Multi-Objective Resource Allocation in Full-Duplex SWIPT Systems
Shiyang Leng, Derrick Wing Kwan Ng, Nikola Zlatanov, Robert Schober
MMulti-Objective Resource Allocation in Full-DuplexSWIPT Systems
Shiyang Leng § , Derrick Wing Kwan Ng ∗ , Nikola Zlatanov † , and Robert Schober ‡ The Pennsylvania State University, USA § The University of New South Wales, Australia ∗ Monash University, Australia † Friedrich-Alexander-University Erlangen-N¨urnberg (FAU), Germany ‡ Abstract —In this paper, we investigate the resource allocationalgorithm design for full-duplex simultaneous wireless informa-tion and power transfer (FD-SWIPT) systems. The consideredsystem comprises a FD radio base station, multiple single-antennahalf-duplex (HD) users, and multiple energy harvesters equippedwith multiple antennas. We propose a multi-objective optimiza-tion framework to study the trade-off between uplink transmitpower minimization, downlink transmit power minimization, andtotal harvested energy maximization. The considered optimiza-tion framework takes into account heterogeneous quality ofservice requirements for uplink and downlink communicationand wireless power transfer. The non-convex multi-objectiveoptimization problem is transformed into an equivalent rank-constrained semidefinite program (SDP) and solved optimally bySDP relaxation under certain general conditions. The solutionof the proposed framework results in a set of Pareto optimalresource allocation policies. Numerical results unveil an interest-ing trade-off between the considered conflicting system designobjectives and reveal the improved power efficiency facilitatedby FD in SWIPT systems compared to traditional HD systems.
I. I
NTRODUCTION
Next generation communication systems aim at providingself-sustainability and high data rates to communication net-works with guaranteed quality of service (QoS). A promisingtechnique for prolonging the lifetime of communication net-works is energy harvesting (EH). Among different EH tech-nologies, wireless power transfer (WPT) via electromagneticwaves in radio frequency (RF) enables comparatively control-lable EH at the receivers compared to conventional naturalsource, such as wind, solar, and tidal. Recent progress in thedevelopment of RF-EH circuitries has made RF-EH practicalfor low-power consumption devices [1]–[3], e.g. wirelesssensors. In particular, RF-EH enables simultaneous wirelessinformation and power transfer (SWIPT) [4]–[7]. Thereby,as a carrier of both information and energy, the RF signalunifies information transmission and power transfer. Besides,RF-EH advocates energy saving by recycling the energy in theRF radiated by ambient transmitters. In SWIPT systems, theamount of harvested energy is an equally important QoS metricas the data rate and the transmit power consumption whichare traditionally considered in communication networks. Thus,resource allocation algorithms for SWIPT systems shouldalso take into account the emerging need for energy transfer[8]–[10]. In [8], energy-efficient SWIPT was investigated inmulticarrier systems, where power allocation, user scheduling,and subcarrier allocation were considered. In [9], the authorsproposed a power allocation scheme for energy efficiency
Robert Schober is also with the University of British Columbia. This workwas supported in part by the AvH Professorship Program of the Alexandervon Humboldt Foundation. maximization of large scale multiple-antenna SWIPT systems.In [10], multi-objective optimization (MOO) was applied tojointly optimize multiple system design objectives to facilitatesecure SWIPT systems. Although SWIPT has been alreadyconsidered for various system setups, the power efficiency ofSWIPT systems [8]–[10], has not been fully investigated andis still unsatisfactory due to the traditional half-duplex (HD)operation.Recently, full-duplex (FD) communication has become aviable option for next generation wireless communicationnetworks. In contrast to conventional HD transmission, FDcommunication allows devices to transmit and receive simul-taneously on the same frequency, thus potentially doublingthe spectral efficiency. In practice, the self-interference (SI)caused by the own transmit signal impairs the simultaneoussignal reception in FD systems severely which has been amajor obstacle for the implementation of FD devices in thepast decades. Fortunately, breakthroughs in analog and digitalself-interference cancellation (SIC) techniques [11] have madeFD communication more practical in recent years. However,various practical implementation issues, such as protocol andresource allocation algorithm design, need to be reinvestigatedin the context of FD communications [12]–[15]. In [12], theauthors proposed a suboptimal beamformer design to maxi-mize the spectral efficiency of FD small cell wireless systems.In [13], resource allocation and scheduling was studied for FDmultiple-input multiple-output orthogonal frequency divisionmultiple access (MIMO-OFDMA) relaying systems. More-over, the energy efficiency of FD-OFDMA relaying systemswas investigated in [14]. The authors of [15] proposed a multi-objective resource allocation algorithm for FD systems byconsidering the trade-off between uplink and downlink trans-mit power minimization. Although FD communication hasdrawn significant research interest [12]–[15], research on FDSWIPT systems is still in its infancy. Lately, the notion of FDcommunication in EH systems has been pursued. Specifically,the combination of FD and WPT was first considered in [16].The authors optimized the resource allocation in a system withWPT in the downlink and wireless information transmission inthe uplink. In [17], the performance of a dual-hop full-duplexrelaying SWIPT system was studied. However, simultaneousuplink and downlink communication has not been studiedthoroughly for SWIPT systems. In fact, uplink and downlinktransmission occurs simultaneously in FD systems and theassociated information signals can also serve as vital energysources for RF energy harvesting. As a result, different trade-off naturally arise in FD-SWIPT systems when considering theaspects of uplink and downlink transmission as well as EH.These observations motivate us to design a flexible resource a r X i v : . [ c s . I T ] J a n nergy signal U p l i n k s i g n a l Self-interference Co-channel interference
