Multi-Objective Resource Allocation in Full-Duplex SWIPT Systems

Transcription

1 Multi-Objective Resource Allocation in Full-Duplex SWIPT Systes Shiyang Leng, Derric Wing Kwan Ng, Niola Zlatanov, and Robert Schober The Pennsylvania State University, USA The University of New South Wales, Australia Monash University, Australia Friedrich-Alexander-University Erlangen-Nürnberg FAU, Gerany Abstract In this paper, we investigate the resource allocation algorith design for full-duplex siultaneous wireless inforation and power transfer FD-SWIPT systes. The considered syste coprises a FD radio base station, ultiple single-antenna half-duplex HD users, and ultiple energy harvesters equipped with ultiple antennas. We propose a ulti-objective optiization fraewor to study the trade-off between uplin transit power iniization, downlin transit power iniization, and total harvested energy axiization. The considered optiization fraewor taes into account heterogeneous quality of service requireents for uplin and downlin counication and wireless power transfer. The non-convex ulti-objective optiization proble is transfored into an equivalent ranconstrained seidefinite progra SDP and solved optially by SDP relaxation under certain general conditions. The solution of the proposed fraewor results in a set of Pareto optial resource allocation policies. Nuerical results unveil an interesting trade-off between the considered conflicting syste design objectives and reveal the iproved power efficiency facilitated by FD in SWIPT systes copared to traditional HD systes. I. INTRODUCTION Next generation counication systes ai at providing self-sustainability and high data rates to counication networs with guaranteed quality of service QoS. A proising technique for prolonging the lifetie of counication networs is energy harvesting EH. Aong different EH technologies, wireless power transfer WPT via electroagnetic waves in radio frequency RF enables coparatively controllable EH at the receivers copared to conventional natural source, such as wind, solar, and tidal. Recent progress in the developent of RF-EH circuitries has ade RF-EH practical for low-power consuption devices [1] [3], e.g. wireless sensors. In particular, RF-EH enables siultaneous wireless inforation and power transfer SWIPT [4] [7]. Thereby, as a carrier of both inforation and energy, the RF signal unifies inforation transission and power transfer. Besides, RF-EH advocates energy saving by recycling the energy in the RF radiated by abient transitters. In SWIPT systes, the aount of harvested energy is an equally iportant QoS etric as the data rate and the transit power consuption which are traditionally considered in counication networs. Thus, resource allocation algoriths for SWIPT systes should also tae into account the eerging need for energy transfer [8] [10]. In [8], energy-efficient SWIPT was investigated in ulticarrier systes, where power allocation, user scheduling, and subcarrier allocation were considered. In [9], the authors proposed a power allocation schee for energy efficiency Robert Schober is also with the University of British Colubia. This wor was supported in part by the AvH Professorship Progra of the Alexander von Huboldt Foundation. axiization of large scale ultiple-antenna SWIPT systes. In [10], ulti-objective optiization MOO was applied to jointly optiize ultiple syste design objectives to facilitate secure SWIPT systes. Although SWIPT has been already considered for various syste setups, the power efficiency of SWIPT systes [8] [10], has not been fully investigated and is still unsatisfactory due to the traditional half-duplex HD operation. Recently, full-duplex FD counication has becoe a viable option for next generation wireless counication networs. In contrast to conventional HD transission, FD counication allows devices to transit and receive siultaneously on the sae frequency, thus potentially doubling the spectral efficiency. In practice, the self-interference SI caused by the own transit signal ipairs the siultaneous signal reception in FD systes severely which has been a ajor obstacle for the ipleentation of FD devices in the past decades. Fortunately, breathroughs in analog and digital self-interference cancellation SIC techniques [11] have ade FD counication ore practical in recent years. However, various practical ipleentation issues, such as protocol and resource allocation algorith design, need to be reinvestigated in the context of FD counications [12] [15]. In [12], the authors proposed a suboptial beaforer design to axiize the spectral efficiency of FD sall cell wireless systes. In [13], resource allocation and scheduling was studied for FD ultiple-input ultiple-output orthogonal frequency division ultiple access MIMO-OFDMA relaying systes. Moreover, the energy efficiency of FD-OFDMA relaying systes was investigated in [14]. The authors of [15] proposed a ultiobjective resource allocation algorith for FD systes by considering the trade-off between uplin and downlin transit power iniization. Although FD counication has drawn significant research interest [12] [15], research on FD SWIPT systes is still in its infancy. Lately, the notion of FD counication in EH systes has been pursued. Specifically, the cobination of FD and WPT was first considered in [16]. The authors optiized the resource allocation in a syste with WPT in the downlin and wireless inforation transission in the uplin. In [17], the perforance of a dual-hop full-duplex relaying SWIPT syste was studied. However, siultaneous uplin and downlin counication has not been studied thoroughly for SWIPT systes. In fact, uplin and downlin transission occurs siultaneously in FD systes and the associated inforation signals can also serve as vital energy sources for RF energy harvesting. As a result, different tradeoff naturally arise in FD-SWIPT systes when considering the aspects of uplin and downlin transission as well as EH. These observations otivate us to design a flexible resource

