Throughput Maximization in Two-hop DF Multiple-Relay Network with Simultaneous Wireless Information and Power Transfer
11 Throughput Maximization in Two-hop DFMultiple-Relay Network with SimultaneousWireless Information and Power Transfer
Qi Gu, Gongpu Wang, Rongfei Fan, Ning Zhang, and Zhangdui Zhong
Abstract
This paper investigates the end-to-end throughput maximization problem for a two-hop multiple-relay network, with relays powered by simultaneous wireless information and power transfer (SWIPT)technique. Nonlinearity of energy harvester at every relay node is taken into account and two models forapproximating the nonlinearity are adopted: logistic model and linear cut-off model. Decode-and-forward(DF) is implemented, and time switching (TS) mode and power splitting (PS) mode are considered.Optimization problems are formulated for TS mode and PS mode under logistic model and linear cut-offmodel, respectively. End-to-end throughput is aimed to be maximized by optimizing the transmit powerand bandwidth on every source-relay-destination link, and PS ratio and/or TS ratio on every relay node.Although the formulated optimization problems are all non-convex. Through a series of analysis andtransformation, and with the aid of bi-level optimization and monotonic optimization, etc., we find theglobal optimal solution of every formulated optimization problem. In some case, a simple yet optimalsolution of the formulated problem is also derived. Numerical results verify the effectiveness of ourproposed methods.
Index Terms
Simultaneous wireless information and power transfer (SWIPT), multiple-relay, throughput maxi-mization, nonlinear energy harvesting model.
Q. Gu and G. Wang are with the School of Computer and Information Technology, Beijing Jiaotong University, Beijing100044, P. R. China (email: { } @bjtu.edu.cn). R. Fan is with the School of Information and Electronics, BeijingInstitute of Technology, Beijing 100081, P. R. China (email:[email protected]). N. Zhang is with the Department of ComputerSciences, Texas A&M University at Corpus Christi, TX, 78412, U.S. ([email protected]). Z. Zhang is with the State KeyLaboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, Beijing, China, 100044 (email:[email protected]). a r X i v : . [ c s . I T ] A p r I. I
NTRODUCTION
Simultaneous wireless information and power transfer (SWIPT) is an emerging technicalsolution for energy-constrained wireless network and Internet of Things (IoT), which enables thetransmitter to transmit power and information simultaneously to receiver via the radio frequency(RF) signal [1]–[3]. To realize SWIPT, there are two modes: 1) Power splitting (PS) mode; 2)Time switching (TS) mode. In PS mode, there is one power splitter at the receiver, which splitsthe received signal into two parts. One part is for energy harvesting (EH) and the other partis for information decoding (ID) [4]. In TS mode, the receiver switches between EH and IDalternatively, in which one round of EH and ID is called as one period [5]. By adjusting thePS ratio or TS ratio between EH and ID, the rate of data transmission and the rate of energyharvesting can be balanced. This topic has been explored in lots of literatures [4]–[15].A special utilization of SWIPT lies in relay network, in which one or more relay nodes withno battery extracts both energy and information from the source signal through SWIPT and thenforward the received signal (in amplify-and-forward (AF) mode) or decoded information (indecode-and-forward (DF) mode) to the destination node by using the harvested energy. TheSWIPT-powered relay network can save relay node from additional power supply, and hasattracted a lot research attentions [14]–[25].Two-hop or multiple-hop relay network are considered and combinations of various systemconfigurations, e.g., PS or TS for implementing the SWIPT, DF or AF for implementing the relay,etc., are investigated in literatures. Categorized by the research goal, two classes of literaturescan be found. The first class of literature focuses on analyzing the system performance, in termsof ergodic capacity [14]–[17], effective throughput [18], or outage probability [14]–[17], [19].Specifically, [14] focuses on the DF relay network under PS mode; [15] considers the AF relaynetwork under PS mode and TS mode; [16] studies the AF and DF relay network under TSmode with full-duplex relay, which brings self-interference into the system; [17] investigatesthe AF relay network under PS mode with multiple-antenna relay and co-channel interference;[18] looks into the AF and DF relay network under TS mode; [19] pays attention to the AFnetwork under PS mode with multiple random distributed relay nodes in space and analyzed theassociated performance under various relay selection strategies. It should be noticed that all thementioned works in the first class investigate a two-hop relay network, among which [15]–[18]assume one relay while [14] and [19] assume multiple relays.
The second class of literature targets at maximizing some utility including the outage capacity[20] or end-to-end throughput [21]–[24], or minimizing some cost such as transmission time forgiven amount of data [25], by optimizing PS ratio, TS ratio, etc. Without specific clarification,two-hop relay network is set up in default in these literatures. In [20], PS ratio and TS ratio areoptimized under PS mode and TS mode in a DF relay network, respectively. In [21], multipleantennas are assumed at an AF relay, and PS ratio and antenna selection strategy are optimizedjointly. In [22], beamforming vector and PS/TS ratio are optimized with multiple antennasimplemented at source node, AF relay, and destination node. In [23], PS ratio is optimizedover multiple channels in PS mode. In [24], PS ratio and TS ratio are optimized respectively fora multi-hop DF relay network. In [25], time for energy harvesting, information decoding, andinformation forwarding at the relay nodes are scheduled jointly.For all the previously surveyed works in SWIPT-powered relay network, linear model isassumed for the energy harvester at the relay node, which indicates that the output power of theenergy harvesting circuit grows linearly with the power of input RF signal. However, measure-ments show that the practical energy harvesting circuit is subject to a non-linear model. Hencethe mismatch of energy harvesting model in surveyed literatures will lead to the degradation ofsystem performance. In [26], a nonlinear EH model based on logistic function is built, whichfits the measurement data well. Some literatures related to SWIPT [12], [27] have also taken useof this non-linear model. For the ease of discussion, we will call this kind of model as logisticmodel. In [28], a linear cut-off model is used to approximate the nonlinear feature of energyharvester, which goes with the power of input RF signal constantly, then linearly, and at lastconstantly in [28]. The linear cut-off model is also shown to be a good approximation. It shouldbe noticed that when logistic model or linear cut-off model is adopted for a SWIPT-poweredrelay network, the methods in existing literatures cannot offer a solution.In this paper, we investigate the two-hop DF relay network with a consideration of nonlinearenergy harvester under TS mode and PS mode for the first time. For the nonlinearity of energyharvester, both