## 3 Proposed Max-SNR-FP Scheme
### Optimize \(\mathbf{v}\) Given \(\mathbf{\Phi}\) and \(\mathbf{\Psi}\)
3:repeat
4:Given \(\mathbf{\mathrm{\Phi}}^{(k)}\) and \(\mathbf{\mathrm{\
abla}}^{(k)}\), solve (17) to obtain \(\mathbf{\mathrm{v}}^{(k+1)}\). 5:Given \(\mathbf{\mathrm{v}}^{(k+1)}\) and \(\mathbf{\mathrm{\Psi}}^{(k)}\), solve (24) to obtain \(\mathbf{\mathrm{\Phi}}^{(k+1)}\). 6:Given \(\mathbf{\mathrm{v}}^{(k+1)}\) and \(\mathbf{\mathrm{\Phi}}^The overall computational complexity of the proposed Max-SNR-FP scheme, i.e., Algorithm 1, is \(\mathcal{O}(L((M+1)^{3}+2 MN^{2}+2M^{2})\text{In}(1/\epsilon)+M^{3}+N^{3}+5M^{2}+2 MN+2M+2 MN^{2})\) float-point operations (FLOPs), where \(L\) represents the numbers of alternating iterations, and \(\epsilon\) denotes the accuracy. ## 4 Proposed Max-SNR-EAR Scheme

In the previous section, we proposed the Max-SNR-FP method to maximize the achievable rate by jointly optimizing beamforming vector \(\mathbf{v}\), IRS phase shift matrices \(\mathbf{\Phi}\), and \(\mathbf{\Psi}\). However, this comes with high computational complexity. Aimed at decreasing complexity, the Max-SNR-EAR scheme with low complexity is proposed in this section. ### Optimizing \(\mathbf{v}\) Given \(\mathbf{\Phi}\) and \(\mathbf{\Psi}\)

Given IRS phase shift matrices \(\mathbf{\Phi}\) and \(\mathbf{\Psi}\), in accordance with the principle of maximizing SLNR in [44], beamforming vector \(\mathbf{v}\) can be optimized by tackling the problem as follows:

\[\underset{\mathbf{v}}{\text{max}} \mathbf{v}^{H}\mathbf{A}\mathbf{\bar{v}} \tag{14a}\]7 (31a) s.t. \[\underset{\mathbf{v}}{\text{max}} \mathbf{v}^{H}\mathbf{A}\mathbf{\bar{v}} \tag{14a}\]8 \[\underset{\mathbf{v}}{\text{max}} \mathbf{v}^{H}\mathbf{A}\mathbf{\bar{v}} \tag{14a}\]9

Similar to Algorithm 1, we calculate \(\mathbf{v}\), \(\mathbf{\Phi}\), and \(\mathbf{\Psi}\) alternately until convergence, i.e., \(|R_{b}^{(k)}-R_{b}^{(k-1)}|\leq\epsilon\). The overall computational complexity of Max-SNR-EAR scheme is \(\mathcal{O}(K(2M^{2}+N^{3}+8N^{2}M+2 MN)\) FLOPs, where \(K\) means the numbers of alternating iterations. ## 5 Proposed Max-SNR-MM Scheme

In Sections 3 and 4, the Max-SNR-FP and Max-SNR-EAR schemes are presented to jointly calculate the transmit beamforming vector and IRS phase shift matrices, where the active and passive IRS phase shift matrices are optimized, respectively. To investigate the effect of the joint design of active and passive IRS phase shift matrices on system performance enhancement, in this section, we propose low-complexity alternating iteration scheme Max-SNR-MM to boost the achievable rate. In the following, on the basis of the criteria of alternate optimization, we optimize \(\mathbf{v}\) and \(\mathbf{\Theta}\). ### Optimizing \(\mathbf{v}\) Given \(\mathbf{\Theta}\)

In accordance with (6), the received signal can be recast as follows:

\[\text{s.t.} \mathbf{v}^{H}\mathbf{v}=1,\eqref{eq:P_r}, \tag{14b}\]0 \[\text{s.t.} \mathbf{v}^{H}\mathbf{v}=1,\eqref{eq:P_r}, \tag{14b}\]1

Then, the SNR can be expressed as

\[\text{s.t.} \mathbf{v}^{H}\mathbf{v}=1,\eqref{eq:P_r}, \tag{14b}\]2

The transmission power of the hybrid IRS in (12b) can be reformulated as follows:

\[\text{s.t.} \mathbf{v}^{H}\mathbf{v}=1,\eqref{eq:P_r}, \tag{14b}\]3 \[\text{s.t.} \mathbf{v}^{H}\mathbf{v}=1,\eqref{eq:P_r}, \tag{14b}\]4 \[\text{s.t.} \mathbf{v}^{H}\mathbf{v}=1,\eqref{eq:P_r}, \tag{14b}\]5

Given the IRS PSM \(\mathbf{\Theta}\), the optimization problem with respect to \(\mathbf{v}\) is expressed as follows:

\[\text{s.t.} \mathbf{v}^{H}\mathbf{v}=1,\eqref{eq:P_r}, \tag{14b}\]6 (43a) s.t. \[\text{s.t.} \mathbf{v}^{H}\mathbf{v}=1,\eqref{eq:P_r}, \tag{14b}\]7

The form of Problem (43) is similar to that of Problem (14), since the denominator is independent of \(\mathbf{v}\), and the same method was taken into account to address Problem (43). For the sake of brevity, this is not repeated here. ### Optimize \(\mathbf{\Theta}\) Given \(\mathbf{v}\)

Given beamforming vector \(\mathbf{v}\), we focus on optimizing the hybrid IRS phase shift matrix \(\mathbf{\Theta}\) in this subsection.