# Regression of Likelihood Probability for Time-Varying MIMO Systems with One-Bit ADCs
## 2 System Model
### Receiver  
The wireless channel follows a Rayleigh fading model and is assumed to be time-variant. When the Gaussian Markov process is used to describe the time-varying nature of the channel, the channel \(\bar{\mathbf{H}}[n]\in\mathbb{C}^{N_{t}\times N_{t}}\) at time \(n\) is given by  
\[\bar{\mathbf{H}}[n]=\epsilon\bar{\mathbf{H}}[n-1]+\sqrt{1-\epsilon^{2}}\bar{ \Delta}[n], \tag{1}\]  
where \(\epsilon\in[0,1]\) is the temporal correlation coefficient, which is defined as \(\epsilon=J_{0}(\frac{2\pi f_{c}\
u T_{c}}{c})\), where \(f_{c}\), \(\
u\), \(T_{s}\), and \(c\) denote the carrier frequency, vehicle velocity, symbol duration, and speed of light, respectively. \(\bar{\Delta}[n]\in\mathbb{C}^{N_{\mathrm{R}_{c}}\times N_{\mathrm{R}_{c}}}\) is a channel evolution matrix. Given a transmitted QAM symbol \(\bar{\mathbf{s}}[n]\in\tilde{\mathcal{X}}\), the received signal \(\bar{\mathbf{r}}[n]\in\mathbb{C}^{N_{t}\times 1}\) can be expressed as  
\[\bar{\mathbf{r}}[n]=\bar{\mathbf{H}}[n]\bar{\mathbf{s}}[n]+\bar{\mathbf{n}}[n], \tag{2}\]  
where \(\bar{\mathbf{n}}[n]\) is an additive white Gaussian noise with \(\mathcal{CN}(0,\sigma_{n}^{2})\). To simplify the model, the complex-valued expressions in (2) are converted into real-valued expressions as follows:  
\[\mathbf{r}[n] =\left[ \begin{array}{c}\mathrm{Re}\{\bar{\mathbf{r}}[n]\}\\ \mathrm{Im}\{\bar{\mathbf{r}}[n]\}\end{array} \right],\mathbf{H}[n]=\left[ \begin{array}{c}\mathrm{Re}\{\bar{\mathbf{H}}[n]\}\\ \mathrm{Im}\{\bar{\mathbf{H}}[n]\}\end{array} \right]\ -\mathrm{Im}\{\bar{\mathbf{H}}[n]\}\ \right],\] \[\mathbf{s}[n] =\left[ \begin{array}{c}\mathrm{Re}\{\bar{\mathbf{s}}[n]\}\\ \mathrm{Im}\{\bar{\mathbf{s}}[n]\}\end{array} \right],\mathbf{n}[n]=\left[ \begin{array}{c}\mathrm{Re}\{\bar{\mathbf{n}}[n]\}\\ \mathrm{Im}\{\bar{\mathbf{n}}[n]\}\end{array} \right], \tag{3}\]  
where \(\mathbf{r}[n]\in\mathbb{R}^{2N_{r}\times 1}\), \(\mathbf{H}[n]\in\mathbb{R}^{2N_{r}\times 2N_{t}}\), \(\mathbf{s}[n]\in\mathbb{R}^{2N_{t}\times 1}\), and \(\mathbf{n}[n]\in\mathbb{R}^{2N_{t}\times 1}\). To detect the transmitted symbol optimally, ML detection is applied as follows:  
\[\hat{k}[n]=\operatorname*{argmax}_{k\in\mathcal{K}}\mathbb{P}(\mathbf{q}[n]| \mathbf{s}[n]=\mathbf{s}_{k}), \tag{5}\]  
where \(\mathbf{s}_{k}=[\operatorname{Re}(\mathbf{s}_{k})^{T},\operatorname{Im}( \mathbf{s}_{k})^{T}]^{T}\) is an element of the real-valued constellation set \(\mathcal{X}=\{\mathbf{s}_{1},\dots,\mathbf{s}_{k}\}\). The likelihood probability for ML detection with one-bit ADCs is given by  
\[\mathbb{P}(\mathbf{q}[n]|\mathbf{s}[n]=\mathbf{s}_{k})=\prod_{r\in\mathcal{R }}\mathbb{P}(q_{r}[n]|\mathbf{s}[n]=\mathbf{s}_{k})=\prod_{r:q_{r}[n]=+1}p_{r, k}[n]\prod_{r:q_{r}[n]=-1}1-p_{r,k}[n], \tag{6}\]  
where \(q_{r}[n]\) is the \(r\)-th row of \(\mathbf{q}[n]\) and the receiver antenna index is \(r\in\mathcal{R}=\{1,2,\dots,2N_{r}\}\). The probability \(p_{r,k}\) in [19] is computed as follows:  
\[p_{r,k}[n]=\Phi\Bigg{(}\sqrt{\frac{2}{\sigma_{n}^{2}}}\mathbf{h}_{r}^{H}[n] \mathbf{s}_{k}\Bigg{)}=\Phi(g_{r,k}[n]), \tag{7}\]  
where \(\mathbf{h}_{r}^{H}[n]\) is the \(r\)-th row of \(\mathbf{H}[n]\) and \(g_{r,k}[n]=\sqrt{\frac{2}{\sigma_{n}^{2}}}\mathbf{h}_{r}^{H}[n]\mathbf{s}_{k}\). Then, the symbol index of the ML estimate is calculated as  
\[\hat{k}[n]=\operatorname*{argmax}_{k\in\mathcal{K}}\prod_{r:q_{r}[n]=1}p_{r,k }[n]\prod_{r:q_{r}[n]=-1}(1-p_{r,k}[n]). \tag{8}\]  
Using this symbol index, the ML estimate is expressed as \(\mathbf{s}_{k[n]}\). Because the true channel \(\mathbf{H}[n]\) is unknown at the receiver, channel estimation is performed during the pilot transmission. After obtaining the initial estimate \(\hat{\mathbf{H}}[0]\) from the channel estimator, the initial likelihood probability \(p_{r,k}[0]\) can be computed using (7). However, the accuracy of the initial probability is gradually degraded because of the time-varying nature of the channel.