logistic model and linear cut-off model will be taken into account. The scenariowith multiple relay nodes is considered, which is more general and beneficial since more copiesof source signal can be utilized. Thus there are multiple links from source to destination througha relay node. End-to-end throughput is targeted to be maximized by optimizing transmit powerand bandwidth on every link and PS ratio or TS ratio on every relay node. Optimization problemsare formulated for TS mode and PS mode, respectively. • For TS mode under two nonlinear models of energy harvester, the associated optimizationproblem is non-convex. To find the global optimal solution, the original optimization prob-lem is decomposed into two levels. In the lower level, with some further transformations andby exploring the special properties of investigated problem, closed-form optimal solution isderived. In the upper level, the associated problem is transformed to be a standard monotonicoptimization problem, whose global optimal solution is achievable. • For PS mode under logistic model, with some transformations, the original optimizationis also transformed to be a standard monotonic optimization problem. Hence the globaloptimal solution is also achievable. • For PS mode under linear cut-off model, the method for PS mode under logistic model alsoapplies. However, to further save the computation complexity, we transform the originaloptimization problem to be an equivalent form and then derive the semi-closed-form solutionfor the transformed problem, which is also global optimal.The rest of this paper is organized as follows. In Section II, the system model is presented andthe research problems are formulated. Section III and Section IV present the optimal solution ofthe formulated problem in TS mode and PS mode, respectively. Section V shows the numericalresults, followed by concluding remarks in Section VI.II. S
YSTEM M ODEL AND P ROBLEM F ORMULATION
Consider a two-hop DF multiple-relay network as shown in Fig. 1, in which the source node S would like to transmit information to the destination node D via relay node R n , who has nopower supply, for n ∈ N (cid:44) { , , ..., N } . The source, destination, and N relays all have singleantenna. Denote the channel gain from S to R n as h n , the channel gain from R n to D as g n ,and the path from node S to node D through node R n as link n , for n ∈ N . A direct link fromthe source node S to the destination node D does not exist due to physical obstacles [15], [29].All the links also constitute the set N (cid:44) { , , ..., N } . In the system, all the channel gains keepstable in one fading block, and are randomly and independently distributed over fading blockswith continuous distribution function.The information is transmitted with the help of relay nodes in the following way. Denote thebandwidth allocated to link n as w n , suppose the transmit power of source node S as p n for S h DR R n R N h n h N g g n g N Fig. 1: Illustration of a two-hop multiple-relay network.link n . By assuming the total system bandwidth as w T , and total transmit power of source node S as p T , there are w n ≥ , ∀ n ∈ N , (1) N (cid:88) n =1 w n ≤ w T , (2) p n ≥ , ∀ n ∈ N , (3)and N (cid:88) n =1 p n = p T . (4)For every relay node, it should be noticed that they all have no power supply. Thus R n for n ∈ N has to harvest energy from the signal transmitted by node S . SWIPT technique is utilized,and two modes are considered: TS mode and PS mode.In TS mode, • Step 1: As shown in Fig. 2, time is divided into multiple frames with equal length T . The T is smaller than the coherence time, hence channel gains h n and g n for n ∈ N within in T keeps invariant. Within one frame, R n first harvests energy from S ’s RF signal in thetime duration between [0 , αT ] , where ≤ α ≤ . In this step, the harvested energy can bewritten as αT φ ( p T h n ) , where φ ( x ) indicates the power of harvested energy of every relaynodes’s energy harvester when the power of received energy is x . • Step 2: In the rest of time of one frame, i.e., within time duration [(1 − α ) T, T ] . The receivedsignal is left for information decoding. Without loss of generality, the feature of φ ( x ) of the every relay node’s energy harvester is assumed to be identical. EH α n T ( n )TT Communication
Fig. 2: Time frame structure of TS mode for relay node R n .In PS mode, time is also divided into multiple frames with equal length T , within which h n and g n for n ∈ N keeps invariant. But different from TS mode, as shown in Fig. 3, a fraction β n where ≤ β n ≤ , of the received signal’s power is left for energy harvesting, and a fraction (1 − β n ) of received signal’s power is left for information decoding. EH β n T Communication n Fig. 3: Time frame structure of PS mode for relay node R n .Denote the transmit power of R n as q n for n ∈ N . In TS mode, the transmit power q n = αφ ( p T h n )(1 − α ) . In PS mode, the transmit power q n = φ ( p T h n β n )) . Thus the end-to-end throughput inTS mode can be written as C t ( α, { p n } , { w n } ) (cid:44) (1 − α ) N (cid:80) n =1 min (cid:18) w n log (cid:16) p n h n σ w n (cid:17) ,w n log (cid:16) αφ ( p T h n ) g n (1 − α ) σ w n (cid:17) (cid:19) . (5)and the end-to-end throughput in PS mode can be written as C p ( { β n } , { p n } , { w n } ) (cid:44) N (cid:80) n =1 min (cid:18) w n log (cid:16) p n h n (1 − β n ) σ w n (cid:17) ,w n log (cid:16) φ ( p T h n β n ) g n σ w n (cid:17) (cid:19) . (6) where σ is the power spectrum density of noise . On the other hand, q n should be also subjectto a limit on the maximal transmit power, denoted as q max , due to the physical limit of the relaynode R n . Hence there is αφ ( p T h n )(1 − α ) ≤ q max , ∀ n ∈ N (7)in TS mode, and φ ( p T h n β n ) ≤ q max , ∀ n ∈ N (8)in PS mode.For the feature of energy harvester, as shown in Fig. 4, experimental measurements in [26]shows that the power of harvested energy first grows with the power of received energy whenthe power of received energy is larger than a threshold, and then the grows slowly and slowlyuntil it reaches up to an upper bound. To approximate this feature, two models are adopted. • Logistic Model: In this model, φ ( x ) = (cid:16) M e − a ( x − b ) − M e ab (cid:17)(cid:16) − e ab (cid:17) (9)where M represents the maximal power the energy harvester can harvest, a and b areparameters for nonlinearity. This model is broadly used when taking into account thenonlinearity of the energy harvester [12], [27]. • Linear Cut-off Model: In this model, φ ( x ) = , when x < x L ,c ( x − x L ) , when x L ≤ x ≤ x U ,c ( x U − x L ) , when x > x U . (10)Note that both the function in (9) and the function in (10) are monotonic increasing functionswith x , which is in coordination with such an intuition: More power can be harvested whenmore power is received. Fig. 4 also plots φ ( x ) versus x under logistic model and linear cut-offmodel under selected parameter setup. It can be seen that both of these two models can achievea good approximation of measurement data. When taking into security issue, a different throughput can be expressed and achieved as shown in [30], [31]. Due to thelimit of space, we will only look into the ideal case without consideration of security in this work, which is also a general casein most of related literatures.
