diff --git "a/2tAzT4oBgHgl3EQf9P4T/content/tmp_files/load_file.txt" "b/2tAzT4oBgHgl3EQf9P4T/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/2tAzT4oBgHgl3EQf9P4T/content/tmp_files/load_file.txt" @@ -0,0 +1,652 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf,len=651 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='01915v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='IT] 5 Jan 2023 CHINA COMMUNICATIONS Sum-Rate Maximization in Active RIS-Assisted Multi-Antenna WPCN Jie Jiang,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Bin Lyu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Pengcheng Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' and Zhen Yang School of Communications and Information Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Nanjing University of Posts and Telecommunications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Nanjing 210003,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' China Abstract: In this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' we propose an active re- configurable intelligent surface (RIS) enabled hybrid relaying scheme for a multi-antenna wireless pow- ered communication network (WPCN),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' where the ac- tive RIS is employed to assist both wireless energy transfer (WET) from the power station (PS) to energy- constrained users and wireless information transmis- sion (WIT) from users to the receiving station (RS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' For further performance enhancement, we propose to employ both transmit beamforming at the PS and re- ceive beamforming at the RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' We formulate a sum- rate maximization problem by jointly optimizing the RIS phase shifts and amplitude reflection coefficients for both the WET and the WIT, transmit and receive beamforming vectors, and network resource alloca- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' To solve this non-convex problem, we propose an efficient alternating optimization algorithm with linear minimum mean squared error criterion, semi-definite relaxation (SDR) and successive convex approxima- tion techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Specifically, the tightness of apply- ing the SDR is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Simulation results demon- strate that our proposed scheme with 10 reflecting el- ements (REs) and 4 antennas can achieve 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='78% and 415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='48% performance gains compared to the single- antenna scheme with 10 REs and passive RIS scheme with 100 REs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Keywords: Wireless powered communication net- work, active reconfigurable intelligent surface, beam- forming, sum-rate maximization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Received: XX Revised: XX Editor: XX I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' INTRODUCTION With the development of the Internet-of-Things (IoT), an intelligent society with ubiquitous interconnections and deep coverage will be truly realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' However, wireless devices (WDs) in IoT networks are generally energy-constrained and suffer from limited lifetime [1], which fundamentally limits the performance of communication networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Traditional ways of chang- ing or recharging the batteries manually are impossi- ble and unacceptable, especially when the number of WDs is numerous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Therefore, how to tackle this issue is a critical problem in the widespread development of IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Wireless powered communication has been pro- posed as a prospective technology for enhancing the energy sustainability of WDs, which can be classified into two directions based on application scenarios [2– 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The first one focuses on investigating simultane- ous wireless information and power transfer (SWIPT), where the base station (BS) simultaneously transfers energy and information signals to energy receivers and information receivers via the common radio fre- quency (RF) signals in the downlink (DL), resulting in a pivotal tradeoff between the achievable rate and harvested energy [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In contrast to the SWIPT, wire- less powered communication network (WPCN) has been proposed as a novel type of wireless network di- agram to improve the lifetime of WDs and enhance the deployment flexibility of IoT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In a WPCN, energy- constrained WDs first harvest energy in the DL and then use the harvested energy to transmit independent information in the uplink (UL) based on the widely used harvest-then-transmit (HTT) protocol [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' WPCN has been widely investigated in the literature China Communications 1 [6–9], which promotes the development of WPCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' However, WPCN generally suffers from the “doubly near-far” phenomenon if the power station (PS) and the receiving station (RS) are co-located at the hybrid access point (HAP) [6, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Specifically, a WD located far away from the HAP harvesting less energy in the DL has to transmit information with more power in the UL, which results in an unfair time and resource allocation among the WDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' To deal with the issue, a promising way is to deploy the PS and RS separately [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In [10], multiple users harvest energy from a dedicated PS and then communicate with an informa- tion RS following the HTT protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In this scenario, a user physically close to the PS is naturally far away from the RS, and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Considering a similar scenario, a user-centric energy-efficient (EE) problem in WPCN is investigated in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' However, the perfor- mance of WPCN is still limited due to the low efficien- cies caused by the severe path-loss, which seriously affects its practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Recently, reconfigurable intelligent surface (RIS), with the unprecedented ability to reshape the wireless transmission environment, has drawn widespread at- tentions from academia and industry [12–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' RIS is comprised of a large number of programmable reflect- ing elements (REs), which can alter the phase shifts and amplitudes of incident signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' As such, RIS can adaptively modify the impinging radio waves towards the appropriate direction [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' According to the re- flection patterns, RIS can be classified as passive RIS and active RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Without the property of power amplifi- cation, the independent diffusive scatterer-based (IDS) model accounts for the basic properties of passive RIS, which has been widely adopted in RIS-assisted wire- less communications [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The passive RIS is only equipped with the phase-shift controller, while the ac- tive RIS contains both the phase-shift controller and the active reflection-type amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Hence, the active RIS can alter both the phase shifts and amplitudes of the incident signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' It is worth noting that the active RIS with its novel hardware structure and signal model has been proposed in [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Different from the full-duplex amplify-and-forward (FD-AF) relay that requires power-consuming RF chains, the active RIS can directly reflect and amplify the incident signals in the EM level in a FD manner without reception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In this way, the active RIS exhibits promising qualities, such as a low power consumption, light weight, conformal geometry and high flexibility for practical deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Currently, the passive RIS has been widely applied in WPCNs for performance enhancement [21–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In [21], the passive RIS is employed between the HAP and users to improve both DL WET and UL WIT effi- ciencies in single-input-single-output (SISO) WPCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' To achieve further performance improvement, the multi-antenna technique is employed in [22], where the HAP with multi-antenna transmits energy signals to users in the DL and receives information from users in the UL by employing transmit bemforming and re- ceive beamforming, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In [23], the fully dynamic RIS beamforming scheme is proposed for WPCN, for which the phase shift vectors are indepen- dently designed over different time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' However, in the above works [21–23], the PS and RS are also co-located at the HAP, which results in performance unfair among users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' To address this issue, the au- thors in [24] consider the scenario where the PS and RS are separately deployed, for which the locations of RIS and users can be carefully considered to achieve a fair performance among users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' However, as mentioned above, the passive RIS can only reflect incident signals without amplification, which leads to the limited per- formance enhancement due to the double-fading ef- fect suffered by the reflecting links.