Relaying Strategies for Wireless-Powered MIMO Relay Networks
aa r X i v : . [ c s . I T ] M a y Relaying Strategies for Wireless-PoweredMIMO Relay Networks
Yang Huang,
Student Member, IEEE , and Bruno Clerckx,
Member, IEEE
Abstract
This paper investigates relaying schemes in an amplify-and-forward multiple-input multiple-outputrelay network, where an energy-constrained relay harvests wireless power from the source informationflow and can be further aided by an energy flow (EF) in the form of a wireless power transfer at thedestination. However, the joint optimization of the relay matrix and the source precoder for the energy-flow-assisted (EFA) and the non-EFA (NEFA) schemes is intractable. The original rate maximizationproblem is transformed into an equivalent weighted mean square error minimization problem andoptimized iteratively, where the global optimum of the nonconvex source precoder subproblem isachieved by semidefinite relaxation and rank reduction. The iterative algorithm finally converges. Then,the simplified EFA and NEFA schemes are proposed based on channel diagonalization, such that thematrices optimizations can be simplified to power optimizations. Closed-form solutions can be achieved.Simulation results reveal that the EFA schemes can outperform the NEFA schemes. Additionally,deploying more antennas at the relay increases the dimension of the signal space at the relay. Exploitingthe additional dimension, the EF leakage in the information detecting block can be nearly separatedfrom the information signal, such that the EF leakage can be amplified with a small coefficient.
Index Terms
Wireless power harvesting, SWIPT, MIMO relay, amplify-and-forward (AF).
A preliminary version of this paper has appeared in the IEEE International Conference on Communications 2015 [1].Y. Huang and B. Clerckx are with the Department of Electrical and Electronic Engineering, Imperial College London, LondonSW7 2AZ, United Kingdom (e-mail: { y.huang13, b.clerckx } @imperial.ac.uk). B. Clerckx is also with the School of ElectricalEngineering, Korea University, Korea. This work has been partially supported by the EPSRC of UK, under grant EP/M008193/1.The work of Y. Huang was supported by China Scholarship Council (CSC) Imperial Scholarship. I. I
NTRODUCTION
Sensor networks have been widely applied to structural monitoring, habitat monitoring, etc.Sensors may be deployed to inaccessible places, which makes replacing the sensor batteriesinconvenient. In such networks, the energy of the nodes frequently selected as relays drainsmore quickly. The lifetime of such energy-constrained relays becomes the bottleneck to prolongthe lifetime of the whole network. As a recent solution, the nodes able to harvest energy fromthe ambient environment are employed as relays [2]. Nevertheless, the relay may harvest powerfrom a more reliable and controllable energy source in the uplink transmission scenario wherethe destination is a collect and process center (which has a sustainable power supply). Motivatedby this scenario, the paper investigates the simultaneous wireless information and power transfer(SWIPT) [3] in an autonomous one-way relay network, where the autonomous relay can extractenergy from the incoming signal from the source to forward information but also can be aided bya dedicated power transfer from the destination. Note that the power required for CSIT sharing,etc. is not supplied by the harvested power [4], and may come from an independent battery.State-of-the-art SWIPT techniques for relaying can be mainly categorized into the powersplitting (PS) relaying and the time switching (TS) relaying [4]–[9]. Ref. [5] proposed a PSrelaying (where the relay extracts power for forwarding from the source information signal) anda TS relaying (where the relay harvests power from an energy signal sent by the source and thenrelays source information in a time-division manner). Another TS relaying, where the energysignal is sent by the destination, was studied in [6]. The PS relaying was also studied in themulti-pair one-way relay networks [7] and the relay interference channels [8], [9]. In [4], therelay employs dedicated antennas to harvest wireless power, while the other antennas perform PSto relay a single data stream. In the above works, the PS relaying reduces the information powerat the relay. The TS relaying consumes more timeslots, though the wireless power is harvestedin a dedicated timeslot. Therefore, these two methods may degrade the rate performance, and arelaying strategy able to harvest sufficient forwarding power without consuming more timeslotswould be appealing.To circumvent those limitations, an energy-flow-assisted (EFA) two-phase amplify-and-forward(AF) one-way relaying can be proposed, where the EFA relay can harvest power from both thesource information signal and a dedicated energy flow (in the form of a wireless power transfer) (a) Energy-flow-assisted two-phase relaying. (b) Two-phase relaying without energy flow.Fig. 1. SWIPT relay network. The source, relay, and destination are designated as S , R , and D , respectively. at the destination, as shown in Fig. 1(a). Thanks to the PS scheme [3], the received superposedsignal at R in phase 1 is split for information detecting (ID) and energy harvesting (EH). Theenergy flow (EF) leaking into the ID receiver is referred to as the EF leakage. Our previouswork [10] shows that EF is beneficial to the EFA relay (with single antenna terminals) only inthe presence of multiple relay antennas. Unfortunately, the method proposed in [10] cannot beexploited in the MIMO case. Here we study a more general scenario where the terminals areequipped with r antennas (where r is no greater than the number of relay antennas r R [11]).The r -antenna terminals can transmit multiple data streams and increase the energy harvested atR through beamforming. As the harvested power at R is also consumed to amplify and forwardthe EF leakage, the information forwarding power would be reduced, which may degrade therate. Thus, we also investigate the non-EFA (NEFA) relaying, which harvests power from theinformation signal, as shown in Fig. 1(b). The autonomous relay makes the EFA and the NEFArelaying different from the conventional relaying (where R has constant energy source). Thelatter allocates the source power to all the diagonalized channels, only to maximize the rate[12]. However, if only aimed at increasing the forwarding power at R , NEFA would performrank-1 beamforming at the source [13]. That is, enhancing information transfer may conflictwith enhancing energy harvesting. For EFA, the relay matrix also has to address the superposedEF leakage in the ID receiver. The main contributions of this paper are listed as follows.Firstly, this paper proposes the EFA scheme for the multiple-input multiple-output (MIMO)relay network, where the autonomous relay is able to harvest EF from the destination andsimultaneously receive the information signal from the source.Secondly, an iterative optimization algorithm is proposed for both the EFA and NEFA schemesto jointly optimize the relay processing matrix and the source precoder. The original problemis nonconvex and intractable, which is then transformed into an equivalent problem [14], suchthat the matrices can be optimized iteratively. The subproblem of source precoding is essentiallya nonconvex quadratically constrained quadratic problem (QCQP). As a prevailing solution, the successive convex approximation [15], [16] cannot guarantee this subproblem yielding a globaloptimum, which may make the overall iterative algorithm fail to converge. To solve this problem,we formulate the nonconvex QCQP as a semidefinite program (SDP) by performing semidefiniterelaxation (i.e. relaxing the nonconvex rank-1 constraint) [17]. We show that there exists a rank-1solution and the relaxation is safe. The global optimum (i.e. the rank-1 solution) of the originalnonconvex QCQP can be derived from the solution to the SDP by performing post-processing.Finally, the iterative algorithm is shown to converge.Thirdly, although the weighted mean-square error (WMSE) criterion and the alternating op-timization (AO) have been exploited in the joint optimization of the relay networks, the issueof convergence has not been well studied. For instance, containing subproblems with multiplesolutions, [18] only conjectures that the algorithms converge to stationary points. In this paper,supposing that tie-breaking strategies [19] are included in solving the subproblems with multiplesolutions, we prove that the minimizers converge to a limit point. This limit point is notnecessarily a stationary point.Fourthly, aiming at less complex EFA relaying algorithms, simplified algorithms are proposed.The original matrices optimization is simplified to a power optimization by performing a channeldiagonalization based on the harvested-power-maximization power-leakage-minimization (HPM-PLM) strategy. Power allocation at R and S are optimized based on an AO. Channel pairingissues introduced in the relay power optimization are solved by an ordering operation. Closed-form solutions can be achieved in the subproblems of relay optimization and source optimization.A simplified NEFA relaying algorithm is also investigated. Simulation results show that the EF isbeneficial to the EFA schemes, such that EFA schemes can outperform rate-wise NEFA schemes.Although the data streams to be forwarded are corrupted by the EF at R in the EFA scheme,the antenna configuration r R > r can make the EF leakage nearly separated from the linearlycombined data streams. Thus, the desired signals can be amplified with a larger coefficient.The remainder of this paper is organized as follows. Section II formulates the system model.Section III proposes the iterative algorithm for the EFA and NEFA schemes. Section IV studiesthe simplified EFA schemes. Then, the simplified NEFA scheme is investigated in Section V.Section VI discusses the simulation results. Finally, conclusions are drawn in Section VII.Notations: Matrices and vectors are in bold capital and bold lower cases, respectively. Thenotations ( · ) T , ( · ) ⋆ , ( · ) ∗ , ( · ) H , Tr {·} , det( · ) , λ i ( · ) and [ · ] i represent the transpose, optimal solution, conjugate, conjugate transpose, trace, determinant, the i th eigenvalue and the i th column of amatrix, respectively. The notation A (cid:23) means that A is positive-semidefinite; π ( a ) denotesthe permutation; k a k denotes the 2-norm. When ≷ and ≶ are used, top cases or bottom casesin the two notations hold simultaneously. The notation ⊗ denotes the Kronecker product.II. S YSTEM M ODEL AND P ROBLEM F ORMULATION
