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MIMO-OFDM-based Wireless-Powered Relaying Communication withan Energy Recycling Interface

Nasir, A. A., Tuan, H. D., Duong, Q., & Poor, H. V. (2020). MIMO-OFDM-based Wireless-Powered RelayingCommunication with an Energy Recycling Interface. IEEE Transactions on Communications.https://doi.org/10.1109/TCOMM.2019.2952897

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MIMO-OFDM-based Wireless-Powered RelayingCommunication with an Energy Recycling Interface

Ali A. Nasir, Hoang D. Tuan, Trung Q. Duong, and H. Vincent Poor

Abstract—This paper considers wireless-powered relayingmultiple-input-multiple-output (MIMO) communication, whereall four nodes (information source, energy source, relay, anddestination) are equipped with multiple antennas. Orthogonalfrequency division multiplexing (OFDM) is applied for infor-mation processing to compensate the frequency selectivity ofcommunication channels between the information source and therelay and between the relay and the destination as these nodes areassumed to be located far apart each other. The relay is equippedwith a full-duplexing interface for harvesting energy not onlyfrom the wireless transmission of the dedicated energy sourcebut also from its own transmission while relaying the sourceinformation to the destination. The problem of designing theoptimal power allocation over OFDM subcarriers and transmitantennas to maximize the overall spectral efficiency is addressed.Due to a very large number of subcarriers, this design problemposes a large-scale nonconvex optimization problem involvinga few thousand variables of power allocation, which is verycomputationally challenging. A novel path-following algorithmis proposed for computation. Based on the developed closed-formcalculation of linear computational complexity at each iteration,the proposed algorithm rapidly converges to an optimal solution.Compared to the best existing solvers, the computational com-plexity of the proposed algorithm is reduced at least 105 times,making it really efficient and practical for online computationwhile that existing solvers are impotent. Numerical results for apractical simulation setting show promising results by achievinghigh spectral efficiency.

Index Terms—Wireless-powered network, full-duplex inter-face, energy recycle, MIMO-OFDM relaying communication,spectral efficiency, power allocation, large-scale nonconvex op-timization, online computation.

I. INTRODUCTION

A. Motivation and Literature Survey

Full-duplexing (FD) is an advanced fifth generation (5G)communication technology for signal transmission and recep-tion at the same communication node, at the same time, and

This work was supported in part by the KFUPM Research Project#SB171005, in part by Institute for Computational Science and Technology,Hochiminh city, Vietnam, in part by the Australian Research Council’sDiscovery Projects under Project DP190102501, in part by an U.K. RoyalAcademy of Engineering Research Fellowship under Grant RF1415\14\22,and in part by the U.S. National Science Foundation under Grants CCF-0939370, CCF-1513915 and CCF-1908308

A. A. Nasir is with the Department of Electrical Engineering, King FahdUniversity of Petroleum and Minerals (KFUPM), Dhahran, Saudi Arabia(email: [email protected]).

H. D. Tuan is with the School of Electrical and Data Engineering,University of Technology Sydney, Broadway, NSW 2007, Australia (email:[email protected]).

T. Q. Duong is with Queen’s University Belfast, Belfast BT7 1NN, UK(email: [email protected])

H. V. Poor is with the Department of Electrical Engineering, PrincetonUniversity, Princeton, NJ 08544, USA (email: [email protected]).

over the same frequency band [1], [2]. On the other hand,wireless power transfer is a promising solution which canassist in powering trillions of devices in Internet of Things(IoTs) in 5G and futuristic communication systems [3]. Thisis because battery replacement is a crucial issue for the massivenumber of wireless sensors in the IoTs [4], [5]. Thus, FD andwireless power transfer technologies can make a promisingblend to improve the spectral efficiency and sensors’ life-timefor futuristic communication systems.

A FD wireless-powered relay is supposed to simultaneouslysplit the received signal for energy-harvesting (EH) and infor-mation decoding (ID) [6]–[12]. Different communication set-tings, e.g., single-input-single-output (SISO) communication[6], SISO communication with multiple relays and relay selec-tion [9], multiple-input-single-output (MISO) communication[7], [8], two-way relaying with multiple-antenna relay andFD source nodes [10], multiple-input multiple-output (MIMO)communication [11], [12], have been considered. However,the power-splitting approach is complicated and inefficient forpractical implementation due to the requirement of variablepower-splitter design [13]. More importantly, the major issuein FD-based communication is the loop self-interference (SI)due to the co-location of transmit and receive antennas. Withthe current state-of-the-art technology of signal isolation andrejection, it is still not practical to mitigate the FD SI to a levelworthy for simultaneous high uplink and downlink throughput[14]. Two-way relaying within one time-slot by a FD relaycannot be more efficient than that by a half-duplex relay in twotime slots [15], [16]. However, such high-powered interferenceof the signal transmission to the signal reception by FDcan open an opportunity for the simultaneous informationtransmission and energy-harvesting [17]. Indeed, unlike theinformation throughput, which is dependent on the signal-to-interference-plus-noise-ratio and thus suffers from the interfer-ence, the harvested energy is critically dependent on how thereceived signal power is, and hence, it can be gained by theinterference as an complement energy source. Therefore, FDis natural for transmitting information while harvesting energywith self-energy recycling in wireless-powered networks [18]–[22]. In relaying communication, the information source sendsthe information signal to the relay in the first phase, and in thesecond phase, the energy source transfers the energy signal tothe relay while the relay simultaneously harvests energy andforwards the information to the destination. Thus, the energyconstrained relay can replenish its energy by harvesting energyfrom the signal composite of the source’s energy signal andits own transmitted signal [19]. Using this protocol, differentcommunication settings, e.g., SISO communication from the

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source-to-relay and MISO communication from the relay-to-destination [23], single-input-multiple-output (SIMO) commu-nication from the source-to-relay and MISO communicationfrom the relay-to-destination [24], [25], SISO communicationover both links [26]–[29], MISO communication over bothlinks [19], [30], [31], SISO communication with multiple re-lays and relay selection [32], MIMO communication from thesource-to-relay and MISO link from the relay-to-destination[33], and MIMO communication over both links [20], havebeen considered.

One of the main issues of long range communication withMIMO setting is its channel frequency selectivity due tomultipath propagation, which can be compensated by the or-thogonal frequency division multiplexing (OFDM) technology.By transforming the frequency selective channel into paral-lel frequency flat sub-channels, MIMO-OFDM provides thedominant air interface for broadband wireless communications[34], [35]. The communication range can be further extendedby deploying cooperative relaying based communication to as-sist the transmission between the two distant ends. Consideringrelay-assisted networks, resource allocation in OFDM basedcommunication with half-duplex (HD) relaying [36], [37] andFD relaying [15], [38] has been investigated. In addition, thereare several recent studies which investigate EH in OFDMbased systems too to assist the needs of energy constrainednodes [39]–[41]. Wireless information and power transfer inHD OFDM relay networks is considered in [42]. However, asmentioned above, FD relaying in energy constrained networkwith the practical two-phase communication has a naturalpreference (see [19], [20], [23]–[30], [32], [33] for non-OFDM systems). This is because during the second phaseof communication, the energy constrained FD relay has theopportunity to harvest energy not only from the wirelessenergy signal from the source but also through the energyrecycled from its own transmission.

