beamforming design for wireless information and power
TRANSCRIPT
Beamforming Design for Wireless Information and Power TransferSystems: Receive Power-Splitting vs Transmit Time-Switching
Nasir, A. A., Tuan, H. D., Duong, T. Q., & Poor, H. V. (2016). Beamforming Design for Wireless Information andPower Transfer Systems: Receive Power-Splitting vs Transmit Time-Switching. IEEE Transactions onCommunications, 65(2), 876. https://doi.org/10.1109/TCOMM.2016.2631465
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Beamforming Design for Wireless Information andPower Transfer Systems: Receive Power-Splitting vs
Transmit Time-SwitchingAli A. Nasir, Hoang D. Tuan, Duy T. Ngo, Trung Q. Duong and H. Vincent Poor
Abstract—Information and energy can be transferred over thesame radio-frequency channel. In the power-splitting (PS) mode,they are simultaneously transmitted using the same signal bythe base station (BS) and later separated at the user (UE)’sreceiver by a power splitter. In the time-switching (TS) mode, theyare either transmitted separately in time by the BS or receivedseparately in time by the UE. In this paper, the BS transmitbeamformers are jointly designed with either the receive PS ratiosor the transmit TS ratios in a multicell network that implementswireless information and power transfer (WIPT). Imposing UEharvested energy constraints, the design objectives include (i)maximizing the minimum UE rate under the BS transmit powerconstraint, and (ii) minimizing the maximum BS transmit powerunder the UE data rate constraint. New iterative algorithms oflow computational complexity are proposed to efficiently solvethe formulated difficult nonconvex optimization problems, whereeach iteration either solves one simple convex quadratic programor one simple second-order-cone-program. Simulation resultsshow that these algorithms converge quickly after only a fewiterations. Notably, the transmit TS-based WIPT system is notonly more easily implemented but outperforms the receive PS-based WIPT system as it better exploits the beamforming designat the transmitter side.
Index Terms—Energy harvesting, power splitting, quadraticprogramming, second-order cone programming, time switching,transmit beamforming, wireless information and power transfer
I. INTRODUCTION
Dense small-cell deployment is identified as one of the ‘bigpillars’ to support the much needed 1, 000× increase in datathroughput for the fifth-generation (5G) wireless networks [1].While there is a major concern with the energy consumptionof such a dense small-cell deployment, recent advances inwireless power transfer allow the emitted energy in the radiofrequency signals to be harvested and recycled [2]–[6]. Thescavenged radio frequency (RF) energy is stored in the device
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 Faculty of Engineering and Information Technology,University of Technology Sydney, Broadway, NSW 2007, Australia (email:[email protected]).
D. T. Ngo is with the School of Electrical Engineering and Comput-ing, The University of Newcastle, Callaghan, NSW 2308, 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 (e-mail: [email protected]).
This work was supported in part by the U.K. Royal Academy of Engineer-ing Research Fellowship under Grant RF1415\14\22
battery and later used to power other signal processing andtransmitting operations. For example, an radio frequency-powered relay can be opportunistically deployed to extendnetwork coverage without the need to access a main powersupply. The wireless power transfer from a base station (BS)to its users (UEs) is viable in a dense small-cell environment,because the close BS-UE proximity enables an adequateamount of radio frequency (RF) energy to be harvested forpractical applications [7]–[9].
The two basic realizable receiver structures for separatingthe received signal for information decoding (ID) and energyharvesting (EH) are power splitting (PS) and time switching(TS) [10]. In the PS approach, information and energy aresimultaneously transmitted using the same signal by the BS.At the UE, a power splitter is employed to divide the receivedsignal into two parts of distinct powers, one for ID andanother for EH. In the receive TS approach, instead of thepower splitter a time switch is applied on the received signal,allowing the UE to decode the information in one portionof time and harvest the energy in the remaining time. In thetransmit TS approach, information and energy are transmittedby BS in different portions of time. The UE then processesthe received signals for ID and EH separately in time. TheTS structure has received considerable research attention (see[3], [11]–[13]) due its simple implementation. Although theperformance of the receive TS approach can be worse thanthe PS approach [3], that of the transmit TS approach has notbeen reported in the literature.
