Energy-Efficiency Optimization for Mutual-Coupling-Aware Wireless Communication System Based on RIS-Enhanced SWIPT

The widespread deployment of the Internet of Things (IoT) is promoting interest in simultaneous wireless information and power transfer (SWIPT), the performance of which can be further improved by employing a reconfigurable intelligent surface (RIS). In this article, we propose a novel RIS-enhanced SWIPT system built on an electromagnetic-compliant framework. The mutual-coupling effects in the whole system are presented explicitly. Moreover, the reconfigurability of RIS is no longer expressed by the reflection-coefficient matrix but by the impedances of the tunable circuit. For comparison, both the no-coupling and the coupling-awareness cases are discussed. In particular, the energy efficiency (EE) is maximized by cooperatively optimizing the impedance parameters of the RIS elements as well as the active beamforming vectors at the base station (BS). For the coupling-awareness case, the considered problem is split into several subproblems and solved alternatively due to its nonconvexity. First, it is transformed into a more solvable form by applying the Neuman series approximation, which can be resolved iteratively. Then, an alternative optimization (AO) framework and semidefinite relaxation (SDR), successive convex approximation (SCA), and Dinkelbach’s algorithm are applied to solve each subproblem decomposed from it. Owning to the similarity between the two cases, the no-coupling one can be viewed as a reduced form of the coupling case and, thus, solved through a similar approach. Numerical results reveal the influence of mutual-coupling effects on the EE, especially in the RIS with closely spaced elements. In addition, physical beam designs are presented to demonstrate how the RIS assists SWIPT through various reflecting states in different conditions.


Energy-Efficiency Optimization for Mutual-Coupling-Aware Wireless Communication
System Based on RIS-Enhanced SWIPT Ruoyan Ma , Jie Tang , Senior Member, IEEE, Xiuyin Zhang , Kai-Kit Wong , Fellow, IEEE, and Jonathon A. Chambers Abstract-The widespread deployment of the Internet of Things (IoT) is promoting interest in simultaneous wireless information and power transfer (SWIPT), the performance of which can be further improved by employing a reconfigurable intelligent surface (RIS).In this article, we propose a novel RISenhanced SWIPT system built on an electromagnetic-compliant framework.The mutual-coupling effects in the whole system are presented explicitly.Moreover, the reconfigurability of RIS is no longer expressed by the reflection-coefficient matrix but by the impedances of the tunable circuit.For comparison, both the no-coupling and the coupling-awareness cases are discussed.In particular, the energy efficiency (EE) is maximized by cooperatively optimizing the impedance parameters of the RIS elements as well as the active beamforming vectors at the base station (BS).For the coupling-awareness case, the considered problem is split into several subproblems and solved alternatively due to its nonconvexity.First, it is transformed into a more solvable form by applying the Neuman series approximation, which can be resolved iteratively.Then, an alternative optimization (AO) framework and semidefinite relaxation (SDR), successive convex approximation (SCA), and Dinkelbach's algorithm are applied to solve each subproblem decomposed from it.Owning to the similarity between the two cases, the no-coupling one can be viewed as a reduced form of the coupling case and, thus, solved through a similar approach.Numerical results reveal the influence of mutual-coupling effects on the EE, especially in the RIS with closely spaced elements.In addition, physical beam designs are presented to demonstrate how the RIS assists SWIPT through various reflecting states in different conditions.

I. INTRODUCTION
T HE Internet of Things (IoT) has ushered in a new era of development with the further application of fifthgeneration (5G) wireless communication technology.However, the performance limit of the 5G system cannot support the unprecedented expansion and more diversified information transmission demands required, such as by high-definition images or via multidimensional sensing information transfer.These demands are expected to be realized through the sixth generation (6G) technology, due to its outstanding performance, e.g., extremely high throughput, ultralow latency, and intelligent autonomous networks [1].All these performance breakthroughs rely on several prospective enabling technologies, such as using a reconfigurable intelligent surface (RIS), terahertz communication, and integrated sensing and communication (ISAC) [1], [2], [3].Among them, since RIS has the controllability of the transmission links between the transmitting and receiving ends, it is regarded as an immense promising technology expected to be employed in 6G systems.
The introduction of a RIS implies that the communication system will no longer passively adapt to the wireless channel, but actively adjust it according to the needs.In particular, a RIS is capable of constructing the virtual Line-of-Sight (LoS) link where the LoS link is obstructed, which is more likely to occur in the high-frequency band used by 6G [4].In addition, the application of the RIS also includes the security reconstruction of the physical layer [5], assisting device-todevice (D2D) communications, suppression of the intercell interference of edge users, and compensation for signal attenuation [4], [6], [7].RIS has not only prominent capabilities and enriched application scenarios but also relatively compact structure and low energy consumption; as such it is anticipated to play a prominent role in the construction of 6G networks and future IoT scenarios [1].
Due to the smaller cells and the more accessible devices, energy consumption becomes an unavoidable problem in 6G.Energy efficiency (EE) therefore has a high priority in performance metrics of the 6G networks [2].Indeed, for the IoT application scenario of 6G, since the terminal nodes will be uncountable and widely distributed, the difficulties of power supply and information exchange are more prominent.Simultaneous wireless information and power transfer (SWIPT), the key green-communication technology, makes it possible for node devices to receive power and exchange information at the same time [8].Specifically, there are two categories of SWIPT technology: 1) the near-field SWIPT and 2) the far-field SWIPT [9].The far-field has attracted much attention from academia and industry because of its flexibility and compatibility at the system level.There is a representative deployment scenario and a novel joint time allocation and power splitting (JTAPS) scheme proposed in [10].With this idea, SWIPT can help the relay node to achieve not only the information but also the power from the source and then transfer information through harvested energy flexibly.Nevertheless, compared to the near-field SWIPT, far-field SWIPT may be frequently impacted by ubiquitous obstructions and undergo tremendous path loss.As a technology that has the ability to control channels and further compensate for these losses, RIS can bring outstanding performance gain to the SWIPT system [11].Next, we will introduce some works on this topic.

