Self-Sustainable Reconfigurable Intelligent Surface Aided Simultaneous Terahertz Information and Power Transfer (STIPT)
Yijin Pan, Kezhi Wang, Cunhua Pan, Huiling Zhu, Jiangzhou Wang
11 Self-Sustainable Reconfigurable IntelligentSurface Aided Simultaneous TerahertzInformation and Power Transfer (STIPT)
Yijin Pan, Kezhi Wang, Cunhua Pan, Huiling Zhu and JiangzhouWang,
Fellow, IEEE
Abstract
This paper proposes a new simultaneous terahertz (THz) information and power transfer (STIPT)system, which is assisted by reconfigurable intelligent surface (RIS) for both the information data andpower transmission. We aim to maximize the information users’ (IUs’) sum data rate while guaranteeingthe energy users’ (EUs’) and RIS’s power harvesting requirements. To solve the formulated non-convexproblem, the block coordinate descent (BCD) based algorithm is adopted to alternately optimize thetransmit precoding of IUs, RIS’s reflecting coefficients, and RIS’s coordinate. The Penalty ConstrainedConvex Approximation (PCCA) Algorithm is proposed to solve the intractable optimization problemof the RIS’s coordinate, where the solution’s feasibility is guaranteed by the introduced penalties.Simulation results confirm that the proposed BCD algorithm can significantly enhance the performanceof STIPT by employing RIS.
Index Terms
Simultaneous terahertz information and power transfer (STIPT), intelligent reflecting surface (IRS),reconfigurable intelligent surface (RIS), terahertz (THz) communications.
Y. Pan is with the National Mobile Communications Research Laboratory, Southeast University, Nanjing 211111, China, andalso with School of Engineering and Digital Arts, University of Kent, UK.K. Wang is with the Department of Computer and Information Sciences, Northumbria University, UK.C. Pan is with the School of Electronic Engineering and Computer Science, Queen Mary, University of London, UK.J. Wang and H. Zhu are with the School of Engineering and Digital Arts, University of Kent, UK. a r X i v : . [ ee ss . SP ] F e b I. I
NTRODUCTION
The current mobile network is experiencing an unprecedented evolution with the increasingnumber of attractive mobile applications, which results in expectations for extremely high datarates for realizing a variety of multimedia services [1]. Thanks to the abundant spectrum resourcesin the terahertz (THz) band, the THz communication is able to realize high transmission ratesfrom hundreds of Gbps to several Tbps. As a result, THz transmission is envisioned as anemerging solution to meet the ultra-high-speed data rate demands of the enhanced mobilebroadband (eMBB) services, such as virtual reality and high-definition data streams [2].At the same time, the wireless power harvesting (WPH) technology provides an attractiveapproach for the Internet of things (IoT) devices with limited battery life to reap power fromexternal radio frequency sources [3]. The WPH technique promises the potentials for replacingbatteries in low power consumption devices or increasing their battery lifespans [4]. With theseproperties, numerous devices and sensors are being manufactured with the WPH power supplymode, including the nanoscale IoT devices [5]. Nevertheless, the traditional low-frequency radiowaves may no longer be suitable for WPH [6]. This is because the low-frequency radio wavenormally requires a large antenna aperture to capture a large portion of radiated electromagneticwave due to the relatively large wavelength [7], which tends to exceed the size limit of the IoTdevices and sensors [8], especially for the nanoscale IoT devices. To tackle this issue, a possiblesolution is to increase the frequency of the radio so that the size of antennas can be miniaturizedand the transmission beam directivity can be improved [6].Fortunately, as a key component of the 6G communication system, the THz band bridgesthe gap between mmWave and optical band. Specifically, THz band ranges from 100 GHz to10 THz such that the wavelength of THz can greatly reduce the required antenna aperture[9]. This makes utilizing the THz electronics for wireless power harvesting a very attractiveapproach. Also, emerging rectennas have been proposed and manufactured which make theenergy harvesting in THz bands becomes possible [10], [11]. As a result, combining the benefitsof the THz transmission in providing high-speed data rates with WPH for IoT devices will be apromising research direction, leading to a new term of Simultaneous Terahertz Information andPower Transfer (STIPT) network.
A. Related Works
In the THz communication system, due to the ultra-high radio frequency, the THz links areeasily blocked by obstacles in transmission paths. This feature of the THz band greatly limits thetransmission distance, which necessitate establishing efficient STIPT communications. To addressthis issue, it has been proposed to utilize the reconfigurable intelligent surface (RIS), also knownas intelligent reflective surface (IRS) to help compensate for the blocked communication links[12]. By adjusting the reflecting coefficients of RIS, the propagation channel condition can besignificantly improved to enhance the system performance [13].Recently, extensive efforts have been devoted to numerous applications of the RIS-assistedtransmissions [14]. A comprehensive survey about these applications of the RIS has been givenin [15], and the potential benefits that can be brought by the RIS has been explained in variousaspects in [16]. In [17], the RIS has been exploited to enhance the cell-edge performance inmulticell MIMO communication systems. Also, the RIS has been utilized to enhance latencyperformance of the mobile edge computing (MEC) system in [18]. Furthermore, the RIS wasalso utilized to enhance the physical layer security by improving the secrecy rate [19]–[23] andreducing the transmit power [24]. The RIS-enhanced orthogonal frequency division multiplexing(OFDM) system and its corresponding transmission protocol were investigated in [25].As for the simultaneous wireless information and power transfer (SWIPT) system, there arealready a few contributions on utilizing the RIS to enhance the performance of SWIPT systems[26]. An RIS-aided MIMO broadcasting SWIPT system was investigated in [27], where thetransmit precoding matrices and passive phase shift matrix of the RIS were jointly optimizedto maximize the weighted sum rate of information users while guaranteeing the users’ energyharvesting requirement. The contributions in [28] and [29] investigated the RIS-aided securetransmission system, where the obtained secure rate and energy efficiency were enhanced byoptimizing the reflecting coefficients of RIS, respectively. In [30], RIS was leveraged to enhancethe performance of non-orthogonal multiple access (NOMA) and the wireless power transfer(WPT) efficiency of SWIPT. The energy harvesting efficiency of the RIS-assisted MIMO broad-casting SWIPT system was investigated in [31], where the total transmit power required at theAP was minimized while satisfying the QoS constraints of the information users and the energyusers. A RIS-assisted wireless power transfer OFDM-based MEC system was investigated in[32], where the total energy consumption was minimized by optimizing the power allocation and computation resource allocation.It is worth pointing out that the above works only considered the micro/millimeter wavecommunications. Compared with traditional micro/millimeter transmission system, the THz bandtransmissions suffer from high molecular absorption and propagation loss [33]. The extremelyshort wavelengths of THz signals makes the obstacles in the path tend to absorb THz signalsrather than reflect them [34]. Therefore, the RIS is envisioned to be a necessity for the futureTHz communications to bypass blockages [35]. Most recently, specific efforts have been devotedto the use of RIS to enhance the THz transmission performance [36], [37]. For instance, thecoverage analysis in [38] has highlighted the impact of the molecular absorption loss on the pathloss of the RIS-assisted THz channel. As the THz band appears to be frequency-selective withmany path loss peaks, the sum rate of an RIS-assisted THz system was maximized in [39], wherea whole band is divided into several sub-bands. The passive reflecting phase shifters of RIS wasinvestigated in [40] to enhance the secrecy rate of THz communication. An RIS-aided multi-userTHz MIMO system with orthogonal frequency division multiple access was investigated in [41],where the weighted sum rate was maximized by jointly optimizing the hybrid beamformingand reflecting matrix of the RIS. As the RIS is able to control the propagation direction ofTHz waves for mitigating the blockage issue, the RIS was utilized in [42] to assist the UAVTHz transmission, where the joint passive beamforming design and trajectory optimization wereinvestigated. In [42], as the available spectrum in the THz band varies with the link distance,the unique channel fading characteristics of the THz channel was exploited to optimize the UAVtrajectory.
