# Regression of Likelihood Probability for Time-Varying MIMO Systems with One-Bit ADCs
## 1 Introduction  
Millimeter-wave (mmWave) and terahertz (THz) communications are critical for advancing wireless standards, especially for applications that demand extremely high data rates, often in the gigabit-per-second range [1; 2; 3; 4; 5; 6]. However, the use of large bandwidths in mmWave bands brings significant technical challenges, particularly around power consumption, which scales linearly with bandwidth and exponentially with the number of quantization bits [7; 8]. This makes conventional systems that typically use 10-14 quantization at the receiver unsuitable for mmWave and THz applications. Communication systems that leverage low-resolution analog-digital-converters (ADCs) have emerged as a solution [9; 10; 11; 12; 13; 14], enabling reduced power consumption with minimal performance degradation.  
Recent studies introduced one-bit ADCs to address the power consumption problem in mmWave and THz systems [15; 16; 17; 18; 19; 20; 21]. Although promising, these ADCs lead to performance degradation in multi-input multi-output (MIMO) systems owing to their nonlinearity, which complicates optimal maximum likelihood (ML) detection [17]. This necessitates practical alternatives, such as convex optimization [18], sphere decoding [19], soft-decision ML detection [20], and deep learning-based detection [21]. However, these methods rely on idealized assumptions of perfect channel information, which is difficult to attain with one-bit ADCs owing to their inherent nonlinearity.  
Channel estimation techniques for one-bit ADCs have been extensively studied [22; 23; 24; 25; 26; 27; 28; 29; 30; 31]. Most studies often assumed a Gaussian quantization error model [24]. Based on the Gaussian error model, standard linear approaches, such as least-squares and minimum mean-squared error (MMSE) estimations can be derived [25; 26]. However, these methods are constrained by inaccuracies in the quantization error model. Instead of linearapproaches, iterative approaches like expectation-maximization [27] and approximate message passing [28] can improve the estimation performance. Furthermore, machine learning approaches have been employed to address the nonlinearity in channel estimation [29; 30; 31]. However, they result in significant computational complexity.  
To address these limitations, recent studies have shifted toward methods that directly calculate likelihood probabilities without relying on explicit channel estimations [32; 33; 34; 35; 36; 37; 38; 39]. For example, supervised learning techniques have been used to estimate likelihood probabilities based on pilot transmissions [35; 36], although their performance suffers when the pilot lengths are insufficient. To resolve this challenge, [37] introduced an efficient learning method that minimizes the mean-squared error (MSE) between the true likelihood probability and learned likelihood probability using input-output samples from the data detector. Because the learning method involves significant complexity, a low-complexity algorithm was introduced in [39], where the pilot- and data-based likelihood probabilities are learned offline. The learning method in [37] was extended to time-varying channels, and a dynamic update model was proposed in [38]. This model captures temporal variations by defining a Markov Decision Process (MDP) aimed at maximizing the accuracy of the likelihood probability. A reinforcement learning (RL) algorithm was used to solve the MDP. Despite its performance benefits, the learning method in [38] involves high computational complexity owing to the intricate update model and a large number of possible state transitions, making it impractical for low-power applications, particularly in the mmWave and THz bands.