based likelihood probability \(p_{r,k}^{\text{channel}}\) using (12) and the sample-based likelihood probability using (16) based on the channel decoder output. Following these calculations, the proposed receiver updates the combined likelihood probabilities \(\hat{p}_{r,k}^{\text{pro}}\) by integrating both the channel- and sample-based likelihood values. The optimal weights \(\mathbf{\Theta}_{d}^{*}\), stored in memory, are selected according to the data block index \(d\) to effectively combine two likelihood probabilities. The complete step-by-step procedure of the proposed receiver is presented in Algorithm 1.  
```
1 Pilot Block
2 Obtain initial channel estimate \(\mathbf{\hat{H}}[0]=\left[\mathbf{\hat{h}}_{1}[0],\cdots,\mathbf{\hat{h}}_{N_ {k}}[0]\right]\).
3 Calculate the initial channel-based likelihood probability \(p_{r,k}^{\text{channel}}[0]\), \(r\in\mathcal{R},k\in\mathcal{K}\) from \(\mathbf{\hat{H}}[0]\).
4 Set the proposed likelihood probability \(\hat{p}_{r,k}^{\text{pro}}\leftarrow p_{r,k}^{\text{channel}}[0]\), \(r\in\mathcal{R},k\in\mathcal{K}\).
5 Data Block
6 for\(d=1\) to \(D\)do
7 for\(n=1\) to \(N_{d}\)do
8 Compute LLR in (24) based on the proposed likelihood probability \(\hat{p}_{r,k}^{\text{pro}}\).
9 Obtain decoded data \(\hat{b}_{i,j}\) from the channel decoder.
10 end for
11 Bring the optimal weight \(\mathbf{\Theta}_{d}^{*}\) in memory trained using (21).
12 Update channel-based likelihood probability \(p_{r,k}^{\text{channel}}\) from (12) and \(\mathbb{E}[p_{r,k}[dN_{d}]]=\hat{p}_{r,k}^{\text{channel}}[dN_{d}]\).
13 The sample-based likelihood probabilities \(p_{r,k}^{\text{sample}}\) are updated from the decoding output in (16).
14 Calculate the proposed likelihood probability \(\hat{p}_{r,k}^{\text{pro}}\leftarrow(\theta_{d}^{0})^{*}+(\theta_{d}^{1})^{*}p_{ r,k}^{\text{channel}}[dN_{d}]+(\theta_{d}^{2})^{*}p_{r,k}^{\text{sample}}[dN_{d}]\) using (22).
15 end for
```  
**Algorithm 1**Procedure of the Proposed Receiver