## 2 System Model
### Communication Model  
Let \(\mathbf{H}_{1}\in\mathbb{C}^{N\times M}\) denote the channel matrix from BS to IRS. \(\mathbf{h}_{k,2}\in\mathbb{C}^{N\times 1}\) represents the channel vector from IRS to user \(k\) with \(\forall k\in\mathcal{K}\). The transmitted signal received by the user \(k\) is given by  
\[\begin{split} y_{c,k}(t)=&\mathbf{h}_{k,2}^{T} \mathbf{\Phi}\mathbf{H}_{1}\mathbf{x}+n_{k}\\ =&\mathbf{h}_{k,2}^{T}\mathbf{\Phi}\mathbf{H}_{1} \mathbf{w}_{k}s_{k}+\sum_{i=1,i\
eq k}^{K}\mathbf{h}_{k,2}^{T}\mathbf{\Phi} \mathbf{H}_{1}\mathbf{w}_{i}s_{i}+n_{k},\end{split} \tag{4}\]  
where \(\mathbf{\Phi}\triangleq\mathrm{diag}\big{\{}\chi_{1}e^{i\phi_{1}},\chi_{2}e^{ i\phi_{2}},\cdots,\chi_{N}e^{i\phi_{N}}\big{\}}\in\mathbb{C}^{N\times N}\) is the diagonal phase shift matrix of the IRS, \(\chi_{n}\in[0,1]\) and \(\phi_{n}\in[0,2\pi)\) indicate the amplitude and phase of element \(n\) with \(\forall n\in\mathcal{N}\), respectively, due to the high overhead of simultaneous implementing of independent control of phase shift and amplitude [13].Therefore, we assume the ideal reflection amplitude of the passive IRS with \(\chi_{n}=1,\forall n\in\mathcal{N}\)[32]. \(n_{k}\sim\mathcal{CN}\big{(}0,\sigma_{c}^{2}\big{)}\) is the additive white Gaussian noise (AWGN).  
We take the Rician fading channel model in this work, and channel \(\mathbf{H}_{1}\) can be formulated as  
\[\mathbf{H}_{1}=\sqrt{\frac{\gamma_{1}}{1+\gamma_{1}}}\mathbf{H}_{\text{LoS}}+ \sqrt{\frac{1}{1+\gamma_{1}}}\mathbf{H}_{\text{NLoS}}, \tag{5}\]  
where \(\gamma_{1}\) denotes the Rician factor. \(\mathbf{H}_{\text{LoS}}\in\mathbb{C}^{N\times M}\) and \(\mathbf{H}_{\text{NLoS}}\in\mathbb{C}^{N\times M}\) are LoS component and NLoS component, respectively. The LoS channel matrix can be expanded as \(\mathbf{H}_{\text{LoS}}=\sqrt{\pi}e^{i\varphi}\mathbf{a}_{r}(\theta_{r}) \mathbf{b}_{t}^{H}(\theta_{t})\), where \(\alpha\) and \(\varphi\) are the large-scale channel gain and a random phase uniformly distributed in the range from \(0\) to \(2\pi\), respectively. Meanwhile, \(\mathbf{a}_{r}(\theta_{r})\in\mathbb{C}^{N\times 1}\) represents the receive steering vector at IRS with the angle of arrival \(\theta_{r}\), \(\mathbf{b}_{t}(\theta_{t})\in\mathbb{C}^{M\times 1}\) indicates the transmit steering vector of BS with the angle of departure \(\theta_{t}\). The steering vector of BS \(\mathbf{b}(\theta)\) can be formulated as  
\[\mathbf{b}(\theta)=\frac{1}{\sqrt{M}}\Big{[}1,e^{-j\frac{2\pi\alpha}{d_{0}} \cos\theta},\cdots,e^{-j\frac{2\pi\alpha}{\lambda}(M-1)d_{0}\cos\theta}\Big{]} ^{T}, \tag{6}\]  
where \(d_{0}\) and \(\lambda\) denote the antennas' spacing and signal wavelength. Similarly, the steering vector of IRS \(\mathbf{a}(v,\theta)\) can be formulated as  
\[\mathbf{a}(v,\theta)=\frac{1}{N}\bigg{[}1,e^{j\frac{2\pi\alpha}{\lambda}(n\cos v \cos\theta+n\sin v\sin\theta)},\cdots,e^{j\frac{2\pi d_{0}}{\lambda}\big{(}( \sqrt{N}-1)\cos v\cos\theta+\big{(}\sqrt{N}-1\big{)}\sin v\sin\theta\big{)}} \bigg{]}^{T}. \tag{7}\]  
We leverage the SINR ratio as the performance indicator of communication. Let \(\rho_{k}\) denote the SINR of user \(k\), which is given by  
\[\rho_{c,k}=\frac{\left|\mathbf{h}_{k,2}^{T}\mathbf{\Phi}\mathbf{H}_{1} \mathbf{w}_{k}\right|^{2}}{\sum_{i=1,i\
eq k}^{K}\left|\mathbf{h}_{k,2}^{T} \mathbf{\Phi}\mathbf{H}_{1}\mathbf{w}_{i}\right|^{2}+\sigma_{c}^{2}}. \tag{8}\]