B. Motivations and Contributions
Most of the above-mentioned RIS-aided transmission approaches are not specifically designedfor STIPT systems, as integrating RIS into the STIPT face many challenges.First of all, in the THz band, many path loss peaks appear and therefore the total band isdivided into many sub-bands [43]. According to [39], the locations of these path loss peaks varywith the carrier frequencies and the transmission distance. Consequently, it becomes challengingto efficiently utilize the frequency-selective THz band, and the precoding for different users needsto be designed carefully on multiple sub-bands. More importantly, the link distance is dependenton the location of the RIS [44], but the impact of the RIS’s location is generally ignored in thecurrent RIS applications. In a typical STIPT application scenario, the energy users are normally
IoT devices or sensors, and the information users are deployed for the monitor tasks such astransmitting high-definition video/figures that entail ultra-high data rate. The locations of energyusers and information users can be available at the network AP. Consequently, in this scenariowhere the positions of users are relatively fixed, optimization of the RIS’s location has thepotential to effectively compensate for the link-distance dependent fading in the THz band. Infact, optimizing the location of RIS with the power constraints is challenging due to multipleperiodic cosine components in the channel expression, and currently, no efficient solutions havebeen reported in the literature, to the best of our knowledge.Secondly, due to the high radio frequencies in the THz band, the number of RIS elements canbe significantly increased within a limited area in order to provide a better reflecting performance.According to [45], the RIS power consumption depends on the type and the number of reflectingelements. Therefore, the energy consumption of RIS cannot be ignored in this case, and it isproportional to the number of RIS elements [46]. In [47], a part of the RIS’s elements areselected to harvest the received energy and the remaining elements help the secure informationtransmission, but this approach was designed for the micro/millimeter transmissions. However,in the current RIS-assisted SWIPT schemes, the power consumption of the RIS is generallyignored, and it is also unclear how to power the RIS in the THz transmission system.As a result, it is imperative to jointly consider the impact of the RIS’s power consumption,reflecting coefficients, and its location on the STIPT’s system. The above mentioned key funda-mental issues need to be resolved, and how to obtain an efficient STIPT communication systemis still unknown.Against the above background, in this paper, we consider the downlink transmission of theSTIPT network, where the information users (IUs) and energy users (EUs) are jointly servedby the RIS-assisted THz links. In our system, the RIS is also equipped with the WPH moduleto harvest energy from the received THz radios to maintain its circuit power consumption.The precoding for IUs, the RIS’s reflecting coefficients, and the RIS’s coordinate are jointlyoptimized to maximize the IUs’ achievable rates while satisfying the EU’s and RIS’s powerharvesting requirements. Overall, our contributions can be summarized as follows: • We propose an RIS-aided STIPT system to simultaneously transmit information and powerfor IUs and EUs, respectively. The RIS-assisted THz channel is modelled as a functionof the RIS reflecting coefficients and the coordinate of RIS. The RIS can harvest powerfrom the received radio, where the harvested power can be adjusted by the amplitude of its reflecting coefficients. • The optimization problem is formulated to maximize the IUs’ sum rates while guaranteeingthe power harvesting requirements of the EUs. The original non-convex problem is firstreformulated by utilizing the equivalence between the weighted minimum mean-squareerror (WMMSE) and the signal-to-noise ratio (SINR). Then, we decouple the optimizationproblem into three subproblems: optimization of the precoding for IUs, RIS’s reflectingcoefficients and RIS’s coordinate. • The precoding for IUs and RIS’s reflecting coefficients are obtained by utilizing the suc-cessive convex approximation method. To deal with the intractable optimization problem ofthe RIS’s coordinate, we propose the Penalty Constrained Convex Approximation (PCCA)Algorithm to guarantee the solution’s feasibility and the convergence of the block coordinatedescent (BCD) algorithm. • Extensive simulation results are provided to show the performance gain achieved by theproposed STIPT system compared with benchmarks. It is shown that the sum rate perfor-mance of IUs is greatly affected by the RIS’s coordinate. By utilizing the proposed BCDalgorithm, the THz channel can be optimized to fully exploit the spatial diversity so thatthe sum rate performance can be significantly enhanced.The reminder of this paper is organized as follows: Section II describes the system modelof the proposed RIS-aided STIPT system and formulates the optimization problem. SectionsIII develops the detailed algorithm to solve the formulated sum-rate maximization problem. InSection IV, the simulation results are presented to show the performance gain and the impact ofsystem parameters, and Section V concludes the paper.
Notation : For a vector x , | x | and ( x ) T respectively denote its Euclidean norm and its transpose. c represents the light speed. For matrix A , A ∗ and A (cid:63) represent the conjugate operator andconverged solution, respectively. C M × denotes the set of M × complex vectors. diag ( X ) represents the vector that is obtained from the diagonal entries of matrix X . a (cid:12) b representsthe Hadamard (point-wise) product of a and b .II. S YSTEM M ODEL AND P ROBLEM F ORMULATION
Consider the downlink of a STIPT system that needs to serve the IoT sensors and videotransmission devices at the same time, as shown in Fig. 1. The network access point (AP)operates in the THz band. The IoT sensors are EUs and they need to harvest power from the
RIS-EU linkZ axisEU X axisRIS-IU linkAP-RIS linkAP-EU link AP-IU linkAP
RIS
Y axis IU
Fig. 1. The RIS-assisted SWIFT system.
THz radio, meanwhile, IUs require high-speed data transmission for transmitting high-definitionvideo/figures.The AP is equipped with N t transmit antennas to serve I IUs and M EUs, and all the IUsand EUs are equipped with N r receive antennas. Let I and M respectively represent the set ofIUs and EUs. Then, the set of total users is given by U = I ∪ M , and the total number of allusers (including EU and IU) is denoted by U = |U | . In the following, the user u can be eitherEU or IU.As the THz channel is frequency-selective, the total THz band is divided into K sub-bands(SBs). Let f k denote the central frequency of SB k , and its wavelength is given by λ k = cf k .Normally, the wireless transmission channel includes the line-of-sight (LOS) link and non-line-of-sight (NLOS) links, where NLOS links consist of reflected, scattered, and diffractedcomponents. As the scattered and diffracted components are shown to play insignificant roles inthe received signal power in [43], [48], similar to [41], [49], the scattered and diffracted rays areneglected in the channel model. In addition, according to [34], the surfaces of walls and ceilingsappear “rough” for the THz signals so that they tend to absorb and scatter the THz signals ratherthan reflect them. As the RIS is specially designed to enable redirecting the incoming signalto the desired directions, the NLOS components in this work are only contributed by reflectedpaths from the RIS. Distance difference Distance difference t t APA Pn nP A s s δ r r uu un n s s δ = s sd u APu s sd AP s t APn s u s u s r un s AP s (a) LOS links from AP’s transmit antenna elements touser u ’s first antenna element Distance difference Distance difference t t APA Pn nP A s s δ r r uu un n s s δ = s sd u APu s sd AP s t APn s u s u s r un s AP s (b) LOS links between user u ’s receive antenna elementsand AP’s first antenna elementFig. 2. The LOS links between AP and user u . A. LOS Links without RIS