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Thus, the energy- constrained users still need to consume much time for harvesting energy in the DL and have less time for in- formation transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Inspired by the amplification characteristic of the active RIS, the active RIS is confirmed to be superior to the passive RIS in terms of performance enhance- ment, and thus has been considered a promising tech- nique for IoT networks [19, 20, 25–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The authors in [19] compare the capacity improvement achieved by the active RIS to the passive RIS, which demon- strats that the active RIS can fundamentally mitigate the double-fading effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In [20], an active RIS is ap- plied in single input multiple output (SIMO) systems, for which the joint optimization of phase shifts ma- trix and receive beamforming is considered to obtain the maximum achievable rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In [25], the placement of the active RIS is optimized to enhance SISO sys- tems’ performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The authors in [26] propose to use the active RIS to achieve secure transmission, which not only establishes the reliable link from the trans- mitter to the receiver but also prevents the confidential information intercepted by the eavesdropper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The ac- 2 China Communications tive RIS-aided multiuser MISO PS-SWIPT is studied in [27] to minimize the base station transmit power, which shows significant improvements compared to the passive RIS-aided system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Similarly, in [28], an active RIS is employed to assist SWIPT to boost the efficiency of both WET and WIT, while the conclu- sions and approaches are inapplicable to the WPCN system because of thire different system models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Although the active RIS has received a lot of inter- ests for wireless communication networks, the appli- cations of active RIS in WPCN is still at the very early stage and has not been well studied in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' To the best of our knowledge, there exists only one paper investigating the usage of active RIS in WPCN [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Specifically, the authors in [29] investigate the weight sum-rate maximization problem in the active RIS assisted single-antenna WPCN, where the PS and RS is co-located at the HAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' As a result, similar to [21–23], a part of WDs in [29] still suffer from the “doubly near-far” phenomenon, which is not suitable for practical applications with high requirement of per- formance fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Moreover, the authors consider a certain simplified communication scenario where both the HAP and the WDs have single antenna each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The transmit beamforming and receive beamforming can- not be exploited, which is also a key technology for performance enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Motivated by the observations above, we propose an active RIS enabled hybrid relaying scheme for the multi-antenna WPCN, where the active RIS is em- ployed to facilitate both the WET from the PS to energy-constrained users and the WIT from users to the RS, which is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Compared with the existing works which used the passive RIS in WPCN [21–24], our proposed active RIS scheme can amplify the energy signals and information signals to achieve a satisfying system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Different from the single-antenna scenario considered in [29], we propose to employ multi-antenna at both the PS and RS, which can construct the transmit beamform- ing and receive beamforming for further performance enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Specifically, the transmit beamforming at the PS can be used to enhance the WPT efficiency from the PS to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In the meanwhile, the receive beamforming at the RS can be used to exploit the an- tenna gain and eliminate the noise caused by the ac- tive RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In the considered system setup, we aim to maximize the sum-rate problem by jointly optimiz- ing the transmit beamforming at the PS, the receive beamforming at the RS, the reflecting coefficients at the RIS, and network resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' It should be noted that compared to [21–24, 29], our formulated problem is much more challenging to solve and the proposed algorithms in [21–24, 29] cannot be used to solve our formulated problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Thus, we propose an efficient algorithm to solve the formulated problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The main contributions of this paper are summarized as follows: We propose an active RIS assisted multi-antenna WPCN for performance enhancement, where the active RIS is served as a hybrid relay to achieve two purposes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=', the first one is to assist the WET from the PS to users, and the second one is to aid the WIT from users to the RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' To fur- ther improve system performance, both trans- mit beamforming and receive beamforming tech- niques are respectively considered at the PS and RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' We investigate the sum-rate maximization prob- lem by jointly optimizing transmit beamforming vector at the PS, receive beamforming vectors at the RS, phase shifts and amplitude reflection coef- ficients at the RIS for both the WET and the WIT, and network resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' To deal with the non-convexity of the formulated problem, we pro- pose an efficient alternating optimization (AO) al- gorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Specifically, the original problem can be divided into four sub-problems, which are solved sequentially in an alternating manner until con- vergence is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' For designing the receive beamforming, we apply the linear minimum mean squared error (MMSE) criterion and obtain the closed-form expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' For the optimization of the transmit beamform- ing and RIS reflecting coefficient matrices for the WET, the semidefinite relaxation (SDR) tech- nique is adopted and the tightness of applying the SDR is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' For the optimization of RIS re- flecting coefficients for the WIT, we obtain the optimal phase shifts in a closed-form and propose a successive convex approximation (SCA) algo- rithm to determine the optimal amplitude reflec- tion coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In addition, the convergence of proposed problem is analyzed and confirmed via numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' China Communications 3 Finally, numerical results are provided to evaluate the performance of proposed scheme, which indi- cates that compared to the single-antenna scheme with 10 REs and the passive-RIS scheme with 100 REs, the proposed scheme with 4 antennas and 10 REs can achieve 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='78% and 