A. Related Works 1) RIS Modeling:
A RIS works mainly in the reflection mode rather than the propagation mode, and its reflection performance is influenced by various internal and external factors, such as element types, array configurations, element spacings, and incident wave features [7], [12], [13].Therefore, it is crucial to present these characteristics as accurately and comprehensively as possible during the modeling.The most commonly used model in network optimizations is the independent-reflection matrix model, where the reflection performance of the RIS is expressed by the reflection-coefficient matrix.Generally, this matrix is diagonal, and each reflection coefficient is independent.Although this model is simple enough to analyze, it does not describe the key physical characteristics of the RIS [7].In addition, the physics-based model is widely studied in the field of electromagnetics (EMs), in which the radar anomalous reflection theory is introduced to construct the model.Indeed, this type of model can reveal more information about the physical characteristics during the RIS reflection than the previous one [14].Moreover, the impedance model is based on the equivalent circuit analysis that describes the physical properties of RIS elements and tunable circuits.Thanks to it, the relationship between the amplitude and the phase shift of the reflection coefficient is presented clearly [15].An end-toend EM system model is proposed to analyze the influences of physical features on the system by combining the ideas of the physical model and the impedance model.In particular, it describes the entire RIS system in the form of the circuit, while the transmissions among various ends are based on mutual impedances [16], [17].
2) Reconfigurability of RIS: A RIS can adjust its reflection characteristics to fit various channel circumstances, which sets it apart from conventional scatterers.As for engineering realizations, there are various ways to adjust the reflection parameters of the RIS elements, which mainly fall into discrete and continuous methods.The number of quantization bits is a worthy tradeoff with regard to the discrete RIS.Concerning the cost of hardware design, although a higher number of quantization bits can improve performance gain, it is harder to deploy.1-bit discrete RIS is realized by PIN diodes working on the on-off states in [18] and the varactor diodes operating in two states with 180 phase difference in [13].Moreover, the design based on the varactors is also able to achieve continuous tunability through controlling input voltages [19].Regarding the theoretical performance analyses, the ideas of tunability design are dependent on the choices of models.As for the independentreflection matrix model, there are several commonly adopted assumptions that the reflection phases of the elements are continuously adjustable from 0 • to 360 • , the reflection magnitudes are set to one and these two parameters about the reflection coefficient are independent.All these assumptions make the optimization problems easier to resolve but detach from physical reality [7].A more practical optimization problem based on the RIS impedance model, which perspicaciously describes the dependence between phases and magnitudes of reflection coefficients, is shown to be solved by the penaltybased algorithm [15].Furthermore, the imaginary parts of the tunable circuits connected to the RIS elements are taken as the optimization variables in the end-to-end EM system model [17], [20].Instead of being idealistic, the hardware limits should be concerned integrated into these physical RIS models.
3) Optimization for the RIS-Assisted System: A RIS has been shown to further enhance the performance of SWIPT-based systems [7] through its channel controllability.In the literature on RIS-assisted SWIPT, the hardware models, problem formulations, and scenario settings are distinct, which makes the optimization strategies different from each other.In addition to the RIS models, the models of the energy harvesting devices, namely, the rectifying circuits, are also worth discussing.Indeed, the nonlinear model is more suitable than the linear model.Specifically, the nonlinear model based on the sigmoidal function is used to describe the nonlinearity of the rectifiers [21].Moreover, a rectenna model is introduced to analyze the effects brought by the waveform and the input power of the signal on the performances of the RIS-aided-SWIPT.Furthermore, the waveform is jointly optimized with active and passive beamforming variables [22].The various objective functions, e.g., maximizing the weighted sum rate (WSR) [23], minimizing the total transmit power [24], maximizing the minimum power received [25], and maximizing the EE [26], are set according to the diverse performance priorities.Unlike the scenarios where information users and energy users are separated [23], [24], [25], [27], co-located information and power receivers based on power splitting are presented in [21], [22], and [28].Differing from the single-RIS system, the performance analysis, adopting multi-RISs to support hot-spot areas with diversified service requirements, is shown in [24].Moreover, the unmanned aerial vehicle (UAV) also can be integrated with the RIS-assisted system to improve the performances, especially computation capacity, under IoT scenarios [29].

B. Motivation and Contribution
Given the above inspirations and the design purposes for analyzing the importance of the hardware characteristics, a Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
RIS-aided-SWIPT network based on the EM is constructed.Then, we propose a system problem orienting to the key hardware parameters and aim to maximize the EE constrained to the QoS requirements of the various users.To address this problem, an effective optimization scheme is proposed.The impacts of the actual hardware characteristics on the system performance are presented, which is expected to guide the design and the deployment of the RIS in diversified scenarios.The main contributions of this article are presented as follows.
1) We build the entire RIS-aided-SWIPT MISO transmission model based on the end-to-end EM transfer model and the nonlinear rectifier model.By using the EM model, the hardware impacts on EE, which include the radiation features of the transmitting antennas, the reflection characteristics of the RIS, and especially the mutualcoupling effects influenced by the array configurations, are presented.2) With the adopted model, the circuit impedances are utilized as the optimization variables to configure the RIS, which is more practical than the ideal assumptions.We propose two optimization schemes for the no-coupling as well as the coupling-awareness cases.Considering simplifying the original problems, the transformations and approximations are adopted for both cases.Concretely, we use the Neuman series approximation to introduce an iterative form of the transfer model for the couplingawareness case.In each iteration, the nonconvex problems can be approximately solved by the proposed optimization framework.Moreover, the no-coupling case can also be resolved by using a similar scheme.
3) The optimization framework is established on the basis of the alternative optimization (AO) strategy.
With its help, we can decompose the problem into two tractable subproblems.Specifically, we propose an approach mainly based on semidefinite relaxation (SDR), Dinkelbackh's algorithm, and successive convex approximation (SCA) for solving each subproblems.Although it is effective for resolving the no-coupling case completely, the coupling-awareness case needs to be invoked multiple times until convergence.4) We employ both the numerical and the full-wave simulations for the performance analysis.As for the numerical simulation, we analyze the influences of QoS requirements and resource budgets on the EE.Moreover, the effects of RIS topologies and the mutual coupling are also considered.Additionally, we reveal the effects of the proposed algorithms on controlling the physical beams in distinct scenarios through full-wave simulations.The results further validate the accuracy of the adopted model and the effectiveness of the algorithms.