Transmit beamforming is beneficial for both PS-based andTS-based WIPT systems. With beamforming, the signal beamsare steered and the radio frequency (RF) energy is focusedat the desired UEs. Beamforming design without energyharvesting has been studied for multicell multi-input-single-output (MISO) [14]–[18] or single-cell MISO [19] networks.Except for [17] and [18], all the formulated problems aresolved in a decentralized manner by applying Lagrangianduality and uplink-downlink duality. In a single-cell energyharvesting MISO network with PS-based receivers, [20]–[24]jointly design transmit beamformers at the BS and receive PSratios at the UEs to minimize the sum beamforming powerunder UE signal-to-interference-plus-noise-ratio (SINR) andEH constraints. Such indefinite quadratic problem is thenrecast as a semidefinite program (SDP) with rank-one matrixconstraints. The rank-one matrix constraints are dropped tohave semidefinite relaxation (SDR) problem.To deal with therank-more-than-one solution given by SDR, [24] proposes us-
2
ing a randomization method after SDR. As shown in [25]–[27],the performance of such a method is inconsistent and couldbe poor in many cases. An approximate rank-one solutionwith compromised performance has been proposed in [28].Suboptimal algorithms based on zero-forcing and maximumratio transmission are proposed in [21] and [24]. As expected,they are outperformed by the SDR solution. Surprisingly, thejoint design of transmit beamformers and TS ratios at thereceivers has not been adequately addressed in the literaturealthough it is much easier to practically implement TS-basedreceivers. The main reason is that even the SDR approach doesnot lead to solutions with tractable computation in this case.Also to the best of our knowledge, such joint design has notbeen previously considered for the transmit TS case.
This paper addresses the joint design of transmit beamform-ing and either PS ratios or transmit TS ratios in a WIPT-enabled MISO multicell network. We choose to investigatethe transmit TS approach instead of the receive TS counterpartbecause of its potential to outperform the receive PS approach.As will be shown later, it is actually the case. Specifically,we consider two important design problems: 1) Maximizingthe minimum UE rate under BS transmit power and UEharvested energy constraints, and 2) Minimizing the maximumBS transmit power under the UE rate and harvested en-ergy constraints. As the considered optimization problems arehighly nonconvex, their global optimality is not theoreticallyguaranteed by any practical methods.
Here we exploit the partial convexity structure of theproblems to propose new algorithms based on either quadraticprogramming iteration or second-order cone iteration. Signif-icantly, our simulation results with practical parameters showthat the proposed algorithms for the receive PS-based WIPTsystem tightly approach the bounds provided by the SDRapproach. This observation demonstrates their ability to locatethe global optimum of the original nonconvex problems in theconsidered numerical examples. While the upper/lower boundis not available for the transmit TS-based WIPT system bythe SDR approach, our practical simulation results reveal thatthis system outperforms the receive PS-based system due itsability to efficiently exploit the transmit beamforming power.It is worth noting that the TS-based WIPT system is typicallysimpler to implement than the PS-based counterpart.
The rest of the paper is organized as follows: Section IIconsiders the optimization of the receive PS-based WIPTsystem whereas Section III considers the optimization of thetransmit TS-based WIPT system. Section IV evaluates the per-formance of our proposed algorithms by numerical examplesand analyzes their computational complexity. Finally, SectionV concludes the paper.
Notation. Standard notation is used throughout the paper.In particular, <{·} denotes the real part of its argument, ∇denotes the first-order differential operator, and 〈x,y〉 , xHy.
II. MAX-MIN RATE AND MIN-MAX POWEROPTIMIZATION FOR RECEIVE POWER-SPLITTING WIPT
SYSTEMS
Consider the downlink of a K-cell network. As shown inFig. 1, the BS of a cell k ∈ K , {1, . . . ,K} is equipped
BS 1 ...
UE (1,1)
UE (1,N1)
...
...
1,1,1h
1,1,1 Nh
Cell 1
BS K ...
UE (K,1)
UE (K,NK)
...
1,,KKh
KNKK ,,h
Cell K
Intended signal
Interference
1,1,Kh
1,1, NKh
Fig. 1. Downlink multiuser multicell interference scenario consisting of Kcells. To keep the drawing clear, we only show the interference scenario incell 1. In general, the interference occurs in all K cells.
with M > 1 antennas and it serves Nk single-antenna UEswithin its cell. By BS k and UE (k, n), we mean the BS thatserves cell k and the UE n ∈ Nk , {1, . . . , Nk} of the samecell, respectively. Assume universal frequency reuse where allUEs in all cells share the same frequency band. While theradio spectrum is best utilized in this approach, the signalinterference situation among multiple UEs in multiple cells ismost severe. Beamforming is then used to mitigate the effectof interference by steering the signal beams in the intendeddirections.