Uplink userDownlink userRoaming user (energy harvester)
Full-duplex base station D o w n li n k s i gna l Fig. 1. Multiuser FD SWIPT system with a FD radio base station, M =1 uplink user, K = 1 downlink user, and J = 1 roaming user (energyharvester). allocation algorithm for FD SWIPT systems which strikes abalance between the different system design objectives.The rest of the paper is organized as follows. In SectionII, we outline the system model for the considered FDSWIPT networks. In Section III, we formulate the multi-objective resource allocation algorithm design as a non-convexoptimization problem and solve this problem by semidefiniteprogramming relaxation. In Section IV, we present numericalperformance results for the proposed optimal algorithm. InSection V, we conclude with a brief summary of our results.II. S YSTEM M ODEL
In this section, we first introduce the notation adopted inthis paper. Then, we discuss the signal model for FD SWIPTnetworks.
A. Notation
We use boldface capital and lower case letters to denotematrices and vectors, respectively. A H , Tr( A ) , and Rank( A ) represent the Hermitian transpose, trace, and rank of matrix A ,respectively; diag( A ) returns a diagonal matrix containing thediagonal elements of matrix A on its main diagonal; A − and A † represent the inverse and Moore-Penrose pseudoinverseof matrix A , respectively; A (cid:23) indicates that A is apositive semidefinite matrix; I N is the N × N identity matrix; C N × M denotes the set of all N × M matrices with complexentries; H N denotes the set of all N × N Hermitian matrices; |·| and (cid:107)·(cid:107) denote the absolute value of a complex scalarand the Euclidean vector norm, respectively;
E{·} denotesstatistical expectation; [ x ] + = max { x, } ; the circularly sym-metric complex Gaussian distribution with mean vector µ andcovariance matrix Σ is denoted by CN ( µ , Σ ) ; and ∼ standsfor “distributed as”. B. Signal Model
We focus on a multiuser wireless communication system.The system consists of an FD radio base station (BS), K HDdownlink users, M HD uplink users, and J roaming users,cf. Figure 1. The BS is equipped with N > antennasthat can simultaneously perform downlink transmission anduplink reception in the same frequency band [11]. All uplinkand downlink users are single-antenna devices to limit thehardware complexity. On the other hand, to facilitate efficient EH, we assume that the roaming users are multiple-antennadevices, which are equipped with N EH > antennas.For downlink FD communication, K independent signalstreams are transmitted simultaneously at the same frequencyto the K downlink users. The transmitted signal at the FDradio BS is given by x = K (cid:88) k =1 w k d DL k + q , (1)where d DL k ∈ C is the information bearing signal intendedfor downlink user k ∈ { , . . . , K } . Without loss of generality,we assume E{| d DL k | } = 1 . Besides, a beamforming vector w k ∈ C N × is employed to assist downlink informationtransmission. On the other hand, in order to facilitate efficientWPT to roaming users, a dedicated energy beam, q ∈ C N × ,is transmitted concurrently with the information signal. Theenergy signal q is modeled as a complex pseudo-randomsequence with covariance matrix Q = E{ qq H } . In general,both pseudo-random signals and constant amplitude signalsare potential candidates for implementing the energy signal.However, pseudo-random energy signals can be shaped moreeasily to satisfy certain requirements on the spectrum maskof the transmit signal and are thus adopted in this paper.In particular, we assume that q is generated at the BS bya pseudo-random sequence generator with a predefined seed.This seed information is known at the downlink users. Thus,the interference caused by the energy signal can be completelycancelled at the downlink users before decoding the desiredsignals. C. Channel Model
We consider a narrow-band slow fading channel. The re-ceived signal at downlink user k is given by y DL k = h Hk x + M (cid:88) m =1 (cid:112) P m f m,k d UL m (cid:124) (cid:123)(cid:122) (cid:125) co-channel interference + n DL k , (2)where h k ∈ C N × denotes the channel vector between theBS and downlink user k . The second term in (2) denotesthe co-channel interference (CCI) caused by simultaneousuplink transmission in the FD system. f m,k ∈ C is thechannel gain between uplink user m and downlink user k . d UL m and P m denote the uplink transmit signal from uplinkuser m and the corresponding transmit power, respectively.We assume E{| d UL m | } = 1 without loss of generality. n DL k ∼ CN (0 , σ ,k ) denotes the additive white Gaussiannoise (AWGN) at downlink user k .At the same time, the FD BS receives signals from M uplinkusers simultaneously. The corresponding received signal isgiven by y UL = M (cid:88) m =1 (cid:112) P m g m d UL m + c (cid:124)(cid:123)(cid:122)(cid:125) self-interference cancellation noise + n UL , (3)where g m ∈ C N × denotes the channel vector between uplinkuser m and the BS. Vector n UL ∈ C N × represents the In this paper, we adopt a normalized unit energy, i.e., Joule-per-second.Thus, the terms “energy” and “power” are interchangeable.