C. Organization
The remainder of this article is organized as follows.In Section II, we present the system model based on the EM analysis and the problem formulation.In Sections III and IV, effective algorithms are proposed to tackle two nonconvex problems, the no-coupling and the coupling-awareness  problems, respectively.Numerical analyses and full-wave simulation results are shown in Section IV.Section V provides the conclusions.The notations adopted in this article are listed in Table I.

A. Signal Model
In this article, we consider a MISO downlink RIS-assisted SWIPT system with separate information decoder user (IDU) and energy harvester user (EHU).We assume the base station (BS) is equipped with N t > 1 antennas, while each IDU and EHU adopts single antenna.Moreover, these two kinds of users, IDUs and EHUs, belong to the set M I = {1, . . ., M I } and M E = {1, . . ., M E }, respectively.Generally, since the power management devices require more energy to operate, the service scopes of the EHUs are smaller than those of IDUs as presented in Fig. 1.This results in that IDUs, which are located further away from the BS, are likely to be blocked and the LoS links of them may not exist.To further improve the harvested power and signal-to-interference plus noise ratio (SINR) at the user ends, a nearly passive RIS, which consists of N i reflecting elements, is introduced into the system.In our scenario, it is assumed that the RIS can be controlled by the continuously adjustable varactors.Particularly, both the information and the energy beamforming are considered at the BS and, therefore, the transmitter signal is given by where w j ∈ C N T ×1 and v l ∈ C N T ×1 denote the transmitting beamforming vectors for the jth IDU and the lth EHU, while s I j and s E l denote the information and energy signals.Particularly, the information signals s I j ∀j are assumed to be independent and identically distributed (i.i.d.) Gaussian random variables, in detail, s I j ∼ CN (0, 1) ∀j.As for the energy signals, they do not transfer any information, so the arbitrary distributions with E{|s E l | 2 } = 1 ∀l are assumed to describe their statistical properties without loss of generality.It is noteworthy that although paper [11] shows that energy signals are not required from the view of optimizing algorithms, we still hope to see their effects on the generations of the actual physical beams in distinct deployment scenarios from the EM perspective.

B. End-to-End EM Transfer Model
We assume that the scenario has two types of multipath propagation scatterers, the reconfigurable scatterer (i.e., the RIS) and the arbitrary scatterer, which coincides with the configuration of [17].Thus, the total transfer links can be divided into the direct link (BS-user), the RIS-assisted link (BS-RISuser) and the scatterer link (BS-scatterer-user).Among them, the BS-user link and the BS-RIS-user link are based on deterministic physical entities (e.g., antennas, reflecting elements, and semiconductor elements).To further analyze the impacts of their key properties on the system performance, we adopt the end-to-end EM physical model described in this article [16] to construct these two links and utilize the clustered channel model in [30] to set random scatterers in the system, hence the total transfer model can be depicted by where h EM is the EM physical transfer model and h Cluster is the clustered model.As for h EM , it is controlled by the tunable component matrix ∈ C N i ×N i .Furthermore, the form of the matrix is = diag( 1 , . . ., N i ) and its entries are the impedances of the back-end circuits of the RIS elements.The diodes or the varactors can be used to configure .In this article, the RIS adopts the varactors to achieve continuously adjustable reactances.In detail, h EM in our scenario can be given by ( 3), shown at the bottom of the page, where Z G ∈ C N t ×N t denotes the source impedance matrix, a diagonal matrix, and Z L is the load impedance.Generally, these two are determined on a case-by-case basis.Moreover, the transfer matrices are described as follows: where T, R, and I signify the transmitter, the receiver, and the RIS, respectively.
and the element numbers of them are N a and N b , represents the mutual-impedance matrix between the transmitter and the receiver, when A and B are diverse.On the contrary, Z AB denotes the self-impedance matrix in the case that they are the same.Moreover, the diagonal and the off-diagonal elements of the self-impedance matrix are the self impedances and the mutual impedances of the elements in the array.Similarly, Z AI ∈ C N A ×N i and Z IB ∈ C N i ×N B refer to the mutual-impedance matrices between these ends and the RIS.
According to [16], when it is presupposed that: all radiators, including antennas and reflectors, are extremely thin dipoles and placed parallel to the z-axis, the entries of these impedance matrices are the mutual or the self impedances of the dipoles in various ends (i.e., the transmitter, the receiver and the RIS).In the above, all impedances can be calculated by ( 8), shown at the bottom of the page, where η 0 = √ (μ 0 / 0 ) and k 0 = (2π/λ) are the characteristic impedance and the wavenumber, and μ 0 , 0 , and λ are the magnetic permeability, the electric permittivity and the wavelength, respectively.According to various impedances, a and b may be the same or the different physical entries.When considering the mutual impedance, In addition, (x a , y a , z a ), (x b , y b , z b ), l a , l b , and r are the center positions, the lengths, and the radii of the dipoles.
dz b dz a (8) Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
We make an assumption that all ends are located in the farfield regions of each other, which is a practical deployment scenario and consistent with the setup in [17]; hence, Z TT in (5) and Z RR in (7) are more dominant compared to other items.Based on this, S TT and S RR can be approximated as Similarly, we can let S RR − S RT (Z G + S TT ) −1 S TR ≈ Z RR in (3) under the far-field condition.Then, plugging approximated ( 6), ( 9), ( 10) into (3), and h EM in (3) can be substituted as To further simplify the form of ( 11), abridged notations can be introduced as follows: Based on ( 12)-( 14), h EM in (11) can be formulated as For clustered channels in (2), in the light of the description in [30], the summation of BS-scatterer-user multipath links can be given by (16) where N cl and N ray are the numbers of scatterers and their rays.γ = N T / N cl k=1 N ray,k denotes the normalization factor and α k,p represents the complex gain of the pth ray in the kth scatterer.PL(r k,p ) is the path loss such that PL(r k,p ) = −(η + ρ log 10 (r k,p ))/10, where θ k,p and φ k,p are the azimuth and elevation angles of departure.The heights of the transmitter and the receiver are represented by h T and h R , whereas r k and d are the distances from the transmitter to the clusters and the receiver, respectively.Finally, the array response vector a t ∈ C 1×N t of a uniform linear array (ULA) on the y-axis can be expressed as a t (φ) = (1/ √ N t )[1, e jk 0 d 0 sin(φ) , . . ., e j(N t −1)k 0 d 0 sin(φ) ], in which d 0 is the interelement spacing of the array.