415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='48% sum-rate gain, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Sec- tion II describes the system model of the active RIS- assisted multi-antenna WPCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The sum-rate maxi- mization problem is formulated in Section III and solved in Section IV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In Section V, per- formance is evaluated by numerical results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Finally, this paper is concluded in Section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Notations: In this paper, vectors and matrices are denoted by boldface lowercase and uppercase letters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The operators ( · )T, ( · )H, | · | and ∥ · ∥ denote the transpose, conjugate transpose, absolute value and the Euclidean norm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Tr( · ) and rank( · ) denote the trace and rank of a matrix, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' X ⪰ 0 represents that X is a positive semidef- inite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' E[ · ] stands for the statistical expecta- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' IN denotes the N-dimensional identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 0 denotes the zero matrix/vector with appropriate size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' CN×M denotes the set of all N × M complex-valued matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' HM denotes the set of all M ×M Hermitian matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' RN×1 represents the set of all N × 1 real- valued vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' CN � µ, σ2� denotes the distribution of a circularly symmetric complex Gaussian random vec- tor with mean µ and variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' arg( · ) denotes the phase extraction operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' diag(x) denotes a diago- nal matrix whose diagonal elements are from vector x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Im( · ) and Re( · ) respectively denotes the imaginary part and real part of a complex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' SYSTEM MODEL As shown in Figure 1, we consider an active RIS-aided multi-antenna WPCN, which consists of a PS with M antennas, a RS with L antennas, an active RIS, and K energy-constrained users each with single antenna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' We assume that each user is equipped with an energy harvesting (EH) circuit for harvesting energy, where a rectifier is used to convert the received RF signals to direct current (DC) signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Then, the net energy harvested from the DC signals can be stored in the rechargeable battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The active RIS consists of N active REs, which can steer the reflected signals in ��,� � ��,� � ��,� ��,� �� �� PS RS �� Active RIS Energy transfer Information transmission EH circuit Rectifier RF signal Rechargeable battery DC signal Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' System model for an active RIS-assisted multi- antenna WPCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' a specific direction and also amplify them by the ac- tive loads (negative resistance) [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In contrast to the passive RE, each active RE is equipped with an addi- tionally integrated active reflection-type amplifier sup- ported by a power supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' By appropriately setting the effective resistance, it is reasonable to assume that the reflection amplitude and phase of each element is inde- pendently [19, 20, 25–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' To power the operations of active RIS and users, the PS is equipped with a stable energy source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In addition, the PS has the capability for performing computational tasks [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In particu- lar, the users first harvest energy from the RF signals transmitted by the PS and then use the harvested en- ergy to deliver information to the RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' To allievate the severe path-loss suffered by the reflecting links, the ac- tive RIS is employed to improve the WET efficiency from the PS to users and the WIT efficiency from the users to the RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The channels are assumed to follow a quasi-static flat-fading model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' That is, all channel coefficients are constant throughout each transmission block but vary from block to block [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The downlink base- band equivalent channels of PS-to-RIS, RIS-to-Uk, and PS-to-Uk links are denoted by Hr ∈ CN×M, hH u,k ∈ C1×N, and hH d,k ∈ C1×M, respectively, where Uk denotes the k-th user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Similarly, the uplink base- band equivalent channels of Uk-to-RIS, RIS-to-RS, and Uk-to-RS links are respectively denoted by gu,k ∈ CN×1, Gr ∈ CL×N, and gd,k ∈ CL×1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Since there have been many efficient channel estima- 4 China Communications 用tion techniques proposed for RIS systems [31–34], we assume the perfect channel state information can be available in advance, which is a common assumption considered in [21, 22, 35] and a prerequisite for inves- tigating the upper-bound of system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' It should be noted that the channel estimation error is generally inevitable [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' However, the effect of chan- nel estimation error on system performance degrada- tion is out the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' According to the HTT protocol [6], the normalized transmission block of interest is divided into K + 1 time slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The first time slot with duration of τ0 ∈ [0, 1] is a dedicated slot for WET, in which all users harvest energy from the PS with the assistance of active RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The remaining K time slots denoted by τ = [τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' , τK], are used for UL WIT via the time di- vision multiple access (TDMA) scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Specifically, during τk, k = 1, · · · , K, Uk delivers its information to the RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Without loss of generality, the whole oper- ation time period is set to a normalized transmission block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The network time scheduling constraint is thus given by τ0 + K � k=1 τk ≤ 1, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (1) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='1 Wireless Energy Transfer Phase In the WET phase, the PS transmits energy signals to all users with the assistance of the active RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Denote the transmitted signal as s = w0s, where s is the pseudo-random baseband signal transmitted by the PS, and w0 ∈ CM×1 is the transmit beamforming vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The energy constraint at the PS is expressed as E � |s|2� = Tr � w0wH 0 � ≤ P0, (2) where P0 denotes the maximum transmit power at the PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The reflecting coefficient matrix of the active RIS in the WET phase is denoted by Φ0 = diag {φ0,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' , φ0,N} ∈ CN×N with φ0,n = a0,nejθ0,n, n = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' , N, where a0,n and θ0,n rep- resent the amplitude reflection coefficient and phase shift of the n-th RE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Without loss of gen- erality, we suppose each active RE has the following constraints a0,n ≤ an,max, 0 ≤ θ0,n ≤ 2π, ∀n, (3) where an,max is the maximum amplitude reflection co- efficient of n-th RE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' It is worth noting that an,max can be greater than 1 [25], which is a main characteris- tic distinguishing the active RIS from the passive RIS since the active load can amplify the reflected signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The received signal at Uk during τ0 is given by yu,k = hH d,ks � �� � direct link + hH u,kΦ0 (Hrs + nv) � �� � RIS-aided link +nu,k, ∀k, (4) where nu,k ∈ C and nv ∈ CN×1 represent the ad- ditive white Gaussian noise (AWGN) at Uk and the RIS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Without