# Improved Likelihood Probability in MIMO Systems Using One-Bit ADCs
## 2 System Model  
This section describes the system model used in this study. The transmitter has \(N_{t}\) antennas and the receiver has \(N_{r}\) antennas. Each transmission frame comprises a pilot block and \(D\) data blocks. The number of symbols in the pilot block is denoted by \(N_{p}\), whereas the number of symbols in the \(d\)-th data block is denoted by \(N_{d}\), where \(d\) is an element of set \(\{1,2,\ldots,D\}\) (Figure 1). The information bits are encoded by a channel encoder with code rate \(R_{c}\), and mapped to \(M\)-ary quadrature amplitude modulation (QAM) symbols \(\mathbf{\tilde{s}}[n]\in\mathbb{C}^{N_{t}\times 1}\) where \(\mathbb{C}\) is a complex space. \(\mathcal{K}\) denotes the index set of the candidate QAM symbols and is defined as \(\{1,2,\ldots,K\}\). \(\mathcal{\hat{X}}\) denotes the set of candidate QAM symbols and is defined as \(\mathcal{\hat{X}}=\{\mathbf{\tilde{s}}_{1},\ldots,\mathbf{\tilde{s}}_{K}\}\). Its size is \(|\mathcal{K}|=M^{N_{t}}\), where the modulation order is \(M\) and \(|\cdot|\) is the cardinality of the set. The wireless channel \(\mathbf{\tilde{H}}\in\mathbb{C}^{N_{r}\times N_{t}}\) is assumed to exhibit Rayleigh fading, and the channel is invariant during each frame trans \begin{table}
\begin{tabular}{c c c} \hline
**References** & **Topics** & **Key Features** \\ \hline
[1; 2; 3; 4; 5; 6] & Millimeter-wave system & Mmwave Channel model and implementation \\
[16; 17; 18; 19; 20; 21] & Low-resolution ADCs & Optimal transceiver design \\
[22; 23; 24; 25; 26; 27; 28] & Channel estimation in one-bit ADCs & Optimal MMSE estimator design \\
[29; 30; 31; 32; 33; 34; 35] & Data detection in one-bit ADCs & Maximum likelihood probability calculation \\
[30; 31; 32; 33] & Machine learning-based data detection & Machine learning-based algorithm \\ \hline \end{tabular}
\end{table} 
Table 1: Related works. symbol \(\mathbf{\hat{s}}[n]\in\mathcal{\bar{X}}\) is transmitted to the wireless channel, the received symbol \(\mathbf{\hat{r}}[n]\in\mathbb{C}^{N_{r}\times 1}\) is expressed as  
\[\mathbf{\hat{r}}[n]=\mathbf{\hat{H}}\mathbf{\hat{s}}[n]+\mathbf{\hat{n}}[n], \tag{1}\]  
where the additive white Gaussian noise is denoted by \(\mathbf{\hat{n}}[n]\), whose distribution follows \(\mathcal{CN}(0,\sigma_{n}^{2})\). For simplicity, the complex value expression in (1) is converted into a real-value expression. Subsequently, the real vectors and matrices \(\mathbf{r}[n]\in\mathbb{R}^{2N_{r}\times 1}\), \(\mathbf{H}\in\mathbb{R}^{2N_{r}\times 2N_{t}}\), \(\mathbf{s}[n]\in\mathbb{R}^{2N_{t}\times 1}\), and \(\mathbf{n}[n]\in\mathbb{R}^{2N_{r}\times 1}\), where \(\mathbb{R}\) is the real space, are defined as  
\[\mathbf{r}[n] =\left[ \begin{array}{c}\mathrm{Re}\{\mathbf{\hat{r}}[n]\}\\ \mathrm{Im}\{\mathbf{\hat{r}}[n]\}\end{array} \right],\mathbf{H}=\left[ \begin{array}{cc}\mathrm{Re}\{\mathbf{\hat{H}}\}&-\mathrm{Im}\{\mathbf{ \hat{H}}\}\\ \mathrm{Im}\{\mathbf{\hat{H}}\}&\mathrm{Re}\{\mathbf{\hat{H}}\}\end{array}  \right],\] \[\mathbf{s}[n] =\left[ \begin{array}{c}\mathrm{Re}\{\mathbf{\hat{s}}[n]\}\\ \mathrm{Im}\{\mathbf{\hat{s}}[n]\}\end{array} \right],\mathbf{n}[n]=\left[ \begin{array}{c}\mathrm{Re}\{\mathbf{\hat{n}}[n]\}\\ \mathrm{Im}\{\mathbf{\hat{n}}[n]\}\end{array} \right]. \tag{2}\]  
where \(\mathrm{Re}(\cdot)\) and \(\mathrm{Im}(\cdot)\) are the real and imaginary components of the complex number, respectively. A one-bit operation is performed on the real and imaginary components of the received symbol \(\mathbf{r}[n]\), which is expressed in [20; 32] as  
\[\mathbf{q}[n]=\mathrm{sign}(\mathbf{r}[n])=\mathrm{sign}(\mathbf{Hs}[n]+ \mathbf{n}[n]), \tag{3}\]  
where \(\mathrm{Sign}(x)\) is a function whose output is \(1\) for \(x\geq 0\) and \(-1\) otherwise.