Wireless Information and Energy Transfer for Two-Hop Non-Regenerative MIMO-OFDM Relay Networks
aa r X i v : . [ c s . I T ] D ec Wireless Information and Energy Transfer forTwo-Hop Non-Regenerative MIMO-OFDM RelayNetworks
Ke Xiong,
Member, IEEE , Pingyi Fan,
Senior Member, IEEE , Chuang Zhang,and Khaled Ben Letaief,
Fellow, IEEE
Abstract —This paper investigates the simultaneous wireless in-formation and energy transfer for the non-regenerative multiple-input multiple-output orthogonal frequency-division multiplexing(MIMO-OFDM) relaying system. By considering two practicalreceiver architectures, we present two protocols, time switching-based relaying (TSR) and power splitting-based relaying (PSR).To explore the system performance limit, we formulate twooptimization problems to maximize the end-to-end achievableinformation rate with the full channel state information (CSI)assumption. Since both problems are non-convex and have noknown solution method, we firstly derive some explicit results bytheoretical analysis and then design effective algorithms for them.Numerical results show that the performances of both protocolsare greatly affected by the relay position. Specifically, PSR andTSR show very different behaviors to the variation of relayposition. The achievable information rate of PSR monotonicallydecreases when the relay moves from the source towards thedestination, but for TSR, the performance is relatively worsewhen the relay is placed in the middle of the source and thedestination. This is the first time to observe such a phenomenon.In addition, it is also shown that PSR always outperforms TSRin such a MIMO-OFDM relaying system. Moreover, the effect ofthe number of antennas and the number of subcarriers are alsodiscussed.
Index Terms —Energy harvesting, wireless power transfer,MIMO-OFDM, non-regenerative relaying
I. I
NTRODUCTION
Energy harvesting (EH) is capable of powering communica-tion devices and networks with energy harvested from environ-ment, which has emerged as a promising approach to prolongthe lifetime of energy constrained wireless communication[1]–[3]. For instance, in wireless sensor networks, when asensor is depleted of energy, it cannot fulfill its role any longerunless the source of energy is replenished. Although replacingor recharging batteries provides a solution to this problem,it may incur a high cost and sometimes even be unavailable
This work was supported by the National Nature Science Foundationof China, no. 61201203, partly by “973” program, no. 2012CB316100(2),and also by the Open Research Fund of National Mobile CommunicationsResearch Laboratory, Southeast University (no. 2014D03).K. Xiong is with the School of Computer and Information Technology, Bei-jing Jiaotong University, Beijing 100044, China and was with the Departmentof Electrical Engineering, Tsinghua University, Beijing 100084, R.P. China,e-mail: [email protected], [email protected]. Y. Fan, and C. Zhang are with the Department of ElectricalEngineering, Tsinghua University, Beijing, R.P. China, 10008, e-mail:[email protected],{Dongyq08@, Taoli}@mails.tsinghua.edu.cn.K. B. Letaief is with the School of Engineering, Hong Kong University ofScience & Technology (HKUST). e-mail: [email protected]. due to some physical or economic limitations (e.g., in a toxicenvironments and for sensors inside the body or embeddedin building structures) [4]. Comparatively, harvesting energyfrom external environment may provide a much safer andmuch more convenient solution for such kinds of scenarios.
A. Background
As for energy harvesting techniques, the primary ones [5]–[8] rely on external energy sources, such as solar, wind,vibration, thermoelectric effects or other physical phenomena.Since these energy sources are not components of the com-munication network, to use primary EH techniques requiresthe deployment of peripheral equipments to harvest externalenergy. Moreover, the external energy source in most casescannot be controlled and thus not be always available. Suchuncertainty is too critical for the practical scenario that hashigh requirements on reliability and stability, so it limits theapplications of conventional EH techniques. Recently, a newbranch of EH techniques has been presented, in which thereceiver is able to collect energy from ambient radio frequency(RF) signals and the wireless signal is used as a media todeliver information and energy simultaneously [9]–[14]. Thus,it potentially provides great convenience to mobile users [14].
B. Previous work
As a matter of fact, the concept of simultaneous wirelessinformation and energy transfer (SWIET) can be traced backto [9], where the tradeoff between the energy and informationrate was also characterized for the point-to-point communica-tion scenario. The extension of SWIET to frequency selectivechannels was studied in [10]. Later, some works, see e.g., [11],[12], investigated the SWIET for other scenarios includingmulti-antenna systems [11] and multi-user systems [12]. Inthese works, the EH receiver was assumed to be able tosimultaneously observe and extract power from the samereceived signal.However, the authors in [14] pointed out that this assump-tion may be not well available in practice, because practicalcircuits for harvesting energy from RF signals are not yetable to decode the carried information directly (power collec-tion and information receiving operating with very differentpower sensitivity, e.g., -10dBm for energy receivers versus -60dBm for information receivers). Therefore, they proposedan implementable design with separated information decoding and energy harvesting receiver for SWIET in [14], where twopractical receiver architectures, namely, time switching (TS)and power splitting (PS) were presented. With TS employedat the receiver, the received signal is either processed byan energy receiver for energy harvesting or processed by aninformation receiver for information decoding (ID). With PSemployed at the receiver, the received signal is split into twosignal flows with a fixed power ratio by a power splitter, whereone stream is to the energy receiver and the other one is tothe information receiver.Due to their implementable features, TS and PS architec-tures for SWIET recently have attracted much attention, seee.g., [15], [16]. In [15], it investigated the joint optimizationof transmit power control and scheduling for information andenergy transfer with the receiver’s mode switching over flatfading channels. In [16], the authors focused on exploring theproblem of throughput optimization for the save-then-transmitprotocol with variable energy harvesting rate.Since in wireless cooperative or sensor networks, the relayor sensor nodes often have very limited battery storage andrequire some external charging mechanisms to remain activein the network. Energy harvesting for such networks seemsparticularly important as it can enable information relayingfor the transmissions. Thus, some recent works began to stressthe SWIET with separated energy harvesting and informationreceiving architecture for relay systems, see e.g., [17]–[19],[23]. In [17], EH policies were designed for one-way relayingsystem, where non-regenerative relaying protocol was involvedand the outage probability, as well as the ergodic capacity, wasanalyzed. In [18], power allocation strategies were investigatedfor multiple source-destination pair cooperative EH relay net-works, where non-regenerative protocol was also considered.In [19], the SWIET was considered for cooperative networkswith spatially random relays, where the outage and diversityperformance were studied by applying stochastic geometry,and in [23], the distributed PS-based SWIET was designed forinterference relay channels by using game theory. However,these works just investigated the SWIET in single-carrier andsingle antenna relay systems.It is well known that multiple-input multiple-output(MIMO) and orthogonal frequency-division multiplexing(OFDM) have emerged as two effective solutions to achievehigh spectral efficiency and throughput for future broadbandwireless systems [20]–[22]. Therefore, some recent works alsobegan to discuss the SWIET for MIMO and OFDM systems,see e.g., [24], [25], [26] and [27]. In [24], a three nodeMIMO broadcasting channel with separate energy harvestingand information decoding receiver was studied, where the rate-energy bound and region were derived for both TS and PSbased schemes. In [25], the transmit beamforming at the multi-antenna base station and the PS strategy at the single-antennausers were jointly optimized for the MISO multi-user system.In [26], the authors optimized the PS for downlink OFDMsystem towards the objective of maximizing the bits/Jouleenergy efficiency, and in [27], the authors investigated theoptimal design of SWIET to maximize the weighted sum-rate for multi-user OFDM systems, where both TS and PSarchitectures were considered. Nevertheless, these works just discussed the SWIET for MIMO systems and OFDM systemsseparately, which did not inherit the benefits of MIMO andOFDM for SWIET in the single system and the MIMO-OFDMchannel was not considered. What’s more, in these works, onlypoint-to-point communication was considered and no relayingwas involved.
C. Motivation
So far, quite a few works have been done for MIMO-OFDM systems, especially for MIMO-OFDM relaying sys-tems. For instance, [28] studied the secure relay beamformingfor SWIET in two-hop non-regenerative relay systems, wherehowever, only the relay was deployed with multiple antennas.The source, the legitimate destination, the power receiver andthe eavesdropper were all assumed with single antenna andnon OFDM channel was considered.To the best of the our knowledge, only two papers (see[29] and [30]) thus far have studied the SWIET for MIMO-OFDM relaying systems. In their work, a two-hop relayingMIMO-OFDM system was considered, where the source andrelay were assumed to be two energy-supplied nodes and thedestination was composed of one information receiver andone energy receiver. The energy receiver could harvest energyfrom the signals transmitted by both the source and the relay,and the information receiver can only collect the informationfrom the signals forwarded by the relay. For such a two-hopMIMO-OFDM relay system, the authors studied its optimumperformance boundaries and measured the rate-energy regions.In this paper, we also focus on the SWIET for two-hopMIMO-OFDM relaying systems. We consider such a scenario,where a source with fixed energy supply desires to transmitits information to a destination. Due to the barrier betweenthe source and the destination or the large distance over theirdirect link, the source cannot directly transmit its signal tothe destination, so it asks a relay to assist its informationtransmission. However, because of the selfish nature (or lackof energy-supply), the relay is not willing to consume itsown energy (or has no available energy) to help the sourceto forward the information. In this case, by employing thefunction of energy harvesting over RF signals, the relayis capable of harvesting energy from the wireless signalstransmitted by the source and then uses its harvested energyto help the information delivering from the source to thedestination. This scenario can be potentially applied in variousenergy-constrained networks. For example, in wireless sensornetworks, where a sink node with energy supply wants to sendits data (e.g., management data or signaling data) to the sensorswhich are too far away from it. So the sink node needs thehelp of some intermediary sensor node. But due to the limitedbattery capacity, the intermediary sensor is not willing to helpthe sink node. For such an application, the SWIET can beemployed to encourage the intermediary sensor to use theharvested energy to help the data transmission. For anotherexample, one source station wants to transmit information toits destination station. There is a hill located between the twostations, so that they cannot communicate with each other. Forsuch an application, a SWIET-aware relay station which has no fixed power supply due to the rugged environment can bedeployed on the top of the hill or in the tunnel of the hill tohelp the information transmission from the source station tothe destination station.
D. Contributions
Compared with previous works, some distinct features ofour work are stressed here. In existing works (e.g., [17]–[19],[23], [24], [29], [30]), both source and relay were assumedwith fixed power supply and the energy harvesting functionwas employed at the destination node and their goal wasto explore the performance region (rate-energy region) ortrade-off for the SWIET, where however, how to efficientlyuse the harvested energy in the same single system wasnot considered. Thus, their considered systems just can bereferred to as the “ harvest-only ” system. Whereas, in ourwork, we investigate the SWIET in a two-hop relaying systemand the energy harvesting function is employed at the relaynode, where all the harvested energy at the relay is used forhelping the information transmission from the source to thedestination. Our investigated system therefore can be referredto as the “ harvest-and-use ” EH system.The contributions of this work are summarized as follows.
Firstly , this is the first work on investigating the “harvest-and-use” SWIET two-hop MIMO-OFDM non-generating relaysystem, where both EH and the consumption of the harvestedenergy are jointly considered in a single system from a system-atic perspective. For this, by adopting the two practical receiverarchitectures presented in [14], we design two protocols, TS-based MIMO-OFDM relaying protocol (TSR) and PS-basedMIMO-OFDM relaying protocol (PSR).