WGN distributed as CN ( , σ I N ) . Due to the concurrentuplink reception and downlink transmission at the FD radioBS, the SI caused by the downlink transmit signal impairsthe uplink signal reception. In practice, different interferencemitigation techniques such as antenna cancellation, baluncancellation, and circulators [18], [19] have been proposedto alleviate the impairment caused by SI. In order to iso-late the resource allocation algorithm design from the spe-cific implementation of self-interference mitigation, we modelthe self-interference cancellation induced noise by vector c ∼ CN ( , (cid:37) diag( E{ H SI ( xx H ) H H SI } )) [19, Eq. (4)], where H SI ∈ C N × N is the self-interference channel and ≤ (cid:37) (cid:28) is a constant indicating the noisiness of the self-interferencecancellation at the FD BS.In the considered system, both downlink and uplink signals act as energy sources to the roaming users (energy harvesters).The received signal at energy harvester j ∈ { , . . . , J } is y EH j = Ω Hj x + M (cid:88) m =1 φ j,m (cid:112) P m d UL m + n EH j , (4)where matrix Ω j ∈ C N × N EH and vector φ j,m ∈ C N EH × denote the channel between the BS and energy harvester j ,and the channel between uplink user m and energy harvester j , respectively. Vector n EH j ∈ C N EH × represents the AWGNat energy harvester j distributed as CN ( , σ I N R ) .We note that all channel variables, i.e., h k , f m,k , g m , H SI , Ω j , and φ j,m , capture the joint effect of path loss and smallscale fading. III. P ROBLEM F ORMULATION
In this section, we first introduce the adopted QoS metrics.Then, from the perspectives of uplink power consumption,downlink power consumption, and EH, we formulate threesingle objective optimization problems. In practice, these threesystem design objectives are all desirable but conflicting. Thus,we apply a MOO framework to study multi-objective resourceallocation algorithm design.
A. Quality of Service Metrics
We assume that full channel state information (CSI) is avail-able at the FD BS for resource allocation. The receive signal-to-interference-plus-noise-ratio (SINR) at downlink user k isgiven by Γ DL k = | h Hk w k | K (cid:88) i (cid:54) = k | h Hk w i | + M (cid:88) m =1 P m | f m,k | + σ ,k , (5)where the interference from the energy beamforming signal,i.e., Tr( h Hk Qh k ) , has already been cancelled since energysignal q is known to the downlink users.For uplink transmission, we adopt zero-forcing beamform-ing (ZF-BF) for detection at the BS. In contrast to optimal min-imum mean square error beamforming (MMSE-BF) detection,ZF-BF facilitates the design of resource allocation algorithmsin the considered problem. Additionally, the performance ofZF-BF converges to the performance of MMSE-BF in the highSINR regime [20], which is the desired operating region of the In general, the adopted system model can be extended to scenarios inwhich the uplink users also transmit energy signal to facilitate EH. However,it may increase the peak-to-average power ratio and is not suitable for uplinkusers equipped with low cost power amplifiers. considered system. Therefore, the receive SINR at the BS withrespect to uplink user m ∈ { , . . . , M } can be expressed as Γ UL m = P m | g Hm z m | M (cid:88) i (cid:54) = m P i | g Hi z m | + S UL m + σ (cid:107) z m (cid:107) , (6)where S UL m = (cid:37) z Hm (cid:16) diag (cid:0) H SI (cid:0) K (cid:88) k =1 w k w Hk + Q (cid:1) H H SI (cid:1)(cid:17) z m (7)is the noise caused by SI cancellation and z m ∈ C N × denotesthe ZF-BF receive vector for decoding the signal of uplink user m . The ZF-BF matrix is given by Z = [ z , . . . , z M ] T = ( G H G ) − G H , (8)where G = [ g , . . . , g M ] . On the other hand, the total amount of harvested energy atenergy harvester j ∈ { , . . . , J } is given by P EH j = η j (cid:20) Tr (cid:16) Ω Hj ( K (cid:88) k =1 w k w Hk + Q ) Ω j (cid:17) + M (cid:88) m =1 P m (cid:107) φ j,m (cid:107) (cid:21) , (9)where ≤ η j ≤ is the energy conversion efficiency ofenergy harvester j . It represents the energy loss in convertingthe received RF energy to electrical energy for storage. Notethat the thermal noise power is ignored in (9) for EH as it isnegligibly small compared to the power of the received signals. B. Optimization Problem Formulation
In FD SWIPT systems, downlink transmit power minimiza-tion, uplink transmit power minimization, and total harvestedenergy maximization are all desirable system design objec-tives. Now, we first propose three single-objective optimizationproblems with respect to these objectives.