2 Self-interference Full-duplex base station Uplin signal Energy signal Downlin signal Roaing user energy harvester Uplin user Co-channel interference Downlin user Fig. 1. Multiuser FD SWIPT syste with a FD radio base station, M = 1 uplin user, K = 1 downlin user, and J = 1 roaing user energy harvester. allocation algorith for FD SWIPT systes which stries a balance between the different syste design objectives. The rest of the paper is organized as follows. In Section II, we outline the syste odel for the considered FD SWIPT networs. In Section III, we forulate the ultiobjective resource allocation algorith design as a non-convex optiization proble and solve this proble by seidefinite prograing relaxation. In Section IV, we present nuerical perforance results for the proposed optial algorith. In Section V, we conclude with a brief suary of our results. II. SYSTEM MODEL In this section, we first introduce the notation adopted in this paper. Then, we discuss the signal odel for FD SWIPT networs. A. Notation We use boldface capital and lower case letters to denote atrices and vectors, respectively. A H, TrA, and RanA represent the Heritian transpose, trace, and ran of atrix A, respectively; diaga returns a diagonal atrix containing the diagonal eleents of atrix A on its ain diagonal; A 1 and A represent the inverse and Moore-Penrose pseudoinverse of atrix A, respectively; A 0 indicates that A is a positive seidefinite atrix; I N is the N N identity atrix; C N M denotes the set of all N M atrices with coplex entries; H N denotes the set of all N N Heritian atrices; and denote the absolute value of a coplex scalar and the Euclidean vector nor, respectively; E{ } denotes statistical expectation; [x] + = ax{x, 0}; the circularly syetric coplex Gaussian distribution with ean vector µ and covariance atrix Σ is denoted by CN µ, Σ; and stands for distributed as. B. Signal Model We focus on a ultiuser wireless counication syste. The syste consists of an FD radio base station BS, K HD downlin users, M HD uplin users, and J roaing users, cf. Figure 1. The BS is equipped with N > 1 antennas that can siultaneously perfor downlin transission and uplin reception in the sae frequency band [11]. All uplin and downlin users are single-antenna devices to liit the hardware coplexity. On the other hand, to facilitate efficient EH, we assue that the roaing users are ultiple-antenna devices, which are equipped with N EH > 1 antennas. For downlin FD counication, K independent signal streas are transitted siultaneously at the sae frequency to the K downlin users. The transitted signal at the FD radio BS is given by x = w d DL + q, 1 where d DL C is the inforation bearing signal intended for downlin user {1,..., K}. Without loss of generality, we assue E{ d DL 2 } = 1. Besides, a beaforing vector w C N 1 is eployed to assist downlin inforation transission. On the other hand, in order to facilitate efficient WPT 1 to roaing users, a dedicated energy bea, q C N 1, is transitted concurrently with the inforation signal. The energy signal q is odeled as a coplex pseudo-rando sequence with covariance atrix Q = E{qq H }. In general, both pseudo-rando signals and constant aplitude signals are potential candidates for ipleenting the energy signal. However, pseudo-rando energy signals can be shaped ore easily to satisfy certain requireents on the spectru as of the transit signal and are thus adopted in this paper. In particular, we assue that q is generated at the BS by a pseudo-rando sequence generator with a predefined seed. This seed inforation is nown at the downlin users. Thus, the interference caused by the energy signal can be copletely cancelled at the downlin users before decoding the desired signals. C. Channel Model We consider a narrow-band slow fading channel. The received signal at downlin user is given by y DL = h H x + P f, d UL } {{ } co-channel interference +n DL, 2 where h C N 1 denotes the channel vector between the BS and downlin user. The second ter in 2 denotes the co-channel interference CCI caused by siultaneous uplin transission in the FD syste. f, C is the channel gain between uplin user and downlin user. d UL and P denote the uplin transit signal fro uplin user and the corresponding transit power, respectively. We assue E{ d UL 2 } = 1 without loss of generality. CN 0, σdl, 2 denotes the additive white Gaussian noise AWGN at downlin user. At the sae tie, the FD BS receives signals fro M uplin users siultaneously. The corresponding received signal is given by n DL y UL = P g d UL + }{{} c +n UL, 3 self-interference cancellation noise where g C N 1 denotes the channel vector between uplin user and the BS. Vector n UL C N 1 represents the 1 In this paper, we adopt a noralized unit energy, i.e., Joule-per-second. Thus, the ters energy and power are interchangeable.