Secondly , to explorethe system performance limits of our proposed TSR andPSR, we mathematically formulate two optimization problemsfor them. The objective is to maximize the E2E achievableinformation rate via joint resource allocation. Specifically, forTSR, the source power allocation, the time switching factor,the subchannel paring between the two hops and the powerassignment of the harvested energy at the relay node arejointly considered and optimized, and for PSR, the sourcepower allocation, the power splitting factors, the subchannelparing over the two hops and the power assignment of theharvested energy at the relay node are jointly consideredand optimized.
Thirdly , since the two optimization problemsare difficult to solve by directly using traditional methods.We adopt a decomposition process to each of them. Bydoing so, for TSR, we derive the closed-form result on theoptimal energy transfer and some closed-form solutions onthe conditional optimal power allocation at the source and therelay. Moreover, for high signal-to-noise ratio (SNR) case,with some approximating operations, we also present theclosed-form solutions on the joint power allocation at thesource and the relay. Based on these results, we design twooptimization schemes for TSR, where one adopts an iterativemanner in terms of the conditional optimal power allocation tofind the final joint optimal power allocation at the source andthe relay, and the other adopts the approximating joint powerallocation. For PSR, we derive the closed-form result on the power splitting factors and design an efficient algorithm tofind the joint optimal power allocation at the source and powersplitting at the relay.
Finally , we provide extensive numericalresults to confirm our theoretical analysis of the proposedPSR and TSR. It is shown that both the optimized PSR andthe optimized TSR can achieve performance gain comparedwith those with only simple resource allocation and systemconfiguration. It is also shown that the relay position greatlyaffects the performance of PSR and TSR protocols in termsof achievable information rate. Specifically, the proposed PSRand TSR show very different reactions to the variation of relayposition. The achievable information rate of the optimizedPSR monotonically decreases when the relay moves fromthe source towards the destination and that of the optimizedTSR firstly decreases and then increases with the incrementof source-relay distance and the relatively worse performanceis obtained when the relay is placed in the middle of thesource and the destination. This is the first time to observesuch a phenomenon. The simulation results also show that theoptimized PSR always outperforms the optimized TSR in thetwo-hop non-regenerative MIMO-OFDM system. Moreover, itis also indicated that increasing either the number of antennasor the number of subcarriers can bring system performancegain to both TSR and PSR.The rest of the paper is organized as follows. Section IIdescribes the system model, where the optimal structure for thetwo-hop non-regenerative MIMO-OFDM system is adopted.Section III presents the proposed TSR and PSR protocols onthe basis of the optimal structure described in Section II andthen formulate an optimization problem for each of them.Section IV and V discuss how to optimize the proposed TSRand PSR, respectively. Section VI presents some numericalresults to discuss the system performance of our optimizedPSR and TSR. Finally, Section VII summarizes this work.
Notations : The lower and upper case bold face letters, e.g., x and X , are used to represent a column vector x and a matrix X , respectively. X H denotes the complex conjugate transposeof X . I and are used to denote an identity matrix and all-zerovector with appropriate dimensions, respectively. k x k denotesthe Euclidean norm of a complex vector x . X ∼ CN ( µ , Σ ) denotes the elements of X following complex Gaussian dis-tribution with mean µ and covariance Σ . E [ X ] denotes thestatistical expectation of matrix X . C x × y denotes the space of x × y matrices with complex entries. For a square matrix X ,tr( X ), | X | , X − , and X denote its trace, determinant, inverse,and square-root, respectively. X (cid:23) means that S is positivedefinite. diag { x , ..., x M } denotes an M × M diagonal matrixwith x , ..., x M being its diagonal elements. Rank( X ) denotesthe rank of matrix X and [ x ] + means max { , x } . x ∗ and x ♯ denote the optimized result and the conditionally optimizedresult of variable x , respectively. To simplify the expressions,we first summarize some commonly used symbols throughoutthe paper in Table I.II. S YSTEM M ODEL
Let us consider a two-hop relay network model, as shown inFigure 1, where source S wants to transmit its information to TABLE I:
Symbol Notations
Notation Representation B : the total system bandwidth; P S : the available transmit power at source; P R : the available transmit power at relay; H q : the channel matrix of the channel with transmit node q ; s i : the source symbol vector at S over subcarrier i with E [ s i s Hi ] = I N S ; F q,i : the processing matrix at node q over subcarrier i ; B i : the processing gain matrix at R associated with subcarrier i ; z q ,i : the received AWGN at node q over subcarrier i ; ˆ x i : the transmit information signal vectorover subcarrier i ; x i : the transmitted signal vector at S forenergy transfer over subcarrier i ; X i : the covariance matrix of x i , i.e., X i = E { x i x Hi } ; X S : the covariance matrix X S = { X , X , · · · , X K } at S for the energy transfer in TSR; p ( i ) S ,n : the transmit power at the source over the i -th subcarrier onthe n -th spatial subchannel; p ( j ) R ,n ′ : the transmit power at the relay over the j -th subcarrier onthe n ′ -th spatial subchannel; ω ℓ : the power allocation factor at S over the ℓ -th subchannel ofthe first hop for both TSR and PSR; ̟ ℓ ′ : the power allocation factor at R over the ℓ ′ -th subchannel ofthe second hop for TSR; θ ℓ,ℓ ′ : the subchannel pairing indicator of the ℓ -th subchannel of thefirst hop and the ℓ ′ -th subchannel of the first hop; α : the time switching factor for TSR; ρ ℓ : the power splitting factor at R over the ℓ -th E2E subchannelfor PSR; S R ... ...... D ... S H R H Energy-harvesting relay
Fig. 1:
System model of the two-hop MIMO-OFDM relaynetwork with an energy-harvesting relay node, where H q = { H q, , H q, , ..., H q,K } and q ∈ { S , R } . destination D . Due to the long distance or the shielding effectcaused by some barrier between S and D , D is not within thecommunication range of S , so that all signals received at D need to be forwarded by the assisting relay R . Such a relayingmodel has been widely adopted to extend the communicationcoverage, which is well-known as the Type-II relaying in IEEEand 3GPP standards [31].To achieve the benefits of MIMO technique, we assume thatall nodes in the system are equipped with multiple antennas,where N S , N R and N D antennas are deployed at the source,the relay and the destination, respectively. Half-duplex modeand non-regenerative relaying are employed at R so that thesignal transmission over the two hops are divided into twophases, i.e., a source phase and a relay phase, are involved incompleting each round of information transmission from S to D , where in the source phase, S transmits its signals to R , andin the relay phase, R amplifies the received signals and thenforwards them to D .With the deployment of OFDM, the total system band-width B (i.e., frequency-selective channel) is divided into K frequency-flat subcarriers (subchannels). Block fading channel model is assumed, so the channel gain over each subcarrier canmaintain constant during each round of two-hop relaying trans-mission. To explore the potential capacity and performancelimit of such a MIMO-OFDM non-regenerative relaying sys-tem, we assume that all nodes have full knowledge of thechannel state information (CSI) and perfect synchronizationover the two hops.The source S is with fixed energy supply while relay R isan energy-constrained/energy-selfish node with EH functiondeployed. That is, R has no energy (or is not willing toconsume its own energy) to help S forward information butit is able to harvest energy from the transmitted signals of S . Thus, R can use the harvested energy to help S transmitthe information to D . Denote P S and P R to be the averageavailable power at S and average harvested power at R ,respectively. So, during each round of two-hop relaying, S first consumes P S power to simultaneously transmit its energyand information to R . Then, R uses up the harvested P R powerto help S transmit the information to D in the same round oftime.In the source phase, S transmits signals to R . Let s i ∈ C N S × be the source symbol vector to be transmitted at source S over subchannel i from S to R . E [ s i s Hi ] = I N S . If we denotethe channel matrix from S to R over subchannel i as H S ,i ,the received signal vector at R over the i -th subcarrier in thesource phase can be expressed as y R ,i = H S ,i F S ,i s i + z R ,i , i ∈ { , , ..., K } (1)where z R ,i ∼ CN (0 , σ R I N R ) represents the received noise at R over subcarrier i . F S ,i ∈ C N S × N S is the precoding matrixat S .In the relay phase, R amplifies the received signals oversubcarrier i of the first hop and then forwards them to D over subcarrier j ( j ∈ { , , ..., K } ) of the second hop. Notethat, without using subcarrier-pairing, the received signal at R over subcarrier i is also forwarded to D over subcarrier i , i.e., i = j , while if subcarrier-paring is employed, j maybe different from i . For clarity, we use ( i, j ) to represent asubcarrier pair composed of the i − th subcarrier over the S − D link and the j − th subcarrier over the R − D link. Let H R ,j and z D ,j ∼ CN (0 , σ D I N D ) be the channel matrix from R to D andthe additive noise received at D over subcarrier j , respectively.Then, the received signal at D over the ( i, j ) subcarrier paircan be given by y D ,j = F D ,j ( H R ,j F R ,i y R ,i + z D ,j ) (2) = F D ,j ( H R ,j F R ,i ( H S ,i F S ,i s i + z R ,i )) + F D ,j z D ,j = F D ,j ( H R ,j F R ,i H S ,i F S ,i s i + H R ,j F R ,i z R ,i ) + F D ,j z D ,j , where F R ,i ∈ C N R × N R and F D ,j ∈ C N D × N D denote theforwarding matrix at R and the processing matrix at D ,respectively.Since each node knows the full CSI, the singular valuedecomposition (SVD) of all channel matrices is available forthe system to determine the transmit-and-receive processingmatrices at all nodes. The SVD of the channel matrices over the two hops can be expressed by ( H S ,i = U S ,i Λ S ,i V H S ,i , H R ,j = U R ,j Λ R ,j V H R ,j , (3)where Λ S ,i ∈ C N R × N S and Λ R ,j ∈ C N D × N R are twodiagonal matrices with non-negative real numbers on thediagonal. Λ S ,i = diag nq λ ( i ) S , , q λ ( i ) S , , · · · , q λ ( i ) S , Rank ( H S ,i ) o with q λ ( i ) S , ≥ q λ ( i ) S , ≥ · · · ≥ q λ ( i ) S , Rank ( H S ,i ) and Λ R ,j = diag nq λ ( j ) R , , q λ ( j ) R , , · · · , q λ ( j ) R , Rank ( H R ,j ) o with q λ ( j ) R , ≥ q λ ( j ) R , ≥ · · · ≥ q λ ( j ) R , Rank ( H R ,j ) . U S ,i ∈ C N R × N R , V S ,i ∈ C N S × N S , U R ,j ∈ C N D × N D and V R ,j ∈ C N R × N R are fourcomplex unitary matrices, so they do not change the statisticsof the channel. This implies that by using SVD, the mutualinformation of the corresponding channels can be preserved[32].Moreover, it was proved that by performing the SVD on H S ,i and H R ,i , the achievable rate of the two-hop non-regenerative MIMO channel can be maximized when theoverall two-hop channel is decomposed into a number ofparallel uncorrelated paths [33]. We therefore adopt sucha formulation to design the SWIET for the two-hop non-regenerative MIMO-OFDM relay channel as follows.Substituting (3) into (2), it can be obtained that y D ,j (4) = F D ,j H R ,j F R ,i H S ,i F S ,i s i + F D ,j H R ,j F R ,i z R ,i + F D ,j z D ,j = F D ,j U R ,j Λ R ,j V H R ,j F R ,i U S ,i Λ S ,i V H S ,i F S ,i s i + F D ,j U R ,j Λ R ,j V H R ,j F R ,i z R ,i + F D ,j z D ,j . Assume that the allocated power at S over subcarrier i is P S ,i satisfying that P Ki =1 P S ,i ≤ P S . Then, we have that F S ,i = p P S ,i V S ,i w ( i ) s , (5)where w ( i ) s , diag nq w ( i ) s, , q w ( i ) s, , ..., q w ( i ) s,N S o is a diagonalmatrix composed of weighting coefficients of the antennasat S over the i − th subcarrier. Then the transmit informationsignal vector from the source over subcarrier i thus can begiven by ˆ x i = p P S ,i V S ,i w ( i ) s s i . F D ,j and F R ,i are chosenas F D ,j = U H R ,j and F R ,i = V R ,i B i U H S ,i , in order to obtainthe parallel single-input single-output (SISO) paths. Such adesign of F R ,i implies a linear processing, which was alsoadopted in some existing works for two-hop amplified-and-forward MIMO systems (e.g., [33]). For simplicity, the parallelSISO paths are referred to as end-to-end (E2E) subchannels inthe sequel. Note that, for a given subcarrier pair ( i, j ), F R ,i and F R ,j actually represent the same processing matrix at R , i.e., F R ,i = F R ,j , resulting in V R ,i = V R ,j . Therefore, substituting(5) and F R ,i = V R ,j B i U H S ,i into (4) yields y D ,j = Λ R ,j B i Λ S ,i ˆ x i + Λ R ,j B i U H S ,i z R ,i + U H R ,j z D ,j . (6)where B i can be regarded as a processing gain matrix at R .With above-mentioned operations, one can see that boththe i -th subcarrier of the first hop and the j -th subcarrierof the second hop are divided into some orthogonal spatial subchannels, and with the one-to-one concatenation betweenthe spatial subchannels over the two hops, a set of orthogonalE2E subchannels are obtained. As a result, it can be deducedthat the number of the E2E subchannels must be bounded tothe minimum number of spatial subchannels of each singlehop. This means that in a two-hop non-regenerative MIMO-OFDM relaying, for each subcarrier, only a subset of spatialsubchannels either in the first or the second hop is used. Let N be the dimension of the subset. It can be inferred that N = min { Rank ( H S ,i ) , Rank ( H R ,j ) } = min { N S , N R , N D } . (7)Let ˆ U q,i , ˆ Λ q,i and ˆ V q,i be N × N matrices, which arecomposed of the first N columns and N rows of U q,i , Λ q,i and U q,i , respectively, where q ∈ { S , R } . Then, (6) is equivalentlytransformed to be y D ,j = ˆ Λ R ,j B i ˆ Λ S ,i ˆ x i + ˆ Λ R ,j B i ˆ U H S ,i z R ,i + ˆ U H R ,j z D ,j . (8)Thus, the mutual information over the ( i, j ) subcarrier pair forthe non-regenerative MIMO-OFDM system can be given by I ( ˆ x i , ˆ y j ) = B K log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I N + ˆ Λ R ,j B i ˆ Λ S ,i ∆ i ˆ Λ H S ,i B Hi ˆ Λ H R ,j σ R ˆ Λ R ,j B i B Hi ˆ Λ H R ,j + σ D I (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (9)where B is the total bandwidth of the OFDM system and is used to describe the time division feature of the two-hopretransmission. ∆ i = E { ˆ x i ˆ x Hi } .From (9), it can be seen that each subcarrier pair ( i, j ) isdivided into N effective E2E subchannels. As there are K subcarriers in the MIMO-OFDM system, the total KN E2Esubchannels can be obtained by using the SVD decompositionover the two hops. In order to maximize I ( ˆ x i , ˆ y j ) , the signaltransmitted over subcarrier i on the n -th spatial subchannel ofthe first hop is allowed to be forwarded by R over the j -thsubcarrier on the spatial subchannel n ′ of the second hop, thisis referred to as subchannel pairing in the sequel. Accordingto [33] and [32], B i satisfies that B i = diag (cid:2) β i, , β i, , · · · , β i,N (cid:3) , (10)with β i,n = s p ( j ) R ,n ′ p ( i ) S ,n λ ( i ) S ,n + σ R , (11)where p ( i ) S ,n , p ( j ) R ,n ′ denote the transmit power of the source overthe i -th subcarrier on the n -th spatial subchannel, the transmitpower of the relay over the j -th subcarrier on the n ′ -th spatialsubchannel, respectively.For notation simplification, we define ℓ , ( i − K + n and ℓ ′ , ( j − K + n ′ , where ≤ n, n ′ ≤ N and ≤ i, j ≤ K . Thus, it can be seen that that ≤ ℓ ≤ KN and ≤ ℓ ′ ≤ KN . Consequently, the subchannel pairing mentionedpreviously can be redescribed as the ℓ -th subchannel of thefirst hop is paired with the ℓ ′ -th subchannel of the secondhop. Therefore, (11) is rewritten as β i,n = β ℓ = r p R ,ℓ ′ p S ,ℓ λ S ,ℓ + σ R , (12) Energy Harvesting at Rover MIMO channel H S,R on each subcarrier Information receiving at R over MIMO channel H S,R on each subcarrier R forwarding informationover MIMO channel H R,D on each subcarrier (1- ) / ) / Energy Harvesting at R over the i- thsubchannel with power splitting factor i Information receiving at R over the i- th subchannel with power splitting factor 1- i R forwarding information with the harvested energy
Fig. 2:
The framework of our proposed two protocols: (a) TSR and(b) PSR. where p S ,ℓ , p ( i ) S ,n , p R ,ℓ , p ( j ) R ,n ′ and λ S ,ℓ , λ ( i ) S ,n . Clearly, P KNℓ =1 p S ,ℓ ≤ P S and P KNℓ ′ =1 p R ,ℓ ′ ≤ P R . By adopting thestructure described above, which was presented to maximizethe instantaneous E2E capacity in [33], the achievable instan-taneous information rate over the subchannel pair ( ℓ, ℓ ′ ) canbe given by R ℓ,ℓ ′ = B K log (cid:18) p S ,ℓ p R ,ℓ ′ λ S ,ℓ σ R λ R ,ℓ ′ σ D p S ,ℓ λ S ,ℓ σ R + p R ,ℓ ′ λ R ,ℓ ′ σ D (cid:19) . (13)III. P ROTOCOLS AND O PTIMIZATION P ROBLEM F ORMULATION
Based on the structure described in Section II, in thissection we shall present two protocols for the two-hop non-regenerative MIMO-OFDM system by considering the twopractical receiver architectures proposed in [14]. Moreover, toexplore the system performance limit, we also formulate twooptimization problems for the two protocols in this Section.
A. Protocol Description and Optimization Problem Formula-tion for TSR1) TSR Protocol:
Firstly, we consider the TS receiverarchitecture at R and then propose a time switching-basedEH non-regenerative MIMO-OFDM Relaying (TSR) protocolas follows.The framework of TSR is illustrated in Figure 2(a), in whicheach time period T is divided into three phases. The first phaseis assigned with a duration of αT , which is used for energytransfer from S to R . α ∈ [0 , denotes the time switchingfactor . The second and the third phases are assigned with equaltime duration of − α T , where the second phase is used forthe information transmission from S to R and the third one isused for R to forward the received information to D .In the first phase of TSR, only energy is transferred. As theEH receiver does not need to convert the received signal fromthe RF band to harvest the carried energy, in order to seeka better system performance, energy transfer can be differentfrom information transmission. Denote the transmitted signalvector at S for energy transfer over subcarrier i to be x i . x i may be different from the information symbol vector ˆ x i defined in Section II. Thus, the received signal at R for energyharvesting can be given by y (EH)R ,i = H S ,i x i + z R ,i , (14)Similar to some existing works (see e.g., [27]), we alsoassumed that the total harvested RF-band energy at R over thesubcarrier i is proportional to that of received baseband signal.Without loss of generality, we normalize the time period T to1 hereafter in this paper. Then the total harvested RF-bandenergy at R over the subcarrier i can be expressed by E R ,i = αη k H S ,i x i k , (15)where η is a constant, which is used to describe the convertingefficiency of the energy transducer in converting the harvestedenergy to electrical energy to be stored. In this paper, weassume η = 1 . Note that such an assumption is also widelyadopted in the exploration of system performance limit for theconvenience of analysis, see e.g., [11].Define X i = E { x i x Hi } as the covariance matrix of x i . Thetotal transmit power over all K subcarriers can be given by P Ki =1 E {k x i k } = P Ki =1 tr ( X i ) , which is constrained bythe available power at S , i.e., X Ki =1 tr ( X i ) ≤ P S , (16)where tr ( X i ) actually can be regarded the transmit powerallocated over subcarrier i for energy transfer.In the second phase of TSR, S transmit the signals overall ℓ ∈ { , , ..., KN } subchannels under the average powerconstraint P S and in the third phase of TSR, R forward thereceived signals over all ℓ ′ ∈ { , , ..., KN } subchannelsunder the available power constraint P R with the optimalstructure described in Section II. Since all energy harvestedin the first phase is used for the information relaying, it canbe deduced that the available transmit power at R for theinformation forwarding is P R = P Ki =1 E R ,i (1 − α )2 = 2 α − α X Ki =1 k H S ,i x i k . (17)Define p S ,ℓ , P S ω ℓ and p R ,ℓ ′ , P R ̟ ℓ ′ , where ≤ ω ℓ ≤ is used to represent the power allocation factor at S forsubcarrier i on the spatial subchannel n (i.e., subchannel ℓ of the first hop) satisfying that P KNℓ =1 ω ℓ ≤ , and ̟ ℓ ′ ∈ [0 , is used to represent the power allocating factor at R over subcarrier j on spatial subchannel n ′ (i.e., subchannel ℓ ′ of the second hop) satisfying that P KNℓ ′ =1 ̟ ℓ ′ ≤ . Note that,since P S ω ℓ = P S ,i w ( i ) s ,n and P S ,i = P S P ( i − K + Nℓ =( i − K +1 ω ℓ , it canbe inferred that w ( i ) s ,n = ω ℓ / P ( i − K + Nb =( i − K +1 ω b , which meansthat if we determine ω ℓ for ℓ ∈ { , , ..., KN } , w ( i ) s ,n can alsobe determined for all n ∈ { , , ..., N } and i ∈ { , , ..., K } .Therefore, we shall discuss how to optimize ω ℓ instead of w ( i ) s ,n in the sequel. Therefore, R ℓ,ℓ ′ in (13) can be rewritten to be R ( TSR ) ℓ,ℓ ′ = B K log (cid:18) P S ω ℓ P R ̟ ℓ ′ λ S ,ℓ σ R λ R ,ℓ ′ σ D P S ω ℓ λ S ,ℓ σ R + P R ̟ ℓ ′ λ R ,ℓ ′ σ D (cid:19) (18)for TSR. Based on (18), the total E2E instantaneous achievableinformation rate of the TSR is given by C TSR = 1 − α X KNℓ =1 X KNℓ ′ =1 θ ℓ,ℓ ′ R ( TSR ) ℓ,ℓ ′ , (19)where θ ℓ,ℓ ′ ∈ { , } is the indicator of the subchannel-pairing.Specifically, θ ℓ,ℓ ′ = 1 means that the ℓ -th subchannel of thefirst hop is paired with the ℓ ′ -th subchannel of the second hop.Otherwise, θ ℓ,ℓ ′ = 0 .
