## 1 Introduction
For the above works, DM synthesis was designed at the radio frequency frontend, which lacks flexibility. To address this problem, the researchers transferred the design of DM synthesis from the radio frequency frontend to the baseband. In [23], on the basis of maximizing the signal-to-artificial noise (AN) ratio and signal-to-leakage-noise ratio approaches designed at the baseband, an AN projection matrix and precoder vector were obtained to maximize the SR of a multi-beam DM network. An AN-aided zero-forcing synthesis scheme that achieved the dynamic characteristics of multi-beam DM through the random varying of AN vector was proposed in [24], which was easier and more effective to implement than traditional dynamic multi-beam DM synthesis methods are in wireless networks with a certain level of performance loss. In [25], the authors investigated a DM system with malicious jamming and proposed three receive beamforming algorithms to boost the SR. In particular, conventional DM networks can only transmit single-bit streams, while the emergence of IRS enabled DM networks to send multiple bit streams. An energy and cost-efficient IRS can be employed to create multiple user-friendly and controllable paths to transmit two-bit streams or increase the rate to enhance the performance of DM systems. For example, in [26], to transmit two-bit streams from Alice to Bob and maximize the SR of an IRS-assisted DM network, the precoder vectors and phase shift matrix (PSM) of the IRS were jointly devised with high-performance general alternating iterative and low-complexity null-space projection algorithms. However, the design of receiver beamforming was not investigated in [26]. To investigate the effect of receiver beamforming on performance improvement on the basis of the system model of [26], the authors in [27] proposed two optimization algorithms to maximize the receiver power sum by jointly optimizing the PSM at IRS and receiver beamforming vectors at the user. In [28], aiming to maximize the SR of an IRS-assisted multiple-input single-output (MISO) DM network, the semi-definite relaxation method was utilized to derive the precoding and IRS PSM when the location information of an eavesdropper is available. Thanks to a combination with IRS, SR performance was significantly boosted. However, all the above work was conducted on the basis of a fully passive IRS, and a satisfactory achievable rate of the system may not be ensured due to the effect of \"double fading\" in cascaded channels. To effectively combat this effect and enhance the performance of passive IRS-aided wireless communication networks, a fully active IRS was recently investigated [29; 30; 31; 32; 33], and the simulation results showed that active IRS achieved significant performance improvements compared to a passive IRS. However, the higher rate achieved by active IRS comes at the price of high hardware cost and power consumption [34]. To overcome the limitations of fully passive and fully active IRSs, a hybrid active-passive IRS was proposed [35]. The main idea of the hybrid IRS is employing some active elements to substitute those of the passive IRS. Active elements with the signal amplification of a hybrid IRS can effectively compensate for the cascade path loss (PL), and increase the achievable rate [36]. In [37], to explore the potential of hybrid relay-IRS in helping single-user mobile edge computing systems in computational offloading, an efficient scheme was proposed to design received beamforming, IRS reflection coefficients, and computational parameters, and the latency was dramatically reduced with the help of the IRS. In [38], a hybrid IRS-assisted covert transmission network was proposed to boost the performance of traditional covert communication networks, and an alternate algorithm with closed-form expression was derived to obtain the transmission power and IRS PSM.