loss of generality, we as- sume nu,k ∼ CN � 0, σ2 u,k � and nv ∼ CN � 0, σ2 vIN � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Denote the equivalent downlink channel as hH k = hH u,kΦ0Hr + hH d,k ∈ C1×M and (4) can be rewritten as yu,k = hH k w0s + hH u,kΦ0nv + nu,k, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (5) Due to the fact that the active RIS not only amplifies the desired signal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=', s, but also amplifies the input noise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=', nv, it is reasonable to consider the second term of (4) for computing the amount of harvested en- ergy accurately [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' However, the noise at Uk is gen- erally quite small and can be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Accordingly, the harvested energy by Uk, denoted by Ek, is given by Ek = β ��hH k w0 ��2τ0 + β ��hH u,kΦ0 ��2σ2 vτ0, ∀k, (6) where β ∈ (0, 1] denotes the energy conversion effi- ciency of each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' It is practical for us to consider the linear EH model here, which is also a common as- sumption in the literature [6, 8, 11, 10, 22, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' It is worth noting that the active RIS can allocate the available reflecting power to amplify the incident signals with active loads [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In the DL WET phase, the amplification power of s and nv is limited by the RIS power budget, which is shown by the following constraint P0 ∥Φ0Hr∥2 + σ2 v ∥Φ0IN∥2 ≤ Pr, (7) where Pr is the maximum reflecting power for ampli- fication at the active RIS and substantially lower than that of an active RF amplifier [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' China Communications 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='2 Wireless Information Transmission Phase In the WIT phase, the users utilize the harvested en- ergy to transmit information to the RS via a TDMA manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Let fk denotes the information-carrying sig- nal of Uk with unit power and then the transmit signal during τk is denoted by xk = √pkfk, where pk is the transmit power at Uk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' We assume that all the harvested energy at Uk in the WET phase is used for delivering its own information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Let p = [p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' , pK] ∈ R1×K, which satisfies pkτk ≤ Ek, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (8) Similarly, the reflecting coefficient matrix at the active RIS for the WIT during τk is denoted by Φk = diag {φk,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' , φk,N} ∈ CN×N, where φk,n = ak,nejθk,n, ak,n and θk,n have the following constraints ak,n ≤ an,max, 0 ≤ θk,n ≤ 2π, ∀k, ∀n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (9) The received signal at the RS from Uk with the as- sistance of active RIS during τk is written as yr,k = gH d,kxk � �� � direct link + GrΦk (gu,kxk + nv) � �� � RIS-aided link +nr, ∀k, (10) where nr ∈ CL×1 represents the AWGN at the RS and satisfies nr ∼ CN � 0, σ2 rIN � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In the UL WIT phase, we also have the amplification power constraint at the active RIS as follows pk ∥Φkgu,k∥2 + σ2 v ∥ΦkIN∥2 ≤ Pr, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (11) Denote the receive beamforming vector at the RS during τk as wk ∈ CL×1, which can be used to ex- tract the desired signal and suppress interference and noises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Let the equivalent uplink channel be gk = gd,k + GrΦkgu,k, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The estimated signal at the RS during τk is expressed as uk =wH k yr,k =wH k gkxk + wH k GrΦknv + wH k nr, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (12) Then, the signal-noise-ratio (SNR) at the RS during τk is written as γk = pk ��wH k (GrΦkgu,k + gd,k) ��2 ��wH k GrΦk ��2 σ2v + ∥wk∥2 σ2r , ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (13) Denote the achievable rate from Uk to the RS as Rk, which is formulated as Rk = τk log (1 + γk) , ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (14) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' PROBLEM FORMULATION In this section, we formulate the system sum-rate max- imization problem by jointly optimizing the reflecting coefficients for both the WET and the WIT at the ac- tive RIS, the transmit beamforming at the PS, the re- ceive beamforming at the RS, the transmit power at each user, and the network time scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The opti- mization problem is formulated as (P1) max τ0,τ,Φ0,Φk,w0,p,wk K � k=1 Rk (15) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (1), (2), (3), (7), (8), (9) and (11), τ0 ≥ 0, τk ≥ 0, pk ≥ 0, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (16) where (16) indicates that the time and power variables are all nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' One can observe that the objective function in (15) is a non-concave function due to the coupled of vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In addition, there exists several non-convex con- straints, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=', (2), (7), (8) and (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' It is thus challeng- ing to solve P1 directly by standard optimization tech- niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In the next section, we propose an AO algo- rithm to solve it efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' ALTERNATING OPTIMIZATION SO- LUTION In this section, an efficient AO algorithm is proposed to solve P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In particular, we decompose P1 into sev- eral subproblems and iteratively solve them in an al- ternating manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' To show the procedure of AO algo- rithm, we summarize a flow chart in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Specif- ically, the variables are partitioned into four blocks, {wk}, {w0, τ, p}, {Φ0, τ0, τ, p}, and {Φk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' the variables in each block are alternately solved by 6 China Communications Linear MMSE-based Receive Beamforming Optimization SDP-based Transmit Beamforming Optimization SDP-based RIS Reflecting Coefficients for the WET and Resource Allocation Optimization SCA-based RIS Reflecting Coefficients Optimization for the WIT System Variables �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' �� ��,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' ��,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' �� �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' {��} {��} �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' ��,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' �� �� �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' � Discarding ��,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' �� ��,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' ��,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' � �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' �� �� AO iteration flow Update Input Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' A flow chart of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' its corresponding sub-problem with the other blocks fixed until the convergence is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='1 Linear MMSE-based Receive Beamform- ing Optimization With the other variables fixed, we first design the re- ceive beamforming vectors {wk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' To cope with the interference caused by nv and nr in (13), we apply the linear MMSE criterion here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Based on this crite- rion, the MMSE-based receive beamforming is given by w∗ k = � gkgH k + σ2 v pk GrΦkΦH k GH r + σ2 r pk IL �−1 gk, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (17) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='2 SDP-based Transmit Beamforming Opti- mization We then proceed to optimize {w0, τ, p} with the other variables {wk, Φ0, τ0, Φk} fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Letting ek = pkτk and e = [e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' , eK] and applying the obtained results in (17), P1 can be simplified as follows (P2) max w0,τ,e K � k=1 τk log � 1 + ǫk ek τk � (18) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' ek ≤ Ek, ∀k, (19) τk ≥ 0, ek ≥ 0, ∀k, (20) (1), (2) and (11), where ǫk = |(w∗ k)H(GrΦkgu,k+gd,k)| 2 ∥(w∗ k)HGrΦk∥ 2σ2v+∥w∗ k∥ 2σ2r , ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' It can