Sum-rate maximization in OFDM system with practical model reconfigurable intelligent surfaces

Telecommunication systems undergo a revolutionary cycle every ten years, where each new cycle introduces significant performance enhancements to accommodate the rapid growth of data transmission demand driven by the continuously increasing number of connected mobile devices (Shafi et al., 2017). The Internet of Things (IoT), a paradigm in which almost any device can be connected to the Internet, is one of the largest contributors to the increasing number of connected devices and has led to an explosive growth in demand for throughput per area (bps/Hz/km²) in recent years (Björnson et al., 2017). However, in addition to increasing the throughput per area, other key performance indicators (KPIs), such as latency, reliability, mobility and energy efficiency, also require further improvement (Andrews et al., 2014).

Many key enabling technologies (referred to as disruptive technologies) have been integrated into communication systems to address these challenges. Typically, these technologies aim to explore new bandwidths, use novel waveforms or enhanced signal-processing methods, or leverage intermediate devices to assist in signal delivery (Letaief et al., 2022). In the latter case, various intermediate devices have been implemented to aid in wireless data delivery, including relays, backscatter devices and reconfigurable intelligent surface (RIS) systems.

Reconfigurable intelligent surface (RIS), also known as intelligent reflecting surface (IRS), is a disruptive technology recognized as a key enabler in transforming the next generation of wireless communication systems (Liu et al., 2021). RIS is a novel paradigm in which wireless data transmission is improved not only through the enhanced design of the transmitter and receiver but also by manipulating the wireless environment itself (Liang et al., 2019). By doing so, RIS opens up the potential for further improving wireless communication throughput, which has stagnated in recent years owing to the limitation of focusing only on the transmitter and receiver sides.

RIS systems consist of multiple reflection elements (REs) that can alter their electric and magnetic fields, and hence, change the phase and magnitude of the incident signal. Unlike conventional intermediary devices such as relays, REs are meta-materials (Basar et al., 2019) and do not incorporate active electronic components for communication, such as operational amplifiers and oscillators. Thus, they are suitable candidates for meeting the emerging need for green communication (Huang et al., 2019). Furthermore, due to the absence of basic data transmission processes, such as modulation, detection, encoding and decoding, beamforming is conducted passively, which has the advantage of significantly reducing power consumption and hardware complexity (Wu and Zhang, 2019).

The literature contains many studies on RIS-assisted wireless communication systems with various system models, including single-input single-output (SISO), multiple-input single-output (MISO), and multiple-input multiple-output (MIMO). SISO communication systems implemented with the assistance of RIS have been widely investigated (Björnson et al., 2020; Ding et al., 2020; Ding and Vincent Poor, 2020; Hou et al., 2020; Lyu and Zhang, 2020; Makarfi et al., 2020; Zhang et al., 2020, 2019). Björnson et al. (2020) compared two systems in which the communication performance was enhanced by RIS or relay performing in the Decode-and-Forward (DF) model, respectively. Makarfi et al. (2020) analyzed the average capacity, BER, and outage probability in an RIS-enabled IoT network with fading channels. Ding et al. (2020) analyzed the performance of a SISO system wherein a source with nonorthogonal multiple access (NOMA) capabilities simultaneously served two users using RIS with either coherent phase shifting or random discrete phase shifting properties. Zhang et al. (2019) proposed a method for minimizing the outage probability in a SISO system with a line-of-sight (LoS) channel model from and to the RIS elements. Ding and Vincent Poor (2020) designed a beamforming approach for a SISO downlink system wherein multiple users were served simultaneously using NOMA. In the considered scenario, some users were assumed to have no direct link to the base station (BS) and were hence assisted by RIS. The outage probability under the proposed beamforming method was calculated for both single-user and multi-user cases. Hou et al. (2020) considered an RIS-aided downlink network in which the BS sent data to several users simultaneously using a single antenna. The reflection design, outage probability, ergodic rate and spectral and energy efficiencies of the network were comprehensively discussed for the case in which the transmissions were handled using a NOMA scheme. Lyu and Zhang (2020) analyzed the achievable rate and spatial throughput of a network in which a BS served multiple users, each of which used an orthogonal resource block. Zhang et al. (2020) investigated the impact of the phase shift resolution on the performance of a RIS-assisted network with limited RIS phase shifts.