Fig. 2 shows the LOS links between the AP and user u . As shown in Fig. 2, the coordinate ofAP’s first antenna element is denoted as s AP ∈ R × , and the coordinate of user u ’s first antennaelement is denoted as s u ∈ R × . Then, the distance between the AP and the users is calculatedas | d u | , where d u = s u − s AP is the transmit vector for user u . In addition, as shown in Fig.2, the transmission distances between different transmit (receive) antenna elements are different,which causes the phase difference between the channel gains in the frequency domain. Thesephase differences are featured by the transmit array vector of the AP and the receive antennavector of the user.Fig. 2 (a) shows the distance differences between the transmit antennas of the AP. In Fig. 2(a), s n t AP ∈ R × denotes the coordinate of AP’s n t -th antenna element. We define directionalvector δ n t AP = s n t AP − s AP , n t = 1 , · · · , N t , and δ AP = [0 , , T . As shown in Fig. 2 (a), thephase difference θ n t k,u between AP’s n t -th antenna element and the first element is evaluated as θ n t k,u = 2 πf k c ( d u ) T δ n t AP | d u | = 2 πλ k ( d u ) T δ n t AP | d u | , n t = 1 , · · · , N t . (1)Then, the transmit array vector from the AP to user u on SB k is then denoted by v k,u = [1 , · · · , exp( − jθ n t k,u ) , · · · , exp( − jθ N t k,u )] T . (2)Fig. 2 (b) shows the distance differences between the receive antennas of the user u . In Fig.2 (b), s n r u ∈ R × denote the coordinate of the n r -th antenna element of user u . Define the directional vector δ n r u = s n r u − s u , n r = 1 , · · · , N r , for the n r -th receive antenna element. Asshown in Fig. 2 (b), the phase difference (cid:37) n r k,u between user’s n r -th antenna element and the firstelement is evaluated as (cid:37) n r k,u = 2 πf k c ( d u ) T δ n r u | d u | = 2 πλ k ( d u ) T δ n r u | d u | , n r = 1 , · · · , N r . (3)Then, the receive array vector from user u to the AP on SB k is r dirk,u = [1 , · · · , exp( − j(cid:37) n r k,u ) , · · · , exp( − j(cid:37) N r k,u )] T . (4)According to the ray tracing techniques [50], the path gain from the AP to user u on SB k isevaluated as h k,u = (cid:18) G r G t λ k π | d u | (cid:19) exp (cid:18) − j π | d u | λ k (cid:19) exp (cid:18) − K ( f k ) | d u | (cid:19) , (5)where G r and G t respectively represents the antenna gain of transmit array and the antenna gainof receiving array, | d u | is the distance from IU u to the AP, and K ( f k ) is the overall absorptioncoefficient of the transmission medium on SB k . Then, the LOS channel from the AP to user u on SB k is denoted by H k,u = h k,u r dirk,u v Hk,u , u ∈ I ∪ M . (6) B. RIS Assisted NLOS Links
The number of reflecting elements of the RIS is N . The coordinate of the RIS’s first reflectingelement is denoted as s RIS ∈ R × .We first consider the link from the AP to the RIS. The transmit distance between the AP andthe RIS is evaluated as | d | , where the transmit vector is d = s RIS − s AP . Similar to (1), thephase difference θ n t k between AP’s n t -th antenna element and the first element is evaluated as θ n t k = 2 πλ k ( d ) T δ n t AP | d | , n t = 1 , · · · , N t . (7)Then, the transmit array vector from the AP to the RIS on SB k is then denoted by v k = [1 , · · · , exp( − jθ n t k ) , · · · , exp( − jθ N t k )] T . (8)In addition, due to different transmit distance between the reflecting elements, we define thereceiving array vector e k to feature the relative phase differences between the signals receivedon SB k at different reflecting elements. Similar to (3), the directional vector is defined as δ nRIS = s nRIS − s RIS , n = 1 , · · · , N , andthe phase difference between RIS’s n -th reflecting element and the first reflecting element is ϑ nk = 2 πλ k ( d ) T δ nRIS | d | , n = 1 , · · · , N. (9)Then, the receive array vector is given by e k = (cid:2) , · · · , exp( − jϑ nk ) , · · · , exp( − jϑ Nk ) (cid:3) T . (10)As the path-loss gain from the AP to RIS on SB k is evaluated as H k = (cid:18) G t λ k π | d | (cid:19) exp (cid:18) − j π | d | λ k (cid:19) exp (cid:18) − K ( f k ) | d | (cid:19) , (11)then the LOS channel from the AP to RIS on SB k is denoted by H k = H k e k v Hk . (12)Then, we consider the links from the RIS to the users. The transmit distance between the RISand the user u is | d ,u | , where d ,u = s u − s RIS . Similarly, the phase difference between the n -th reflecting element and the first reflecting element is ϑ nk,u = 2 πλ k ( d ,u ) T δ nRIS | d ,u | , n = 1 , · · · , N, (13)and the transmit array vector from the RIS to user u on SB k is expressed as e k,u = (cid:2) , · · · , exp( − jϑ nk,u ) , · · · , exp( − jϑ Nk,u ) (cid:3) T . (14)As the power consumption of the RIS cannot be ignored, we assume that the WPH moduleis equipped in the RIS so that the RIS can also harvest energy from the radios sent by AP. As aresult, the reflecting coefficient is denoted by Φ n = β n exp( jφ n ) , where β n and φ n respectivelyrepresent the amplitude and the phase shift of the n -th reflecting element. Then, the phase shiftmatrix of the RIS is denoted by Φ = diag { β n exp( jφ n ) , n = 1 , · · · , N } . (15)We then have the following constraints for the reflecting coefficients as C | β n exp( jφ n ) | ≤ , n = 1 , · · · , N. (16)In addition, the phase difference at SB k between user u ’s first receive antenna element andthe n r -th element is ζ n r k,u = 2 πλ k ( d ,u ) T δ n r u | d ,u | , n r = 1 , · · · , N r . (17) Then, the receive array vector from user u to RIS on SB k is expressed as r k,u = (cid:2) , · · · , exp( − jζ n r k,u ) , · · · , exp( − jζ N r k,u ) (cid:3) T . (18)The cascaded channel gain of the AP-RIS-user u link on SB k can be expressed as [51] g k,u = (cid:32) G t G r λ k √ π | d ,u || d | (cid:33) exp (cid:18) − j π | d ,u | + | d | λ k (cid:19) exp (cid:18) − K ( f k )( | d ,u | + | d | ) (cid:19) , (19)where | d ,u | and | d | represent the distance from the RIS to the user u and the AP, respectively.Overall, the AP-RIS-user u link on SB k is given by G k,u = g k,u r k,u e Hk,u Φ e k v Hk , u ∈ I ∪ M . (20) C. Information Transfer
The signal vector transmitted from the AP to IU i on SB k is s k,i ∈ C d × . Suppose that thedata symbol s k,i satisfies E [ s k,i s ∗ k,i ] = I d and E [ s k,i s ∗ k,j ] = for i (cid:54) = j . Let F k,i ∈ C N t × d denotethe precoding matrix used by the AP for IU i on SB k . Then, the transmitted signal x k ∈ C N t × from the AP on SB k is x k = I (cid:88) i =1 F k,i s k,i . (21)With the aid of the RIS, the received signal at the IU i on SB k is y k,i = ( H k,i + G k,i ) x k + n k,i = Z k,i x k + n k,i , (22)where Z k,i = H k,i + G k,i , and n k,i is the additive Gaussian noise.Then, the achievable data rate of IU i on SB k is given by R k,i = log (cid:12)(cid:12) I N r + F Hk,i Z Hk,i Z k,i F k,i J − k,i (cid:12)(cid:12) , (23)where J k,i = (cid:80) Iu (cid:54) = i Z k,i F k,u F Hk,u Z Hk,i + σ k,i I N r , and σ k,i is the noise power.Furthermore, as the transmit power is limited, we have the following constraints for theprecoding matrices: C K (cid:88) k =1 I (cid:88) i =1 (cid:107) F k,i (cid:107) F ≤ P maxT . (24) D. Energy Harvesting
As the RIS also harvests energy from the AP, so that the reflecting coefficients can be adjustedto satisfy the energy harvesting requirement. That is to say, a part of the AP’s energy is reflectedby the RIS, and the remaining part is fed into the RIS’s WPH unit for harvesting. The powerreceived by RIS on SB k is q ink = I (cid:88) i =1 tr (cid:0) F Hk,i H Hk H k F k,i (cid:1) . (25)The reflected power by RIS on SB k is q outk = I (cid:88) i =1 tr (cid:0) F Hk,i H Hk Φ H Φ H k F k,i (cid:1) . (26)Then, the harvested power by RIS is calculated by P RIS = K (cid:88) k =1 η k ( q ink − q outk ) . (27)where η k denotes the power harvesting efficiency on SB k , since the RF-DC conversion efficiencyis dependent on the carrier’s frequency. Let P I denote the required power for RIS, then we have C K (cid:88) k =1 I (cid:88) i =1 η k tr (cid:0) F Hk,i H Hk ( I − Φ H Φ ) H k F k,i (cid:1) ≥ P I . (28)Similarly, the power harvested by EU m should satisfy the following constraint: C K (cid:88) k =1 I (cid:88) i =1 η k tr (cid:0) Z k,m F k,i F Hk,i Z Hk,m (cid:1) ≥ P Um , m ∈ M , (29)where P Um is the required power of EU m , and Z k,m is the composite channel gain between theAP and the EU m on SB k . E. Problem Formulation
For simplicity, we define β = [ β , · · · , β N ] as the amplitude vector of the reflecting coeffi-cients, and define φ = [ φ , · · · , φ N ] as the phase shifts of the reflecting coefficients. For easeof presentation, in the following, we utilize the notation L to represent the coordinate of RIS,which is optimized in the following section. In the system model, we take the coordinate ofRIS’s first reflecting element, i.e., s RIS , as the reference coordinate in the system model. Thatis to say, L is equivalent to s RIS in the following. It is observed that the sum rate of IUs and the harvested power for EUs are dependent on thecoordinate of RIS L , the transmit precoding matrices and the reflecting coefficients of the RIS.Then, we can formulate the problem as:max β , φ , L , F k,i R s = K (cid:88) k =1 I (cid:88) i =1 R k,i s.t. C − C . (30)It is observed that Problem (30) is non-convex and difficult to solve due to the followingreasons. First of all, the optimization variables are coupled together and the objective functionis intractable. Moreover, according to the RIS-assisted channel model in Section II-B, there isa complicated relationship between the position of RIS, the RIS’s reflecting coefficients andthe channel gain. Therefore, effective reformulation and simplification are required to tackle theabove optimization problem. III. S OLUTION A NALYSIS
