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MIMO Energy Harvesting in Full-Duplex Multi-User Networks
Tam, H. H. M., Tuan, H. D., Nasir, A. A., Duong, Q., & Poor, H. V. (2017). MIMO Energy Harvesting in Full-Duplex Multi-User Networks. IEEE Transactions on Wireless Communications, 16(5), 3282-3297.https://doi.org/10.1109/TWC.2017.2679055
Published in:IEEE Transactions on Wireless Communications
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Download date:13. Aug. 2021
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MIMO Energy Harvesting in Full-DuplexMulti-user Networks
H. H. M. Tam, H. D. Tuan, A. A. Nasir, T. Q. Duong and H. V. Poor
Abstract—The paper aims at the efficient design of precodingmatrices for the sum throughput maximization under throughputQoS constraints and energy harvesting (EH) constraints forenergy-constrained devices in a full-duplex (FD) multicell multi-user multiple-input-multiple-output (MU-MIMO) network. Bothtime splitting (TS) and power splitting (PS) are consideredto ensure practical EH and information decoding (ID). Theseproblems are quite complex due to highly non-concave objec-tives and nonconvex constraints. Especially, with TS, which isimplementation-wise quite simple, the problem is even morechallenging because time splitting variable is not only coupledwith the downlink (DL) throughput function but also coupledwith the self-interference (SI) in the uplink (UL) throughputfunction. New path-following algorithms are developed for theirsolutions, which just require a single convex quadratic programfor each iteration and ensure fast convergence too. Finally, theFD EH maximization problem under throughput QoS constraintsin TS is also considered. The performance of the proposedalgorithms is also compared with that of the modified problemsassuming half-duplex systems. In the end, the merit of theproposed algorithms is shown through extensive simulations.
Index Terms—Full-duplexing transceiver, energy harvesting,information precoder, energy precoder, path-following algorithm,matrix inequality
I. INTRODUCTION
Recently, wireless energy harvesting (i.e. energy constraineddevices scavenge energy from the surrounding RF signals)is gaining more and more attraction from both industry andacademia [1], [2]. Since the amount of energy opportunis-tically harvested from the ambient/natural energy sources isuncertain and cannot be controlled, base stations (BSs) insmall-cell networks can be configured to become dedicatedand reliable wireless energy sources [3]. The small cell sizenot only gives the benefit of efficient resource reuse acrossa geographic area [4] but also provides an adequate amountof RF energy to battery powered user equipments (UEs) for
This work was supported in part by the Australian Research CouncilsDiscovery Projects under Project DP130104617, in part by the U.K. RoyalAcademy of Engineering Research Fellowship under Grant RF1415/14/22and U.K. Engineering and Physical Sciences Research Council under GrantEP/P019374/1, and in part by the U.S. National Science Foundation underGrants CNS-1456793 and ECCS-1647198.
Ho Huu Minh Tam and Hoang Duong Tuan are with the Faculty ofEngineering and Information Technology, University of Technology Sydney,Broadway, NSW 2007, Australia (email: [email protected],[email protected]).
Ali Arshad Nasir is with the Department of Electrical Engineering, KingFahd University of Petroleum and Minerals (KFUPM), Dhahran, 31261, KSA(Email: [email protected]).
Trung Quang Duong is with Queen’s University Belfast, Belfast BT7 1NN,U.K. (Email: [email protected])
H. Vincent Poor is with the Department of Electrical Engineering, PrincetonUniversity, Princeton, NJ 08544 USA (e-mail: [email protected]).
practical applications [1], [2], [5] due to the close BS-UEproximity. In order to transfer both energy and informationby the same communication channel, UEs are equipped withboth information decoding receiver and energy harvestingreceiver. Since the received signal cannot be used for energyharvesting after being decoded, there are two available im-plementations for wireless energy harvesting and informationdecoding: (i) receive power splitting in which a receiver splitsthe received signal into two streams of different power fordecoding information and harvesting energy separately and(ii) transmit time splitting to enable the receiver to decodeinformation for a portion of a time frame and harvest energyfor the rest. Beamforming can be applied to focus the RFsignal to energy harvesting receiver or enhance throughput atinformation decoding receiver [5].
Most of the previous works (see e.g. [6], [7] and referencestherein) only focus on beamforming power optimization sub-ject to ID throughput and EH constraints with PS in multi-input single-output (MISO) networks. The ID throughputconstraints are equivalent to signal-to-interference-plus-noiseratio (SINR) constraints, which are indefinite quadratic inbeamforming vectors. The harvested energy constraints arealso indefinite quadratic constraints. Thus, [6], [7] used semi-definite relaxation (SDR) to relax such indefinite quadraticoptimization problems to semi-definite programs (SDP) bydropping the matrix rank-one constraints on the outer productsof beamforming vectors. The variable dimension of SDPis explosively large, and the beamforming vectors that arerecovered based on the matrix solution of SDR perform poorly[8]. Moreover, SDR cannot be applied to throughput or EHmaximization as the problems resultant by SDR are still highlynonconvex. Only recently there was an effective developmentto address these problems in [9] and [10].
Considering multi-input multi-output (MIMO) interferencechannels, information throughput and harvested energy, i.e.,rate-energy (R-E) trade-off, was investigated in [11] and [12],assuming that any UE either acts as an ID receiver or an EHreceiver. In case that UEs can operate both as an ID receiverand EH receiver (namely co-located cases), the R-E region ofpoint-to-point MIMO channel was studied in [13]. Note thatin MIMO networks, the information throughput function isinvolved with the determinant operation of a matrix and canno longer be expressed in the form of SINR. Consequently,the throughput constraints are always very challenging inprecoding signals. [14], [15] used zero-forcing or interference-alignment to cancel all interferences, making the throughputfunctions concave in the signal covariance. The covarianceoptimization becomes convex but it is still computationally
2
difficult with no available algorithm of polynomial time.Moreover, there is no known method to recover the pre-coder matrices from the signal covariance. Only recently, theMIMO throughput function optimization has been successfullyaddressed for non-EH system in our previous work via asuccessive convex quadratic programming [16]. The result of[16] can be adapted to MIMO networks that employ EH byPS approach. However, there is almost no serious research forthe systems employing TS in MIMO networks. Though TS-based system is practically easier to implement, the relatedformulated problem is quite complex because the throughputfunction in such case is coupled with the TS variable thatdefines the portion of time slot dedicated to EH and ID.This renders the aforementioned precoder design [14]–[16]for PS inapplicable. To the best of our knowledge, both thethroughput maximization problem and the harvested energymaximization problem with TS are still very open.
All aforementioned works only assume that UEs only har-vest energy arriving from BSs’ downlink (DL) transmission.In reality, UEs can also opportunistically harvest energy fromother UEs’ signals during their uplink (UL) transmission.Furthermore, by allowing the BSs to simultaneously transmitand receive information, both the spectral efficiency and theamount of transferred energy will be improved. With the recentadvances in antenna design and RF circuits in reducing self-interference (SI) [17]–[20], which is the interference from aBS’s DL transmission to its UL receiver, the full duplex (FD)technology is recently proposed as one of the key transceivingtechniques for the fifth generation (5G) networks [20]–[24].In this paper, we are interested in a network in which eachFD multi-antenna BS simultaneously serves a group of ULUEs (ULUs) and a group of DL UEs (DLUs). In the sametime, the BS also transfers energy to DLUs via TS or PS.FD transmission introduces even more interferences into thenetwork by adding not only SI but also the interference fromUL users (ULUs) toward downlink users (DLUs) and theinterference from DL transmission of other BSs. Consequently,the UL and DL precoders are coupled in both DL andUL throughput functions, respectively, which makes the op-timization problems for UL transmission and DL transmissioninseparable.
In literature, [14], [25], [26] proposed covariance matricesdesign in (non-EH) FD MU-MIMO networks using D.C.iterations [27], which are still very computationally demandingas they require log-determinant function optimizations as men-tioned above. Our previous work [16] has recently proposed aframework to directly find the optimal precoding matrices forthe sum throughput maximization under throughput constraintsin FD MU-MIMO multi-cell networks, which requires only aconvex quadratic program of moderate size in each iterationand thus is very computationally efficient.
In this paper, we propose the design of efficient precodingmatrices for the network sum throughput maximization underQoS in terms of MIMO throughput constraints and EH con-straints in an FD EH-enabled multicell MU-MIMO network.Both PS and TS are considered for the precoder designs andcalled by PS problem and TS problem, respectively. They arequite challenging computationally due to nonconcave objective
function and nonconvex constraints. However, we will seethat the PS problem can be efficiently addressed by adaptingthe algorithm of [16]. On the other hand, the TS problem ismuch more challenging because the TS variable ααα is not onlycoupled with the DL throughput function but also coupledwith the SI in the UL throughput function. It is nontrivial toextend [16] to solve the problem for the TS problem. Towardthis end, we develop a new inner approximation of the originalproblem and solve the problem by a path-following algorithm.Finally, we also consider the FD EH maximization problemwith throughput QoS constraints with TS. This problem alsohas a nonconvex objective function and nonconvex constraintsand will be addressed by applying an approach similar to thatof proposed for the TS problem.