Input RF power (mW) H a r v e s t ed po w e r ( m W ) Measurement dataNonlinear modelLinear cut-off
Fig. 4: Harvested power vs. input power.Collecting the formulated constraints, the associated optimization problem under TS modeand PS mode can be given as follows.In TS mode, the associated optimization problem is
Problem 1: max α, { p n } , { w n } C t ( α, { p n } , { w n } ) s.t. ≤ α ≤ , (11a) p n ≥ , ∀ n ∈ N , (11b) w n ≥ , ∀ n ∈ N , (11c) αφ ( p T h n )(1 − α ) ≤ q max , ∀ n ∈ N , (11d) N (cid:88) n =1 w n ≤ w T , (11e) N (cid:88) n =1 p n = p T . (11f)In PS mode, the associated optimization problem is Problem 2: max { β n } , { p n } , { w n } C p ( { β n } , { p n } , { w n } ) s.t. ≤ β n ≤ , ∀ n ∈ N , (12a) p n ≥ , ∀ n ∈ N , (12b) w n ≥ , ∀ n ∈ N , (12c) φ ( p T h n β n ) ≤ q max , ∀ n ∈ N , (12d) N (cid:88) n =1 w n ≤ w T , (12e) N (cid:88) n =1 p n = p T . (12f)In the following, we will show how to solve Problem 1 and Problem 2 under two energyharvester models, i.e., logistic model and linear cut-off model, respectively.III. O PTIMAL S OLUTION IN
TS M
ODE
In this section, Problem 1 will be solved. Note that Problem 1 is a non-convex optimizationproblem given that the function C t ( α, { p n } , { w n } ) is a non-concave function with the vector of α , { p n } , and { w n } . Thus the global optimal solution of Problem 1 is hard to achieve. In thefollowing, we will do some transformation and simplification on Problem 1, and find the globaloptimal solution of Problem 1. Attention that the presented solution in this section works forboth the case under logistic model and the case under cut-off model.To solve Problem 1 optimally, we decompose it into two levels . In the lower level, α isfixed, and the following optimization problem need to be solved This method is referred to as bi-level optimziation . Problem 3: F ( α ) (cid:44) max { p n } , { w n } N (cid:88) n =1 min (cid:18) w n log (cid:18) p n h n σ w n (cid:19) ,w n log (cid:18) αφ ( p T h n ) g n (1 − α ) σ w n (cid:19) (cid:19) s.t. p n ≥ , ∀ n ∈ N , (13a) w n ≥ , ∀ n ∈ N , (13b) N (cid:88) n =1 w n ≤ w T , (13c) N (cid:88) n =1 p n = p T (13d)For the upper level, look into the constraint (11d), which is equivalent with α ≤ q max q max + φ ( p T h n ) , ∀ n ∈ N . (14)Define α max (cid:44) min n ∈N q max q max + φ ( p T h n ) . Note that α max ≤ . Thus the constraint (11d) and constraint(11a) can be combined to be ≤ α ≤ α max . (15)In the upper level, we need to optimize α so as to solve the following optimization problem Problem 4: max α (1 − α ) F ( α ) s.t. ≤ α ≤ α max . (16a)It can be checked that Problem 1 is equivalent with the upper level optimization problem, i.e.,Problem 4. A. Optimal Solution for the Lower Level Optimization Problem
In this subsection, we will solve the lower level optimization problem, i.e., Problem 3. Tosimplify the solving of Problem 3, we impose one additional constraint p n ≤ αφ ( p T h n ) g n (1 − α ) h n , ∀ n ∈ N , (17)then the objective function of Problem 3 reduces to N (cid:88) n =1 w n log (cid:18) p n h n σ w n (cid:19) . In addition, by relaxing the equality constraint (13d) in Problem 3 to be an inequality, Problem3 turns to be the following optimization problem
Problem 5: max { p n } , { w n } N (cid:88) n =1 w n log (cid:18) p n h n σ w n (cid:19) s.t. p n ≥ , ∀ n ∈ N , (18a) w n ≥ , ∀ n ∈ N , (18b) N (cid:88) n =1 p n ≤ p T , (18c) N (cid:88) n =1 w n ≤ w T , (18d) p n ≤ αφ ( p T h n ) g n (1 − α ) h n , ∀ n ∈ N . (18e)It should be noticed that maximal achievable utility of Problem 5 equals the maximal achiev-able utility of Problem 3. The reason is as follows: Even the optimal solution of Problem3 does not obey the constraint (17), i.e., p n > αφ ( p T h n ) g n (1 − α ) h n , the throughput on link n is still w n log (cid:16) αφ ( p T h n ) g n (1 − α ) σ w n (cid:17) , which can be achieved by setting p n = αφ ( p T h n ) g n (1 − α ) h n . In other words, inthe feasible region such that constraint (17) holds, the maximal achievable utility of Problem3 is also achievable. To be consistent with the constraint (17), the equality constraint (13d) inProblem 3 is relaxed to be the inequality constraint (18c), which has no influence on equalitybetween the maximal achievable utility of Problem 3 and the maximal achievable utility ofProblem 5. Therefore solving Problem 3 is equivalent with solving Problem 5.It should be also noticed that the solution of Problem 5 may not serve as the optimal solutionof Problem 3 directly, since the optimal solution of Problem 5 may have (cid:80) Nn =1 p n < p T . Inthe real application, to get the optimal solution of Problem 3, we only need to find the optimalsolution of Problem 5 in the first step, and then keeps w n unchanged for n ∈ N , and enlarge p n for n ∈ N calculated by solving Problem 5 such that constraint (13d) holds.Next we turn to solve Problem 5. It can be checked that Problem 5 is a convex opti-mization problem since the constraints of Problem 5 are all linear and the objective function w n log (cid:16) p n h n σ w n (cid:17) is concave with ( w n , p n ) T . Although existing method can help to find theglobal optimal solution, in the next we will explore some special property of Problem 5’s optimalsolution so as to simplify the solving of Problem 5. It can be checked that Problem 5 satisfies the Slater’s condition. Hence the KKT conditionof Problem 5 can serve as the sufficient and necessary condition of its optimal solution [32],which can be given as follows w n h n p n h n + w n σ + η n − λ n − δ = 0 , (19a) ln (cid:18) p n h n w n σ (cid:19) − p n h n p n h n + w n σ + µ n − ν = 0 , (19b) λ n (cid:18) p n − αφ ( p T h n ) g n (1 − α ) h n (cid:19) = 0 , ∀ n ∈ N , (19c) η n p n = 0 , ∀ n ∈ N , (19d) µ n w n = 0 , ∀ n ∈ N , (19e) δ (cid:32) p T − (cid:88) n ∈N p n (cid:33) = 0 , (19f) ν (cid:32) w T − (cid:88) n ∈N w n (cid:33) = 0 , (19g) η n ≥ , λ n ≥ , µ n ≥ , ∀ n ∈ N , (19h) ν ≥ , δ ≥ , (19i)Constraints (18 a ) , (18 b ) , (18 c ) , (18 d ) , (18 e ) . (19j)where η n , µ n , δ , ν , and λ n are the Lagrange multipliers associated with the constraints (18a),(18b), (18c), (18d), (18e), respectively.Before we start the investigation on the KKT condition listed in (19), two facts about theoptimal solutions of Problem 5 are claimed. • Define A (cid:44) { n | p n > , w n > } and B (cid:44) { n | p n = 0 , w n = 0 } . Then there is N = A ∪ B .This fact indicates that the case with p n > , x n = 0 (or the case p n = 0 , x n > ) will nothappen for the optimal solution of Problem 5. This is because the case with p n > , x n = 0 (or the case p n = 0 , x n > ) indicates a wasteful use of power resource p n (or spectrumresource x n ). Higher utility can be achieved by transferring the wasted resources to theother links with positive bandwidth allocation or power allocation. • The constraint (18d) is active, which means that (cid:80) Nn =1 w n = w T , for the optimal solution ofProblem 5. This is due to the fact that the objective function of Problem 5 is an increasingfunction with w n for n ∈ N . So it is better to increase w n for n ∈ N as much as possible. Then we turn to investigate the KKT condition in (19), which can help to prove the followinglemma.