C. Problem Formulation
Based on the above transfer model, the received signal of the jth IDU is + n I AWGN (17) where n I ∼ CN (0, σ 2 ) is the additive white Gaussian noise (AWGN) at the jth IDU with the noise power σ 2 .We assume that the IDUs have the ability to cancel the interference from the energy signals and, therefore, the jth IDU only suffers the interference from other IDUs (e.g., s 1 , . . ., s j−1 , s j+1 , . . ., s M I ).Then, the SINR of the IDU can be presented by Similarly, the received model of the nth EHU is given by where n E ∼ CN (0, σ 2 ) is the AWGN of the nth EHU and can be ignored.Moreover, its received radio-frequency (RF) power is expressed as The RF power in (20) needs to be rectified as direct current (dc) power for serving EHUs.Furthermore, when the input power is excessively high, due to the reverse bias of the diode, the nonlinear characteristic of the rectifying circuit should be considered.To characterize this feature, the dc output power of the nth EHU calculated by a nonlinear EH model can be presented as (21), shown at the bottom of the page, where ν and are determined by the circuit parameters.B is the maximum dc power represented by B = ([V 2  br ]/[4R L ]), where V br and R L denote the reverse breakdown voltage and the load resistance.Considering the convenience of the optimization, we can define an inverse function G(x) of (21) as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Moreover, the total required power of the system is given as follows: Here, denoting the transmitting power as . P c and P I are the power consumption of the front-end modules (e.g., mixers and filters) at the BS and the control circuits of the RIS elements.
In this article, we concentrate on both the sum rate of the system and the corresponding dissipated energy; thus, the EE is introduced to tradeoff them.The fractional form of EE can be presented as For the RIS-enhanced SWIPT network, we set the problem to maximize the EE subject to the maximal transmitting power, the minimal rate of the IDU, the minimal dc output power of the EHU, and the hardware restrictions of the control circuit at the RIS.The above optimization problem has the form of the following P 0 : Re q = R 0 ≥ 0 ∀q (25e) Im q ∈ R ∀q. (25f) denotes the rate requirement of the jth IDU.As for (25c), P Max is the maximal transmitting power budget.The E (D)   n in (25d) is the minimal dc output power demand of the nth EHU.Equations (25e) and (25f) represent the limitations of the control circuit.R 0 in (25e) denotes the loss of the circuit and R 0 ≥ 0 is the guarantee that the RIS operates in the passive mode instead of the active mode [16].In this article, the reactances of the circuits in (25f) are reconfigurable and we assume that their scopes of tunability are the set of real numbers.
The proposed system EE optimization problem P 0 is quite challenging to resolve owing to the fractional objective function and the nonconvex constraints.Particularly, to further show the influences of the hardware features on the system performances, the problem is separated into the no-coupling and the coupling-awareness situations for comparison.In addition, the transformations of the problems and the optimization schemes are introduced to solve them.

III. SOLUTION APPROACH FOR NO MUTUAL COUPLING
Before starting to solve the problem, the form of ( 15) can be simplified under the no-coupling assumption for the RIS, since Z II only keeps the self impedances of the RIS elements and turns into a diagonal matrix.It should be noted that the assumption is an ideal condition.Then, the total transfer model is reformulated as where ) and r q + jx q for q = 1, 2, . . ., N i is presented by r q = Re(Z II (q, q)) + R 0 (27) x q = Im(Z II (q, q) + (q, q)). (28) In our work, Z II (q, q) is calculated based on the dipole antenna and its real part is positive, thus r q > 0. Furthermore, the entries of can be replaced as follows: in which tan θ = −(x q /r q ).Then, plugging ( 29) into ( 26), we can obtain the transfer model for the users where shorthand notations are g = e − (rL/2r q ) + h cluster and According to cos θ = ([r q ]/[ r 2 q + x 2 q ]) > 0, the angle constraints can be θ q ∈ (−π/2, π/2) and φ q = 2θ q ∈ (−π, π).
Relying on the above assumption without mutual-coupling effects of the RIS, the system problem P 0 turns into P NC−0 as follows: Even though the structure of the channel model (30) has been further reduced and is similar to the cascaded channel model based on the independent diffusive scatterer (IDS) assumption in [7], the problem P NC−0 remains a nonconvex problem that is difficult to solve.To effectively address it, we adopt the SDR approach and the AO strategy.Specifically, for the beamforming and the circuit parameter subproblems, Dinkelbach's algorithm and SCA are introduced to tackle them.

