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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
Published in:IEEE Transactions on Communications
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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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2
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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4
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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5
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)
-
6
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)
1 + ak,np(κ)ID,k,n + bk,np
(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)
1 + ak,np(κ)ID,k,n + bk,np
(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
1 + ak,np(κ)ID,k,n + bk,np
(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
1 + ak,np(κ)ID,k,n + bk,np
(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
1 + ak,np(κ)ID,k,n + bk,np
(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
1 + ak,np(κ)ID,k,n + bk,np
(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
γ(κ)k,n + λ2(1− ηγLI)
= 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
γ(κ)k,n + λ2(1− ηγLI)
) . (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)
Proposed Alg. 1Equal Power Allocation
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)
Proposed Alg. 1Equal Power Allocation
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
Proposed Alg. 1, P = 28 dBm
Proposed Alg. 1, P = 30 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.