Lemma 1:
Define A (cid:44) { n | p n > , w n > } , the term p n h n w n σ equals a constant for n ∈ A . Proof:
For n ∈ A , there is w n > , thus it can be inferred that µ n = 0 from (19e). Define γ n = p n h n w n σ , then (19b) can be rewritten as ln(1 + γ n ) − γ n γ n = ν, ∀ n ∈ N . (20)The function ln (1 + x ) − x x is actually a strictly increasing function with x for x > . Hencefrom (20) it can be concluded that γ n for n ∈ A equals a common value, which is denoted as γ for the ease of presentation in the following.This completes the proof.According to the claim in Lemma 1, there is w n = p n h n γσ . Combining with the two claimedfacts for the optimal solution of Problem 5, it can be derived that w T = N (cid:88) n =1 w n = (cid:88) n ∈A w n = (cid:88) n ∈A p n h n γσ = N (cid:88) n =1 p n h n γσ (21)which further indicates that γ = N (cid:88) n =1 p n h n w T σ . (22)Therefore the objective function of Problem 5 can be rewritten as N (cid:80) n =1 w n log (cid:16) p n h n σ w n (cid:17) = (cid:80) n ∈A w n log (cid:16) p n h n σ w n (cid:17) = (cid:80) n ∈A w n log (1 + γ )= w T log (cid:18) N (cid:80) n =1 p n h n w T σ (cid:19) . (23)Since maximizing w T log (cid:18) N (cid:80) n =1 p n h n w T σ (cid:19) is equivalent with maximizing (cid:80) Nn =1 p n h n , then solv-ing Problem 5 is equivalent with solving the following optimization problem Problem 6: max { p n } N (cid:88) n =1 p n h n s.t. p n ≥ , ∀ n ∈ N , (24a) N (cid:88) n =1 p n ≤ p T , (24b) p n ≤ αφ ( p T h n ) g n (1 − α ) h n , ∀ n ∈ N . (24c)For Problem 6, it is straightforward to see that the optimal policy is to allocate more powerresource to the link with higher channel gain, i.e., to set the p n with higher h n as large aspossible. Specifically, the optimal allocation of p n for n ∈ N can be found as follows. Algorithm 1
Searching procedure for the optimal solution of Problem 6. Order h n for n ∈ N in descending order, such that h s ≥ h s ≥ ... ≥ h s N . Define i ∗ = arg min i (cid:80) ij =1 αφ ( p T h sj ) g sj (1 − α ) h sj > p T . Set p s i = αφ ( p T h si ) g si (1 − α ) h si for i = 1 , , ..., ( i ∗ − , p s i = p T − (cid:80) i ∗ − j =1 αφ ( p T h sj ) g sj (1 − α ) h sj for i = i ∗ , and p s i = 0 for i = i ∗ + 1 , i = i ∗ + 2 , ..., i = N . Note that when (cid:80) Nj =1 αφ ( p T h sj ) g sj (1 − α ) h sj ≤ p T , i ∗ does not exist and the optimal solutionis p n = αφ ( p T h n ) g n (1 − α ) h n , ∀ n ∈ N .In the end of this subsection, the optimal solution of the lower level optimization problem,i.e., Problem 3, can be summarized as follows. Algorithm 2
Searching procedure for the optimal solution of Problem 3. By following Algorithm 1, find the optimal p n for n ∈ N of Problem 6. Set w n = w T p n h n (cid:80) Nn =1 p n h n where p n is calculated in Step 1 of Algorithm 2 for n ∈ N . Increase p n calculated in Step 1 to be p (cid:48) n for n ∈ N such that (cid:80) Nn =1 p (cid:48) n = p T . Output p (cid:48) n and w n for n ∈ N . B. Optimal Solution for the Upper Level Optimization Problem
In this subsection, we will solve the upper level optimization problem, i.e., Problem 4. In thefirst step, there is such a lemma.
Lemma 2:
The function F ( α ) , which is defined in Problem 3, is monotonically increasingwith α . Proof:
Suppose there is ≤ α † ≤ α ‡ ≤ . Define the optimal solution of Problem 3associated with α † and α ‡ are p † n and w † n , and p ‡ n and w ‡ n , respectively, for n ∈ N . Then there is F ( α † ) = N (cid:80) n =1 min (cid:18) w † n log (cid:16) p † n h n σ w † n (cid:17) ,w † n log (cid:16) α † φ ( p T h n ) g n (1 − α † ) σ w † n (cid:17) (cid:19) ( a ) ≤ N (cid:80) n =1 min (cid:18) w † n log (cid:16) p † n h n σ w † n (cid:17) ,w † n log (cid:16) α ‡ φ ( p T h n ) g n (1 − α ‡ ) σ w † n (cid:17) (cid:19) ( b ) ≤ N (cid:80) n =1 min (cid:18) w ‡ n log (cid:16) p ‡ n h n σ w ‡ n (cid:17) ,w ‡ n log (cid:16) α ‡ φ ( p T h n ) g n (1 − α ‡ ) σ w ‡ n (cid:17) (cid:19) = F ( α ‡ ) where ( a ) holds is due the fact that the coefficient α † − α † ≤ α ‡ − α ‡ for α † ≤ α ‡ , and ( b ) holds sincethe set of p ‡ n and w ‡ n for n ∈ N is the optimal solution of Problem 3 when α = α ‡ .This completes the proof.With Lemma 2, the objective function of Problem 4 is actually the difference between twomonotonically increasing function with α , i.e., the difference between F ( α ) and αF ( α ) . Thussolving Problem 4 is equivalent with solving the following optimization problem Problem 7: max α,z F ( α ) + z s.t. αF ( α ) + z ≤ F ( α max ) , (25a) ≤ α ≤ α max . (25b)For Problem 7, since both F ( α ) in its objective function and z in its objective function areincreasing functions with α and z respectively, the maximum of Problem 7 can be achieved byincreasing both α and z as large as possible. Looking into the constraint (25a), both αF ( α ) and z are increasing functions with α and z respectively, thus the maximum of Problem 7 willbe achieved when both α and z reach their maximal allowable value in the feasible region ofProblem 7, in which case there is αF ( α ) + z = F ( α max ) , (26) which indicates that z = F ( α max ) − αF ( α ) . (27)Replace z with the expression in (27), the objective function of Problem 7 turns to be F ( α ) − αF ( α ) + F ( α max ) . Since maximizing F ( α ) − αF ( α ) + F ( α max ) is equivalent with maximizing F ( α ) − αF ( α ) , solving Problem 7 is equivalent with solving Problem 4.Then we focus on solving Problem 7. Although being non-convex, Problem 7 actually fallsinto the standard form of Monotonic Optimization Problem , whose standard form can be givenas follows.