3 AWGN distributed as CN 0, σul 2 I N. Due to the concurrent uplin reception and downlin transission at the FD radio BS, the SI caused by the downlin transit signal ipairs the uplin signal reception. In practice, different interference itigation techniques such as antenna cancellation, balun cancellation, and circulators [18], [19] have been proposed to alleviate the ipairent caused by SI. In order to isolate the resource allocation algorith design fro the specific ipleentation of self-interference itigation, we odel the self-interference cancellation induced noise by vector c CN 0, ϱ diage{h SI xx H H H SI } [19, Eq. 4], where H SI C N N is the self-interference channel and 0 ϱ 1 is a constant indicating the noisiness of the self-interference cancellation at the FD BS. In the considered syste, both downlin and uplin signals 2 act as energy sources to the roaing users energy harvesters. The received signal at energy harvester j {1,..., J} is y EH j = Ω H j x + φ j, P d UL + n EH j, 4 where atrix Ω j C N N EH and vector φ j, C N EH 1 denote the channel between the BS and energy harvester j, and the channel between uplin user and energy harvester j, respectively. Vector n EH j C N EH 1 represents the AWGN at energy harvester j distributed as CN 0, σeh 2 I N R. We note that all channel variables, i.e., h, f,, g, H SI, Ω j, and φ j,, capture the joint effect of path loss and sall scale fading. III. PROBLEM FORMULATION In this section, we first introduce the adopted QoS etrics. Then, fro the perspectives of uplin power consuption, downlin power consuption, and EH, we forulate three single objective optiization probles. In practice, these three syste design objectives are all desirable but conflicting. Thus, we apply a MOO fraewor to study ulti-objective resource allocation algorith design. A. Quality of Service Metrics We assue that full channel state inforation CSI is available at the FD BS for resource allocation. The receive signalto-interference-plus-noise-ratio SINR at downlin user is given by = h H w 2, 5 h H w i 2 + P f, 2 + σdl, 2 i where the interference fro the energy beaforing signal, i.e., Trh H Qh, has already been cancelled since energy signal q is nown to the downlin users. For uplin transission, we adopt zero-forcing beaforing ZF-BF for detection at the BS. In contrast to optial iniu ean square error beaforing MMSE-BF detection, ZF-BF facilitates the design of resource allocation algoriths in the considered proble. Additionally, the perforance of ZF-BF converges to the perforance of MMSE-BF in the high SINR regie [20], which is the desired operating region of the 2 In general, the adopted syste odel can be extended to scenarios in which the uplin users also transit energy signal to facilitate EH. However, it ay increase the pea-to-average power ratio and is not suitable for uplin users equipped with low cost power aplifiers. considered syste. Therefore, the receive SINR at the BS with respect to uplin user {1,..., M} can be expressed as where S UL Γ UL = i P g H z 2 P i g H i z 2 + S UL + σ 2 UL z 2, 6 = ϱz H diag K H SI w w H + Q H H SI z 7 is the noise caused by SI cancellation and z C N 1 denotes the ZF-BF receive vector for decoding the signal of uplin user. The ZF-BF atrix is given by Z = [z 1,..., z M ] T = G H G 1 G H, 8 where G = [g 1,..., g M ]. On the other hand, the total aount of harvested energy at energy harvester j {1,..., J} is given by Pj EH =η j [Tr Ω H j w w H +QΩ j + P φ j, ], 2 9 where 0 η j 1 is the energy conversion efficiency of energy harvester j. It represents the energy loss in converting the received RF energy to electrical energy for storage. Note that the theral noise power is ignored in 9 for EH as it is negligibly sall copared to the power of the received signals. B. Optiization Proble Forulation In FD SWIPT systes, downlin transit power iniization, uplin transit power iniization, and total harvested energy axiization are all desirable syste design objectives. Now, we first propose three single-objective optiization probles with respect to these objectives. Proble 1: Downlin Transit Power Miniization: s.t. C1 : iniize Q H N,w,P w 2 + TrQ w 2 + TrQ P DL ax, C2 : P Pax,, UL, C3 :,, C4 : Γ UL Γ UL