## 2 Multiuser Hybrid Beamforming System Model  
The proposed multiuser system model is depicted in Figure 1, wherein a mmWave MIMO base station, equipped with \(N_{BS}\) antennas and \(N_{RF}\) RF chain, is in communication with \(M\) simultaneous users each having \(N_{M}\) antennas and one RF chain through \(N_{t}\) streams. Since each UE is assumed to be served by only one downlink stream and contains only one RF chain, analog combining is applied at the UE. This configuration is similar to works in [4]. The base station utilizes hybrid beamforming, employing a network of r-bit quantized phase shifters. The decision to employ a single RF chain per UE is motivated by practical considerations, aiming for lower complexity, cost, and power consumption. Conversely, the BS is equipped with advanced digital signal processing (DSP) capabilities designed to effectively handle multiple data streams. BS attaches with each UE via one stream. This leads to a total of \(N_{t}=M\) streams, where M represents the maximum number of simultaneous users the BS can serve at once. This aligns with the count of RF chains at the BS (\(M\leq N_{RF}\)), possible through hybrid schemes enabling spatial multiplexing and multi-user MIMO. This grants the BS the ability to communicate concurrently with multiple UEs using several beams. Design of the end-to-end communication system is shown in Figure 2. Parameter for each processing block in Figure 2 is shown in Table 1.  
Figure 1: Transceiver architecture for multiuser hybrid beamforming. In consideration of operational efficiency and hardware constraints, beamforming codebooks are commonly resorted to in mmWave and massive MIMO systems to effectively accommodate users. The sum rate achievable across all UEs is optimized using MADDPG in the proposed approach. Through MADDPG, the RF codebook at the base station is estimated. Represented by \(W\), the beam codebook chosen by the base station consists of N beamforming and combining vectors, each crafted in accordance with the structure outlined in Equation (1). \[w=\frac{1}{\sqrt{N_{BS}}}\Big{[}e^{i\theta_{1}},e^{i\theta_{2}},...,e^{i\theta _{N_{BS}}}\Big{]}^{T} \tag{1}\]  
In this context, each phase shift \(\theta_{m}\) is chosen from a finite set \(S\) containing \(2^{r}\) discrete values, uniformly selected from the range of \((-\pi,\pi)\). Here, the parameter \(r\) represents the number of quantization bit used in phase shifters. The BS employs baseband precoding denoted by \(F_{BB}=[f_{BB_{1}},f_{BB_{2}},...,f_{BB_{M}}]\in\mathbb{C}^{N_{RF}\times M}\) to process the transmit signal \(s=[s_{1},s_{2},...,s_{M}]^{T}\in C^{M}\) in compliance with \(E\{ss^{H}\}=\frac{p}{M}I_{M}\), assuming uniform power distribution among users. Notably, \(P\) signifies the average power. RF precoders \(F_{RF}\in C^{N_{BS}\times N_{RF}}\), constructed using phase shifters are utilized to direct the signal to \(N_{BS}\) transmit antennas. Moreover, considering \(F_{RF}\) consists of analog phase shifters, constant equal-norm components in the RF precoder is assumed, i.e., \(\left|[F_{RF}]_{ij}\right|^{2}=\frac{1}{N_{BS}}\). Furthermore, the power constraint \(||F_{RF}F_{BB}||_{F}^{2}=M\) is maintained through \(F_{BB}\) normalization. Consequently, the transmitted signal comprising \(N_{BS}\times 1\) elements is given as  
\[x=F_{RF}F_{BB}s. \tag{2}\]  
considering \(H_{k}\in C^{N_{M}\times N_{BS}}\) be the channel matrix between \(BS\) and \(k\)th user, the received signal for \(k\)th user for a narrowband block fading channel is given as  
\[\vec{y_{k}}=H_{k}\sum_{n=1}^{M}F_{RF}F_{BB_{n}}s_{n}+n_{k}, \tag{3}\]