# Regression of Likelihood Probability for Time-Varying MIMO Systems with One-Bit ADCs
## 3 Proposed Problem
### Channel-Based Likelihood Probability  
This subsection introduces the channel-based likelihood probability, defined as the expected variation in the likelihood probability. Thus, the channel-based likelihood probability dynamically adjusts changes in the channel over time. Using (1), the likelihood probability is formulated as a function of the time index \(m\), given by  
\[p_{r,k}[n]=\Phi(\epsilon g_{r,k}[n-1]+\delta_{r,k}[n])=\Phi\big{(}\epsilon^{n -m}g_{r,k}[m]+\delta_{r,k}[n,m]\big{)}, \tag{9}\]  
where \(\delta_{r,k}[u]=\sqrt{\frac{2(1-\epsilon^{2})}{\sigma_{n}^{2}}}\mathbf{A}_{r, k}^{H}[n]\mathbf{s}_{k}\) for \(u\in\{m+1,\dots,n\}\) and \(\delta_{r,k}[n,m]=\sum_{u=m+1}^{n}\epsilon^{n-u}\delta_{r,k}[u]\). Thus, \(\delta_{r,k}[n,m]\sim N(0,\
u_{k}[n-m])\) with \(\
u_{k}[u]=(1-\epsilon^{2u})\frac{||\mathbf{s}_{k}||^{2}}{\sigma^{2}}\).  
Assuming the channel evolves according to a time-variant Gaussian Markov process with \(\epsilon\approx 1\), the difference in likelihood probabilities between times \(n\) and \(m\) is calculated as follows:\[p_{r,k}[n]-p_{r,k}[m] =\Phi\big{(}e^{n-m}g_{r,k}[m]+\delta_{r,k}[n,m]\big{)}-\Phi(g_{r,k}[ m])\] \[\stackrel{{(a)}}{{\approx}}-((1-e^{n-m})g_{r,k}[m]- \delta_{r,k}[n,m])\phi\bigg{(}\frac{(1-e^{n-m})g_{r,k}[m]+\delta_{r,k}[n,m]}{2 }\bigg{)}\] \[=-2((1-e^{n-m})g_{r,k}[m]-\delta_{r,k}[n,m])\frac{1}{\sqrt{2\pi \times 4}}e^{\frac{-\big{(}\delta_{r,k}[n,m]+(1+e^{n-m})g_{r,k}[m]\big{)}^{2}}{2 \times 4}}, \tag{10}\]  
where the approximation in \((a)\) is valid as \(\Phi(x)-\Phi(a)\approx(x-a)\Phi^{\prime}(\frac{x+a}{2})\) when \(x\to a\) under the assumption \(e\approx 1\). Here, \(\Phi^{\prime}(x)=\phi(x)=\frac{1}{\sqrt{2\pi}}\exp(-\frac{x^{2}}{2})\).  
With an assumed expected value of \(\mathbb{E}[p_{r,k}[m]]=\hat{p}_{r,k}[m]\), the expectation of (10) can be further simplified to  
\[\mathbb{E}[p_{r,k}[n]] \approx\hat{p}_{r,k}^{\text{channel}}[m]-2\big{(}(1-e^{n-m})\hat{ g}_{r,k}[m]\big{)}\mathbb{E}\bigg{[}\frac{1}{\sqrt{2\pi\times 4}}e^{-\big{(} \delta_{r,k}[n,m]+(1+e^{n-m})g_{r,k}[m]\big{)}^{2}}\bigg{]}\] \[\stackrel{{(a)}}{{\approx}}\hat{p}_{r,k}^{\text{ channel}}[m]-\frac{2(1-e^{n-m})\hat{g}_{r,k}[m]}{\sqrt{2\pi(v_{k}[n-m]+4)}}e^{-\frac{ \big{(}(1+e^{n-m})\hat{g}_{r,k}[n]\big{)}^{2}}{2(v_{k}[n-m]+4)}}, \tag{11}\]  
where \(\mathbb{E}\bigg{\{}\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(X-p)^{2}}{2 \sigma^{2}}}\bigg{\}}=\frac{1}{\sqrt{2\pi(v^{2}+\sigma^{2})}}e^{-\frac{p^{2}} {2(v^{2}+\sigma^{2})}}\) is applied in \((a)\) when \(X\sim\mathcal{N}(0,v^{2})\).  
The derivation in (11), which removes the term \(\delta_{r,k}[n,m]\), differs from that in [38], resulting in reduced computational complexity.  
Consequently, the channel-based likelihood probability under the assumption that \(\mathbb{E}[p_{r,k}[m]]=\hat{p}_{r,k}^{\text{channel}}[m]\), and tracking from time \(m\), is given by  
\[p_{r,k}^{\text{channel}}[n]=\hat{p}_{r,k}^{\text{channel}}[m]-f(\Phi^{-1}(\hat {p}_{r,k}^{\text{channel}}[m]),v_{k}^{n-m},e^{n-m}), \tag{12}\]  
where \(f(g,\
u,\epsilon)=\frac{2(1-\epsilon)g}{\sqrt{2\pi(v+4)}}e^{-\frac{((1+ \epsilon)g)^{2}}{2(v+4)}}\).