The rest of this paper is organized as follows: SectionII presents the system model the SCP algorithm of the PSproblem. The main contribution of the paper is Section IIIand Section IV, which develop algorithms for the TS problemand FD EH maximization problem. Section V evaluates theperformance of our devised solutions by numerical examples.Finally, Section VI concludes the paper.
Notation. All variables are boldfaced. In denotes the identitymatrix of size n×n. The notation (.)H stands for the Hermitiantranspose. |A| denotes the determinant of a square matrix Aand 〈A〉 denotes the trace of a matrix A. (A)2 is Hermitiansymmetric positive definite AAH . The inner product 〈X,Y 〉 isdefined as 〈XHY 〉 and therefore the Frobenius squared normof a matrix X is ||X||2 = 〈(X)2〉. The notation A � B(A � B, respectively) means that A − B is a positivesemidefinite (definite, respectively) matrix. E[.] denotes theexpectation operator and <{.} denotes the real part of acomplex number.
The following concept of function approximation [28] playsan important in our development.
Definition. A function f is called a (global) minorant of afunction f at a point x in the definition domain dom(f) of fif f(x) = f(x) and f(x) ≥ f(x) ∀ x ∈ dom(f).
The following result [16] is used.Theorem 1: For function f(V,Y) = ln |In + VHY−1V|
in matrix variable V ∈ Cn×m and positive definite matrixvariable Y ∈ Cm×m, the following quadratic function is itsminorant at (V , Y )
f(V,Y) = a+ 2<{〈A,V〉} − 〈B,VVH + Y〉,
where 0 > a , f(V , Y ) − 〈V H Y −1V 〉, A = Y −1V and0 � B = Y −1 − (Y + V V H)−1.
II. EH-ENABLED FD MU-MIMO NETWORKS
We consider an MU-MIMO EH-enable network consistingof I cells. In cell i ∈ {1, . . . , I}, a group of D DLUs in thedownlink (DL) channel and a group of U ULUs in uplink(UL) channel are served by a BS i as illustrated in Fig. 1.Each BS operates in the FD mode and is equipped withN , N1 + N2 antennas, where N1 antennas are used totransmit and the remaining N2 antennas to receive signals.In cell i, DLU (i, jD) and ULU (i, jU) operate in the HDmode and each is equipped with Nr antennas. In the DL,
3
(1,1D)
(1,2D)
(1,1U)(2,1D)
(2,2U)(2,1U)
(3,1D)
(3,1U)Base stationDLU
ULUIntended signalInterference
BS 1
BS 2
BS 3
SI
SI
SI
ITI1
ITI2
ITI2
ITI2
ITI2
ITI3
ITI3 ITI3
ITI3
ITI1
ITI1
ITI1
Fig. 1. Interference scenario in an FD multicell network, where SI denotesthe self-interference and ITIi denotes the interference from the BS and ULUsof cell i.
let si,jD ∈ Cd1 be the symbol intended for DLU (i, jD)where E
[si,jD(si,jD)H
]= Id1 , d1 is the number of concurrent
data streams and d1 ≤ min{N1, Nr}. The vector of symbolssi,jD is precoded and transmitted to DLU (i, jD) through theprecoding matrix Vi,jD ∈ CN1×d1 . Analogously, in the UL,si,jU ∈ Cd2 is the information symbols sent by ULU (i, jU)and is precoded by the precoding matrix Vi,jU ∈ CNr×d2 ,where E
[si,jU(si,jU)H
]= Id2 , d2 is the number of concurrent
data streams and d2 ≤ min{N2, Nr}. For notational conve-nience, let us define
I , {1, 2, . . . , I}; D , {1D, 2D, . . . , DD};U , {1U, 2U, . . . , UU};S1 , I × D; S2 , I × U ;
VD = [Vi,jD ](i,jD)∈S1 ;VU = [Vi,jU ](i,jU)∈S2 ;
V , [VD VU];
In the DL channel, the received signal at DLU (i, jD) isexpressed as:
yi,jD ,Hi,i,jDVi,jDsi,jD︸ ︷︷ ︸desired signal
+∑
(m,`D)∈S1\(i,jD)
Hm,i,jDVm,`Dsm,`D︸ ︷︷ ︸DL interference
+∑`U∈U
Hi,jD,`UVi,`Usi,`U︸ ︷︷ ︸UL intracell interference
+ni,jD , (1)
where Hm,i,jD ∈ CNr×N1 and Hi,jD,`U ∈ CNr×Nr arethe channel matrices from BS m to DLU (i, jD) and fromULU (i, `U) to DLU (i, jD), respectively. Also, ni,jD is theadditive white circularly symmetric complex Gaussian noisewith variance σ2
D. In this work, the UL intercell interferenceis neglected since it is very small compared to the DL intercellinterference due to the much smaller transmit power of ULUs.Nevertheless, it can be incorporated easily in our formulation.
Assuming that DLUs are equipped by both devices for IDand EH, the power splitting technique is applied at each DLUto simultaneously conduct information decoding and energyharvesting. The power splitter divides the received signal yi,jDinto two parts in the proportion of αααi,jD : (1 − αααi,jD) whereαααi,jD ∈ (0, 1) is termed as the PS ratio for DLU (i, jD). Inparticular, the signal split to the ID receiver of DLU (i, jD) isgiven by √
αααi,jDyi,jD + zci,jD , (2)
where each r-th element of zci,jD (i.e. |zci,jD,r|2 = σ2
c , r =1, .., Nr) is additional noise introduced by the ID receivercircuity. An EH receiver processes the second part of the splitsignal
√1−αααi,jDyi,jD for the harvested energy√
ζi,jD(1−αααi,jD)yi,jD ,
where ζi,jD ∈ (0.4, 0.6) is the efficiency of energy conversion.It follows from the receive equation (1) and the split
equation (2) that the downlink information throughput at DLU(i, jD) is
fi,jD(VD,VU,αααi,jD) ,
ln∣∣∣INr
+ (Li,jD(Vi,jD))2Ψ−1i,jD(VD,VU,αααi,jD)∣∣∣, (3)
where Li,jD(Vi,jD) , Hi,i,jDVi,jD and
Ψi,jD(VD,VU,αααi,jD) , Ψi,jD(VD,VU)+(σ2c/αααi,jD)INr (4)
with the downlink interference covariance mapping
Ψi,jD(VD,VU) ,∑
(m,`D)∈S1\(i,jD)
(Hm,i,jDVm,`D)2
+∑`U∈U
(Hi,jD,`UVi,`U)2 + σDINr . (5)
The harvested energy at UE (i, jD) is
Ei,jD(VD,VU,αααi,jD) = ζi,jD(1−αααi,jD)〈Φi,jD(VD,VU)〉, (6)
with the downlink signal covariance mapping
Φi,jD(VD,VU) ,∑
(m,`D)∈S1
(Hm,i,jDVm,`D)2
+∑`U∈U
(Hi,jD,`UVi,`U)2 + σ2DINr
. (7)
In the UL channel, the received signal at BS i is expressed as
yi ,∑`U∈U
Hi,`U,iVi,`Usi,`U︸ ︷︷ ︸desired signal
+∑
m∈I\{i}
∑`U∈U
Hm,`U,iVm,`Usm,`U︸ ︷︷ ︸UL interference
+∑
m∈I\{i}
HBm,i∑jD∈D
Vm,jDsm,jD︸ ︷︷ ︸DL intercell interference
+ nSIi︸︷︷︸residual SI
+ni,(8)
where Hm,`U,i ∈ CN2×Nr and HBm,i ∈ CN2×N1 are thechannel matrices from ULU (m, `U) to BS i and from BS
4
m to BS i, respectively; ni is the additive white circularlysymmetric complex Gaussian noise with variance σ2
U; nSIiis the residual SI (after self-interference cancellation) at BSi and depends on the transmit power of BS i. Specifically,nSIi is modelled as the additive white circularly symmetriccomplex Gaussian noise with variance σ2
SI
∑`D∈D ||Vi,`D ||2
[29], where the SI level σ2SI is the ratio of the average SI
powers after and before the SI cancellation process.Following [14], [16], [26], the optimal minimum mean
square error - Successive interference cancellation (MMSE-SIC) decoder is applied at BSs. Therefore, the achievableuplink throughput at BS i is given as [30]
fi(VD,VU) , ln∣∣IN2
+ (Li(VUi))2Ψ−1i (VD,VU)
∣∣ , (9)
where VUi , [Vi,`U ]`U∈U and Li(VUi) ,[Hi,1U,iVi,1U , Hi,2U,iVi,2U , . . . ,Hi,UU,iVi,UU
], which means
that (Li(VUi))2 =
U∑`=1
(Hi,`U,iVi,`U)2, and
Ψi(VD,VU) , ΨUi (VU) + ΨSI
i (VD) (10)
with uplink interference covariance mapping
ΨUi (VU) ,
∑m∈I\{i}
∑`U∈U
(Hm,`U,iVm,`U)2
+∑
m∈I\{i}
HBm,i
∑jD∈D
(Vm,jD)2
(HBm,i)H + σ2
UIN2(11)
and SI covariance mapping
ΨSIi (VD) , σ2
SI
∑`D∈D
||Vi,`D ||2IN2. (12)
We consider the design problem
maxVD,VU,ααα
P1(VD,VU,ααα) ,∑i∈I
fi(VD,VU)
+∑
(i,jD)∈S1
fi,jD(VD,VU,αααi,jD) s.t. (13a)
0 < αααi,jD < 1, (i, jD) ∈ S1, (13b)∑(i,jD)∈S1
||Vi,jD ||2 +∑
(i,jU)∈S2
||Vi,jU ||2 ≤ P, (13c)
∑jD∈D
||Vi,jD ||2 ≤ Pi,∀i ∈ I, (13d)
||Vi,jU ||2 ≤ Pi,jU ,∀(i, jU) ∈ S2, (13e)〈Φi,jD(VD,VU)〉 ≥ emin
i,jD/ζi,jD(1−αααi,jD),
∀(i, jD) ∈ S1, (13f)
fi(VD,VU) ≥ rU,mini ,∀i ∈ I (13g)
fi,jD(VD,VU,αααi,jD) ≥ rD,mini,jD
,∀(i, jD) ∈ S1. (13h)
In the formulation (13), all channel matrices in the downlinkequation (1) and uplink (8) are assumed to be known by usingthe channel reciprocity, feedback and learning mechanisms(see e.g. [31]). The convex constraints (13d) and (13e) spec-ify the maximum transmit power available at the BSs andthe ULUs whereas (13c) limits the total transmit power ofthe whole network. The nonconvex constraints (13f), (13g)
and (13h) represent QoS guarantee, where emini,jD
, rU,mini and
rD,mini,jD
are the minimum harvested energy required by DLU(i, jD), the minimum data throughput required by BS i andthe minimum data throughput required by DLU (i, jD). Incomparison to [16] for FD non-EH-enable networks, the ULthroughput function fi(VD,VU) in (9) is the same, where theDL throughput function fi,jD(VD,VU,αααi,jD) is now addition-ally dependent on the SP variable αααi,jD , is decoupled in (5)and thus does not add more difficulty as we will show now.We also show that the nonconvex EH constraints (13f) caneasily be innerly approximated.
