Beamforming for PIN Diode-Based IRS-Assisted Systems Under a Phase Shift-Dependent Power Consumption Model

Consider a downlink narrowband multi-user multiple-input single-output (MISO) system as shown in Fig. 1. The system consists of a BS equipped with an $N$-antenna uniform linear array (ULA) and $K$ single-antenna users. The direct links between the BS and the users are assumed to be blocked, and thus an IRS is deployed to establish virtual line-of-sight (LoS) links to the users. The IRS is designed as a uniform planar array (UPA) of dimension $M_{\mathrm{x}}\times M_{\mathrm{y}}$. The IRS operates with 1-bit phase control resolution, such that the phase shift of each scattering element is binary-switchable between two states through the on/off-state of a PIN diode. Thus, we define ${\mathbf{b}}\in\{0,1\}^{M}$ as the state vector of the employed PIN diodes, where $M=M_{\mathrm{x}}\times M_{\mathrm{y}}$. For the $m$-th element, $[\mathbf{b}]_{m}=1$ ($[\mathbf{b}]_{m}=0$) indicates that the corresponding PIN diode is in the on-state (off-state). Without loss of generality, we set the phase shift of the $m$-th element to zero when $[\mathbf{b}]_{m}=1$ and $\pi$ when $[\mathbf{b}]_{m}=0$. In this way, the IRS phase shift matrix $\boldsymbol{\Phi}\in\mathbb{C}^{M\times M}$ is given by

Therefore, the equivalent baseband received signal $y_{k}$ at the $k$-th user can be represented as

where $\mathbf{s}=[s_{1},\dots,s_{K}]^{T}\in\mathbb{C}^{K\times 1}$ is the BS transmitted symbol vector with $\mathbb{E}\{\mathbf{s}\mathbf{s}^{H}\}=\mathbf{I}_{K}$, $\mathbf{F}=\big{[}\mathbf{f}_{1},\dots,\mathbf{f}_{K}\big{]}\in\mathbb{C}^{N\times K}$ is the active beamforming at the BS, and $\mathbf{B}=\mathrm{diag}(\mathbf{b})$. $n_{k}\sim\mathcal{CN}(0,\sigma^{2})$ represents the additive Gaussian noise at the $k$-th user. $\mathbf{G}\in\mathbb{C}^{M\times N}$ denotes the BS-IRS channel matrix, and $\mathbf{h}_{k}^{H}\in\mathbb{C}^{1\times M}$ is the channel vector between the IRS and the $k$-th user.

In this paper, we assume the BS-IRS channel $\mathbf{G}$ to be Rician distributed comprising one LoS and a number of non-LoSs (NLoSs) links. Hence, denoting the Rician factor by $\kappa_{\mathrm{G}}$, $\mathbf{G}$ can be modeled as [20]

where $\mathbf{G}_{\mathrm{LoS}}\in\mathbb{C}^{M\times N}$ and $\mathbf{G}_{\mathrm{NLoS}}\in\mathbb{C}^{M\times N}$ are the LoS and NLoS components, respectively. Specifically, the LoS component can be characterized by the plane-wave model as $\mathbf{G}_{\mathrm{LoS}}=\sqrt{\mathrm{PL}_{\mathrm{G}}MN}\mathbf{a}_{\mathrm{I}}(\theta_{\mathrm{r}},\varphi_{\mathrm{r}})\mathbf{a}^{H}_{\mathrm{BS}}(\theta_{\mathrm{t}})$, where $\mathrm{PL}_{\mathrm{G}}$, $\theta_{\mathrm{t}}$, $\theta_{\mathrm{r}}$, $\varphi_{\mathrm{r}}$, and $\lambda$ denote the path loss, the angle of departure (AoD), the elevation angle of arrival (AoA), the azimuth AoA of the LoS link, and the carrier wavelength, respectively. The array response vectors for BS and IRS are denoted by $\mathbf{a}_{\mathrm{BS}}$ and $\mathbf{a}_{\mathrm{I}}$, respectively. By defining $\mathbf{a}(N,x)=\frac{1}{\sqrt{N}}\left[1,e^{\mathrm{j}\pi{x}},\dots,e^{\mathrm{j}\pi(N-1){x}}\right]^{T}$, the array response vector of the ULA at the BS can be expressed as $\mathbf{a}_{\mathrm{BS}}(\theta_{\mathrm{t}})=\mathbf{a}\big{(}N,\mathrm{cos}(\theta_{\mathrm{t}})\big{)}$. For the UPA at the IRS, the array response vector is given by $\mathbf{a}_{\mathrm{I}}(\theta_{\mathrm{r}},\varphi_{\mathrm{r}})=\mathbf{a}\big{(}M_{\mathrm{x}},-\mathrm{sin}(\theta_{\mathrm{r}})\mathrm{sin}(\varphi_{\mathrm{r}})\big{)}\otimes\mathbf{a}\big{(}M_{\mathrm{y}},-\mathrm{sin}(\theta_{\mathrm{r}})\mathrm{cos}(\varphi_{\mathrm{r}})\big{)}$. The NLoS component $\mathbf{G}_{\mathrm{NLoS}}$, on the other hand, can be expressed as $\mathbf{G}_{\mathrm{NLoS}}=\sqrt{\mathrm{PL}_{\mathrm{G}}}\mathbf{G}_{\mathrm{SS}},$ where the elements of matrix $\mathbf{G}_{\mathrm{SS}}\in\mathbb{C}^{M\times N}$ are complex Gaussian distributed with mean zero and unit covariance, and the distribution of each element is independent of the other elements.