req,,, C5 : Pj EH P in,j, j, C6 : P 0,, C7 : Q 0, 10 where w = {w, } and P = {P, } denote the downlin beaforing vector policy and the uplin transit power policy, respectively. In 10, we iniize the total downlin transit power by jointly optiizing downlin inforation beaforing vectors w,, the covariance atrix of energy signal, Q, and uplin transit power P,. Constants Pax DL and Pax, UL in C1 and C2 denote the axiu downlin transit power for the FD BS and the axiu transit power of uplin user, respectively. QoS requireents of reliable counication are taen into account in C3 and C4. In particular, > 0,, and ΓUL req, > 0,, are the iniu required SINRs for the downlin and uplin

4 users, respectively. P in,j, j, in C5 is the iniu required aount of harvested energy for energy harvester j. In addition, C6 and C7 enforce the non-negative uplin transit power constraints and the positive seidefinite Heritian atrix constraint for covariance atrix Q, respectively. On the other hand, for the syste designs with the objectives of uplin transit power iniization and total harvested energy axiization, respectively, we have the sae constraint set as for Proble 1. Therefore, the proble forulations for these two other syste design objectives are given as, respectively, Proble 2: Uplin Transit Power Miniization: iniize Q H N,w,P P s.t. C1 C7, 11 Proble 3: Total Harvested Energy Maxiization: axiize P Q H N j EH,w,P s.t. C1 C7. 12 The interdependency between the aforeentioned objectives is non-trivial in the considered FD SWIPT syste. For instance, although a large transit power ensures high received SINRs at the downlin users, the strong SI ipairs the reception of the uplin signals at the FD BS. Siilarly, increasing the uplin transit power to satisfy a ore stringent uplin SINR requireent will lead to severe CCI which degrades the downlin signal reception. On the other hand, the EH QoS requireent has to be fulfilled by transferring a sufficient aount of power in both uplin and downlin. Yet, iniizing either uplin or downlin transit power conflicts with the objective of having a higher power for EH. Hence, a nontrivial trade-off between these three syste design objectives naturally arises in the considered FD SWIPT syste. Thus, a flexible resource allocation algorith which can accoodate diverse syste design preferences is desired. To this end, we apply MOO to systeatically address this resource allocation proble. In the literature, MOO is coonly adopted as a atheatical fraewor to study the trade-off between ultiple desirable but conflicting syste design objectives. The optial solution of a MOO progra MOOP is defined by a Pareto optial set; a set of points that satisfy the concept of Pareto optiality [10]. In the following, we forulate a MOOP based on the weighted Tchebycheff ethod [10], in which the preferences for the aforeentioned single syste design objectives are quantified by a set of pre-specified weights. In fact, copared to other approaches to forulate MOOPs, the weighted Tchebycheff ethod can provide a coplete Pareto optial set by varying the weights, even if the MOOP is non-convex. For the sae of notational siplicity, we denote the objective functions of Probles 1 3 as F n Q, w, P, n {1, 2, 3}, respectively. Then, the MOOP is given by Proble 4: Multi-Objective Optiization: iniize Q H N,w,P ax n=1,2,3 } {λ n F n Q, w, P Fn s.t. C1 C7, 13 where Fn is the optial objective value with respect to Proble n {1, 2, 3}. In order to represent the three single syste design objective functions in a unified anner, without loss of generality, the axiization in Proble 3 was rewritten as an equivalent iniization. As a result, F 3 Q, w, P in Proble 4 is given by F 3 Q, w, P = J P j EH. Constant λ n is a weight iposed on the n-th objective function subject to 0 λ n 1 and n λ n = 1, which indicates the preference of the syste designer for the n-th objective function over the others. We can obtain a set of resource allocation policies by solving Proble 4 for different predefined weights. In the extree case, when λ n = 1 and λ l = 0, l n, Proble 4 is equivalent 3 to the n-th single-objective optiization proble. C. Optial Solution Probles 1-4 are non-convex optiization probles due to the non-convex constraints C3 and C4. To overcoe the non-convexity, we recast these probles as SDPs via SDP relaxation. To this end, we define new variables W = w w H, H = h h H, G = g g H, 14 Z = z z H, and Φ j, = φ j, φ H j,. 