be found that P2 is highly non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' To solve it efficiently, the SDR technique [36] is em- ployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Define Hk = hkhH k , and Gu,k = diag � |gu,k,1|2 , |gu,k,2|2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' , |gu,k,N|2� , ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Let W0 = w0wH 0 , which satisfies W0 ⪰ 0 and rank(W0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Then, (19) is rewritten as ek ≤ βτ0[Tr (HkW0) + ��hH u,kΦ0 ��2 σ2 v], ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (21) Denote the RIS reflecting coefficient vector for the WET as ϕk = [φk,1, φk,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' , φk,N]T, ∀k, (11) can be recast as ekϕH k Gu,kϕk + τkσ2 vϕH k ϕk ≤ τkPr, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (22) Then, P2 can be equivalently transformed into (P2-1) max W0,τ,e K � k=1 τk log � 1 + ǫk ek τk � (23) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Tr (W0) ≤ P0, (24) W0 ⪰ 0, (25) rank(W0) = 1, (26) (1), (20), (21) and (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Since the rank-one constraint in (26) is non-convex, we employ the SDR technique to relax it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Thus, P2-1 becomes to be a convex semidefinite program (SDP) and can be solved with the interior-point method [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The optimal transmit beamforming matrix obtained by solving the relaxed version of P2-1, denoted by W ∗ 0 , is rank-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Please refer to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' According to Proposition 1, the tightness of SDR is guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Hence, we can employ Cholesky de- composition to obtain the optimal energy beamform- ing vector w∗ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='3 SDP-based RIS Reflecting Coefficients for the WET and Resource Allocation Opti- mization In this sub-section, we focus on optimizing the re- flecting beamforming at the RIS in the WET phase, the transmit power at each user, and the network time scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Since τ0 and Φ0 are coupled, we first optimize {Φ0, τ, p} with τ0 given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Define Ψ0 = ˜ϕ0 ˜ϕH 0 with Ψ0 ⪰ 0 and rank(Ψ0) = 1, where ˜ϕ0 = [ϕH 0 , 1]H and ϕ0 = [φ0,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' , φ0,N]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' China Communications 7 Let Hu,k = diag {hu,k,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' , hu,k,N} and Qu,k = diag � |hu,k,1|2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' , |hu,k,N|2 , 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Then, (7) and (19) are respectively reformulated as P0Tr( ˜ HrΨ0) + σ2 vTr(Ψ0) ≤ Pr, (27) ek ≤ βτ0Tr[(V + σ2 vQu,k)Ψ0] − βτ0σ2 v, ∀k, (28) where V = �HH u,kHrW0HH r Hu,k HH u,kHrW0hd,k hH d,kW0HH r Hu,k hH d,kW0hd,k � , and ˜ Hr = �HrHH r 0 0 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' With the obtained solutions in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='2, P1 can be equivalently written as (P2-2) max Ψ0,τ,e K � k=1 τk log � 1 + ǫk ek τk � (29) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Ψ0 ⪰ 0, rank(Ψ0) = 1, (30) [Ψ0]n,n ≤ a2 max, ∀n, (31) [Ψ0]N+1,N+1 = 1, (32) (1), (20), (22), (27) and (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Similarly, after the relaxation of the rank-one con- straint in (30), P2-2 is also an SDP and can be solved by the interior-point method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Recall that the tightness of optimizing W0 by SDR can be guaranteed, we can also prove that the obtained solution Ψ0 is rank-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Then, ˜ϕ0 can be recovered by implementing Cholesky decomposition of Ψ0, and the optimal reflection co- efficient vector for the WET ϕ∗ 0 can be obtanied by linear operation from ˜ϕ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Subsequently, the optimal RIS reflecting coefficient matrix Φ∗ 0 can be obtained by Φ∗ 0 = diag((ϕH 0 )∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Finally, we continue to update the optimal energy transmission time τ0 ∈ [0, 1] by the one-dimensional search method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Thus, the maximum sum-rate of this sub-problem is achieved with the optimal solution {Φ∗ 0, τ ∗ 0 , τ ∗, p∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The procedure is summarized in Al- gorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' SDP-based RIS reflecting coefficients for the WET and resource allocation optimization Input: w0, {wk}, {Φk}, ∀k Output: Φ∗ 0, τ ∗ 0 , τ ∗, p∗ 1: Initialization: The maximum objective function value Rmax = 0 and the step size δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 2: for τ0 = 0 : δ : 1 do 3: Given w0, {wk}, {Φk}, we obtain τ ′ 0, Ψ′ 0, τ ′ k, e′ k, ∀k by solving P2-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 4: Calculate R = �K k=1 τ ′ k log � 1 + ǫk e′ k τ ′ k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 5: if R > Rmax then 6: Update Rmax ← R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 7: Update τ0 ← τ ′ 0, Ψ0 ← Ψ′ 0, τk ← τ ′ k, ek ← e′ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 8: end if 9: end for 10: Obtain ˜ϕ0 from Ψ0 by Cholesky decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 11: Obtain ϕ0 from ˜ϕ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 12: Set Φ∗ 0 = diag((ϕH 0 )∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 13: Calculate p∗ k = e∗ k/τ ∗ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 14: return Φ∗ 0, τ ∗ 0 , τ ∗, p∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='4 SCA-based RIS Reflecting Coefficients Optimization for the WIT In this sub-section, we investigate the optimization of RIS reflecting coefficient matrix Φk in the WIT phase, which is given by (P3) max Φk K � k=1 Rk (33) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (9) and (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Note that P3 is still a non-convex optimization prob- lem as the active RIS introduces additional noise term in the denominator of the objective function, which re- sults in a quadratic fractional programming problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In fact, the RIS reflecting coefficients include ampli- tude reflection coefficients and phase shifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' To sim- plify this problem, we derive the optimal phase shifts in the closed-form and then exploit an SCA algorithm to obtain the near-optimal amplitude reflection coeffi- cients according to [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' It is worth noting that P3 can be decomposed into K 8 China Communications independent subproblems, each of which maximizes the SNR of Uk at the RS during τk with respect to the RIS reflection coefficient vector ϕk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Specifically, with w∗ k obtained in (17) and introducing some new aux- iliary variables, the k-th SNR maximization problem can be formulated as γk = pk ��bH k ϕk + gd,k ��2 ϕH k Qrϕkσ2v + σ2r , (34) where gd,k = wH k gd,k, gH r,k = wH k Gr, bH k = gH r,kdiag (gu,k), Qr = diag � |gr,k,1|2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' , |gr,k,N|2� , ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Let Fk = pkGu,k + σ2 vIN, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The subproblem can be expressed as (P4) max ϕk γk (35) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' ϕH k Fkϕk ≤ Pr, ∀k, (36) (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' To solve P4, we decompose the optimization of RIS reflecting coefficient vector ϕk into two sub-problems for the amplitude reflection coefficient design and the optimal phase shift design, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Let ϕk = Θk ¯ϕk, where Θk = diag � ejθk,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' , ejθk,N� ∈ CN×N and ¯ϕk = [ak,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' , ak,N]T ∈ RN×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='1 Optimization of phase shifts for the WIT The optimal design of phase shifts is given in the fol- lowing proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The optimal RIS phase shift of the n-th RE for the WIT during τk is derived as θ∗ k,n = arg(gd,k) − arg(gu,k,n) + arg(gr,k,n), (37) ∀k, ∀n, where gu,k,n and gr,k,n denote the n-th element of the vector gu,k and gr,k, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Please refer to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='2 Optimization of amplitude reflection coeffi- cients for the WIT For P4, the optimal