The original Problem (30) is non-convex and challenging to solve, we first reformulate theproblem by leveraging the equivalence between the minimum mean-square error (MMSE) andthe signal-to-noise ratio (SINR). At IU i , the receive decoding matrix U k,i ∈ C N r × d is appliedto the received signal on SB k so that ˆ s k,i = U Hk,i y k,i . Then, the received mean square error(MSE) of IU i on SB k is given by E k,i = E s , n (cid:2) ( ˆ s k,i − s k,i )( ˆ s k,i − s k,i ) H (cid:3) = ( U Hk,i Z k,i F k,i − I d ) + I (cid:88) u (cid:54) = i U Hk,i Z k,i F k,u F Hk,u Z Hk,i U k,i + U k,i σ k,i . (31)The optimal MMSE decoding matrix { U k,i } is given by U k,i = ( Z k,i F k,i F Hk,i Z Hk,i + J k,i ) − Z k,i F k,i . (32)Then, substituting (32) into (31), we have E min k,i = I d − F Hk,i Z Hk,i ( Z k,i F k,i F Hk,i Z Hk,i + J k,i ) − Z k,i F k,i . (33) According to the relationship between the E min k,i and the SINR shown in [52], the originalProblem (30) can be reformulated as Problem (34) by introducing a set of auxiliary variables { W k,i } together with the receiving matrices { U k,i } .min β , φ , L , F k,i , W k,i , U k,i O tot = K (cid:88) k =1 I (cid:88) i =1 ( tr ( W k,i E k,i ) − log | W k,i | ) s.t. C − C . (34)Although Problem (34) has more optimization variables, the objective function of Problem (34)is more tractable. Consequently, Problem (34) can be solved by employing the BCD algorithm,where the optimization variables can be iteratively obtained while keeping the others fixed. Thatis to say, we decouple the optimization problem into three subproblems: optimization of theprecoding for IUs, RIS’s reflecting coefficients and RIS’s coordinate. Note that the receivingmatrices { U k,i } and the auxiliary matrices W k,i can be directly solved according to the aboveanalysis. Then, the optimal decoding matrix { U k,i } is given by (32), and the optimal W ∗ k,i isgiven by W ∗ k,i = ( E min k,i ) − . (35) A. Precoding Matrices Design
Given the coordinate of RIS L , RIS’s reflecting coefficients, auxiliary matrices W k,i and { U k,i } , the precoding matrices are optimized in this section. By substituting the MSE E k,i in(31) into (34) and discarding the constant terms, the precoding matrices F k,i are determined bythe following problemmin F k,i K (cid:88) k =1 I (cid:88) i =1 tr (cid:0) F Hk,i ¯ W k F k,i (cid:1) − K (cid:88) k =1 I (cid:88) i =1 (cid:60) (cid:2) tr ( ¯ Z k,i F k,i ) (cid:3) s.t. C − C , (36)where ¯ W k = (cid:80) Ii =1 Z Hk,i U k,i W k,i U Hk,i Z k,i and ¯ Z k,i = W k,i U Hk,i Z k,i . Although the objectivefunction of Problem (36) is convex, the energy harvesting constraints C and C are non-convex. Note that I − Φ Hn Φ n is positive definite. As a result, we adopt the successive convexapproximation method by leveraging the first-order Taylor expansions with the given precodingmatrix ¯ F k,i as tr ( F Hk,i B k F k,i ) ≥ (cid:60){ tr ( ¯ F Hk,i B k F k,i ) } − tr ( ¯ F Hk,i B k ¯ F k,i ) , (37)tr ( F Hk,i C k,m F k,i ) ≥ (cid:60){ tr ( ¯ F Hk,i C k,m F k,i ) } − tr ( ¯ F Hk,i C k,m ¯ F k,i ) , (38) where B k = η k H Hk ( I − Φ H Φ ) H k and C k,m = η k Z Hk,m Z k,m . Then, C and C can berespectively reformulated as K (cid:88) k =1 I (cid:88) i =1 (cid:60){ tr ( ¯ F Hk,i B k F k,i ) } ≥ K (cid:88) k =1 I (cid:88) i =1 tr ( ¯ F Hk,i B k ¯ F k,i ) + P RIS , (39) K (cid:88) k =1 I (cid:88) i =1 (cid:60){ tr ( ¯ F Hk,i C k,m F k,i ) } ≥ K (cid:88) k =1 I (cid:88) i =1 tr ( ¯ F Hk,i C k,m ¯ F k,i ) + P UEm . (40)Then, by substituting C and C with (39) and (40) respectively, Problem (36) can be transformedinto a series of convex problems, which can be solved by standard tools, such as the CVX. B. RIS Reflecting Coefficient Optimization
Given W k,i , { U k,i } , { F k,i } and the RIS’s coordinate L , we consider the optimization of RIS’sreflecting coefficients ϕ n = β n exp( jφ n ) , where the reflecting matrix is Φ = diag { [ ϕ n ] Nn =1 } .According to the MSE given in (31), we havetr ( W k,i E k,i ) = tr ( W k,i U Hk,i Z k,i F sk Z Hk,i U k,i ) − (cid:60) (cid:2) tr ( Z k,i F k,i W k,i U Hk,i ) (cid:3) + const , (41)where F sk = (cid:80) Iu =1 F k,u F Hk,u . The term “const” denotes the constant that is irrelevant with thereflecting coefficients ϕ n . As Z k,i = H k,i + G k,i , by removing the irrelevant terms in (41), thereflecting coefficient optimization problem is formulated asmin ϕ K (cid:88) k =1 I (cid:88) i =1 O k,i ( ϕ ) s.t. C , C , C , (42)where ϕ = [ ϕ , · · · , ϕ N ] T . In the objective function of (42), for simplicity, we define ¯ U k,i = U k,i W k,i U Hk,i , ¯ F k,i = F k,i W k,i U Hk,i , and then we have O k,i ( ϕ ) = 2 (cid:60) [ tr ( G k,i F sk H Hk,i ¯ U k,i )] + tr ( G k,i F sk G Hk,i ¯ U k,i ) − (cid:60) [ tr ( G k,i ¯ F k,i )] . (43)Note that ¯ U k,i is hermitian, but it is still difficult to solve Problem (42) with this formulation.To obtain a more tractable problem formulation, we define u k,i = e Hk,i (cid:12) e Tk . Then, the RISassisted channel gain is represented as G k,i = g k,i ( u k,i ϕ ) r k,i v Hk . (44)Substituting (44) into (43), we have O k,i ( ϕ ) = 2 (cid:60){ g k,i ξ k,i u k,i ϕ } + A k,i ( g k,i ) ( u k,i ϕ ) , (45) where A k,i = tr ( r k,i v Hk F sk v k r Hk,i ¯ U k,i ) , and ξ k,i = ( tr ( r k,i v Hk F sk H Hk,i ¯ U k,i ) − tr ( r k,i v Hk ¯ F k,i )) .Similarly, according to (12), constraint C can be reformulated astr (cid:0) F Hk,i H Hk ( I − Φ H Φ ) H k F k,i (cid:1) = ( H k ) tr (cid:0) F Hk,i v k v Hk F k,i (cid:1) ( N − ϕ H ϕ ) , (46)where e Hk e k = N is utilized.Similarly, for constraint C , by substituting Z k,m = H k,m + G k,m and (44) into C , we havethe following reformulation as I (cid:88) i =1 tr (cid:0) Z k,m F k,i F Hk,i Z Hk,m (cid:1) = ( g k,m ) Λ k,m ( u k,m ϕ ) + 2 (cid:60){ g k,m w k,m u k,m ϕ } + Q k,m , (47)where Λ k,m = tr (cid:0) r k,i v Hk F sk v k r Hk,i (cid:1) , w k,m = tr (cid:0) r k,i v Hk F sk H Hk,m (cid:1) , and Q k,m = tr (cid:0) H k,m F sk H Hk,m (cid:1) .According to (44), (46) and (47), the RIS’s reflecting coefficient problem can be reformulatedas min ϕ ϕ H Aϕ + (cid:60){ ξϕ } (48a)s.t. ϕ H Λ m ϕ + (cid:60){ ω m ϕ } ≥ ˜ P Um . (48b) ( N − ϕ H ϕ ) C RIS ≥ P I (48c) | ϕ | ≤ . (48d)where A = K (cid:88) k =1 I (cid:88) i =1 A k,i ( g k,i ) u Hk,i u k,i , ξ = K (cid:88) k =1 I (cid:88) i =1 g k,i ξ k,i u k,i , Λ m = K (cid:88) k =1 η k ( g k,m ) Λ k,m u Hk,m u k,m , ω m = 2 K (cid:88) k =1 η k g k,m w k,m u k,m , ˜ P Um = P Um − K (cid:88) k =1 Q k,m , and C RIS = K (cid:88) k =1 I (cid:88) i =1 η k ( H k ) tr (cid:0) F Hk,i v k v Hk F k,i (cid:1) . However, it is observed that constraint (48b) is non-convex. Note that Λ m is positive-definite sothat we adopt the first-order Taylor expansion for convex approximation. At given ¯ ϕ , we have ϕ H Λ m ϕ ≥ (cid:60){ ϕ H Λ m ¯ ϕ } − ¯ ϕ H Λ m ¯ ϕ . (49)By utilizing (49) to simplify the (48b), Problem (48) can be transformed into a series of simpleconvex problems, which can be easily solved by CVX. C. Optimization of RIS’s Coordinate
We consider the optimization of RIS coordinate with given W k,i and { U k,i } , { F k,i } and thephase shift matrix Φ . In this case, based on the formulations given in (45), (46) and (47), theoriginal Problem (34) with respect to the RIS coordinate is formulated asmin L O tot ( L ) = K (cid:88) k =1 I (cid:88) i =1 g k,i ( L ) E k,i ( L ) + (cid:60){ g k,i ( L ) F k,i ( L ) } + Cst ( W k,i , U k,i , F k,i ) (50a)s.t. K (cid:88) k =1 ( g k,i ( L ) λ k,m ( L ) + (cid:60){ g k,m ( L ) χ k,m ( L ) } + η k Q k,m ) ≥ P Um , (50b) K (cid:88) k =1 H k ( L ) D k ( L ) ≥ P I , (50c)where Cst ( W k,i , U k,i , F k,i ) is the constant term, and E k,i ( L ) = A k,i ( L )( u k,i ( L ) ϕ ) , (51) F k,i ( L ) = 2 ξ k,i ( L ) u k,i ( L ) ϕ , (52) λ k,m ( L ) = η k Λ k,m ( L )( u k,m ( L ) ϕ ) , (53) χ k,m ( L ) = 2 η k w k,m ( L ) u k,m ( L ) ϕ , (54) D k ( L ) = I (cid:88) i =1 η k tr (cid:0) F Hk,i v k ( L ) v Hk ( L ) F k,i (cid:1) ( N − ϕ H ϕ ) . (55)According to the channel model, many periodic cosine components with respect to the SB’sindex and UE’s index are involved in E k,i ( L ) , F k,i ( L ) , λ k,m ( L ) , χ k,m ( L ) and D k ( L ) . However,they are all dependent on the RIS’s coordinate L = [ X, Y, Z ] . Their complex expressions makeit very difficult to directly optimize the objective function given in (50a). Therefore, we seek tofind a tractable formulation of the coordinate optimization problem by regarding these intractableterms as constants.First of all, for ease of exposition, we define auxiliary variables r u and d , which are dependenton the RIS’s coordinate L by d = | L − s AP | , r u = | L − s u | , u ∈ I ∪ M . (56)Then, we define the function f k,u ( r u , d ) with respect to ( r u , d ) as f k,u ( r u , d ) = µ k r u d exp ( − K k ( r u + d )) , u ∈ I ∪ M , (57)where µ k = G t G r λ k √ π , and K k = K ( f k )2 . According to (19), the cascaded channel gain g k,i ( L ) can be represented as a function withrespect to ( r i , d ) as g k,i ( r i , d ) = f k,i ( r i , d ) exp (cid:18) − j π r i + d λ k (cid:19) , u ∈ I ∪ M . (58)Next, the objective function and constraints are investigated step by step. As the solution isobtained by the iterative algorithm, we adopt a given RIS’s coordinate obtained at the ( n ) -thiteration denoted as L ( n ) to help find the tractable formulation of Problem (50).
