## 2 System Model
### ML Detection  
The ML detection rule is revisited in the context of one-bit quantized MIMO systems inspired by the formulation in [27]. Let \(\mathcal{G}=\{i:q_{ab,i}\geq 0,i\in\mathcal{I}\}\) and \(\mathcal{L}=\{i:q_{ab,i}<0,i\in\mathcal{I}\}\). Given a candidate signal vector \(\mathbf{x}_{a}\), the probability of observing a quantized output \(\mathbf{q}_{ab}\) is computed under the assumption of statistically independent components as follows:  
\[\mathbb{P}\{\mathbf{q}_{ab}|\mathbf{x}_{a}\} =\mathbb{P}\{q_{ab,i}\geq 0|\forall i\in\mathcal{G}\}\mathbb{P}\{q _{ab,i}<0|\forall i\in\mathcal{L}\}\] \[=\mathbb{P}\{q_{ab,i}\mathbf{h}_{ab,i}^{T}\mathbf{x}_{a}\geq-n_{ ab,i}|\forall i\in\mathcal{G}\}\mathbb{P}\{q_{ab,i}\mathbf{h}_{ab,i}^{T} \mathbf{x}_{a}\geq n_{ab,i}|\forall i\in\mathcal{L}\}\] \[\overset{(a)}{=}\mathbb{P}\{q_{ab,i}\mathbf{h}_{ab,i}^{T}\mathbf{ x}_{a}\geq-n_{ab,i}|\forall i\in\mathcal{G}\}\mathbb{P}\{q_{ab,i}\mathbf{h}_{ab,i}^{T} \mathbf{x}_{a}\geq-n_{ab,i}|\forall i\in\mathcal{L}\}\] \[\overset{(b)}{=}\prod_{i=1}^{2N_{b}}1-Q\Big{(}\sqrt{2}q_{ab,i} \mathbf{h}_{ab,i}^{T}\mathbf{x}_{a}\Big{)}, \tag{5}\]  
where step \((a)\) holds because \(n_{ab,i}\) and \(-n_{ab,i}\) are identically distributed, and step \((b)\) holds because \(\mathbf{n}_{ab}\) follows Gaussian distribution with zero mean and \(\frac{1}{2}\) variance. Then, the likelihood probability can be formulated as follows. \[\mathbb{P}\{\mathbf{q}_{ab}|\mathbf{x}_{a}\} =\prod_{q_{ab,j}=c_{ab,j}(\mathbf{x}_{a})}\Bigl{(}1-Q\Bigl{(}\sqrt{ 2}q_{ab,j}\mathbf{h}_{ab,j}^{T}\mathbf{x}_{a}\Bigr{)}\Bigr{)}\prod_{q_{ab,j} \
eq c_{ab,j}(\mathbf{x}_{a})}\Bigl{(}1-Q\Bigl{(}\sqrt{2}q_{ab,j}\mathbf{h}_{ab,j}^{T}\mathbf{x}_{a}\Bigr{)}\Bigr{)}\] \[\overset{(a)}{=}\prod_{q_{ab,j}=c_{ab,j}(\mathbf{x}_{a})}\Bigl{(} 1-Q\Bigl{(}\sqrt{2}q_{ab,j}\mathbf{h}_{ab,j}^{T}\mathbf{x}_{a}\Bigr{)}\Bigr{)} \prod_{q_{ab,j}\
eq c_{ab,j}(\mathbf{x}_{a})}Q\Bigl{(}-\sqrt{2}q_{ab,j}\mathbf{ h}_{ab,j}^{T}\mathbf{x}_{a}\Bigr{)}\] \[=\prod_{i=1}^{2N_{b}}\mathbb{P}\{q_{ab,j}|c_{ab,j}(\mathbf{x}_{a} )\}, \tag{6}\]  
where \(c_{ab,j}(\mathbf{x}_{a})\) represents the \(i\)-th entry of the noiseless decision vector \(\mathbf{c}_{ab}(\mathbf{x}_{a})\triangleq\text{sign}(\mathbf{H}_{ab}\mathbf{ x}_{a})\) in \((a)\). This vector captures the quantized version of the channel output in the absence of noise. The conditional probability \(\mathbb{P}\{q_{ab,j}|c_{ab,j}(\mathbf{x}_{a})\}\) of an individual quantized symbol \(q_{ab,j}\) depends on whether it matches the ideal decision bit \(c_{ab,j}\), and it is given by  
\[\mathbb{P}\{q_{ab,j}|c_{ab,j}(\mathbf{x}_{a})\}=\left\{ \begin{array}{ll}p_{ ab,j}(\mathbf{x}_{a})&,q_{ab,j}\
eq c_{ab,j}(\mathbf{x}_{a})\\ 1-p_{ab,j}(\mathbf{x}_{a})&,q_{ab,j}=c_{ab,j}(\mathbf{x}_{a}),\end{array} \right. \tag{7}\]  
where the error probability \(p_{ab,j}(\mathbf{x}_{a})\) associated with the \(i\)-th bit is characterized using the Q-function \(Q\Bigl{(}\sqrt{2}|\mathbf{h}_{ab,j}^{T}\mathbf{x}_{a}|\Bigr{)}\). Taking the logarithm of the likelihood probability in (6), the product of individual conditional probabilities is converted into a sum, which enables a more tractable metric for detection. \[\log(\mathbb{P}\{\mathbf{q}_{ab}|\mathbf{x}_{a}\})\] \[=\sum_{i=1}^{2N_{b}}\Bigl{[}\log(p_{ab,i}(\mathbf{x}_{a}))\|q_{ ab,i}-c_{ab,j}(\mathbf{x}_{a})\|_{0}+\log(1-p_{ab,i}(\mathbf{x}_{a}))(1-\|q_{ ab,i}-c_{ab,j}(\mathbf{x}_{a})\|_{0})\Bigr{]}\] \[=-\biggl{(}\sum_{i=1}^{2N_{b}}\Bigl{[}w_{ab,j}(\mathbf{x}_{a})\|q_ {ab,i}-c_{ab,j}(\mathbf{x}_{a})\|_{0}+\bar{w}_{ab,j}(\mathbf{x}_{a})(1-\|q_{ ab,i}-c_{ab,j}(\mathbf{x}_{a})\|_{0})\Bigr{]}\biggr{)}, \tag{8}\]  
where \(w_{ab,i}(\mathbf{x}_{a})=-\log(p_{ab,i}(\mathbf{x}_{a}))\) and \(\bar{w}_{ab,j}(\mathbf{x}_{a})=-\log(1-p_{ab,i}(\mathbf{x}_{a}))\). The above expression is simplified to a negative-weighted sum.