Under the definitions,
Mi,jD(VD,VU,αααi,jD) ,(Li,jD(Vi,jD))2
+ Ψi,jD(VD,VU,αααi,jD) (14)�Ψi,jD(VD,VU,αααi,jD), (15)
Mi(VD,VU) ,(Li(VUi))2 + Ψi(VD,VU) (16)
�Ψi(VD,VU), (17)
by applying Theorem 1 as in [16], we obtain the follow-ing concave quadratic minorants of throughput functionsfi,jD(VD,VU,αααi,jD) and fi(V
(κ)D ,V
(κ)U ) at (V
(κ)D , V
(κ)U , α(κ))
, ([V(κ)i,jD
](i,jD)∈S1 , [V(κ)i,`U
](i,`U)∈S2 , [α(κ)i,jD
](i,jD)∈S1):
Θ(κ)i,jD
(VD,VU,αααi,jD) ,
a(κ)i,jD
+ 2<{〈A(κ)
i,jD,Li,jD(Vi,jD)〉
}−〈B(κ)i,jD
,Mi,jD(VD,VU,αααi,jD)〉 (18)
and
Θ(κ)i (VD,VU) ,
a(κ)i + 2<
{〈A(κ)
i ,Li(VUi)〉}− 〈B(κ)i ,Mi(VD,VU)〉, (19)
where
0 > a(κ)i,jD
, fi,jD(V(κ)D , V
(κ)U , α
(κ)i,jD
)
−<{〈Ψ−1i,jD(V
(κ)D , V
(κ)U )Li,jD(V
(κ)i,jD
),Li,jD(V(κ)i,jD
)〉},
A(κ)i,jD
= Ψ−1i,jD(V(κ)D , V
(κ)U , α
(κ)i,jD
)Li,jD(V(κ)i,jD
),
0 � B(κ)i,jD= Ψ−1i,jD(V
(κ)D , V
(κ)U , α
(κ)i,jD
)
−M−1i,jD(V(κ)D , V
(κ)U , α
(κ)i,jD
),(20)
and
0 > a(κ)i = fi(V
(κ)D , V
(κ)U )
−<{〈Ψ−1i (V
(κ)D , V
(κ)U )Li(V (κ)
i ),Li(V (κ)i )〉
},
A(κ)i = Ψ−1i (V
(κ)D , V
(κ)U )Li(V (κ)
Ui),
0 � B(κ)i = Ψ−1i (V(κ)D , V
(κ)U )
−M−1i (V(κ)D , V
(κ)U ).
(21)
To handle the nonconvex EH constraints (13f), we define anaffine function φ(κ)i,jD
(VD,VU) as the first-order approximation
5
Algorithm 1 Path-following algorithm for PS sum throughputmaximization (13)
Initialization: Set κ := 0, and choose a feasible point(V
(0)D , V
(0)U , α(0)) that satisfies (13b)-(13h).
κ-th iteration: Solve (23) for an optimal solution(V ∗D , V
∗U , α
∗) and set κ := κ + 1, (V(κ)D , V
(κ)U , α(κ)) :=
(V ∗D , V∗U , α
∗) and calculate P1(V(κ)D , V
(κ)U , α(κ)). Stop if∣∣∣ (P1(V
(κ)D , V
(κ)U , α(κ))− P1(V
(κ−1)D , V
(κ−1)U , α(κ−1))
)/P1(V
(κ−1)D , V
(κ−1)U , α(κ−1))
∣∣∣ ≤ ε.of the convex function 〈Φi,jD(VD,VU)〉 at (V
(κ)D , V
(κ)U ):
φ(κ)i,jD
(VD,VU) , −〈Φi,jD(V(κ)D , V
(κ)U )〉
+2<{∑
(m,`D)∈S1
〈Hm,i,jDV(κ)m,`D
Vm,`DHHm,i,jD
〉}
+2<{∑`U∈U
〈Hi,iD,`UV(κ)i,`U
VHi,`U
HHi,iD,`U
〉}+ 2σ2DNr, (22)
which is an minorant of 〈Φi,jD(VD,VU)〉 at (V(κ)D , V
(κ)U ) [28].
We now address the nonconvex problem (13) by succes-sively solving its following inner approximation:
maxVD,VU,ααα
P(κ)1 (VD,VU,ααα) ,
∑i∈I
Θ(κ)i (VD,VU)
+∑
(i,jD)∈S1
Θ(κ)i,jD
(VD,VU,αααi,jD) (23a)
s.t. (13b)− (13e) (23b)
φ(κ)i,jD
(VD,VU) ≥ emini,jD
/ζi,jD(1−αααi,jD),
∀(i, jD) ∈ S1, (23c)
Θ(κ)i (VD,VU) ≥ rU,min
i ,∀i ∈ I, (23d)
Θ(κ)i,jD
(VD,VU,αααi,jD) ≥ rD,mini,jD
,∀(i, jD) ∈ S1. (23e)
Initializing from (V(κ)D , V
(κ)U , α(κ)) being feasible point to
(13), the optimal solution (V(κ+1)D , V (κ+1)
U , α(κ+1)) of convexprogram (23) is feasible to the nonconvex program (13) andit is better than (V
(κ)D , V (κ)
U , α(κ)):
P1(V(κ+1)D , V
(κ+1)U , α(κ+1)) ≥
P(κ)1 (V
(κ+1)D , V
(κ+1)U , α(κ+1)) ≥ (24)
P(κ)1 (V
(κ)D , V
(κ)U , α(κ)) = (25)
P1(V(κ)D , V
(κ)U , α(κ)), (26)
where the inequality (24) and the equality (26) follow fromthe fact that P(κ)
1 is a minorant of P1 while the inequal-ity (25) follows from the fact that (V
(κ+1)D , V
(κ+1)U , α(κ+1))
and (V(κ)D , V
(κ)U , α(κ)) are the optimal solution and feasi-
ble point of (23), respectively. This generates a sequence{(V (κ)
D , V(κ)U , α(κ))} of feasible and improved points which
converge to a local optimum of (13) after finitely manyiterations [16].
The proposed path-following procedure that solves problem(13) is summarized in Algorithm 1. To find a feasible initial
point (V(0)D , V
(0)U , α(0)) meeting the nonconvex constraints
(13f)-(13h) we consider the following problem:
maxVD,VU,ααα
P1,f (VD,VU,ααα) ,
min(i,jD)∈S1
{Φi,jD(VD,VU)−
emini,jD
ζi,jD (1−αααi,jD) ,
fi,jD(VD,VU,ααα)− rmini,iD, fi(VD,VU)− rmin
i
}s.t. (13b)− (13e).