1) Simplification of Objective:
To simplify objective (50a), substituting the coordinate L ( n ) into (51) and (52), we can obtain the following constants as E k,i (cid:44) E k,i ( L ( n ) ) , F k,i (cid:44) (cid:60) (cid:40) exp (cid:32) − j π ˜ r i + ˜ d λ k (cid:33) F k,i ( L ( n ) ) (cid:41) , i ∈ I , where the constants ˜ r i and ˜ d are obtained by leveraging the coordinate L ( n ) in ( n ) -th iterationas ˜ r i = | L ( n ) − s i | , ˜ d = | L ( n ) − s AP | . The Hessian matrix of f k,i ( r i , d ) with respect to ( r i , d ) is given by (cid:53) f k,i = µ k r i d exp ( − K k ( r u + d )) ( K k r u +1) +1 r i ( K k r u +1)( K k d +1) r i d ( K k r u +1)( K k d +1) r i d ( K k d +1) +1 d . (59)Then, it is observed that (cid:53) f k,i is positive-definite so that f k,i ( r i , d ) is convex with respect to ( r i , d ) . However, note that the calculated coefficient F k,i is not necessarily positive. As a result,the first-order Taylor expansion of f k,i ( r i , d ) is adopted for simplification as f k,i ( r i , d ) ≥ f k,i (˜ r i , ˜ d ) + ∇ r i f k,i (˜ r i , ˜ d )( r i − ˜ r i ) + ∇ d f k,i (˜ r i , ˜ d )( d − ˜ d ) , (60)where the derivative is ∇ x f k,i ( x, y ) = − µ k K k x + 1 x y exp ( − K k ( x + y )) . (61)In addition, the Hessian matrix of f k,i ( r i , d ) with respect to ( r i , d ) is (cid:53) f k,i = µ k r i d exp ( − K k ( r u + d )) (2 K k r u +2) +2 r i (2 K k r u +2)(2 K k d +2) r i d (2 K k r u +2)(2 K k d +2) r i d (2 K k d +2) +2 d . (62)As (cid:53) f k,i is positive definite, f k,i ( r i , d ) is convex with respect to ( r i , d ) . Furthermore, it isobserved that the calculated coefficient E k,i is always positive. Consequently, by leveraging (58) and (60), the calculated E k,i , F k,i and discarding the irrelevant constants, a simplified version ofthe objective (50a) with a given coordinate L ( n ) is given as O L ( n ) ( r i , d ) = K (cid:88) k =1 I (cid:88) i =1 ( f k,i ( r i , d ) E k,i + F k,i ( ∇ r i f k,i (˜ r i , ˜ d ) r i + ∇ d f k,i (˜ r i , ˜ d ) d )) . (63)It is observed that the objective O L ( n ) ( r i , d ) is convex with respect to ( r i , d ) .
2) Simplification of Constraints for EUs:
Then, we deal with constraint (50b) for the EU m .By substituting the coordinate L ( n ) into (53) and (54), the following constants can be obtainedas λ k,m (cid:44) λ k,m ( L ( n ) ) , χ k,m (cid:44) (cid:60) (cid:40) exp (cid:32) − j π ˜ r m + ˜ d λ k (cid:33) χ k,m ( L ( n ) ) (cid:41) , m ∈ M , where the constant ˜ r m = | L ( n ) − s m | is calculated with the coordinate L ( n ) .Then, by employing the function f k,m ( r m , d ) , m ∈ M in (57), the constants λ k,m and χ k,m ,constraint (50b) is rewritten as K (cid:88) k =1 ( f k,m ( r m , d ) λ k,m + f k,m ( r u , d ) χ k,m + η k Q k,m ) ≥ P Um , m ∈ M . (64)However, (64) is still non-convex. As f k,m ( r u , d ) and f k,m ( r u , d ) are both convex functions,their first-order Taylor expansions are f k,m ( r m , d ) ≥ f k,m (˜ r m , ˜ d ) + ∇ r m f k,m (˜ r m , ˜ d )( r m − ˜ r m ) + ∇ d f k,m (˜ r m , ˜ d )( d − ˜ d ) , (65) f k,m ( r m , d ) ≥ f k,i (˜ r m , ˜ d ) + ∇ r m f k,m (˜ r i , ˜ d )( r m − ˜ r m ) + ∇ d f k,m (˜ r m , ˜ d )( d − ˜ d ) , (66)where ∇ x f k,m ( x, y ) is given in (61), and ∇ x f k,m ( x, y ) = 2 f k,m ( x, y ) ∇ x f k,m ( x, y ) .Then, substituting (65) and (66) into (64), constraint (50b) for the EU m can be furtherreformulated as A m ( L ( n ) , ˜ r m , ˜ d ) r m + B m ( L ( n ) , ˜ r m , ˜ d ) d + C m ( L ( n ) , ˜ r m , ˜ d ) ≥ P Um , m ∈ M , (67)where A m ( L ( n ) , ˜ r m , ˜ d ) = (cid:80) Kk =1 (cid:16) λ k,m ( L ( n ) ) ∇ r m f k,m (˜ r m , ˜ d ) + χ k,m ( L ( n ) ) ∇ r m f k,m (˜ r m , ˜ d ) (cid:17) , B m ( L ( n ) , ˜ r m , ˜ d ) = (cid:80) Kk =1 (cid:16) λ k,m ( L ( n ) ) ∇ d f k,m (˜ r m , ˜ d ) + χ k,m ( L ( n ) ) ∇ d f k,m (˜ r m , ˜ d ) (cid:17) , and C m ( L ( n ) , ˜ r m , ˜ d ) = (cid:80) Kk =1 ( λ k,m ( L ( n ) ) f k,m (˜ r m , ˜ d )+ χ k,m ( L ( n ) ) f k,m (˜ r m , ˜ d )) − A m ( L ( n ) , ˜ r m , ˜ d )˜ r m − B m ( L ( n ) , ˜ r m , ˜ d ) ˜ d + (cid:80) Kk =1 η k Q k,m .
3) Simplification of Constraints for RIS:
Similarly, the constraint (50c) for the RIS is simpli-fied in the following. We first define function h k ( d ) as h k ( d ) = ρ k d exp ( − K k d ) , (68)where ρ k = (cid:0) λ k π (cid:1) . By checking the Hessian matrix of h k ( d ) , it can be verified that h k ( d ) isconvex with respect to d . Also, by substituting the coordinate L ( n ) into (55), constant D k canbe obtained as D k (cid:44) D k ( L ( n ) ) .However, by employing the function h k ( d ) in (68), and substituting D k into (50b), thefollowing constraint is still non-convex: K (cid:88) k =1 h k ( d ) D k ≥ P I . (69)Therefore, the first-order Taylor expansion of h k ( d ) is utilized for convex approximation,which is h k ( d ) ≥ h k ( ˜ d ) + ∇ h k d ( ˜ d )( d − ˜ d ) , (70)where ∇ h k d ( ˜ d ) represents the first-order derivative of h k ( d ) given by ∇ h k d ( d ) = − ρ k K k d + 2 d exp ( − K k d ) . Then, substituting (70) into (50c), we have A RIS ( L ( n ) , ˜ d ) d + B RIS ( L ( n ) , ˜ d ) ≥ P I , (71)where the constants are evaluated as A RIS ( L ( n ) , ˜ d ) = K (cid:88) k =1 D k ( L ( n ) ) ∇ h k d ( ˜ d ) , (72) B RIS ( L ( n ) , ˜ d ) = K (cid:88) k =1 D k ( L ( n ) )( h k ( ˜ d ) − ˜ d ∇ h k d ( ˜ d )) . (73)
4) Simplification of Problem (50):
Finally, given RIS’s coordinate obtained at the ( n ) -thiteration denoted as L ( n ) , by replacing O tot ( L ) with O L ( n ) ( r i , d ) in (63), replacing (50b) and(50c) with (67) and (71) respectively, Problem (50) is reformulated asmin L ,r u ,d O L ( n ) ( r i , d ) (74a)s.t. | L − s u | ≤ r u , u ∈ I ∪ M (74b) | L − s AP | ≤ d . (74c) (67) , (71) . Constraints (74b) and (74c) are introduced for the auxiliary variables r u and d . Then, it can beverified that Problem (74) is convex, which can be readily solved by CVX.However, note that Problem (74) is the simplified version of Problem (50). The optimalsolution to Problem (74) denoted by ( L ∗ , r ∗ u , d ∗ ) may not satisfy all the constraints of Problem(50). Consequently, we need to add the following procedure to ensure that the obtained solution ( L ∗ , r ∗ u , d ∗ ) is a feasible solution to Problem (50).