Problem 8: max x f ( x ) s.t. g ( x ) ≤ , (28a) x L ≤ x ≤ x U , (28b)where the variable x is a multiple dimensional vector, x L and x U represent the lower bound andupper bound of x respectively, and both f ( x ) and g ( x ) are monotonically increasing functionswith x . For a standard monotonic optimization problem, there is a polyblock algorithm to achievethe ε -optimal solution of a standard monotonic optimization problem, where ε indicates the gapbetween the achieved utility and the global optimal utility is bounded by ε . The ε > isa predefined parameter before running the polyblock algorithm. By following the polyblockalgorithm, the detailed procedure for solving Problem 7 is given as follows. Algorithm 3 ε -optimal solution for Problem 7. Initialize a point set Z by a two-dimensional point z = ( α max , F ( α max )) T , where the α max and F ( α max ) indicate the maximal achievable value of the variable α and z , respectively. while |Z| > do for l = 1 , , ..., |Z| do Find the u l such that u l z l (1) F ( u l z l (1)) + u l z l (2) = F ( α max ) by utilizing the bisectionsearch method, where z l (1) and z l (2) are the 1st and 2nd element of the vector z l respectively. Set Ω l = u l z l . Find l ∗ = arg max ≤ l ≤|Z| [ F (Ω l (1)) + Ω l (2)] , where Ω l (1) and Ω l (2) are the 1st and 2nd elementof the vector Ω l respectively. For ∀ z ∈ Z , if there is [ F ( z (1)) + z (2)] ≤ [ F (Ω l ∗ (1)) + Ω i ∗ (2)] + ε , then delete the point z from the set Z . if |Z| > then Search j ∗ = arg max ≤ j ≤|Z| [ F ( z j (1)) + z j (2)] Find u j ∗ such that u j ∗ z j ∗ (1) F ( u j ∗ z j ∗ (1)) + u j ∗ z j ∗ (2) = F ( α max ) by utilizing thebisection search method. Set Ω j ∗ = u j ∗ z j ∗ . Generate two new points z (cid:48) = z j ∗ + (Ω j ∗ − z j ∗ ) ◦ (1 , and z (cid:48)(cid:48) = z j ∗ + (Ω j ∗ − z j ∗ ) ◦ (0 , ,where ◦ is the operation of Hadamard product. Add z (cid:48) and z (cid:48)(cid:48) into Z , delete z j ∗ from the set Z . Output the last Ω l ∗ before Z is subtracted to be an empty set.By following Algorithm 3, the optimal solution of Problem 7 can be achieved, which alsopaves the way for working out the optimal solution of Problem 4. To this end, Problem 4 canbe solved optimally, which also indicates the optimal solving of Problem 1.IV. O PTIMAL S OLUTION IN
PS M
ODE
In this section, Problem 2 will be solved. It can be checked that Problem 2 is a non-convexoptimization problem either, considering the non-convexity of φ ( x ) under both logistic modeland linear cut-off model in the objective function of Problem 2. In the following, we will showhow to find the global optimal solution of Problem 2 under logistic model and linear cut-offmodel, respectively. A. The Case under Logistic Model
In this subsection, logistic model is adopted for the energy harvester, i.e., φ ( x ) is set tobe the function in (9). To solve Problem 2, look into the objective function of Problem 2,the term w n log (cid:16) p n h n (1 − β n ) σ w n (cid:17) is monotonically decreasing function with β n , and the term w n log (cid:16) φ ( p T h n β n ) g n σ w n (cid:17) is monotonically increasing function with β n considering the increasingmonotonicity of the function φ ( x ) . Hence the maximal value of the term w n log (cid:16) p n h n (1 − β n ) σ w n (cid:17) and the term w n log (cid:16) φ ( p T h n β n ) g n σ w n (cid:17) is achieved when these two terms are equal, equivalently,there is p n h n (1 − β n ) = φ ( p T h n β n ) g n , (29)which further indicates p n = φ ( p T h n β n ) g n h n (1 − β n ) . (30)Taking into account the fact that p n ≤ p T for n ∈ N , and combine the constraint (30), thereis an implicit constraint φ ( p T h n β n )(1 − β n ) ≤ p T h n g n , ∀ n ∈ N , (31)which imposes an upper bound on β n , denoted as β U Y n , for n ∈ N . The β U Y n can be found byfollowing bi-section search method such that φ ( p T h n β U Y n )(1 − β U Y n ) = p T h n g n , ∀ n ∈ N . (32)It can be easily derived that β U Y n < since p T h n g n is bounded for n ∈ N . Combining with theconstraint (12d), which indicates that β n ≤ φ − ( q max ) p T h n , and the constraint (12a), which indicatesthat β n ≤ , define β U Y n = min( β U Y n , φ − ( q max ) p T h n , for n ∈ N , β n should satisfy ≤ β n ≤ β U Y n , ∀ n ∈ N . (33)Then by following the similar discussion for Problem 5 in Section III, solving Problem 2 isequivalent with solving the following optimization problem Problem 9: max { β n } , { w n } N (cid:88) n =1 w n log (cid:18) φ ( p T h n β n ) g n σ w n (cid:19) s.t. ≤ β n ≤ β U Y n , ∀ n ∈ N , (34a) w n ≥ , ∀ n ∈ N , (34b) N (cid:88) n =1 w n ≤ w T , (34c) N (cid:88) n =1 φ ( p T h n β n ) g n h n (1 − β n ) ≤ p T . (34d)By following the similar transformation from Problem 5 to Problem 6, Problem 9 is equivalentwith the following optimization problem Problem 10: max { β n } N (cid:88) n =1 φ ( p T h n β n ) g n s.t. ≤ β n ≤ β U Y n , ∀ n ∈ N , (35a) N (cid:88) n =1 φ ( p T h n β n ) g n h n (1 − β n ) ≤ p T . (35b)Recalling φ ( x ) defined in (9) is a monotonically increasing function, thus both the objectivefunction of Problem 10 and the left-hand side function of (35b) are increasing functions withthe vector β (cid:44) ( β , β , ..., β N ) T . Hence when N (cid:80) n =1 φ ( p T h n β UYn ) g n h n (1 − β UYn ) ≤ p T , the optimal solution isjust set β n as large as possible, i.e., set β n = β U Y n for n ∈ N . In general case, i.e., when N (cid:80) n =1 φ ( p T h n β UYn ) g n h n (1 − β UYn ) > p T , Problem 10 also falls into the standard form of monotonic optimizationproblem. Then by following the similar procedure in Algorithm 3 , the ε -optimal solution ofProblem 10 can be achieved.In summary, the optimal solution of the original optimization problem in PS mode, i.e.,Problem 2, can be achieved by following the steps in Algorithm 4, i.e., Algorithm 3 works for a two-dimensional vector. The general solving algorithm can be found in [33] and is omitted due tothe limit of space. Algorithm 4
Searching procedure for the optimal solution of Problem 2 under logistic model. if N (cid:80) n =1 φ ( p T h n β UYn ) g n h n (1 − β UYn ) ≤ p T then Set β n = β U Y n for n ∈ N . else By following the similar procedure in Algorithm 3, find the optimal β n for n ∈ N . Set w n = w T φ ( p T h n β n ) g n (cid:80) Nn =1 φ ( p T h n β n ) g n , where β n is calculated in Step 2 or Step 4 of Algorithm 4 for n ∈ N . Calculate p n for n ∈ N according to (30). Increase p n calculated in Step 6 to be p (cid:48) n for n ∈ N such that (cid:80) Nn =1 p (cid:48) n = p T . Output β n , w n , and p (cid:48) n for n ∈ N . B. The Case under Linear Cut-off Model