Problem 1: Downlink Transmit Power Minimization: minimize Q ∈ H N , w , P K (cid:88) k =1 (cid:107) w k (cid:107) + Tr( Q )s . t . C1 : K (cid:88) k =1 (cid:107) w k (cid:107) + Tr( Q ) ≤ P DLmax , C2 : P m ≤ P ULmax ,m , ∀ m, C3 : Γ DL k ≥ Γ DLreq ,k , ∀ k, C4 : Γ UL m ≥ Γ ULreq ,m , ∀ m, C5 : P EH j ≥ P min ,j , ∀ j, C6 : P m ≥ , ∀ m, C7 : Q (cid:23) , (10)where w = { w k , ∀ k } and P = { P m , ∀ m } denote the down-link beamforming vector policy and the uplink transmit powerpolicy, respectively. In (10), we minimize the total downlinktransmit power by jointly optimizing downlink informationbeamforming vectors w k , ∀ k , the covariance matrix of energysignal, Q , and uplink transmit power P m , ∀ m . Constants P DLmax and P ULmax ,m in C1 and C2 denote the maximum downlinktransmit power for the FD BS and the maximum transmitpower of uplink user m , respectively. QoS requirements ofreliable communication are taken into account in C3 and C4.In particular, Γ DLreq ,k > , ∀ k, and Γ ULreq ,m > , ∀ m, are theminimum required SINRs for the downlink and uplink users,espectively. P min ,j , ∀ j, in C5 is the minimum required amountof harvested energy for energy harvester j . In addition, C6 andC7 enforce the non-negative uplink transmit power constraintsand the positive semidefinite Hermitian matrix constraint forcovariance matrix Q , respectively.On the other hand, for the system designs with the objectivesof uplink transmit power minimization and total harvested en-ergy maximization, respectively, we have the same constraintset as for Problem 1. Therefore, the problem formulationsfor these two other system design objectives are given as,respectively, Problem 2: Uplink Transmit Power Minimization: minimize Q ∈ H N , w , P M (cid:88) m =1 P m s . t . C1 − C7 , (11) Problem 3: Total Harvested Energy Maximization: maximize Q ∈ H N , w , P J (cid:88) j =1 P EH j s . t . C1 − C7 . (12)The interdependency between the aforementioned objectives isnon-trivial in the considered FD SWIPT system. For instance,although a large transmit power ensures high received SINRsat the downlink users, the strong SI impairs the receptionof the uplink signals at the FD BS. Similarly, increasingthe uplink transmit power to satisfy a more stringent uplinkSINR requirement will lead to severe CCI which degradesthe downlink signal reception. On the other hand, the EHQoS requirement has to be fulfilled by transferring a sufficientamount of power in both uplink and downlink. Yet, minimizingeither uplink or downlink transmit power conflicts with theobjective of having a higher power for EH. Hence, a non-trivial trade-off between these three system design objectivesnaturally arises in the considered FD SWIPT system. Thus, aflexible resource allocation algorithm which can accommodatediverse system design preferences is desired. To this end, weapply MOO to systematically address this resource allocationproblem.In the literature, MOO is commonly adopted as a math-ematical framework to study the trade-off between multipledesirable but conflicting system design objectives. The optimalsolution of a MOO program (MOOP) is defined by a Paretooptimal set; a set of points that satisfy the concept of Paretooptimality [10]. In the following, we formulate a MOOPbased on the weighted Tchebycheff method [10], in whichthe preferences for the aforementioned single system designobjectives are quantified by a set of pre-specified weights. Infact, compared to other approaches to formulate MOOPs, theweighted Tchebycheff method can provide a complete Paretooptimal set by varying the weights, even if the MOOP isnon-convex. For the sake of notational simplicity, we denotethe objective functions of Problems 1–3 as F n ( Q , w , P ) , n ∈ { , , } , respectively. Then, the MOOP is given by Problem 4: Multi-Objective Optimization: minimize Q ∈ H N , w , P max n =1 , , (cid:110) λ n (cid:16) F n ( Q , w , P ) − F ∗ n (cid:17)(cid:111) s . t . C1 − C7 , (13)where F ∗ n is the optimal objective value with respect toProblem n ∈ { , , } . In order to represent the three single system design objective functions in a unified manner, withoutloss of generality, the maximization in Problem 3 was rewrittenas an equivalent minimization. As a result, F ( Q , w , P ) inProblem 4 is given by F ( Q , w , P ) = − (cid:80) Jj =1 P EH j . Constant λ n is a weight imposed on the n -th objective function subjectto ≤ λ n ≤ and (cid:80) n λ n = 1 , which indicates the preferenceof the system designer for the n -th objective function over theothers. We can obtain a set of resource allocation policiesby solving Problem 4 for different predefined weights. In theextreme case, when λ n = 1 and λ l = 0 , ∀ l (cid:54) = n , Problem 4 isequivalent to the n -th single-objective optimization problem. C. Optimal Solution