B. Research Gap and Contribution:

It must be realized that the physical conditions for wirelessinformation and energy transfers are distinct. In fact, asenergy-harvesting (EH) is possible only when the power ofthe received energy signal passes a threshold that is very largecompared to the power of the received information signal [17],[43], the energy transfer can be implemented only when thesource is located really near to the relay. With the source nodelocated near to the relay as in [24], [26], [27], [30], [32], thedistances from the source to the relay and from the relay to thedestination are not much different. The direct communicationfrom the source to the destination is then preferred in anyaspect, and the energy transfer to the relay is superfluous.

Relaying communication is in need when the source nodeis located farther away from the relay as in [19], [42],but certainly the energy transfer from this node cannot beimplemented because the received signal power will not passthe threshold for EH at the relay.

This paper considers a practical two-phase MIMO-OFDMcommunication assisted by a wireless-powered FD relay. Toavoid the aforementioned unpractical assumptions, it proposes

a dedicated energy source in the network, which is placed in aclose vicinity of the relay node to transmit wireless energy sig-nal to the relay during the second communication phase. Thus,the relay harvests energy both from the recycled energy fromits own transmission and from the wireless energy signal fromthe energy source. It is noteworthy that the dedicated energysource is definitely required for EH because the received signalpower at the FD relay from its own transmission (loop SI) doesnot pass the threshold for EH at the relay. Under this practicalsetup with a dedicated energy source, the information sourceis located quite far from the destination, which transmits theinformation signal to the relay during the first communicationphase. The key contributions of this work are as follows:

• This is the first paper to consider a practical two-phaseMIMO-OFDM communication assisted by a wireless-powered FD relay. The MIMO-OFDM relaying is re-ally needed to assist the communication between theinformation source and destination, which are located farapart each other. The energy source is set to be placedsufficiently near to the relay to enable practical EH bythe latter, which moreover uses FD to recycle the energyexerted from its signal transmission. The problem ofdesigning the precoding and relaying matrices to max-imize the overall spectral efficiency is formulated. UnderMIMO-OFDM, the design of precoding and relayingmatrices is simplified to that of power allocation overmultiple subcarriers and transmit antennas.

• The optimization problem of power allocation is not onlynonconvex but is large-scale due to a large number ofsubcarriers (up to thousands), making all the existingnonconvex solvers useless. This is the first paper topropose a path-following algorithm, which is practicalfor large-scale nonconvex optimization of thousands ofvariables (subcarriers). The algorithm is based on severalinnovative approximations for both concave and non-concave functions, and for both convex and nonconvexsets, which enable the iterative optimization problem toadmit the optimal solution in a closed-form. Certainly,approximating a concave function by another concavefunction and a convex set by another convex set for aclosed form of the optimal solution is truly a new ideain optimization.

• The proposed algorithm guarantees a computational so-lution, around in one-tenth of a second, even in thepresence of thousands of subcarriers, due to rapid conver-gence and linear computational complexity. The detailedcomputational complexity analysis clearly shows hugecomputational gain of the proposed algorithm comparedto the existing algorithms, which already struggle towork for small-scale systems with a few subcarriersdue to their inherent high computational complexity.Particularly, comparing to the best existing solvers, thecomputational complexity of the proposed algorithm isreduced at least 105 times, making it really efficient andpractical for online computation. The provided numericalresults with practical simulation setup show promisingresults by achieving high spectral efficiency.

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C. Organization and Notation:

Organization: The paper is organized as follows. After abrief system model, Section II presents the formulation ofrate maximization problem. Section III describes the proposedsolution to the problem. Section IV evaluates the performanceof our proposed algorithm by assuming practical simulationsetup. Finally, Section V concludes the paper.

Notation: Bold-face upper-case letters, e.g., X are used formatrices. Bold-faced lower-case letters, e.g., x, are used forvectors and non-bold lower-case letters, e.g., x, are used forscalars. [X]a,b represents (a, b)th entry (ath row, bth column)of the matrix X. Hermitian transpose and normal transpose ofthe vector x are denoted by xH , and xT , respectively. C isthe set of all complex numbers and R is the set for all realnumbers. ∂f(x)∂x is the derivative of function f(·) with respectto its variable x. Finally, E(X) is the expectation operatorapplied to any random variable X .

II. SYSTEM MODEL AND PROBLEM FORMULATION

Consider a cooperative communication system in which anenergy constrained relay node (R) assists the communicationbetween a source node (S) and a destination node (D). Weassume that D is out-of-reach from S and there is no directcommunication link between them. As shown in Fig. 1, adedicated energy source (E) is placed in a close vicinity ofthe relay node to assist in fulfilling the energy requirementsof the relay node. S, D, and E are equipped with N antennasbut R is equipped with N transmit antennas and N receiveantennas.1

As shown in Fig. 2, the communication from S to Dtakes place in two-phases. During the first phase, S sends aninformation signal to R. During the second phase, E sendsan energy signal to R and at the same time, R forwards thesource information to D. Note that during the second stage, Rharvests energy not only from the dedicated wireless energysignal from E but also from its own transmission. In otherwords, the self-interfering link at R enables the self-energyrecycling.

Due to multipath propagation, the MIMO channel from Sto R and from R to D is frequency selective. To combat suchmultipath channel distortion, multi-carrier modulation, i.e.,OFDM is employed for transmitting the information-bearingsignal from S to R in the first phase and also for forwardingit by R to D in the second phase. The channel between E andR is assumed frequency flat due to the short distance betweenthem and thus the energy signal is transmitted by a singlecarrier.

Let the L-tap multipath channel from the antenna n at Sto the antenna n̄ at R be denoted by gn,n̄S ∈ CL and thatfrom the antenna n at R to the antenna n̄ at D be denoted bygn,n̄R ∈ CL, n, n̄ ∈ {1, . . . , N}. Due to the line-of-sight linkbetween E and R, the channel between them can be modeledby a single-tap channel, which is denoted by gn,n̄E ∈ C. Byincorporating the effect of large scale and small scale fading,

1Under general assumption of Nt transmit antennas and Nr receiveantennas, N will refer to the number of spatially independent paths or therank of the channel matrix, i.e., N = min(Nt, Nr).

the channel gain gn,n̄E is assumed to be Rician distributed,while all the channel taps of gn,n̄S and g

n,n̄R are assumed to be

Rayleigh distributed. It is also assumed that the full channelstate information (CSI) is available by some high-performingchannel estimation mechanism, and a central processing unitaccesses that information to optimize resource allocation underperfect timing and carrier frequency synchronization. Particu-larly, using standard channel estimation techniques, the relaynode can estimate the S-to-R channel gn,n̄S , and the E-to-Rchannel gn,n̄E , and the destination node can estimate the R-to-D channel, gn,n̄R , which are sent to the central processingunit to solve the resource allocation problem. Note that theperformance of the proposed resource allocation algorithm willbe later analyzed under imperfect CSI in Section IV-B.

In the following the system model for information andenergy processing is presented, which is followed by theproblem formulation.

