# Improved Likelihood Probability in MIMO Systems Using One-Bit ADCs
## 5 Simulation Results
\begin{table}
\begin{tabular}{c c c c} \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{2}{c}{**Estimated Likelihood Probability Calculation**} & \multirow{2}{*}{**Total 1**} \\ \cline{2-2} \cline{4-4}  & **Optimal Policy Calculation** & & \\ \hline RL method [32] & \(N_{r}N_{d}K+N_{r}N_{d}^{2}K+8N_{r}(K+1)(2K+1)N_{d}\) & \(4N_{r}KD\) & \(2.30\times 10^{9}\) \\ Proposed method & - & \(4N_{r}KD\) & \(3.27\times 10^{5}\) \\ \hline \end{tabular}  
* The number of real multiplications is counted when \(N_{t}=4\), \(N_{r}=8\), \(M=2\), \(N_{d}=128\), and \(D=40\).  
\end{table} 
Table 3: Complexity comparison: the number of real multiplications. convergence speed of the conventional RL method is higher than that of the proposed method. This is because the weights for the conventional RL method are obtained with respect to the instantaneous MSE not the expected MSE. Thus, despite an SNR of \(-4\) dB, the weights \((\alpha,\beta)\) almost saturated. The FER and MSE were compared using the above-mentioned weights, as shown in Figures 6 and 7. The performance of the proposed method depends on the number of antennas used. To verify the effect of the number of antennas, the proposed method was simulated based on the number of transmit antennas \(N_{r}\). As shown in Figure 8, the weight \(\beta\) for the data-based likelihood probability was higher when \(N_{16}\) was used compared with when \(N_{8}\) was used. This is because increasing the number of receive antennas improves the reliability of the detected data symbols. Thus, the accuracy of the data-based likelihood probability increases. Because the optimal weights for the proposed method differs from the system parameters, various values of the weights are saved in memory based on the system environment. Subsequently, based on the system input parameters, the weights are used to the combine pilot- and data-based likelihood probabilities. Figure 4: Combined weights for the proposed method based on SNR. Figure 3: Combined weights for the proposed method based on number of data blocks. Figure 5: Combined weights for conventional RL based on number of data blocks. Figure 6: Combined weights for different number of antennas for \(N_{r}=8\) and 16. Figure 7: FER for the proposed method based on different detection methods. The FERs for different detection methods are illustrated in Figure 7 to demonstrate the effectiveness of the proposed method. First, to achieve an FER of \(10^{-1}\) at \(4\times 8\) antennas, the proposed method exhibited an SNR gain of approximately 0.2 and 0.3 dB compared to conventional RL and CE methods, respectively. In addition, the FER of the proposed method was similar to that of the optimal case. This is attributed to the fact that the proposed method improves the likelihood probability toward the optimal likelihood probability. Second, the FER decreases as the number of receive antennas increases. All the detection methods show an improvement in FER due to the diversity gain associated with a large number of receive antennas. In particular, at 4 antennas, the proposed method showed an SNR gain of approximately 0.2 dB compared to CE method, while achieving a similar FER to the conventional RL. Note that the proposed method can obtain the weights more easily than the conventional RL. Figure 8 shows the MSE between the true and estimated likelihood probabilities, i.e., \(\mathbb{E}\{p_{r,k}-\bar{p}_{r,k}\}\) for \(4\times 8\) MIMO systems. First, the proposed method achieved a lower MSE than the conventional RL. In particular, to achieve an MSE of \(-18\) dB, the proposed method requires approximately 15 number of data blocks, while the conventional RL requires approximately 35 number of data blocks. This indicates that the proposed method achieves more accurate likelihood probability with fewer data blocks. Moreover, the MSE of the proposed method approached that of the optimal case. This is because the proposed method determines weights to minimize the expected MSE between the true and estimated likelihood probabilities.