(27)
Initialized by a (V(0)D , V
(0)U , α(0)) feasible to the
convex constraints (13b)-(13e), an iterative point(V
(κ+1)D , V
(κ+1)U , α(κ+1)) for κ = 0, 1, . . . , is generated
as the optimal solution of the following convex maximinprogram:
maxVD,VU,ααα
P(κ)1,f (VD,VU,ααα) ,
min(i,jD)∈S1
{φ(κ)i,jD
(VD,VU)−emini,jD
ζi,jD(1−αααi,jD),
〈Θ(κ)i,jD
(VD,VU,ααα)− rmini,iD,Θ
(κ)i (VD,VU)− rmin
i ,}
s.t (13b)− (13e).(28)
which terminates upon reaching
fi,jD(V(κ)D , V
(κ)U , α(κ)) ≥ rmin
i,iD, fi(V
(κ)D , V
(κ)U ) ≥ rmin
i ,
〈Φi,jD(V(κ)D , V
(κ)U )〉 ≥
emini,jD
ζi,jD(1−αααi,jD), ∀(i, jD) ∈ S1
to satisfy (13b)-(13h).In parallel, we consider the following transmission strategy
to configure FD BSs to operate in the HD mode. Here, allantennas N = N1 + N2 at each BS are used to serve all theDLUs in the downlink and all the ULUs in the uplink usinghalf time slots, where DLUs are allowed to harvest energyfrom ULUs. The problem can be formulated as
maxVD,VU,ααα
1
2
[ ∑(i,jD)∈S1
fi,jD(VD, 0U,αααi,jD)
+∑i∈I
fi(0D,VU)]
(29a)
s.t. (13b), (13c), (13d), (13e),1
2(Ei,jD(VD, 0U,αααi,jD) + Ei,jD(0D,VU, 0)) ≥
emini,jD
,∀(i, jD) ∈ S1, (29b)1
2fi(0D,VU) ≥ rU,min
i ,∀i ∈ I (29c)
1
2fi,jD(VD, 0U,αααi,jD) ≥ rD,min
i,jD,∀(i, jD) ∈ S1, (29d)
where 0D and 0U are zero quantity of the same dimensionwith VD and VU. In (29), DLU (i, jD) uses (1−αααi,jD) of thereceived signal during DL transmission and the whole receivedsignal during UL transmission for EH as formulated in (29b).The main difference between (13) and (29) is in (29b) wherethe harvested energy from UL transmission at DLU (i, jD)does not multiply with αααi,jD . The constraint (29b) can be recastas
〈Φi,jD(VD, 0U)〉+〈Φi,jD(0D,VU)〉
(1−αααi,jD)≥
2emini,jD
ζi,jD(1−αααi,jD).
6
Define the following convex function:
Λi,jD(VU,αααi,jD) ,〈Φi,jD(0D,VU)〉
(1−αααi,jD)
=〈∑`U∈U (Hi,jD,`UVi,`U)2 + σ2
DINr〉
1−αααi,jD, (30)
with its first-order approximation
Λ(κ)i,jD
(VU, 1−αααi,jD) ,
2<{〈∑`U∈U (Hi,jD,`UVi,`U)(Hi,jD,`UV
(κ)i,`U
)H〉}
1− α(κ)i,jD
−〈∑`U∈U (Hi,jD,`UV
(κ)i,`U
)2 + σ2DINr 〉
(1− α(κ)i,jD
)2(1−αααi,jD), (31)
which is its minorant at (V(κ)D , V
(κ)U , α(κ)).
Algorithm 1 can be used with the following convex programsolved for κ-iteration:
maxVD,VU,ααα
1
2
[ ∑(i,jD)∈S1
Θ(κ)i,jD
(VD, 0U,αααi,jD)
+∑i∈I
Θ(κ)i (0D,VU, 0)
](32a)
s.t. (13b), (13c), (13d), (13e),
φ(κ)i,jD
(VD, 0U) + Λ(κ)i,jD
(VU, 1−αααi,jD) ≥2emini,jD
ζi,jD(1−αααi,jD),∀(i, jD) ∈ S1, (32b)
1
2Θ
(κ)i (0D,VU) ≥ rU,min
i ,∀i ∈ I (32c)
1
2Θ
(κ)i,jD
(VD, 0U,αααi,jD) ≥ rD,mini,jD
,∀(i, jD) ∈ S1, (32d)
where φ(κ)i,jD(VD, 0U) and Θ
(κ)i,jD
(VD, 0U,αααi,jD) are defined by(22) and (18) with both VU and V (κ)
U replaced by 0U, whileΘ
(κ)i (0D,VU) is defined by (19) with both VD and V
(κ)D
replaced by 0D.Problems (23), (28) and (32) involve n = 2(N1d1ID +
Nrd2IU) + ID scalar real decision variables and m =5ID+IU+2I+1 quadratic constraints so their computationalcomplexity is O(n2m2.5 +m3.5).
III. EH-ENABLED FD MU-MIMO BY TS
A much easier implementation is time splitting 0 < ααα < 1in downlink transmission where (1−ααα) time is used for DL en-ergy transfer and ααα time is used for DL information transmis-sion. In this section, we define VI
D , [VIi,jD
](i,jD)∈S1 ,VED ,
[VEi,jD
](i,jD)∈S1 and redefine the notation VD , [VID,V
ED ]
where VIi,jD
and VEi,jD
are the information precoding matrixfor ID and energy precoding matrix for EH, respectively. Thereceived signal at DLU (i, jD) for EH is
yEi,jD ,∑
(m,`D)∈S1
Hm,i,jDVEm,`D
sEm,`D
+∑`U∈U
Hi,jD,`UVi,`Usi,`U︸ ︷︷ ︸UL intracell interference
+njD , (33)
where sEm,`D is the energy signal sent for (1−ααα) time. Withthe definition (6), the harvested energy is
Ei,jD(VED ,VU,ααα) = ζi,jD(1−ααα)〈Φi,jD(VE
D ,VU)〉,
where the downlink signal covariance mapping Φi,jD(., .) isdefined from (7).Similarly to (1), the signal received at DLU (i, jD) during theinformation transmission in time fraction ααα is
yIi,jD ,Hi,i,jDVi,jDsIi,jD︸ ︷︷ ︸
desired signal
+∑
(m,`D)∈S1\(i,jD)
Hm,i,jDVm,`DsIm,`D︸ ︷︷ ︸
DL interference
+∑`U∈U
Hi,jD,`UVi,`Usi,`U︸ ︷︷ ︸UL intracell interference
+ni,jD , (34)
where sIm,`D is the information signal intended for DLU(m, `D). The ID throughput at DLU (i, jD) is then given asαααfi,jD(V), where
fi,jD(VID,VU) = ln
∣∣∣INr+ (Li,jD(VI
i,jD))2Ψ−1i,jD(VI
D,VU)∣∣∣,
(35)
with the downlink interference covariance mapping Ψ(., .)defined from (5).The uplink throughput at the BS is
fi(VD,VU,ααα) , ln∣∣∣IN2 + (Li(VUi))
2sΨ−1i (VD,VU,ααα)∣∣∣,
(36)
where Li(VUi) is already defined from (9) but
Ψi(VD,VU,ααα) , ΨUi (VU) + ΨTSI
i (VD,ααα) (37)
with the uplink interference covariance mapping ΨUi (.) defined
by (11) and the time-splitting SI covariance mapping
ΨTSIi (VD,ααα) , σ2
SI
∑jD∈D
((1−ααα)||VE
i,jD||2 +ααα||VI
i,jD||2)IN2
.
(38)
The problem of maximizing the network total throughputunder throughput QoS and EH constraints is the following:
7
maxVD,VU,ααα
P2(VD,VU,ααα) ,∑(i,jD)∈S1
(αααfi,jD(VI
D,VU) + fi(VD,VU,ααα))
(39a)
s.t. 0 < ααα < 1, (39b)||Vi,jU ||2 ≤ Pi,jU ,∀(i, jU) ∈ S2, (39c)∑
(i,jD)∈S1
((1−ααα)||VE
i,jD||2 +ααα||VI
i,jD||2)
+∑
(i,jU)∈S2
||Vi,jU ||2 ≤ P, (39d)
∑jD∈D
((1−ααα)||VE
i,jD||2 +ααα||VI
i,jD||2)
≤ Pi,∀i ∈ I, (39e)fi(VD,VU,ααα) ≥ rU,min
i ,∀i ∈ I, (39f)
αααfi,jD(VID,VU) ≥ rD,min
i,jD,∀(i, jD) ∈ S1, (39g)
Ei,jD(VED ,VU,ααα) ≥ emin
i,jD,∀(i, jD) ∈ S1. (39h)
Constraints (39c), (39d) and (39e) limits the transmit powerof each ULU, the whole network and each BS, respectively.Constraints (39h) ensures that each DLUs harvest more thana threshold whereas constraints (39f) and (39g) guaranteethe throughput QoS at BSs and DLUs, respectively. Thekey difficulty in problem (39) is to handle the time splittingfactor ααα that is coupled with the objective functions and othervariables. Using the variable change ρρρ = 1/ααα, which satisfiesthe convex constraint
ρρρ > 1, (40)
problem (39) is equivalent to
maxVD,VU,ρρρ>0
P2(VD,VU, ρρρ) ,∑(i,jD)∈S1
fi,jD(VID,VU)/ρρρ+
∑i∈I
fi(VD,VU, 1/ρρρ) (41a)
s.t. (40), (39c),∑(i,jD)∈S1
(||VE
i,jD||2 + ||VI
i,jD||2/ρρρ
)+
∑(i,jU)∈S2
||Vi,jU ||2 ≤ P +∑
(i,jD)∈S1
||VEi,jD||2/ρρρ, (41b)
∑jD∈D
(||VE
i,jD||2 + ||VI
i,jD||2/ρρρ
)≤
Pi +∑jD∈D
||VEi,jD||2/ρρρ, ∀i ∈ I, (41c)
Ei,jD(VED ,VU, 1/ρρρ) ≥ emin
i,jD,∀(i, jD) ∈ S1, (41d)
fi(VD,VU, 1/ρρρ) ≥ rU,mini ,∀i ∈ I, (41e)
fi,jD(VID,VU)/ρρρ ≥ rD,min
i,jD,∀(i, jD) ∈ S1. (41f)
Problem (41) is much more difficult computationally than (13).Firstly, the DL throughput is now the multiplication of datathroughput and the portion of time 1/ρρρ. Secondly, the SI inUL throughput is also coupled with 1/ρρρ. Finally, the powerconstraints (41b), (41c) are also coupled with 1/ρρρ. Therefore,the objective function (41a) and constraints (41b)-(41f) are
all nonconvex and cannot be addressed as in (13). In thefollowing, we will develop the new minorants of the DLthroughput function and UL throughput function.