As shown in Fig. 1(a), it is considered that there is no direct link between S and D dueto barriers (which causes huge shadow fading), such that the communication between S and D has to rely on the autonomous relay R . The D -to- R , S -to- R , and R -to- D channels arerespectively designated as H R,D ∈ C r R × r , H R,S ∈ C r R × r , and H D,R ∈ C r × r R , which are independentand identically distributed Rayleigh flat fading channels. Channel reciprocity is assumed suchthat H D,R = H TR,D . It is assumed that each node has perfect full CSIT, following similar systems[4], [11]. At each antenna of the relay, a fraction of the received power, denoted as the PS ratio ρ , is conveyed to the EH receiver. The noise at the ID receiver (at R ) and D are respectivelydenoted by n R ∼ CN (0 , σ n I ) and n D ∼ CN (0 , σ n I ) , while the effect of noise at the EH receiver issmall and neglected [3]. The relay node works in a half-duplex mode.In phase 1, the received signal at the EH receiver is given by y R, EH = ρ / ( H R,D B D x D + H R,S B S x S ) , where B D and B S represent precoders at D and S , respectively; x D and x S aretransmitted signals from D and S , respectively. Assuming an RF-to-DC conversion efficiency of 1,the harvested power equals Tr (cid:8) ρ H R,D Q D H HR,D + ρ H R,S Q S H HR,S (cid:9) , where Q D = B D B HD , Q S = B S B HS , Tr { Q D } ≤ P D , and Tr { Q S } ≤ P S . Meanwhile, the baseband signal input to the ID receiver forforwarding is given by y R, ID = (1 − ρ ) / ( H R,D B D x D + H R,S B S x S )+ n R . In phase 2, the informationreceived at D is given by y D = (1 − ρ ) / H D,R F ( H R,S B S x S + H R,D B D x D ) + H D,R Fn R + n D , where F denotes the relay processing matrix. With perfect CSI at D , following the similar systems[11], we assume that the self-interference in y D , i.e. the term related to x D , can be canceled, butpower at the relay is consumed to forward this self-interference (i.e. the EF leakage). Defining R = σ n H D,R FF H H HD,R + σ n I , the rate maximization problem can be formulated as max B S , F
12 log det (cid:0) I +(1 − ρ ) H D,R FH R,S B S B HS H HR,S F H H HD,R R − (cid:1) (1a)s.t. Tr (cid:8) (1 − ρ ) (cid:0) FH R,S B S B HS H HR,S F H + FH R,D Q D H HR,D F H (cid:1) + σ n FF H (cid:9) ≤ ρ Tr (cid:8) H R,D Q D H HR,D + H R,S B S B HS H HR,S (cid:9) , (1b)Tr { B S B HS } ≤ P S . (1c) In this problem, the optimization is not performed over Q D , due to the high difficulty . In thefollowing sections, the iterative algorithm for both the EFA and NEFA schemes is proposed,where the corresponding algorithms are designated as EFA-OPT and NEFA-OPT. The algorithmiteratively optimizes B S and F . Then, to avoid the complexity of matrices optimization, asimplified EFA algorithm (designated as EFA-S1) is proposed based on channel diagonalization,such that the problem is simplified to a power optimization. This algorithm is further simplified asEFA-S2 and provide closed-form solutions of the relay and source strategies. Finally, a simplifiedNEFA algorithm (designated as NEFA-S) is proposed by using channel diagonalization.III. I TERATIVE A LGORITHM
This section proposes an iterative algorithm to solve the joint optimization problem (1), wherethe value of the fixed Q D depends on the relaying scheme and differs in EFA-OPT and NEFA-OPT. Take the singular value decomposition (SVD) of H R,D = V ∗ D,R Σ D,R U TD,R (due to thechannel reciprocity). In EFA-OPT, to maximize the amount of power harvested at R , Q D = P D [ U ∗ D,R ] max [ U ∗ D,R ] H max , where [ U ∗ D,R ] max is the right singular vector (RSV) corresponding to themaximum singular value λ / D,R, max of H R,D (see Proposition 1 in [13]). In NEFA-OPT, Q D = .Because the optimization variable F within the matrix inversion R − makes problem (1)intractable, we introduce an auxiliary variable A (cid:23) to transform problem (1) into an equivalentWMSE minimization problem [14] given by min A (cid:23) , W , F , B S Tr { A E ( W , F , B S ) } − log det ( A ) (2a)s.t. Tr (cid:8) (1 − ρ ) FH R,S B S B HS H HR,S F H + (1 − ρ ) FH R,D Q D H HR,D F H + σ n FF H (cid:9) ≤ ρ Tr (cid:8) H R,D Q D H HR,D + H R,S B S B HS H HR,S (cid:9) , (2b)Tr { B S B HS } ≤ P S , (2c) where W denotes the receive filter at D , while E ( · ) represents the MSE matrix defined in theMSE E{ ( W H y D − x S ) H ( W H y D − x S ) } = Tr {E{ ( W H y D − x S )( W H y D − x S ) H }} = Tr { E ( W , F , B S ) } ,and E ( W , F , B S ) is given by E = (1 − ρ ) W H H D,R FH R,S B S B HS H HR,S F H H HD,R W + W H H D,R FF H H HD,R W σ n + W H W σ n − (1 − ρ ) / B HS H HR,S F H H HD,R W − (1 − ρ ) / W H H D,R FH R,S B S + I r . (3) Since Q D is absent from (1a), it cannot be optimized iteratively. Alternatively, if the coupled F and Q D are optimized ina subproblem, the subproblem is essentially a bilinear problem, which is NP-hard and hard to yield the global optimum [20].This means that the value of (1a) may not monotonically increases over iterations, and the iterative algorithm cannot converge. The proof of the equivalence between problems (1) and (2) is similar to the Appendix A in [14].The details are omitted here. Since the optimization variables A , W , F , and B S are coupled in(2a) and (2b), problem (2) is still intractable. Subsequently, the original problem is decoupledinto four subproblems of A , W , F , and B S . The variable corresponding to each subproblemis alternatively optimized by fixing the others. A. Subproblems of A and W Fixing the variables ( W , F , B S ) , the subproblems of A can be written as min A (cid:23) Tr { A E } − log det( A ) . (4) Similarly, fixing the variables ( A , F , B S ) , the subproblem of W can be formulated as min W Tr { A E ( W ) } . (5) Since the above two subproblems are strictly convex, an unique optimal solution can be obtainedfor each subproblem by the first-order condition of optimality. Calculating the derivatives ofobjective functions of the two subproblems [21], the optimal A ⋆ and W ⋆ (which is the minimummean square error receiver) are given by A ⋆ = E − , (6) W ⋆ = W mmse = (cid:2) (1 − ρ ) H D,R FH R,S B S B HS H HR,S F H H HD,R + H D,R FF H H HD,R σ n + I r σ n (cid:3) − · (1 − ρ ) / H D,R FH R,S B S . (7) Substituting (6) into (2a) yields Tr { I } − log det( E − ( W , F , B S )) , where log det( E − ) is equal to twicethe end-to-end achievable rate, i.e. (1a). This reveals the physical meaning of the quantity of theobjective function (2a) and the equivalence between problems (1) and (2). B. Subproblem of F Fixing the variables ( A , W , B S ) , the subproblem of F (where r R ≥ r > ) can be formulated as min F Tr (cid:8) (1 − ρ ) F H H HD,R WA W H H D,R FH R,S B S B HS H HR,S + σ n F H H HD,R WA W H H D,R F − (1 − ρ ) / F H H HD,R WA B HS H HR,S − (1 − ρ ) / H R,S B S A W H H D,R F o (8)s.t. (2b) . By applying the manipulation Tr { ABC } = vec ( A H ) H ( I ⊗ B ) vec ( C ) , vec ( AB ) = ( B T ⊗ I ) vec ( A ) , and Tr { ABCD } = vec ( A H ) H ( D T ⊗ B ) vec ( C ) , problem (8) can be equivalently written as min f f H A f − f H a − a H f (9a)s.t. f H A f ≤ C f , (9b) where f = vec ( F ) , A = (((1 − ρ ) H R,S B S B HS H HR,S ) T + I r R σ n ) ⊗ ( H HD,R WA W H H D,R ) , a = vec ((1 − ρ ) / H HD,R WA B HS H HR,S ) , A = ((1 − ρ ) H R,S B S B HS H HR,S + (1 − ρ ) H R,D Q D H HR,D + I r R σ n ) T ⊗ I r R , and C f = ρ Tr { H R,D Q D H HR,D + H R,S B S B HS H HR,S } . Because of the positive-semidefinite A and A ,problem (9) is a convex QCQP. Although the numerical result can be achieved by solvingthe problem with an convex optimization toolbox such as CVX [22], a closed-form solutioncan be obtained by analyzing the Karush-Kuhn-Tucker (KKT) conditions. Letting ξ denotethe Lagrangian multiplier associated to (9b), the KKT conditions of problem (9) are listedas [ f ⋆ ] H A f ⋆ ≤ C f , ξ ⋆ ≥ , ξ ⋆ ([ f ⋆ ] H A f ⋆ − C f ) = 0 , and ( A + ξ ⋆ A ) f ⋆ = a . It follows that if [ f ⋆ ] H A f ⋆ < C f , ξ ⋆ = 0 and A f ⋆ = a ; if [ f ⋆ ] H A f ⋆ = C f , ( A + ξ ⋆ A ) f ⋆ = a . Thus, if a is withinthe column space of A and a H A † A A † a < C f (where A † denotes the pseudo inverse of A ),the closed-form solution is obtained as f ⋆ = A † a + N ( A ) , (10) where N ( A ) denotes the null space of A . Otherwise, the optimal solution is given by f ⋆ =( A + ξ ⋆ A ) − a , where the optimal ξ ⋆ can be achieved by solving a H ( A + ξ ⋆ A ) − A ( A + ξ ⋆ A ) − a = C f . In the scenario where r R ≥ r = 1 , the rate maximization problem is equivalentto the signal-to-noise ratio (SNR) maximization, which boils down to the problem solved in [10]. C. Subproblem of B S Fixing the variables ( A , W , F ) , the subproblem of B S can be written as min B S Tr (cid:8) (1 − ρ ) B HS H HR,S F H H HD,R WA W H H D,R FH R,S B S (cid:9) − Tr n (1 − ρ ) / B HS H HR,S F H H HD,R WA + (1 − ρ ) / A W H H D,R FH R,S B S o (11)s.t. (2b) and (2c) . Similarly to the linear algebra manipulation of problem (8), problem (11) can be further formu-lated as an equivalent QCQP problem given by min b b H A b − b H a − a H b (12a)s.t. b H A b ≤ C b , (12b) b H b ≤ P S , (12c) where b = vec ( B S ) , A = I r ⊗ ((1 − ρ ) H HR,S F H H HD,R WA W H H D,R FH R,S ) , a = vec ((1 − ρ ) / H HR,S F H · H HD,R WA ) , A = I r ⊗ ((1 − ρ ) H HR,S F H FH R,S − ρ H HR,S H R,S ) , and C b = Tr { ρ H R,D Q D H HR,D − (1 − ρ ) FH R,D Q D H HR,D F H − σ n FF H } . Because A is indefinite, (12b) is nonconvex and problem (12) isnonconvex, which makes it hard to find the global optimum solution. Since the critical problemis the nonconvex (12b), in the following section, we transform problem (12) into an equivalentform such that (12b) can be written in a linear form and the reformulated (12b) can be convex.
1) Convex Relaxation:
By introducing auxiliary variables t and b ′ (subject to b = b ′ /t and | t | = 1 ), problem (12) is transformed into an equivalently homogenized form given by min b ′ ,t Tr { B Φ ( b ′ , t ) } (13a)s.t. Tr { B Φ ( b ′ , t ) } ≤ C b , (13b)Tr { B Φ ( b ′ , t ) } ≤ P S . (13c)Tr { B Φ ( b ′ , t ) } = 1 , (13d) where Φ ( b ′ , t ) = ( b ′ [ b ′ ] H , t ∗ b ′ ; t [ b ′ ] H , | t | ) , B = ( A , − a ; − a H , , B = ( A , ; H , , B = ( I , ; H , ,and B = ( r × r , ; H , . In problem (13), the optimal b ⋆ of problem (12) can be achieved bycalculating b ⋆ = [ b ′ ] ⋆ /t ⋆ . In order to solve problem (13), by replacing the variables Φ ( b ′ , t ) withone matrix variable X b , the problem can be linearized as an equivalent form given by min X b (cid:23) Tr { B X b } (14a)s.t. Tr { B X b } ≤ C b , (14b)Tr { B X b } ≤ P S , (14c)Tr { B X b } = 1 , (14d)rank ( X b ) = 1 . (14e) Note that problem (14) is still nonconvex and intractable, due to the rank constraint (14e). Inorder to obtain the solution of (14), we relax (14e), achieving a SDP given by min X b (cid:23) Tr { B X b } (15)s.t. (14b), (14c), and (14d) . Problem (15) is convex and can be solved by CVX. However, the minimized value of theobjective function Tr { B X b } may only provide a lower bound of the original problem, becausethe achieved minimizers of problem (15) may violate the rank constraint (14e) in the originalproblem. Fortunately, as proved in Proposition 3.5 in [23], for a separable SDP with m x matrix variables and m c linear constraints, if m c ≤ m x + 2 , an optimal solution to the SDP exists witheach minimizer of rank one. It can be shown that (15) satisfies all the conditions required byProposition 3.5 in [23]. Thus, problem (15) has among others a rank-1 solution. This meansthat with such a rank-1 solution, (14e) can be safely relaxed and the achieved rank-1 solutionturns out to be the global optimum of problem (14). Thereby, the global optimal solutions ofproblems (13) and (12) can be achieved.