Problems 1-4 are non-convex optimization problems dueto the non-convex constraints C3 and C4. To overcome thenon-convexity, we recast these problems as SDPs via SDPrelaxation. To this end, we define new variables W k = w k w Hk , H k = h k h Hk , G m = g m g Hm , (14) Z m = z m z Hm , and Φ j,m = φ j,m φ Hj,m . (15)Thus, the considered problems can be equivalently transformedas follows: Transformed Problem 1: minimize W , Q ∈ H N , P Tr (cid:16) K (cid:88) k =1 W k + Q (cid:17) s . t . C2 , C6 , C7 , C1 : Tr (cid:0) K (cid:88) k =1 W k + Q (cid:1) ≤ P max , C3 : Tr( H k W k )Γ DLreq ,k ≥ I DL k + σ ,k , ∀ k, C4 : P m Tr (cid:0) G m Z m (cid:1) Γ ULreq ,m ≥ I UL m + σ Tr( Z m ) , ∀ m, C5 : P EH j ≥ P min ,j , ∀ j, C8 : W k (cid:23) , ∀ k, C9 : Rank( W k ) ≤ , ∀ k, (16)where I DL k = K (cid:88) i (cid:54) = k Tr( H k W i ) + M (cid:88) m =1 P m | f m,k | , (17) I UL m = M (cid:88) i (cid:54) = m P i Tr( G i Z m )+ (cid:37) Tr (cid:16) Z m diag (cid:0) H SI (cid:0) K (cid:88) k =1 W k + Q (cid:1) H H SI (cid:1)(cid:17) , (18) P EH j = η j (cid:104) Tr (cid:16) Ω Hj (cid:0) K (cid:88) i = k W k + Q (cid:1) Ω j (cid:17) + M (cid:88) m =1 P m Tr( Φ j,m ) (cid:105) , (19)and W = { W k , ∀ k } is the set of downlink beamforming ma-trices to be optimized. Constraints C8, C9, and W k ∈ H N areintroduced due to the definition of W k . Similarly, Problems2-4 are equivalently transformed to Transformed Problem 2: minimize W , Q ∈ H N , P M (cid:88) m =1 P m s . t . C1 − C9 . (20) Here, equivalent means that both problems have the same solution. ransformed Problem 3: maximize W , Q ∈ H N , P J (cid:88) j =1 P EH j s . t . C1 − C9 . (21) Transformed Problem 4: maximize W , Q ∈ H N , P ,τ τ s . t . C1 − C9 , C10 : λ n (cid:16) F n ( Q , W , P ) − F ∗ n (cid:17) ≤ τ, n ∈{ , , } , (22)where τ is an an auxiliary optimization variable [21].Evidently, Transformed Problem 4 is a generalization ofTransformed Problems 1-3. Hence, we focus on the method-ology for solving Transformed Problem 4 in the following.Transformed Problem 4 is non-convex due to the rank-one ma-trix constraint C9. To obtain a tractable problem formulation,we apply SDP relaxation. Specifically, we relax constraint C9in (22) by removing it from the problem. Then, the consideredproblem becomes maximize W , Q ∈ H N , P ,τ τ s . t . C1 − C8 , C10 : λ n (cid:16) F n ( Q , W , P ) − F ∗ n (cid:17) ≤ τ, n ∈{ , , } . (23)We note that the rank constraint relaxed problem in (23) isa convex SDP which can be solved by standard numericalconvex program solvers such as CVX [22]. In particular, ifthe obtained solution of the relaxed problem satisfies the rank-one constraint, i.e., Rank( W ∗ k ) ≤ , then the solution of (23)is the optimal solution of the original Problem 4. Thus, theoptimal beamforming vector w ∗ k of the original problem canbe retrieved by solving the relaxed problem. Now, we revealthe tightness of the SDP relaxation by the following theorem. Theorem 1:
Assuming that the channels Ω j , H SI , and h k ,are statistically independent, the optimal beamforming matrixfor (23) is a rank-one matrix, i.e., Rank( W ∗ k ) = 1 , ∀ k , andthe energy beamforming matrix satisfies Rank( Q ∗ ) ≤ withprobability one for Γ DLreq k > . Proof:
Please refer to the Appendix.In other words, whenever the channels satisfy the generalcondition stated in Theorem 1, the adopted SDP relaxation istight. Hence, the optimal solution of the original MOOP can beobtained by solving the relaxed SDP problem in (23). Besides,information beamforming, i.e.,