A. Information Processing

Using the fast Fourier transform (FFT), the received vectorỹR,k ∈ CN at the relay on subcarrier k ∈ {1, 2, . . . ,K} isgiven by

ỹR,k = HS,ks̃ID,k + w̃R,k, (1)

where• HS,k ∈ CN×N is the MIMO channel matrix between

S and R on subcarrier k such that any element ofHS,k, e.g., (n, n̄)th element, [HS,k]n,n̄, represents thesub-channel k between the transmit antenna n of S andthe receive antenna n̄ of R, i.e., we can obtain [HS,k]n,n̄,k = {1, . . . ,K} by evaluating the K-point FFT of gn,n̄S ,

• s̃ID,k = ΨksID,k, Ψk ∈ CN×N is the transmit precod-ing matrix on subcarrier k, sID,k ∈ CN is the modulatedinformation data (ID) on subcarrier k,

• w̃R,k ∈ CN is the additive zero-mean Gaussian noisewith covariance RR = σRIN , σR is the noise varianceat each receive antenna and IN is the N × N identitymatrix.2

During the second phase, R amplifies the received signalvector on subcarrier k by a matrix Fk ∈ CN×N and for-wards the processed signal vector to D. The received vectorỹD,k ∈ CN , after FFT, on subcarrier k ∈ {1, 2, . . . ,K} at thedestination is given by

ỹD,k = HR,kFk (HS,ks̃ID,k + w̃R,k) + w̃D,k, (2)

where• HR,k ∈ CN×N is the MIMO channel matrix between R

and D on subcarrier k such that any element of HR,k, e.g.,(n, n̄)th element, [HR,k]n,n̄, represents the sub-channelk between the transmit antenna n of R and the receiveantenna n̄ of D, i.e. [HR,k]n,n̄, k = {1, . . . ,K} isobtained by evaluating the K-point FFT of gn,n̄R ,

• w̃D,k ∈ CN is the additive zero-mean Gaussian noisewith covariance RD = σDIN , and σD is the noisevariance at each receive antenna of D.

2FFT operation at the receiver does not change the noise covaraince.

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Fig. 1. Wireless-powered relaying with self-energy recycling.

Energy Transfer

E −→ RInformation Transfer

R −→ D

Information Transfer

S −→ R

T

T/2 T/2

Fig. 2. Protocol for full-duplex wireless-powered relaying.

TABLE ILIST OF FREQUENTLY USED SYMBOLS WITH THEIR DEFINITIONS

Symbols DefinitionhS,k,n channel power gain from S to R over subcarrier k and spatial channel nhR,k,n channel power gain from R to D over subcarrier k and spatial channel nhE,n channel power gain from E to R over spatial channel npR,k,n power allocated to R to forward the source information over subcarrier k and spatial channel npID,k,n power allocated to S to transmit source information data over subcarrier k and spatial channel npEH,n power allocated to E to transmit energy signal to R over spatial channel n

Without loss of generality, the channel matrices HS,k andHR,k are assumed nonsingular, which admit the singular valuedecomposition (SVD)

HS,k = VS,kΛS,kUS,k, (3a)HR,k = UR,kΛR,kVR,k, (3b)

where

ΛS,k = diag{√hS,k,n}Nn=1, (4a)

ΛR,k = diag{√hR,k,n}Nn=1, (4b)

and

• VS,k, VR,k, US,k, and UR,k are unitary matrices ofdimension N ×N ;

• {hS,k,n}Nn=1 and {hR,k,n}Nn=1 are the eigenvalues ofHS,kH

HS,k and H

HR,kHR,k, respectively. The factors√

hS,k,n ∈ R and√hR,k,n ∈ R can be seen as channel

gains from S to R and from R to D over subcarrier k andspatial channel n.

Based on the channel decomposition in (3), the precodingmatrix and the relay processing matrix can be set as follows:

Ψk = UHS,kΓS,k (5a)

Fk = VHR,kΓR,kV

HS,k, (5b)

where

ΓS,k , diag{√pID,k,n}Nn=1, (6a)

ΓR,k , diag{√ζk,n√pR,k,n}Nn=1, (6b)

and

• pID,k,n is the power allocated to transmit the sourcesignal sID,k,n over subcarrier k and spatial channel n,n ∈ {1, . . . , N};

• pR,k,n is the power allocated to the relay to forward thesource signal over subcarrier k and spatial channel n;

• ζk,n is the scaling (normalization) factor at R to ensuretransmit power of pR,k,n over subcarrier k and spatialchannel n.

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The list of frequently used symbols with their definition isprovided in Table I. Use the matrix decompositions in (3) and(5) to rewrite (1) to

yR,k = ΛS,kΓS,ksID,k + wR,k, (7)

where yR,k = VHS,kỹR,k, and wR,k = VHS,kw̃R,k is a noise

vector with zero mean and covariance = VHS,kRRVS,k, whichis equal to the covariance RR = σRIN due to unitary matrixVS,k.

Similarly, (2), can be written to

yD,k = ΛR,kΓR,kΛS,kΓS,ksID,k + ΛR,kΓR,kwR,k + wD,k(8)

where yD,k = UHR,kỹD,k, and wD,k = UHR,kw̃D,k is a noise

vector with zero mean and covariance = UHR,kRDUR,k, whichis equal to the covariance RD = σDIN due to unitary matrixUR,k.

Using (4) and (6) and the fact that the MIMO channel onthe subcarrier k is decomposed to N spatial parallel channels,(7) can be further simplified to

yR,k,n =√hS,k,n

√pID,k,nsID,k,n +

√σRwR,k,n, (9)

where yR,k,n is the received signal at R (during the firststage) over subcarrier k and spatial channel n, sID,k,n is theinformation data on the subcarrier k and the transmit antennan such that E(|sID,k,n|2) = 1, wR,k,n is the normalized noisesuch that E(|wR,k,n|2) = 1, pID,k,n is defined below (6), and√hS,k,n ∈ R is defined from (4).Similarly, (8) can be further simplified to

yD,k,n =√hR,k,n

√pR,k,n

hS,k,npID,k,n + σR

×(√

hS,k,npID,k,nsID,k,n +√σRwR,k,n

)+√σDwD,k,n, (10)

where yD,k,n is the received signal at D over subcarrier k andspatial channel n, wD,k,n is the normalized noise such thatE(|wD,k,n|2) = 1, pR,k,n is defined from (6), and

√hR,k,n ∈

R is defined from (4). In (10), ζk,n = 1hS,k,npID,k,n+σR is therelay power normalization factor.

Using (10), the signal-to-noise ratio (SNR) at the destinationis given by

SNRD =hR,k,npR,k,nhS,k,npID,k,n

(hS,k,npID,k,n + σR)(

hR,k,npR,k,nσRhS,k,npID,k,n+σR

+ σD

)=

hR,k,npR,k,nhS,k,npID,k,nhR,k,npR,k,nσR + (hS,k,npID,k,n + σR)σD

=(hS,k,n/σR)pID,k,n(hR,k,n/σD)pR,k,n

1 + (hS,k,n/σR)pID,k,n + (hR,k,n/σD)pR,k,n.

B. Energy Processing

During the second stage, E transmits a dedicated energysignal to R, while R also receives interference from its owntransmission. Let HE ∈ CN×N be the time-domain MIMOchannel matrix between E and R, i.e., [HE ]n,n̄ = g

n,n̄E , and

HLI ∈ CN×N be the time-domain self-loop interference

channel matrix at R. The received time-domain signal at Rduring the second communication phase is given by

ỹEH,i = HE s̃EH,i + HLIxR,i + w̃R,i, (11)

where• i = {1, 2, . . . ,K} denotes the time index such that

[ỹEH,1, ỹEH,2, . . . , ỹEH,K ] represents the received time-domain signal over one OFDM symbol duration,

• s̃EH,i = ΨEHsEH,i such that ΨEH ∈ CN×N is thetransmit precoding matrix at the E, sEH,i ∈ CN is thetransmitted energy signal such that E(|sEH,i|2) = 1N×1

• w̃R,i ∈ CN is the additive zero-mean Gaussian noisewith covariance RR = σRIN , σR is the noise varianceat each receive antenna at R, and