2) Postprocessing to Obtain the Rank-1 Solution:
However, it is worth noting that problem(15) does not only have a rank-one solution, and the contemporary interior-point algorithms (IPA)(which are exploited to obtain numerical results for SDPs) usually yield highest-rank solutions[24]. That is, the optimal X ⋆b for (15) achieved by CVX (or other optimization solvers basedon the interior-point algorithm) is always high-rank. Fortunately, the rank reduction procedureproposed in [23] can be applied to find the optimal rank-1 solution. Let R x = rank ( X b ) and X b = V x V Hx (for V x ∈ C ( r +1) × R x ). The optimal solution X b is updated by X b, = V x ( I − /δ ∆ ) V Hx , (16) where ∆ is a R x -by- R x Hermitian matrix satisfying Tr (cid:8) V Hx B m V x ∆ (cid:9) = 0 , m = 2 , , . (17) The coefficient δ in (16) is calculated by δ = arg max { δ k } Rxk =1 | δ k | , where δ k denote the eigenval-ues of ∆ . The updated solution X b, (whose rank is at least one less than rank ( X b ) ) preservesthe primal feasibility and the complementary slackness such that it is optimal for the originalproblem [23]. The optimal rank-1 solution can be found by repeating (16) and (17). Then, theoptimal b ⋆ can be extracted from the rank-1 X b, . D. Convergence of the Iterative Algorithm
Algorithm 1 shows the proposed iterative algorithm, where C iter ( A , W , F , B S ) = Tr { A E ( W , F , B S ) } − log det( A ) . In each iteration (from Lines 3 to 7), the above four subproblems aresolved, rank reduction is performed and the stopping criterion is checked. Theorem 1:
The iterative algorithm as shown in Algorithm 1 converges, as κ tends to infinity. Proof:
Let x ( κ ) , ( x ( κ )1 , . . . , x ( κ )4 ) denote the sequence of the minimizers at the κ th iter-ation, where x , vec ( A ) T , x , vec ( W ) T , x , vec ( F ) T , and x , vec ( B S ) T . Let y ( κ +1) i , ( x ( κ +1)1 , . . . , x ( κ +1) i , x ( κ ) i +1 , . . . , x ( κ )4 ) . Since the subproblems (6), (7), and (9) are convex and B ( κ +1) S isthe global optimal solution of problem (12), it is shown that C iter ( x ( κ ) ) ≥ C iter ( y ( κ +1)1 ) ≥ C iter ( y ( κ +1)2 ) ≥ C iter ( y ( κ +1)3 ) ≥ C iter ( x ( κ +1) ) . (18) Algorithm 1
The proposed iterative algorithm Initialize A (0)0 , W (0) , F (0) , and B (0) S ; set κ ← ; repeat Given ( W ( κ ) , F ( κ ) , B ( κ ) S ) , update A ( κ +1)0 by calculating (6); Given ( A ( κ +1)0 , F ( κ ) , B ( κ ) S ) , update W ( κ +1) by calculating (7); Given ( A ( κ +1)0 , W ( κ +1) , B ( κ ) S ) , obtain f ⋆ by solving problem (9); F ( κ +1) ← reshape ( f ⋆ , r R , r R ) ; Given ( A ( κ +1)0 , W ( κ +1) , F ( κ +1) ) , obtain X ⋆b by solving problem (15) with CVX; Perform the rankreduction procedure (i.e. Algorithm 1 in [23]) for X ⋆b and obtain the optimal rank-1 X ′ b ; b ⋆ ← [ b ′ ] ⋆ /t ⋆ ; B ( κ +1) S ← reshape ( b ⋆ , r, r ) ; κ ← κ + 1 ; until (cid:12)(cid:12)(cid:12) C iter ( A ( κ )0 , W ( κ ) , F ( κ ) , B ( κ ) S ) − C iter ( A ( κ )0 , W ( κ ) , F ( κ ) , B ( κ − S ) (cid:12)(cid:12)(cid:12) < ε Thus, C iter ( x ( κ ) ) monotonically decreases as κ increases. Additionally, C iter ( · ) is lower-bounded.Hence, C iter ( x ( κ ) ) converges. Note that the stopping criterion of Algorithm 1 is related to theconvergence of C iter ( · ) but not the convergence of minimizers as in [19], [25]. Therefore, weconclude that Algorithm 1 converges. Theorem 2:
Suppose that tie-breaking strategies [19] are included in solving problems (9) and(15), as well as the rank reduction procedure, such that f ⋆ , X ⋆b and the rank-1 X b, are uniquelyobtained. Then, the sequences { ( vec ( A ( κ )0 ) T , vec ( W ( κ ) ) T , vec ( F ( κ ) ) T , vec ( B ( κ ) S ) T ) } ∞ κ =0 converge to aunique limit point. Proof:
A tie-breaking strategy is a rule to select a solution from multiple solutions, e.g. toachieve the unique solution to problem (9), the term N ( A ) in (10) can be omitted to yield theunique closed-form solution. To prove the theorem, we show that x ( κ ) and x ( κ +1) converge tothe same limit point by contradiction [25], [26]. For details, please see Appendix A. Theorem 3:
The limit point in Theorem 2 is not necessarily a stationary point of problem (2).
Proof:
See Appendix B for details.As a summary, Theorem 1 indicates that Algorithm 1 can always converge, when the stoppingcriterion is designed as the difference of the objective functions (as shown in Line 7), althoughthe solution to each subproblem may not be unique. Alternatively, the criterion can also berelated to the convergence of the minimizers [19], [25] such as k x ( κ +1) − x ( κ ) k / k x ( κ +1) k < ε ′ . (19) Intuitively, if a subproblem has multiple global solutions, the minimizer may not converge.Theorem 2 illustrates that with such a criterion, the algorithm still converges (i.e. the minimizer converges) provided tie-breaking strategies are applied. Replacing Line 7 in Algorithm 1 with(19), to make Algorithm 1 converge, in the κ th iteration, (9) can be solved by a numericalalgorithm (e.g. an optimization solver) with a uniquely specified initial point (of f ) given ( A ( κ )1 , a ( κ )1 , A ( κ )2 , C ( κ ) f ) . The uniqueness means that if ( A ( κ )1 , a ( κ )1 , A ( κ )2 , C ( κ ) f ) = ( A ( κ ′ )1 , a ( κ ′ )1 , A ( κ ′ )2 , C ( κ ′ ) f ) at two iterations κ and κ ′ , the initial points must be identical in those two iterations. Therefore, f ⋆ can be uniquely attained for a specific (9). Similarly, to uniquely achieve X ⋆b for a specific(15), a unique initial point of X b should be specified for the specific ( B m , C b , P S ) . Then, in therank reduction procedure, solving the system of (17) numerically with a uniquely specified initialpoint of ∆ for the specific ( V x , B m ) , a unique rank-1 X ⋆b, can be finally obtained. Thus, X ⋆b, is uniquely attained for a specific problem of (15). In our implementation, CVX is exploited for(9) and (15), where the default solver SDPT3 solves a specific problem with a uniquely specifiedinitial point [27]. The system of (17) is solved by the fsolve function in MATLAB. Simulationresults confirm the convergence.IV. S IMPLIFIED
EFA R
ELAYING A LGORITHMS
To avoid the relatively high complexity caused by the matrices optimization , considering thescenario r = r R , this section proposes the simplified EFA schemes. Specifically, by taking thesingular value decomposition (SVD) of H D,R and the joint decomposition of ˜H R,S = H R,S Q / S (i.e. the S -to- R effective channel) and H R,D based on the HPM-PLM strategy (which is discussedin Section IV-A2), the arguments (matrices) in det( · ) and Tr ( · ) in (1) can be diagonalized, suchthat the matrices optimization can be simplified to the power optimization. The proposed Algorithm 1 solves four subproblems, among which three subproblems can yield closed-formsolutions. The complexity of the IPA for solving (15) is upper bounded by O (1) (cid:0) r + 1) (cid:1) / ( r +1) (cid:0) r + 1) + 8( r + 1) + 12( r + 1) + 24 (cid:1) [28], [29]. The number of iterations consumed by the following rankreduction procedure is upper bounded by r + 1 . In the scenario r < r R , precoder at S can be the RSV (corresponding to non-zero singular values) of H R,S . The receiver at R can be the conjugate transpose of the corresponding left singular vectors. The precoder at D can be the linear combinationof the RSV of H R,D such that the EF leakage in ID can be close to a certain vector lying in the null space of H R,S . By thismeans, the power consumption of the retransmitted EF leakage can be well controlled. For the space constraint, here we do notdiscuss this scenario, while the case of r = r R is more non-trivial. A. Channel Diagonalization1) Structure of Relay Matrix:
Take the SVD of H D,R = U D,R Σ D,R V HD,R and the SVD of ˜H R,S = ˜U R,S ˜Σ R,S ˜V HR,S . Applying the matrix inversion lemma to (1a) yields / · log det( I + (1 − ρ ) σ n ˜Σ R,S ( I − ( I + ˜U HR,S F H H HD,R H D,R
F ˜U
R,S ) − ) ˜Σ R,S ) , where the matrix between the two ˜Σ R,S equals a positivesemidefinite matrix ˜U HR,S F H H HD,R ( I + H D,R FF H H HD,R ) − H D,R
F ˜U
R,S . Hence, the matrix in det( · ) ispositive-definite. According to Hadamard’s inequality [30], the above log det( · ) is maximizedprovided ˜U HR,S F H V D,R Σ D,R V HD,R
F ˜U
R,S is diagonal. Hence, F = V D,R Σ F ˜U HR,S for Σ F ∈ C r × r andthe argument of the det( · ) in (1a) is diagonalized. Note that in the simplified EFA relaying, tomaximize (1a) with diagonalized argument, all the power harvested at R is used for forwarding(i.e. the inequality in (1b) is converted to an equality). The structure of F indicates that R couples a given receive eigenmode of ˜U R,S with a given transmit eigenmode of V D,R with anamplification factor given by the corresponding diagonal entry of Σ F .