Rank( W ∗ k ) = 1 , and energybeamforming, i.e., Rank( Q ∗ ) ≤ , is optimal for optimizingthe considered conflicting objective functions. On the otherhand, the optimal solutions of the single-objective problemscan be achieved by solving special cases of (23). For instance,the solution of single-objective Problem 1 can be obtained bysolving (23) with λ = 1 , λ = 0 , and λ = 0 .IV. R ESULTS
In this section, we investigate the performance of theproposed multi-objective resource allocation algorithm. Theimportant simulation parameters are summarized in Table I.We evaluate a system with an FD radio BS located at thecenter of a cell. Furthermore, K = 3 downlink users and M = 8 uplink users located in the range between the referencedistance of meters and the maximum distance of meters. TABLE IS
IMULATION P ARAMETERS .Carrier center frequency 915 MHzBandwidth 200 kHzAntenna gain at FD BS 10 dBiAntennas gain at users 0 dBiDownlink user noise power -71 dBmBS noise power -83 dBmSI cancellation constant (cid:37) -110 dBEnergy conversion efficiency, η j J = 2 energy harvesters are located close to the FD BS at adistance of between to meters in order to facilitate EH.Each energy harvester is equipped with N EH = 3 antennas.The small scale fading of the uplink and downlink channels ismodeled as independent and identically distributed Rayleighfading. The EH channel and the SI channel are modeled asRician fading channels with Rician factor dB. The maximumtransmit power supply in downlink and uplink are P DLmax = 46 dBm and P ULmax ,m = 23 dBm, ∀ m , respectively. Without loss ofgenerality, we assume that the required SINRs at all downlinkusers are identical. Besides, we specify Γ ULreq ,m = 15 dBm, ∀ m , for uplink users. At the energy harvesters, the minimumrequired harvested energy is P min ,j = − dBm, ∀ j . A. Trade-off Region of Multiple Design Objectives
Figure 2 depicts the trade-off region for uplink transmitpower minimization, downlink transmit power minimization,and total harvested energy maximization achieved by theproposed optimal scheme. There are N = 8 transmit antennasat the BS and the minimum required downlink SINR is Γ DLreq ,k = 21 dBm, ∀ k . The points shown for the trade-offregion were obtained by solving the SDP relaxed problem fordifferent sets of weights ≤ λ n ≤ , n = 1 , , subjectto (cid:80) n λ n = 1 . As can be seen, there is a nontrivial trade-off between uplink and downlink transmit power minimizationand total harvested energy maximization. In particular, for afixed weight λ for EH maximization, the downlink trans-mit power monotonically decreases for an increasing uplinktransmit power which suggests that downlink transmit powerminimization and uplink transmit power minimization conflictwith each other. On the other hand, the objective of totalharvested energy maximization does not align with the ob-jectives of uplink and downlink transmit power minimization.It can be seen that the amount of harvested energy can only beincreased by transmitting with higher uplink and/or downlinktransmit power. In particular, the resource allocation policymaximizes the harvested energy using the maximum downlinkand uplink transmit power allowances, which corresponds tothe top corner point in Figure 2. In fact, this is the optimalsolution of single objective optimization Problem 3 whichcan be found by solving (23) with λ = λ = 0 and λ = 1 . Similarly, the other two extreme points in the left andright corners correspond to the solutions of single-objectiveProblems 1 and 2, which are obtained from the extreme casesof (23) for λ = 1 and λ = 1 , respectively. B. Average Uplink and Downlink Transmit Powers
In Figure 3, we show the trade-off between uplink anddownlink transmit power minimization for different minimumrequired downlink SINRs, Γ DLreq ,k . In particular, we selectresource allocation policies with λ = 0 and λ + λ = 1 . Thepoints are obtained by solving (23) for different pairs of λ and λ . For comparison, we adopt a baseline scheme based