15 Thus, the considered probles can be equivalently transfored as follows: Transfored Proble 1: K iniize W,Q H N,P Tr W + Q s.t. C2, C6, C7, where I DL = I UL = C1 : Tr K W + Q P ax, C3 : TrH W I DL C4 : P Tr G Z Γ UL req, + σ 2 DL,,, I UL + σ 2 UL TrZ,, C5 : Pj EH P in,j, j, C8 : W 0,, C9 : RanW 1,, 16 TrH W i + i P i TrG i Z i P f, 2, 17 + ϱ Tr Z diag K H SI W + Q H H SI, 18 Pj EH = η j [Tr Ω H j K W +Q ] Ω j + P TrΦ j,, 19 i= and W = {W, } is the set of downlin beaforing atrices to be optiized. Constraints C8, C9, and W H N are introduced due to the definition of W. Siilarly, Probles 2-4 are equivalently transfored to Transfored Proble 2: iniize W,Q H N,P P s.t. C1 C Here, equivalent eans that both probles have the sae solution.

5 Transfored Proble 3: axiize W,Q H N,P Transfored Proble 4: axiize W,Q H N,P,τ τ P EH j s.t. C1 C9. 21 s.t. C1 C9, C10 : λ n F n Q, W, P F n τ, n {1,2,3},22 where τ is an an auxiliary optiization variable [21]. Evidently, Transfored Proble 4 is a generalization of Transfored Probles 1-3. Hence, we focus on the ethodology for solving Transfored Proble 4 in the following. Transfored Proble 4 is non-convex due to the ran-one atrix constraint C9. To obtain a tractable proble forulation, we apply SDP relaxation. Specifically, we relax constraint C9 in 22 by reoving it fro the proble. Then, the considered proble becoes axiize W,Q H N,P,τ τ s.t. C1 C8, C10 : λ n F n Q, W, P F n τ, n {1,2,3}.23 We note that the ran constraint relaxed proble in 23 is a convex SDP which can be solved by standard nuerical convex progra solvers such as CVX [22]. In particular, if the obtained solution of the relaxed proble satisfies the ranone constraint, i.e., RanW 1, then the solution of 23 is the optial solution of the original Proble 4. Thus, the optial beaforing vector w of the original proble can be retrieved by solving the relaxed proble. Now, we reveal the tightness of the SDP relaxation by the following theore. Theore 1: Assuing that the channels Ω j, H SI, and h, are statistically independent, the optial beaforing atrix for 23 is a ran-one atrix, i.e., RanW = 1,, and the energy beaforing atrix satisfies RanQ 1 with probability one for req > 0. Proof: Please refer to the Appendix. In other words, whenever the channels satisfy the general condition stated in Theore 1, the adopted SDP relaxation is tight. Hence, the optial solution of the original MOOP can be obtained by solving the relaxed SDP proble in 23. Besides, inforation beaforing, i.e., RanW = 1, and energy beaforing, i.e., RanQ 1, is optial for optiizing the considered conflicting objective functions. On the other hand, the optial solutions of the single-objective probles can be achieved by solving special cases of 23. For instance, the solution of single-objective Proble 1 can be obtained by solving 23 with λ 1 = 1, λ 2 = 0, and λ 3 = 0. IV. RESULTS In this section, we investigate the perforance of the proposed ulti-objective resource allocation algorith. The iportant siulation paraeters are suarized in Table I. We evaluate a syste with an FD radio BS located at the center of a cell. Furtherore, K = 3 downlin users and M = 8 uplin users located in the range between the reference distance of 10 eters and the axiu distance of 50 eters. J = 2 energy harvesters are located close to the FD BS at a TABLE I SIMULATION PARAMETERS. Carrier center frequency 915 MHz Bandwidth 200 Hz Antenna gain at FD BS 10 dbi Antennas gain at users 0 dbi Downlin user noise power -71 db BS noise power -83 db SI cancellation constant ϱ -110 db Energy conversion efficiency, η j 0.8 distance of between 2 to 10 eters in order to facilitate EH. Each energy harvester is equipped with N EH = 3 antennas. The sall scale fading of the uplin and downlin channels is odeled as independent and identically distributed Rayleigh fading. The EH channel and the SI channel are odeled as Rician fading channels with Rician factor 6 db. The axiu transit power supply in downlin and uplin are