Similarly, the IRS-user channel $\mathbf{h}^{H}_{k}$ is also modeled as Rician fading, i.e.,

where $\mathbf{h}^{H}_{\mathrm{LoS},k}=\sqrt{\mathrm{PL}_{\mathrm{h}_{k}}M}\mathbf{a}^{H}_{\mathrm{I}}(\vartheta_{\mathrm{t,}k},\psi_{\mathrm{t,}k})$ and $\mathbf{h}_{\mathrm{NLoS},k}\sim\mathcal{CN}(0,\mathrm{PL}_{\mathrm{h}_{k}}\mathbf{I}_{M})$. $\kappa_{\mathrm{h}_{k}}$, $\mathrm{PL}_{\mathrm{h}_{k}}$, $\vartheta_{\mathrm{t,}k}$, and $\psi_{\mathrm{t,}k}$ are the Rician factor, the path loss, the LoS elevation AoD, and the LoS azimuth AoD, respectively. In this paper, we assume that $\mathbf{G}$ and $\mathbf{h}_{k}$ are perfectly known at the BS.

The total power consumption of a PIN-diode-based IRS comprises two components [16], [17]. The first one is the static power consumption, which encompasses the energy used by control devices and circuits. The second component is the PS-DPC caused by the PIN diodes, which varies dynamically with respect to the beamforming design at the IRS. Therefore, the total power consumption of the IRS can be modeled as

where $P_{\mathrm{IRS,static}}$ and $P_{\mathrm{IRS,PS}}$ are the static power consumption and the PS-DPC, respectively. As $P_{\mathrm{IRS,static}}$ is fixed once the IRS is manufactured, in this paper, we mainly focus on the PS-DPC. Specifically, we consider a 1-bit IRS, where the phase shift of each element is switched between two states according to the on/off-states of a PIN diode. Therefore, the PS-DPC of the IRS can be expressed as follows

where $P_{\mathrm{PIN}}$ is the power consumption of an on-state PIN diode.

In the literature, the transmit power at the BS is typically the only power consumption considered for the wireless system design, regardless of whether [21] or not [22] an IRS is deployed. In contrast, if the PS-DPC of the IRS is also considered, the power consumption of the entire system includes the power consumption of the BS and the IRS, and is given by

where $P_{\mathrm{BS,t}}=\mathrm{Tr}(\mathbf{F}^{H}\mathbf{F})$ is the BS transmit power consumption. $P_{\mathrm{BS,C}}$ and $P_{\mathrm{BS,RF}}$ are the powers consumed by control circuits and each radio frequency chain at the BS, respectively. Note that the power consumption terms in (7) are independent of the beamforming design at BS and IRS, except for $\mathrm{Tr}(\mathbf{F}^{H}\mathbf{F})$ and $P_{\mathrm{PIN}}\mathbf{1}_{M}^{T}\mathbf{b}$. Thus, we shall focus on the terms $\mathrm{Tr}(\mathbf{F}^{H}\mathbf{F})+P_{\mathrm{PIN}}\mathbf{1}_{M}^{T}\mathbf{b}$ for beamforming design in the following.