2) Maximize Harvested Power and Minimize Power Leakage:
Take the eigenvalue decompo-sitions (EVD) of Q D = V D Σ D V HD and H R,D Q D H HR,D = ˜U R,D ˜Σ R,D ˜U HR,D . Recall that the inequality in(1b) has been converted to an equality. With the SVD of H R,D , rearranging (1b) yields Tr n (1 − ρ ) Σ HF Σ F ˜Σ R,S + σ n Σ HF Σ F − ρ ˜Σ R,S o (20a) = Tr n(cid:0) ρ I − (1 − ρ ) Σ HF Σ F (cid:1)(cid:16) ˜U HR,S V ∗ D,R Σ D,R U TD,R V D Σ D V HD U ∗ D,R Σ D,R V TD,R ˜U R,S (cid:17)o (20b) = Tr n(cid:0) ρ I − (1 − ρ ) Σ HF Σ F (cid:1)(cid:16) ˜U HR,S ˜U R,D ˜Σ R,D ˜U HR,D ˜U R,S (cid:17)o . (20c) Eq. (20c) describes the difference between the power of the EF harvested at the EH receiverand the EF leaking into the ID receiver. Thus, a strategy can be proposed to maximize theharvested power at R and minimize the power leakage. Recall that the harvested power hasbeen maximized by letting Q D = P D [ U ∗ D,R ] max [ U ∗ D,R ] H max , such that ρ Tr { H R,D Q D H HR,D } = ρP D λ D,R, max .To minimize the power leakage, the EF leaking into the ID receiver should be paired withthe minimum amplification coefficient, such that the power of the retransmitted leakage equals (1 − ρ ) λ f, min P D λ D,R, max , where λ f, min denotes the minimum diagonal entry of Σ HF Σ F . With theHPM-PLM strategy, (20c) should equal ( ρ − (1 − ρ ) λ f, min ) P D λ D,R, max , which is shown to be anupper bound of (20c) by applying Lemma II.1 in [31]. To make (20c) equal to the upper bound, ˜U HR,S V ∗ D,R = P π in (20b), where P π permutates the unique non-zero diagonal entry P D λ D,R, max in the diagonal matrix Σ D,R U TD,R V D Σ D V HD U ∗ D,R Σ D,R to the same position as ρ − (1 − ρ ) λ f, min in ρ I − (1 − ρ ) Σ HF Σ F . By this means, (20c) achieves the upper bound, and the matrices in the traces of (1b) is diagonalized. In summary, the argument of the det( · ) in (1a) is diagonalized with thedecomposed H D,R and the structure of F . The argument of the traces in (1b) is diagonalizedwith the HPM-PLM strategy, i.e. the rank-1 Q D and ˜U R,S = V ∗ D,R P Tπ . Since Q S = H e ˜Σ R,S H He where H e = ( ˜U HR,S H R,S ) − , (1c) becomes Tr { Q S } = P rm =1 k h e,m k ˜ λ R,S,m ≤ P S , where h e,m = [ H e ] m and ˜ λ R,S,m denotes the m th diagonal entry of ˜Σ R,S . Hence, problem (1) reduces to the poweroptimization.
3) Discussion:
Substituting ˜U HR,S = P π V TD,R into the structure of F , we have F = V D,R Σ F P π · V TD,R . Namely, the RSV of F matches the left singular vectors (LSV) of ˜H R,S and a permutationof the LSV of H D,R . Thus, the power of y D is given by E{k y D k } = Tr { (1 − ρ ) Σ D,R Σ F Σ HF ( ˜Σ R,S + P π Σ D,R U TD,R Q D U ∗ D,R P Tπ )+( Σ D,R Σ F Σ HF + I ) σ n } , (21) where U TD,R Q D U ∗ D,R is diagonal. In (21), because ˜H R,S and H R,D share the same (but permutated)LSV, the channel power gains of the effective S -to- R and D -to- R channels in phase 1 are over-lapped at the ID receiver, i.e. (1 − ρ )( ˜Σ R,S + P π Σ D,R U TD,R Q D U ∗ D,R P Tπ ) . Although the retransmitted en-ergy flow can be canceled at D (i.e. P π Σ D,R U TD,R Q D U ∗ D,R P Tπ in (21) is eliminated), the overlappedchannel power gains in phase 1 still impact the rate, because the EF leakage is retransmittedand consume power. Denote the diagonal entries of ˜Σ R,S and (1 − ρ ) P π Σ D,R U TD,R Q D U ∗ D,R P Tπ as ˜ λ R,S , [˜ λ R,S, , . . . , ˜ λ R,S,r ] T and β , [ β , . . . , β r ] T (where the unique non-zero β m = (1 − ρ ) P D λ D,R, max , c ), respectively. In the diagonalized relay power constraint (1b) (i.e. the following (22e)), thediagonal entries of Σ F Σ HF , denoted as λ f , [ λ f, , . . . , λ f,r ] T , are multiplied by the entries of theoverlapped channel power gains, i.e. (1 − ρ )˜ λ R,S,m + β m for m = 1 , . . . , r . Thus, the pairings ofthe diagonal entries of Σ D,R (which is also coupled with Σ F Σ HF ) and the overlapped channelpower gains in phase 1 affect the optimization of λ f and thereby the rate. Additionally, in phase1, the value of each (1 − ρ )˜ λ R,S,m + β m is affected by P π (which determines the pairing of each ˜ λ R,S,m and β m ). B. Joint Power Allocation Optimization
To make further calculation and analysis tractable, the power optimization problem only maxi-mizes the achievable rate at high receive SNR. Substituting the previous channel decompositions into problem (1), the problem can be reformulated as min λ f , ˜ λ R,S − r X m =1 log (1 − ρ )˜ λ R,S,m λ f,m λ D,R,m σ n (1 + λ f,m λ D,R,m ) ! (22a)s.t. λ f, , λ f, , . . . , λ f,r > , (22b) ˜ λ R,S, , ˜ λ R,S, , . . . , ˜ λ R,S,r > , (22c)Tr { Q S } = r X m =1 k h e,m k ˜ λ R,S,m ≤ P S , (22d) r X m =1 λ f,m (cid:16) (1 − ρ )˜ λ R,S,m + σ n + β m (cid:17) = r X m =1 ρ ˜ λ R,S,m + ρP D λ D,R, max , (22e) where β m is constrained by β m = c for m = index ( λ f, min ) (where index ( λ f, min ) returns the indexof λ f, min ); otherwise, β m = 0 . Problem (22) is not convex due to the non-affine (22e). Then,problem (22) is solved by performing power optimizations for R and D alternatively.
1) Relay Optimization with Fixed Source Power Allocation:
With given ˜ λ R,S,m , the poweroptimization problem at R is formulated as max λ f r X m =1 log (1 − ρ )˜ λ R,S,m λ f,m λ D,R,m σ n (1 + λ f,m λ D,R,m ) ! (23)s.t. (22b) and (22e) . The challenge in solving problem (23) is that c of β m is required to be paired with λ f, min , but theposition of λ f, min in λ f is unknown before solving the problem; the rate is affected by the pairingsof the elements of λ D,R , [ λ D,R, , . . . , λ D,R,r ] T and ˜ λ R,S (i.e. the pairings of the eigenmodes ofthe forwarding channel H D,R and the eigenmodes of the effective S -to- R channel ˜H R,S ), evenif the constraint on c is relaxed and β m is fixed. To avoid the high complexity of searchingthe best pairings, we then reveal that the pairing issues can be solved by ordering operations.Firstly, we relax the constraint on c , i.e. P π becomes an arbitrary permutation matrix ˜P π , andassume columns of V D,R are arranged in certain orders, such that the elements of λ D,R and ˜ λ R,S are paired in certain ways and c is paired with a certain ˜ λ R,S,m . Problem (23) then becomes aconvex problem regardless of the pairing issues. By analyzing KKT conditions, a closed-formsolution can be obtained by λ ⋆f,m = − λ D,R,m + 12 vuut λ D,R,m + 4 ν ⋆ λ D,R,m (cid:16) (1 − ρ )˜ λ R,S,m + σ n + β m (cid:17) , (24) where ν ⋆ denotes the Lagrange multiplier for (22e) and is greater than 0 (because λ ⋆f,m > ).It can be calculated by solving P rm =1 λ ⋆f,m (cid:16) (1 − ρ )˜ λ R,B,m + σ n + β m (cid:17) = ρP D λ D,R, max + P rm =1 ρ ˜ λ R,S,m with bisection. Based on (24), two lemmas are revealed. For notational simplicity, let z m , (1 − ρ )˜ λ R,S,m + σ n + β m , z , [ z , . . . , z r ] T ; l m , (1 − ρ )˜ λ R,S,m + σ n , l , [ l , . . . , l r ] T . Lemma 1:
Suppose that the elements in π ( z ) are arranged in the same order as anotherpermutation π ( z ) except that z i and z j (where z i ≤ z j for i < j ) in π ( z ) are swapped in π ( z ) .Namely, z i and z j in π ( z ) are respectively paired with λ D,R,p and λ D,R,q (where λ D,R,p ≤ λ D,R,q for p < q ), while z j and z i in π ( z ) are respectively paired with λ D,R,p and λ D,R,q . Then, thevalue of the objective function of (23) with π ( z ) is no less than that with π ( z ) . Proof:
This lemma is proved by respectively substituting π ( z ) and π ( z ) into (24) with λ D,R and comparing the values of the objective function of (23). See Appendix C for details.Lemma 1 addresses the pairings of the transmit eigenmodes of F (i.e. V D,R ) and the overlappedchannel power gains. With fixed pairings of ˜ λ R,S,m and β m for m = 1 . . . r , the values of theentries of the overlapped channel power gains, i.e. (1 − ρ )˜ λ R,S,m + β m for m = 1 . . . r , are fixed.Lemma 1 reveals that, for two pairs of the transmit eigenmodes of V D,R and the overlappedchannel power gains (while other pairings are fixed), the strongest eigenmode of V D,R and thestrongest overlapped channel power gain should be paired together. The following Lemma 2addresses the pairings of the channel power gains of the effective S -to- R channel and the non-zero channel power gain of the effective D -to- R channels, i.e. the pairings of ˜ λ R,S,m and c (i.e.the unique non-zero β m ) for m = 1 . . . r . It is shown that, for two channel power gains in ˜ λ R,S,m ,the strongest ˜ λ R,S,m should be paired with c . Lemma 2:
Assume two permutations π ( l ) and π ( l ) . In π ( l ) , positions of l i and l j followsthat min { l i + c, l j } and max { l i + c, l j } are respectively paired with λ D,R,i and λ D,R,j (where i < j , l i ≤ l j and λ D,R,i ≤ λ D,R,j ). In π ( l ) , positions of l i and l j follows that min { l i , l j + c } and max { l i , l j + c } are respectively paired with λ D,R,i and λ D,R,j . Other pairings between λ D,R,m and l m (for m = i, j ) in π ( l ) are the same as π ( l ) . Then, the objective function of (23) with c paired with l j yields a higher value than that with c paired with l i . Proof:
Lemma 2 is proved based on Lemma 1. The proof strategy is similar to Lemma 1.See Appendix D for details.