In this subsection, linear cut-off model is adopted for the energy harvester, i.e., φ ( x ) is set to bethe function in (10). Since the function φ ( x ) in (10) is also a monotonically increasing functionwith x , by following Algorithm 4, the optimal solution of Problem 2 can be also achieved.However, in this subsection, we will develop a simpler solution.Looking into the expression of φ ( x ) in (10), to guarantee positive energy harvested, thereshould be p T h n β n ≥ x L , ∀ n ∈ N , (36)which indicates β n ≥ x L p T h n (cid:44) β Ln , ∀ n ∈ N . (37)On the other hand, when more than a power of x U is received at the energy harvester, the energyharvester will become saturated. Thus there is no need to set the power of received energy tobe larger than x U , i.e., we have p T h n β n ≤ x U , ∀ n ∈ N , (38)which implies β n ≤ x U p T h n , ∀ n ∈ N . (39)With the holding of constraints (37) and (39), there is φ ( x ) = c ( x − x L ) . (40) By following the similar discussion as in Section IV-A, we also have p n h n (1 − β n ) = φ ( p T h n β n ) g n = c ( p T h n β n − x L ) g n . (41)which indicates p n = c ( p T h n β n − x L ) g n h n (1 − β n ) . (42)Still following the similar discussion as in Section IV-A, the constraint (31) also holds forlinear-cutoff model, which indicates that β n is upper bounded by β U Z n ( β U Z n < ), such that c ( p T h n β U Z n − x L )(1 − β U Z n ) = p T h n g n , ∀ n ∈ N . (43)In addition, combine the constraint (12a) and constraint (12d),define β U Z n (cid:44) min (cid:16) β U Z n , ( q max + cx L ) cp T h n , x U p T h n , (cid:17) , β n should be subject to the following constraint, β n ≤ β U Z n , ∀ n ∈ N . (44)Then combining the constraints (37) and (44) and the expression of φ ( x ) in (40), with the samediscussion for the transformation from Problem 2 to Problem 9, to find the optimal solution ofProblem 2 under linear cut-off model for energy harvester, we only need to solve the followingoptimization problem Problem 11: max { β n } , { w n } N (cid:88) n =1 w n log (cid:18) c ( p T h n β n − x L ) g n σ w n (cid:19) s.t. β Ln ≤ β n ≤ β U Z n , ∀ n ∈ N , (45a) w n ≥ , ∀ n ∈ N , (45b) N (cid:88) n =1 w n ≤ w T , (45c) N (cid:88) n =1 φ ( p T h n β n ) g n h n (1 − β n ) ≤ p T , (45d)which can be simplified to be the following optimization problem by following the discussionmethod from Problem 9 to Problem 10 Problem 12: max { β n } N (cid:88) n =1 c ( p T h n β n − x L ) g n s.t. β Ln ≤ β n ≤ β U Z n , ∀ n ∈ N , (46a) N (cid:88) n =1 c ( p T h n β n − x L ) g n h n (1 − β n ) ≤ p T . (46b)For Problem 12, it can be checked that the objective function of Problem 12 is a linear functionwith β n for n ∈ N , and the left-hand side function in (46b) is convex with β n when p T h n x L > .Therefore, Problem 12 is a convex optimization problem. It can be checked that Problem 12satisfies Slater’s condition. Thus the KKT condition of Problem 12 can serve as the sufficientand necessary condition of its optimal solution [32], which can be written as cp T h n g n − Ξ cg n h n ( p T h n − x L )( β n − + Γ n − ∆ n = 0 , (47a) Γ n (cid:0) β n − β Ln (cid:1) = 0 , ∀ n ∈ N , (47b) ∆ n (cid:0) β n − β U Z n (cid:1) = 0 , ∀ n ∈ N , (47c) Ξ (cid:32) p T − N (cid:88) n =1 c ( p T h n β n − x L ) g n h n (1 − β n ) (cid:33) = 0 , (47d) Γ n ≥ , ∆ n ≥ , ∀ n ∈ N , (47e) Ξ ≥ , (47f)Constraints (37) , (44) , (46 b ) . (47g)According to (47b) and (47c), when β n > β Ln and β n < β U Z n , there is Γ n = 0 and ∆ n ,respectively. So when β Ln < β n < β U Z n , Γ n = ∆ n = 0 , there is β n = 1 − (cid:115) Ξ ( p T h n − x L ) p T h n (48)according to (47a).For a given Ξ , if the calculated β n by following (48) is larger than its upper bound β U Z n , then β n = β U Z n by checking (44), (47a) and (47c). Similarly, if the calculated β n by following (48) issmaller than its lower bound β Ln , then β n = β Ln . Hence β n can be expressed by Ξ in a preciseway as follows β n (Ξ) = (cid:34) − (cid:115) Ξ ( p T h n − x L ) p T h n (cid:35) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) β UZn β Ln , ∀ n ∈ N , (49) where the operation [ x ] | ba = max( a, min( x, b )) . The β n (Ξ) is actually a monotonically decreasingfunction with Ξ for n ∈ N .On the other hand, it should be noticed both the left-hand side function of (46b) and theobjective function of Problem 10 are increasing functions with β n for n ∈ N , so it is betterto set β n as large as possible, which indicates that the optimal solution of Problem 12 happenswhen the constraint (46b) become active, i.e., N (cid:88) n =1 c ( p T h n β n (Ξ) − x L ) g n h n (1 − β n (Ξ)) = p T . (50)Given that the left-hand side function of constraint (50) is increasing with β n , and the mono-tonicity of β n (Ξ) with Ξ . The left-hand side function of constraint (50) is also monotonic with Ξ .Hence the Ξ such that the equality (50) holds can be searched by following bi-section method.Note that when N (cid:88) n =1 c ( p T h n β n (0) − x L ) g n h n (1 − β n (0)) , (51)which is the maximal value of the left-hand side of (50) is less than p T , the optimal configurationof β n is β n = β U Z n for n ∈ N .In the end of this subsection, the simple solution for PS mode under linear cut-off model issummarized as follows Algorithm 5
Searching procedure for the optimal solution of Problem 2 under linear cut-offmodel. if N (cid:80) n =1 c ( p T h n β n (0) − x L ) g n h n (1 − β n (0)) ≤ p T then Set β n = β U Z n for n ∈ N . else Use bi-section search method to find the Ξ such that equality (50) holds. Calculate β n (Ξ) according to (49) for n ∈ N . Calculate p n for n ∈ N according to (42). Set w n = w T c ( p T h n β n − x L ) g n (cid:80) Nn =1 c ( p T h n β n − x L ) g n , where β n is calculated in Step 2 or Step 5 of Algorithm 5for n ∈ N . Increase p n calculated in Step 3 to be p (cid:48) n for n ∈ N such that (cid:80) Nn =1 p (cid:48) n = p T . Output β n , w n , and p (cid:48) n for n ∈ N . V. N
UMERICAL R ESULTS