A. Semidefinite Relaxation
The introduction of the SDR method will further enhance the solvability of the problem and pave the way for the solution of the subproblems.For beamforming vectors w j and v l at the BS, denoting W j = w j w H j and V l = v l v H l while satisfying W j 0, rank(W j ) = 1, V l 0 and rank(V l ) = 1.Concerning r L in (30), it can be replaced by fL u , where f [(exp(jφ 1 ))/(2r 1 ), . . ., (exp(jφ N i ))/(2r N i )] and L u −diag(r)L.Moreover, we let f = [f, 1] and Lu = [L u ; g].SDR can also applied in f and we denote F = fH f subject to F 0 and rank( F) = 1.Then, the rate R j (W j , F) of the jth IDU can be transformed as Tr F Lu,j Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
The received RF power P RF,n (W j , V l , F) of the nth EHU can be substituted by The transmitting power P t (W j , V l ) is relpaced by After SDR with dropping rank-one constraints, the problem P NC−0 is transformed into P NC−1 Tr F Lu,j Fq,q = 1 4r 2 q ∀q, Fq+1,q+1 = 1 (35e) However, the variables remain coupled; hence, even if the SDR approach was adopted, the problem P NC−1 is still nonconvex.Nevertheless, the problem can be further transformed into a more solvable form using the AO strategy.More specifically, fix F to optimize W j and V l first, then use the obtained W * j and V * l to obtain F * , iteratively.

B. AO Strategy 1) Optimization of {W j , V l }:
As for the beamforming subproblem with fixed F * , problem P NC−1 can be simplified as follows: First, the objective function of problem P NC−2 has the form of the fraction, which makes the problem tough to optimize.The idea is converting the fraction into a more concise one.Luckily, the effective fractional programming approach, Dinkelbach's algorithm proposed in [31], can be adopted to transform it into a solvable form.

Algorithm 1 Beamforming Vectors Optimization Framework Based on Dinkelbach's Algorithm
Initialize: (0) = 0, EE(0) = 0, ε; ; Achieve w * j and v * l by the eigenvector of W * j (i) and v * l (i), if the rank-one condition is met.Otherwise, Gaussian randomization is adopted [26]; Output: the optimal w * j and v * l .
Lemma 1: Under the condition that * is the unique zero, problem P NC−2 can be equivalently transformed as the auxiliary problem P NC−3 with the subtraction form of the objective function (37) Proof: The proof was presented in [31], we will not go into detail here.
For tackling the problem P NC−3 , we utilize the steps in Algorithm 1 to iteratively solve the auxiliary problem until the objective function of the problem converges to a threshold value ε.Indeed, according to the algorithm, the crux of the algorithm is addressing the following problem P NC−4 in each iteration i: Then, to further solve P NC−4 , the auxiliary variable τ is introduced to replace the fractional equation of the rate and, thus, the problem can be reformulated in the equivalent form as follows: where ( 39) is an essential constraint, which is utilized to ensure that the rate of IDU is not less than τ .Moreover, the objective Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
equation (τ j (i), W j (i), V l (i)) of P NC−4 is presented as So far, the fractional parts of the objective are eliminated.Then, for handling the introduced constraint (39), we define the other two variables a j (i) and b j (i) which satisfy the following equalities: With the help of ( 41) and ( 42), problem P NC−4 can be rewritten as problem P NC−5 The first-order approximation can be adopted to linearize the left hand of (43b) and (43c), due to the following inequalities: e a j (i) ≥ e āj (i) + e āj (i) a j (i) − āj (i) (44) where āj (i) and bj (i) are the feasible values of a j (i) and b j (i).
Up to this point, problem P NC−4 is transformed into a convex problem P NC−6 after a series of substitutions and approximations.The convex programming solver, i.e., CVX, can be utilized to solve it.Nevertheless, the approximate solution can be more accurate through the SCA algorithm as presented in Algorithm 2. In particular, problem P NC−6 will be solved continuously until the final condition is satisfied and the results of ā(t) Output: the optimal W * (t) into the next solution process.Specifically, ā(t) j (i) and b(t) j (i) can be updated by where W * (t) k (i) is obtained from the tth solution of the SCA iteration and τ * (t) j (i) can be replaced continuously by Tr (50) 2) Optimization of F: After achieving the optimized W * j = w * j w * H j and V * l = v * l v * H l , problem P NC−1 can be introduced with some auxiliary variables and reformulated as follows: Tr F Lu,j W j LH u,j ≥ e p j +q j + σ 2 e q j ∀j (51a) Similar to problem P NC−5 , the above problem is a convex problem and the SCA strategy adopted in Algorithm 2 can be used to solve it iteratively.For achieved F * , eigenvalue decomposition or Gaussian randomization is introduced to get f * .Through the AO strategy, the beamforming and the circuit parameters subproblems can be resolved iteratively until the termination condition of the total algorithm is reached.