design of phase shifts shown in Proposition 2 holds because the value of the ampli- fication power in (9) and (36) and the noise power in the denominator of (34) are independent with the phase shift of each RE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In addition, optimizing θk,n is equivalent to maximizing the objective function in (34) [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' With the optimal phase shifts in Proposition 2, we proceed to optimize the RIS amplitude reflec- tion coefficients for the WIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In particular, P4 can be simplified as (P4-1) max ¯ϕk ¯γk = pk ��¯bH k ¯ϕk + |gd,k| ��2 ¯ϕH k Qr ¯ϕkσ2v + σ2r (38) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' ¯ϕH k Fk ¯ϕk ≤ Pr, ∀k, (39) ak,n ≤ an,max, ∀k, ∀n, (40) where ¯γk = |γk|, and ¯bk is element-wise modulus of bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' To deal with the non-convexity of the objec- tive function (38), we introduce a new auxiliary vari- able nk = ¯ϕH k Qr ¯ϕkσ2 v + σ2 r, which denotes the noise power received at the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Then, P4-1 can be con- verted into the following equivalent form (P4-2) max ¯γk,nk, ¯ϕk ¯γk (41) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='√pk �¯bH k ¯ϕk + |gd,k| � ≥ √nk¯γk, ∀k, (42) ¯ϕH k Qr ¯ϕkσ2 v + σ2 r ≤ nk, ∀k, (43) (39) and (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' However, the constraint (42) is still non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' To solve P4-1 efficiently, we exploit the SCA algorithm to approximate the square root by a convex upper-bound in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Define ¯γk(t) and nk(t) as the iter- ative optimization variables after the t-th step itera- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In terms of {¯γk (t) , nk (t)}, the first-order Tay- lor polynomial is used to approximate √nk¯γk, which is given by √nk¯γk ≤G(¯γk, nk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' t) = � ¯γk(t)nk(t) + 1 2 �nk(t) ¯γk(t) � 1 2 [ ¯γk − ¯γk(t)] + 1 2 � ¯γk(t) nk(t) � 1 2 [n − nk(t)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (44) Based on (44), (42) can be rewritten as √pk �¯bH k ¯ϕk + |gd,k| � ≥ G(¯γk, nk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' t), ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (45) China Communications 9 Then, P4-2 can be reformulated as the following problem (P4-3) max ¯γk,nk, ¯ϕk ¯γk, (46) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (39), (40), (43) and (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' As P4-3 is convex and can be solved by the inter- point method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' We then discuss the initialization of ¯γk(t) and nk(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' First, we propose an initial solution ¯ϕk(0) by solving a simple feasible version of problem P4-1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=', ¯ϕk(0), satisfying the constraints (39) and (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Then, the reasonable initialization of ¯γk(0) and nk(0) is given by ¯γk(0) = pk ��¯bH k ¯ϕk(0) + |gd,k| ��2 σ2v ¯ϕH k (0)Qr ¯ϕk(0) + σ2r , (47) nk(0) = σ2 v ¯ϕH k (0)Qr ¯ϕk(0) + σ2 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (48) With the initialization described in (47) and (48), the optimal amplitude reflection coefficients for the WIT, denoted by ¯ϕ∗ k, can be obtained by iteratively solving P4-3 until the convergence is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' As a result, the RIS reflecting coefficients during τk can be calculated by Φ∗ k = diag (Θ∗ k ¯ϕ∗ k) , ∀k, (49) where Θ∗ k = diag{ejθ∗ k,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' , ejθ∗ k,N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The detailed description of optimizing the RIS re- flecting coefficients in P3 is summarized in the Algo- rithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' SCA-based RIS reflecting coefficients for the WIT Input: {wk}, p, ∀k Output: {Φ∗ k}, ∀k 1: Initialization: ¯γk(t), nk(t), ¯ϕk(t), and t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 2: Obtain θ∗ k,n in Proposition 2 and have Θ∗ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 3: repeat 4: t = t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 5: Update ¯γk(t), nk(t), ¯ϕk(t) by solving P4-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 6: until the convergence is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 7: Obtain Φ∗ k = diag (Θ∗ k ¯ϕ∗ k), where ¯ϕ∗ k = ¯ϕk(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 8: return {Φ∗ k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='5 Algorithm Summarization and Analysis Based on the above analysis, the algorithm for solving P1 is summarized in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Based on the opti- mality analysis,the objective function of P1 is a non- decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Due to the power budget con- straint (2), (7) and (11), the optimal objective value of problem P1 is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Hence, the convergence of Algorithm 2 can be thus guaranteed, which will be also confirmed by numerical simulations in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='1 Complexity Analysis The computational complexity of our proposed AO algorithm is analyzed as follows, which contains the linear MMSE-based receive beamforming cal- culation, the SDR algorithm and the SCA algo- rithm in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' For the linear MMSE- based receive beamforming optimization, we derive a closed-form solution to the (17), and the approx- imate worst-case computational complexity is given by O � KL max(N, L)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' According to [36], for the subproblem of SDP-based transmit beamform- ing optimization, the worst-case computational com- plexity is O � max(K, M)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='5 log(1/ǫ) � , where ǫ is the computational accuracy of the interior-point method in CVX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Similarly, for the SDP-based RIS reflect- ing coefficients for the WET and resource alloca- tion optimization, the worst-case computational com- plexity is O � Iτ max(N, K)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='5 log(1/ǫ) � , where Iτ is the iteration number for updating τ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' For the SCA-based RIS reflecting coefficients optimization in the WIT, the computational complexity is less than O � ISKN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='5 log(N/ǫ) � , where IS is the iteration number for the SCA algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Thus, the computa- tional complexity of the overall AO algorithm is given by O � IA � KL max(N, L)2 + max(K, M)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='5 log(1/ǫ) +Iτ max(N, K)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='5 log(1/ǫ) + ISKN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='5 log(N/ǫ) �� , (50) where IA denotes the number of iterations required for convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='2 Optimality Analysis As the formulated problem P1 is extremely non- convex, it is very difficult to obtain the globally op- timal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' To solve P1 efficiently, we propose 10 China Communications an efficient AO algorithm to obtain the suboptimal so- lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Firstly, we obtain the optimal receive beam- forming {wk} in a closed-form, which are the globally optimal solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' With the obtained receive beam- forming solutions, the formulated problem can be sim- plified but is still non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Secondly, the SDR technique is adopted to optimize the transmit beam- forming, the RIS reflecting coefficient matrices for the WET phase, the transmit power at each user, and the network time scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In particular, we prove the obtained solutions of P2-1 and P2-2 are rank-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Since the tightness of applying SDR can be guaran- teed, the obtained solutions {w0, Φ0, τ, p} are glob- ally optimal [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Then, we use the one-dimensional search method to exploit the optimal energy trans- mission time τ0 by setting an appropriate step size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Thirdly, for the optimization of RIS reflecting coef- ficients for the WIT phase, we decompose P4 into two sub-problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' On the one hand, the