• xR,i ∈ CN is the transmitted signal by the R duringsecond communication phase, which is given by the K-point IFFT of the signal Fk (HS,ks̃ID,k + w̃R,k) (see eq.(2)), i.e.,

xR,i =1√K

K∑k=1

Fk (HS,ks̃ID,k + w̃R,k) ej 2πK (k−1)(i−1)

(12)

Let the SVD of the channel matrix HE be given by

HE = VEΛEUE

where ΛE , diag{√hE,n}Nn=1, VE and UE are unitary

matrices of dimensions and N × N , and {hE,n}Nn=1 are theeigenvalues of HEHHE . The transmit precoding matrix at E isgiven by

ΨEH = UHEΓE

where ΓE = diag{√pEH,n}Nn=1 and pEH,n is the powerallocated to transmit the energy signal over spatial channeln. Thus, the (11) of the received signal at R during secondcommunication phase can be written as

yEH,i = ΛEΓEsEH,i︸︷︷︸from energy source

+ VHEHLIxR,i︸︷︷︸SI from relay

+ VHE w̃R,i︸︷︷︸noise

, (13)

for yEH,i = VHE ỹEH,i. Given that the transmit power atR over subcarrier k and spatial channel n is pR,k,n, thepower of the transmitted signal at the relay xR,i is given by∑Kk=1

∑Nn=1 pR,k,n. Thus, the power harvested at the R due

to the self-loop interference from its own transmission (secondfactor in (13) can be expressed as

ηγLI

K∑k=1

N∑n=1

pR,k,n,

where γLI is the self-loop path gain or in other words, it isthe inverse of the path loss of the self-loop channel [19] and ηis the energy harvesting efficiency. Thus, the recycled energydue to self-loop interference depends on γLI and η. Using(13), the power harvested by the relay (combined from all thereceive antennas) during the second communication phase isgiven by

e = η

N∑n=1

(hE,npEH,n + γLI

K∑k=1

pR,k,n

). (14)

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Note that in (14), the noise factor (third factor in (13) has beenignored as it results in negligible harvested energy. In order toensure that the total transmitted power by the relay does notexceed the total harvested power, the following constraint isimposed:

K∑k=1

N∑n=1

pR,k,n ≤ e. (15)

C. Problem Formulation

By setting

ak,n =hS,k,nσR

, bk,n =hR,k,nσD

, cn =hE,nσR

, (16)

the sum-rate maximization problem at the destination can beformulated as

max(pID,pEH ,pR)

f(pID,pR)

,K∑k=1

N∑n=1

ln

(1 +

ak,npID,k,nbk,npR,k,n1 + ak,npID,k,n + bk,npR,k,n

)(17a)

s.t.

K∑k=1

N∑n=1

pID,k,n +

N∑n=1

pEH,n ≤ P, (17b)

pID,k,n ≥ 0, pEH,n ≥ 0, pR,k,n ≥ 0, (17c)K∑k=1

N∑n=1

pR,k,n ≤ ηN∑n=1

(σRcnpEH,n + γLI

K∑k=1

pR,k,n

),

(17d)

where P is the total transmit power budget, pID , {pID,k,n :k = 1, . . . ,K;n = 1, . . . , N}, and similarly, pEH , {pEH,n :n = 1, . . . , N} and pR , {pR,k,n : k = 1, . . . ,K;n =1, . . . , N}. Note that it is due to the use of OFDM andMIMO channel decomposition that the design of precodingand relaying matrices in a MIMO communication setup issimplified to that of scalar power allocation problem overmultiple subcarriers and spatial channels.

Apart from the non-concave objective function (17a), themain issue is the high dimension of (17). Since the numberK of subcarriers is up to 4096, the number of optimizationvariables in (17) can easily exceed ten thousands, making (17)large-scale nonconvex optimization problem. Our main goalto develop a method that admits a closed-form solution at eachiteration.

III. PROPOSED SOLUTIONIn this section, we propose a solution to solve the non-

convex optimization problem (17). In what follows, define

fk,n (pID,k,n, pR,k,n) ,

ln

(1+

ak,npID,k,nbk,npR,k,n1 + ak,npID,k,n + bk,npR,k,n

)(18)

and decompose it as

fk,n (pID,k,n, pR,k,n) =

ln(1 + ak,npID,k,n) + ln(1 + bk,npR,k,n)

− ln(1 + ak,npID,k,n + bk,npR,k,n). (19)

The first two terms in (19) are concave while the last termis convex, so the objective function (17a) is a d.c. (differenceof two convex) function [44]. As such in principle (17) canbe addressed by d.c. iterations (DCI) [45] as follows. Let(p

(κ)ID,p

(κ)EH ,p

(κ)R ) be a feasible point for (17) that is found

from the (κ−1)th iteration. By linearizing the last term in (19)around

(p

(κ)ID,p

(κ)R

)while keeping the first two terms (19) one

can easily obtain a lower bounding concave approximation ofthe objective function (17a) as

f(κ)DC(pID,pR) ,

K∑k=1

N∑n=1

[ln(1 + ak,npID,k,n) + ln(1 + bk,npR,k,n)

− ln(1 + ak,np(κ)ID,k,n + bk,np(κ)R,k,n)

−ak,n(pID,k,n − p(κ)ID,k,n) + bk,n(pR,k,n − p

(κ)R,k,n)

1 + ak,np(κ)ID,k,n + bk,np

(κ)R,k,n

]. (20)

DCI [45] solves the following problem of a logarithmic func-tion optimization under convex constraints at the κth iterationto generate the next feasible point (p(κ+1)ID ,p

(κ+1)EH ,p

(κ+1)R ) for

(17):

max(pID,pEH ,pR)

f(κ)DC(pID,pR) s.t. (17b), (17c), (17d).

(21)However, there is no solver of polynomial complexity for (21).Furthermore, using the following lower bounding concaveapproximation of the objective function (17a) [46]:

f(κ)QD(pID,pR) ,

K∑k=1

N∑n=1

[ak,n

1 + ak,np(κ)ID,k,n

(p

(κ)ID,k,n −

(p(κ)ID,k,n)

2

pID,k,n

)

+bk,n

1 + bk,np(κ)R,k,n

(p

(κ)R,k,n −

(p(κ)R,k,n)

2

pR,k,n

)+fk,n(p

(κ)ID,k,np

(κ)R,k,n)

−ak,n(pID,k,n − p(κ)ID,k,n) + bk,n(pR,k,n − p

(κ)R,k,n)


(κ)R,k,n

](22)

leads to solving the following convex quadratic optimizationat the κth iteration instead of (21) to generate the next feasible(p

(κ+1)ID ,p

(κ+1)EH ,p

(κ+1)R ) for (17):

max(pID,pEH ,pR)

f(κ)QD(pID,pR) s.t. (17b), (17c), (17d).

(23)Both (21) and (23) are convex but very large scale problems.The computational complexity of (23) is [47]:

O(((2KN +N)3(2KN +N + 2))

), (24)

i.e. it is more computationally tractable than (21). However,this computational complexity is still too high for practicalapplication. Even with only K = 64 subcarriers and N =4 antennas, it takes more than half an hour for (21)-basediterations and around 3 minutes for (23)-based iterations ona core-i5 machine (8 GB RAM) using MATLAB and CVXsolver to find the optimal solution.