Denote by wk,n ∈ CM×1 the beamforming vector by BSk ∈ K for its UE (k, n) where n ∈ Nk , {1, . . . , Nk}. Lethk,k,n ∈ CM×1 be the flat fading channel vector betweenBS k and UE (k, n), which includes large-scale pathloss andsmall-scale fading. Denote xk,n as the information signal tobe transmitted by BS k to UE (k, n) where E{|xk,n|2} = 1.The complex baseband signal received by UE (k, n) is thenexpressed as:
yk,n =∑k∈K
hHk,k,n
∑n∈Nk
wk,nxk,n + zak,n, (1)
where zak,n ∼ CN (0, σ2a) is the zero-mean circularly complex
Gaussian noise with variance σ2a at the receive antenna of UE
(k, n). To show the effect of interference at UE (k, n), let usexplicitly write (1) as:
yk,n = hHk,k,nwk,nxk,n + hHk,k,n∑
n∈Nk\{n}
wk,nxk,n
+∑
k∈K\{k}
hHk,k,n
∑n∈Nk
wk,nxk,n + zak,n. (2)
The first term in (2) is the intended signal for UE (n, k), thesecond term is the intracell interference from within cell k,
3
and the third term is the intercell interference from other cellsk ∈ K \ {k}.
The short BS-UE distances allow the UEs to practicallyimplement the wireless information and power transfer. Thus,the UE (k, n) applies the power splitting (PS) technique tocoordinate both information decoding (ID) and energy harvest-ing (EH). Specifically, the power splitter divides the receivedsignal yk,n into two parts in the proportion of αk,n : (1−αk,n),where αk,n ∈ (0, 1) is termed as the PS ratio for UE (k, n).The first part √αk,nyk,n forms an input to the ID receiver as:√αk,nyk,n + zck,n =
√αk,n
∑k∈K
hHk,k,n
∑n∈Nk
wk,nxk,n + zak,n
+ zck,n, (3)
where zck,n ∼ CN (0, σ2c ) is the additional noise introduced by
the ID receiver circuitry. Upon denoting w , [wk,n]k∈K,n∈Nkand α , [αk,n]k∈K,n∈Nk , the signal-to-interference-plus-noiseratio (SINR) at the input of the ID receiver of UE (k, n) isgiven by:
SINRk,n ,|hHk,k,nwk,n|2
ϕk,n(w, αk,n), (4)
where
ϕk,n(w, αk,n) ,∑
n∈Nk\{n}
|hHk,k,nwk,n|2︸ ︷︷ ︸intracell interference
+∑
k∈K\{k}
∑n∈Nk
|hHk,k,nwk,n|2︸ ︷︷ ︸intercell interference
+σ2a + σ2
c/αk,n,
Assuming a normalized time duration of one second, theenergy of the second part
√1− αk,nyk,n of the received
signal yk,n is harvested by the EH receiver of UE (k, n) as
Ek,n(w, αk,n) , ζk,n(1− αk,n)(pk,n(w) + σ2
a
), (5)
where the constant ζk,n ∈ (0, 1) denotes the efficiency ofenergy conversion at the EH receiver,1 and
pk,n(w) ,∑k∈K
∑n∈Nk
|hHk,k,nwk,n|2.
Ek,n can be stored in a battery and later used to power theoperations of UE (k, n) (e.g., processing the received signalsin the downlink, or transmitting data to the BS in the uplink).
A. Max-Min Rate Iterative Optimization
First, we aim to consider max-min rate optimization prob-lem, which provides fairness in allocating the radio resourcesto the most disadvantaged user, especially that at the celledges. As this user suffers from severe interference andonly achieves low throughput, it is sensible to maximize itsthroughput for an acceptable quality of service. We aim tojointly optimize the transmit beamforming vectors wk,n and
1The value of ζk,n is typically in the range of 0.4−0.6 for practical energyharvesting circuits [5].