IV. SOLUTION APPROACH FOR MUTUAL-COUPLING AWARENESS
When considering the mutual coupling among the RIS elements, T = Z II + in ( 15) cannot be treated as a diagonal matrix and the matrix inverse must be retained, which makes Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
the system problem difficult to be tackled.However, thanks to the Neuman series approximation [32], the inverse of the matrices can be simplified.Before adopting it, we assume the problem can be solved iteratively and the problem in the th iteration is The optimized tunable component matrix * ( ) is introduced to update T; thus, T ( +1) satisfies where G ( ) = j Im( * ( ) ) and T (0) = Z II + R 0 for fixing the resistances and adjusting the reactances of the circuits.Based on the Neuman series approximation, the inverse of T ( +1) in ( 15) can be approximated by The transformation from (54) to (55) can be traced to [32].Furthermore, the above approximation (56) only maintains the first and the second terms of the Neuman expansion.However, its accuracy needs to be analyzed.The difference D between (54) and (56) satisfies the below inequality It is obvious that T (k) −1 G ( ) should be sufficiently small to ensure accuracy.For achieving this goal, we can let Then, the th updated total transfer model for the users is simplified as In terms of simplicity, we introduce some shorthand variables as follows: Algorithm 3 AO Based on Dinkelbach's Algorithm and Neuman Series Approximation Achieve h ( ) through (58) for every user; 5: Adopt AO strategy, SDR approach, Dinkelbach's algorithm and SCA method proposed in Algorithm 1 and Algorithm 2 to solve problem P ( ) Output solved w * ( ) j (j ), v * ( ) l (j ), G * ( ) (j ) to calculate EE (j ); Return G * ( ) (j ) and EE ( ) = EE (j ); Achieve T ( +1) = T ( ) + G * ( ) ; 14: Update j = 1, = + 1; 15: end while Output: the optimal w * ( ) j According to ( 59)-( 61), ( 58) can be further substituted by The above equation also has a similar cascade form, which is relatively easy to fix.With its help, we can reformulate the system problem in each iteration P ( )  CA−1 : max Equation ( 63c) is adopted to ensure only the imaginary part of the tunable circuit impedance is adjustable as the same setting in problem P 0 .In addition, (63d) is the guarantee of the Neuman approximation's accuracy.It is worth noting that the choices of ζ need to be careful.If it is too small, it makes the approximation imprecise, while the convergence rate may be slow with large ζ .
As to resolve problem P ( ) CA−1 , the SDR method can be utilized again.Then, the beamforming and the circuit parameter subproblems are tackled iteratively.The optimized variable G ( ) is used to update the transfer model until convergence.The overall structure of the optimization frameworks is shown in Algorithm 3 detailedly.We can further find that the optimization scheme for solving the inner layer of the coupling-awareness problem can fully cope with the nocoupling case.In a sense, this also proves that the no-coupling case is indeed less complex and just like the choice of the IDS-based RIS model.Nevertheless, the cost of this simplification is to ignore the key hardware feature, the mutual coupling, of the RIS, and hence may cause optimization bias.The inaccuracy of this ideal assumption will be discussed in Section V.

A. Semidefinite Relaxation
For beamforming vectors, the SDR forms of them coincide with the previous section.Whereas, u ( ) B ( ) u can be adopted to substitute z ( ) G ( ) E ( ) in (62), where u ( ) [j ( ) 1 , . . ., j ( ) where unit imaginary j is introduced for recovering the original vector easily after achieving the optimized vectors.Moreover, according to SDR, Û( ) = û( ) H û( ) is introduced with Û( ) H 0 and rank( Û( ) ) = 1.With the help of the relaxed variables, P ( ) CA−1 can be transformed into the following form: where (64e) is the SDR form of accuracy constraint.
Particularly, the rate of jth IDU R j (W ( ) j , Û( ) ) and total power consumption P(W Tr V ( ) Compared to the no-coupling case, although the core of the problem is varied, problem P ( ) CA−2 is nonconvex and has a similar form as P NC−1 ; thus, the same approach used in the previous section can be adopted to resolve it.For decomposing the above problem into several subproblems, the AO strategy is presented in the next section.

B. AO Strategy 1) Optimization of {W
In each iteration, beamforming vectors and the reactances of the control elements are optimized iteratively until the objective convergences.When solving the beamforming subproblem, the Dinkelbach algorithm based on SCA proposed in Algorithms 1 and 2 still works under the condition that Û( ) is known.Since the solution approach is identical, the detailed process may not be repeated here.
2) Optimization of Û( ) : When obtaining resolved W * ( ) j and V * ( ) l , the equivalent subproblem is presented as ≥ e ṗj +q j + σ 2 e qj ∀j (67a) The above problem is solved by the SCA method iteratively until the stop condition is satisfied.Moreover, when using eigenvalue decomposition or Gaussian randomization to recover û( ) from Û( ) , the output vectors are real vectors and then every element of the vectors must be transformed into the pure imaginary value for coinciding with the assumption in (63c).