5) Feasibility guarantee:
First we define the following penalty indicators as α = A RIS ( L ∗ , d ∗ ) d ∗ + B RIS ( L ∗ , d ∗ ) − P I , (75) α m = A m ( L ∗ , r ∗ m , d ∗ ) r ∗ m + B m ( L ∗ , r ∗ m , d ∗ ) d ∗ + C m ( L ∗ , r ∗ m , d ∗ ) − P Um . (76)If α < , the obtained coordinate L ∗ does not satisfy constraint (50c). This implies thatthe distance between RIS and AP is too large, i.e., the power harvested by the RIS does notexceed the requirement. By checking the first-order derivative of h k ( d ) with respect to d , itis verified that ∇ h k d ( d ) < , ∀ d ≥ . According to (72), as D k ( L ∗ ) ≥ for all coordinates, sothat A RIS ( L ∗ , d ∗ ) < . Consequently, to ensure that (50c) is satisfied, the RIS should be closerto the AP, i.e., d should be reduced. To this end, the required harvested power of RIS, i.e., P I is modified as P I (cid:48) = P I + (cid:15), if α < , (77)where (cid:15) > is the introduced penalty. Also, if α m < , the obtained coordinate L ∗ does not satisfy constraint (50b) for EU m . Notethat ∇ x f k,m ( x, y ) < , ∇ y f k,m ( x, y ) < and λ k,m ( L ) ≥ , ∀ L . Similarly, according to (67), thisimplies that d or r m should be decreased to satisfy constraint (50b). Given that d is modifiedby adjusting P I , therefore, we only adjust the P Um to reduce d m , which should be modified as P Um (cid:48) = P Um + (cid:15) m , if α m < , (78)where (cid:15) m > is the penalty for EU m .Overall, constraints (71) and (67) are updated by replacing P I and P Um with P I (cid:48) and P Um (cid:48) in(77) and (78), respectively. The solution to Problem (74) with the modified constraints (71) and(67) should be updated accordingly. Then, the finally obtained solution ( L ∗ , r ∗ u , d ∗ ) is guaranteedto satisfy all the constraints of Problem (50) when α ≥ and α m ≥ , ∀ m . In practise, penalties are set as of the required powers for satisfactory performance, i.e., (cid:15) = 1% P I and (cid:15) m = 1% P Um . Remark
1: Once the obtained ( L ∗ , r ∗ u , d ∗ ) is a feasible solution to the original Problem (50). Wedenote it as the ( n + 1) -th coordinate of the RIS, i.e., L ( n +1) = L ∗ when α ≥ and α m ≥ , ∀ m .Then, Problem (74) can be formulated based on L ( n +1) , and the feasible coordinate of RIS for the ( n + 2) -th iteration can be obtained by leveraging the penalties for power harvesting constraints. Remark
2: In addition, as the objective function O L ( n ) ( r i , d ) is the approximation of (50a),we need to check the original objective value O tot ( L ( n +1) ) in (50a) at each iteration. Note thatthe receiving matrix U nk,i aims to minimize the MSE for a given channel Z ( L ( n ) ) , and weight W ( n ) k,i is dependent on the current MMSE matrix. As the channel is optimized by adopting theRIS’s new coordinate L ( n +1) , the receiving matrix U k,i matched to this new channel and theobtained MMSE should be updated according to (32) and (35), respectively. That is to say, theobjective value at the ( n + 1) -th iteration is evaluated as O tot ( L ( n +1) , W n +1 k,i , U n +1 k,i ) . Remark
3: Denote the initial coordinate of RIS as L (0) . To guarantee the monotonicity ofRIS’s position optimization, we need to find the optimized coordinate L ( n (cid:48) ) which can reducethe objective O tot compared with the initial coordinate L (0) . That is to say , O tot ( L ( n (cid:48) ) , W n (cid:48) k,i , U n (cid:48) k,i ) < O tot ( L (0) , W k,i , U k,i ) . (79)Although the obtained objective in ( n + 1) -th iteration may be larger than that of the ( n ) -thiteration, we still update Problem (74) by utilizing L ( n +1) for the next feasible solution whenthe coordinate L ( n (cid:48) ) satisfying (79) have not be found. Remark
4: Note that Problem (74) may become infeasible due to the penalties added inconstraints (71) and (67). Consequently, it may exist the case that the coordinate L ( n (cid:48) ) satisfying(79) cannot be found. In this case, the RIS should keep the initial coordinate L (0) , i.e., L ( n (cid:48) ) = L (0) .Overall, the above analysis can be summarized as the following Penalty Constrained ConvexApproximation (PCCA) Algorithm 1 to optimize the RIS’s coordinate. D. BCD Algorithm to Solve Problem (34)
Based on the above analysis, a BCD based alternating optimization algorithm is proposed foralternately optimizing the precoding matrices of the AP, the phase shifts of the RIS and thecoordinate of RIS. The detailed algorithm is presented as in Algorithm 2. As O tot consists of two parts with different physical meanings: weighted MSE and rate, their numerical values may have morethan 3 orders of magnitude difference. In this case, it is better to check these two parts separately for satisfactory performance. Algorithm 1
Penalty Constrained Convex Approximation Algorithm (PCCA) Initialize coordinate L (0) , and the objective O tot ( L (0) , W k,i , U k,i ) ; Initialize iterative number n = 0 and maximum number of iterations N max ; repeat Obtain ( L ∗ , r ∗ u , d ∗ ) by solving Problem (74); if α ≥ and α m ≥ then Update L n +1 = L ∗ ; Calculate W n +1 k,i , U n +1 k,i according to (32) and (35), respectively. Calculate O tot ( L n +1 , W n +1 k,i , U n +1 k,i ) ; if O tot ( L n +1 , W n +1 k,i , U n +1 k,i ) < O tot ( L (0) , W k,i , U k,i ) then L ( n (cid:48) ) = L n +1 ; end if else Update P Um for all m, α m < according to (76) ; Update P I according to (75) if α < ; if Problem (74) is not feasible then
Set n = N max and L ( n (cid:48) ) = L ; end if end if until n = N max ; Output: L ( n (cid:48) ) , U n (cid:48) k,i , W n (cid:48) k,i ; Algorithm 2
The BCD Algorithm to Solve Problem (34) Initialize feasible L , ϕ , and F k,i . Initialize U (0) k,i and W (0) k,i according to (32) and (35), respectively. Initialize maximum number of iterations S max and the iterative number s = 0 . repeat Calculate F ( s +1) k,i by solving the convex approximations of Problem (36); Calculate ϕ ( s +1) by solving the convex approximations of Problem (48); Calculate { L ( s +1) , U ( s +1) k,i , W ( s +1) k,i } according to the PCCA Algorithm; until s = S max . The proposed BCD algorithm is guaranteed to converge. Specifically, the generated solutions { L ( s ) , r ( s ) u , d ( s )0 } by the PCCA algorithm are always feasible to the coordinate optimizationproblem (50) by updating P Um and P I . Furthermore, according to the steps 15-17 and steps 9-11of the PCCA algorithm, the obtained objective value { O tot ( L s , W sk,i , U sk,i ) } is monotonicallydecreasing. Then, according to Section III-A and Section III-B, Problem (36) and Problem (48)are reformulated based on the Taylor-expansion based convex optimization. It can be readilyverified that the sequence of solutions generated by the BCD Algorithm is always feasible forProblem (34). The monotonic property of the BCD Algorithm can be similarly proved by usingthe method in [53]. IV. S IMULATION R ESULTS
Simulation results are presented in this section to evaluate the performance of the proposedalgorithm. In the simulation, THz frequency range at the AP is - GHz, the bandwidthof each sub-band is GHz, and the molecular absorption coefficients are generated accordingto [44]. The AP is located along the Y-axis with height of m. The AP’s transmit antenna ismodelled as a uniform planar array (UPA) with size of N t = 5 × , and P maxt = 10 W. TheIUs and EUs are both equipped with receive antennas. The antenna gain is set as G t = 15 and G r = 6 . The separations between the transmit/receive antennas are set to be . mm.As shown in Fig. 3, there are IUs and EUs which are randomly distributed in a squarearea with width m. The RIS is installed on the X-axis, and the separations between the RISreflecting elements are . mm. In Fig. 3, the number of reflecting elements is set to . Theinitial coordinate of the RIS in X-axis is marked by “*” and the optimized coordinate in X-axisis marked by “ × ”. It is observed that in the layout shown in Fig. 3, the optimized coordinateof the RIS is updated by the proposed BCD algorithm.The proposed BCD algorithm given in Algorithm 2 is labelled as “PropBCD”. For performancecomparison, we consider two benchmark schemes: • The first scheme which is labelled as “BeamOpt” only optimizes the transmit precodingmatrices with the fixed RIS’s phase shift and coordinate; This scheme can be obtained byremoving step 6 and step 7 of Algorithm 2. • The scheme labelled as “FixedLoc” optimizes both the transmit precoding matrices and theRIS’s phase shift, where the RIS’s coordinate is kept fixed. This scheme is obtained byremoving step 7 of Algorithm 2. X-axis Y - a x i s AP locationIU locationsEU locationInitial RIS locationOptimized RIS location
Fig. 3. The simulation scenario of the STIPT system.
Number of iterations S u m r a t e o f I U s : G bp s PropBCDFixedLocBeamOpt
Fig. 4. The convergence performance of different algorithms.