2) Optimization Problem Formulation for TSR:
From thedescription in Section III-A1, one can see that for the TSR,only the available transmit power at the source P S and allchannel matrices are known, while the rest parameters suchas the covariance matrix X S = { X , X , · · · , X K } at S forthe energy transfer, the time switching factor α , the powerallocating matrix ω = { ω ℓ } KN × at S , the power allocatingvector ̟ = { ̟ ℓ ′ } KN × at R and the subchannel pairingmatrix θ = { θ ℓ,ℓ ′ } KN × KN are all required to be determinedand configured. Since all these parameters may affect thesystem performance, it is necessary to jointly design them forachieving the optimal performance. In this subsection, we for-mulate an optimization problem for it and we shall investigatehow to solve the optimization problem in Section IV. Our goalis to find the optimal ω ∗ , ̟ ∗ , θ ∗ and α ∗ to maximize the end-to-end achievable information rate of TSR. The correspondingoptimization problem can be mathematically expressed as max X S , ω , ̟ , θ ,α C TSR (20)s.t. X KNℓ =1 ω ℓ ≤ , X KNℓ ′ =1 ̟ ℓ ′ ≤ X KNℓ =1 θ ℓ,ℓ ′ = 1 , ∀ ℓ ′ ; X KNℓ ′ =1 θ ℓ,ℓ ′ = 1 , ∀ ℓ X Ki =1 tr ( X i ) ≤ P S , X i (cid:23) , ≤ α ≤ , θ ℓ,ℓ ′ ∈ { , } where the constraints P KNℓ =1 ω ℓ ≤ and P KNℓ ′ =1 ̟ ℓ ≤ implythat the available transmit power at the source and the relayare limited by P S and P R in (17), respectively. The constraints P KNℓ =1 θ ℓ,ℓ ′ = 1 , ∀ ℓ ′ and P KNℓ ′ =1 θ ℓ,ℓ ′ = 1 , ∀ ℓ mean that eachsubchannel of the first hop is only allowed to be pairedone subchannel of the second hop, vise verse. The constraint P Ki =1 tr ( X i ) ≤ P S indicates that the energy transfer in thefirst phase of TSR is also constrained by the available powerat the source. B. Protocol Description and Optimization Problem Formula-tion for PSR1) PSR Protocol:
In this subsection, we shall presentanother relaying protocol, power splitting-based EH non-regenerative MIMO-OFDM Relaying (PSR) by consideringthe PS receiver architecture presented in [14].The framework of PSR is illustrated in Figure 2(b), in whichthe total time period T is equally divided into two parts, wherein the first T , energy and information are simultaneouslytransferred from S to R and in the rest T , R uses theharvested energy and forwards the received information to D .Specifically, the receiver at R can be explained as follows. (cid:1)(cid:2) (cid:1) (cid:1) (cid:1) (cid:2) (cid:3) (cid:1) (cid:3) (cid:1)(cid:2) (cid:1) (cid:4) (cid:1)(cid:2) (cid:2)(cid:1) (cid:5) (cid:1) (cid:6) (cid:1)(cid:2) (cid:1) (cid:7) (cid:4)(cid:2) (cid:3) (cid:4) (cid:2) (cid:4) (cid:3) (cid:1) (cid:4)(cid:2) (cid:1) (cid:1) (cid:5)(cid:5)(cid:5) (cid:6)(cid:7)(cid:8) (cid:1) (cid:8) (cid:6)(cid:7)(cid:8) (cid:9) (cid:10) (cid:1) − (cid:1) (cid:8) (cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:15)(cid:17)(cid:18)(cid:15)(cid:14)(cid:19)(cid:20)(cid:21)(cid:22)(cid:13)(cid:14)(cid:15)(cid:16)(cid:15)(cid:17)(cid:18)(cid:15)(cid:14) (cid:1) (cid:4) (cid:23) Fig. 3:
The structure of our proposed PSR.
The received signal over the ℓ -th subchannel at R is firstlycorrupted by a Gaussian noise at the RF-band, which isassumed to have zero mean and equivalent baseband power.The RF-band signal is then fed into a power splitter, whichis assumed to be perfect without any noise induced. Afterthe power splitter, a portion of signal power is allocated tothe EH receiver. Suppose ρ ℓ be the power splitting factorfor the EH receiver. Then the rest − ρ ℓ part is input intothe information receiver. The signal split to the informationreceiver then goes through a sequence of non-regenerativerelaying with the system structure described in Section II.For clarity, the structure of our proposed PSR is illustrated inFigure 3, where at R the received signals are firstly processedby the matrix U H S ,i and then split into two flows with a powersplitting factor matrix ρ . After this, the ( I − ρ ) part is inputinto the “info receiver” and the rest ρ part is input into the“EH receiver”. The harvested energy by the EH receiver isthen allocated to the information flow via the amplifying gainmatrix B i . The detailed process is described as follows.Let ρ i,n be the portion of the power split to the EH receiverover the n -th spatial subchannel of the i -th subcarrier at R , n ∈ { , , ..., N } . Thus, the total harvested RF-band energyat R over the subcarrier i is proportional to that of receivedbaseband signal, which can be expressed by E R ,i = η k ρ i U H S ,i H S ,i F S ,i s i k , (21)where ρ i = diag { ρ i, , ρ i, , ..., ρ i,N } . With the assumption of η = 1 , we have that E R ,i = k p P S ,i ρ i U H S ,i H S ,i V S ,i w ( i ) s s i k , = P S ,i tr ( ρ i U H S ,i U S ,i Λ S ,i V H S ,i V S ,i w ( i ) s s i (22) × s Hi w ( i ) s V H S ,i V S ,i Λ S ,i U H S ,i U S ,i ρ ) , = P S ,i k ρ i Λ S ,i w ( i ) s k (23)Further, it is rewritten to be E R ,i = X Nn =1 ρ i,n λ ( i ) S ,n P S ,i w ( i ) s ,n , (24)where λ ( i ) S ,n P S ,i w ( i ) s ,n can be treated as the harvested energy onthe n -th spatial subchannel over the i -th subcarrier, i.e., theharvested energy over the ℓ -th subchannel of the first hop.Therefore, the total energy harvested at R can be given by E R = X Ki =1 E R ,i = X Ki =1 X Nn =1 ρ i,n λ ( i ) S ,n P S ,i w ( i ) s ,n , = X KNℓ =1 ρ ℓ λ S ,ℓ P S ω ℓ , (25)where ℓ = ( i − K + n and ω ℓ ∈ [0 , denote the powerallocating factor at S over subchannel ℓ .In the meantime, the rest (1 − ρ i,n ) power is split to theinformation receiver of R at the n -th spatial subchannel over the i -th subcarrier. Thus, the signal collected at the “inforeceiver” is y (IF) R ,i = ( I − ρ ) ˆ Λ S ,i ˆ x i + ˆ U H S ,i z R ,i . (26)As a result, the received signal at D can be given by y D ,j = ( I − ρ ) ˆ Λ R ,j B i ˆ Λ S ,i ˆ x i + ˆ Λ R ,j B i ˆ U H S ,i z R ,i + ˆ U H R ,j z D ,j . In this paper, we assume that the energy harvested on the ℓ -th subchannel over the S − R link is only used for theinformation transmission on its paired subchannel, i.e., the ℓ ′ -th subchannel of the R − D link. Therefore, the availabletransmit power for the ℓ ′ -th subchannel at R is p R ,ℓ ′ = ρ ℓ λ S ,ℓ P S ω ℓ . Since p S ,ℓ , (1 − ρ ℓ ) P S ω ℓ , R ℓ,ℓ ′ in (13) thencan be reexpressed as R ( PSR ) ℓ,ℓ ′ (27) = B K log (cid:18) − ρ ℓ ) ρ ℓ P S ω ℓ λ S ,ℓ λ S ,ℓ σ R λ R ,ℓ ′ σ D − ρ ℓ ) P S ω ℓ λ S ,ℓ σ R + ρ ℓ λ S ,ℓ P S ω ℓ λ R ,ℓ ′ σ D (cid:19) for PSR. Consequently, the instantaneous achievable informa-tion rate of PSR can be given by C PSR = KN X ℓ =1 KN X ℓ ′ =1 θ ℓ,ℓ ′ R (PSR) ℓ,ℓ ′ , (28)where θ ℓ,ℓ ′ ∈ { , } is the indicator for the subchannel-pairing,which has the same definition with that below (19).
2) Optimization Problem Formulation for PSR:
To opti-mally design the proposed PSR, in this subsection, we for-mulate an optimization problem to jointly optimize the powersplitting factor vector ρ = { ρ ℓ } KN × , the power allocatingmatrix ω = { ω ℓ } KN × at S and the subchannel pairing θ = { θ ℓ,ℓ ′ } KN × KN over the two hops. The objective is alsoto find the optimal ω ∗ , θ ∗ and ρ ∗ to maximize the E2Eachievable information rate. Thus, the optimization problemcan be expressed as max ω , θ , ρ C PSR (29)s.t. X KNℓ =1 ω ℓ ≤ , θ ℓ,ℓ ′ ∈ { , } X KNℓ =1 θ ℓ,ℓ ′ = 1 , ∀ ℓ ′ ; X KNℓ ′ =1 θ ℓ,ℓ ′ = 1 , ∀ ℓ ≤ ρ ℓ ≤ , ∀ ℓ ∈ { , , ..., KN } , where the constraint P KNℓ =1 ω ℓ ≤ implies that the availabletransmit power at the source is constrained by P S . Theconstraints P KNℓ =1 θ ℓ,ℓ ′ = 1 , ∀ ℓ ′ and P KNℓ ′ =1 θ ℓ,ℓ ′ = 1 , ∀ ℓ meanthat each subchannel of the first hop is allowed to be pairedwith only one subchannel of the second hop, vise verse. Weshall discuss how to solve Problem (29) in Section V.IV. O PTIMAL D ESIGN OF
TSRIn this section, we shall investigate how to solve theoptimization problem (20) for TSR. It can be observed thatProblem (20) is a combinatorial optimization problem withdiscrete variables θ ℓ,ℓ ′ ∈ { , } . Even if we remove thediscrete variables θ ℓ,ℓ ′ , it is still a non-convex optimization problem, which cannot be solved by using conventional meth-ods. Thus, we solve it as follows.Firstly, it can be seen that, in the first phase of TSR, onlyenergy is transferred and accordingly only X S is requiredto be optimized, which means that X S is independent withother variables. Thus, it can be independently designed at firstwithout loss of the global optimality. Secondly, we found thatthe separation principle designed in [35] for joint channelpairing and power allocation optimization still holds in oursystem and θ is also independent with α (We shall prove thisin Section IV-B). Therefore, θ can also be optimized separatelywithout jointly considering other variables. Based on these, wepresent a solution to Problem (20) as shown in Algorithm 1.Then, we shall describe the detailed processing associated witheach step of Algorithm 1 in the successive subsections. Algorithm 1
Optimization Framework for TSR Calculate the optimal X ∗ S ; Calculate the optimal θ ∗ ; With the obtained X ∗ S and θ ∗ , find the optimal { ω ∗ , ̟ ∗ } and α ∗ to maximize C TSR . A. Optimal X ∗ S for TSR From (17), it can be seen that for a given α , the larger P Ki =1 E R ,i , the higher P R , which means more available powerfor R to assist the information transmission from S to D . Thismotivates that the objective of the optimal design of X S is tomaximize P Ki =1 E R ,i . For a given α , the optimization problemcan be expressed by max X S k H S ,i x i k (30) X Ki =1 tr ( X i ) ≤ P S X i (cid:23) As mentioned in Section II, by using SVD, H S ,i = U S ,i Λ S ,i V H S ,i . Let v ( i ) s, be the first column of V S ,i . With a giventr ( X i ) , we can obtain the following Lemma 1 for solvingProblem (30). Lemma 1.
In TSR, for a given tr ( X i ) , the optimal X ♯i = tr ( X i ) v ( i ) s, v ( i ) Hs, . Proof:
The proof of Lemma 1 can be found in AppendixA of this paper.Based on Lemma 1, we can further derive the followingLemma 2.
Lemma 2.