V. NUMERICAL RESULTS AND FULL-WAVE SIMULATION
To analyze the effectiveness of our proposed algorithm and the impacts of relevant hardware attributes on system performance in more detail, we first conduct a series of numerical simulation experiments for different QoS requirements of the users (i.e., energy harvesting for EHUs and minimum transmission rate for IDUs), various transmitting power levels, and diverse RIS topologies (i.e., various RIS element spacings with the same element number and different numbers of RIS elements under the uniform surface area).Specifically, simulations are all conducted under different situations including the ideal case, the mutual-coupling awareness, the mutualcoupling unawareness, the no RIS case, and the nonoperating RIS case.Among them, the mutual-coupling unawareness means that the optimization results from the no-coupling case are deployed in practical condition.In addition, the nonoperating case denotes that the reactances of control circuits are not optimized.Besides, the full-wave simulation is performed to analyze the physical active beams at the BS and the reflecting beams of the RIS in several typical scenarios.
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A. Numerical Results
The 2-D experimental scene is shown in Fig. 2. The center location of the 8-element-array BS is located at (0, 0) and the RIS with 2 × 8 array configuration is located at (1,1).Two EHUs and two IDUs are generated at the rectangular regions of (x IDU ,y IDU ) and (x EHU ,y EHU ) with [4,15] and y IDU ∈ [−5, −24].Moreover, all dipoles are perpendicular to the plane.
As for the transfer model in (2), it has two submodels: 1) the EM transfer model and 2) the clustered model.Concerning the EM transfer model, the system operates at 28 GHz.All antennas and RIS elements are dipoles with λ/500 radii and λ/2 lengths [16], [17].The element spacings of the BS and the RIS are λ/2 and λ/4, respectively.The reason for the choice is that the RIS generally has more compact spacing for deployment.The characteristic impedance η 0 is set by 377 .The real part of the tunable circuit, the source impedance and the load impedance are 73 , which coincides with the resistance part of the antenna impedance as in [16] and [17].Regarding the clustered model in (16), η and ρ in the path-loss equation are set by η = 72 and ρ = 2.92, respectively.The number of clusters follows N cl ∼ max{Poisson(1.8),1} and the ray number in them follows uniform distribution N ray ∼ U(0, 30).Complex gain α k,j is a complex standard normal random variable, which follows: α k,p ∼ CN (0, 1).Azimuth and elevation angles (θ t k,p , φ t k,p ) of departure are generated from the Laplacian distribution.The mean values of them uniformly distributed in [−π/2, π/2] and standard deviations are set as 15 • .The above parameter settings for the clustered model are set at 28 GHz and can be traced back to [30], [33], and [34].
The energy harvesting requirement for EHUs is E  1) Convergence Behaviors of the Proposed Algorithms in Coupling Awareness: Both for the cases of no coupling and coupling awareness, the cores of the optimization framework are similar.The difference is that an additional iteration algorithm based on the Neuman series approximation should be introduced for tackling the coupling-awareness case.Interestingly, the algorithm used to resolve the no-coupling case can be seen as the inner solution of the algorithm adopted to solve the coupling-awareness case; hence, the convergence behaviors of the algorithm in the case are exhaustive and the effectiveness of the proposed algorithm can be seen obviously from Fig. 4.
2) QoS Requirements and Resource Allocation Analysis: In the study of the EE performance under the different energy harvesting requirements in Fig. 5, the benchmark is set to the circumstance that the mutual-coupling effect does not exist, which is the case of Section III.Furthermore, compared to the other situations, our proposed coupling-awareness algorithm is closest to it.Especially, the performance is better than the coupling-unawareness case, which indicates that the coupling attributes among the RIS elements cannot be ignored.In addition, the performances of the above conditions with the operating RIS are better than those in the no-RIS case and the nonoperating RIS case, which demonstrates that RIS can improve EE performance effectively.Besides, the EE indicators of the above situations have the same downward trends with the increasing power demand of the EHU, as the transmitting power needs to be increased further to cope with the energy demands without enhancing the overall rate level.
As for the research that various rate thresholds impact EE performances in Fig. 6, the differences among all comparative   situations are identical to those in the previous analysis.Specifically, the trends are the same and gradually declining.The reason for this phenomenon is that the power may be assigned to the IDUs with better channel conditions, while the rate constraints are accessible for all IDUs.Nevertheless, more power resources need to be allocated to the IDUs with poor channel conditions due to the continuously increasing rate requirements.Indeed, the cost of the enhancement is greater, especially in its later stage.In detail, the improvement of the sum rate cannot keep up with the increment of resource consumption, which leads to decreasing EE levels.
In our work, the transmitting power is a guarantee of the QoS requirements and needs to be analyzed.From Fig. 7, we can find that the EE level rises as the transmitting power increases in every condition, however, the rate of growth decreases and levels off eventually.This is because at high transmit power values, the interference term grows noticeably and negatively affects the system's EE.Moreover, when in the later period of the power increment, the rate growth and power consumption are locked in a stalemate and, thus, the EE values maintain at the same levels.The above phenomenon shows that the transmitting power should be weighed carefully because of its dual character on EE.
3) RIS Topology Analysis: For further improving the system performance, the RIS often contains more elements.Nevertheless, the array may be too cumbersome to deploy with a large number of elements.Therefore, reducing the element spacing turns out to be a direct and effective method.From Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.the perspective of engineering practice, the element spacing is an important design factor for the RIS, which is generally less than λ/2 [13].Whereas, the smaller the spacing, the greater the mutual coupling.We research this influence on the EE in Fig. 8.After comparing the coupling-unawareness and the coupling-awareness case, it can be found that the tighter spacing of horizontal elements results in a larger performance gap between them.Additionally, the EE performances of all cases, including the ideal benchmark and the nonoperating RIS case, improve as the spacing enlarges.We believe this is mainly owing to the increased effective area of the RIS.
Fixing the number of vertical elements is 2 and the total effective area of the RIS, we provide more horizontal elements to the RIS.From Fig. 9, as the increased mutual coupling caused by the high number of elements is not considered, the EE performance of the ideal benchmark has approximately linear enhancement with more elements.However, considering the condition of the actual coupling awareness, the EE improvement is slowing down, since more elements induce more serious mutual coupling.In detail, performance enhancements and coupling effects will work concurrently, resulting in later stages of sluggish growth.The mutual-coupling effect may be more obvious in the coupling-unawareness case, since this case ignores its effect and, thus, leads to optimization bias.Furthermore, its performance degrades as the aggravated coupling brought by more elements.Finally, since the effective area of the RIS remains at the same level, the nonoperating RIS case has nearly unchanged EE performance.In short, the results show that the RIS configuration should be carefully taken into account as it affects EE performance.