Fig. 4 shows the convergence performance of the proposed algorithm and the benchmarks,where its simulation scenario is shown in Fig. 3. In Fig. 4, the required harvest power by RISand the EU are set as . mW. It is observed that the achieved sum rate of IUs increases with thenumber of iterations for all considered cases, and all the considered schemes converge within 7iterations. As expected, the proposed BCD algorithm achieves the best performance. Significantrate improvement can be obtained by optimizing the RIS’s coordinate.Fig. 5 shows the achieved sum rates of IUs by different schemes versus the number of RISreflecting elements. The simulation results are averaged by 100 random realizations, where theinitial coordinates of the RIS, the IUs, and the EUs are randomly generated. As shown in Fig.
80 90 100 110 120 130
Number of RIS reflecting elements S u m r a t e o f I U s : G bp s PropBCDFixedLocBeamOpt
Fig. 5. The sum rate versus the number of reflecting elements.
Number of IUs S u m r a t e o f I U s : G bp s PropBCDFixedLocBeamOpt
Fig. 6. The sum rate versus the number of IUs.
5, the sum rates of IUs achieved by the proposed BCD algorithm and the “FixedLoc” algorithmincrease with the number of reflecting elements. However, the proposed BCD algorithm canachieve a higher sum rate, and the performance gap increases with the number of reflectingelements. This implies that the proposed BCD algorithm can fully exploit the potential benefitsprovided by the RIS, especially in this STIPT system. In addition, it is observed that the sum rateof the “BeamOpt” algorithm remains the same as the number of reflecting elements increasessince the RIS related parameters are not optimized in this scheme.Fig. 6 shows the achieved sum rates of the IUs by different schemas versus the numberof IUs in the system. It is interesting to see that the achieved sum rates by the “PropBCD” Number of EUs S u m r a t e o f I U s : G bp s PropBCDFixedLocBeamOpt
Fig. 7. The sum rate versus the harvested power of EUs. algorithm increases with the number of IUs. Meanwhile the sum rate obtained by the “BeamOpt”algorithm slightly increases, and that of the “FixedLoc” algorithm keeps fixed. This result clearlyvalidates the benefits provided by the RIS in the STIPT system. By utilizing the RIS, thetransmission channel can also be optimized to fully exploit the spatial diversity so that thesum rate performance can be enhanced. In particular, in the THz system, where the channelgain is very sensitive to the transmit distance, optimizing the RIS’s coordinate can help provideconsiderable performance gain.Fig. 7 shows the achieved sum rates of the IUs versus the number of EUs in the system, andthe number of IUs is fixed to be 2. Then, it is observed that the sum rate of the IUs decreaseswith the number of EUs for all cases, as more power needs to be harvested for the EUs. Inaddition, the performance gap between the “PropBCD” algorithm and the other benchmarksincreases with the number of EUs, which also shows the superiority of proposed algorithm.Fig. 8 shows the impact of the harvest power required by the EUs on the sum rate performance.In Fig. 8, the required power of the RIS is fixed to . mW, and the number of EUs is . Itis observed that the proposed BCD algorithm outperforms the other two algorithms. As shownin Fig. 8, the achieved sum rate of IUs by the BCD algorithm decreases with the requiredharvested power of EUs for all considered cases. In addition, the performance gap between the“PropBCD” algorithm and the other schemes keeps stable with the required power. This impliesthat the proposed algorithm still has stable performance advantages when the power harvestingdemand from EUs increases. Harested power of EUs: mW S u m r a t e o f I U s : G bp s PropBCDFixedLocBeamOpt
Fig. 8. The sum rate versus the harvested power of EUs.
Harested power of RISs: mW S u m r a t e o f I U s : G bp s Prop BCDFixedLocBeamOpt
Fig. 9. The sum rate versus the harvested power of RISs
Finally, Fig. 9 shows the impact of the harvest power required by the RIS on sum rateperformance. In Fig. 9, the required power of the EU is . mW, and the other simulationparameters are the same as those of Fig. 8. As expected, the proposed BCD algorithm outperformsthe other two algorithms, but the performance gap slightly decreases with the required powerto harvest. It becomes more difficult to find the new coordinate of RIS to improved IU’s ratewhile guaranteeing the more stringent harvested power constraints of RIS. Also, compared withthe impact of the EU’s harvested power shown in Fig. 8, RIS’s required harvest power has aslighter impact on rate performance. V. C
ONCLUSIONS
In this paper, we have investigated a new simultaneous THz information and power transfersystem, named as STIPT, where the RIS is utilized to support the THz transmission. In thissystem, the RIS can utilize the power harvesting technology to self-sustain its power consumption.The optimization problem has been formulated to maximize the IUs’ sum rate while guarantee-ing the EU’s and RIS’s power harvesting requirements. A BCD-based alternating optimizationalgorithm has been proposed to optimize the transmit precoding for IUs, the RIS’s reflectingcoefficients and the RIS’s coordinate. Simulation results have shown that the proposed algorithmcan achieve considerable performance gain in terms of the sum rate. With the assistance ofRIS, the transmission channel can be optimized to fully exploit the spatial diversity so that thesum rate performance can be enhanced. As the channel gain of the THz transmission is verysensitive to the transmission distance, optimizing the RIS’s coordinate in the STIPT system canhelp provide considerable performance gain.R
EFERENCES [1] X. You and et al., “Towards 6G wireless communication networks: vision, enabling technologies, and new paradigm shifts,”
Science China Information Sciences , vol. 64, no. 1, pp. 1–74, 2021.[2] M. T. Barros, R. Mullins, and S. Balasubramaniam, “Integrated terahertz communication with reflectors for 5G small-cellnetworks,”
IEEE Transactions on Vehicular Technology , vol. 66, no. 7, pp. 5647–5657, 2017.[3] R. Zhang and C. K. Ho, “MIMO broadcasting for simultaneous wireless information and power transfer,”
IEEE Transactionson Wireless Communications , vol. 12, no. 5, pp. 1989–2001, 2013.[4] Z. Zhang, H. Pang, A. Georgiadis, and C. Cecati, “Wireless power transfer—an overview,”
IEEE Transactions on IndustrialElectronics , vol. 66, no. 2, pp. 1044–1058, 2019.[5] N. Akhtar and Y. Perwej, “The internet of nano things (IoNT) existing state and future prospects,”
GSC AdvancedResearch and Reviews , vol. 5, no. 2, pp. 131–150, 2020. [Online]. Available: https://gsconlinepress.com/journals/gscarr/content/internet-nano-things-iont-existing-state-and-future-prospects[6] S. Mizojiri and K. Shimamura, “Wireless power transfer via subterahertz-wave,”
Applied Sciences
Micro and Nano Systems Letters , vol. 5, no. 1, pp. 1–16, 2017.[8] Z. Rong, M. S. Leeson, M. D. Higgins, and Y. Lu, “Simultaneous wireless information and power transfer for AF relayingnanonetworks in the terahertz band,”
Nano Communication Networks , vol. 14, pp. 1–8, 2017.[9] J. Tan and L. Dai, “THz precoding for 6G: Applications, challenges, solutions, and opportunities.” [Online]. Available:https://arxiv.org/pdf/2005.10752[10] S. Mizojiri, K. Shimamura, M. Fukunari, S. Minakawa, S. Yokota, Y. Yamaguchi, Y. Tatematsu, and T. Saito, “Subterahertzwireless power transmission using 303-GHz rectenna and 300-kW-class gyrotron,”
IEEE Microwave and WirelessComponents Letters , vol. 28, no. 9, pp. 834–836, 2018. [11] S. Mizojiri, K. Takagi, K. Shimamura, S. Yokota, M. Fukunari, Y. Tatematsu, and T. Saito, “Demonstration of sub-terahertzcoplanar rectenna using 265 GHz gyrotron,” in . Piscataway,NJ: IEEE, 2019.[12] Q. Wu, S. Zhang, B. Zheng, C. You, and R. Zhang, “Intelligent reflecting surface aided wireless communications: Atutorial.” [Online]. Available: http://arxiv.org/pdf/2007.02759v2[13] M. Di Renzo, A. Zappone, M. Debbah, M.-S. Alouini, C. Yuen, J. de Rosny, and S. Tretyakov, “Smart radio environmentsempowered by reconfigurable intelligent surfaces: How it works, state of research, and the road ahead,” IEEE Journal onSelected Areas in Communications , vol. 38, no. 11, pp. 2450–2525, 2020.[14] E. Basar, M. Di Renzo, J. de Rosny, M. Debbah, M.-S. Alouini, and R. Zhang, “Wireless communications throughreconfigurable intelligent surfaces,”