10 15 20 25 30203040−15−10−5051015 A v e r a g e t o t a l u p li n k t r a n s m i t p o w e r ( d B m ) A v e r age t o t a l do w n li n k t r an s m i t po w e r ( d B m ) A v e r age ha r v e s t ed ene r g y ( d B m ) Fig. 2. Trade-off region between uplink transmit power minimization, down-link transmit power minimization, and total harvested energy maximizationfor N = 8 . on HD communication, where a HD radio BS is employedfor transmission and reception in alternating time slots. Inother words, for a given time interval, the required datarates for uplink and downlink transmissions in each HD slotare given by Rate HD − UL m = 2 log(1 + Γ ULreq ,m ) , ∀ m , and Rate HD − DL k = 2 log(1 + Γ DLreq ,k ) , ∀ k , respectively. Thus, therequired uplink and downlink SINRs in HD transmission aregiven by Γ HD − ULreq ,m = (1 + Γ ULreq ,m ) − and Γ HD − DLreq ,k =(1 + Γ DLreq ,k ) − , respectively. Additionally, both SI and CCIcan be avoided in the HD scenario. The baseline schemeis designed to achieve the optimal trade-off between thethree considered objectives in a HD system with identicalsets of weights as for the proposed FD algorithm. In thebaseline scheme, we optimize the same variables { Q , w , P} and impose the same QoS requirements as in the FD case,and also apply ZF-BF detection. As shown in Figure 3,significant power savings can be achieved by the proposedFD resource allocation algorithm compared to the HD system,as indicated by the double-sided arrows. Furthermore, whenthe downlink SINR required becomes less stringent, e.g. from Γ DLreq ,k = 21 dB to Γ DLreq ,k = 15 dB, both the uplink anddownlink transmit powers can be reduced simultaneously. Thisis due to the following two reasons. First, a smaller downlinktransmit power is required to satisfy the less stringent downlinkSINR requirements. Second, the decrease in downlink transmitpower reduces the self-interference impairing the uplink signalreception which improves the uplink transmit power efficiency. C. Average Total Harvested Power
In Figure 4, we show a trade-off between total harvestedpower maximization and downlink transmit power minimiza-tion. In particular, we select resource allocation policies with λ = 0 and λ + λ = 1 . The points are obtained by solving(23) for different pairs of λ and λ . Besides, the HD baselinescheme is adopted again for comparison. As can be observed,the proposed FD scheme is able to provide a larger trade-off region compared to the baseline scheme. In particular,although the FD scheme suffers from self-interference, it canfacilitate power-efficient SWIPT via the proposed resourceallocation optimization. Besides, a more stringent downlink A v e r age do w n li n k t r an s m i t po w e r ( d B m ) FD, SINR
DLreq,k = 21 dBFD, SINR
DLreq,k = 15 dBBaseline HD, SINR
DLreq,k = 21 dBBaseline HD, SINR
DLreq,k = 15 dBPerformance gainPerformance gain
Fig. 3. Average downlink transmit power (dBm) versus average uplinktransmit power (dBm). The double-sided arrows indicate the power savingdue to FD communication.
10 15 20 25 30 35 40 45−15−10−5051015 Average downlink transmit power (dBm) A v e r age t o t a l ha r v e s t ed po w e r ( d B m ) FD, SINR
DLreq,k = 21 dBFD, SINR
DLreq,k = 15 dBBaseline HD, SINR
DLreq,k = 21 dBBaseline HD, SINR
DLreq,k = 15 dBPerformance gain
Fig. 4. Average total harvested power (dBm) versus the average downlinktransmit power (dBm). The double-sided arrows indicate the system perfor-mance gain due to FD communication. minimum SINR requirement reduces the size of the trade-offregion achieved by the proposed FD communication scheme.In fact, the more stringent downlink minimum SINR require-ment reduces the feasible solution set of optimization problem(23) which yields a less flexible resource allocation.V. C
ONCLUSION
In this paper, we designed a resource allocation algorithmfor multiuser FD SWIPT systems. We proposed a MOO frame-work based on the weighted Tchebycheff method to studythe trade-off between uplink transmit power minimization,downlink transmit power minimization, and total harvestedenergy maximization. The non-convex optimization problemwas transformed into an equivalent rank-constrained SDP andsolved optimally by SDP relaxation. The proposed algorithmprovided a set of resource allocation policies and demon-strated a remarkable performance gain in power consumptioncompared to a baseline algorithm employing conventional HDtransmission.