7

Leaving aside its practical issue as mentioned in the In-troduction, the work [20] considered the design of robustnon-linear transceivers (source precoding, MIMO relaying,and receiver matrices) to minimize the mean-squared errorat the destination end in the presence of channel uncertainty.Alternating optimization, which optimizes one of these threematrices with other two held fixed, is used to address theposed optimization problem. The alternating optimization inthe source precoding matrix is still nonconvex, for whichthe DCI [45] is also revoked. It should be noted that thesolution founded by alternating optimization approach doesnot necessarily satisfy a necessary optimality condition. Suchan optimization problem as posed in [20] should be much moreefficiently addressed by the matrix optimization techniquesin [48], [49], which simultaneously optimize all the matrixvariables in each iteration, avoiding alternating optimization.

We now develop a new lower bounding concave approxi-mation for the objective function (17a) as well as a new innerconvex approximation for the polytopic constraints (17b)-(17d), which lead to a closed-form of the optimal solutionat each iteration.

As before, let (p(κ)ID,p(κ)EH ,p

(κ)R ) be a feasible point for (17)

that is found from the (κ− 1)th iteration.

A. Lower bounding concave approximation for the objectivefunction (17a) at the κth iteration

By using the inequality (45) in the appendix:

ln(1 + ak,npID,k,n) ≥ ln(1 + ak,np(κ)ID,k,n)

+ak,np

(κ)ID,k,n

1 + ak,np(κ)ID,k,n

(ln pID,k,n − ln p(κ)ID,k,n

)(25)

and

ln(1 + bk,npR,k,n) ≥ ln(1 + bk,np(κ)R,k,n)

+bk,np

(κ)R,k,n

1 + bk,np(κ)R,k,n

(ln pR,k,n − ln p(κ)R,k,n

), (26)

while by using the inequality (43) in the appendix

ln(1 + ak,npID,k,n + bk,npR,k,n) ≤ln(1 + ak,np

(κ)ID,k,n + bk,np

(κ)R,k,n)

+ak,n(pID,k,n − p(κ)ID,k,n) + bk,n(pR,k,n − p

(κ)R,k,n)


(κ)R,k,n

. (27)

Therefore,

fk,n(pID,k,n, pR,k,n) ≥f

(κ)k,n(pID,k,n, pR,k,n) ,

α(κ)k,n + β

(κ)k,n ln pID,k,n − χ

(κ)k,npID,k,n

+δ(κ)k,n ln pR,k,n − γ

(κ)k,npR,k,n, (28)

where

α(κ)k,n = ln

(1 +

ak,np(κ)ID,k,nbk,np

(κ)R,k,n


(κ)R,k,n

)

−ak,np

(κ)ID,k,n

1 + ak,np(κ)ID,k,n

ln p(κ)ID,k,n −

bk,np(κ)R,k,n

1 + bk,np(κ)R,k,n

ln p(κ)R,k,n

+1


(κ)R,k,n

×(ak,np

(κ)ID,k,n + bk,np

(κ)R,k,n)

), (29a)

β(κ)k,n =

ak,np(κ)ID,k,n

1 + ak,np(κ)ID,k,n

> 0, (29b)

χ(κ)k,n =

1


(κ)R,k,n

ak,n > 0, (29c)

δ(κ)k,n =

bk,np(κ)R,k,n

1 + bk,np(κ)R,k,n

> 0, (29d)

γ(κ)k,n =

1


(κ)R,k,n

bk,n > 0. (29e)

Note that f (κ)k,n(pID,k,n, pR,k,n) is a concave function, whichmatches with fk,n(pID,k,n, pR,k,n) at (p

(κ)ID,k,n, p

(κ)R,k,n), i.e.

fk,n(p(κ)ID,k,n, p

(κ)R,k,n) = f

(κ)k,n(p

(κ)ID,k,n, p

(κ)R,k,n). Consequently,

the function

f (κ)(pID,pR) ,K∑k=1

N∑n=1

f(κ)k,n(pID,k,n, pR,k,n)

is a lower bounding concave approximation of f(pID,pR):

f(pID,pR) ≥ f (κ)(pID,pR) ∀ (pID,pR), (30)

and matches with f(pID,pR) at (p(κ)ID,p

(κ)R ):

f(p(κ)ID,p

(κ)R ) = f

(κ)(p(κ)ID,p

(κ)R ). (31)

B. Inner approximation of polytopic constraints at the κthiteration

Observe that pEH,n appears only in the polytopic constraints(17b) and (17d). Now, note that

p2EH,n + (p(κ)EH,n)

2 − 2pEH,np(κ)EH,n = (pEH,n − p(κ)EH,n)

2

≥ 0

leading to

pEH,n ≤ 0.5(p2EH,n/p

(κ)EH,n + p

(κ)EH,n

),

we innerly approximate the polytopic constraint (17b) by theconvex quadratic constraint

K∑k=1

N∑n=1

pID,k,n + 0.5

N∑n=1

(p2EH,n

p(κ)EH,n

+ p(κ)EH,n

)≤ P. (32)

Indeed, it is obvious that any feasible point for (32) is alsofeasible for (17b).

8

C. Closed-form solution at the κth iteration

At the κth iteration, we solve the following convex opti-mization problem to generate the next iterative feasible point(p

(κ+1)ID ,p

(κ+1)EH ,p

(κ+1)R ) for (17):

max(pID,pEH ,pR)

f (κ)(pID,pR) s.t. (17d), (17c), (32). (33)

The difference between the convex problem (33) and the DCI(21) is that the concave functions ln(1 + ak,npID,k,n) andln(1 + bk,npR,k,n) in the objective function of the formerare further approximated by other concave functions, andthe polytopic constraint (17b) is innerly approximated by theconvex constraint (32). These approximations help to ease theprocess of finding a closed form solution of (33).

The Lagrangian of problem (33) is given by

L(pID,pEH ,pR, λ1, λ2) =K∑k=1

N∑n=1

f(κ)k,n(pID,k,n, pR,k,n)

− λ1(

K∑k=1

N∑n=1

pID,k,n +1

2

N∑n=1

(p2EH,n

p(κ)EH,n

+ p(κ)EH,n

)− P

)

− λ2(

(1− ηγLI)K∑k=1

N∑n=1

pR,k,n − ηN∑n=1

σRcnpEH,n

)+ νk,npID,k,n + ν̄k,npR,k,n + ν̂npEH,n,

where λ1 ≥ 0 and λ2 ≥ 0 are the Lagrange multiplierscorresponding to the constraints (32) and (17d), respectively,while νk,n ≥ 0, ν̄k,n ≥ 0, and ν̂n ≥ 0 are the Lagrange mul-tipliers corresponding to the power constraints pID,k,n ≥ 0,pR,k,n ≥ 0, and pEH,n ≥ 0, respectively, in (17c). We setthe Lagrange multipliers νk,n = 0, ν̄k,n = 0, and ν̂n = 0to satisfy the complementary slackness, νk,npID,k,n = 0,ν̄k,npR,k,n = 0, and ν̂npEH,n = 0, for all k = 1, . . . ,K,n = 1, . . . , N , in Karush-Kuhn-Tucker (KKT) conditions.