IEEE Access , vol. 7, pp. 116 753–116 773, 2019.[15] S. Gong, X. Lu, D. T. Hoang, D. Niyato, L. Shu, D. in Kim, and Y.-C. Liang, “Toward smart wireless communicationsvia intelligent reflecting surfaces: A contemporary survey,”
IEEE Communications Surveys & Tutorials , vol. 22, no. 4, pp.2283–2314, 2020.[16] C. Pan, H. Ren, K. Wang, J. F. Kolb, M. Elkashlan, M. Chen, M. Di Renzo, Y. Hao, J. Wang, A. L. Swindlehurst,X. You, and L. Hanzo, “Reconfigurable intelligent surfaces for 6G and beyond: Principles, applications, and researchdirections.” [Online]. Available: https://arxiv.org/pdf/2011.04300[17] C. Pan, H. Ren, K. Wang, W. Xu, M. Elkashlan, A. Nallanathan, and L. Hanzo, “Multicell MIMO communications relyingon intelligent reflecting surfaces,”
IEEE Transactions on Wireless Communications , vol. 19, no. 8, pp. 5218–5233, 2020.[18] T. Bai, C. Pan, Y. Deng, M. Elkashlan, A. Nallanathan, and L. Hanzo, “Latency minimization for intelligent reflectingsurface aided mobile edge computing,”
IEEE Journal on Selected Areas in Communications , vol. 38, no. 11, pp. 2666–2682,2020.[19] M. Cui, G. Zhang, and R. Zhang, “Secure wireless communication via intelligent reflecting surface,”
IEEE WirelessCommunications Letters , vol. 8, no. 5, pp. 1410–1414, 2019.[20] L. Dong and H.-M. Wang, “Secure MIMO transmission via intelligent reflecting surface,”
IEEE Wireless CommunicationsLetters , vol. 9, no. 6, pp. 787–790, 2020.[21] K. Feng, X. Li, Y. Han, S. Jin, and Y. Chen, “Physical layer security enhancement exploiting intelligent reflecting surface,”
IEEE Communications Letters , p. 1, 2020.[22] S. Hong, C. Pan, H. Ren, K. Wang, and A. Nallanathan, “Artificial-noise-aided secure MIMO wireless communicationsvia intelligent reflecting surface,”
IEEE Transactions on Communications , vol. 68, no. 12, pp. 7851–7866, 2020.[23] X. Yu, D. Xu, and R. Schober, “Enabling secure wireless communications via intelligent reflecting surfaces,” in . IEEE, 122019, pp. 1–6.[24] Z. Chu, W. Hao, P. Xiao, and J. Shi, “Intelligent reflecting surface aided multi-antenna secure transmission,”
IEEE WirelessCommunications Letters , vol. 9, no. 1, pp. 108–112, 2020.[25] Y. Yang, B. Zheng, S. Zhang, and R. Zhang, “Intelligent reflecting surface meets OFDM: Protocol design and ratemaximization,”
IEEE Transactions on Communications , vol. 68, no. 7, pp. 4522–4535, 2020.[26] Z. Feng, B. Clerckx, and Y. Zhao, “Waveform and beamforming design for intelligent reflecting surface aided wirelesspower transfer: Single-user and multi-user solutions.” [Online]. Available: https://arxiv.org/pdf/2101.02674[27] C. Pan, H. Ren, K. Wang, M. Elkashlan, A. Nallanathan, J. Wang, and L. Hanzo, “Intelligent reflecting surface aidedMIMO broadcasting for simultaneous wireless information and power transfer,”
IEEE Journal on Selected Areas inCommunications , vol. 38, no. 8, pp. 1719–1734, 2020.[28] N. Hehao and L. Ni, “Intelligent reflect surface aided secure transmission in MIMO channel with SWIPT,”
IEEE Access ,vol. 8, pp. 192 132–192 140, 2020. [29] J. Liu, K. Xiong, Y. Lu, D. W. K. Ng, Z. Zhong, and Z. Han, “Energy efficiency in secure IRS-aided SWIPT,” IEEEWireless Communications Letters , vol. 9, no. 11, pp. 1884–1888, 2020.[30] Z. Li, W. Chen, and Q. Wu, “Joint beamforming design and power splitting optimization in IRS-assisted SWIPT NOMAnetworks.” [Online]. Available: https://arxiv.org/pdf/2011.14778[31] Q. Wu and R. Zhang, “Joint active and passive beamforming optimization for intelligent reflecting surface assisted SWIPTunder qos constraints,”
IEEE Journal on Selected Areas in Communications , vol. 38, no. 8, pp. 1735–1748, 2020.[32] T. Bai, C. Pan, H. Ren, Y. Deng, M. Elkashlan, and A. Nallanathan, “Resource allocation for intelligentreflecting surface aided wireless powered mobile edge computing in OFDM systems,” 2020. [Online]. Available:https://arxiv.org/pdf/2003.05511[33] J. M. Jornet and I. F. Akyildiz, “Channel modeling and capacity analysis for electromagnetic wireless nanonetworks inthe terahertz band,”
IEEE Transactions on Wireless Communications , vol. 10, no. 10, pp. 3211–3221, 2011.[34] M. Pengnoo, M. T. Barros, L. Wuttisittikulkij, B. Butler, A. Davy, and S. Balasubramaniam, “Digital twin for metasurfacereflector management in 6G terahertz communications,”
IEEE Access , vol. 8, p. 1, 2020.[35] X. Ma, Z. Chen, W. Chen, Y. Chi, L. Yan, C. Han, and S. Li, “Joint hardware design and capacity analysis for intelligentreflecting surface enabled terahertz MIMO communications.” [Online]. Available: https://arxiv.org/pdf/2012.06993[36] B. Ning, Z. Chen, W. Chen, and Y. Du, “Channel estimation and transmission for intelligent reflecting surface assistedTHz communications.” [Online]. Available: https://arxiv.org/pdf/1911.04719[37] X. Ma, Z. Chen, Y. Chi, W. Chen, L. Du, and Z. Li, “Channel estimation for intelligent reflecting surface enabled terahertzMIMO systems,” in . IEEE, 2020/6/7- 2020/6/11, pp. 1–6.[38] A.-A. A. Boulogeorgos and A. Alexiou, “Coverage analysis of reconfigurable intelligent surface assisted THz wirelesssystems,”
IEEE Open Journal of Vehicular Technology , p. 1, 2021.[39] Y. Pan, K. Wang, C. Pan, H. Zhu, and J. Wang, “Sum rate maximization for intelligent reflecting surface assistedterahertz communications.” [Online]. Available: https://arxiv.org/pdf/2008.12246[40] B. Ning, Z. Chen, W. Chen, and L. Li, “Improving security of THz communication with intelligent reflecting surface,” in . IEEE, 122019, pp. 1–6.[41] W. Hao, G. Sun, M. Zeng, Z. Zhu, Z. Chu, O. A. Dobre, and P. Xiao, “Robust design for intelligent reflecting surfaceassisted MIMO-OFDMA terahertz communications.” [Online]. Available: https://arxiv.org/pdf/2009.05893[42] Y. Pan, K. Wang, C. Pan, H. Zhu, and J. Wang, “UAV-assisted and intelligent reflecting surfaces-supported terahertzcommunications.” [Online]. Available: http://arxiv.org/pdf/2010.14223v1[43] C. Han and I. F. Akyildiz, “Distance-aware bandwidth-adaptive resource allocation for wireless systems in the terahertzband,”
IEEE Transactions on Terahertz Science and Technology , vol. 6, no. 4, pp. 541–553, 2016.[44] A.-A. A. Boulogeorgos, E. N. Papasotiriou, and A. Alexiou, “A distance and bandwidth dependent adaptive modulationscheme for THz communications,” in
IEEE 19th International Workshop on Signal Processing Advances in WirelessCommunications . IEEE, 2018, pp. 1–5.[45] C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah, and C. Yuen, “Reconfigurable intelligent surfaces for energyefficiency in wireless communication,”
IEEE Transactions on Wireless Communications , vol. 18, no. 8, pp. 4157–4170,2019.[46] B. Lyu, P. Ramezani, D. T. Hoang, S. Gong, Z. Yang, and A. Jamalipour, “Optimized energy and information relaying inself-sustainable IRS-empowered WPCN,”
IEEE Transactions on Communications , vol. 69, no. 1, pp. 619–633, 2021.[47] S. Hu, Z. Wei, Y. Cai, C. Liu, D. W. K. Ng, and J. Yuan, “Robust and secure sum-rate maximization for multiuser misodownlink systems with self-sustainable irs.” [Online]. Available: http://arxiv.org/pdf/2101.10549v1 [48] S. Priebe, M. Kannicht, M. Jacob, and T. Kurner, “Ultra broadband indoor channel measurements and calibrated ray tracingpropagation modeling at THz frequencies,” Journal of Communications and Networks , vol. 15, no. 6, pp. 547–558, 2013.[49] J. Du, F. R. Yu, G. Lu, J. Wang, J. Jiang, and X. Chu, “MEC-assisted immersive vr video streaming over terahertz wirelessnetworks: A deep reinforcement learning approach,”
IEEE Internet of Things Journal , p. 1, 2020.[50] C. Han, A. O. Bicen, and I. F. Akyildiz, “Multi-ray channel modeling and wideband characterization for wirelesscommunications in the terahertz band,”
IEEE Transactions on Wireless Communications , vol. 14, no. 5, pp. 2402–2412,2015.[51] W. Tang and et al., “Wireless communications with reconfigurable intelligent surface: Path loss modeling and experimentalmeasurement.” [Online]. Available: https://arxiv.org/pdf/1911.05326[52] D. P. Palomar, J. M. Cioffi, and M. A. Lagunas, “Joint tx-rx beamforming design for multicarrier MIMO channels: a unifiedframework for convex optimization,”
IEEE Transactions on Signal Processing , vol. 51, no. 9, pp. 2381–2401, 2003.[53] C. Pan, H. Zhu, N. J. Gomes, and J. Wang, “Joint precoding and RRH selection for user-centric green MIMO C-RAN,”