Firstly, we address a lower approximation for eachfi,jD(VI
D,VU)/ρρρ in (41a) and (41f). Recalling the definition(35) of fi,jD(VI
D,VU) we introduce
Mi,jD(VID,VU) , (Li,jD(Vi,jD))2 + Ψi,jD(VD,VU),
to have its following minorant at (V(κ)D , V
(κ)U ):
Θ(κ)i,jD
(VID,VU) ,a(κ)i,jD
+ 2<{〈A(κ)
i,jD,Li,jD(VI
i,jD)〉}
− 〈B(κ)i,jD,Mi,jD(VI
D,VU)〉, (42)
where similarly to (20)
0 > a(κ)i,jD
= fi,jD(V(κ)D , V
(κ)U )
−<{〈Ψ−1i,jD(V
(κ)D , V
(κ)U )Li,jD(V
(κ)i,jD
),Li,jD(V(κ)i,jD
)〉},
A(κ)i,jD
= Ψ−1i,jD(V(κ)D , V
(κ)U )Li,jD(V
(κ)i,jD
),
0 � B(κ)i,jD= Ψ−1i,jD(V
(κ)D , V
(κ)U )−M−1i,jD(V
(κ)D , V
(κ)U ).
(43)A minorant of fi,jD(VI
D,VU)/ρρρ is Θ(κ)i,jD
(VID,VU)/ρρρ but it is
still not concave. As fi,jD(VID,VU) > 0 it is obvious that its
lower bound Θ(κ)i,jD
(VID,VU) is meaningful for (VI
D,VU) suchthat
Θ(κ)i,jD
(VID,VU) ≥ 0, (i, jD) ∈ S1 (44)
which particularly implies
<{⟨
(A(κ)i,jD
,Li,jD(VIi,jD
)⟩}≥ 0, (i, jD) ∈ S1. (45)
Under (45), we have
<{〈A(κ)
i,jD,Li,jD(VI
i,jD)〉}
ρρρ≥
2b(κ)i,jD
√<{〈A(κ)
i,jD,Li,jD(VI
i,jD)〉}− c(κ)i,jD
ρρρ (46)
for
0 < b(κ)i,jD
=
√〈A(κ)
i,jD,Li,jD(V
I,(κ)i,jD
)〉ρ(κ)
,
0 < c(κ)i,jD
= (b(κ)i,jD
)2. (47)
Therefore, the following concave function
g(κ)i,jD
(VID,VU, ρρρ) ,
a(κ)i,jD
ρρρ+ 4b
(κ)i,jD
√<{〈A(κ)
i,jD,Li,jD(VI
i,jD)〉}− 2c
(κ)i,jDρρρ
−〈B(κ),Mi,jD(VI
D,VU)〉ρρρ
(48)
is a minorant of fi,jD(VID,VU)/ρρρ at (V
I,(κ)D , V
(κ)U , ρ(κ)).
Next, we address a lower approximation offi(VD,VU, 1/ρρρ) in (41a), (41e). Recalling the definition (36)of fi(VD,VU, 1/ρρρ) we introduce
Mi(VD,VU, ρρρ) ,(Li(VUi))2
+ ΨUi (VU) + ΨTSI
i (VD, 1/ρρρ), (49)
8
for ΨTSIi (VD, 1/ρρρ) defined from (38) as
ΨTSIi (VD, 1/ρρρ) =σ2
SI
∑jD∈D
(||VE
i,jD||2
+1
ρρρ||VI
i,jD||2 − 1
ρρρ||VE
i,jD||2)IN2
, (50)
to have its following minorant at (V(κ)D , V
(κ)U , ρ(κ)):
Θ(κ)i (VD,VU, ρρρ) ,a(κ)i + 2<
{〈A(κ)
i ,Li(VUi)〉}
− 〈B(κ)i ,Mi(VD,VU, ρρρ)〉, (51)
where similarly to (21)
0 > a(κ)i = fi(V
(κ)D , V
(κ)U )
−<{〈Ψ−1i (V
(κ)D , V
(κ)U )Li(V (κ)
i ),Li(V (κ)i )〉
},
A(κ)i = Ψ−1i (V
(κ)D , V
(κ)U )Li(V (κ)
Ui),
0 � B(κ)i = Ψ−1i (V(κ)D , V
(κ)U )−M−1i (V
(κ)D , V
(κ)U ).
(52)Function Θ
(κ)i (VD,VU, ρρρ) is not concave due to the term
ΨTSIi (VD, 1/ρρρ) defined by (50). However, the following ma-
trix inequality holds true:
1
ρρρ||VE
i,jD||2IN2 �(
2
ρ(κ)<{〈V E,(κ)i,jD
,VEi,jD〉}
−||V E,(κ)i,jD
||2
(ρ(κ))2ρρρ)IN2
, (53)
which yields the matrix inequality
Mi(VD,VU, ρρρ) �M(κ)
i (VD,VU, ρρρ) ,
(Li(VUi))2 + ΨU
i (VU) + σ2SI
(||VE
i,jD||2 +
1
ρρρ||VI
i,jD||2
− 2
ρ(κ)<{〈V E,(κ)i,jD
,VEi,jD〉}+
||V E,(κ)i,jD||2
(ρ(κ))2ρρρ)IN2
.
As B(κ)i � 0 by (52), we then have
〈B(κ)i ,Mi(VD,VU, ρρρ)〉 ≥ 〈B(κ)i ,M(κ)
i (VD,VU, ρρρ)〉
so a concave minorant of both fi(VD,VU, 1/ρρρ) andΘ(κ)(VD,VU, ρρρ) is
Θ(κ)i (VD,VU, ρρρ) ,a(κ)i + 2<
{〈A(κ)
i ,Li(VUi)〉}
− 〈B(κ)i ,M(κ)i (VD,VU, ρρρ)〉. (54)
Concerned with ||VEi,jD||2/ρρρ in the right hand side (RHS) of
(41b) and (41c), it follows from (53) that
||VEi,jD||2/ρρρ ≥ γ
(κ)i,jD
(VEi,jD
, ρρρ)
, 2<{〈V E,(κ)i,jD,VE
i,jD〉}/ρ(κ)
−ρρρ||V E,(κ)i,jD||2/(ρ(κ))2.
We also have φ(κ)i,jD
(VED ,VU) defined in (22) as a minorant
of 〈Φi,jD(VED ,VU)〉. We now address the nonconvex problem
Algorithm 2 Path-following algorithm for TS optimizationproblem (41)
Initialization: Set κ := 0, and choose a feasible point(V
(0)D , V
(0)U , α(0)) that satisfies (39b)-(39g). Set ρ(0) :=
1/α(0).κ-th iteration: Solve (55) for an optimal solution(V ∗D , V
∗U , ρ
∗) and set κ := κ + 1, (V(κ)D , V
(κ)U , ρ(κ)) :=
(V ∗D , V∗U , ρ
∗) and calculate P2(V(κ)D , V
(κ)U , 1/ρ(κ)). Stop if∣∣(P2(x(κ))− P2(x(κ−1))
)/P2(x(κ−1))
∣∣ ≤ ε, where x(κ) ,(V
(κ)D , V
(κ)U , 1/ρ(κ))
(41) by successively solving its following innerly approxi-mated convex program at κ-iteration:
maxVD,VU,ρρρ>0
P(κ)2 (VD,VU, ρρρ) (55a)
s.t. (40), (39c), (55b)∑(i,jD)∈S1
(||VE
i,jD||2 +
1
ρρρ||VI
i,jD||2)
+
∑(i,jU)∈S2
||Vi,jU ||2 ≤ P +∑
(i,jD)∈S1
γ(κ)i,jD
(VEi,jD
, ρρρ), (55c)
∑jD∈D
(||VE
i,jD||2 +
1
ρρρ||VI
i,jD||2)≤
Pi +∑jD∈D
γ(κ)i,jD
(VEi,jD
, ρρρ),∀i ∈ I, (55d)
φ(κ)i,jD
(VED ,VU) ≥ emin
i,jD(1 +
1
ρρρ− 1)/ζi,jD ,
∀(i, jD) ∈ S1, (55e)
Θ(κ)i (VD,VU, ρρρ) ≥ rU,min
i ,∀i ∈ I, (55f)
g(κ)i,jD
(VID,VU, ρρρ) ≥ rD,min
i,jD.∀(i, jD) ∈ S1. (55g)
where P(κ)2 (VD,VU, ρρρ) ,
∑i∈I Θ
(κ)i (VD,VU, ρρρ)
+∑
(i,jD)∈S1 g(κ)i,jD
(VID,VU, ρρρ).
