# Regression of Likelihood Probability for Time-Varying MIMO Systems with One-Bit ADCs
## 5 Simulation Results
This improvement is attributed to the decoder's output, which provides more accurate information about the channel. In Figure 3, the optimal weights of the linear regression for the proposed method are shown as a function of \(d\). As the number of data blocks increases, the accuracy of the sample-based likelihood improves. Thus, the weight \(\theta_{d}^{2}\) for the sample-based likelihood probability increases, while the weight \(\theta_{d}^{1}\) for the channel-based likelihood probability decreases. In contrast, the constant weight \(\theta_{d}^{0}\) approaches zero with the increase in \(d\) in the slow-fading channels. In addition, when the SNR increases, the convergence speed increases because the correctly decoded output improves the accuracy of the sample-based likelihood probability. Compared to slow-fading channels, the increase in the sample-based likelihood probability is smaller in the fast-fading channels, where the channel-based likelihood probability dominantly affects the overall likelihood probability. Figure 2: Channel- and sample-based likelihood probabilities compared to conventional methods when \(v=40\) km/h and SNR = 0 dB. The optimal weights for the LASSO regression are simulated with \(\lambda=10^{-2}\) for slow fading channels and \(\lambda=10^{-1}\) for fast fading channels, as shown in Figure 4. Both figures show a similar overall trend to the linear regression in Figure 3. However, in the LASSO regression, the optimal weight \(\theta_{d}^{0}\) is exactly zero, indicating that the constant weight is less significant than the weights for the channel- and sample-based likelihood probabilities. Compared to slow-fading channels, the weights for the channel-based likelihood probability in fast-fading channels are larger, as the channel-based likelihood probability can more effectively track changes in the likelihood probability for the subsequent block. Figure 3: Optimal weights for the linear regression according to the number of data blocks in time-varying channels. The FERs for the different detection methods are plotted in Figure 5 to demonstrate the effectiveness of the proposed method in both the slow and fast fading channels. For the LASSO method, the optimal weights from Figure 4 are applied in both channel conditions. The Conv. and Robust methods exhibit poor performance owing to their inability to track the true likelihood probability. The FER of the RL-TV exhibits a slight improvement, indicating that it struggles to follow the channel variations. This result contrasts with [38], where the sample-refinement scheme primarily addresses temporal channel variation compensation. The proposed (Reg) and proposed (LASSO) methods achieve SNR gains of approximately 0.8 dB and 0.7 dB over the RL-TV in slow fading channels at an FER of \(10^{-1}\), while the SNR gain in fast fading channels is substantial compared to conventional methods. This improvement is attributed to the proposed methods' ability to track changes in the likelihood probability by optimally combining both channel- and sample-based likelihood probabilities. Note that the proposed (LASSO) method uses only two weights to calculate the proposed likelihood probability, while the proposed (Reg) method uses three weights. Figure 4: Optimal weights for the LASSO regression according to the number of data blocks in time-varying channels. Figure 6 presents the MSE between the true and proposed likelihood probabilities, i.e., \(\mathbb{E}\{\|p_{r,k}-\hat{p}_{r,k}\|^{2}\}\). The proposed method achieves a lower MSE compared to the conventional methods in both slow and fast fading channels. The MSEs of the perfect CSI and Conv. methods are gradually degraded in time-varying channels because they do not update the likelihood probability and fail to adapt to channel variations. In contrast, the MSEs of RL-TV and proposed methods improve as \(d\) increases in slow-fading channels, as both methods derive the likelihood probabilities based on time-varying channel models.