In this paper, we aim to maximize the sum rate of all users under a system power constraint, which leads to the following optimization problem:

where $R_{k}=\mathrm{log}\left(1+\frac{\left|\mathbf{h}_{k}^{H}(2\mathbf{B}-\mathbf{I}_{M})\mathbf{G}\mathbf{f}_{k}\right|^{2}}{\gamma_{k}}\right)$ denotes the rate (bps per Hertz) of the $k$-th user, with ${\gamma}_{k}=\sum_{k^{\prime}\neq k}|\mathbf{h}_{k}^{H}(2\mathbf{B}-\mathbf{I}_{M})\mathbf{G}\mathbf{f}_{k^{\prime}}|^{2}+\sigma^{2}$ representing the interference plus noise. Furthermore, $P_{0}=P-P_{\mathrm{BS,C}}-NP_{\mathrm{BS,RF}}-P_{\mathrm{IRS,static}}$ is the effective system power budget.

Due to the binary constraint on $\mathbf{b}$, problem (8) is a mixed integer nonlinear programming (MINLP) problem, which is nondeterministic polynomial-time (NP)-hard [23]. Moreover, as we account for the PS-DPC, $\mathbf{F}$ and $\mathbf{b}$ are coupled in both the objective function and the constraint $\mathrm{C1}$. This makes the beamforming optimization more challenging than for conventional IRS-assisted rate maximization problems in the literature, where only the power consumption at the BS is considered [6], [21], [24].

Remark 1: In the literature, classical methods such as the alternating optimization (AO) were adopted to tackle the coupling of $\mathbf{F}$ and $\mathbf{b}$ [21]. However, this approach cannot be directly applied to solve problem (8) due to the total system power constraint C1. In particular, if the conventional AO algorithm is adopted to solve problem (8), optimizing $\mathbf{F}$ for a fixed $\mathbf{b}$ will always lead to an equality in power constraint $\mathrm{C1}$, i.e., $\mathbf{F}^{H}\mathbf{F}+P_{\mathrm{PIN}}\mathbf{1}_{M}^{T}\mathbf{b}=P_{0}$. However, with this condition, the feasible set during the subsequent optimization of $\mathbf{b}$ is exceedingly small. Specifically, during the $\mathbf{b}$ optimization in the $i$-th iteration, C1 is reformulated as $\mathbf{1}_{M}^{T}\mathbf{b}\leq\mathbf{1}_{M}^{T}\mathbf{b}_{(i-1)}$, where $\mathbf{b}_{(i-1)}$ denotes the solution of $\mathbf{b}$ in the last iteration. This means that during the optimization of $\mathbf{b}$, we need to seek a larger achievable rate with fewer on-state PIN diodes, which significantly restricts the IRS’s beamforming DoF. Therefore, applying the conventional AO algorithm for sum rate maximization under the PS-DPC model results in a quick convergence to highly sub-optimal local maxima. To tackle this challenge, in the following sections, we introduce several novel beamforming methods for both the single-user and multi-user cases, which can jointly optimize the power allocation and the beamforming at both the BS and the IRS.

The main idea of the GBD algorithm is to decompose the original MINLP problem into two subproblems, i.e., a primal problem and a master problem, which provide upper and lower bounds for the original problem, respectively [23], [25], [26]. By iteratively solving these two subproblems, the gap between the upper and lower bounds can be progressively reduced, thereby converging towards the optimal solution. To guarantee the convergence of the GBD algorithm, the following two conditions have to be satisfied [23]:

• •

  The problem is convex with respect to the involved continuous variables if the discrete variables are fixed.

• •

  The problem is linear with respect to the involved discrete variables if the continuous variables are fixed.