MISO communication systems using RIS to boost the system performance have been widely investigated (Guo et al., 2020; Kundu and McKay, 2021; Ma et al., 2020; Papazafeiropoulos et al., 2022; Wu and Zhang, 2019; Zhang and Zhao, 2021; Zhou et al., 2020). Wu and Zhang (2019) considered the problem of minimizing the total transmit power at the access point of an RIS-enabled MISO broadcast channel subject to satisfying quality of service (QoS) constraints. Separate optimization problems were formulated for both single-user and multiuser cases. Ma et al. (2020) considered a MISO single-user channel wherein the RIS not only assisted the BS data transmissions but also transmitted secondary additional data to the receiver. A method was proposed for determining the transmit precoder and reflection coefficient such that the transmit power at the BS was minimized while simultaneously ensuring the QoS of the primary data transmission. Kundu and McKay (2021) performed channel estimation for a RIS-enhanced MISO system serving a single user. The MMSE channel estimates obtained from several deep-learning approaches were compared with those obtained using conventional linear estimators. Guo et al. (2020) proposed a novel, low-complexity method for maximizing the sum rate of a MISO broadcast system with RIS. The analysis considered not only the case where the channel state information (CSI) was accurately known but also that in which the CSI was imperfect, resulting in the need for probabilistic optimization. Zhang and Zhao (2021) used the block minorization-maximization (BMM) method to obtain the transmit precoder at the BS and phase shift at the RIS that collectively maximized the sum rate of a MISO downlink system with the help of RIS. Zhou et al. (2020) minimized the total transmit power at the BS in a MISO system by optimizing the transmit precoder and RIS phase shift coefficient subject to QoS constraints for each user. A robust beamforming design was formulated to deal with the assumption of imperfect CSI, and a novel convex-concave procedure (CCP) was proposed to find a stationary solution for all possible channel-error realizations. Papazafeiropoulos et al. (2022) considered the problem of channel estimation and sum rate maximization in MISO networks with hardware impairments.

MIMO wireless systems with RIS have also attracted attention (Ning et al., 2020; Perović et al., 2021; You et al., 2021). Perović et al. (2021) maximized the achievable rate of a MIMO single-point RIS-assisted system by jointly optimizing the precoder at the transmitter and the reflection coefficient at the RIS using the projected gradient method to tackle the unit circle constraint. Ning et al. (2020) proposed a new metric called sum-path-gain maximization (SPGM) as an alternative to the conventional sum-rate maximization metric to find the optimal reflection coefficient of the RIS in a single-point MIMO system. You et al. (2021) aimed to find the optimal transmit precoders for each user and the RIS reflection coefficient that maximized the global energy efficiency in a multi-user MIMO system in which several users with multiple antennas simultaneously transmitted their uplink signals to a base station with the assistance of RIS.

All of the aforementioned studies assume that the reflection magnitude in the RIS system is constant. However, in real-world RIS systems, the magnitude of the reflection varies with the phase shift. Therefore, a growing number of researchers have recently adopted more realistic RIS reflection models (Abeywickrama et al., 2020; Cai et al., 2020; Cai et al., 2022; Li et al., 2021; Sun et al., 2023; Wang et al., 2021; Zhang and Li, 2022). Abeywickrama et al. (2020) applied a phase-shift-dependent magnitude reflection method to design an accurate beamforming approach for narrowband RIS-aided communications systems. Following (Abeywickrama et al., 2020), the authors in (Cai et al., 2020; Li et al., 2021) designed a more general framework for a practical reflection model that could be applied to wideband systems. The authors in (Sirojuddin et al., 2024; Sun et al., 2023; Wang et al., 2021; Zhang and Li, 2022) analyzed various aspects of RIS-assisted narrowband systems using the phase shift-dependent magnitude reflection model proposed in (Abeywickrama et al., 2020). Finally, the authors in (Cai et al., 2022) applied the practical reflection model presented in (Li et al., 2021) to an RIS-assisted multicell wideband system.