Proposition 1:
When the elements in ˜ λ R,S and λ D,R are arranged in increasing orders and thenon-zero β m is paired with the maximum ˜ λ R,S,m , the value of the objective function of (23) ismaximized and the optimal λ ⋆f,m are arranged in a decreasing order. Proof:
Applying Lemma 1 and Lemma 2, the orderings of ˜ λ R,S and λ D,R and the pairing of β m and ˜ λ R,S,m are proved by induction. The decreasing order of λ ⋆f,m requires that λ ⋆f,m − λ ⋆f,m +1 = − a m + q a m + ν ⋆ a m z m − ( − a m +1 + q a m +1 + ν ⋆ a m +1 z m +1 ) ≥ , where a m = λ D,R,m . Rearrangingthe above inequality, the proof ends up showing ν ⋆ ≥ − ( a m z m − a m +1 z m +1 ) z m ( z m − z m +1 ) z m +1 ( a m − a m +1 ) . The provedordering and pairing show that a m ≤ a m +1 and z m ≤ z m +1 . Since optimal solution (24) isachieved only when ν ⋆ > , there always exists λ ⋆f,m ≥ λ ⋆f,m +1 , i.e. λ ⋆f,m are arranged in adecreasing order. So far, Proposition 1 has been proved.Proposition 1 illustrates that the constraint on β m (i.e. c is paired with λ f, min ) can be safely re-laxed. Following the ordering operation in Proposition 1, entries ( ρ − (1 − ρ ) λ f, min ) and P D λ D,R, max are at lower-right corners of matrices ρ I − (1 − ρ ) Σ HF Σ F and Σ D,R U TD,R V D Σ D V HD U ∗ D,R Σ D,R in (20),respectively. Hence, the permutation matrix ˜P π = I = P π .
2) Source Optimization with Fixed Relay Power Allocation:
According to Proposition 1, λ f,m are arranged in a decreasing order, and index ( λ f, min ) = r . Thus, the source power optimizationproblem is formulated as min ˜ λ R,S − r X m =1 log (1 − ρ )˜ λ R,S,m λ f,m λ D,R,m σ n (1 + λ f,m λ D,R,m ) ! (25a)s.t. < ˜ λ R,S, ≤ ˜ λ R,S, ≤ . . . ≤ ˜ λ R,S,r , (25b)(22d) and (22e) . Problem (25) is convex and can be solved by an optimization solver. Analytical solutions arestill attractive due to its low complexity. The challenge in deriving a closed-form solution is theordering constraint in (25b). However, when the ordering constraint in (25b) is relaxed, the output ˜ λ ⋆R,S,m can still be in an increasing order if ˜ λ R,S,m in (22d) are uniformly weighted (otherwise,the Lagrange multiplier for (22d) would be non-uniformly weighted in the derived closed-formsolution, which may violate the ordering of ˜ λ R,S ). Therefore, problem (25) can be simplifiedby respectively replacing constraints (25b) and (22d) with ˜ λ R,S, , ˜ λ R,S, , . . . , ˜ λ R,S,r > and P rm =1 h e, max ˜ λ R,S,m ≤ P S (where h e, max = max {k h e,m k } ). This simplified problem is referredto as the simplified source power optimization . The simplified source power optimization isconvex, and its KKT conditions are listed as follows. ˜ λ ⋆R,S,m > , m = 1 , . . . , r (26a) r X m =1 ˜ λ ⋆R,S,m ≤ P S /h e, max , (26b) r X m =1 ( λ f,m (1 − ρ ) − ρ ) ˜ λ ⋆R,S,m = − r X m =1 ( σ n + β m ) λ f,m + ρP D λ D,R, max , (26c) γ ⋆ ,m ≥ , γ ⋆ ,m ˜ λ ⋆R,S,m = 0 , (26d) γ ⋆ ≥ , γ ⋆ r X m =1 ˜ λ ⋆R,S,m − P S /h e, max ! = 0 , (26e) Algorithm 2
EFA-S1 Initialize λ (0) f and ˜ λ (0) R,S repeat Update λ ( κ +1) f by calculating (24); Update ˜ λ ( κ +1) R,S by solving (25) with an opti-mization solver; κ ← κ + 1 ; until (cid:12)(cid:12)(cid:12) C ( λ ( κ ) f , ˜ λ ( κ ) R,S ) − C ( λ ( κ ) f , ˜ λ ( κ − R,S ) (cid:12)(cid:12)(cid:12) < ǫ Algorithm 3
EFA-S2 Initialize λ (0) f and ˜ λ (0) R,S repeat Update λ ( κ +1) f by calculating (24); if (28) is satisfied then update ˜ λ ( κ +1) R,S by cal-culating (27); else update ˜ λ ( κ +1) R,S by calculating (29); κ ← κ + 1 ; until (cid:12)(cid:12)(cid:12) C ( λ ( κ ) f , ˜ λ ( κ ) R,S ) − C ( λ ( κ ) f , ˜ λ ( κ − R,S ) (cid:12)(cid:12)(cid:12) < ǫ − / ˜ λ ⋆R,S,m − γ ⋆ ,m + γ ⋆ + µ ⋆ ( λ f,m (1 − ρ ) − ρ ) = 0 , (26f) where γ ⋆ ,m , γ ⋆ , and µ ⋆ denote the optimal Lagrange multipliers. Eq. (26a) and (26d) reveal that γ ⋆ ,m = 0 . If γ ⋆ = 0 , according to (26f), it is obtained that ˜ λ ⋆R,S,m = 1 / ( µ ⋆ ( λ f,m (1 − ρ ) − ρ )) , (27) where µ ⋆ is obtained by solving r/µ ⋆ = ρP D λ D,R, max − P rm =1 ( σ n + β m ) λ f,m . Since ˜ λ ⋆R,S,m > ∀ m and µ ⋆ also conforms to (26b), (27) is obtained provided λ f,m (1 − ρ ) − ρ ≷ , ∀ m ≶ µ ⋆ ⋚ P S /h e, max P rm =1 1 λf,m (1 − ρ ) − ρ (28) is satisfied. On the other hand, if λ > , the optimal ˜ λ ⋆R,S,m is achieved by ˜ λ ⋆R,S,m = 1 / ( γ ⋆ + µ ⋆ ( λ f,m (1 − ρ ) − ρ )) , (29) where γ ⋆ and µ ⋆ can be obtained by solving the non-linear system composed of (26b) and (26c).As a summary, the simplified EFA algorithms are outlined in Algorithms 2 and 3. Thealgorithm solving (25) with an optimization solver is referred to as EFA-S1, while the other(which solves the simplified (25), i.e. the simplified source optimization) is referred to as EFA-S2.The function C ( λ f , ˜ λ R,S ) denotes the objective function (22a). Since the optimization problems(23), (25) and the simplified source optimization are convex, C ( λ f , ˜ λ R,S ) monotonically decreasesover iterations. Because (22a) is lower-bounded, the two algorithms finally converge.V. S IMPLIFIED
NEFA S
CHEME
This section proposes a simplified NEFA relaying (i.e. NEFA-S), considering uniform sourcepower allocation. Similar to the simplified EFA schemes, the optimization problem is simplified to a power optimization by channel diagonalization. The original design problem of an NEFAscheme (i.e. problem (1), where Q D = and the inequality (1b) is converted to equality) can beformulated as max Q ′ S , F ′
12 log det (cid:16) I + (1 − ρ ) H D,R F ′ H R,S Q ′ S H HR,S [ F ′ ] H H HD,R [ R ′ ] − (cid:17) (30a)s.t. Tr n (1 − ρ ) F ′ H R,S Q ′ S H HR,S [ F ′ ] H + σ n F ′ [ F ′ ] H o = ρ Tr (cid:8) H R,S Q ′ S H HR,S (cid:9) , (30b)Tr { Q ′ S } ≤ P S , Q ′ S (cid:23) , (30c) where R ′ = σ n H D,R F ′ [ F ′ ] H H HD,R + σ n I , and F ′ denotes the relay processing matrix. Due to theabsence of the energy flow Q D , the S -to- R channel can be decomposed by SVD, such that H R,S = U R,S Σ R,S V HR,S , where Σ R,S = diag { λ R,S, , . . . , λ R,S,r } . Take the EVD of H R,S Q ′ S H HR,S = ˜U ′ R,S ˜Σ ′ R,S [ ˜U ′ R,S ] H , where ˜Σ ′ R,S = diag { ˜ λ ′ R,S, , . . . , ˜ λ ′ R,S,r } . It follows that Q ′ S = V R,S Σ ′ S V HR,S , where Σ ′ S = diag { λ ′ S, , . . . , λ ′ S,r } , ˜U ′ R,S = U R,S , and ˜Σ ′ R,S = Σ ′ S Σ R,S . Due to the uniform source powerallocation, λ ′ S,m = P S /r ∀ m . Applying the above decomposition and omit the coefficient / , thepower optimization of problem (30) at high receive SNR is formulated as min λ ′ f − r X m =1 log (1 − ρ )˜ λ ′ R,S,m λ ′ f,m λ D,R,m σ n (cid:16) λ ′ f,m λ D,R,m (cid:17) (31a)s.t. λ ′ f, , λ ′ f, , . . . , λ ′ f,r ≥ , (31b) r X m =1 (cid:16) (1 − ρ ) λ ′ f,m ˜ λ ′ R,S,m + σ n λ ′ f,m (cid:17) = r X m =1 ρ ˜ λ ′ R,S,m (31c) where λ ′ f , [ λ ′ f, , . . . , λ ′ f,r ] T . The pairings of ˜ λ ′ R,S,m and λ D,R,m for m = 1 , . . . , r can be solved byLemma 1 with P D = 0 and β = . Hence, if ˜ λ ′ R,S,m = P S /r · [ λ R,S, min , . . . , λ
R,S, max ] T and λ D,R,m are arranged in an increasing order, the pairing problem is solved. Thus, the optimal λ ′ f,m isobtained by (cid:2) λ ′ f,m (cid:3) ⋆ = − λ D,R,m + 12 vuut λ D,R,m + 4[ ν ′ ] ⋆ λ D,R,m (cid:16) (1 − ρ )˜ λ ′ R,S,m + σ n (cid:17) , (32) where [ ν ′ ] ⋆ satisfies (31c). VI. S IMULATION R ESULTS
In the simulations, we assume broadside arrays are exploited, such that the channel matrix H i,j = Λ − i,j ( q K K + q K ¯H i,j ) , where (i.e. all-ones matrix) is the line-of-sight component,and ¯H i,j is the Rayleigh component. The large-scale fading is given by Λ − i,j = d − / ij , where d ij is the distance between nodes i and j and d DS , d DR + d RS . The noise power σ n = 1 µ W, d DS = 10 m, the numbers of antennas at the terminals and R are set as r = r R = 4 , the Rician A c h i e v ab l e r a t e [ bp s / H z ] EFA−OPTEFA−S1EFA−S2Converges at iteration 34Converges at iteration 3Converges at iteration 7
Fig. 2. Convergence example of EFA-OPT, EFA-S1, and EFA-S2. A v e r age r a t e [ bp s / H z ] EFA−OPT NEFA−OPTEFA−S1 EFA−S2 NEFA−S