In TSR, to achieve the maximum energy transfer,all power at S should be allocated to the subcarrier with themaximum k ˜ h ( i ) S , k for all i ∈ { , , ..., K } . Proof:
The proof of Lemma 1 can be found in AppendixB of this paper.With Lemma 1 and Lemma 2, we can easily arrive at thefollowing Theorem 1.
Theorem 1.
The optimal solution of Problem (30) is X ∗ S = { X ∗ i } , where X ∗ i = P S v ( i ) s, v ( i ) Hs, , i = arg max b =1 ,...,K k ˜ h ( b ) S , k , , otherwise , (31) and for a given α , the optimal P R is P ♯ R = 2 α − α P S k ˜ h ( c ) S , k , (32) where c = arg max b =1 ,...,K k ˜ h ( b ) S , k and i = 1 , ..., K . Proof:
By combining Lemma 1 and (50), Theorem 1 canbe easily proved.Theorem 1 indicates that the optimal power allocation forthe energy transfer in TSR should be performed by projectingall the transmit energy contained in the space of the eigen-vector with the largest eigenvalue of the channel matrix. Notethat although the results in Lemma 1, Lemma 2 and Theorem1 are derived for TSR in MIMO-OFDM relaying system,they just involve the energy transfer over the first hop, sothese results also hold for multi-channel single-hop systemsincluding MIMO and OFDM systems. Some similar resultscan also be seen in [24].
B. Optimal θ ∗ for TSR As described in (13), the signal transmitted over the ℓ -thsubchannel of the first hop is allowed to be forwarded by R over the ℓ ′ -th subcarrier of the second hop, which inspiressubchannel pairing between the two hops. Substituting X ∗ S into Problem (20), for a given α , the optimization problem isrewritten to be max ω , ̟ , θ B K − α KN X ℓ =1 θ ℓ,ℓ ′ log (cid:18) P S ω ℓ P ♯ R ̟ ℓ λ S ,ℓσ R λ R ,ℓ ′ σ D P S ω ℓ λ S ,ℓσ R + P ♯ R ̟ ℓ λ R ,ℓ ′ σ D (cid:19) s.t. X KNℓ =1 ω ℓ ≤ , X KNℓ ′ =1 ̟ ℓ ≤ X KNℓ =1 θ ℓ,ℓ ′ = 1 , ∀ ℓ ′ , X KNℓ ′ =1 θ ℓ,ℓ ′ = 1 , ∀ ℓ (33)Problem (33) can be regarded as a joint power allocationand subchannel pairing (JPASP) problem [34], [35]. In [35],it was proved that the JPASP problem can be solved in aseparated manner without losing the global optimality, wherethe channel pairing can be determined with sorted channelgain. That is, the subchannel with the i -th largest channel gainover the first hop should be paired with the subchannel withthe i -th largest channel gain over the second hop. Therefore,by optimally allocating the power over the paired subchannels,the obtained result is the same with that obtained by jointlyoptimized power allocation and subchannel pairing. Based onthis, we can derive the following Lemma 3. Lemma 3.
In TSR, the optimal θ ∗ satisfies that θ ∗ ℓ,ℓ ′ = ( , if Order ( λ S ,ℓ ) = Order ( λ R ,ℓ ′ ) , , otherwise . (34) where Order ( λ u,ℓ ) represents the rank position of λ u,ℓ among λ u,i for all i = 1 , ..., KN with a descending sorting order atnode u ∈ { S , R } . Proof:
With the decoupling policy presented in [35], wecan easily prove that the optimal θ for Problem (33) is (34).Moreover, it can be inferred that the change of α only affectsthe available power P ♯ R and the time duration for informationtransmission, which does not affect the channel gain of allsubchannels, so the result in (34) is also the global optimalsolution for the original Problem (20). C. Joint optimal ω ∗ , ̟ ∗ and α ∗ With the optimal subchannel pairing θ ∗ obtained in Lemma3 and the optimal design of X ∗ S described in Theorem 1, C TSR can be considered as the sum of rate over KN independentE2E paths. However, it is still neither joint convex nor jointconcave w.r.t. ω , ̟ and α . Therefore, we adopt the followingmethod to solve it. Firstly, for a given α , we find the optimal ω ∗ , ̟ ∗ . Then, we consider ω ∗ and ̟ ∗ as two functions of α , ω ∗ ( α ) and ̟ ∗ ( α ) . By substituting ω ∗ ( α ) and ̟ ∗ ( α ) for ω and ̟ , respectively, we then calculate the optimal α ∗ . Thedetailed operations are described in the following subsections.
1) Optimal { ω ∗ , ̟ ∗ } for a given α : For a given α , theoptimization problem (20) can be rewritten to be max ω , ̟ − α X KNℓ =1 R ℓ s.t. X KNℓ =1 ω ℓ ≤ , X KNℓ =1 ̟ ℓ ≤ , (35)where R ℓ = B K log (cid:18) P S ω ℓ P ♯ R ̟ ℓ λ S ,ℓ σ R λ R ,ℓ σ D P S ω ℓ λ S ,ℓ σ R + P ♯ R ̟ ℓ λ R ,ℓ σ D (cid:19) .It can be proved that the objective function of Problem(35) is not jointly concave in ω ℓ and ̟ ℓ . Therefore, it isnot possible to obtain the global optimal solution analytically.However, we find that if ω ℓ is fixed, the objective function ofProblem (35) is concave in ̟ ℓ , vice versa. Therefore, Karush-Kuhn-Tucker (KKT) [36] can be applied to derive the optimalsolutions. As a result, we arrive at Lemma 4. Lemma 4.
In TSR, for a given α and ̟ ℓ ( ℓ = { , ..., KN } ) , the optimal ω ℓ is ω ♯ℓ = σ R P S · λ S ,ℓ " P ♯ R ̟ ℓ λ R ,ℓ σ D s λ S ,ℓ σ D σ R λ R ,ℓ P ♯ R ̟ ℓ µ − ! − + , (36) while for given ̟ ℓ ( ω = { , ..., KN } ) , the optimal ̟ ℓ is ̟ ♯ℓ = σ D P ♯ R · λ R ,ℓ " P S ω ℓ λ S ,ℓ σ R s λ R ,ℓ σ R σ D λ S ,ℓ P S ω ℓ ν − ! − + , (37) where [ x ] + = max { , x } . µ and ν are defined as non-negative Lagrange multipliers, which have to be chosen suchthat P KNℓ =1 ω ♯ℓ ≤ , and P KNℓ =1 ̟ ♯ℓ ≤ , respectively. According to Lemma 4, we design such an iterative method,as shown in Algorithm 2, to get the near optimal solution forProblem (35) with a given bias error ǫ .Since C cur in Algorithm 2 is concave w.r.t in ω ℓ and ̟ ℓ ,each round of iteration Algorithm 2 can improve C cur . As P KNℓ =1 ω ℓ ≤ and P KNℓ =1 ̟ ℓ ≤ , C cur cannot be increasedwithout limit. This implies the convergence of Algorithm 2. Algorithm 2
Finding the local or global optimal { ω ∗ , ̟ ∗ } for each ℓ ∈ [1 , KN ] do Initialize ω ℓ = KN ; Calculate ̟ ℓ in terms of (37); end for Initialize C pre = 0 ; Calculate C cur = − α P KNℓ =1 R ℓ ; while | C cur − C pre | > ǫ do Update C pre = C cur ; for each ℓ ∈ [1 , KN ] do Update ω ℓ in terms of (40); Update ̟ ℓ in terms of (37); Update C cur = − α P KNℓ =1 R ℓ ; end for end while Return { ω , ̟ } .Moreover, it also can be observed that Algorithm 2 dependson the initialization of ω ℓ , although we adopt the equalweight of ω ℓ for it, Algorithm 2 cannot always guaranteethe global optimality. Therefore, we shall also investigate anasymptotically global optimal solution of { ω ∗ , ̟ } ∗ as followsfor high SNR case.At high SNR region, we can approximate R ℓ as R ℓ = B K log (cid:18) P S ω ℓ P ♯ R ̟ ℓ λ S ,ℓ σ R λ R ,ℓ σ D P S ω ℓ λ S ,ℓ σ R + P ♯ R ̟ ℓ λ R ,ℓ σ D (cid:19) ≃ B K log (cid:18) P S ω ℓ P ♯ R ̟ ℓ λ S ,ℓ σ R λ R ,ℓ σ D P S ω ℓ λ S ,ℓ σ R + P ♯ R ̟ ℓ λ R ,ℓ σ D (cid:19) (38)Such an approximation leads to the Hessian Matrix (cid:2) ∂ R ℓ ∂ω ℓ , ∂ R ℓ ∂̟ ℓ (cid:3) T (cid:22) , which indicates a jointly concave R ℓ in ω ℓ and ̟ ℓ . In this case, we can give the optimal solution asdescribed in Lemma 5 by using KKT conditions. Lemma 5.
In TSR, at high SNR regime, for a given α , theoptimal ω ∗ ℓ and ω ∗ ℓ can be approximated by ω ⋆ℓ = 1 P S (cid:16) r λ S ,ℓ σ D σ R λ R ,ℓ (cid:17) " µ − (cid:0) r λ S ,ℓ σ R + r λ R ,ℓ σ D (cid:1) λ S ,ℓ σ R λ R ,ℓ σ D + (39) and ̟ ⋆ℓ = 1 P ♯ R (cid:16) r λ R ,ℓ σ R σ D λ S ,ℓ (cid:17) " ν − (cid:0)q λ S ,ℓ σ R + q λ R ,ℓ σ D (cid:1) λ S ,ℓ σ R λ R ,ℓ σ D + , (40) respectively, where µ and ν are two positive Lagrangianparameters, which have to be chosen such that P KNℓ =1 ω ⋆ℓ ≤ , and P KNℓ =1 ̟ ⋆ℓ ≤ .2) Optimal α ∗ : From Lemma 4 and Lemma 5, we can seethat each of ω ∗ , ̟ ∗ , ω ⋆ and ̟ ⋆ has close relationship with α . Thus, each of them can be regarded as a function w.r.t α . Substituting { ω ∗ , ̟ ∗ } into (35), we have that C TSR ( α ) = B K − α KN X ℓ =1 log (cid:18) P S ω ∗ ℓ ( α ) P ♯ R ( α ) ̟ ∗ ℓ ( α ) λ S ,ℓσ R λ R ,ℓ σ D P S ω ∗ ℓ ( α ) λ S ,ℓσ R + P ♯ R ( α ) ̟ ∗ ℓ ( α ) λ R ,ℓσ D (cid:19) . (41)Let G ( α ) = KN X ℓ =1 log (cid:18) P S ω ∗ ℓ ( α ) P ♯ R ( α ) ̟ ∗ ℓ ( α ) λ S ,ℓσ R λ R ,ℓσ D P S ω ∗ ℓ ( α ) λ S ,ℓσ R + P ♯ R ( α ) ̟ ∗ ℓ ( α ) λ R ,ℓσ D (cid:19) G ′ ( α, α ′ ) = KN X ℓ =1 log (cid:18) P S ω ∗ ℓ ( α ) P ♯ R ( α ′ ) ̟ ∗ ℓ ( α ) λ S ,ℓσ R λ R ,ℓσ D P S ω ∗ ℓ ( α ) λ S ,ℓσ R + P ♯ R ( α ′ ) ̟ ∗ ℓ ( α ) λ R ,ℓσ D (cid:19) . (42)We can derive the following Lemma 6. Lemma 6. G ( α ) is a monotonically increasing function w.r.tvariable α ∈ [0 , . Proof:
Suppose the two variables α and α ′ , ≤ α < α ′ ≤ , then we have G ( α ) < G ( α ′ ) . As α < α ′ , we can deducethat P ♯ R ( α ) < P ♯ R ( α ′ ) , which implies that G ( α ) < G ′ ( α, α ′ ) .Moreover, for given α ′ , ω ∗ ℓ ( α ′ ) and ̟ ∗ ℓ ( α ′ ) are optimized toincrease G ( α ′ ) , so it can be inferred that G ( α ′ ) ≥ G ′ ( α, α ′ ) .With the definition of G ( α ) , one can rewritten C TSR ( α ) as C TSR ( α ) = B K − α G ( α ) , which is a product of a monotonicdecreasing function and a monotonic increasing function.Besides, it can be easily observed that C TSR (0) = C TSR (1) = 0 and C TSR ( α ) > for α ∈ (0 , . Thus, there exists amaximum value of C TSR ( α ) within the interval α ∈ (0 , .Nevertheless, it is still very difficult to analytically discussthe convexity of C TSR ( α ) due to the implicit expression of ω ∗ ℓ ( α ) and ̟ ∗ ℓ ( α ) . As our goal is to explore the potentialcapacity of the TSR, we adopt a numerical method to searchthe maximum C TSR ( α ∗ ) over α ∈ (0 , with an updating step ∆ α , where the computational complexity is about O ( α ) .Note that in our simulations, we found that C TSR ( α ) is alwaysa firstly increasing and then decreasing function of α , whichindicates that C TSR ( α ) has only one peak (or maximum) within α ∈ [0 , . Therefore, some conventional algorithms with fastconvergence such as hill climbing algorithm [37] also can beadopted to find α ∗ .V. O PTIMAL D ESIGN OF
PSR P
ROTOCOL
In this Section, we shall investigate how to solve theoptimization problem (29) for PSR. It also can be seen thatthe Problem (29) is with discrete variables θ ℓ,ℓ ′ ∈ { , } ,which leads to a combinatorial optimization problem with highcomputational complexity. Thus, it cannot be easily solved byusing conventional methods. In this subsection, we shall solveit as follows.Firstly, we found that the optimal subchannel pairing strat-egy for TSR also holds for PSR, so that the optimal subchan-nel pairing can also be performed separately, which greatlyreduces the complexity. Moreover, for a given ω ℓ , we derivethe explicit expression of optimal ρ ∗ ℓ , which can be regarded asa function w.r.t ω ℓ , i.e., ρ ∗ ℓ ( ω ℓ ) . Thus, we replace the variable ρ ℓ with ρ ∗ ℓ ( ω ℓ ) to reduce the number of variables required tobe optimized and then derive the optimal ω ∗ ℓ . For clarity, we present the optimization framework of PSRas shown in Algorithm 3 at first. Then we shall explainthe detailed operation of each step of the framework in thesuccessive subsections.