optimal phase shifts have been derived in a closed-form which has been proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' On the other hand, P4-1 is solved by Algorithm 2, which obtains the reflection amplitudes of {Φk} are near-optima of the original problem [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Hence, Algorithm 3 can be used to obtain the near- optimal solutions to P1 with a high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' AO algorithm for P1 1: Initialization: w0, {wk}, Φ0, {Φk}, τ0, τ, p, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 2: repeat 3: Given Φk and p, update {wk} by (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 4: Given Φ0, τ0, {Φk}, {wk}, update w0 with by solving P2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 5: Given w0, {wk}, {Φk}, update Φ0, τ0, τ and p by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 6: Given {wk} and p, update {Φk} by Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 7: until �K k=1 Rk converged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 8: return w∗ 0, {w∗ k}, Φ∗ 0, {Φ∗ k}, τ ∗ 0 , τ ∗, p∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' NUMERICAL RESULTS In this section, numerical results are presented to eval- uate the performance of the proposed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' As shown in Figure 3, we consider that the simulated net- work deployment is a 2-D coordinate system, where ��� �� (!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=') (0,0) ( ", 0) ( #, 0) $(!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=') ( %, &) ��������� �� Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Placement model of simulation setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' the coordinates of the PS, the RIS, and the RS are given as (0,0), (xr, 0), and (xs,0), respectively, the users are randomly deployed within a circular area centered at (xu, xh) with radius 1m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' We follow the channel model considered in [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In particular, the large-scale path-loss is modeled as L = A(d/d0)−α, where A is the path-loss at the reference distance d0 = 1m and set as A = −30dB, d denotes the dis- tance between two nodes, and α is the path-loss expo- nent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' For the RIS related links, the path-loss exponent is set as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='2 since the location of RIS can be carefully designed to avoid the severe signal blockage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' While the path-loss exponents for the RIS unrelated links are set as 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='5 due to the users’ random deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' We assume the direct link channels follow Rayleigh fad- ing but the RIS related channels follow Rician fading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Specifically, the small-scale channel from the PS to the RIS can be expressed as Hr = �� βr βr + 1 ¯ HLoS r + � 1 βr + 1 ¯ HNLoS r � (51) where βr is the Rician factor for the PS-RIS link, ¯ HLoS r denotes the deterministic line of sight (LoS) component, and ¯ HNLoS r denotes the non-LoS compo- tent with circularly symmetric complex Gaussian ran- dom variables with zero mean and unit variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The other channels can be similarly defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Unless oth- erwise stated, other parameters are given as follows: βr = 10 [39], ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='8, σ2 v = σ2 r = −90dBm, P0 = 20dBm [20], Pr = 20dBm, amax = 25dB [40], N = 10, K = 4, M = 4, L = 4, xr = 10m, xu = 10m, xs = 20m, and xh = 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' For comparisons, we also evaluate the performance of the following benchmark schemes: (1) Active RIS-aided single-antenna WPCN scheme (Active-SA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' China Communications 11 (2) Passive RIS-aided multi-antenna WPCN scheme (Passive-MA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (3) Active RIS-aided multi-antenna WPCN with uni- form energy beamforming scheme (Active-MA- UEBF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Notice that for the multi-antenna schemes, the num- ber of antennas is set as 4 in the PS and RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In addition, we set the number of REs in the Passive-MA scheme as N = 100 to show the superiority of the proposed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Before performance comparisons, we first show the convergence performance of the proposed AO algo- rithm in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' One can observe that as the num- ber of iterations increases, the sum-rate first increases but finally converges to a constant after nearly 8 iter- ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' This demonstrates that the convergence of the proposed scheme can be achieved quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The other observation is that the effect of the parameter setting on convergence is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 2 4 6 8 10 12 Number of iterations 13 14 15 16 17 18 19 20 21 22 23 Sum rate (bps/Hz) N=10, M=L=4 N=10, M=L=8 N=20, M=L=4 N=20, M=L=8 N=30, M=L=4 N=30, M=L=8 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Convergence behavior of the proposed scheme under different parameter settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Figure 5 shows the impact of the transmit power at the PS (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=', P0) on the sum-rate when the RIS’s maximum reflecting power Pr = 10, 20 dBm and the RIS’s maximum amplitude reflection coefficient amax = 10, 25 dB, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In general, the pro- posed scheme outperforms the Active-SA scheme with the same parameters, which confirms that the assis- tance of multiple antennas can achieve a significant performance gain by constructing the transmit beam- forming at the PS and the receive beamforming at the RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' For a given amax = 25 dB, our proposed scheme with 10 REs can achieve 415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='48% performance gain 5 10 15 20 25 30 35 40 45 Transmit power at the PS(dBm) 0 5 10 15 20 25 30 35 Sum rate (bps/Hz) Proposed, Pr=20, amax=25 Proposed, Pr=20, amax=10 Proposed, Pr=10, amax=25 Proposed, Pr=10, amax=10 Active-SA, Pr=20, amax=25 Active-SA, Pr=20, amax=10 Passive-MA, N=100 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Sum-rate versus the transmit power at the PS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' compared to the passive RIS scheme with 100 REs when P0 = 20 dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Indeed, the active RIS can con- siderably make use of its amplification characteristic to amplify the energy signals at low transmit power and thereby realize a superior capability at the cost of additional power consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' For a given amax = 10 dB, it can be seen that the sum-rates achieved by the proposed scheme with Pr = 20 dBm and Pr = 10 dBm are almost the same, which implies that the am- plification power constraints defined in (7) and (11) are inactive since amax is limited for the small trans- mit power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Note that, the performance gap is signif- icant between the scheme with amax = 25 dB and amax = 10 dB because amax directly limits the ampli- tude reflection coefficient of the active RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In addi- tion, the sum-rate of the passive-MA scheme is gener- ally lower than the active RIS schemes with the same REs and the passive RIS needs to be equipped with more REs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=', 100 REs) to achieve the similar per- formance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In Figure 6, we evaluate the sum-rate versus the number of reflecting elements at the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' It can be seen that the proposed schemes can achieve a higher performance gain compared with the other benchmark schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' With an increasing number of reflecting ele- ments, the sum-rate increases due to the fact that more transmission links can be provided for both the WET and the WIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In addition, to investigate the best system performance, the maximum number of users is set to be equal to the number of REs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Since the active RIS can amplify the incident