In this section, numerical results are presented to verify the effectiveness of our proposedmethods. The system parameters are set as follows in default. There are 4 relay nodes, i.e., N = 4 and N = { , , , } . The total system bandwidth w T = 1 MHz. The total transmitpower of source node p T = 1 W. The power spectrum density of noise σ = 1 × − W/Hz.The maximal transmit power of every relay node q max = 50 mW. The carrier frequency is set as1 GHz. Both g n and h n for n ∈ N are uniformly distributed between -40dB and -50dB, whichapproximately correspond to the attenuation in free space between 2m and 10m, respectively.For the energy harvester, by utilizing the curve fitting tool on the measured data points in Fig.4,it is calculated that M = 2 . × − , a = 170 , and b = 1 . × − for logistic model, and c = 0 . , x L = 0 , and x U = 3 × − for linear cut-off model. When running the polyblockalgorithm, ε is set as × − . As a comparison, a relay selection method, which usually appearsin literatures [19], is implemented. In the relay selection method, all the allowable transmit powerand bandwidth are imposed on one link from the source node through some relay node to thedestination node. The link associated with the maximal end-to-end throughput is selected. Withthe implementation of relay selection method, there are also two modes: TS mode and PS mode.For the ease of presentation, we will denote the relay selection method under TS mode and PSmode as “TS-select” and “PS-select”, respectively.In Fig. 5, the end-to-end throughput from the source node to the destination node is plottedversus the transmit power p T for TS mode, TS-select mode, PS mode, and PS-select moderespectively, under logistic model (in Fig. 5a) and under linear cut-off model (in Fig. 5b). Itcan be observed that as p T grows, the end-to-end throughput under every mode grows, which isin coordination with intuition. Additionally, it can be seen that PS mode outperforms PS-selectmode and TS mode outperforms TS-select mode, which verifies the effectiveness of our proposedmethod. Moreover, it can be also observed that PS mode always outperforms TS mode. Thisis consistent with the results in existing literatures [6], [24] on SWIPT and provides helpfulsuggestion for the implementation in real application.In Fig. 6, the end-to-end throughput is plotted versus system bandwidth w T for TS mode,TS-select mode, PS mode, and PS-select mode respectively, under logistic model (in Fig. 6a)and under linear cut-off model (in Fig. 6b). Similar observations can be obtained as for Fig.5. The only difference from Fig. 5 lies in that the end-to-end throughput grows with w T at a p T E nd - t o - end t h r oughpu t unde r l og i s t i c m ode l TSTS-selectPSPS-select (a) The case under logistic model. p T E nd - t o - end t h r oughpu t unde r li nea r c u t - o ff m ode l TSTS-selectPSPS-select (b) The case under linear cut-off model.
Fig. 5: End-to-end throughput versus transmit power p T . w T E nd - t o - end t h r oughpu t unde r l og i s t i c m ode l TSTS-selectPSPS-select (a) The case under logistic model. w T E nd - t o - end t h r oughpu t unde r li nea r c u t - o ff m ode l TSTS-selectPSPS-select (b) The case under linear cut-off model.
Fig. 6: End-to-end throughput versus system bandwidth w T . -80 -70 -60 -50 -40 -30 -20 Mean of g n and h n (dB) -2 E nd - t o - end t h r oughpu t unde r l og i s t i c m ode l TSTS-selectPSPS-select (a) The case under logistic model. -80 -70 -60 -50 -40 -30 -20
Mean of g n and h n (dB) -2 E nd - t o - end t h r oughpu t unde r li nea r c u t - o ff m ode l TSTS-selectPSPS-select (b) The case under linear cut-off model.
Fig. 7: End-to-end throughput versus channel gain g n and h n .decreasing rate, rather than a nearly constant rate. This indicates that increasing total transmitpower p T will play a more significant effect on improving end-to-end throughput compared withincreasing the system bandwidth w T .In Fig. 7, the end-to-end throughput is plotted versus the mean of g n and h n for TS mode,TS-select mode, PS mode, and PS-select mode respectively, under logistic model (in Fig. 7a)and under linear cut-off model (in Fig. 7b). Note that when the mean of g n and h n are set as x dB. Then the g n and h n are uniformly distributed between [ x − , x + 5] dB. Similar observationscan be obtained as for Fig. 5 as well. It can be also seen that the value of g n and h n havegreat influence on the end-to-end throughput. This indicates such a suggestion: We should trythe best to place relay nodes at the locations close to source node and destination node withlittle shadowing and fading. VI. C ONCLUSION
In this paper, end-to-end throughput is maximized for a two-hop DF multiple-relay networkimplemented with SWIPT under TS mode and PS mode. Transmit power and bandwidth on everylink from source to destination, and the PS ratio or TS ratio on every relay node are optimized.Two types of nonlinear model are adopted for the energy harvester. For every combinational case in terms of working mode and nonlinear model, an optimization problem is formulated,all of which are non-convex. With a series of analysis and transformation, and with the aid ofbi-level optimization and monotonic optimization, etc., we find the global optimal solution forthe optimization problem in every case. In some case, the offered optimal solution is closed-form or semi-closed-form. Our findings can provide helpful suggestion for the application ofSWIPT-powered relay network in the future.R EFERENCES [1] X. Lu, P. Wang, D. Niyato, D. I. Kim, and Z. Han, “Wireless networks with RF energy harvesting: A contemporarysurvey,”
IEEE Commun. Surveys Tuts. , vol. 17, no. 2, pp. 757-789, Second Quarter, 2015.[2] W. Lu, Y. Gong, X. Liu, J. Wu, and H. Peng, “Collaborative energy and information transfer in green wireless sensornetworks for smart cities,”
IEEE Trans. Ind. Electron. , vol. 14, no. 4, 1585-1593, Apr. 2018.[3] X. Liu, F. Li, and Z. Na, “Optimal resource allocation in simultaneous cooperative spectrum sensing and energy harvestingfor multichannel cognitive radio,”
IEEE Access , vol. 5, pp. 3801-3812, 2017.[4] L. Liu, R. Zhang, and K. C. Chua, “Wireless information and power transfer: A dynamic power splitting approach,”
IEEETrans. Commun. , vol. 61, no. 9, pp. 3990-4001, Sept. 2013.[5] L. Liu, R. Zhang, and K. C. Chua, “Wireless information transfer with opportunistic energy harvesting.”
