## 2 System Model
### MIMO With Low-Precision ADC  
Consider an uplink single-cell scenario, where one small cell base station (BS) has \(N\) antennas serving \(K\) single-antenna users (UEs). In this case, the performance is dominated by the multiuser interference from the same cell. Following [15], we assume a Rayleigh fading channel for each coherent block, whose size depends on the coherence bandwidth and the coherent time of the channel. The received signal from all antennas are modeled as  
\[\mathbf{y}=\mathbf{H}\sqrt{\mathbf{P}}\mathbf{x}+\mathbf{n}, \tag{1}\]  
where \(\mathbf{x}\in\mathbb{C}^{K\times 1}\) is the transmitted signal vector of all UEs, and \(x_{ij}\) has zero mean and unit power, i.e., \(\mathbb{E}\left[x_{ij}\right]=0\) and \(\mathbb{E}\left[\left|x_{ij}\right|^{2}\right]=1\). The power of the UEs are specified by \(\sqrt{\mathbf{P}}\triangleq\text{diag}\left\{\sqrt{p_{1}},\cdots,\sqrt{p_{K}}\right\}\), \(p_{k}\) in Joule/symbol represents the power allocated to the \(k\)-th user. The entries of a given column of the channel matrix \(\mathbf{H}\), which corresponds to the channel vector of UE-\(k\), are independent and identically distributed (i.i.d), complex Gaussian following \(\mathcal{CN}(0,\varphi_{k})\), with \(\varphi_{k}\) being the corresponding large-scale fading for UE-\(k\). The noise vector \(\mathbf{n}\) is additive white Gaussian noise (AWGN), in which all elements independently follow the distribution \(\mathcal{CN}\left(0,\sigma^{2}\right)\). The total transmitted SNR of the \(K\) UEs is defined as  
\[\text{SNR}_{\text{TX}}=10\text{log}_{10}\left(\frac{\sum_{k=1}^{K}p_{k}}{ \sigma^{2}}\right), \tag{2}\]  
and the received SNR of UE-\(k\) is defined as  
\[\text{SNR}_{\text{RX}k}=10\text{log}_{10}\left(\frac{\varphi_{k}p_{k}}{\sigma ^{2}}\right). \tag{3}\]  
Note that, under the i.i.d. assumption of the small scale fading of all the UEs, the BER performance of the system averaged over small-scale fading depends on \(\{\text{SNR}_{\text{RX}k}\}\). This paper assumes that the received SNRs of the different UEs are the same, achieved by reasonable power allocations according to the UEs' large-scale fadings, as will be discussed in detail later. This amounts to assume that the UEs have symmetric requirements of quality of service (QoS) in terms of transmission rates and error rates, following [34; 35; 33]. In practical mobile communication systems, the elements in \(\mathbf{y}\) need to be scaled and quantized. From [36; 37], the scaling factor \(\rho_{n}\) for the \(n\)-th antenna can be chosen as  
\[\rho_{n}=3\left(\left\|\mathbf{h}_{n}\sqrt{\mathbf{P}}\right\|_{2}^{2}+\sigma ^{2}\right)^{\frac{1}{2}},n\in\left\{1,\cdots,N\right\}, \tag{4}\]  
where \(\mathbf{h}_{n}\) stands for the \(n\)-th row of \(\mathbf{H}\). Let \(\mathbf{\Psi}=\text{diag}\left\{\rho_{1},\cdots,\rho_{N}\right\}\). The received signal scales to  
\[\mathbf{y}^{\prime}=\mathbf{\Psi}^{-1}\mathbf{y}. \tag{5}\]  
As illustrated in Figure 1, each antenna is equipped with two ADCs to uniformly quantize the real and imaginary parts of the signal. After quantization, the quantized received signal can be re-scaled and modeled as [15; 38]\[\mathbf{y}_{q}=\boldsymbol{\
abla}Q\left(\mathbf{y}^{\prime}\right)=\boldsymbol{ \
abla}Q\left(\mathcal{R}\left(\mathbf{y}^{\prime}\right)\right)+j\boldsymbol{ \
abla}Q\left(\mathcal{I}\left(\mathbf{y}^{\prime}\right)\right), \tag{6}\]  
where \(\mathcal{R}\left(\cdot\right)\) and \(\mathcal{I}\left(\cdot\right)\) denote the real and imaginary part of a complex scalar, respectively. The nonlinear quantization function \(Q\left(\cdot\right)\) assumes uniform quantization with step \(\Delta=2^{-b}\) as [39]  
\[Q(y^{\prime})= \begin{cases}\left[\frac{y^{\prime}}{\Delta}-\frac{1}{2}\right] \Delta+\frac{\Delta}{2},&\text{if }-\frac{2^{b}\Delta}{2}\leq y^{\prime}\leq\frac{2^{b}\Delta}{2},\\ \frac{2^{b}\Delta}{2}-\frac{\Delta}{2},&\text{if }\ y^{\prime}>\frac{2^{b} \Delta}{2},\\ -\frac{2^{b}\Delta}{2}+\frac{\Delta}{2},&\text{if }\ y^{\prime}<-\frac{2^{b} \Delta}{2},\end{cases}