signals, a limited number of REs is sufficient to reach the desired SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Therefore, the size of active RIS can be reduced, making it ap- 12 China Communications 5 10 15 20 25 30 35 Number of REs 10 12 14 16 18 20 22 24 Sum rate (bps/Hz) Proposed, K=4 Proposed, K=N Active-SA, K=4 Active-SA, K=N Active-MA-UEBF, K=4 Active-MA-UEBF, K=N Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Sum-rate versus number of reflecting elements at the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' 2 3 4 5 6 7 8 9 10 Number of Users 2 4 6 8 10 12 14 16 18 20 Sum rate (bps/Hz) Proposed Active-SA Active-MA-UEBF Passive-MA Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Sum-rate versus the number of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' plicable to the scenario where the space for the RIS deployment is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In Figure 7, we study the effect of number of user on the sum-rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' As the number of users increases, the total amount of harvested energy by users im- proves, which results in a higher sum-rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Nonethe- less, when the number of users reaches a threshold, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' K = 8, the sum-rate achieved by our pro- posed scheme becomes to be saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' This is due to the fact that the increment of number of users re- duces the energy transfer duration and the time allo- cated to each user for information transmission, which makes the sum-rate converge to a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Again, our proposed scheme notably outperforms the other benchmark schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' For example, when the num- ber of users is K = 4, our proposed scheme can achieve 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='78% and 415.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='48% performance gain com- 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='5 4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='5 7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='5 10 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='5 13 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='5 16 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='5 19 Location of RIS 0 5 10 15 20 25 Sum rate (bps/Hz) Proposed Active-SA Active-MA-UEBF Passive-MA Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Sum-rate versus x-coordinate of the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' pared with the Active-SA scheme and the Passive-MA scheme with 100 REs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In Figure 8, we plot the sum-rate versus the hori- zontal ordinate of the RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' As xr varies, the sum-rates of all schemes first increase but then decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Com- pared to the scenario that the RIS is close to the RS, by deploying the RIS near the PS, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=', xr = 1, the sum-rate can be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' It is because the users can harvest more energy assisted by the active RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Moreover, we can observe that the sum-rate is maxi- mized at xr = 10, where the reflecting link between the active RIS and each user is strongest so the users can benefit from a larger amplification and reflection gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' However, when the RIS is neither close to the PS nor the users, both the PS-RIS link and the RIS- users links become weak, which results in the reduce of harvested energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Furthermore, since the Active- MA-UEBF scheme adopts the uniform energy beam- forming, the energy signals cannot adaptively align with the direction of the desired channels, which re- sults in a low WET efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' In addition, the schemes with the active RIS can achieve a much better perfor- mance than the passive RIS scheme, which demon- strates that the active RIS with the amplification func- tionality can significantly mitigates the double-fading effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The above observation demonstrates that the lo- cation of the active RIS should be carefully designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' CONCLUSIONS In this paper, we have proposed an active RIS as- sisted relaying scheme to enhance the performance of multiuser multi-antenna WPCN, which is involved China Communications 13 in both the WET from the PS to users and the WIT from users to the RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' To further enhance system per- formance, both transmit beamforming at the PS and receive beamforming at the RS have been designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' We have formulated a system sum-rate maximization problem by jointly optimizing the RIS reflection coef- ficients for both the WET and the WIT, transmit and receive beamforming vectors, transmit power at each user, and network time scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' As the formulated problem is non-convex, we have proposed an AO al- gorithm with linear MMSE, SDR and SCA techniques to solve it efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Finally, numerical results have been provided to confirm the performance superiority of the proposed scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Proof of Proposition 1 The Lagrangian function of P2-1 can be expressed as L = K � k=1 λkβTr(Hd,kW0) − ξTr(W0) + Tr(ΩW) + δ, ∀k, (52) where λk ≥ 0, ξ ≥ 0, and Ω ∈ HM are the Lagrange multipliers associated with constraints (21), (22), and (24), respectively, δ denotes the term unrelated with W0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' The Karush-Kuhn-Tucker (KKT) conditions of P2-1 are given as follows ∂L ∂W0 = K � k=1 λ∗ kβHd,k − ξ∗IM + Ω∗ = 0, (53) Ω∗W ∗ 0 = 0, (54) where λ∗ k, ξ∗ and Ω∗ are the optimal Lagrangian mul- tipliers for the dual problem of P2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' It can be proved that λ∗ k > 0 and ξ∗ > 0 since the constraints (21) and (22)are equalities in the optimal condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Based on (53) and (54), it is straightforward to obtain the fol- lowing equality (ξ∗IM − K � k=1 λ∗ kβHd,k)W ∗ 0 = 0 (55) According to [41], rank(ξ∗IM −�K k=1 λ∗ kβHd,k) ≥ M − 1 due to the fact that Hd,k for ∀k are indepen- dently distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Thus, from (55), we can obtain that rank(W0) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' It is obvious that W0 = 0 is not the optimal solution to P2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Hence, we derive that rank(W0) = 1, which thus proves Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Proof of Proposition 2 Since Qr and Fk are diagonal matrices, we observe that the noise power in the denominator of (35) and the amplification power in (9) and (36) are independent of the phase shift of each RE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Therefore, maximizing γk with respect to Θk is equivalent to the following optimization problem (P4-4) max Θk ��bH k Θk ¯ϕk + gd,k ��2 (56) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' |Θk,n| = 1, ∀k, ∀n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (57) We rewrite the objective function as ��bH k Θk ¯ϕk ��2 + |gd,k|2 + 2 ��bH k Θk ¯ϕk �� |gd,k| cos α, (58) where α = arctan Im(bH k Θk ¯ϕk) Re(bH k Θk ¯ϕk) − arctan Im(gd,k) Re(gd,k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Obviously, the maximum of ��bH k Θk ¯ϕk + gd,k ��2 is achieved when arg(bH k Θk ¯ϕk) = arg(gd,k) ≜ ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Let vk = [vk,1, vk,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=', vk,N]T ∈ RN×1 and ξk = diag(bH k ) ¯ϕk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' As bH k Θk ¯ϕk = vH k ξk, P4-4 can be rewritten as (P4-5) max vk ��vH k ξk ��2 (59) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' |vk,n| = 1, ∀k, ∀n, (60) arg(vH k ξk) = ω, ∀k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' (61) Based on [12], the optimal solution to P4- 5 can be expressed as v∗ k = ej(ω−arg(ξk)) = ej(��−arg(diag(bH k ) ¯ϕk)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Then, the optimal RIS phase shift for the n-th RE is expressed as θk,n = arg(gd,k) − arg(bH k,n) − arg( ¯ϕk,n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tAzT4oBgHgl3EQf9P4T/content/2301.01915v1.pdf'} +page_content=' Finally, we ob- tain that θk,n = 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