IEEETrans. Wireless Commun. , vol. 12, no. 1, pp. 288-300, Jan. 2013.[6] X. Zhou, R. Zhang, and C. K. Ho, “Wireless information and power transfer: Architecture design and rate-energy tradeoff,”
IEEE Trans. Commun. , vol. 61, no. 11, pp. 4754-4767, Nov. 2013.[7] X. Zhou, “Training-based SWIPT: Optimal power splitting at the receiver,”
IEEE Trans. Veh. Technol. , vol. 64, no. 9,pp. 4377-4382, Sep. 2015.[8] I. M. Kim and D. I. Kim, “Wireless information and power transfer: Rate-energy tradeoff for equi-probable arbitrary-shapeddiscrete inputs,”
IEEE Trans. Wireless Commun. , vol. 15, no. 6, pp. 4393-4407, June 2016.[9] S. Li, W. Xu, Z. Liu, and J. Lin, “Independent power splitting for interference-corrupted SIMO SWIPT systems,”
IEEECommun. Lett. , vol. 20, no. 3, pp. 478-481, Mar. 2016.[10] R. Zhang and C. K. Ho, “MIMO broadcasting for simultaneous wireless information and power transfer,”
IEEETrans. Wireless Commun. , vol. 12, no. 5, pp. 1989-2001, May 2013.[11] X. Zhu, W. Zeng, and C. Xiao, “Precoder design for simultaneous wireless information and power transfer systems withfinite-alphabet inputs,”
IEEE Trans. Veh. Technol. , vol. 66, no. 10, pp. 9085-9097, Oct. 2017.[12] K. Xiong, B. Wang, and K. J. R. Liu, “Rate-energy region of SWIPT for MIMO broadcasting under nonlinear energyHarvesting Model,”
IEEE Trans. Wireless Commun. , vol. 16, no. 8, pp. 5147-5161, Aug. 2017.[13] Q. Gu, G. Wang, R. Fan, Z. Zhong, K. Yang, and H. Jiang, “Rate-energy tradeoff in simultaneous wireless information andpower transfer over fading channels with uncertain distribution,”
IEEE Trans. Veh. Technol. , vol. 67, no. 4, pp. 3663-3668,Apr. 2018.[14] Z. Chen, P. Xu, Z. Ding, and X. Dai, “Cooperative transmission in simultaneous wireless information and power transfernetworks,”
IEEE Trans. Veh. Technol. , vol. 65, no. 10, pp. 8710-8715, Oct. 2016.[15] A. A. Nasir, X. Zhou, S. Durrani, and R. A. Kennedy, “Relaying protocols for wireless energy harvesting and informationprocessing,”
IEEE Trans. Wireless Commun. , vol. 12, no. 7, pp. 3622-3636, Jul. 2013. [16] C. Zhong, H. A. Suraweera, G. Zheng, I. Krikidis, and Z. Zhang, “Wireless information and power transfer with fullduplex relaying,” IEEE Trans. Commun. , vol. 62, no. 10, pp. 3447-3461, Oct. 2014.[17] G. Zhu, C. Zhong, H. A. Suraweera, G. K. Karagiannides, Z. Zhang, and T. A. Tsiftsis, “Wirelss information and powertransfer in relay systems with multiple antennas and interference,”
IEEE Trans. Commun. , vol. 63, no. 4, pp. 1400-1418,Apr. 2015.[18] A. A. Nasir, X. Zhou, S. Durrani, and R. A. Kennedy, “Wireless-powered relays in cooperative communications: Time-switching relaying protocols and throughput analysis,”
IEEE Trans. Commun. , vol. 63, no. 5, pp. 1607-1622, May 2015.[19] Z. Ding, I. Krikidis, B. Sharif, and H. V. Poor, “Wireless information and power transfer in cooperative networks withspatially random relays,”
IEEE Trans. Wireless Commun. , vol. 13, no. 8, pp. 4440-4453, Aug. 2014.[20] M. Ju, K.-M. Kang, K.-S. Hwang, and C. Jeong, “Maximum transmission rate of PSR/TSR protocols in wireless energyharvesting DF-based relay networks,”
IEEE J. Select. Areas Commun. , vol. 33, no. 12, pp. 2701-2717, Dec. 2015.[21] Z. Zhou, M. Peng, Z. Zhao, and Y. Li, “Joint power splitting and antenna selection in energy harvesting relay channels,”
IEEE Signal Process. Lett. , vol. 22, no. 7, pp. 823-827, Jul. 2015.[22] K. Xiong. P. Fan, C. Zhang, and K. B. Letaief, “Wireless information and energy transfer for two-hop non-regenerativeMIMO-OFDM relay networks,”
IEEE J. Select. Areas Commun. , vol. 33, no. 8, pp. 1595-1611, Aug. 2015.[23] Y. Liu, and X. Wang, “Information and energy cooperation in OFDM relaying: Protocols and optimization,”
IEEE Trans.Veh. Technol. , vol. 65, no. 7, pp. 5088-5098, Jul. 2016.[24] R. Fan, S. Atapattu, W. Chen, Y. Zhang, and J. Evans, “Throughput maximization for multi-hop decode-and-forward relaynetwork with wireless energy harvesting,”
IEEE Access , vol. 6, pp. 24582-24595, 2018.[25] L. Tang, X. Zhang, and X. Wang, “Joint data and energy transmission in a two-hop network with multiple relays,”
IEEECommun. Lett. , vol. 18, no. 11, pp. 2015-2018, Nov. 2014.[26] E. Boshkovska, D. W. K. Ng, N. Zlatanov, and R. Schober, “Practical non-linear energy harvesting model and resourceallocation for SWIPT systems,”
IEEE Commun. Lett. , vol. 19, no. 12, pp. 2082-2085, Dec. 2015.[27] E. Boshkovska, D. W. K. Ng, N. Zlatanov, A. Koelpin, and R. Schober, “Robust resource allocation for MIMO wirelesspowered communication networks based on a non-linear EH model,”
IEEE Trans. Commun. , vol. 65, no. 5, pp. 1984-1999,May 2017.[28] P. N. Alevizos and A. Bletsas, “Sensitive and nonlinear far-field RF energy harvesting in wireless communications,”
IEEETrans. Wireless Commun. , vol. 17, no. 6, pp. 3670-3675, Jun. 2018.[29] I. Krikidis, S. Timotheou, and S. Sasaki, “Energy transfer for cooperative networks: Data relaying or energy harvesting?”
IEEE Commun. Lett. , vol. 16, no. 11, pp. 1772-1775, Nov. 2012.[30] W. Tu and L. Lai, “Keyless authentication and authenticated capacity,
IEEE Trans. Inf. Theory , vol. 64, no. 5, pp. 3696-3714,May 2016.[31] W. Tu, M. Goldenbaum, L. Lai, and H. V. Poor, “On simultaneously generating multiple keys in a joint source-channelmodel,
IEEE Trans. Inf. Forensics Security , vol. 12, no. 2, pp. 298-308, Feb. 2017.[32] S. P. Boyd and L. Vandenberghe,
Convex Optimization.
Cambridge University Press, 2004.[33] C. Floudas and P. M. Pardalos,