Fig. 3. Average rate as a function of PS ratio with P D = 0 . W, P S = 0 . W, and d DR /d DS = 0 . . DR /d DS ratio A c h i e v ab l e r a t e [ bp s / H z ] EFA−OPTNEFA−OPTEFA−S1NEFA−SEFA−S2 (a) Achievable rate vs. d DR /d DS , K = 0 . DR /d DS ratio P o w e r s p li tt i ng r a t i o EFA−OPTNEFA−OPTEFA−S1NEFA−SEFA−S2 (b) Best PS ratio vs. d DR /d DS , K = 0 . DR /d DS ratio A c h i e v ab l e r a t e [ bp s / H z ] EFA−OPTNEFA−OPTEFA−S1NEFA−S (c) Achievable rate vs. d DR /d DS , K = 0 . .Fig. 4. Rate performance under different d DR /d DS ratios with P D = 0 . W and P S = 0 . W. factor K = 0 , unless otherwise stated. In the following Figs. 4, 5, 7 and 6, the PS ratio ρ isexhaustively searched among 0.02:0.02:0.98 to maximize the average rate.Fig. 2 illustrates the convergence behaviors of EFA-OPT, EFA-S1, and EFA-S2, when r = 4 ,SNR = 20 dB ( d DR = d RS = 1 m, P S = P D = 0 . W, σ n = 10 − W). Starting at the same initialpoint, EFA-OPT, EFA-S1, and EFA-S2 can converge after 34, 7, and 3 steps, respectively.Fig. 3 investigates the average rate as a function of the PS ratio for a certain d DR /d DS . Itis shown that the average rate curves of the five relaying schemes are concave over the PSratios, reaching the maximum rates at PS ratios of 0.8, 0.8, 0.88, 0.74, and 0.72, respectively.The concave trend can be explained because a low PS ratio results in less available forwardingpower, while a high PS ratio reduces the receive SNR at R . Both the above two factors candecrease the receive SNR at D .Fig. 4(a) shows the achievable rate as a function of d DR /d DS ratio with symmetric powerbudgets at D and S , when K = 0 . In general, the rates of the NEFA schemes (including NEFA-OPT and NEFA-S) decrease as R moves towards D , because R only extracts forwarding powerfrom the information flow, and the reduced forwarding power degrades the rate. Different fromthe NEFA schemes, thanks to the EF, the rates of the EFA schemes (including EFA-OPT, EFA-S1, and EFA-S2) increase as R moves towards D . When R is close to S , similarly to the NEFA schemes, the rates of the EFA schemes can also increase as d DR /d DS increases, because theEFA schemes can also harvest power from the source information flow. It is also observed inFig. 4(a) that the rate of EFA-OPT can be significantly higher than those of the NEFA schemes,when d DR /d DS < . ; while when d DR /d DS ≥ . , the rate of EFA-OPT is slightly lower thanthat of NEFA-OPT, the reason is discussed in the explanation of Fig. 6. The suboptimality ofEFA-S1 and EFA-S2 makes the rate of EFA-S1 always lower than that of EFA-OPT and therate of EFA-S2 always lower than that of EFA-S1. This is because in EFA-S1, the LSV of ˜H R,S is restricted to be V ∗ D,R , which reduces the searching area of the original feasible set of (1b).Further, the source optimization (25) of EFA-S1 requires the eigenvalues of ˜H R,S ˜H HR,S (recallthat ˜H R,S is the S -to- R effective channel) to be ordered in an increasing order, which tightensthe constraint (22c). These reasons lead to a reduced SNR at R . Thus, EFA-S1 is inferior toNEFA-OPT. As a further simplified version, the source power constraint (22d) of EFA-S1 isfurther simplified to P rm =1 max {| h e,m | } ˜ λ R,S,m ≤ P S in EFA-S2, which limits the receivedinformation signal power at R . The rate of NEFA-S is always lower than that of NEFA-OPT,because of the uniform source power allocation in NEFA-S. As d DR /d DS increases, the gapbetween the rates of the NEFA schemes decreases. This is because with high-quality S -to- R link, S prefers to uniformly allocate the source power.Fig. 4(b) demonstrates that when R is close to D , the S -to- R link is the critical link. Toachieve high SNR at D , a low PS ratio should be selected. On the contrary, when R is closeto S , the R -to- D link becomes the critical link, and the best PS ratios in this case are higherthan those at low d DR /d DS ratios. In general, the best PS ratios of the NEFA schemes are muchhigher than those of the EFA schemes at d DR /d DS = 0 . , because the NEFA schemes onlyharvests power from the information flow. When R is close to S (e.g. d DR /d DS is . or . ),the gap between the best PS ratios of the EFA schemes and those of the NEFA schemes arenegligible. This is because the EF heavily attenuates in these d DR /d DS regions, both the EFAschemes and the NEFA schemes have to severely rely on the information flow for gaining theforwarding power. Fig. 4(b) also depicts that the best PS ratios of NEFA-S are higher than thatof NEFA-OPT. This is because the uniform source power allocation makes NEFA-S suffer froma low-performance S -to- R link. Thus, in order to improve the rate, the fraction of the signalpower allocated to the EH receiver can be increased to enhance the forwarding power, such thatthe SNR at D can be improved. DR /d DS ratio A c h i e v ab l e r a t e [ bp s / H z ] EFA−OPTNEFA−OPT NEFA−SEFA−S1 (a) Achievable rate vs. d DR /d DS . DR /d DS ratio P o w e r s p li tt i ng r a t i o EFA−OPTNEFA−OPTEFA−S1NEFA−S (b) Best PS ratio vs. d DR /d DS .Fig. 5. Rate performance under different d DR /d DS ratios with P D = 5 W and P S = 0 . W. As shown in Fig. 4(c), we also investigate the scenario where K = 0 . [32]. The averageachievable rates are similar to the scenario where K = 0 . The best PS ratios in this scenario arealso similar to Fig. 4(b). Thus, the plot on PS ratio is omitted, due to the space constraint.Fig. 5 studies the asymmetric power budgets scenario where P D is much greater than P S .Intuitively, in such a scenario, the effect of the S -to- R link on the end-to-end rate may not beas significant as in the symmetric case. Thus, as shown in Fig. 5(a), EFA-S1 can outperformthe NEFA schemes rate-wise, when d DR /d DS ≤ . , although it suffers from the low S -to- R link performance. It is also observed that the rate of EFA-OPT is higher than those of all theNEFA schemes, when d DR /d DS ≤ . . Different from the symmetric case, the rates of the EFAschemes always increase as d DR /d DS increases, but not decrease and then increase as in Fig.4(a). Because of the asymmetric power budgets, although the forwarding power decreases as R moves towards S , the forwarding power can still make the rates scale with d DR /d DS . Asshown in Fig. 5(b), even when d DR /d DS = 0 . , the best PS ratios of the EFA schemes can besignificantly lower than those of the NEFA schemes (which is different from Fig. 4(b)). Thisis because P D is much higher than P S , and the power of the attenuated EF signal is still largeenough to affect the forwarding power. It is shown that the gaps between the best PS ratio ofthe EFA schemes are significant at d DR /d DS of . and . . This is because EFA-OPT have abetter S -to- R link performance (i.e. the receive SNR at the ID receiver can be much higher),such that the rate can be improved by increasing the fraction of the signal power allocated tothe ID receiver.In the scenario where r R ≥ r = 1 , our previous study [10] reveals that the EF is beneficial tothe EFA scheme (i.e. the rate of the EFA scheme is significantly higher than that of the NEFAscheme) only when r R > . In the MIMO relay system, although the EFA schemes can stillbenefit from the EF (i.e. the rates of EFA schemes can increase as R moves towards D ) when DR /d DS ratio A c h i e v ab l e r a t e [ bp s / H z ] EFA−OPT, r R = 8NEFA−OPT, r R = 8EFA−OPT, r R = 4NEFA−OPT, r R = 4 (a) Achievable rate vs. d DR /d DS . DR /d DS ratio P c t. o f Q S w / o λ i ( Q S ) c l o s e t o Blue: r R = 4 Red: r R = 8 (b) EFA-OPT: Percentage of the Q S w/oeigenvalues (i.e. λ i ( Q S ) ) close to 0. DR /d DS ratio P o w e r s p li tt i ng r a t i o EFA−OPT, r R = 8NEFA−OPT, r R = 8EFA−OPT, r R = 4NEFA−OPT, r R = 4 (c) Best PS ratio vs. d DR /d DS .Fig. 6. Rate performance with different numbers of antennas at R with P D = 0 . W and P S = 0 . W. r R = r , Fig. 6 reveals that the presence of more antennas at R (i.e. r R > r ) can further enhancethe rate of the EFA scheme. It is observed in Fig. 6(a) that when d DR /d DS = 0 . , the rate ofEFA-OPT with r R = 4 (i.e. . ) is slightly lower than that of NEFA-OPT with r R = 4 (i.e. . ). A similar phenomenon can also be observed in Fig. 5(a). However, at d DR /d DS = 0 . ,the rate of EFA-OPT with r R = 8 (i.e. . ) is slightly higher than that of NEFA-OPT(i.e. . ). Analyzing Figs. 6(a) and 6(b) reveals the reason. Fig. 6(b) studies the percentageof the Q S (achieved by EFA-OPT) without eigenvalues (i.e. λ i ( Q S ) for i = 1 , . . . , r ) close to0 at d DR /d DS ratios of . and . . Such a Q S without eigenvalues close to 0 indicates thatno data stream is allocated with power close to 0. As shown in Fig. 6(b), when r R = r and d DR /d DS = 0 . , in most cases, all the r data streams at S are allocated with considerably largepower. Thus, in most cases, all the r linearly combined data streams at R are allocated withconsiderably high power. However, in EFA-OPT, except the r data streams, there is one moreEF signal being input into R . Since the dimension of the signal space at the ID receiver of R is r , the EF leakage is totally combined with the r data streams and amplified considerably.Although the retransmitted EF leakage can be canceled at D , it consumes lots of forwardingpower. Nevertheless, when r R > r , the dimension of the signal space at R is r R ≥ r +1 , such thatthe EF leakage can be nearly aligned with a vector direction orthogonal to those of the