PPENDIX -P ROOF OF T HEOREM
Rank( Q ∗ ) ≤ . First of all, we introduce the Lagrangian ofthe problem as follows L ( W , Q , P , τ, α, β , γ , δ , µ , ν , X , Y , ρ , ρ , ρ ) (24) = τ + α (cid:104) Tr (cid:0) K (cid:88) k =1 W k + Q (cid:1) − P DLmax (cid:105) + M (cid:88) m =1 ν m ( P m − P ULmax ,m ) − K (cid:88) k =1 β k (cid:16) Tr( H k W k )Γ DLreq ,k − I DL k − σ ,k (cid:17) − K (cid:88) k =1 Tr( X k W k ) − Tr( YQ ) − M (cid:88) m =1 γ m (cid:16) P m Tr (cid:0) G m Z m (cid:1) Γ ULreq ,m − I UL m − σ Tr( Z m ) (cid:17) − M (cid:88) m =1 µ m P m − J (cid:88) j =1 δ j (cid:16) P EH j − P min ,j (cid:17) + ρ (cid:104) λ (cid:16) Tr (cid:0) K (cid:88) k =1 W k + Q (cid:1) − F ∗ (cid:17) − τ (cid:105) + ρ (cid:104) λ (cid:16) M (cid:88) m =1 P m − F ∗ (cid:17) − τ (cid:105) + ρ (cid:104) λ (cid:16) J (cid:88) j =1 P EH j − F ∗ (cid:17) − τ (cid:105) , where α, β , γ , δ , µ , ν , X , Y , ρ , ρ , and ρ are dual variablescorresponding to the associated constraints. β k , γ m , δ j , µ m , and ν m are the elements of dual variables β , γ , δ , µ , and ν , respectively. Since the SDP relaxed problem satisfiesSlater’s constraint qualification and is convex with respect tothe optimization variables, strong duality holds. Denote theoptimal primal solution as { W ∗ , Q ∗ , P ∗ } , and the optimaldual variables as { α ∗ , β ∗ , γ ∗ , δ ∗ , µ ∗ , ν ∗ , X ∗ , Y ∗ , ρ ∗ , ρ ∗ , ρ ∗ } .Then, the KKT conditions used for the proof are given by: Y ∗ = ( α ∗ + ρ ∗ λ ) I − V , where (25) V = η J (cid:88) j =1 ( δ ∗ j + ρ ∗ λ ) Ω j Ω Hj − M (cid:88) m =1 γ ∗ m H H SI diag( Z m ) H SI , (26) X ∗ k = Y ∗ + K (cid:88) i (cid:54) = k β ∗ i H i − β ∗ k Γ DLreq ,k H k , ∀ k, (27) Y ∗ Q ∗ = , X ∗ k W ∗ k = , ∀ k. (28)Since we have for the Lagrangian multiplier Y ∗ (cid:23) , inequal-ity α ∗ + ρ ∗ λ ≥ ξ max must hold, where ξ max is the largesteigenvalue of V . If α ∗ + ρ ∗ λ = ξ max , then Rank( Y ∗ ) = N − . According to the complementary slackness conditionin (28), Q ∗ lies in the null space spanned by the columnvectors of Y ∗ . Thus, Rank( Q ∗ ) ≤ . On the other hand,when α ∗ + ρ ∗ λ > ξ max holds, we have Rank( Y ∗ ) = N and Q ∗ = . As a result, Rank( Q ∗ ) ≤ must be satisfied. Inother words, at most one energy beam is needed to achievethe system design objectives.Next, we prove the second part, i.e., Rank( W ∗ k ) = 1 , ∀ k .It can be verified that β ∗ k > for Γ DLreq ,k > . Besides, asproved in the first part, we have Rank( Y ∗ ) ≥ N − . Since allchannel variables in the system are statistically independent, Y ∗ and (cid:80) Ki (cid:54) = k β ∗ i H i span the whole signal space leading to Rank( Y ∗ + (cid:80) Ki (cid:54) = k β ∗ i H i ) = N . Then, based on the basicproperty of the rank of matrices, we obtain Rank( X ∗ k )+Rank( β ∗ k Γ DLreq ,k H k ) ≥ Rank( Y ∗ + K (cid:88) i (cid:54) = k β ∗ i H i )= ⇒ Rank( X ∗ k ) ≥ N − . (29)As W ∗ k lies in the null space of X ∗ k according to (28), Rank( W ∗ k ) ≤ holds. Considering W ∗ k (cid:54) = must hold tofulfill the downlink SINR requirement, we finally concludethat Rank( W ∗ k ) = 1 holds for the optimal solution. (cid:4) R EFERENCES[1] I. Krikidis, S. Timotheou, S. Nikolaou, G. Zheng, D. W. K. Ng, andR. Schober, “Simultaneous Wireless Information and Power Transferin Modern Communication Systems,”
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