The present study is essentially a combination of the two works presented in (Yang et al., 2020) and (Abeywickrama et al., 2020). In particular, the study considers a SISO OFDM communication system, in which the achievable rate is improved by enhancing the signal transmissions using RIS. However, in contrast to the ideal RIS model considered in (Yang et al., 2020), the present work uses the more practical model proposed in (Abeywickrama et al., 2020), in which the reflection magnitude of each reflecting element (RE) varies with the phase shift. The main contributions of this study are as follows:

• The study considers the problem of maximizing the achievable rate in the considered OFDM system. This is achieved by maximizing the sum rate of each subcarrier by optimizing the power allocation on each subcarrier and the phase shift of each RE in the practical RIS model.

• An alternating optimization (AO) method is proposed in which the considered problem is solved by optimizing the power allocation and phase shifts of the REs in turn. For each RE, the optimal phase shifts are obtained using the gradient descent (GD) method, in which the first differentiation of the achievable rate with respect to the phase shift is analytically derived.

• The AO algorithm generally leads to a sub-optimal solution. Thus, to corroborate the optimality of the AO approach, a joint optimization method is proposed, wherein the power allocation and phase shifts are optimized at the same time with the assistance of the GD method once again.

• As a lower-complexity alternative to the achievable rate maximization problem, the study also considers a channel power maximization (CPM) problem, in which the aim is to determine the sub-optimal phase shift of the RE. The sub-optimal solution obtained for the RE phase shift is then considered as the initial point for the main rate maximization problem.

• The convergence rate of the GD method used in the two optimization procedures is heavily dependent on the step size. Accordingly, a simple method is proposed for managing the step size in such a way as to ensure the convergence of the GD method without significantly degrading the quality of the optimization results.

z, $z$ and $Z$ denote scalar, vector, and matrix, respectively. For each matrix, the subscript $[Z]m,n$ denotes the entry of $Z$ in row m column n. Notations $(·)T$⁠, $(·)*$⁠, $(·)H$⁠, $|·|$⁠, and $‖·‖$ denote the transpose, complex conjugate, conjugate transpose, absolute value and matrix norm operators, respectively. Symbols $Rm×n$ and $Cm×n$ represent real and complex matrices, respectively, with size $m×n$⁠, while $S++m$ denotes the positive definite matrix. $CN(0,Z)$ indicates a complex Gaussian random variable with zero-mean and covariance matrix $Z$⁠. Finally, * denotes convolution.

This study considers an OFDM communication system in which a single-antenna source (S) transmits a signal to a destination (D) with the assistance of a RIS system with M reflecting elements (REs), as illustrated in Figure 1. Assume that the OFDM system comprises N orthogonal subcarriers (SCs) that equally divide the whole bandwidth. Let $P≜diag(p1,p2,⋯,pN)T∈RN×N$⁠, where $pn≥0,∀n∈N={1,⋯,N}$⁠, denotes the $nth$SC transmit power. Suppose the total power at S is $PS$⁠. The total power allocation to the SCs must therefore satisfy $tr (P)≤PS$⁠.

Let $f=[f1,f2,⋯,fU,01×(N−U)]T∈CN×1$ denote the U-tap zero-padded discrete-time baseband-equivalent multipath channel for the S-D direct link. The S-D link through the RIS can be modeled as a concatenation of the S-RIS link and the RIS-D link for the OFDM wideband channel. Specifically, let $hm=[h1,h2,⋯,hU¯]T∈CU¯×1$ and $gm=[g1,g2,⋯,gU⌣]T∈CU⌣×1$⁠, $∀m∈M={1,…,M}$ denote the $U¯$-tap baseband equivalent multipath channel from S to the $mth$RE and the $U⌣$-tap multipath channel from the $mth$RE to D, respectively. Each RE reflects the incident signal, resulting in not only a phase shift but also an amplitude attenuation. Let $ϕ=[ϕ1,⋯,ϕM]T∈CM×1$ be a column vector containing all the reflection coefficients of the RIS, where $ϕm=βmejθm$ is the reflection coefficient of the $mth$RE with magnitude $βm∈[0,1]$ and phase shift $θm∈[−π,π)$⁠. The S-D link through the $mth$RE is then given by the concatenation of the S-$mth$RE link, $mth$RE reflection coefficient, and $mth$ RE-D link. That is, $hm ∗ ϕm ∗ gm=ϕmhm ∗ gm∈CU^×1$⁠, where $U^=U¯+U⌣−1$⁠. For ease of derivation, let the following notations be introduced:

i.e.$vm$⁠, $∀m∈M$⁠, are the zero-padded concatenated S-RIS and RIS-D channels through the $mth$RE. The superposed channel impulse response (CIR) of S-D channel $h^∈CN×1$ is the composite of the S-D (direct) channel and the S-$mth$ RE-D channel, which is given by:

where $ϕ=[ϕ1,⋯,ϕM]T$⁠. Let $QN$ be the $N×N$ discrete Fourier transform matrix. The channel frequency response (CFR) of the superposed S-D CIR channel, denoted as $ν∈CN×1$⁠, is then expressed as $ν=Q(f+Vϕ)$⁠, while the CFR at each $nth$SC is expressed as:

where $qn∈CN×1$ is the $nth$ column of $Q$⁠. Assume that the OFDM symbol contains a cyclic prefix of length $NCP≥max(U,U^)$⁠. The achievable rate of the considered OFDM system is thus expressed as (Yang et al., 2020):

where $Γ$ is a coefficient representing the shortfall between the maximum theoretical channel capacity and the actual capacity achieved using a practical modulation and coding scheme (MCS). In addition, $σ2$ is the noise power at each SC.

This study considers a practical RIS model wherein the reflection amplitude from each RE varies with its phase shift. In practical RIS systems, the REs have an electronic circuit arrangement in which a variable capacitor can be adjusted in such a way as to yield the desired phase shift of the incident signal, as shown in [1, Figure 2]. Based on the figure, the phase shift of the REs is given in (Abeywickrama et al., 2020; Cai et al., 2020) as:

where $Zm(Cm)$ is the impedance of the $mth$RE whose value is controlled by adjusting the capacitor capacitance $Cm$⁠, given by:

where the constants $L1$⁠, $L2$⁠, R, and $ω$ are the inductances of inductors 1 and 2, the resistance of the RE resistor, and the frequency of the incident signal in rad/s, respectively. Note that $Cm$ affects not only the value of the phase shift reflection $θm≜∠ϕm∈R$⁠, but also the amplitude of the reflection $βm≜|ϕm|∈R$⁠. Given a set of values of $Cm$ ranging from $Cm(min)$ to $Cm(max)$⁠, equation (5) outputs a set of values of $ϕm$ in which $L1$⁠, $L2$⁠, R, and $ω$ are taken as constants. After obtaining these $ϕm$ values, a set of paired values of $θm=∠ϕm$ and $βm=|ϕm|$ can be extracted. Plotting $βm$ versus $θm$ yields a distorted U-shaped curve depicting the relationship between $θm$ and $βm$⁠, as shown in Figure 2. As described in (Abeywickrama et al., 2020), the relationship between $βm$ and $θm$ can be accurately approached using the equation:

where the parameters $βmin$⁠, $α$⁠, and k are obtained using the regression technique, and $θ(min)$ and $θ(max)$ are obtained when $C=Cm(max)$ and $C=Cm(min)$⁠, respectively. Note that the parameters in (2) reflect the inherent characteristics of each RE in the RIS system and are independent of the channel statistics. In other words, their values are determined in a one-off procedure prior to RIS installation and are then applied in future practical implementations without any further consideration.