The optimal solution of (33) thus satisfies

∂L(pID,pEH ,pR, λ1, λ2)∂pEH,n

= 0

⇔ −λ1pEH,np

(κ)EH,n

+ λ2ησRcn = 0 (34)

⇔ pEH,n = λ2ησRcnp(κ)EH,n/λ1, (35)

and

∂L(pID,pEH ,pR, λ1, λ2)∂pID,k,n

= 0

⇔β

(κ)k,n

pID,k,n− χ(κ)k,n − λ1 = 0

⇔ pID,k,n =β

(κ)k,n

χ(κ)k,n + λ1

, (36)

and

∂L(pID,pEH ,pR, λ1, λ2)∂pR,k,n

= 0

⇔δ

(κ)k,n

pR,k,n− γ(κ)k,n − λ2(1− ηγLI) = 0

⇔ pR,k,n =δ

(κ)k,n

γ(κ)k,n + λ2(1− ηγLI)

, (37)

where (35) shows that λ1 should be strictly positive. To satisfythe complementary slackness under KKT conditions for theconstraints (32) and (17d), λ1 > 0 and λ2 > 0 are chosensuch that the constraints (32) and (17d) are met with equality.Thus,

K∑k=1

N∑n=1

pID,k,n + 0.5

N∑n=1

(p2EH,n

p(κ)EH,n

+ p(κ)EH,n

)≤ P ⇔

K∑k=1

N∑n=1

β(κ)k,n

χ(κ)k,n + λ1

+ 0.5

N∑n=1

p(κ)EH,n

(λ22η

2σ2Rc2n

λ21+ 1

)= P ,

(38)

and

(1− ηγLI)K∑k=1

N∑n=1

pR,k,n = η

N∑n=1

σRcnpEH,n

⇔(1− ηγLI)K∑k=1

N∑n=1

δ(κ)k,n


= η

N∑n=1

λ2σRcnησRcnp(κ)EH,n

λ1. (39)

Simultaneously solving (38) and (39) will yield the optimalvalues of λ2 and λ1. From (39), we can get λ1 expressed asa function of λ2:

λ1 =ηλ2

1− ηγLI

∑Nn=1 ησ

2Rc

2np

(κ)EH,n

K∑k=1

N∑n=1

(δ

(κ)k,n


) . (40)

Substituting this value of λ1 into (38), we can solve for λ2 byusing bisection search and finally get λ1 from (40).

Remark 1. In (25) and (26), we approximate concavefunctions by other concave functions, which lead to the simpleclosed forms (36) and (37) and consequently, the analyticalformula (40) for determining the Lagrange multiplier λ1.

Remark 2. It can be easily seen that using the polytopicconstraint (17b) cannot reveal a closed-form of pEH,n becausethe derivative of the corresponding Lagrangian with respect topEH,n as in (34) is independent of pEH,n. That is why weneed the inner approximation (32) for (17b) that leads to theclosed-form (35) for pEH,n.

9

Algorithm 1 Resource Allocation Algorithm for Rate Maxi-mization Problem (17).Initialization: Set κ := 0. Take the initial feasible point

(p(0)EH ,p

(0)ID,p

(0)R ) by using (41).

1: repeat2: Solve for λ1 and λ2 using (38) and (40) (the details are

given below (39)).3: Update

(p

(κ+1)EH ,p

(κ+1)ID ,p

(κ+1)R

)using the closed-

forms (35), (36), and (37).4: Set κ := κ+ 1.5: until Convergence

D. Algorithm and convergence

The following feasible point (p(0)EH ,p(0)ID,p

(0)R ) is taken as

the initial point:

p(0)EH,n =

0.5P

N(41a)

p(0)ID,k,n =

0.5P

KN(41b)

p(0)R,k,n =

ηhE,np(0)EH,n

K(1− ηγLI). (41c)

It can be easily seen that all the constraints (17b), (17c), and(17d) are met. Algorithm 1 outlines the steps to solve the ratemaximization problem (17). The computational complexity ofAlgorithm 1 is

O(KN), (42)

as it just involves 5N , 2KN , and 5KN addition or multi-plication operations to update

(p

(κ+1)EH ,p

(κ+1)ID ,p

(κ+1)R

)using

(35), (36), and (37), respectively, over the κth iteration, whilethe algorithm converges within few iteration (around 15− 20)and we are dealing with K = 1024 subcarriers.

In the following, we show that Algorithm 1 generates asequence {p(κ)EH ,p

(κ)ID,p

(κ)R } of improved feasible points for

(17). It is clear from (31) that

f(p(κ)ID,p

(κ)R ) = f

(κ)(p(κ)ID,p

(κ)R ).

Furthermore,

f (κ)(p(κ+1)ID ,p

(κ+1)R ) > f

(κ)(p(κ)ID,p

(κ)R )

as far as(p

(κ+1)EH ,p

(κ+1)ID ,p

(κ+1)R

)6=(p

(κ)EH ,p

(κ)ID,p

(κ)R

),

because (p(κ+1)EH ,p(κ+1)ID ,p

(κ+1)R ) is the optimal solution of

(33) while (p(κ)EH ,p(κ)ID,p

(κ)R ) is a feasible point for (33).

Therefore, by (30),

f(p(κ+1)ID ,p

(κ+1)R ) ≥ f (κ)(p

(κ+1)ID ,p

(κ+1)R )

> f (κ)(p(κ)ID,p

(κ)R )

= f(p(κ)ID,p

(κ)R ),

i.e. (p(κ+1)EH ,p(κ+1)ID ,p

(κ+1)R ) is a better feasible point

(p(κ)EH ,p

(κ)ID,p

(κ)R ) for the original nonconvex optimization

problem (17). As such, Algorithm 1 converges at least to alocally optimal solution of (17) [13].

26 28 30 32total power budget, P (dBm)

0

5

10

15

20

avera

ge s

pectr

al effic

iency (

bps/H

z)

Proposed Alg. 1Equal Power Allocation

Fig. 3. Average spectral efficiency versus the total power budget P .

IV. NUMERICAL RESULTS

This section evaluates the performance of the proposed Alg.1 and also analyses its computational efficiency.

A. The simulation setup

We assume N = 4 antennas at each node. To modellarge scale fading, each spatial L = 16-tap source-to-relaychannel, gn,n̄S , between any transmit antenna n and receiveantenna n̄, follows the path loss model 30 + 10β log10(dSR)and each spatial L = 16-tap R-to-S channel, gn,n̄R , betweenany transmit antenna n and receive antenna n̄, follows thepath loss model 30 + 10β log10(dRD), where the path lossexponent β = 3, the distance from S-to-R, dSR, is setto 100 meters and unless specified otherwise, the distancefrom R-to-D, dRD, is also set to 100 meters. On the otherhand, the channel between E and R, gn,n̄E follows the pathloss model 30 + 10βE log10(dER), where path loss exponentβE = 2, and the distance between E and R, dER is set to10 meters. Note that different values for path-loss exponentshave been proposed for different type of channel conditionsin the literature too [50], [51]. To ensure meaningful wirelesspower transfer to energy harvesting node, smaller distance withline-of-sight component and hence, smaller value of path-lossexponent are adopted in the literature [51].

To model small scale fading, gn,n̄S and gn,n̄R follow Rayleigh

fading, while gn,n̄E is assumed to be Rician distributed withRician factor K = 6 dB. To simulate the effect of frequencyselectivity in each spatial multipath channel gn,n̄S and g

n,n̄R ,

we assume an exponential power delay profile with root-mean-square delay spread of σRMS = 3Ts, for the symbol time Ts =1/B. In addition, the spatial correlation among the MIMOchannels are modeled according to Case B of the 3 GPP I-METRA MIMO channel model [35].

The time-domain multipath channels gn,n̄S and gn,n̄R are

converted to frequency domain via a K-point FFT. Unlessspecified otherwise, the number of OFDM subcarriers is con-sidered to be K = 1024 and the total transmit power budget

10

80 100 120 140relay-to-destination distance dRD (meters)

0

2

4

6

8

10

12

14avera

ge s

pectr

al effic

iency (

bps/H

z)


Fig. 5. Average spectral efficiency versus the relay-to-destination distancedRD .

