# Improved Likelihood Probability in MIMO Systems Using One-Bit ADCs
## 5 Simulation Results  
The performance of the proposed method was evaluated in terms of the frame error rate (FER) and MSE. The antenna configuration is \((N_{t},N_{r})=(4,8)\) as a practical case of 3 GPP specifications. Considering that the maximum number of streams is 8 in 3 GPP specifications, \(N_{t}=4\) is a practical choice. For the given transmit antennas, \(N_{r}=8\) provides a satisfactory FER performance within a practical SNR range in MIMO systems using one-bit ADCs [29; 30; 31; 32; 33; 34; 35]. Each link between the transmit and receive antennas was assumed to be quasi-static Rayleigh fading. The information bits were encoded using a turbo encoder at a rate \(R_{c}=\frac{1}{2}\). To improve the performance, a 16-bit cyclic redundancy check was applied, as explained in [32]. A modulation order of four (\(4-QAM\)) was applied to the encoded bits, which resulted in a constellation size of \(K=M^{N_{t}}=256\). The frame structure comprised one pilot with \(N_{p}=32\) and \(D=40\) data blocks with \(N_{d}=128\). A linear MMSE method was applied to obtain the pilot-based likelihood probability during the pilot transmission. The SNR is defined as \(\frac{1}{\log_{2}|\mathcal{X}|\varphi_{x}^{2}}\). For performance benchmarking, the following methods were considered:  
* **PCSI:** As an optimal case, the likelihood probability \(p_{r,k}\) is obtained based on perfect channel information. * **CE:** This is a conventional method in which the likelihood probability \(p_{r,k}^{pi}\) is computed based on the channel estimates obtained during the pilot transmission. In this method, the likelihood probability remains unchanged during data transmission. * **Optimal:** This method obtains the optimal weights \((\alpha,\beta)\) in the optimization problem (15) via exhaustive search, where the resolution of the weights is set to 0.01. * **Conv RL:** Weights are obtained using the RL method presented in [32]. In this RL method, the sum of the weights is always 1, i.e., \(\beta=1-\alpha\). This is expressed as \[\hat{p}_{r,k}=\alpha p_{r,k}^{pi}+(1-\alpha)p_{r,k}^{de},\] (22) where \(\alpha=\frac{1}{|\mathcal{R}||\mathcal{K}|}\sum_{r\in\mathcal{R}}\sum_{k\in \mathcal{K}}\alpha_{r,k}\). Unlike the method presented in [32], the Conv RL method obtains the weights via an offline manner. Subsequently, the obtained weights are applied in the weighted combination method expressed in (14). In Figure 3, the combined weights for the proposed method are shown based on \(d\). As the number of data blocks increases, the accuracy of the data-based likelihood probability increases. Thus, the weight \(\beta\) for the data-based likelihood probability increases, whereas the weight \(\alpha\) for the pilot-based likelihood probability decreases. The sum of weights remains as approximately 1 as the number of data blocks increases. In addition, when the SNR increases, the convergence speed increases because the correctly detected data symbols improve the accuracy of the data-based likelihood probability. Thus, the weight of the data-based likelihood probability has a higher value than that at a lower SNR. The combined weights for the proposed method were simulated based on the SNR, as shown in Figure 4. As shown in this figure, the weight \(\beta\) for the data-based likelihood probability increases with the SNR. Compared with Figure 3, the slope of the weights in Figure 4 increases linearly with the SNR. In Figure 5, the weights of the conventional RL method are compared with those of the proposed method. Similar to the weights in Figure 3, the weights for the data-based likelihood probability in the conventional RL method increases with \(d\). However, the