the PS ratios αk,n for all k ∈ K, and n ∈ Nk by solving thefollowing max-min rate optimization problem:
maxwk,n∈CM×1,αk,n∈(0,1),∀ k∈K, n∈Nk
mink∈K,n∈Nk
ln
(1 +|hHk,k,nwk,n|2
ϕk,n(w, αk,n)
)(6a)
s.t.∑n∈Nk
‖wk,n‖2 ≤ Pmaxk , ∀k ∈ K (6b)∑
k∈K
∑n∈Nk
‖wk,n‖2 ≤ Pmax, (6c)
Ek,n(w, αk,n) ≥ emink,n , ∀k ∈ K, n ∈ Nk. (6d)
Constraint (6b) caps the total transmit power of each BSk at a predefined value Pmax
k . Constraint (6c) ensures thatthe total transmit power of the network will not exceed theallowable budget Pmax, which helps limit any potential undueinterference from the considered multicell network to anothernetwork. Constraint (6d) requires that the minimum energyharvested by UE (k, n) exceeds some target emin
k,n for usefulEH. It is obvious that (6) is equivalent to the following max-min SINR problem:
maxwk,n∈CM×1,αk,n∈(0,1),∀ k∈K, n∈Nk
mink∈K,n∈Nk
fk,n(w, αk,n) ,|hHk,k,nwk,n|2
ϕk,n(w, αk,n)
s.t. (6b)− (6d). (7)
While (6b) and (6c) are convex, the objective in (7) is notconcave and the constraint (6d) is not convex due to the strongcoupling between wk,n and αk,n in both the SINR and EHexpressions [see (4) and (5)]. Moreover, the objective in (7)is also nonsmooth due to the minimization operator. Indeed,(7) is a nonconvex nonsmooth function optimization problemsubject to nonconvex constraints. If one fixes αk,n at someconstants, problem (7) would still be nonconvex in wk,n. It isnot straightforward to even find a feasible solution that satisfiesconstraints (6b)-(6d).
In principle, both problems (6) and (7) could be solvedby the d.c. optimization framework of [29] and [30], whereeach function fk,n(w, αk,n) in the objective (6a) would berecast as a d.c. (difference of two convex functions) functionin numerous constrained additional variables. The objective
mink∈K,n∈Nk
fk,n(w, αk,n) in (6a) would then be represented as
a difference of a convex nonsmooth function and a smoothconvex function for the d.c. iteration technique of [31] toapply. In this paper, we will develop a new and more efficientapproach to solve problem (7).
As observed in [32], for wk,n = e−.arg(hHk,k,nwk,n)wk,n,one has |hHk,k,nwk,n| = hHk,k,nwk,n = <{hHk,k,nwk,n} ≥ 0
and |hHk′,k,n′wk,n| = |hHk′,k,n′wk,n| for (k′, n′) 6= (k, n) and ,√−1. The original problem (7) is thus equivalent to the
4
following optimization problem:
maxwk,n∈CM×1,αk,n∈(0,1),∀ k∈K, n∈Nk
mink∈K,n∈Nk
fk,n(w, αk,n) ,(<{hHk,k,nwk,n})2
ϕk,n(w, αk,n)(8a)
s.t.<{hHk,k,nwk,n} ≥ 0, ∀ k ∈ K, n ∈ Nk, (8b)(6b), (6c), (6d) .(8c)
Since the function fk,n(wk,n, t) , (<{hHk,k,nwk,n})2/t isconvex in wk,n ∈ CM×1 and t > 0 [26], it is true that [33]
fk,n(wk,n, t) ≥ fk,n(w(κ)k,n, t
(κ))
+ 〈∇fk,n(w(κ)k,n, t
(κ)), (wk,n, t)− (w(κ)k,n, t
(κ))〉
= 2<{hHk,k,nw
(κ)k,n
}<{hHk,k,nwk,n
}/t(κ)
−(<{hHk,k,nw
(κ)k,n
})2
t/(t(κ))2 (9)
for all wk,n ∈ CM×1,w(κ)k,n ∈ CM×1, t > 0, t(κ) > 0.
Therefore, given (w(κ), α(κ)) from κ-th iteration, substitutingt := ϕk,n(w, αk,n) and t(κ) := ϕk,n(w(κ), α
(κ)k,n) into the
above inequality (9) gives
fk,n(w, αk,n) ≥ f (κ)k,n(w, αk,n), ∀(w, αk,n) (10)
where
f(κ)k,n(w, αk,n) ,
2<{
(w(κ)k,n)Hhk,k,n
}<{hHk,k,nwk,n
}ϕk,n(w(κ), α
(κ)k,n)
−
(<{hHk,k,nw
(κ)k,n
})2
ϕk,n(w, αk,n)
ϕ2k,n(w(κ), α
(κ)k,n)
(11)
The function f(κ)k,n(w, αk,n) is concave quadratic and agrees
with fk,n(w, αk,n) at (w(κ), α(κ)k,n) as:
fk,n(w(κ), α(κ)k,n) = f
(κ)k,n(w(κ), α
(κ)k,n). (12)
Next, the nonconvex energy harvesting constraint (6d) canbe expressed as
emink,n
ζk,n(1− αk,n)− pk,n(w) ≤ 0, ∀k ∈ K, n ∈ Nk, (13)
which is still nonconvex. From
|hHk,k,nwk,n|2 ≥ −|hHk,k,nw(κ)
k,n|2
+ 2<{(
w(κ)
k,n
)Hhk,k,nh
Hk,k,nwk,n
},
∀wk,n,w(κ)
k,n(14)
it follows that
pk,n(w) ≥ p(κ)k,n(w), ∀w and pk,n(w(κ)) = p
(κ)k,n(w(κ))
(15)where
p(κ)k,n(w) , −pk,n(w(κ))
+ 2∑k∈K
∑n∈Nk
<{
(w(κ)
k,n)Hhk,k,nh
Hk,k,nwk,n
}.