A path-following procedure similar to Algorithm 1 can beapplied to solve (41) as summarized in Algorithm 2. Thanksto the following relation, which is similar to (26):
P2(V(κ+1)D , V
(κ+1)U , ρ(κ+1)) ≥ P2(V
(κ)D , V
(κ)U , ρ(κ)), (56)
Algorithm 2 improves feasible point at each iteration and thenbring a local optimum after finitely many iterations.
To find an initial feasible point for Algorithm 2, we considerthe following problem:
maxVD,VU,ρρρ
min(i,jD)∈S1
{fi(VD,VU, 1/ρρρ)− rmin
i ,
Ei,jD(VED ,VU, 1/ρρρ)− emin
i,jD,
fi,jD(VID,VU)/ρρρ− rmin
i,iD
}: (41b)− (41c)
(57)
which can be addressed by successively solving the followingconvex maximin program:
maxVD,VU,ρρρ
min(i,jD)∈S1
{g(κ)i,jD
(VID,VU, ρρρ)− rmin
i,iD,
φ(κ)i,jD
(VED ,VU)− emin
i,jD(1 +
1
ρρρ− 1)/ζi,jD ,
Θ(κ)i (VD,VU, ρρρ)− rmin
i
}: (55b)− (55d),
(58)
9
upon reaching fi,jD(VI,(κ)D , V
(κ)U , α(κ)) ≥
rmini,iD
, fi(V(κ)D , V
(κ)U , α(κ)) ≥ rmin
i andEi,jD(V
E,(κ)D , V
(κ)U , 1/ρ(κ)) ≥ emin
i,jD, ∀(i, jD) ∈ S1.
For the system operating in HD mode, we apply the sametransmission strategy as in Section II. Specifically, we considerthe following problem:
maxVD,VU,ρρρ
1
2[∑
(i,jD)∈S1
1
ρρρfi,jD(VD, 0U)
+∑i∈I
fi(0D,VU, 1)] s.t. (39b)− (39e) (59a)
1
2(Ei,jD(VD, 0U, 1/ρρρ) + Ei,jD(0D,VU, 1)) ≥,
emini,jD
(i, jD) ∈ S1, (59b)1
2ρρρfi,jD(VD, 0U) ≥ rD,min
i,jD,∀(i, jD) ∈ S1, (59c)
1
2fi(0D,VU, 1) ≥ rU,min
i ,∀i ∈ I. (59d)
In (59), DLUs harvest energy for (1 − ααα) of 1/2 time slotduring DL transmission and for the whole 1/2 time slot duringUL transmission as formulated in (59b). The constraint (59b)can be written by
Ξi,jD(VD,VU, ρρρ) ≥2emini,jD
ζi,jD(1 +
1
ρρρ− 1), (60)
for
Ξi,jD(VD,VU, ρρρ) ,
〈Φi,jD(VD, 0U)〉+ 〈Φi,jD(0D,VU)〉+〈Φi,jD(0D,VU)〉
ρρρ− 1.
As Ξi,jD is convex, its minorant is its first-order approximationat (V
(κ)D , V
(κ)U , ρ(κ)):
Ξ(κ)i,jD
(VD,VU, ρρρ) = φ(κ)i,jD
(VD, 0U) + φ(κ)i,jD
(0D,VU)
+Λ(κ)i,jD
(VU, ρρρ− 1),
for Λ(κ)i,jD
(., .) defined by (31) and φ(κ)i,jD(0D,VU) defined from
(22) with both VU and V (κ)U replaced by 0U.
The problem (39) thus can be addressed via a path-followingprocedure similar to Algorithm 2 where the following convexprogram is solved for κ-iteration:
maxVD,VU,ρρρ
1
2[∑
(i,jD)∈S1
g(κ)i,jD
(VID, 0U, ρρρ)
+∑i∈I
Θ(κ)i (0D,VU, 1)] : (39b)− (39e) (61a)
Ξ(κ)i,jD
(VD,VU, ρρρ) ≥2emini,jD
ζi,jD(1 +
1
ρρρ− 1),
(61b)(i, jD) ∈ S1, (61c)
1
2g(κ)i,jD
(VD, 0U, ρρρ) ≥ rD,mini,jD
,∀(i, jD) ∈ S1, (61d)
1
2Θ
(κ)i (0D,VU, 1) ≥ rU,min
i ,∀i ∈ I, (61e)
where g(κ)i,jD(VI
D, 0U, ρρρ) is defined by (48) with both VU andV
(κ)U replaced by 0U, while Θ
(κ)i (0D,VU, 0) is defined by (54)
with both VD and V (κ)D replaced by 0D and both ρρρ and ρ(κ)
replaced by 1.
IV. THROUGHPUT QOS CONSTRAINEDENERGY-HARVESTING OPTIMIZATION
We will justify numerically that TS is not only easier imple-mented but performs better than PS for FD EH-enabled MUMIMO networks. This motivates us to consider the followingEH optimization with TS, which has not been previouslyconsidered at all:
maxVD,VU,ααα
P3(V,ααα) ,∑
(i,jD)∈S1
Ei,jD(VED ,VU,ααα)
s.t. (39b)− (39g).(62)
By defining ρρρ = 1/ααα, we firstly recast Ei,jD(VED ,VU, 1/ρρρ) as
Ei,jD(VED ,VU, 1/ρρρ) =ζi,jD
(〈Φi,jD(VE
D ,VU)〉
−Qi,jD(VED ,VU, ρρρ)
), (63)
where Qi,jD(VED ,VU, ρρρ) , 1
ρρρ 〈Φi,jD(VED ,VU)〉 is a convex
function. Recalling that φ(κ)i,jD(VE
D ,VU) defined in (22) isa minorant of 〈Φi,jD(VE
D ,VU)〉, we can now address thenonconvex problem (62) by successively solving the followingconvex program at κ-iteration:
maxV,ρρρ
∑(i,jD)∈S1
ζi,jD
(φ(κ)i,jD
(VED ,VU)
−Qi,jD(VED ,VU, ρρρ)
)s.t. (39c), (40), (55c), (55d), (55f), (55g).
(64)
A path-following procedure similar to Algorithm 2 can beapplied to solve (62).
For the system operating in HD mode, the same trans-mission strategy as in Section II is applied. Specifically, weconsider the following problem:
maxV,ρρρ
∑(i,jD)∈S1
1
2(Ei,jD(VE
D , 0U, 1/ρρρ) + Ei,jD(0D,VU, 0))
s.t. (39c), (40), (39d), (39e), (59c), (59d).(65)
The problem (65) can be addressed via a path-followingprocedure similar to Algorithm 2 where the following convexprogram is solved for κ-iteration:
maxV,ρρρ
∑(i,jD)∈S1
ζi,jD2
(φ(κ)i,jD
(VED , 0U)−Qi,jD(VE
D , 0U, ρρρ)
+φ(κ)i,jD
(0D,VU))
s.t. (39c), (40), (39d), (39e), (61d), (61e),(66)
where φ(κ)i,jD(VE
D , 0U) (φ(κ)i,jD(0D,VU), resp.) is defined by (22)
with both VU and V(κ)U (both VD and V
(κ)D , resp.) replaced
by 0U (0D, resp.).Problems (55), (58), (61), (64) and (66) involve n =
2(N1d1ID + Nrd2IU) + 3 scalar real decision variablesand m = ID + IU + 2I + 3 quadratic constraints so itscomputational complexity is O(n2m2.5 +m3.5).
10
r1
DLU (i,jD)
ULU (i,jU)
r1
r1
BS i
r2
Fig. 2. A three-cell network with 3 DLUs and 3 ULUs. DLUs are randomlylocated on the circles with radius of r1 centered at their serving BS. ULUsare uniformly distributed within the cell of their serving BS.
