# Improved Likelihood Probability in MIMO Systems Using One-Bit ADCs
## 1 Introduction
Although the RL methods in [32; 33] significantly improve performance, the complexity associated with computing the optimal policy is not conducive to the receivers. To address this complexity, this study investigates a new method for enhancing likelihood probability without calculating the optimal policy. This study proposes an improved likelihood probability method for multi-input multi-output (MIMO) systems that use one-bit ADCs. To achieve this, the proposed method utilizes the detected data symbols to calculate the data-based likelihood probability, which is combined with the pilot-based likelihood probability obtained during pilot transmission. An optimization problem is presented herein for the combined process and a computationally efficient algorithm for solving it is described. The key contributions of this study are summarized as follows:  
* An optimization problem is formulated to improve the pilot-based likelihood probability. In particular, a data-based likelihood probability is defined using detected data symbols, and a weighted combination method is proposed between the pilot- and data-based likelihood probabilities. The optimization problem minimizes the expected mean squared error (MSE) between the true and weighted combined likelihood probabilities. Unlike the likelihood probability in [32], the data-based likelihood probability considers all detected data symbols in its calculation, and the weights are unconstrained rather than summed up to one. * A computationally efficient method for solving the optimization problem is proposed. The optimization problem is challenging to solve because the non-linear function in the likelihood probability and the expression of the data-based likelihood probability are mathematically intractable. Hence, the method exploits realized samples to obtain weights for the optimization problem. The expectation of optimization problem is replaced with the average of the realized samples. The solution to the optimization problem is derived using the least-squares method. Because the derived solution is obtained offline, its weights are saved in memory. * The effectiveness of the proposed method is demonstrated through simulations. The combined weights for the proposed method are updated from the memory at each end of the data block because updating the weights of every data symbol is burdensome to the memory. Based on simulation results, the proposed method indicates improved performance compared with conventional methods without sacrificing complexity. This is because the proposed method improves the reliability of the likelihood probability by exploiting the detected symbols. **Related works:** This paper focuses on improving the likelihood probability in MIMO systems using one-bit ADCs. The investigation of likelihood probability in one-bit ADCs was initially conducted in [18; 20]. Although the likelihood probability offers optimal performance in one-bit ADCs, its performance remains unsatisfactory. To address this performance limitation, the authors [32; 33] introduced the concept of data-based likelihood probability. These methods exploit RL algorithms to obtain the optimal policy. However, due to considerable complexity involved in calculating the optimal policy, the methods in [32; 33] are still impractical for wireless applications. This study considers a new method to leverage the data-based likelihood probability without the need for complex calculation. To achieve this, an optimization problem is defined that minimizes the expected MSE between the weighted combination of pilot-based and data-based likelihood probabilities and the true likelihood probability. One key advantage of the proposed method is that it does not require weight calculation during the transmission. Thus, the complexity is significantly reduced compared to [32; 33]. To improve the comprehensiveness of this study, the related works are summarized in Table 1. The focus of this study is on data detection in one-bit ADCs.