To be able to employ the GBD algorithm, we first transform the rate maximization problem in (9) into an equivalent problem that maximizes the power of the received signal at the user. This transformation leads to the following problem

Unfortunately, the objective function of (10) is still not suitable for applying the GBD algorithm, as it is non-linear with respect to $\mathbf{b}$. However, assuming the optimal solution of (10) is $\mathbf{f}^{\star}$, we notice that $\mathrm{e}^{\mathrm{j}\psi}\mathbf{f}^{\star}$ is also an optimal solution of (10) for any phase shift $\psi$. This intrinsic property allows us to choose $\psi$ such that the phase of $\mathbf{f}^{H}\mathbf{H}^{H}_{\mathrm{c}}(2\mathbf{b}-\mathbf{1}_{M})$ is rotated to zero. Consequently, we can apply this phase rotation to transform (10) into a more tractable equivalent form:

which satisfies the conditions required for application of the GBD algorithm.

In the $i$-th iteration, for the primal problem, we fix the discrete variable $\mathbf{b}$ to the value of the last iteration, i.e., $\mathbf{b}_{(i-1)}$. Thus, problem (11) reduces to an optimization problem with respect to $\mathbf{f}$, and is given as

Problem (12) is clearly a convex problem, and the optimal solution, i.e., $\mathbf{f}_{(i)}$, can be computed by standard convex program solvers such as CVX [27]. Noticing that $\left(\mathbf{f}_{(i)},\mathbf{b}_{(i-1)}\right)$ is a feasible solution of problem (11), the primal problem can provide an upper bound $\eta_{\mathrm{U},(i)}$ for problem (11). Furthermore, the Lagrangian function of primal problem (12) can be constructed as

where $\xi$ and $\mu$ are the dual variables for constraints $\mathrm{C1}$ and $\mathrm{C3}$, respectively.

However, problem (12) may be infeasible for the given $\mathbf{b}_{(i-1)}$, e.g., when $P_{\mathrm{PIN}}\mathbf{1}_{M}^{T}\mathbf{b}_{(i-1)}>P_{0}$. In this case, we consider the following feasibility check problem

where $\delta\in\mathbb{R}^{+}_{0}$ is an auxiliary variable. By denoting $\overline{\xi}$ and $\overline{\mu}$ as the dual variables for constraints $\mathrm{C1}$ and $\mathrm{C3}$, respectively, we can express the Lagrangian for problem (14) as follows

This Lagrangian will be used in the formulation of the master problem, which will be discussed in the next subsection.

The master problem of the GBD algorithm is obtained based on the nonlinear convex duality theory [25]. By respectively denoting $\xi_{(i)}$, $\mu_{(i)}$ and $\overline{\xi}_{(i)}$, $\overline{\mu}_{(i)}$ as the optimal dual solutions of problems (12) and (14) in the $i$-th iteration, the master problem can be formulated as follows

where constraints $\mathrm{C4}$ and $\mathrm{C5}$ represent the optimality and feasibility cuts, respectively. Here, sets $\mathcal{F}{(i)}$ and $\mathcal{I}{(i)}$ contain the iteration indices for which the primal problem was feasible and infeasible, respectively. The optimal value of the master problem, $\eta_{\mathrm{L},(i)}$, establishes a lower bound for problem (10) [25]. However, it is still challenging to directly solve the master problem in (16), because constraints $\mathrm{C4}$ and $\mathrm{C5}$ involve an inner optimization problem related to $\mathbf{f}$. To tackle this issue, we further relax the master problem into the variant 2 (V-2) form of GBD [25]. Specifically, constraints $\mathrm{C4}$ and $\mathrm{C5}$ are relaxed by substituting the support functions (the right-hand side parts of constraints $\mathrm{C4}$ and $\mathrm{C5}$ in problem (16)) with their local linear approximations at point $\mathbf{f}_{(i)}$. This leads to

According to (13) and (15), $L$ and $\overline{L}$ are linear with respect to $\mathbf{b}$. Thereby, problem (17) is a mixed integer linear programming (MILP) problem, which can be directly solved by MILP solvers such as MOSEK [27]. The solution obtained, i.e., $\mathbf{b}_{(i)}$, is then employed in the primal problem (12) in the next iteration.