0 2 4 6 8 10Rician factor, K (dB)

0

2

4

6

8

10

12

avera

ge s

pectr

al effic

iency (

bps/H

z)


Fig. 6. Average spectral efficiency versus the Rician factor K of the timedomain channel between energy source and relay.

-18 -14 -10 -6 -2self-loop path gain (SI), 10 log10(γLI) (dB)

0

2

4

6

8

10

12

ave

rag

e s

pe

ctr

al e

ffic

ien

cy (

bp

s/H

z)

Proposed Alg. 1No self-energy recylcing (γLI = 0)Equal Power Allocation

improvement in the spectral efficiencydue to self-energy recycling

Fig. 4. Average spectral efficiency versus the self-loop path gain (SI) γLI(evaluated in dB).

P is set to 26 dBm. The system bandwidth is set to B = 1-MHz and the subcarrier bandwidth is B/K. This subcarrierbandwidth is quite smaller than the coherence bandwidth of0.02/σRMS to ensure flat fading over each subcarrier. In eachsub-channel, the power spectral density (noise per unit BW) ofadditive white Gaussian noise, σRB/K and

σDB/K at each antenna

is set to −174 dBm/Hz. Therefore, the total noise powerover each subcarrier, e.g., at the relay-receiver, can be foundas σR = 10(−174+30)/10 × (B/K). The correlation betweennoise samples from different antennas is set to 0.2. The carrierfrequency is assumed to be 1 GHz. Unless specified otherwise,the self-loop path gain or SI is set to γLI = −10 dB [20], [28],which is justified since there is no need of self-interferenceattenuation at R for energy recycle at R. The energy harvestingefficiency η is set to 0.5 [17, Table III]. Note that using theaforementioned path loss model and practical simulation setup,the power of the received signal at the relay during second

communication phase passes the threshold minimum powerrequirement (−21 dBm with 13 nm CMOS technology [17])to carry out EH at the relay.

The tolerance level for the convergence of Alg. 1 is set to0.001 and to calculate the average spectral efficiency, we run1000 independent simulations and average the results to getthe final figures. In the simulation results, we compare theperformance of the proposed Alg. 1 with that obtained by theequal power allocation where the latter assumes the solutionobtained in (41) because in (41), same power is allocatedover each spatial channel and subcarrier at the energy andinformation source nodes.

B. Achievable spectral efficiency performance

Fig. 3 plots the average spectral efficiency for differentvalues of power P . The optimization by the proposed Alg. 1achieves significant gain in the spectral efficiency compared tothe equal power allocation approach. As expected, increasingthe power budget increases the achievable spectral efficiency.

Fig. 4 plots the average spectral efficiency versus the self-loop path gain or SI, γLI . It is interesting to note thatunlike in the non-EH based systems, where SI hurts thesimultaneous signal transmission and reception, the SI in Fig.4 enhances the achievable spectral efficiency. For example, itcan be seen from Fig. 4 that the improvement in the spectralefficiency due to self-energy recycling is {0.14, 0.44} bps/Hzat γLI = {−10,−6} dB. Indeed, increase in the self-looppath gain results in more harvested energy from the loop SI,which is added to the available transmission power from Rand that would improve the spectral efficiency at S. It is alsoclear from Fig. 4 that the optimization by the proposed Alg.1 achieves significant gain in the spectral efficiency comparedto the equal power allocation approach.

Fig. 5 plots the average spectral efficiency versus the relay-to-destination distance dRD. As expected, the increase in therelay-to-destination distance decreases the achievable spectral

11

10-3

10-2

10-1

normalized variance (ǫo) of error in CSI

0

2

4

6

8

10

12avera

ge s

pectr

al effic

iency (

bps/H

z)

Proposed Alg. 1 (ǫo = 0)Proposed Alg. 1 with imperfect CSIEqual power Allocation (ǫo = 0)

decrease in the spectral efficiency dueto imperfect channel estimation

Fig. 7. Average spectral efficiency versus the normalized variance ofchannel estimation error.

0 5 10 15 20 25iterations

6

7

8

9

10

11

12

13

14

spectr

al effic

iency (

bps/H

z)

Proposed Alg. 1, P = 26 dBm



Fig. 8. The convergence of Proposed Alg. 1.

TABLE IICOMPUTATIONAL TIME OF PROPOSED ALG. 1 VERSUS (21) AND (23) BASED ITERATIONS.

Solving Methodology K = 16 K = 64 K = 256 K = 1024 K = 4096Proposed. Alg. 1 0.01 sec 0.012 sec. 0.017 sec. 0.046 sec. 0.1602 sec.

(21) based iterations 8 mins 30 mins. - - -(23) based iterations 1.2 mins 3.1 mins 12.5 75 mins. -

efficiency due to increase in the path-loss. Fig. 6 plots theaverage spectral efficiency versus the Rician factor K of thetime-domain channel gn,n̄E . Fig. 6 shows that increase in theRician factor results in increase in the achievable spectralefficiency of the proposed Alg. 1 only. The equal powerallocation fails to show improvement in the spectral efficiency.This is because though the E-to-R channel gets strongerby increasing its Rician factor which increases the R-to-Dpower under equal power allocation approach (see (41c)), suchapproach cannot guarantee an improved spectral efficiency asit allocates same power, no matter small or large, over allsubcarriers of the R-to-D channel and thus, does not use theresources intelligently. It is also clear from Figs. 5-6 thatthe optimization by the proposed Alg. 1 achieves significantgain in the spectral efficiency compared to the equal powerallocation approach. This significant gain is due to nature ofthe OFDM channel, where equal power allocation over allsubcarriers suffers a lot compared to the optimal resourceallocation.

Fig. 7 shows the impact of erroneous channel state infor-mation (CSI) on the achievable spectral efficiency. The resultsare plotted with respect to the normalized variance, �o, of thechannel estimation error. By normalized variance, we meanthat the variance is normalized with respect to the squaredchannel magnitude, e.g., if δn,n̄S,` =

[gn,n̄S

]`−[ĝn,n̄S

]`

is thechannel estimation error for the `-th tap of the S-to-R channeland

[ĝn,n̄S

]`

is its estimate, then δn,n̄S,` ∼ N(

0, �o∣∣[gn,n̄S ]`∣∣2).

Similarly, the channel estimation error for the R-to-D channel,gn,n̄R , and the E-to-R channel, g

n,n̄E , subject to the same

normalized variance �o. The results shows that the spectral

efficiency is very mildly affected due to the erroneous channelestimation which shows the robustness of the proposed algo-rithm even in the presence of imperfect channel estimation.

C. Computational Complexity and Performance Comparison

Fig. 8 shows the convergence of the proposed Alg. 1for different values of the power budget P for a partic-ular simulation. We can see that the algorithm convergesquickly after 20-25 iterations. On average, Alg. 1 requires{21.0, 19.46, 18.26, 17.14} iterations before convergence forP = {26, 28, 30, 32} dBm. Fig. 8 shows that the requirednumber of iterations decreases with an increase in the powerbudget, which means that the Alg. 1 quickly figures out theoptimal resource allocation in the presence of a relatively largepower budget.