Pax DL = 46 db and Pax, UL = 23 db,, respectively. Without loss of generality, we assue that the required SINRs at all downlin users are identical. Besides, we specify Γ UL req, = 15 db,, for uplin users. At the energy harvesters, the iniu required harvested energy is P in,j = 20 db, j. A. Trade-off Region of Multiple Design Objectives Figure 2 depicts the trade-off region for uplin transit power iniization, downlin transit power iniization, and total harvested energy axiization achieved by the proposed optial schee. There are N = 8 transit antennas at the BS and the iniu required downlin SINR is = 21 db,. The points shown for the trade-off region were obtained by solving the SDP relaxed proble for different sets of weights 0 λ n 1, n = 1, 2, 3 subject to n λ n = 1. As can be seen, there is a nontrivial tradeoff between uplin and downlin transit power iniization and total harvested energy axiization. In particular, for a fixed weight λ 3 for EH axiization, the downlin transit power onotonically decreases for an increasing uplin transit power which suggests that downlin transit power iniization and uplin transit power iniization conflict with each other. On the other hand, the objective of total harvested energy axiization does not align with the objectives of uplin and downlin transit power iniization. It can be seen that the aount of harvested energy can only be increased by transitting with higher uplin and/or downlin transit power. In particular, the resource allocation policy axiizes the harvested energy using the axiu downlin and uplin transit power allowances, which corresponds to the top corner point in Figure 2. In fact, this is the optial solution of single objective optiization Proble 3 which can be found by solving 23 with λ 1 = λ 2 = 0 and λ 3 = 1. Siilarly, the other two extree points in the left and right corners correspond to the solutions of single-objective Probles 1 and 2, which are obtained fro the extree cases of 23 for λ 1 = 1 and λ 2 = 1, respectively. B. Average Uplin and Downlin Transit Powers In Figure 3, we show the trade-off between uplin and downlin transit power iniization for different iniu required downlin SINRs,. In particular, we select resource allocation policies with λ 3 = 0 and λ 1 +λ 2 = 1. The points are obtained by solving 23 for different pairs of λ 1 and λ 2. For coparison, we adopt a baseline schee based on HD counication, where a HD radio BS is eployed

6 Average harvested energy db Average total downlin transit power db Average total uplin transit power db Average downlin transit power db Perforance gain FD, SINR DL = 21 db FD, SINR DL = 15 db Baseline HD, SINR DL = 21 db Baseline HD, SINR DL = 15 db Perforance gain Average uplin transit power db Fig. 2. Trade-off region between uplin transit power iniization, downlin transit power iniization, and total harvested energy axiization for N = 8. Fig. 3. Average downlin transit power db versus average uplin transit power db. The double-sided arrows indicate the power saving due to FD counication. for transission and reception in alternating tie slots. In other words, for a given tie interval, the required data rates for uplin and downlin transissions in each HD slot are given by Rate HD UL = 2 log1 + Γ UL req,,, and = 2 log1 +,, respectively. Thus, the Rate HD DL required uplin and downlin SINRs in HD transission are given by Γ HD UL req, = 1 + Γ UL req, 2 1 and Γ HD DL 1 + = 2 1, respectively. Additionally, both SI and CCI can be avoided in the HD scenario. The baseline schee is designed to achieve the optial trade-off between the three considered objectives in a HD syste with identical sets of weights as for the proposed FD algorith. In the baseline schee, we optiize the sae variables {Q, w, P} and ipose the sae QoS requireents as in the FD case, and also apply ZF-BF detection. As shown in Figure 3, significant power savings can be achieved by the proposed FD resource allocation algorith copared to the HD syste, as indicated by the double-sided arrows. Furtherore, when the downlin SINR required becoes less stringent, e.g. fro = 21 db to ΓDL = 15 db, both the uplin and downlin transit powers can be reduced siultaneously. This is due to the following