## 2 System Model  
In this work, we consider a hybrid mmWave system in which an \(M\)-antenna \(N_{\text{RF}}\)-RF-chain base station (BS) wishes to communicate with \(N\) single-RF-chain single-antenna users, e.g., \(M\geqslant N_{\text{RF}}\geqslant N\geqslant 1\). We assume that the MU mmWave system is operating in a time division duplex (TDD) mode. In addition, the BS and users are fully synchronized [31]. The BS is equipped with a fully connected hybrid architecture and a uniform linear array (ULA), where each RF chain is connected to the \(M\) antennas. We also consider that 1-bit A/Ds are equipped at both transceivers, as illustrated in Figure 1.  
We consider a narrowband block fading mmWave channel with one dominate path and \(N_{\text{cl}}\) scattering components [32; 33; 34; 35; 36]. With this model, the uplink channel \(\mathbf{h}_{k}\) can be represented as  
\[\mathbf{h}_{k}=\underbrace{\sqrt{\frac{\xi_{k}}{\xi_{k}+1}}\mathbf{h}_{k, \text{IOS}}}_{\text{Strongest component}}+\underbrace{\sqrt{\frac{1}{\xi_{k}+1}} \sqrt{\frac{1}{N_{\text{cl}}}}\widehat{\alpha}_{k,j}\mathbf{h}_{k,j}}_{ \text{Scattering components}}, \tag{1}\]  
where \(\widehat{\alpha}_{k,l},l\in\{1,\cdots,N_{\text{cl}}\}\), represents the complex small-scale fading for the \(l\)-th scattering component [37], which we assume to follow a complex Gaussian distribution with zero mean and unit variance (Note that, the power ratio of the strongest path power over the sum of other scattered paths' power, \(\xi_{k}=1/\left[\sum_{i=1}^{N_{\text{d}}}|\tilde{\alpha}_{k,i}|^{2}\right]\), is usually larger than 1 [37]. In an outdoor scenario, the power ratio for line-of-sight (LOS) can be 10 dB and the power ratio for none-line-of-sight (NLOS) can be 6 dB [37]). In addition, \(\mathbf{h}_{k,l}\in\mathbb{C}^{M\times 1}\) are the array response vectors of user \(k\) at the BS associated to the \(l\)-th propagation path, which can be expressed as  
\[\mathbf{h}_{k,l}=\left[1,e^{-j\pi\frac{d}{\lambda}\cos\left(\theta_{k,l}\right) },\ldots,e^{-j\pi(M-1)\frac{d}{\lambda}\cos\left(\theta_{k,l}\right)}\right]^{ T}, \tag{2}\]  
where \(d\) is the antenna spacing at the BS and \(\lambda\) is the carrier wavelength. Variable \(\theta_{k,l}\) is the AoA of the \(l\)-th path at the antenna array of the BS from user \(k\), which is assumed to be uniformly distributed between \([0,\ \pi]\). To estimate the AoAs at the BS, each user transmits pilots respectively according to a fixed \"0-1\" codebook while the BS performs periodic beam scan to measure pilot signals [27]. The goal of this process is to find the AoAs of the dominant paths such that beams can be steered to these directions to achieve the best effective SNR. Denoting \(\mathcal{F}=\left\{f_{\text{RF},k,1},\ldots,f_{\text{RF},k,\mathcal{F}}\right\}\) as the BS codebook [28], the best beamforming vector for the strongest AoA estimation can be expressed as  
\[f_{\text{RF},k}^{H}=\arg\max_{f_{\text{RF},k,i}\in\mathcal{F}}\left|f_{\text{ RF},k,i}^{H}\mathbf{h}_{k}\right|, \tag{3}\]  
where \(f_{\text{RF},k}^{H}\) is the best beam of the BS, and \(f_{\text{RF},k,i}\) is the beam vector for spatial scanning of the \(i\)-th direction. With ULA, the beamformer, \(f_{\text{RF},k,i}\), \(k\in\left\{1,\ldots,N\right\}\), can be represented as  
\[f_{\text{RF},k,i}=\frac{1}{M}\Big{[}\,1,e^{-j\pi\frac{d}{\lambda}\mu},\ldots, e^{-j2\pi(M-1)\frac{d}{\lambda}\mu}\Big{]}^{T}, \tag{4}\]  
where \(\mu=\cos\big{(}\overline{\theta}_{i}\big{)}=-1+\frac{2i-1}{M}\in(-1,1),i\in \left\{1,\ldots,M\right\}\). Denote further \(h_{k,i}=f_{\text{RF},k,i}^{H}\mathbf{h}_{k}\) as the effective channel after receiving beamforming \(f_{\text{RF},k,i}^{H}\).