V. NUMERICAL RESULTS
In this simulation study, we use the example network inFig. 2 to study the total network throughput in the presenceof SI. The HD system is also implemented as a base line forboth time splitting mechanism and power splitting mechanism.DLUs are randomly located on the circles with radius ofr1 = 20 m centered at their serving BSs whereas ULUsare uniformly distributed within the cell of their serving BSswhose radius are r2 = 40 m. There are two DLUs andtwo ULUs within each cell. We set the path loss exponentβ = 4. For small-scale fading, we generate the channelmatrices Hm,i,jD from BS m to UE (i, jD), matrices Hi,jD,`U
from ULU (i, `U) to DLU (i, jD), matrices Hm,`U,i from ULU(m, `U) to BS i and matrices HBm,i from BS m to BS i usingthe Rician fading model as follows:
H =
√KR
1 +KRHLOS +
√1
1 +KRHNLOS , (67)
where KR = 10 dB is the Rician factor, HLOS is the line-of-sight (LOS) deterministic component and each elementof Rayleigh fading component HNLOS is the circularly-symmetric complex Gaussian random variable CN (0, 1). Here,we use the far-field uniform linear antenna array model [32]with
HLOS = [1, ejθr , ej2θr , . . . , ej(Nr−1)θr ]×[1, ejθt , ej2θt , . . . , ej(N1−1)θt ]H ,
(68)
for θr = 2πd sin(φr)λ , θt = 2πd sin(φt)
λ , where d = λ/2 isthe antenna spacing, λ is the carrier wavelength and φr, φtis the angle-of-arrival, the angle-of-departure, respectively. Inour simulations, φr and φt are randomly generated between0 and 2π. Unless stated otherwise, the number of transmitantennas and the number of receive antennas at a BS are set
as N1 = N2 = 4. The numbers of concurrent downlink datastreams and the numbers of concurrent uplink data streamsare equal and d1 = d2 = Nr. To arrive at the final figures,we run each simulation 100 times and average out the result.In all simulations, we set P = 23 dBW, Pi = 16 dBW∀i ∈ I, Pi,jU = 10 dBW ∀(i, jU) ∈ S2, ∀i ∈ I ,ζ = 0.5,σ2c = −90 dBW, σ2 = −90 dBW, rmin
i,jD= rD = 1 bps/Hz and
rmini = rU = UrD bps/Hz. We further assume that the required
harvested energies of all DLUs are the same and emini,jD
= emin,∀(i, jD). Unless stated otherwise, we set emin = −20 dBm asin [7], [33]. According to the current state-of-the-art-electroniccircuitry, the sensitivity level of a typical energy harvester isaround -20 dBm (0.01mW) [34], which means that we canactivate the EH circuitry with that much amount of receivedpower. The SI level σ2
SI is choosen within the range of[−150,−90] dB 1 as in [14], [16], [26] where σ2
SI = −150dB represents the almost perfect SI cancellation.
A. Single cell network
Firstly, we consider the sum throughput maximization prob-lem and the total harvested energy in the single cell networks.This will facilitate the analysis of the impact of SI to thenetwork performance since there is no intercell interference.The network setting in Fig. 2 is used but only one cell isconsidered.
Fig. 3 illustrates the comparison of total network throughputbetween the power splitting mechanism and the time splittingmechanism in both FD and HD systems. Though FD providesa substantial improvement in comparison to HD in both powersplitting mechanism (25.8%) and time splitting mechanism(26.1%) for Nr = 2 2 at σ2
SI = −150 dB. Note that wecannot expect a FD system to achieve twice the throughputof that is achieved by a HD system. This is because evenwhen the SI cancellation is perfect, DLUs in FD are stillvulnerable to the intracell interference from the ULUs of thesame cell. Moreover, DLUs and ULUs in HD are served withmore BS’s antennas, resulting in a larger spatial diversity.Consequently, FD cannot double HD’s throughput even withthe almost perfect SI cancellation.
When we reduce the number of antennas at UEs fromNr = 2 to Nr = 1, the total network throughput of FD issignificantly reduced by 42% for the time splitting mechanismand by 41% for the power splitting mechanism at σ2
SI = −150dB. Notably, since UEs in FD are exposed to more sourcesof interference than UEs in HD, reducing the number ofantennas of UEs degrades the performance of FD more thanthe counterpart of HD. Consequently, the improvement of FDin comparison to HD reduces to 16% at σ2
SI = −150 dB forboth time splitting mechanism and power splitting mechanism.
Fig. 4 further illustrates how the total throughput is dis-tributed into the downlink and uplink channels in the timesplitting mechanism. The behaviour of the power splittingmechanism is similar and omitted here for brevity. With the
1At σ2SI = −90 dB, if a BS transmits at full power (i.e. 16 dBW), the SI
power is 16 dB stronger than the background AWGN.2Nr has been defined in the begining of section II as the number of antennas
of UEs (DLUs and ULUs).
11
−150 −140 −130 −120 −110 −100 −9019
20
21
22
23
24
25
26
27
28
29
σSI2 (dB)
Sum
thro
ughp
ut (
bps/
Hz)
FD − TSFD − PSHD − TSHD − PS
(a) Nr = 1
−150 −140 −130 −120 −110 −100 −9035
40
45
50
σSI2 (dB)
Sum
thro
ughp
ut (
bps/
Hz)
FD − TSFD − PSHD − TSHD − PS
(b) Nr = 2
Fig. 3. Effect of SI on the sum throughput performance in the single-cellnetworks.
increase of σ2SI , the UL throughput consistently decreases.
Moreover, since the UL transmission becomes less efficient,ULUs reduce their transmission power to reduce the inter-ference toward DLUs. Consequently, a slight increase in FDDL throughput is observed as σ2
SI increases. Another note isthat since the distance between ULU-DLU in small cell canbe quite small due to the random deployment of ULUs andDLUs, DLUs’ throughput can be severely degraded by theinterference from ULUs. In fact, the FD DL throughput is60% less than the counterpart of HD at Nr = 1, σ2
SI = −150dB. By implementing multiple antenna at UEs (i.e. Nr = 2),DLUs in FD can handle the interference better and the FDDL throughput at σ2
SI = −150 dB is only 10% less than thecounterpart of HD.
To analyze the effect of energy harvesting constraint, we fixNr = 2, σ2
SI = −110 dB and vary emin. Fig. 5 illustrates aconsistent decreasing trend of all schemes as emin increases.The time splitting scheme outperforms the power splittingscheme in the considered range of emin for both FD and HD.A similar conclusion can be drawn from Fig. 3. By using two
−150 −140 −130 −120 −110 −100 −906
8
10
12
14
16
18
20
22
24
σSI2 (dB)
Sum
thro
ughp
ut (
bps/
Hz)
FD − ULFD − DLHD − ULHD − DL
(a) Nr = 1
−150 −140 −130 −120 −110 −100 −9016
18
20
22
24
26
28
30
32
σSI2 (dB)
Sum
thro
ughp
ut (
bps/
Hz)
FD − ULFD − DLHD − ULHD − DL
(b) Nr = 2
Fig. 4. Effect of SI on the UL/DL throughput performance in the single-cellnetworks.
different precoder matrices VI and VE for data transmissionand energy transferring, the time splitting scheme can exploitthe spatial diversity better than the power splitting schemewhich only uses one type of precoder matrix for both purposes.Thus, the time splitting scheme is more efficient than thepower splitting scheme in term of performance.
The comparison of maximum harvested energy of timesplitting scheme in both FD and HD systems is studied inFig. 6. Interestingly, in case of Nr = 1, FD roughly harvestsas much as HD. The reason of this is two folds. Firstly, it hasbeen reported in [16], [26], [35] that FD not always harnessperformance gain over HD if the distance between ULUs andDLUs are not large enough. Since we consider small cellnetworks with randomly deployed ULUs and DLUs, the ULU-DLU distance can be very small, which creates significantinterference to DLUs. Secondly, with Nr = 1, DLUs can notexploit the spatial diversity to mitigate the interference fromULUs. Consequently, ULUs must reduce its transmit powerto ensure the QoS at the DLUs, which lowers the amount ofharvested energy at DLUs. In contrast, the results show that
12
−20 −17.5 −15 −12.5 −1020
25
30
35
40
45
emin (dBm)
Sum
thro
ughp
ut (
bps/
Hz)
FD − TSFD − PSHD − TSHD − PS
(a) Pi = 30 dBm
−20 −17.5 −15 −12.5 −1036
38
40
42
44
46
48
50
emin (dBm)
Sum
thro
ughp
ut (
bps/
Hz)
FD − TSFD − PSHD − TSHD − PS
(b) Pi = 46 dBm
Fig. 5. Effect of energy harvesting constraints on the total harvested energyperformance in the single-cell networks.
FD harvests more energy than HD given that σ2SI ≤ −90 dB
for Nr = 2. All this implies that having multiple antennas atUEs is important to combat with extra interferences in FD.
B. Three-cell network
Now, we consider the sum throughput maximization prob-lem and the total harvested energy in the three-cell networksas depicted in Fig. 2. In this scenario, DLUs and BSs areexposed to additional intercell interferences. According to Fig.7, FD now only provides a marginal improvement regardingHD in both power splitting scheme (11.7%) and time split-ting scheme (11.8%) for Nr = 2, σ2
SI = −150 dB. ForNr = 1, σ2
SI = −150 dB, the improvement is even lower with4.1% in case of the power splitting scheme and 4.4% in case oftime splitting scheme. Therefore, FD can give marginal gainscompared to HD in the multi-cell networks with high level ofinterference.
−150 −140 −130 −120 −110 −100 −90 −80 −70−4
−2
0
2
4
6
8
10
σSI2 (dB)
Tot
al h
arve
sted
ene
rgy
(dB
m)
FD − TS, Nr=2
FD − TS, Nr=1
HD − TS, Nr=2
HD − TS, Nr=1
Fig. 6. Effect of SI on the total harvested energy performance in the single-cell networks.