linearlycombined data streams. Thus, the EF leakage can be amplified with a smaller coefficient, andmore power is consumed for the desired signal. Therefore, when r R = 8 , the rate of EFA-OPTis higher than that of NEFA-OPT. Despite the increased dimension of the signal space at R , theincrease of r R also improves the information signal power, as well as the EF power, at R . Thus,as shown in Fig. 6(a), the rates of EFA-OPT and NEFA-OPT with r R = 8 are higher than those DR /d DS ratio A c h i e v ab l e r a t e [ bp s / H z ] EFA−OPTNEFA−OPTEFA−S1NEFA−S (a) Rate of relay schemes when r R = 4 . DR /d DS ratio A c h i e v ab l e r a t e [ bp s / H z ] NEFA−OPT, r R =4EFA−OPT, r R =4EFA−OPT, r R =8NEFA−OPT, r R =8 (b) Effect of number of antennas at R .Fig. 7. Achievable Rate vs. d DR /d DS . For EFA schemes, P D = 0 . W and P S = 0 . W. For NEFA schemes, P ′ S = 0 . W. of the schemes with r R = 4 , respectively. Fig. 6(b) also implies that an increase of r R enhancesthe S -to- R link performance. Fig. 6(c) indicates that the best PS ratios of NEFA-OPT when r R = 8 is lower than those with r R = 4 , due to the increased information signal power at R .Fig. 7 studies the scenario where the EFA and the NEFA systems have the same total powerbudget. When the relay is close to S , the power of the harvested EF at R is tiny in the EFAschemes, because of the high path loss . The forwarding power at R mainly comes from the sourcesignal. Therefore, as shown in Fig. 7(a), the achievable rates of the EFA schemes are less thanthat of the NEFA schemes. When R is close to D , the amount of the harvested EF power is highenough, such that the EFA-OPT can outperform NEFA-OPT rate-wise when d DR /d DS = 0 . .Fig. 7(b) depicts that by increasing the number of antennas at R , harvested power at R can beefficiently used to amplify the desired signal (as discussed in the explanation for Fig. 6), suchthat the rate difference between EFA-OPT and NEFA-OPT at d DR /d DS = 0 . when r R = 8 islarger than that when r R = 4 . VII. C ONCLUSION
In this paper, we have proposed the energy-flow-assisted (EFA) relaying protocol for theMIMO autonomous relay network, where the wireless-power relay node can relay the multiplesource data streams and harvest the power for forwarding by processing the superposition of theenergy flow (EF) from the destination and the source information signal. It is shown that contraryto the non-energy-flow-assisted (NEFA) relaying (where the relay only extracts power from thesource signal for forwarding), the EF can significantly improve the rate of the EFA schemes,when the relay is close to the destination. It is also revealed that the additional antennas at therelay (i.e. number of antennas at the relay is greater than that at the terminals) can increase thedimension of the signal space at the information detecting receiver of the relay. By making use ofthe additional dimension, the information signal can be less interfered with the EF leakage, such that more power can be used to amplify and forward the desired information signal. Althoughthe EFA scheme in this paper is studied from a communication theory and signal processingperspective and relies on several assumptions, the outcome of the research can be used asbenchmarks for future studies, e.g. robust design for imperfect CSIT and practical impairments.A PPENDIX
A. Proof of Theorem 2
Eq. (10) implies that problem (9) has infinite number of solutions. To achieve the uniquesolution f ⋆ to (9), a tie-breaking rule can be included. Problem (15) and the system of (17)also have multiple solutions. Applying tie-breaking strategies, X ⋆b can be uniquely obtained bysolving (15), while the optimal rank-1 solution X b, can be uniquely derived from X ⋆b by therank reduction. Hence, the global optimal solution b ⋆ to (12) is uniquely attained.Due to the compactness of x , there exists a limit point ¯x = ( ¯x , ¯x , ¯x , ¯x ) such that x ( κ ) converges to ¯x as κ tends to infinity (i.e. x ( κ ) → ¯x ). Because of the convergence shown inTheorem 1, we have C iter ( x ( κ ) ) → C iter ( ¯x ) . Proving the convergence of { x ( κ ) } ∞ κ =0 is to show that if x ( κ ) → ¯x , x ( κ +1) → ¯x . Due to (6) and (7), problems (4) and (5) have unique solutions. Thanksto the tie-breaking rule, problem (9) also has an unique optimal solution. Thus, by using thecontradiction method in [25], it can be easily shown that if x ( κ ) → ¯x , y ( κ +1)1 → ¯x ; if y ( κ +1)1 → ¯x , y ( κ +1)2 → ¯x ; if y ( κ +1)2 → ¯x , y ( κ +1)3 → ¯x . Then, it remains to show that if y ( κ +1)3 → ¯x , x ( κ +1) → ¯x . Recallthat when solving the subproblem of B S , x ( κ )4 and x ( κ +1)4 are extracted from the optimal rank-1 matrices X ( κ ) b, and X ( κ +1) b, , respectively. Let y ( κ +1) B, , ( x ( κ +1)1 , x ( κ +1)2 , x ( κ +1)3 , vec ( X ( κ ) b, ) T ) , y ( κ +1) B, , ( x ( κ +1)1 , x ( κ +1)2 , x ( κ +1)3 , vec ( X ( κ +1) b, ) T ) and ¯X b, = [ ¯x T ¯x ∗ , ¯x T ; ¯x ∗ , . Proving the above claim ends upshowing that if y ( κ +1) B, → ( ¯x , ¯x , ¯x , vec ( ¯X b, ) T ) , y ( κ +1) B, → ( ¯x , ¯x , ¯x , vec ( ¯X b, ) T ) . This is then provedby contradiction. Assuming that the above claim is not true, there always exists a non-zero scalar e such that k X ( κ +1) b, − X ( κ ) b, k F ≥ e . Let Z = ( X ( κ +1) b, − X ( κ ) b, ) / k X ( κ +1) b, − X ( κ ) b, k F such that Z → ¯Z . Byfixing a θ ∈ [0 , e ] , we can obtain a point X ( κ ) b, + θ Z lying in the segment of X ( κ +1) b, and X ( κ ) b, . Sincethe feasible set of problem (15) is convex, the point X ( κ ) b, + θ Z is within this feasible set. Denotethe objective function of (15) as C B ( X b ; x , x , x ) . Since ( x , x , x ) in y ( κ +1) B, and y ( κ +1) B, are fixedas ( x ( κ +1)1 , x ( κ +1)2 , x ( κ +1)3 ) , the notation of this objective function is simplified as C B ( X b ) . Due to theoptimality of the rank-1 X ( κ +1) b, , we have C B ( X ( κ +1) b, ) ≤ C B ( X ( κ ) b, + θ Z ) . Meanwhile, because C B ( X b ) is convex, C B ( X ( κ ) b, + θ Z ) ≤ C B ( X ( κ ) b, ) [33]. Thus, C B ( X ( κ +1) b, ) ≤ C B ( X ( κ ) b, + θ Z ) ≤ C B ( X ( κ ) b, ) . (A.1) Because C iter ( x ( κ ) ) → C iter ( ¯x ) and (18), C iter ( y ( κ +1) B, ) = C B ( X ( κ ) b, ) + C ( x ( κ +1)1 , x ( κ +1)2 , x ( κ +1)3 ) → C iter ( ¯x ) = C B ( ¯X b, )+ C ( ¯x , ¯x , ¯x ) , where C ( · ) denotes the other terms not containing ¯X b, . Hence, C B ( X ( κ ) b, ) → C B ( ¯X b, ) . Because of (A.1), the value of C B ( X ( κ +1) b, ) also converges to C B ( ¯X b, ) . Taking the limit of(A.1) as κ tends to infinity yields C B ( ¯X b, ) ≤ C B ( ¯X b, + θ ¯Z ) ≤ C B ( ¯X b, ) , i.e. C B ( ¯X b, + θ ¯Z ) = C B ( ¯X b, ) .This means that given vec ( A ) T = ¯x , vec ( W ) T = ¯x , vec ( F ) T = ¯x , both the high-rank ¯X b, + θ ¯Z and the rank-1 ¯X b, are the optimal solutions of problem (15). Next, making use of contradiction,we show that the optimal rank-1 solution derived from ¯X b, + θ ¯Z is different from ¯X b, . Thus,assume the contrary, i.e. ¯X b, is the rank-1 solution derived from ¯X b, + θ ¯Z . Since the rank updaterules of (16) and (17) preserve the primal feasibility (i.e. Tr { B m X b } = Tr { B m X b, } for m = 2 , , )[23], it follows that Tr { B m ( ¯X b, + θ ¯Z ) } = Tr { B m ¯X b, } , namely, Tr { B m ¯Z } = 0 . Recall that we haveassumed that X ( κ ) b, and X ( κ +1) b, converge to different limit points. Let X ( κ +1) b, converge to anotherrank-1 matrix ¯X ′ b, . It follows that Tr { B m ¯X ′ b, } = Tr { B m ¯X b, } , which implies that two rank-1solutions can be derived from one high-rank optimal solution by the rank reduction procedure.This contradicts to the hypothesis that (with a tie-breaking strategy) the rank reduction procedureyields an unique rank-1 solution. Thus, the rank-1 solution derived from ¯X b, + θ ¯Z is differentfrom ¯X b, . However, this claim contradicts to the hypothesis that solving problem (15) only yieldan unique rank-1 solution. This contradiction illustrates that if y ( κ +1) B, → ( ¯x , ¯x , ¯x , vec ( ¯X b, ) T ) , y ( κ +1) B, → ( ¯x , ¯x , ¯x , vec ( ¯X b, ) T ) . Consequently, we conclude that if x ( κ ) → ¯x , x ( κ +1) → ¯x . B. Proof of Theorem 3