Algorithm 3
Optimization Framework for PSR Calculate the optimal θ ∗ ; Calculate the optimal ρ ∗ for a given ω ; Calculate the optimal ω ∗ with the obtained ρ ∗ . A. Optimal θ ∗ for PSR We find that Lemma 3 still holds for PSR. The reasonis that, with the decoupling policy presented in [35], theoptimal subchannel pairing for PSR can also be obtained withthe sorted channel gain of the subchannels. Moreover, sinceboth ρ and ω do not affect the result of the sorted channelgain of the subchannels over the two hops, the optimal θ ∗ can be independently obtained without considering the othervariables. Thus, the optimal θ ∗ for PSR also can be determinedaccording to Lemma 3. B. Optimal ρ ∗ for a given ω Substituting θ ∗ into problem (29), it can be rewritten as max ω , ρ B K KN X ℓ =1 log (cid:18) (1 − ρ ℓ ) ρ ℓ P S ω ℓ λ S ,ℓσ R λ R ,ℓ ′ σ D − ρ ℓ ) P S ω ℓ λ S ,ℓσ R + ρ ℓ λ S ,ℓ P S ω ℓ λ R ,ℓ ′ σ D (cid:19) s.t. X KNℓ =1 ω ℓ ≤ , ≤ ρ ℓ ≤ , ∀ ℓ ∈ { , , ..., KN } (43)Since Problem (43) is also neither joint concave nor convexw.r.t ω and ρ , we shall discuss it with a given ω at first. Theorem 2.
Given a power allocation weight vector ω atsource S , the conditional optimal power splitting vector ρ ♯ satisfies that ρ ♯ℓ = A ℓ ω ℓ + 1 − p ( A ℓ ω ℓ + 1)( Q ℓ ω ℓ + 1) A ℓ ω ℓ − Q ℓ ω ℓ , A ℓ = Q ℓ , A ℓ = Q ℓ , (44) where A ℓ = P S λ S ,ℓ σ R and Q ℓ = P S λ S ,ℓ λ R ,ℓ ′ σ D . Proof:
The proof of Theorem 2 can be found in AppendixC.For a special case, where the system is with a single carrierand each node is with only one antenna, we can easily deducethe following corollary 1 from Theorem 2.
Corollary 1.
For a single-carrier single-antenna two-hopnon-regenerative PSR relaying system, if the channel gain-tonoise ratio (CNR) of the two hops satisfies that γ S , R = γ R , D , theoptimal power splitting ratio is ρ ∗ = 0 . , where γ u,v = h u,v σ v . C. Optimal ω ∗ for PSR By substituting (44) into Problem (43), we then obtain that max ω B K KN X ℓ =1 log (cid:18) (1 − ρ ♯ℓ ) ρ ♯ℓ P S ω ℓ λ S ,ℓσ R λ R ,ℓ ′ σ D − ρ ♯ℓ ) P S ω ℓ λ S ,ℓσ R + ρ ♯ℓ λ S ,ℓ P S ω ℓ λ R ,ℓ ′ σ D (cid:19) s.t. X KNℓ =1 ω ℓ ≤ . (45)From (44), although ρ ♯ℓ can be regarded as a function of ω ℓ , it is still not a simple expression, which makes problem(45) too difficult to be solved by using conventional methods.Thus, we firstly discuss analytically and then design efficientalgorithm to solve it. Lemma 8.
Let r ℓ ( ω ℓ ) = log (cid:18) − ρ ℓ ( ω ℓ )) ρ ℓ ( ω ℓ ) ω ℓ A ℓ Q ℓ − ρ ℓ )( ω ℓ ) ω ℓ A ℓ + ρ ℓ ( ω ℓ ) ω ℓ Q ℓ (cid:19) , where ρ ℓ ( ω ℓ ) is a function w.r.t. ω ℓ , as shown in (44). r ℓ is amonotonically increasing function of ω ℓ . Proof:
We can derive ∂r ℓ ∂ω ℓ with the consideration of cases A ℓ = Q ℓ and A ℓ = Q ℓ , respectively, as follows. ∂r ℓ ∂ω ℓ = ( Z ℓ ( C ℓ −G ℓ ) B ℓ , for A ℓ = Q ℓA ℓ Q ℓ ω ℓ ( A ℓ ω ℓ + Q ℓ ω ℓ +4)( A ℓ ω ℓ +2)( Q ℓ ω ℓ +2)( A ℓ ω ℓ + Q ℓ ω ℓ +2) , for A ℓ = Q ℓ (46)where Z ℓ = AQ ( A − Q ) ( A ℓ ω ℓ + 1)( Q ℓ ω ℓ + 1)( A ℓ + Q ℓ + AQ ℓ ω ℓ ) C ℓ = ( A ℓ ω ℓ + Q ℓ ω ℓ + AQ ℓ ω ℓ + 1) G ℓ = ( Q ℓ ω ℓ + 1) ( A ℓ ω ℓ + 1) B ℓ = − ( Q ℓ ω ℓ + 1) ( A ℓ ω ℓ + 1) ( A ℓ − Q ℓ ) ( A ℓ + Q ℓ + A ℓ Q ℓ ω ℓ ) . (47) With some algebraic manipulation, Lemma 8 can be easilyproved.
Lemma 9.
Let r ℓ ( ω ℓ ) = log (cid:18) − ρ ℓ ( ω ℓ )) ρ ℓ ( ω ℓ ) ω ℓ A ℓ Q ℓ − ρ ℓ )( ω ℓ ) ω ℓ A ℓ + ρ ℓ ( ω ℓ ) ω ℓ Q ℓ (cid:19) . It can be observed that r ℓ (0) = 0 .Although we have obtained some disciplines on Problem(45), it is still very difficult to prove the convexity of P KNℓ =1 r ℓ w.r.t ω ℓ . Therefore, by using Lemma 8 and Lemma 9, we shalldesign the Algorithm 4 to find the optimal ω ∗ . The basic ideaof Algorithm 4 is that a larger weight should be assigned tothe E2E subchannel with higher increasing rate of achievablerate, because larger weight implies higher power efficiency.The accuracy of Algorithm 4 relies on the step size △ ω . Thesmaller △ ω is, the more accurate the obtained results are, butthe convergence time may become longer. We set △ ω as 0.001in the following discussion.VI. N UMERICAL R ESULTS
In this section, some numerical results are presented tovalidate our analysis and discuss the performance of theproposed TSR and PSR. In the simulations, we consider atypical three node relaying network as shown in Figure 4, in Algorithm 4
Finding the optimal ω ∗ for each ℓ ∈ [1 , KN ] do Initialize ω ℓ = 0 ; end for while P ω ℓ < do Find q = arg max ℓ ∂r ℓ ∂ω ℓ for all ℓ ∈ { , , ..., KN } interms of (46); Update ω q = ω q + △ ω ; end while Return ω . S R D
S,R d R,D d S,D d h barrier - Fig. 4:
Relay position model of two-hop MIMO-OFDM relaynetwork with a barrier between the source and its destination. which there is a barrier between S and D . R is located on thetop of the barrier to assist the information delivering from S to D .The distance between S and D is regarded as a referencedistance, which is denoted as d S , D . The height of the barrieris h . The variable φ ∈ (0 , is used to describe the ratio ofthe distance between S and the barrier to d S , D . In this case,the distance between S and R and the distance between S and R can be respectively expressed by d S , R = p h + ( φd S , D ) and d R , D = p h + ((1 − φ ) d S , D ) . Note that when h = 0 , R is located on the direct line between S and D , which has beenwidely adopted as a model to discuss the relay position fortwo-hop relay systems, where the relay is moving along the S − D direct line. The total system bandwidth is assumed tobe B = 5 MHz, so each subcarrier is allocated with × K Hz.The power spectral density of the receiving noise at both R and D is set as − dBm. Besides, the path loss effect isalso considered, where the path loss factor is set to 2. Thedistance between S and D is regarded as a reference distance,which is set to be 100m. Note that all configuration parametersmentioned above will not change in the following simulationsunless specified otherwise.To show the performance gain of the optimized PSR andTSR, we consider two schemes with simple configurations asbenchmarks, i.e., the simple TSR and the simple PSR. In thesimple TSR, the optimal sub-channel pairing is involved, but α is set to a constant with α = . ω ℓ and ̟ ℓ is assigned by usingsuch a simple strategy, in which the value of each element of ω and ̟ is proportional to the eigenvalue of correspondingsub-channel, i.e., ω ℓ = λ S ,ℓ P KNj =1 λ S ,j and ̟ ℓ ′ = λ R ,ℓ ′ P KNj =1 λ R ,j . In thesimple PSR, the optimal sub-channel paring is also involvedand ω ℓ is determined with a simple method similar to thatused in the simple TSR, i.e., ω ℓ = λ S ,ℓ P KNj =1 λ S ,j . Moreover, forthe optimized TSR, we also consider two different methodsfor it, i.e., the optimized TSR-I and the optimized TSR-II. The P s (dBm) A c h i e v ab l e i n f o r m a t i on r a t e ( na t/ s ) Optimized PSR (numerical)Optimized PSR (simulation)Simple PSROptimized TSR−I (numerical)Optimized TSR−II (numerical)Optimized TSR (simulation)Simple TSR
Fig. 5:
System performance vs P S with N = 2 , K = 4 configuration. P s (dBm) A c h i e v ab l e i n f o r m a t i on r a t e ( na t/ s ) Optimized PSR (numerical)Optimized PSR (simulation)Simple PSROptimized TSR−I (numerical)Optimized TSR−II (numerical)Optimized TSR (simulation)Simple TSR
Fig. 6:
System performance vs P S with N = 4 , K = 64 configura-tion. optimized TSR-I denotes the TSR scheme optimized with thealternative updating of ω and ̟ described in Lemma 4 andthe optimized TSR-II denotes the TSR scheme optimized withthe approximating optimal ω ⋆ and ̟ ⋆ described in Lemma5. A. Performance vs P S In Figure 5 and Figure 6, we present the performance ofvarious schemes versus the available transmission power P S for N = 2 , K = 4 and N = 4 , K = 64 , respectively. In thesimulations, h and φ are set to 0 and 0.3, respectively, whichmeans that the relay is located on the straight line between S and D and R is closer to S than D . To validate our theoreticalanalysis, the simulation results obtained by using computersearch are also plotted. P S is changed from 0dBm to 50dBm.From the two figures, firstly, it can be seen that the nu-merical results match the simulation ones very well, whichindicates the validation of our theoretical analysis and theproposed algorithms. Moreover, it shows that all schemes P s (dBm) α Optimized TSR−IOptimized TSR−II N =4, K =64 N =2, K =4 α is around 0.34 Fig. 7:
The optimal α ∗ of the optimized TSR vs P S . achieve higher achievable information rate with the incrementof P S , because high P S will lead to high SNR for thesystem. It is also shown that the performance of the optimizedTSR is lower than the optimized PSR. The reason may beexplained as follows. Under the same channel conditions, theperformances of both TSR and PSR depend on the energyharvested and information received at the relay. In order towell match the harvested energy and collected information, inTSR, besides the power allocation, the time duration for energytransfer and information transmission over all subchannelsis adjusted by the same factor α . But in PSR, besides thepower allocation, each E2E subchannel has its own factor ρ i ( i = 1 , ..., K ) to adjust the ratio between the energyharvesting and information collecting. Compared with TSR,PSR provides more flexibility to adjust the system resources,so it may yield higher performance gain than TSR.In Figure 5, it also can be observed that when P S iswithin the interval of 30dBm to 40dBm, the performance ofthe simple TSP is very close to that of the optimized ones.The similar results also can be seen in Figure 6 between P S = P S = α ∗ of TSR isplotted versus P S for both N = 2 , K = 4 and N = 4 , K = 64 .It can be seen that for K = N = 2 when P S is within theinterval of 30dBm to 40dBm, the value of the optimal α ∗ isaround 0.34 and for N = 2 , K = 4 when P S is within theinterval of 20dBm to 30dBm, the value of the optimal α ∗ isalso around 0.34. Since in the simple TSR, we set α to , itis very similar to 0.34, which approximates the optimal ones.Therefore, it makes the performance of the simple TSR veryclose to that of the optimized one between P S = P S = P S = P S = K = N = 2 and K = N = 4 , respectively. The intervals of30dBm to 40dBm and 20dBm to 30dBm can also be regardedas the efficiently workable intervals for the simple TSR for N = 2 , K = 4 and N = 4 , K = 64 systems, respectively. φ A c h i e v ab l e i n f o r m a t i on r a t e ( na t/ s ) Optimized PSROptimized TSR−IOptimized TSR−IISimple PSRSimple TSR
Fig. 8:
System performance vs φ with h = 25 m , N=3 and N=16. B. Performance vs φ In this subsection, we shall discuss the performance of ouroptimized schemes versus the relay location. As illustratedin Figure 4, the relay location is described with the factor φ ∈ (0 , , in the simulations of Figure 8, φ is varied from 0.1to 0.9. h is set to m . K = 2 and N = 2 . P S is 20dBm. FromFigure 8, it can be observed that when the relay moves wayfrom the source, the achievable information rate is decreased.The reason is that the farther the distance between the sourceand the relay, the less the energy harvesting efficiency at therelay due to the path loss effect. As a result, a relativelylower performance of the system can be achieved when R is relatively farther away from S .In the simulations of Figure 9, we set h = 0 . In this case, R moves on the straight line between S and D . It can beobserved the achievable information rate of the PSR schemesfirstly decrease with the growth of φ , while the achievableinformation rate of the TSR schemes firstly decreases andthen increases with the growth of φ . This is the first time toobserve such a phenomenon. The reason may be that when φ issmall, R is closer to S , which yields a relatively higher energyharvesting efficiency. When φ is relatively large, R is closer to D . In this case, although a relatively lower energy harvestingefficiency can be achieved, a relatively better channel qualityover the R − D link is brought, which may improve the systemperformance.We also plot the optimal α and ρ in Figure 10 for the casewhen K = N = 1 and h = 0 . One can see that with theincrement of φ , the value of optimal ρ monotonically decreaseswhile that of α first increases and then decreases. C. Performance vs the number of antennas N In this subsection, we shall discuss the impact of the numberof antennas N on the system performance. In the simulations, K is set to 4 and h = 0 . We increase N from 2 to 10.Figure 11 plots the results averaged over 100 simulations.It can be observed that the achievable information rates ofall schemes increase with the increment of the number ofantennas. The reason is that more antennas can yield more φ A c h i e v ab l e i n f o r m a t i on r a t e ( na t/ s ) Optimized PSROptimized TSR−IOptimized TSR−IISimple PSRSimple TSR
Fig. 9:
System performance vs φ with h = 0 m , N=3 and K=16. φ α o r ρ Optimized PSROptimized TSR−IOptimized TSR−II α ρ Fig. 10:
System performance vs φ with h = 0 m . spatial subchannels. As a result, higher multiplex gain overthe subchannels can be achieved. Moreover, it also shows thatPSR achieves the highest achievable information among allschemes and it is also with the highest increasing rate thanother ones. D. Performance vs the number of subcarriers K In this subsection, we shall discuss the impact of thenumber of subcarriers K on the system performance. In thesimulations, N is set to 2 and h = 0 . K is gradually increasedfrom 20 to 100. Figure 12 plots the averaged results over 100simulations. It can be observed that the achievable informationrates of all schemes increase with the increment of the numberof subcarrier. However, the increasing rate of each curve goesslower and slower with the increment of K . The reason is thatmore subcarriers may yield more subchannels and bring moreflexible configuration to increase the system capacity, but witha fixed total system bandwidth, more subcarrier may cause asmaller bandwidth allocated to each subcarrier. In this case,there exists a trade-off between the number of subcarriers K and system achievable information rate. Number of antennas N A c h i e v ab l e i n f o r m a t i on r a t e ( na t/ s ) Optimized PSROptimized TSR−IOptimized TSR−IISimple PSRSimple TSR7 8 91.41.51.61.71.8 x 10 Fig. 11:
System performance vs the number of antennas N with K = 4 subcarriers.
20 30 40 50 60 70 80 90 10000.511.522.5 x 10 The number of subcarriers K A c h i e v ab l e i n f o r m a t i on r a t e ( na t/ s ) Optimized PSROptimized TSR−IOptimized TSR−IISimple PSRSimple TSR
70 80 9066.16.26.36.4 x 10 Fig. 12:
System performance vs the number of subcarriers K with N = 2 antenna configuration. VII. C
ONCLUSION
This paper studied the simultaneous wireless energy harvest-ing and information transfer for the non-regenerative MIMO-OFDM relaying system, where both the energy harvestingand energy consumption were considered in a single system.We presented two protocols, TSR and PSR for the system.In order to investigate the system performance limits, weformulated two optimization problems for them to jointlyoptimize the multiple system configuration parameters so thatthe end-to-end achievable information rate of each protocolcan be maximized. To the optimized problems, we derivedsome explicit theoretical results and designed some effectivealgorithms for them. Various numerical results were presentedto confirm our analytical results and to show the performancegain of our optimized PSR and TSR. In addition, it is alsoshown that the performances of both protocols are greatlyaffected by the relay position. The achievable informationrate of PSR monotonically decreases with the increment ofsource-relay distance and that of TSR firstly decreases and then increases with the increment of source-relay distance andthe relatively worse performance is obtained when the relayis placed in the middle of the source and the destination.The simulation results also show that PSR always outperformsTSR in the two-hop non-regenerative MIMO-OFDM system.In addition, increasing either the number of antennas or thenumber of subcarriers can bring system performance gain tothe two protocols.This work also suggests that in a non-regenerative MIMO-OFDM system, it is better to adopt PSR if the CSI is perfect.For the imperfect CSI case, we will consider it in the future.A PPENDIX AT HE P ROOF OF L EMMA X i , we havethat X i = V ( i ) x Ξ ( i ) x V ( i ) Hx , where V ( i ) x V ( i ) Hx = I and Ξ ( i ) x = diag n ξ ( i ) x, , ξ ( i ) x, , · · · , ξ ( i ) x,N E o with ξ ( i ) x, ≥ ξ ( i ) x, ≥· · · ≥ ξ ( i ) x, Rank ( H S ,i ) ≥ and P N E q =1 ξ ( i ) x,q ≤ tr ( X i ) , where N E = Rank ( H S ,i ) . Let V ( i ) x = [ v ( i ) x, , v ( i ) x, , ..., v ( i ) x,N E ] T and ˜ H S ,i = H S ,i V ( i ) x = [˜ h ( i ) S , , ˜ h ( i ) S , , · · · , ˜ h ( i ) S ,N E ] . It can be deducedthat k H S ,i x i k = tr ( H S ,i X i H H S ,i ) = tr ( ˜ H S ,i Ξ ( i ) x ˜ H H S ,i ) (48) = X N E q =1 ξ ( i ) x,q k ˜ h ( i ) S ,q k ≤ tr ( X i ) k ˜ h ( i ) S , k , where the equality holds if k ˜ h ( i ) S , k = max q k ˜ h ( i ) S ,q k and ξ ( i ) x,q = ( tr ( X i ) , q = 1 , , q = 2 , , ..., N E . Moreover, since H S ,i = U S ,i Λ S ,i V H S ,i , it can be inferred that V ( i ) x = V S ,i and k ˜ h ( i ) S , k = max q k ˜ h ( i ) S ,q k only if v ( i ) x, isthe first column of V S ,i , which is corresponding to the largestsingular value of H S ,i , i.e., λ ( i ) S , . So, X ♯i = v ( i ) s, ξ ( i ) x, v ( i ) Hs, = tr ( X i ) v ( i ) s, v ( i ) Hs, . Lemma 1 is therefore proved.A PPENDIX BT HE P ROOF OF L EMMA max tr ( X ) , tr ( X ) ,..., tr ( X K ) X Ki =1 tr ( X i ) k ˜ h ( i ) S , k (49) X Ki =1 tr ( X i ) ≤ P S , tr ( X i ) ≥ . By doing so, it can be easily derived that the optimal solutionof Problem (49) istr ( X i ) ∗ = P S , i = arg max b =1 ,...,K k ˜ h ( b ) S , k , , otherwise . (50)Lemma 2 thus is proved. A PPENDIX CT HE P ROOF OF T HEOREM F = P KNℓ =1 r ℓ , where r ℓ = log (cid:18) (1 − ρ ℓ ) ρ ℓ P S ω ℓ λ S ,ℓσ R λ R ,ℓ ′ σ D − ρ ℓ ) P S ω ℓ λ S ,ℓ σ R + ρ ℓ λ S ,ℓ P S ω ℓ λ R ,ℓ ′ σ D (cid:19) . (51)It can be calculated that ∂ F ∂ρ ℓ = ∂r ℓ ∂ρ ℓ = A ℓ ω ℓ Q ℓ ω ℓ ( A ℓ ω ℓ − ρ ℓ − A ℓ ω ℓ ρ ℓ + A ℓ ω ℓ ρ ℓ − Q ℓ ω ℓ ρ ℓ +1)( Q ℓ ω ℓ ρ ℓ +1)( A ℓ ω ℓ (1 − ρ ℓ )+1)( A ℓ ω ℓ (1 − ρ ℓ )+ Q ℓ ω ℓ ρ ℓ +1) . (52)Since ≤ ρ ℓ ≤ , it can be easily observed that thedenominator of (52) is always larger than 0. Assume ∂ F ∂ρ ℓ = 0 ,for the case A ℓ = Q ℓ , we obtain that ρ ℓ = A ℓ ω ℓ +1 − √ ( A ℓ ω ℓ +1)( Q ℓ ω ℓ +1) A ℓ ω ℓ − Q ℓ ω ℓ . (53)It can be easily seen that (53) satisfies that ≤ ρ ℓ ≤ . Moreover, it also can be seen that if <ρ ℓ < A ℓ ω ℓ +1 − √ ( A ℓ ω ℓ +1)( Q ℓ ω ℓ +1) A ℓ ω ℓ − Q ℓ ω ℓ , ∂ F ∂ρ ℓ > and if A ℓ ω ℓ +1 − √ ( A ℓ ω ℓ +1)( Q ℓ ω ℓ +1) A ℓ ω ℓ − Q ℓ ω ℓ < ρ ℓ < , then ∂ F ∂ρ ℓ < . Besides,as r ℓ (0) = r ℓ (1) = 0 , then we can deduce that F only has onemaximum value which can be achieved only when ρ ℓ meetsthe equation (53). For the case that A ℓ = Q ℓ , (51) can besimplified as r ℓ = log (cid:0) (1 − ρ ℓ ) ρ ℓ A ℓ ω ℓ Q ℓ ω ℓ − ρ ℓ ) A ℓ ω ℓ + ρ ℓ Q ℓ ω ℓ (cid:1) = log (cid:0) (1 − ρ ℓ ) ρ ℓ A ℓ ω ℓ A ℓ ω ℓ (cid:1) . In this case, it can be easily seen that if and only if (1 − ρ ℓ ) ρ ℓ achieves the maximum, r ℓ will be maximal. Obviously, when (1 − ρ ℓ ) = ρ ℓ , i.e., ρ ℓ = 0 . , (1 − ρ ℓ ) ρ ℓ achieves its maximumvalue. Therefore, Theorem 2 is proved.R EFERENCES[1] W. K. G. Seah, Z. A. Eu, H. P. Tan, “Wireless sensor networks poweredby ambient energy harvesting (WSN-HEAP) - Survey and challenges,”in
Proc. Wireless VITAE 2009 , pp. 1 - 5, May 2009.[2] C. Huang, R. Zhang, S. G. Cui, “Throughput maximization for theGaussian relay channel with energy harvesting constraints,”
IEEE J. Sel.Areas in Commun. , vol. 31, no. 8, pp. 1469-1479, Aug. 2013.[3] D. T. Hoang, D. Niyato, P. Wang, D.I. Kim, “Opportunistic ChannelAccess and RF Energy Harvesting in Cognitive Radio Networks,”
IEEEJ. Sel. Areas in Commun. , vol. 32, no. 11, pp. 2039-2052, Nov. 2014.[4] J. Xu and R. Zhang, “Throughput Optimal Policies for Energy HarvestingWireless Transmitters with Non-Ideal Circuit Power,”
IEEE J. Sel. Areasin Commun. , vol. 32, no. 2, pp. 322-332 , Feb. 2014.[5] V. Raghunathan, S. Ganeriwal and M. Srivastava, “Emerging techniquesfor long lived wireless sensor networks,”
IEEE Commun. Mag. , vol. 44,no. 4, pp. 108 - 114, Apr. 2006.[6] O. Ozel, K. Tutuncuoglu, J. Yang, S. Ulukus, A. Yener, “Transmissionwith energy harvesting nodes in fading wireless channels: Optimalpolicies,”
IEEE J. Sel. Areas in Commun. , vol. 29, no. 8, pp. 1732-1743,Sept. 2011[7] J. Yang, S. Ulukus, “Optimal Packet Scheduling in an Energy HarvestingCommunication System,”
IEEE Trans. Commun. , vol. 60, no. 1, pp. 220-230, Jan. 2012.[8] K. Tutuncuoglu, A. Yener, “Optimum transmission policies for batterylimited energy harvesting nodes,”
IEEE Trans. Wireless Commun. , vol.11, no. 3, pp. 1180-1189, March 2012[9] L. R. Varshney, “Transporting information and energy simultaneously,”in
Proc. IEEE ISIT , 2008.[10] P. Grover and A. Sahai, “Shannon meets Tesla: wireless information andpower transfer,” in
Proc. IEEE ISIT , 2010. [11] Z. Xiang and M. Tao, “Robust beamforming for wireless informationand power transmission,” IEEE Wireless Commun. Lett. , vol. 1, no. 4, pp.372-375, 2012.[12] A. M. Fouladgar and O. Simeone, “On the transfer of information andenergy in multi-user systems,”
IEEE Commun. Lett. , vol. 16, no. 11, pp.1733-1736, Nov. 2012.[13] P. Popovski, A. M. Fouladgar and O. Simeone, “Interactive joint transferof energy and information,”
IEEE Trans. Commun. , vol. 61, no. 5, pp.2086-2097, May 2013.[14] X. Zhou, R. Zhang, and C. K. Ho, “Wireless information and powertransfer: Architecture design and rate-energy tradeoff,”
IEEE Trans.Commun. , vol. 61, no. 11, pp. 4754-4767, Nov. 2013.[15] L. Liu, R. Zhang, K. C. Chua, “Wireless information transfer withopportunistic energy harvesting,”
IEEE Trans. Wireless Commun. , vol.12, no. 1, pp. 288-300, Jan. 2013.[16] S. X. Yin, E. Q. Zhang, J. Li, L. Yin, “Throughput optimization for self-powered wireless communications with variable energy harvesting rate,”in
Proc. IEEE WCNC’13 , Shanghai, April 2013.[17] A. A. Nasir, X. Zhou, S. Durrani, and R. A. Kennedy, “Relayingprotocols for wireless energy harvesting and information processing,”
IEEE Trans. Wireless Commun. , vol. 12, no. 7, pp. 3622-36, Jul. 2013[18] Z. G. Ding, S. M. Perlaza, I. Esnaola, H. Vincent Poor, “Power allocationstrategies in energy harvesting wireless cooperative networks,”
IEEETrans. Wireless Commun. , vol.13, no. 2, pp. 846-860, Feb. 2014.[19] Z. G. Ding, I. Esnaola, B. Sharif, H. Vincent Poor, “Wireless informationand power transfer in cooperative networks with spatially random relays,”
IEEE Trans. Wireless Commun. , vol.13, no. 8, pp. 4400-4453, Aug. 2014.[20] K. Xiong, P. Y. Fan, Z. F. Xu, H. C. Yang, K. B. Letaief, “Optimalcooperative beamforming design for MIMO decode-and-forward relaychannels,”
IEEE Trans. Signal Process. , vol.62, no. 6, pp. 1476-1489,March 2014.[21] D. Zhang, P. Y. Fan, Z. G. Cao, “Interference cancellation for OFDMsystems in presence of overlapped narrow band transmission system,”
IEEE Trans. Consumer Electronics , vol.50, no. 1, pp. 108-114, Feb. 2004.[22] D. Zhang, P. Y. Fan, Z. G. Cao, “A novel narrowband interferencecanceller for OFDM systems,”
Proc. IEEE WCNC , vol. 3, pp. 1426-1430,2004.[23] H. Chen, Y. H. Li, Y. X. Jiang, Y. Y. Ma, B. Vuccetic, “Distributed powersplitting for SWIPT in relay interference channles using game theory,”
IEEE Trans. Wireless Commun. , early access, 2014.[24] R. Zhang and C. K. Ho, “MIMO broadcasting for simultaneous wirelessinformation and power transfer,”
IEEE Trans. Wireless Commun. , vol. 12,no. 5, pp. 1989 - 2001, May 2013.[25] Q. J. Shi, L. Liu, W. Q. Xu and R. Zhang, “Joint transmit Beamformingand receieve power splitting for MISO SWIPT,”
IEEE Trans. WirelessCommun. , vol. 13, no. 6, pp. 3269 - 3280, June 2014.[26] D. W. Kwan Ng, E. S. Lo and R. Schober, “Wireless information andpower transfer: energy efficency optimization in OFDMA systems,”
IEEETrans. Wireless Commun. , vol. 12, no. 12, pp. 6352 - 6370, Dec. 2014.[27] X. Zhou, R. Zhang, and C. K. Ho, “Wireless information and powertransfer in multiuser OFDM systems,”
IEEE Trans. Wireless Commun. ,vol. 13, no, 4, pp. 2282-2294, April 2014.[28] Q. Z. Li, Q. Zhang and J. Y. Qin, “Secure relay beamforming forsimultaneous wireless information and power transfer in nonregenerativerelay networks,”
IEEE Trans. Veh. Technology , vol. 63, no. 5, pp. 2462-2467, June 2014.[29] B. K. Chalise, Y. D. Zhang, and M. G. Amin, “Energy harvesting inan OSTBC based non-regenerative MIMO relay system,” in
Proc. IEEEICASSP , pp. 3201-3204, March 2012.[30] B. K. Chalise, W. K. Ma, Y. D. Zhang, H. Suraweera, M. G.Amin,“Optimum performance boundaries of OSTBC based AF-MIMOrelay system with energy harvesting receiver,”
IEEE Trans. Signal Pro-cess. , vol. 61, no. 17, pp. 4199 - 4213, Sept. 2013.[31] K. Xiong, P. Y. Fan, M. Lei, S. Yi, “Network coding-Aware cooperativerelaying for downlink cellular relay networks,”
China Communications ,vol. 10, no. 7, pp. 44 - 56, July, 2013.[32] I. Hammerstr´’om and A. Wittneben, “Power allocation schemes foramplify-and-forward MIMO-OFDM relay links,”
IEEE Trans. WirelessCommun. ,vol. 6, no. 8, pp. 2798-2802, Aug. 2007.[33] X. J. Tang and Y. B. Hua, “Optimal design of non-regenerative MIMOwireless relays,”
IEEE Trans. Wireless Commun. , vol. 6, no. 4, pp. 1398-1470, April 2007.[34] K. Xiong, T. Li, P. Y. Fan, Z. D. Zhong, K. B. Letaief, “Outage proba-bility of space-time network coding with amplify-and-forward relays,” in
Proc. IEEE GLOBECOM , pp. 3571-3576, Dec. 2013. [35] M. Hajiaghayi, M. Dong and B. Liang, “Jointly optimal channel pairingand power allocation for multichannel multihop relaying,”
IEEE Trans.signal Process. , vol. 59, no. 10, pp. 4998-5012, Oct. 2011.[36] S. Boyd and L. Vandenberghe,
Convex Optimization , Cambridge Uni-versity Press, 2004.[37] S. J. Russell, P. Norvig, Peter,