The overall GBD-BF scheme is summarized in Algorithm 1. In each iteration, we first fix $\mathbf{b}$ to the optimal solution in the last iteration, and solve the primal problem or the feasibility check problem to obtain the optimal beamforming vector $\mathbf{f}$ as well as the optimal dual variables $\xi$ and $\mu$ for the primal problem ($\overline{\xi}$ and $\overline{\mu}$ for the feasibility check problem). Then, the V-2 master problem is solved for fixed $\mathbf{f}$, $\xi$ ($\overline{\xi}$), and $\mu$ ($\overline{\mu}$) to obtain $\mathbf{b}$ for the next iteration. The computational complexity of the GBD-BF method in each iteration is $\mathcal{O}(2^{M})$.

We first focus on the online beamforming design at the BS and the IRS for given power allocation, i.e., $P_{\mathrm{BS,t}}$ at the BS and $P_{\mathrm{IRS,PS}}$ at the IRS. Specifically, as IRSs are generally deployed such that the channels are dominated by the LoS link [3], we ignore the NLoS component for algorithm design for simplicity, thus approximating $\mathbf{G}$ and $\mathbf{h}^{H}$ as

where ${\alpha}_{\mathrm{G}}=\sqrt{\frac{\kappa_{\mathrm{G}}}{1+\kappa_{\mathrm{G}}}\mathrm{PL}_{\mathrm{G}}}$ and ${\alpha}_{\mathrm{h}}=\sqrt{\frac{\kappa_{\mathrm{h}}}{1+\kappa_{\mathrm{h}}}\mathrm{PL}_{\mathrm{h}}}$. In this case, the optimal $\mathbf{f}$ for problem (9) for given $\mathbf{\Phi}$ is obtained as

where $\epsilon=\sqrt{\frac{{P_{\mathrm{BS,t}}}}{\mathbf{h}^{H}\boldsymbol{\Phi}\mathbf{G}\mathbf{G}^{H}\boldsymbol{\Phi}^{H}\mathbf{h}}}$. Therefore, the received SNR at the user can be expressed as:

where $\mathbf{h}_{\mathrm{o}}^{H}=\mathbf{a}^{H}_{\mathrm{I}}(\vartheta_{\mathrm{t}},\psi_{\mathrm{t}})\circ\mathbf{a}^{T}_{\mathrm{I}}(\theta_{\mathrm{r}},\varphi_{\mathrm{r}})$ and $\boldsymbol{\phi}=\mathrm{diag}(\mathbf{\Phi})$. To maximize the SNR, the continuous phase shift of the $m$-th IRS element without considering the PS-DPC is given by

However, for a 1-bit IRS, the phase shifts of the IRS elements are constrained to binary states, i.e., $0$ or $\pi$. Moreover, due to the PS-DPC, the number of elements with phase shift $\pi$ (${M}_{\mathrm{on}}$) is limited by the available power budget for the IRS, i.e., ${M}_{\mathrm{on}}=\lfloor\frac{P_{\mathrm{IRS,PS}}}{P_{\mathrm{PIN}}}\rfloor$. In this case, by rounding (21) to the binary phase shift states and restricting ${M}_{\mathrm{on}}\leq\lfloor\frac{P_{\mathrm{IRS,PS}}}{P_{\mathrm{PIN}}}\rfloor$, the practical phase shift configuration of the $m$-th scattering element is given by

where the set $\mathcal{M}$ is defined by the following process: First, we sort the elements of $\mathbf{h}^{H}_{\mathrm{o}}$ in descending order based on their real parts. Then, we select the indices of the top $\widetilde{M}_{\mathrm{on}}$ elements from this sorted list to form the set $\mathcal{M}$. Here, $\widetilde{M}_{\mathrm{on}}$ is defined as $\mathrm{min}(M_{\mathrm{p}},M_{\mathrm{on}})$, where $M_{\mathrm{p}}$ represents the number of positive elements in $\mathcal{R}\big{\{}{\mathbf{h}_{\mathrm{o}}}\big{\}}$.

Based on the online beamforming design presented in Section IV-A, in this subsection, we reformulate the received SNR as a function of the power consumption at the BS and the IRS by exploiting S-CSI. Based on this relationship, the power allocation design that satisfies the system power constraint can be optimized offline without frequent updates based on instantaneous CSI.