The closed-form solution proposed in Alg. 1 with thelinear computational complexity (42), which is less than(2KN +N)3 times compared with the polynomial computa-tional complexity (24) for computing (23), is computationallyvery efficient. The computational time of the proposed Alg.1, which is calculated on a 2.7 GHz Intel core-i5 machinewith 8 GB RAM, is shown in Table II for various systemimplementations (different numbers of subcarriers). In orderto simulate small-scale system with K = 16 subcarriersor K = 64 subcarriers, we generate gn,n̄S and g

n,n̄R with

an exponential power delay profile having root-mean-squaredelay spread of σRMS = Ts, which results in L = 4-tapchannel. Table II shows that the Alg. 1 quickly locates theoptimal solution (in less than 110 seconds). The computational

12

26 28 30 32total power budget, P (dBm)

0

5

10

15

20avera

ge s

pectr

al effic

iency (

bps/H

z)

Proposed Alg. 1(21)/(23) based iterationsEqual Power Allocation

Fig. 9. Comparison of average spectral efficiency versus the total powerbudget P for different algorithms.

time increases very slightly as the scale of the problem grows(as the number of subcarriers K increases). Note that the (23)-based or (21)-based iterations already consume a lot of timeeven for small-scale problems (with K = 64 or K = 256).

By Fig. 9, we compare the average spectral efficiency ofAlg. 1 with that invoking (21) or (23). To ensure reasonablesimulation time for solving (21) or (23) using CVX, weconsider only a small-scale problem for K = 64. It can be seenfrom Fig. 9 that the proposed Alg. 1 achieves similar spectralefficiency performance as that obtained by invoking (21) or(23). However, Table II particularly indicates that the proposedAlg. 1 is almost 150, 000 or 15, 500 times computationallyfaster than that invoking (21) or (23), respectively.

V. CONCLUSIONS

This paper has considered a MIMO-OFDM based wireless-powered relaying communication, in which the cooperativerelay forwards the source information to the destination whileharvesting energy not only from wireless signals from adedicated energy source but also through energy recyclingfrom its own transmission. The high values of residual self-interference are helpful for wireless-powering the relay node.The objective is to maximize the spectral efficiency of sucha system by power allocation over each subcarrier and eachtransmit antenna. This optimization problem, which is largescaled in the presence of a large number of subcarriers, is verycomputationally challenging. The paper has proposed a newpath-following algorithm, which is practical for large-scaleoptimization as it requires only a few closed-form calculationsof linear computational complexity. The provided simula-tions under a practical assumptions show promising resultsby achieving high spectral efficiency, which is quite highcompared to the simple “equal power allocation” approach.

APPENDIX: BASIC INEQUALITIES

As the function γ(x) = ln(1 +x) is concave in the domaindom(γ) = {x > 0}, it is true that [44]

ln(1 + x) ≤ γ(x̄) + ∂γ(x̄)∂x

(x− x̄)

= ln(1 + x̄) +x− x̄1 + x̄

∀ x > 0, x̄ > 0.(43)

On the other hand, function β(y) , ln(1 + ey) is convexin the domain dom(β) = {y > 0} because ∂2f(y)/∂y2 =ey/(1 + ey)2 > 0 ∀ y ∈ dom(β). Therefore [44]

ln(1 + ey) ≥ β(ȳ) + ∂β(ȳ)∂y

(y − ȳ)

= ln(1 + eȳ) +eȳ

1 + eȳ(y − ȳ). (44)

Substituting x = ey and x̄ = eȳ into (44) yields the followinginequality

ln(1+x) ≥ ln(1+ x̄)+ x̄1 + x̄

(lnx− ln x̄) ∀ x > 0, x̄ > 0.(45)

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Ali Arshad Nasir (S’09-M’13) is an Assistant Pro-fessor in the Department of Electrical Engineering,King Fahd University of Petroleum and Minerals(KFUPM), Dhahran, KSA. Previously, he held theposition of Assistant Professor in the School of Elec-trical Engineering and Computer Science (SEECS)at National University of Sciences & Technology(NUST), Paksitan, from 2015-2016. He received hisPh.D. in telecommunications engineering from theAustralian National University (ANU), Australia in2013 and worked there as a Research Fellow from

2012-2015. His research interests are in the area of signal processing inwireless communication systems. He is an Associate Editor for IEEE CanadianJournal of Electrical and Computer Engineering.

14

Hoang Duong Tuan received the Diploma (Hons.)and Ph.D. degrees in applied mathematics fromOdessa State University, Ukraine, in 1987 and 1991,respectively. He spent nine academic years in Japanas an Assistant Professor in the Department ofElectronic-Mechanical Engineering, Nagoya Univer-sity, from 1994 to 1999, and then as an AssociateProfessor in the Department of Electrical and Com-puter Engineering, Toyota Technological Institute,Nagoya, from 1999 to 2003. He was a Professor withthe School of Electrical Engineering and Telecom-

munications, University of New South Wales, from 2003 to 2011. He iscurrently a Professor with the School of Electrical and Data Engineering,University of Technology Sydney. He has been involved in research with theareas of optimization, control, signal processing, wireless communication, andbiomedical engineering for more than 20 years.

Trung Q. Duong (S’05, M’12, SM’13) received hisPh.D. degree in Telecommunications Systems fromBlekinge Institute of Technology (BTH), Swedenin 2012. Currently, he is with Queen’s UniversityBelfast (UK), where he was a Lecturer (AssistantProfessor) from 2013 to 2017 and a Reader (As-sociate Professor) from 2018. His current researchinterests include Internet of Things (IoT), wirelesscommunications, molecular communications, andsignal processing. He is the author or co-author of290 technical papers published in scientific journals

(165 articles) and presented at international conferences (125 papers).Dr. Duong currently serves as an Editor for the IEEE TRANSACTIONS

ON WIRELESS COMMUNICATIONS, IEEE TRANSACTIONS ON COMMUNI-CATIONS, IET COMMUNICATIONS, and a Lead Senior Editor for IEEECOMMUNICATIONS LETTERS. He was awarded the Best Paper Award atthe IEEE Vehicular Technology Conference (VTC-Spring) in 2013, IEEEInternational Conference on Communications (ICC) 2014, IEEE GlobalCommunications Conference (GLOBECOM) 2016, and IEEE Digital SignalProcessing Conference (DSP) 2017. He is the recipient of prestigious RoyalAcademy of Engineering Research Fellowship (2016-2021) and has won aprestigious Newton Prize 2017.

H. Vincent Poor (S’72, M’77, SM’82, F’87) re-ceived the Ph.D. degree in EECS from PrincetonUniversity in 1977. From 1977 until 1990, he wason the faculty of the University of Illinois at Urbana-Champaign. Since 1990 he has been on the facultyat Princeton, where he is currently the MichaelHenry Strater University Professor of Electrical En-gineering. During 2006 to 2016, he served as Deanof Princetons School of Engineering and AppliedScience. He has also held visiting appointments atseveral other universities, including most recently at

Berkeley and Cambridge. His research interests are in the areas of informationtheory and signal processing, and their applications in wireless networks,energy systems and related fields. Among his publications in these areas isthe recent book Multiple Access Techniques for 5G Wireless Networks andBeyond. (Springer, 2019).

Dr. Poor is a member of the National Academy of Engineering and theNational Academy of Sciences, and is a foreign member of the ChineseAcademy of Sciences, the Royal Society, and other national and internationalacademies. He received the Marconi and Armstrong Awards of the IEEECommunications Society in 2007 and 2009, respectively. Recent recognitionof his work includes the 2017 IEEE Alexander Graham Bell Medal, the 2019ASEE Benjamin Garver Lamme Award, a D.Sc. honoris causa from SyracuseUniversity awarded in 2017, and a D.Eng. honoris causa from the Universityof Waterloo awarded in 2019.