B. Full-Wave Simulation
For demonstrating the properties of the physical beams more clearly and to further prove the effectiveness of the algorithm, we perform a full-wave simulation by Ansys Electronics 2019 software.In detail, the radiation patterns and radar cross section (RCS) patterns are simulated for generating the active beams and reflecting beams according to the optimized parameters.In this section, we analyze the impacts of QoS performances, end locations and the role of the energy beamforming on the characteristics of the physical beams (i.e., active beams of the BS and reflecting beams of the RIS).To further demonstrate the beams' differences in various cases, we set the default power requirement satisfying E (D) = 1.0 μW, and keep only one pair of the EHU and the IDU with positions, (0.5, −2) and (10, −11), while leaving other simulation parameters unchanged.Without loss of generality, we remove the clustered model and assume that there are no barriers blocking transmissions for simulation convenience.Moreover, we define the angles between two of these ends as ϑ BS-RIS , ϑ BS-IDU , ϑ BS-EHU , ϑ RIS-IDU , and ϑ RIS-IDU .The default settings will be used unless stated otherwise.
In Fig. 10, every subfigure includes two parts, the active beams of the BS and the reflecting beams of the RIS.All active beams have the primary and the secondary sub-beams for serving the IDU and the EHU.However, the disparities between the two are various in different situations because of the varied QoS requirements of the users.In Fig. 10(b), as E (D) increases, the gain level of the IDU beam shrinks and it performs oppositely for the EHU beam compared to the default case in Fig. 10(a).Nevertheless, due to the intense demands of the IDU, this situation is reversed in Fig. 10(c).Furthermore, the differentiations of reflecting beams among them are insignificant, which point at 36 • .This can be explained by the presence of the BS-IDU transfer link and the ease of covering users by the BS under the default position setting.This also means that the active beams play more essential roles and the reflecting beams only act as an aid to enhance the system performance.
When altering the positions of the ends as in Fig. 11, almost all the beams change significantly.As it can be seen in Fig. 11(a), adjusting the IDU to locate at (10, −4) causes the difference between ϑ BS-IDU and ϑ BS-EHU to increase, thus the BS is difficult to generate two sub-beams with a large gap for serving the users simultaneously.As a consequence, the RIS will share most of the responsibility for covering the IDU.Indeed, the form of the active beam is almost unaltered compared with the default case, but the angle of the main lobe of the reflecting beam changes to 46 • in order to support the IDU with a large ϑ RIS-IDU .In 11(b), while placing the BS on  (0, −0.5), the two sub-beams of the active beam turn into one.The reason for that is the closer difference between ϑ BS-IDU and ϑ BS-EHU .Moreover, the reflecting beam points at 30 • for assisting it to serve them.In 11(c), we put the EHU on the other side of the BS, while keeping the IDU unchanged.In line with expectations, the BS generates two opposite sub-beams at −42 • and 14 • , respectively.In this case, the reflecting beam is only produced for the IDU, which remains at 36 • .
In the above cases, with or without energy beamforming does not show any difference from the results of EM simulation. it also indicates that the energy beamforming is not essential and the absence of it will bring convenience to system design and algorithm optimization, which coincides with statements of [11] and [35].

VI. CONCLUSION
The hardware features and the performance analyses of an IRS-assisted-SWIPT MISO downlink system from the EM perspective were presented in this article.Particularly, the transfer model based on the end-to-end EM transmission and multipath propagation was introduced to analyze the physical characteristics of the system.The problem was divided into two categories: 1) the no coupling and 2) the coupling awareness, and both of which were formulated as EE maximization problems subject to the QoS requirements and the hardware constraints.The effective algorithm frameworks were proposed to tackle the above nonconvex problems.For the couplingawareness case, the Neuman series approximation was adopted to simplify the transfer model.In each iteration of the approximation, the SDR transformation and the AO framework based on Dinkelbach's algorithm, and SCA approach were utilized to resolve the subproblems.Moreover, the no-coupling case can also adopt a similar scheme.The effectiveness of the proposed frameworks and the importance of mutual coupling were presented clearly in the numerical simulation.Specifically, the full-wave simulation results showed that the reflecting beams of the RIS may play different roles under various conditions and the energy beamforming is unnecessary.As for future research points, a more general end-to-end EM transfer system will be considered to reveal the deeper hardware features of the Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
system.In detail, it will be compatible with more antenna configurations.Furthermore, the research about the more complex system settings with the co-located receivers will be conducted further.
Ruoyan Ma received the B.Eng. degree in human-computer interaction (innovative experiment major) from South China University of Technology, Guangzhou, China, in 2020, where he is currently pursuing the Ph.D. degree with the School of Electronic and Information Engineering, under the supervision of Prof. J. Tang.
His research interests include intelligent antenna system, reconfigurable intelligent surface, simultaneous wireless information and power transfer, electromagnetic theory, optimization theory, and 6G networks.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

1 . 3
μW ∀n and the rate threshold for IDUs is R (D) j = 1 bit/s/Hz ∀j.P c at the BS and P I at the control circuit are set by P c = 1 W and P I = 10 mW.Noise power is σ 2 = 1 × 10 (−12) W. Transmitting power budget is configured to P Max = 4 W.Moreover, we select the HSMS2850 rectifier diode as the core device to design a rectifier circuit through the advanced design system (ADS) 2017 software.The actual ADS simulation result is fit by(21) as in Fig.3.The fit circuit

Fig. 9 .
Fig. 9. EE versus number of RIS elements under the same surface area.