This study aims to determine the phase shift vector $θ≜[θ1,⋯,θM]∈RM×1$ and power allocation matrix over N subcarriers $P≜diag(p1,⋯,pN)∈RN×N$ such that the sum rate over all the subcarriers is maximized. This problem can be mathematically formulated as follows:

Remark 1: For clarity of exposition and to isolate the impact of the practical RIS model, this work focuses on a SISO OFDM system, which is representative of low-complexity and energy-constrained scenarios such as IoT and edge devices, as reflected in (8). The formulation can be extended to MISO or MIMO systems by replacing the scalar effective channel term $pn|qnTf+qnTVϕ|2$ with its vector- or matrix-valued counterpart that incorporates transmit beamforming and/or receive combining. In such extensions, conventional beamforming strategies, such as maximum ratio transmission (MRT) for MISO systems and singular value decomposition (SVD)–based precoding for MIMO systems, can be naturally incorporated. Correspondingly, the diagonal power allocation matrix $P$ would be generalized to a transmit covariance matrix, leading to a log-determinant rate expression. Although these extensions increase the problem dimensionality, the joint optimization structure and the alternating optimization with gradient-based updates developed in this work remain applicable, albeit with higher computational complexity.

Because the joint optimization (8) over $θ$ and $P$ is challenging, this study proposes an alternating optimization (AO) algorithm (Abeywickrama et al., 2020) which optimizes each target alternately. Given $θ$⁠, problem (8) reduces to:

where $ζn=(Γσ2)/(|qnTf+qnTVϕ|2)$ is a constant w.r.t. $P$⁠. The convex problem (9) is solved by using the KKT conditions, where the solution is given by:

where $p^$ is a constant set to meet the sum power constraint $∑n=1N(p^−ζn)+=PS$⁠. The value of $p^$ can be found using the water-filling (WF) algorithm or using the Newton-Raphson Method proposed in (Sirojuddin et al., 2024).

Given $P$⁠, problem (8) can be recast as:

Note that by substituting the constraint (11 b) into the objective (11a), and always keeping $θm,∀m$ between $θ(min)$ and $θ(max)$ (by executing (29)), problem (11) becomes an unconstrained maximization problem where the optimal value $θ$ can be found using the gradient descent (GD) method. In particular, the optimal $θ$ can be obtained by iteratively executing the following equation:

where $υ1>0$ is the step size, and i is the iteration index.

To obtain the value of vector $∂R^sum/∂θ∈RM×1$⁠, it is first necessary to analytically determine the value of its elements, which can be expressed using the chain rule as follows:

where $R^n$⁠, $γn$ and $∂γn/∂θm$ are defined as:

The second equality in (13) holds because $γn,∀n$⁠, and $θm,∀m$⁠, are real. Based on (13), the next task is to analytically derive the values of $∂R^/∂γn$ and $∂γn/∂θm$⁠.

It follows simply from (14) that:

The problem of determining $∂γn/∂θm$ can be broken down into that of computing the values of $∂γn/∂ϕm$ and $∂ϕm/∂θm$⁠, respectively. The value of $∂γn/∂ϕ$ is given by:

where from (18), each component of $∂γn/∂ϕ$ can be obtained as:

Because $ϕm=βmejθm$⁠, the value of $∂ϕm/∂θm$ is expressed as:

where, from (2), it can be verified that:

From (19) and (20), the value of $∂γn/∂θm$ can be calculated as:

Given (17) and (22), (13) can be calculated. However, the calculation process involves a loop computation over N subcarriers. Furthermore, (13) needs to be repeated M times to yield the derivation of $∂R^sum/∂θ$⁠. This serial computation spends a lot of time, and hence, a kind of parallel computation needs to be structured as a substitution. To facilitate parallel computation, it is necessary to construct a compact form of $∂R^sum/∂θ$ that is structured from matrix multiplication. To achieve this, let the following vector be defined: $R^=[R^1,⋯,R^N]T$⁠. Furthermore, let us introduce a Jacobian matrix $∂R^/∂θ∈RN×M$⁠, whose entry in row n column m is the first derivative of $R^n$ w.r.t. $θm$⁠, i.e.$∂R^n/∂θm$⁠. The value of $∂R^sum/∂θ$ can then be generalized from (13) for all $θm$ as follows:

i.e. the first derivative of $R^sum$ over all $θ$s, i.e.$∂R^sum/∂θ$⁠, is a (column) vector whose $mth$ entry is the sum of all elements in column m of $∂R^/∂θ$ divided by $N+NCP$⁠.