Specifically, according to the channel model in Section II-B, we can further rewrite $\mathbf{h}_{\mathrm{o}}^{H}$ as follows

where $\widehat{\theta}=-\sin{(\vartheta_{\mathrm{t}})}\sin{(\psi_{\mathrm{t}})}+\sin{(\theta_{\mathrm{r}})}\sin{(\varphi_{\mathrm{r}})}$ and $\widehat{\varphi}=-\sin{(\vartheta_{\mathrm{t}})}\cos{(\psi_{\mathrm{t}})}+\sin{(\theta_{\mathrm{r}})}\cos{(\varphi_{\mathrm{r}})}$. In this case, the elements in $\mathbf{h}_{\mathrm{o}}^{H}$ have the following property.

Proof: Without loss of generality, we assume $\widehat{\varphi}$ to be an irrational number. In this case, according to the Weyl’s Equidistribution Theorem [28], the sequence $\mathbf{v}^{H}_{[x]}=[\mathrm{mod}_{2}(-x),\mathrm{mod}_{2}\big{(}-x-\widehat{\varphi}),\dots,\mathrm{mod}_{2}(-x-(M_{\mathrm{y}}-1)\widehat{\varphi}\big{)}]$ is uniformly distributed in $[0,2)$ for any $x\in\mathbb{R}$ when $M_{\mathrm{y}}\rightarrow\infty$. Furthermore, note that $\mathbf{h}_{\mathrm{o}}$ in (23) can be represented by $\mathbf{v}^{H}_{[x]}$ as

As $\mathbf{v}_{[0]}^{H},\ \mathbf{v}_{[\widehat{\theta}]}^{H},\ \dots,\ \mathbf{v}_{[(M_{\mathrm{x}}-1)\widehat{\theta}]}^{H}$ are uniformly distributed in $[0,2)$, we can prove that the sequence of all elements of $\mathbf{h}_{\mathrm{o}}^{H}$ approaches to the distribution described by the random variable $h_{\mathrm{o}}$ in (24), which completes the proof.$\hfill\blacksquare$

Therefore, by assuming $M_{\mathrm{x}}\rightarrow\infty$ and $M_{\mathrm{y}}\rightarrow\infty$, we can replace $\mathbf{h}_{\mathrm{o}}^{H}\boldsymbol{\phi}^{\star}$ by an expectation as follows³³3Although the number of IRS elements in practical systems is always finite, assuming an infinite number of elements is an efficient asymptotic method to simplify analysis and provide valuable insights. Hence, this approach has been widely applied in previous works [21], [29].

where

with $\tau\in[0,\frac{1}{M})$ denoting the value corresponding to $P\left(\mathcal{R}\left\{h_{\mathrm{o}}\right\}>\tau\right)=\frac{\widetilde{M}_{\mathrm{on}}}{M}$. Specifically, the probability density function (PDF) of $\mathcal{R}\left\{{h}_{\mathrm{o}}\right\}$ is given by

By further defining $t=\mathrm{arcsin}(M\tau)\in[0,\frac{\pi}{2})$, the probability of $\mathcal{R}\left\{h_{\mathrm{o}}\right\}>\tau$ can be formulated as

which implies that $\frac{\widetilde{M}_{\mathrm{on}}}{M}=\frac{1}{2}-\frac{1}{\pi}t$. In this case, we can denote the power consumption at IRS and BS as a function of $t$:

Therefore, the essential part of power allocation optimization is to express the SNR as a function of $t$, and then find the optimal $t$ that maximizes the SNR. To achieve this, we further establish two key properties of $h_{\mathrm{o}}\varphi^{\star}$ in the following lemmas.

