## 1 Introduction  
With the explosive growth of wireless communication devices, the scarcity of available spectrum resources has become increasingly acute. Consequently, effectively utilizing existing spectrum resources has emerged as a pivotal factor in advancing wireless communication technology [1; 2; 3]. Recently, in-band full-duplex (FD) systems have drawn notable interest in the design of next-generation wireless communication systems due to their capability of transmitting and receiving data on the same frequency band concurrently. Compared to traditional half-duplex (HD) systems, such as time division duplex or frequency division duplex, FD systems can significantly enhance spectrum utilization and reduce the time delay of wireless communication systems [4; 5]. Additionally, FD systems can significantly improve confidentiality while maintaining the same data rate [4]. Therefore, FD technology is critical for communication in ultra-dense network environments in the future. Despite the advantages mentioned above, a prominent problem in FD systems is the inevitable self-interference (SI) emanating from the transmitter to the receiver, as its high intensity significantly interferes with uplink (UL) signals. To tackle this issue, three typical categories of SI cancellation (SIC) methods are discussed in the literature: transmission-domain processing, analog-domain processing, and digital-domain signal processing [4]. The transmission-domain SIC scheme involves suppressing SI through hardware design,such as physical isolation, shielding materials, and antennas with different polarization directions, thereby minimizing the power of SI signals before they reach the receiver [6; 7; 8]. The analog-domain SIC scheme focuses on suppressing SI in the radio frequency (RF) chain by adding isolators or designing circuits to generate signals that counteract the SI signals [9; 10; 11; 12; 13]. In [9], the authors added an additional RF chain at the transmitter, connected to the receiver. They estimated the SI channel and used the results to digitally preprocess the transmitted signal, generating a counteracting signal with the same amplitude but opposite phase to the SI, thus canceling it at the receiver. Similarly, in [10], an active RF chain was utilized to generate a signal compensating for the inevitable leakage from passive analog SIC. The digital-domain SIC scheme employs digital signal processing techniques on baseband signals to cancel residual SI, exploiting the spatial degrees of freedom provided by multi-antenna systems. Typically, digital SIC methods involve beamforming techniques such as zero-forcing (ZF), minimum mean square error (MMSE), and least squares [14; 15; 16; 17]. In [14], the authors used ZF precoding at the transmitter and maximum ratio combining at the receiver to effectively cancel SI. In [15], the authors employed the MMSE algorithm to jointly design transmitting and receiving beamforming, aiming to minimize the mean square error (MSE) and reduce the SI. However, despite numerous attempts, these SIC methods come with high hardware expenses, substantial complexity, and constrained performance. Thus, the reconfigurable intelligent surface (RIS) is considered a promising technology and has attracted increasing attention in the design of FD systems. Specifically, RIS is a two-dimensional (2D) metasurface consisting of programmable, low-cost, and passive elements [18; 19]. Due to its flexibility and low-cost nature, RIS has been extensively applied in wireless communication systems [20; 21; 22; 23; 24]. For the FD systems, RIS has been proven a promising technology to tackle the SIC and thus is widely investigated in the literature [25; 26; 27; 28; 29]. In [26], intelligent reflecting surface (IRS), as the most famous type of RIS, was exploited in FD systems to achieve efficient SIC.