The analytical expression of $∂R^/∂θ$ can be determined by breaking it down into the product of two Jacobian matrices. Because $∂R^n/∂θm=(∂R^n/∂γn)(∂γn/∂θm)$⁠, the Jacobian of $∂R^/∂θ∈RN×M$ can be expressed as:

where $γ=[γ1,⋯,γN]T$⁠, and $∂R^/∂γ∈RN×N$ is given by:

i.e.$∂R^/∂γ$ is a Jacobian matrix in which the entry in row n column n is given by (17). Note that $∂R^/∂γ$ is diagonal because $∂R^n/∂γn^=0$ for all $n ≠ n^$⁠.

Next, the analytical expression of the Jacobian matrix $∂γ/∂θ∈RN×M$ must be calculated. Through an appropriate matrix structure manipulation, (3.1) can be restructured as:

where $∂ϕ/∂θ∈CM×M$ is given by:

i.e.$∂ϕ/∂θ$ is a Jacobian matrix whose entry in row m column m is given by (20). Note that $∂ϕ/∂θ$ is diagonal because $∂ϕm/∂θm^=0$ for all $m ≠ m^$⁠. From (26), the Jacobian matrix $∂γ/∂θ$ is then expressed as:

At this point, (25) and (28) have both been analytically calculated. Therefore, $∂R^/∂θ$ in (24) can be easily obtained, and $∂R^sum/∂θ$ in (23) is ready to be implemented in (12) to seek an optimal solution for $θ$ by using the GD method.

Algorithm 1 presents the pseudo-code of the proposed AO algorithm, where $ξ$ denotes the convergence threshold and $imax$ represents the maximum number of iterations. As shown, the algorithm alternately optimizes $P$ using the WF algorithm (line 4) and $θ$ using the GD algorithm (lines 6–10). Because $θm,∀m∈M$ is bounded by $θ(min)$ and $θ(max)$⁠, line 9 updates $θm,∀m$ using the following equation:

i.e. if $θm<θ(min)$⁠, then $θm$ is set to $θ(min)$⁠; if $θm>θ(max)$⁠, then $θm$ is set to $θ(max)$⁠; otherwise, $θm$ is kept as it is.

Algorithm 1.AO procedure for solving problem (8)

Because an AO algorithm separates the joint variables and attempts to optimize each variable in turn, it may require many iterations to converge (Perović et al., 2021). Thus, the AO algorithm proposed in Subsection 3.1 might require a long computational time. In addition, the optimality of its solutions is not guaranteed to be better than that of a joint optimization algorithm. Accordingly, this section proposes an alternative approach, in which problem (8) is solved by optimizing $P$ and $θ$ jointly rather than separately. The purpose of this alternative method is to assess the optimality of the main proposed AO algorithm and to evaluate its computational time.

As stated in Subsec. 3.1, the problem of solving $P$ given $θ$ (problem (9)) can be tackled using the iterative water-filling (WF) algorithm, which is a simple algorithm with linear complexity (Khakurel et al., 2014). Accordingly, this section commences by defining an open-form function that takes $θ$ as an input (along with other constants such as the channel, transmit, and noise powers), computes the optimal power allocation $P$⁠, and then outputs the rate.

Definition 1. Function $R˜(θ)$ is an open-form function comprising three steps. In particular, it 1) takes $θ$ as one of its inputs, 2) calculates the power allocation given $θ$ using the WF algorithm, and 3) outputs the rate using (4).

As explained in Definition 1, $R˜$ is open form because it uses the iterative, linear complexity WF algorithm to calculate the optimal $P$⁠. Note that $R˜$ takes only $θ$ as its input, and is hence different from (4), which takes both $θ$ and $P$ as inputs. In other words, function $R˜$ has only one independent variable, $θ$⁠, and thus the main problem (8) takes the following form:

By substituting the equality constraint (30 b) into the objective (30a) (and always keeping $θm,∀m$ between $θ(min)$ and $θ(max)$⁠, similar to the case described in Subsec. 3.1), problem (30) becomes an unconstrained maximization problem, for which the optimal solution can be determined using a conventional method such as GD. In particular, $θ$ is iteratively updated using the following equation:

where $υ2>0$ is the step size. Because $R˜(θ)$ is not a closed-form function (owing to the iterative WF algorithm), $∂R˜/∂θ$ cannot be found analytically. Nevertheless, its value can be approximated using the following limit equation approach:

Due to the round-off error in digital machines, the value of $δ$ cannot be set too low in practical applications. As a result, the value of $∂R˜/∂θ$ obtained when (32) is implemented in an algorithm is approximate rather than exact. Nonetheless, a tiny value of approximately $10−9$ is typically sufficient to closely approximate the true derivative value. In addition, the approximation error of (32) is proportional to $δ2$ (Chapra and Canale, 2015). The overall procedure for obtaining the optimal value of $R˜$ using the GD algorithm is shown in Algorithm 2, where $ξ$ denotes the convergence threshold and $imax$ represents the maximum number of iterations.

Algorithm 2. Joint procedure for solving problem (30)

Alg. 1 requires an initial RIS phase-shift vector $θ(init)$ as a starting point for the iterative optimization. The quality of this initial phase-shift configuration has a significant impact on the convergence behavior and performance of the algorithm. To this end, following (Yang et al., 2020), this study uses the channel power maximization (CPM) problem as a low-complexity approach to obtain an effective initial estimate of $θ$⁠.

The underlying idea of CPM can be explained as follows. Referring to the system model in Figure 1, increasing the number of reflecting elements (REs) at the RIS creates multiple propagation paths between the source and destination. By appropriately adjusting the RIS phase shifts, the reflected signals can be constructively combined at the receiver, resulting in a higher effective channel power, denoted by $||h^||2$⁠. As discussed in (Yang et al., 2020), a larger effective channel power generally provides favorable conditions for achieving a higher achievable rate. More rigorously, Proposition 1 in (Ning et al., 2020) shows that, under equal power allocation, maximizing the effective channel power corresponds to maximizing a lower bound of the achievable rate. In this sense, CPM can be viewed as a computationally efficient surrogate objective for RIS phase-shift initialization.

Motivated by both this physical intuition and the aforementioned theoretical insight, CPM is formulated to maximize $||h^||2$ with respect to the RIS phase shifts, as given by:

Similar to problem (8), problem (33) can be rendered essentially unconstrained by substituting constraint (33b) into objective (33a). Problem (8) can then be solved by using the GD method once again. In particular, the sub-optimal value of $θ$ can be obtained by iteratively executing the following equation:

where $υ3>0$ is the step size.

To determine the value of $∂PCH/∂θ$⁠, let us first analytically find the values of its elements, which, by using the chain rule, can be expressed as:

The derivation of $∂ϕm/∂θm$ has already been given in (20). The value of $∂PCH/∂ϕm$ is expressed as:

Substituting (20) and (36) into (35) yields:

From (37), the vector form of $∂PCH/∂θ∈RM×1$ is expressed as:

where $∂ϕ/∂θ$ is given by (27). With the analytical expression of $∂PCH/∂θ$ given by (38), $θ$ can be updated using the GD method stated in (34). Algorithm 3 presents the pseudo-code of the CPM-based phase-shift initialization, where $ξ$ denotes the convergence threshold and $imax$ represents the maximum number of iterations.

Algorithm 3. Procedure for finding sub-optimal θ using CPM metric

It is worth noting that CPM is not intended as an alternative to the rate maximization problem, because the CPM objective is independent of the power allocation matrix $P$⁠. Consequently, the resulting phase-shift solution depends only on $θ$⁠. Compared with the rate maximization problem, CPM involves fewer optimization variables and simpler operations, leading to substantially lower computational complexity. The obtained phase-shift vector is then used to initialize Algs. 1 and 2, where joint optimization of $P$ and $θ$ is subsequently performed.