Proof: According to (24), we have $\mathcal{R}\left\{{h}_{\mathrm{o}}\right\}=\frac{1}{M}\cos{(v)}$ and $\mathcal{I}\left\{{h}_{\mathrm{o}}\right\}=\frac{1}{M}\sin{(v)}$. In this case, if $\mathcal{R}\left\{{h}_{\mathrm{o}}\right\}>\tau$, it means $v\in[0,\widehat{\tau})\cup(2\pi-\widehat{\tau},2\pi)$ with $\widehat{\tau}=\mathrm{arccos}(M\tau)$. Therefore, we can further rewrite $\mathcal{I}\left\{{h}_{\mathrm{o}}{\varphi}^{\star}\right\}$ as

which means $\mathbb{E}\left\{\mathcal{I}\left\{{h}_{\mathrm{o}}{\varphi}^{\star}\right\}\right\}=0$ as $v\sim\mathcal{U}(0,2\pi)$.$\hfill\blacksquare$

Proof: According to (27) and (28), the PDF of $\mathcal{R}\left\{{h}_{\mathrm{o}}\right\}\varphi^{\star}$ can be written as

therefore, $\mathbb{E}\left\{\mathcal{R}\left\{{h}_{\mathrm{o}}{\varphi}^{\star}\right\}\right\}$ can be calculated by

which completes the proof of Lemma 3.$\hfill\blacksquare$

By exploiting the properties in Lemma 2 and Lemma 3, we ignore the term $\mathbb{E}\left\{\mathcal{I}\left\{{h}_{\mathrm{o}}{\varphi}^{\star}\right\}\right\}$, and further reformulate the SNR as a function of $t$, which leads to

the maximum of which is given in the following Lemma.

Proof: To find the maximum value of the SNR in (34), we first calculate the derivative of the SNR with respect to $t$ as

As $\frac{\mathrm{d}\big{(}\mathrm{SNR}\big{)}}{\mathrm{d}t}\Big{|}_{t=0}=\frac{4|{\alpha}_{\mathrm{G}}{\alpha}_{\mathrm{h}}|^{2}M^{2}N}{\pi^{2}\sigma^{2}}\frac{P_{\mathrm{PIN}}M}{\pi}>0$ and $\frac{\mathrm{d}\big{(}\mathrm{SNR}\big{)}}{\mathrm{d}t}\Big{|}_{t\rightarrow\frac{\pi}{2}}=\frac{4|{\alpha}_{\mathrm{G}}{\alpha}_{\mathrm{h}}|^{2}M^{2}N}{\pi^{2}\sigma^{2}}(-2P_{0}\sin{t}\cos{t})$ tends to zero from below, the derivative in (36) must have at least one zero point, which can be calculated from

For $t\in[0,\frac{\pi}{2})$ and $\left(\frac{2\pi P_{0}}{P_{\mathrm{PIN}}M}-\pi+2t\right)^{-1}>0$, the left-hand side of (37) is a decreasing function, while the right-hand side is an increasing function. Therefore, there can only be one zero point of (37), which is the maximum point of (34), i.e., $t^{\star}$.$\hfill\blacksquare$

Once $t^{\star}$ is found according to Lemma 4, the power allocation strategy that satisfies the system power constraint can be determined based on (30). Since (35) does not depend on instantaneous CSI, the power allocation can be optimized offline without frequent updates.

The S-CSI-BF method is summarized as follows. In the first step, we obtain the $t^{\star}$ that maximizes the received SNR by solving (35), thus determining the power allocation between BS and IRS offline. With the determined power allocation, the beamforming at the BS and the IRS can then be respectively optimized according to (19) and (22). The complexity of the S-CSI-BF method is $\mathcal{O}\left(MN+M\mathrm{log}M\right)$, which is mainly attributed to the calculation of $\mathbf{f}$ and the sorting of $\mathcal{R}\big{\{}\mathbf{h}_{\mathrm{o}}\big{\}}$.

It should be noted that during the derivation of the S-CSI-BF method, certain approximations were applied, e.g., ignoring the NLoS components of channels (in (18)), assuming $M_{\mathrm{x}}\rightarrow\infty$ and $M_{\mathrm{y}}\rightarrow\infty$ (in (26)), and ignoring the term $\mathbb{E}\left\{\mathcal{I}\left\{{h}_{\mathrm{o}}{\varphi}^{\star}\right\}\right\}$ (in (34)). Despite these simplifications, our numerical results in Section VI shall verify the effectiveness of the S-CSI-BF method. More importantly, the computational complexity of the S-CSI-BF method is much lower than that of the GBD-BF method, as it avoids the high-complexity iteration processes and the MILP master problem in (17).

By referring to [30], we can equivalently transform the sum rate maximization problem (8) into the following WMMSE problem