# Regression of Likelihood Probability for Time-Varying MIMO Systems with One-Bit ADCs
## 5 Simulation Results  
Simulations are performed using Matrix Laboratory (MATLAB). The optimal weights in (19) and (23) are obtained through the statistics and machine learning toolbox. The obtained optimal weights are then used in simulations to evaluate the frame error rate (FER) and MSE over 10,000 iterations. The antenna configuration used is \((N_{t},N_{r})=(4,8)\). For the time-variant channel, a carrier frequency \(f_{c}=3.5\) GHz, and a sampling period of \(T_{s}=16.67\) us is considered, as specified in [40]. Velocities of \(v=5\) km/h and \(v=40\) km/h, represent slow and fast time-varying channels, respectively. The information bits are encoded using a turbo encoder at a rate \(R_{c}=\frac{1}{2}\). A modulation order of 4-QAM is applied to the encoded bits, resulting in a constellation size of \(K=M^{N_{t}}=256\). The frame structure consists of one pilot with \(N_{p}=32\) and \(D=20\) data blocks with \(N_{d}=128\). A linear MMSE method was applied to obtain the initial channel estimation during pilot transmission. The signal-to-noise ratio (SNR), denoted as \(E_{b}/N_{0}\), is defined as \(\frac{1}{\log_{2}|\mathcal{X}|\sigma_{n}^{2}}\). For performance benchmarking, the following methods are considered:  
* **Perfect CSI:** This represents the optimal case in time-invariant channels, where the likelihood probability is obtained using perfect channel state information (CSI) of \(\mathbf{H}[0]\). Note that the resulting likelihood probability remains constant over time, and thus is not optimal in time-varying channels. * **Conv:** This conventional method calculates the probability \(p_{r,k}^{\text{channel}}[0]\) based on channel estimates \(\hat{\mathbf{H}}[0]\) obtained during pilot transmission. In this method, the likelihood probability remains fixed during data transmission. * **Robust:** Developed in [37] for time-invariant MIMO channels. This method calculates the likelihood probability using an RL algorithm, where the initial likelihood probability is optimally combined with the empirical-based likelihood probability obtained from the detected symbol. * **RL-TV:** Introduced in [38] for time-varying MIMO channels, this method calculates the likelihood probability using an RL algorithm, where the model-based likelihood probability and the quantized received signal are optimally combined. Notably, the sample refinement scheme is not used in this study where all input-output samples are used to update the likelihood probability when the cyclic redundancy check (CRC) is correct. This exclusion is intended to ensure a fair comparison with other methods. * **Proposed:** The proposed method exploits the likelihood probability from (22), where both the channel- and the sample-based likelihood probabilities are learned via linear regression. The proposed (Reg) denotes the case when the optimization problem in (19) is applied, while the proposed (LASSO) refers to when the regularization technique in (23) is applied. The precise definitions for the channel- and sample-based likelihood probabilities differ from those in [37] and [38], respectively. In Figure 2, the channel- and sample-based likelihood probabilities are compared with true likelihood probability in (7) under fast-fading channels. In Figure 2a, the conventional model-based likelihood probability from [38] is compared with the channel-based likelihood probability in (12). While the conventional model-based likelihood probability shows a gradual response to changes in the true likelihood probabilities in (7), the channel-based likelihood probability more efficiently captures these changes. Note that the channel-based likelihood probability is derived in (10) under the assumption of slow-fading channels \(\epsilon\approx 1\). As a result, it exhibits slower changes compared to the true likelihood probability in fast-fading channels. In Figure 2b, the sample-based likelihood probability in (16) outperforms the conventional empirical-based likelihood probability from [37] in capturing rapid changes in the true likelihood probability.