−150 −140 −130 −120 −110 −100 −9052
54
56
58
60
62
64
66
68
σSI2 (dB)
Sum
thro
ughp
ut (
bps/
Hz)
FD − TSFD − PSHD − TSHD − PS
(a) Nr = 1
−150 −140 −130 −120 −110 −100 −9085
90
95
100
105
110
σSI2 (dB)
Sum
thro
ughp
ut (
bps/
Hz)
FD − TSFD − PSHD − TSHD − PS
(b) Nr = 2
Fig. 7. Effect of SI on the sum throughput performance in the three-cellnetworks.
The effect of energy harvesting constraint to the network
13
−20 −17.5 −15 −12.5 −1030
40
50
60
70
80
90
100
emin (dBm)
Sum
thro
ughp
ut (
bps/
Hz)
FD − TSFD − PSHD − TSHD − PS
(a) Pi = 30 dBm
−20 −17.5 −15 −12.5 −1090
92
94
96
98
100
102
104
106
108
110
emin (dBm)
Sum
thro
ughp
ut (
bps/
Hz)
FD − TSFD − PSHD − TSHD − PS
(b) Pi = 46 dBm
Fig. 8. Effect of energy harvesting constraints on the total harvested energyperformance in the three-cell networks.
sum throughput is also investigated in Fig. 8 for the three-cell networks with Nr = 2, σ2
SI = −110 dB. As in Fig. 5, aconsistent decreasing trend of all schemes is observed as emin
increases. Since DLUs can also harvest energy from the signalsarriving from other BSs in multicell networks, the FD networkthroughput only decreases by about 3% for both harvestingscheme when emin increases from -20 dBm to -10 dBm. Thecounterpart throughput decrease in single-cell scenarios wasabout 8%.
Fig. 9 also illustrates the comparison of total harvestedenergy per cell of the EH maximization problem in both FDand HD systems in three-cell network. For Nr = 1, FD evenharvests lesser amount of energy than HD given σ2
SI > −150dB due to the increasing level of interference when comparedto a single-cell network. Similar to the single-cell network,FD outperforms HD for σ2
SI ≤ −90 dB if more antennasare equipped at UEs (i.e. Nr = 2). This observation againemphasizes the importance of having multiple antenna at UEs
−150 −140 −130 −120 −110 −100 −90 −80 −70−10
−5
0
5
10
15
σSI2 (dB)
Tot
al h
arve
sted
ene
rgy
per
cell
(dB
m)
FD − TS, Nr=2
FD − TS, Nr=1
HD − TS, Nr=2
HD − TS, Nr=1
Fig. 9. Effect of SI on the total harvested energy performance in the three-cellnetworks.
in FD to mitigate interference. Another note is that givenNr = 2 the amount of energy harvested per cell in three-cellnetworks (i.e. 10.09 dBm at σ2
SI = −150 dB) is much higherthan the harvested energy of single cell in Fig. 6 (i.e. 8.5 dBmat σ2
SI = −150 dB), thanks to the extra energy harvested fromthe intercell interference.
C. Convergence behaviour
Finally, the convergence behavior of the proposed Algo-rithm 1 is illustrated in Fig. 10. For brevity, we only present thecase of the three-cell network at σ2
SI = −110 dB and Nr = 2.Fig. 10(a) plots the convergence of the objective functionsof the sum throughput maximization problem for the timesplitting scheme and the power splitting scheme, whereas Fig.10(b) plots the convergence of the objective function of the EHmaximization problem. As can be seen, the sum throughputmaximization problem achieve 90% of its final optimal valuewithin 40 iterations whereas the EH maximization problemneeds 10 iterations. Table I shows the average number of iter-ations required to solve each program. Note that each iterationof the proposed algorithms invokes a convex subproblem togenerate a new feasible point (V
(κ+1)D , V
(κ+1)U , α(κ+1)) that is
better than the incumbent (V(κ)D , V
(κ)U , α(κ)). Such a convex
subproblem can be solved efficiently by the available convexsolvers of polynomial complexity such as CVX [36]. To savethe computational time, it is recommended to input the in-cumbent (V
(κ)D , V
(κ)U , α(κ)) as the initial point for the process
of solving this subproblem. Also, the high dimensionalityand the nonconvexity of the considered problems imply thatchecking the global optimality of the computed solution isboth theoretically and practically prohibitive. Nevertheless, ourrecent results in [9] and [10] for the particular multi-inputsingle output (MISO) case of the HD optimization problem(29) show that both Algorithm 1 and Algorithm 2 are capableof delivering the global optimal solutions.
VI. CONCLUSION
14
0 20 40 60 80 10065
70
75
80
85
90
95
100
105
110
Number of iterations
Sum
thro
ughp
ut (
bps/
Hz)
FD−TSFD−PSHD−TSHD−PS
(a) Sum throughput maximization
0 5 10 15 20 25 300
1
2
3
4
5
6
Number of iterations
To
tal h
arve
sted
en
erg
y (m
W)
FDHD
(b) EH maximization
Fig. 10. Convergence of the proposed algorithm for ε = 10−4.
TABLE ITHE AVERAGE NUMBER OF ITERATIONS REQUIRED BY THE PROPOSED
ALGORITHMS
Programs Throughput max., Throughput max., EH max.TS PSFD 74 65.4 24.1HD 67.5 50.6 20.2
We have proposed new optimal precoding designs for EH-enabled FD multicell MU-MIMO networks. Specifically, sumthroughput maximization under throughput QoS constraintsand EH constraints for energy-constrained devices under eitherTS or PS has been considered. The FD EH maximizationproblem under throughput QoS constraints in TS has alsobeen addressed. Toward this end, we have developed new path-following algorithms for their solution, which require a convexquadratic program for each iteration and are guaranteed tomonotonically converge at least to a local optimum. Finally,we have demonstrated the merits of our proposed algorithmsthrough extensive simulations. Note that an interesting topicfor further research is this area is robust precoder/beamformer
design in the presence of channel estimation errors.
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Ho Huu Minh Tam was born in Ho Chi Minh City,Vietnam. He received the B.S. degree in electricalengineering and telecommunications from the HoChi Minh City University of Technology in 2012,and he is currently pursuing the Ph.D. degree withthe University of Technology Sydney, NSW, Aus-tralia, under the supervision of Prof. H. D. Tuan.His research interest is in optimization techniquesin signal processing for wireless communications.
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 Profes-sor with the School of Electrical Engineering and
Telecommunications, University of New South Wales, from 2003 to 2011.He is currently a Professor with the Faculty of Engineering and InformationTechnology, University of Technology Sydney. He has been involved inresearch with the areas of optimization, control, signal processing, wirelesscommunication, and biomedical engineering for more than 20 years.
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.
Trung Q. Duong (S’05, M’12, SM’13) received hisPh.D. degree in Telecommunications Systems fromBlekinge Institute of Technology (BTH), Sweden in2012. Since 2013, he has joined Queen’s Univer-sity Belfast, UK as a Lecturer (Assistant Profes-sor). His current research interests include small-cellnetworks, physical layer security, energy-harvestingcommunications, cognitive relay networks. He isthe author or co-author of 240 technical paperspublished in scientific journals (125 articles) andpresented at international conferences (115 papers).
Dr. Duong currently serves as an Editor for the IEEE TRANSACTIONSON WIRELESS COMMUNICATIONS, IEEE TRANSACTIONS ON COMMUNI-CATIONS, IET COMMUNICATIONS, and a Senior Editor for IEEE COMMU-NICATIONS LETTERS. He was awarded the Best Paper Award at the IEEEVehicular Technology Conference (VTC-Spring) in 2013, IEEE InternationalConference on Communications (ICC) 2014, and IEEE Global Communi-cations Conference (GLOBECOM) 2016. He is the recipient of prestigiousRoyal Academy of Engineering Research Fellowship (2016-2021).
H. Vincent Poor (S72, M77, SM82, F87) receivedthe Ph.D. degree in EECS from Princeton Universityin 1977. From 1977 until 1990, he was on the facultyof the University of Illinois at Urbana-Champaign.Since 1990 he has been on the faculty at Princeton,where he is currently the Michael Henry StraterUniversity Professor of Electrical Engineering. Dur-ing 2006 to 2016, he served as Dean of PrincetonsSchool of Engineering and Applied Science. Hisresearch interests are in the areas of informationtheory, statistical signal processing and stochastic
analysis, and their applications in wireless networks and related fields such assmart grid and social networks. Among his publications in these areas is thebook Mechanisms and Games for Dynamic Spectrum Allocation (CambridgeUniversity Press, 2014).
Dr. Poor is a member of the National Academy of Engineering, the NationalAcademy of Sciences, and is a foreign member of the Royal Society. He isalso a fellow of the American Academy of Arts and Sciences, the NationalAcademy of Inventors, and other national and international academies. Hereceived the Marconi and Armstrong Awards of the IEEE CommunicationsSociety in 2007 and 2009, respectively. Recent recognition of his workincludes the 2016 John Fritz Medal, the 2017 IEEE Alexander GrahamBell Medal, Honorary Professorships at Peking University and TsinghuaUniversity, both conferred in 2016, and a D.Sc. honoris causa from SyracuseUniversity awarded in 2017.