two reasons. First, a saller downlin transit power is required to satisfy the less stringent downlin SINR requireents. Second, the decrease in downlin transit power reduces the self-interference ipairing the uplin signal reception which iproves the uplin transit power efficiency. C. Average Total Harvested Power In Figure 4, we show a trade-off between total harvested power axiization and downlin transit power iniization. In particular, we select resource allocation policies with λ 2 = 0 and λ 1 + λ 3 = 1. The points are obtained by solving 23 for different pairs of λ 1 and λ 3. Besides, the HD baseline schee is adopted again for coparison. As can be observed, the proposed FD schee is able to provide a larger tradeoff region copared to the baseline schee. In particular, although the FD schee suffers fro self-interference, it can facilitate power-efficient SWIPT via the proposed resource allocation optiization. Besides, a ore stringent downlin iniu SINR requireent reduces the size of the trade-off Average total harvested power db FD, SINR DL = 21 db FD, SINR DL = 15 db Baseline HD, SINR DL = 21 db Baseline HD, SINR DL = 15 db Perforance gain Average downlin transit power db Fig. 4. Average total harvested power db versus the average downlin transit power db. The double-sided arrows indicate the syste perforance gain due to FD counication. region achieved by the proposed FD counication schee. In fact, the ore stringent downlin iniu SINR requireent reduces the feasible solution set of optiization proble 23 which yields a less flexible resource allocation. V. CONCLUSION In this paper, we designed a resource allocation algorith for ultiuser FD SWIPT systes. We proposed a MOO fraewor based on the weighted Tchebycheff ethod to study the trade-off between uplin transit power iniization, downlin transit power iniization, and total harvested energy axiization. The non-convex optiization proble was transfored into an equivalent ran-constrained SDP and solved optially by SDP relaxation. The proposed algorith provided a set of resource allocation policies and deonstrated a rearable perforance gain in power consuption copared to a baseline algorith eploying conventional HD transission.

7 APPENDIX-PROOF OF THEOREM 1 Theore 1 can be proved by following a siilar approach as in [7] via investigating the Karush-Kuhn-Tucer KKT optiality conditions of the SDP relaxed proble 23. The proof can be divided into two parts. In the first part, we prove that the optial energy beaforing signal satisfies RanQ 1. First of all, we introduce the Lagrangian of the proble as follows LW, Q, P, τ, α, β, γ, δ, µ, ν, X, Y, ρ 1, ρ 2, ρ 3 24 = τ + α [Tr K W +Q ] Pax DL + ν P Pax, UL TrH W β P Tr G Z γ Γ UL req, I DL σdl, 2 TrX W TrYQ I UL σ 2 ULTrZ [ δ j Pj EH P in,j +ρ 1 λ 1 Tr K [ M +ρ 2 λ 2 P F2 τ ]+ρ 3 [λ 3 µ P W +Q F 1 ] τ ] Pj EH F3 τ, where α, β, γ, δ, µ, ν, X, Y, ρ 1, ρ 2, and ρ 3 are dual variables corresponding to the associated constraints. β, γ, δ j, µ, and ν are the eleents of dual variables β, γ, δ, µ, and ν, respectively. Since the SDP relaxed proble satisfies S- later s constraint qualification and is convex with respect to the optiization variables, strong duality holds. Denote the optial prial solution as {W, Q, P }, and the optial dual variables as {α, β, γ, δ, µ, ν, X, Y, ρ 1, ρ 2, ρ 3}. Then, the KKT conditions used for the proof are given by: Y = α + ρ 1λ 1 I V, where 25 V = η δj+ρ 3λ 3 Ω j Ω H j γh H SI diagz H SI, 26 X = Y + βi H i β H,, 27 i Y Q = 0, X W = 0,. 28 Since we have for the Lagrangian ultiplier Y 0, inequality α + ρ 1λ 1 ξ ax ust hold, where ξ ax is the largest eigenvalue of V. If α + ρ 1λ 1 = ξ ax, then RanY = N 1. According to the copleentary slacness condition in 28, Q lies in the null space spanned by the colun vectors of Y. Thus, RanQ 1. On the other hand, when α + ρ 1λ 1 > ξ ax holds, we have RanY = N and Q = 0. As a result, RanQ 1 ust be satisfied. In other words, at ost one energy bea is needed to achieve the syste design objectives. Next, we prove the second part, i.e., RanW = 1,. It can be verified that β > 0 for ΓDL > 0. Besides, as proved in the first part, we have RanY N 1. 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