The GD method is used several times in the previous sections, including in (12), (31) and (34). One of the most critical parameters in the GD method is the step size, which directly impacts the convergence rate. Thus, in the present study, a simple mechanism is incorporated into the GD method used in Algorithms 1, 2 and 3 to improve its convergence. Let the $θ$ update equations be denoted as $θ(i+1)=θ(i)+υ·∂s/∂θ$⁠, where $υ$ represents $υ1$⁠, $υ2$ and $υ3$⁠, and s represents $R^sum$⁠, $R˜$ and $PCH$ in (12), (31) and (34), respectively. Notably, the value of $υ$ is not determined at the beginning of Algorithms 1, 2, and 3, but in the first iteration (⁠$i=1$⁠). In particular, after calculating $∂s/∂θ$⁠, $υ$ is determined as:

i.e.$θ^$ is divided by the largest entry of vector $|∂s/∂θ|$⁠. In other words, at $i=1$⁠, the largest step taken to translate $θ(i)$ to $θ(i+1)$ is limited to $θ^$⁠, i.e.$maxm(|θm(i+1)−θm(i)|)=θ^$⁠. By leveraging the fact that the phase shift is bounded by $θ(min)$ and $θ(max)$⁠, the value of $θ^$ can be intuitively chosen as a particular positive number, for example, between 0 and $(θ(max)−θ(min))/2$⁠.

Besides initializing $υ$⁠, it is also necessary to monitor the monotonicity of the iterations. In the present study, this property is addressed using the simple step-size control mechanism proposed in (Sirojuddin and Huang, 2022). Suppose the value of s at the $(i)$th iteration, i.e. $s(θ(i))$⁠, has been calculated. At the $(i+1)$th iteration, after calculating $∂s/∂θ$ and updating $θ(i+1)$⁠, the value of $s(θ(i+1))$ is computed and compared with $s(θ(i))$⁠. If the value of $Δs≜s(θ(i+1))−s(θ(i))$ is negative (indicating that the step size is too large), $υ$ is reduced by $υ=τυ$⁠, where $0<τ<1$⁠. $θ(i+1)$ is then re-updated and $Δs$ is re-evaluated. This loop terminates once $Δs$ is non-negative. This simple procedure ensures the monotonicity of s over the iterations with minimal computational cost.

In practice, the computational complexities of Algs. 1, 2 and 3 must be analyzed to properly understand the effects of increasing the values of M, N, and the other size-representing variables on the growth of the required computational resources (e.g. time or memory). In Alg. 1, the computational complexities of lines 1, 4, 7, 8 and 9 are $O(U¯U⌣M)$⁠, $O(N)$ (Khakurel et al., 2014), $O(M2N2)$⁠, $O(M)$ and $O(M)$⁠, respectively. Because evaluating the objective function (4) as a termination criterion has a complexity of $O(MN2)$⁠, the overall complexity of Alg. 1 after incorporating the step-size control mechanism is given as: $O(U¯U⌣M+iAO1{N+iAO2[M2N2+tAO(M+M+MN2)]+MN2}) ≈ O(iAO1iAO2(M2N2+tAOMN2))$⁠, where $iAO1$⁠, $iAO2$⁠, and $tAO$ are the values of $i1$ at the end of Alg. 1, the average value of $i2$ in each $i1$⁠, and the average iteration number of the step-size control mechanism, respectively.

In Alg. 2, the computational complexity of line 4 is $O(M(2M)N2)$⁠, where the factor 2 M comes from repeating the calculation of $R˜$ 2 M times in (32). Consequently, the overall complexity of Alg. 2 after incorporating the step-size control strategy is equal to: $O(U¯U⌣M+iJO{M(2M)N2+tJO[M+M+MN2]}) ≈ O(iJO(M2N2+tJOMN2))$⁠, where $iJO$ and $tJO$ are the value of i at the end of Alg. 2 and the average iteration number of the step-size control strategy, respectively.

In Alg. 3, the computational complexity of line 4 is $O(MN)$⁠. Because the evaluation of the objective function (33a) as a termination criterion has a complexity of $O(MN)$⁠, the overall complexity of Alg. 3 is $O(U¯U⌣M+iCP[MN+tCP(M+M+MN)]) ≈ O(iCPtCPMN)$⁠, where $iCP$ and $tCP$ are the value of i at the end of Alg. 3 and the average iteration number of the step-size control mechanism, respectively.