# Improved Likelihood Probability in MIMO Systems Using One-Bit ADCs
## 1 Introduction  
Millimeter-wave (mmWave) systems are key technologies in current wireless applications [1; 2; 3; 4; 5; 6; 7]. A large bandwidth is easily available in mmWave bands, thus enabling data rates in the order of gigabits-per second to be achieved. However, technical challenges exist when exploiting this large bandwidth, such as increased power consumption. Power consumption increases linearly with bandwidth and exponentially with the number of quantization bits [8; 9]. Thus, conventional systems that use approximately 12 quantization bits at the receiver are unsuitable for mmWave systems. To overcome this limitation, communication systems using low-resolution analog-digital converters (ADCs) have been introduced [10; 11; 12; 13; 14; 15]. These systems enable the use of mmWave bands by considerably reducing the power consumption with only slight degradation in system performance. In recent studies, one-bit ADCs have been introduced, which accelerate the realization of mmWave systems by addressing the power-consumption problem [16; 17; 18; 19; 20; 21]. However, the MIMO systems using one-bit ADCs suffers from performance degradation. Maximum-likelihood (ML) detection, which was originally introduced in [18], has been considered to achieve optimal performance. However, owing to the non-linearity of one-bit ADCs, the original ML detection exhibits unfavorable complexity at the receiver, thus necessitating a convex optimization solution [19]. Hence, practical algorithms were introduced [20; 21]. Sphere decoding is a conventional low-complexity algorithm for ML detection and its application in one-bit ADCs was proposed [20]. Meanwhile, soft decisions for ML detection have been investigated in [21]. However, these methods assume perfect channel information at the receiver, which cannot be achieved in one-bit ADCs due to their non-linear properties. Despite its difficulty in improving estimation performance, channel estimation in one-bit ADCs has been widely investigated [22; 23; 24; 25; 26; 27; 28]. Most investigations assumed that the quantization error model is Gaussian [24]. Similar to conventional linear approaches, least-squares and minimum mean-squared error (MMSE) estimations can be easily derivedbased on the quantization error model in one-bit ADCs [25; 26]. However, the linear approach exhibits unsatisfactory performance because the quantization error model is inaccurate. Iterative approaches can improve estimation performance. The expectation-maximization [27] and approximate message passing [28] algorithms are representative methods for one-bit ADCs. However, iterative approaches exhibit considerable inherent complexities. Consequently, the accuracy of the likelihood probability is limited owing to complexity. Recent investigations have focused on improving the performance of one-bit ADCs by directly obtaining the likelihood probability [29; 30; 31; 32; 33; 34; 35]. These studies do not rely on obtaining channel estimates; instead, they learn the likelihood probability using machine-learning algorithms, such as supervised learning and reinforcement learning (RL). In previous studies [30; 31], a supervised-learning algorithm was applied to pilot transmission; however, its performance was dependent on the pilot length and it yields unsatisfactory performance when extended to wireless standards. This problem can be mitigated by using detected data symbols in data transmission [32]; however, only correct symbols can improve the likelihood probability because the detected data symbol may contain errors. Hence, a Markov decision process problem was defined [32], and a corresponding RL algorithm was introduced to calculate the optimal policy. However, the computation of the optimal policy involved significant computational complexity, and the number of candidate symbols increased exponentially with the number of modulation orders and transmit antennas. In [33], a low-complexity algorithm for RL was introduced that exploited a reliability-based policy, which significantly reduced the complexity.