# GAO-FCNN-Enabled Beamforming of the RIS-Assisted Intelligent Communication System
## 2 System Model  
As depicted in Figure 1, a downlink intelligent MIMO wireless communication system is considered in this paper. The BS has \(N_{t}\) BS antennas, the RIS has \(M\) reflection elements, and the receiver has \(N_{r}\) antennas. During the process of collaborative beamforming, both the transmission beamforming array associated with the BS and a reflection beamforming array belonging to the RIS should be refined. The emission direction of the signal is controlled by adjusting its emission beamforming vector [29; 30]. And each unit of the beamforming vector represents a unique weight assigned to each BS antenna. These weights are utilized to adjust both phase and magnitude of the transmitted signal, known as the transmit precoding. The incoming signal received at the RIS is given by  
\[\mathbf{y}_{1}=\mathbf{H}_{1}\mathbf{s}+\mathbf{n}_{r}, \tag{1}\]  
where \(\mathbf{s}\in\mathbb{C}^{N_{t}\times 1}\) is the transmitted signal vector, \(\mathbf{y}_{1}\in\mathbb{C}^{M\times 1}\) represents the received signal, \(\mathbf{H}_{1}\in\mathbb{C}^{M\times N_{t}}\) is the channel response between the BS and RIS, and \(\mathbf{n}_{r}\in C^{M\times 1}\) represents the noise vector at the RIS. Therefore, the reflection signal matrix on the RIS (i.e., the signal reflected by the RIS) can be expressed as  
\[\mathbf{x}=\mathbf{H}_{1}\mathbf{s}, \tag{2}\]  
The signal transmitted by the BS passes through the channel \(\mathbf{H}_{1}\) to the RIS. Subsequently, it is bounced back by the RIS in accordance with this reflection phase matrix \(\mathbf{\Phi}\), where \(\mathbf{\Phi}\in C^{M\times M}\) represents the matrix of reflection phases associated with the RIS. It is expressed as  
\[\mathbf{\Phi}=diag(\beta_{1}e^{i\varphi_{1}},\beta_{2}e^{j\varphi_{2}},\dots, \beta_{m}e^{i\varphi_{m}},\dots,\beta_{M}e^{i\varphi_{M}}), \tag{3}\]  
where \(e^{i\varphi_{m}}(m=1,2,\dots,M)\) represents the elements on the diagonal, \(j\) is the imaginary unit, \(\varphi_{m}\) represents the phase of the \(m\)-th reflective component in radians, and \(\beta_{m}\) is the magnitude reflection factor of the \(m\)-th unit of an RIS. They are respectively given by in the following equation:  
\[\varphi_{m}\in(0,2\pi],m=1,2,\dots,M_{r}\beta_{m}\in[0,1],m=1,2,\dots,M, \tag{4}\]  
Next, the signal \(\mathbf{y}_{2}\in C^{N_{t}\times 1}\) obtained at the receiver through different channel propagation links is expressed as  
\[\mathbf{y}_{2}=(\mathbf{H}_{0}+\mathbf{H}_{2}\mathbf{\Phi}\mathbf{H}_{1}) \mathbf{g}\
u+\mathbf{n}, \tag{5}\]  
Figure 1: The RIS–assisted intelligent downlink MIMO communication system. where \(\mathbf{H}_{0}\in C^{N_{t}\times N_{t}}\) represents the direct channel link matrix, \(\mathbf{H}_{1}\in C^{M\times N_{t}}\) represents the channel matrix between the BS and the RIS, \(\mathbf{H}_{2}\in C^{N_{r}\times M}\) is the channel matrix between the RIS and the receiver, \(\mathbf{g}\in C^{N_{t}\times 1}\) is the beamforming vector of the BS, and \(\
u\) is the transmitted symbol. It is assumed that the channel responses in the downlink follow an independent and identical distribution with zero mean, and \(\mathbf{n}\in C^{N_{r}\times 1}\) represents the additive white Gaussian noise (AWGN) vector. This joint beamforming problem is focused on optimizing system performance by adjusting transmission parameters collaboratively. These parameters belong to the following multiple nodes: the transmitter, the receiver, and the RIS. Additionally, maximizing signal-to-noise ratio (SNR) or minimizing BER is the key metric of performance improvements. However, minimizing BER directly is often very complex, since the expression of the BER may contain intractable integrals or summations. As a consequence, our technique is aimed to optimize the SNR of the received signal to achieve the optimization goal. It can be expressed as follows:  
\[