Let A = Ψ A ( W , F , B S ) , W = Ψ W ( A , F , B S ) , F = Ψ F ( A , W , B S ) , and B S = Ψ B S ( A , W , F ) represent subproblems (4), (5), (8), and (11), respectively. It is shown in Appendix A that y ( κ +1)1 , y ( κ +1)2 , y ( κ +1)3 and x ( κ +1) converge to ¯x . Hence, we have ¯A = Ψ A ( ¯W , ¯F , ¯B S ) , ¯W = Ψ ¯A , W ( ¯F , ¯B S ) , ¯F = Ψ F ( ¯A , ¯W , ¯B S ) , and ¯B S = Ψ B S ( ¯A , ¯W , ¯F ) . Thus, in the subproblems of A and W , ¯A and ¯W respectively satisfy corresponding Karush-Kuhn-Tucker (KKT) conditions such that ∇ A ( Tr { ¯A E ( ¯W , ¯F , ¯B S ) } − log det( ¯A )) = 0 and ∇ W ( Tr { ¯A E ( ¯W , ¯F , ¯B S ) } ) = 0 . Let g R ( F , B S ) , Tr { (1 − ρ ) FH R,S B S B HS · H HR,S F H + (1 − ρ ) FH R,D Q D H HR,D F H + σ n FF H } − ρ Tr { H R,D Q D H HR,D + H R,S B S B HS H HR,S } and g S ( B S ) , Tr { B S B HS }− P S . The Lagrangian of problem (8) is given by L F ( F , ξ ) = Tr { ¯A E ( ¯W , F , ¯B S ) } + ξ g R ( F , ¯B S ) . Thus, ¯F and the associated optimal Lagrangian multiplier ¯ ξ must satisfy theKKT conditions given by ∇ F ∗ ( Tr { ¯A E ( ¯W , ¯F , ¯B S ) } ) + ¯ ξ ∇ F ∗ g R ( ¯F , ¯B S ) = 0 and ¯ ξ ≥ , g R ( ¯F , ¯B S ) ≤ , ¯ ξ g R ( ¯F , ¯B S ) = 0 . The Lagrangian of problem (11) is: L B S ( B S , ξ , ǫ ) = Tr { ¯A E ( ¯W , ¯F , ¯B S ) } + ξ g R ( ¯F , B S ) + ǫ g S ( B S ) . Hence, ¯B S and the associated optimal multipliers ¯ ξ and ¯ ǫ must satisfythe KKT conditions listed as follows. ∇ B ∗ S ( Tr { ¯A E ( ¯W , ¯F , ¯B S ) } )+ ¯ ξ ∇ B ∗ S g R ( ¯F , ¯B S )+ ¯ ǫ ∇ B ∗ S g S ( ¯B S ) = 0 , ¯ ξ ≥ , g R ( ¯F , ¯B S ) ≤ , ¯ ξ g R ( ¯F , ¯B S ) = 0 , and ¯ ǫ ≥ , g S ( ¯B S ) ≤ , ¯ ǫ g S ( B S ) = 0 . The complementaryslackness conditions in the KKT conditions of ¯F = Ψ F ( ¯A , ¯W , ¯B S ) and ¯B S = Ψ B S ( ¯A , ¯W , ¯F ) implies that when g R ( ¯F , ¯B S ) < , ∇ F ∗ g R ( ¯F , ¯B S ) and ∇ B ∗ S g R ( ¯F , ¯B S ) are inactive in the Lagrangianfunctions for F and B S . Thus, under this condition, combining the KKT conditions of ¯A =Ψ A ( ¯W , ¯F , ¯B S ) , ¯W = Ψ ¯A , W ( ¯F , ¯B S ) , ¯F = Ψ F ( ¯A , ¯W , ¯B S ) and ¯B S = Ψ B S ( ¯A , ¯W , ¯F ) shows that ( ¯A , ¯W , ¯F , ¯B S , ¯ ǫ ) satisfies the KKT conditions of problem (2). C. Proof of Lemma 1
In the subsequent part, it is defined that a m , λ D,R,m and h ( z m , ν, a m ) , a m λ ⋆f ( z m , ν, a m ) / (1+ λ ⋆f ( z m ,ν, a m ) a m ) , where λ ⋆f ( · ) denotes (24). Since the objective function of (23) equals log[ (cid:0) (1 − ρ ) /σ n (cid:1) r · Q (˜ λ R,S,m h ( z m , ν, a m ))] and log( · ) monotonically increases, proving Lemma 1 ends up showing h ( z i , ν , a p ) h ( z j , ν , a q ) Y [ h ( z m , ν , a n )] ≥ h ( z j , ν , a p ) h ( z i , ν , a q ) Y [ h ( z m , ν , a n )] , (C.1) where m = i, j , n = p, q ; ν and ν optimal multipliers corresponding to π ( z ) and π ( z ) . Sincethe power allocation at the source is fixed, the r.h.s. of the equality constraint (22e) equals aconstant. Thus, the l.h.s. of (22e) (which is a function of ν and π ( z ) ) with ν and π ( z ) is equal tothat with ν and π ( z ) . That is, ν and ν conform to z i λ ⋆f ( z i , ν , a p )+ z j λ ⋆f ( z j , ν , a q ) − z j λ ⋆f ( z j , ν , a p ) − z i λ ⋆f ( z i , ν , a q ) = P [ √ z m ( √ ν ν z m + 4 a n ν − √ ν ν z m + 4 a n ν ) / (2 a n √ ν ν )] , where m = i, j and n = p, q . The above equality reveals a constraint on ν and ν : if ν ≤ ν , the l.h.s. of the aboveequality is no greater than 0; otherwise, its l.h.s. is no less than 0.When < ν ≤ ν , ∂h ( z m , ν, a n ) /∂ν < . Therefore, the proof of Lemma 1 ends up show-ing h ( z i , ν , a p ) h ( z j , ν , a q ) ≥ h ( z j , ν , a p ) h ( z i , ν , a q ) . After manipulation, proving the aboveinequality becomes to show ν p ν z i + 4 a q √ z j √ z i p ν z j + 4 a p − ν √ z j √ z i p ν z i + 4 a p p ν z j + 4 a q − ν z i z j + ν z i z j − a p ν z j + 2 a p ν z i − a q ν z i + 2 a q ν z j ≥ . Since ν ( a p z i + a q z j ) − ν ( a p z j + a q z i ) ≥ ν ( a p z i + a q z j ) − ν ( a p z j + a q z i ) = 4 ν ( a p − a q )( z i − z j ) ≥ , z i z j ( ν − ν ) ≥ and a p a q ( ν − ν ) ≥ ,we have ν p ν z i + 4 a q √ z j √ z i p ν z j + 4 a p − ν √ z j √ z i p ν z i + 4 a p p ν z j + 4 a q ≥ . Similarly, we alsohave − a p ν z j +2 a p ν z i − a q ν z i +2 a q ν z j ≥ − a p ν z j +2 a p ν z i − a q ν z i +2 a q ν z j = 2 ν ( a p − a q )( a i − a j ) ≥ . Hence, Lemma 1 is proved in the region < ν ≤ ν . Verified by numerous numerical results,we conjecture that in the region ν > ν , (C.1) still holds provided the aforementioned constraintis satisfied. The mathematical proof is not shown, because of high complexity and difficulty. D. Proof of Lemma 2
In the following proof, it is still defined that a m , λ D,R,m . Since l m + β m = z m , (24) is definedas λ ⋆f ( l m + β m , ν, a m ) . The non-zero β m is denoted by c . Thereby, h ( l m + β m , ν, a m ) , λ ⋆f ( l m + β m , ν, a m ) a m / (1 + λ ⋆f ( l m + β m , ν, a m ) a m ) .
1) Case of l i + c ≤ l j : According to Lemma 1, l i + c and l j in z are paired with a i and a j ,while l i and l j + c in z are paired with a i and a j . Proving Lemma 2 ends up showing h ( l i , ν , a i ) h ( l j + c, ν , a j ) Y h ( z m , ν , a n ) ≥ h ( l i + c, ν , a i ) h ( l j , ν , a j ) Y h ( z m , ν , a n ) , (D.1) where m, n = i, j . Similar to the proof of Lemma 1, according to (22e), ν and ν conform to: d i λ ⋆f ( l i , ν , a i )+( l j + c ) λ ⋆f ( l j + c, ν , a j ) − ( l i + c ) λ ⋆f ( l i + c, ν , a i ) − l j λ ⋆f ( l j , ν , a j ) = P [ √ z m ( √ ν ν z m + 4 a n ν −√ ν ν z m + 4 a n ν ) / (2 a n √ ν ν )] , where m, n = i, j . This equality indicates constraints on ν and ν : when ν ≤ ν , the l.h.s. of the above equality is no greater than 0; otherwise, the l.h.s. is noless than 0.When ν ≤ ν , (D.1) always holds, if h ( l i , ν , a i ) h ( l j + c, ν , a j ) ≥ h ( l i + c, ν , a i ) h ( l j , ν , a j ) . Aftermanipulation, proving the above inequality ends up showing − cl i ν + cl j ν − l i l j ν + l i l j ν − a i cν − a i l j ν +2 a i l j ν +2 a j cν − a j l i ν +2 a j l i ν − ν √ l i ν + 4 a i √ l i p cν + l j ν + 4 a j p l j + c + ν √ cν + l i ν + 4 a i ·√ l i + c p l j ν + 4 a j p l j ≥ . It is easy to prove that − cl i ν + cl j ν − l i l j ν + l i l j ν − a i cν − a i l j ν +2 a i l j ν + 2 a j cν − a j l i ν + 2 a j l i ν ≥ . Then, since ν √ l i ν + 4 a i p cν + l j ν + 4 a j p l i l j + l i c ≤ ν ·√ l i ν + 4 a i p cν + l j ν + 4 a j p l i l j + l i c = ν √ l i ν + 4 a i p cν / ( l j ν + 4 a j ) + 1 p l j ν + 4 a j p l i l j + l i c and ν √ cν + l i ν + 4 a i p l j ν + 4 a j p l i l j + l j c ≥ ν √ cν + l i ν + 4 a i p l j ν + 4 a j p l i l j + l j c = ν √ l i ν + 4 a i · p cν / ( l i ν + 4 a i ) + 1 p l i l j + l j c p l j ν + 4 a j , it is obtained that ν √ cν + l i ν + 4 a i √ l i + c p l j ν + 4 a j · p l j − ν √ l i ν + 4 a i √ l i p cν + l j ν + 4 a j p l j + c ≥ . Thereby, (D.1) is proved, and the aforemen-tioned constraint on ν and ν is actually relaxed.
2) Case of l i + c ≥ l j : According to Lemma 1, l j and l i + c in z are paired with a i and a j , respectively; while l i and l j + c in z are paired with a i and a j , respectively. Thus, provingLemma 2 ends up showing h ( l i , ν , a i ) h ( l j + c, ν , a j ) Y h ( z m , ν , a n ) ≥ h ( l j , ν , a i ) h ( l i + c, ν , a j ) Y h ( z m , ν , a n ) , (D.2) where m, n = i, j . According to (22e), ν and ν in (D.2) conform to: l i λ ⋆f ( l i , ν , a i ) + ( l j + c ) λ ⋆f ( l j + c, ν , a j ) − l j λ ⋆f ( l j , ν , a i ) − ( l i + c ) λ ⋆f ( l i + c, ν , a j ) = P [ √ z m ( √ ν ν z m + 4 a n ν − √ ν ν z m + 4 a n ν ) / (2 a n ·√ ν ν )] , where m, n = i, j . Thus, ν and ν conform to: when ν ≤ ν , the l.h.s of the aboveequality is no greater than 0; otherwise, the l.h.s. is no less than 0. When ν ≤ ν , proving (D.2) ends up showing h ( l i , ν , a i ) h ( l j + c, ν , a j ) ≥ h ( l j , ν , a i ) h ( l i + c, ν , a j ) . After manipulation, proving the above inequality becomes to show − cl i ν + cl j ν − l i l j ν + l i l j ν − a i cν + 2 a i cν + 2 a i l i ν − a i l j ν − a j l i ν + 2 a j l j ν − ν p cν + l j ν + 4 a j · p l j + c √ l i ν + 4 a i √ l i + ν p l j ν + 4 a i p l j p cν + l i ν + 4 a j √ l i + c ≥ . In the above formula, it is clear that − cl i ν + cl j ν − l i l j ν + l i l j ν − a i cν + 2 a i cν ≥ and a i l i ν − a i l j ν − a j l i ν + 2 a j l j ν ≥ a i l i ν − a i l j ν − a j l i ν +2 a j l j ν = 2 ν ( a i − a j )( l i − l j ) ≥ . Additionally, ( l j ν + 4 a i )( l i ν + 4 a j ) − ( l i ν + 4 a i )( l j ν + 4 a j ) = l i l j ( ν − ν )+4 a i l i ν +4 a j l j ν − a i l j ν − a j l i ν ≥ l i l j ( ν − ν )+4 ν ( a i − a j )( l i − l j ) ≥ and cν / ( l i ν +4 a j ) ≥ cν / ( l j ν + 4 a j ) . Therefore, ν p l j ν + 4 a i p l j p cν + l i ν + 4 a j √ l i + c − ν p cν + l j ν + 4 a j p l j + c √ l i ·√ l i ν +4 a i = ν p ( l j ν +4 a i )( l i ν +4 a j ) p cν / ( l i ν + 4 a j ) + 1 p l i l j + cl j − p cν / ( l j ν +4 a j ) + 1 p l i l j + cl i · ν p ( l i ν + 4 a i )( l j ν + 4 a j ) ≥ . Hence, (D.2) is proved. Similar to Appendix C, when ν > ν and ν > ν , we conjecture that (D.1) and (D.2) hold, respectively.R EFERENCES [1] Y. Huang and B. Clerckx, “Joint wireless information and power transfer in a three-node autonomous MIMO relay network,”in
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