Algorithm 1 QP-based Iterative Optimization to Solve Prob-lem (7)
1: Initialize κ := 0.2: Choose a feasible point (w
(0)k,n, α
(0)k,n), ∀k ∈ K, n ∈ Nk of
(7).3: repeat4: Solve QP (17) for w
(κ+1)k,n and α
(κ+1)k,n , ∀k ∈ K, n ∈
Nk.5: Set κ := κ+ 1.6: until convergence of the objective in (7).
Therefore, whenever (w(κ), α(κ)) is feasible to (6d), thenonconvex constraint (6d) is inner-approximated by the convexconstraint
emink,n
ζk,n(1− αk,n)− p(κ)
k,n(w) ≤ σ2a, ∀k ∈ K, n ∈ Nk. (16)
From (12) and (16), for a given (w(κ)k,n, α
(κ)k,n) the following
convex quadratic program (QP) provides minorant maximiza-tion for the nonconvex program (7):
maxwk,n∈CM×1,αk,n∈(0,1),∀k∈K,n∈Nk
mink∈Kn∈Nk
f(κ)k,n(w, αk,n)
s.t. (6b), (6c), (8b), (16). (17)
Using (17), we propose in Algorithm 1 a QP-based iterativealgorithm that solves the max-min SINR problem (7). Here,the initial point w(0) , [w
(0)k,n]k∈K,n∈Nk can be found by
randomly generating M × 1 complex vectors followed bynormalizing them to satisfy (6b) and (6c). For a given w(0),α(0) , [α
(0)k,n]k∈K,n∈Nk is then generated by solving (6d) with
an equality sign. In each iteration of Algorithm 1, only onesimple QP (17) needs to be solved. The solution of which isthen used to improve the objective value in the next iteration.
Proposition 1: Algorithm 1 generates a sequence{(w(κ),α(κ))} of improved points for (7), which convergesto a Karush-Kuhn-Tucker (KKT) point.
Proof: Let us define
F (w,α) , mink∈Kn∈Nk
fk,n(w, αk,n) and
F (κ)(w,α) , mink∈Kn∈Nk
f(κ)k,n(w, αk,n),
which satisfies [cf. (10) and (12)]
F (κ)(w,α) ≥ F (κ)(w,α) ∀ w,α and
F (κ)(w(κ),α(κ)) = F (κ)(w(κ),α(κ)).
Hence,
F (w(κ+1),α(κ+1)) ≥ F (κ)(w(κ+1),α(κ+1))
> F (κ)(w(κ),α(κ)) = F (w(κ),α(κ)),
where the second inequality follows from the fact that(w(κ+1),α(κ+1)) and (w(κ),α(κ)) are the optimal solution
5
and a feasible point of (17), respectively. This result shows that(w(κ+1),α(κ+1)) is a better point to (7) than (w(κ),α(κ)).
Furthermore, the sequence {(w(κ),α(κ))} is bounded byconstraints (6b) and (6c). By Cauchy’s theorem, there is aconvergent subsequence {(w(κν),α(κν))} with a limit point(w, α), i.e.,
limν→+∞
[F (w(κν),α(κν))− F (w, α)
]= 0.
For every κ, there is ν such that κν ≤ κ ≤ κν+1, and so
0 = limν→+∞
[F (w(κν),α(κν))− F (w, α)]
≤ limκ→+∞
[F (w(κ),α(κ))− F (w, α)]
≤ limν→+∞
[F (w(κν+1),α(κν+1))− F (w, α)] = 0, (18)
which shows that limκ→+∞
F (w(κ),α(κ)) = F (w, α). Each
accumulation point {(w, α)} of the sequence {(w(κ),α(κ))}is indeed a KKT point according to [34, Th. 1].
It is noteworthy that our simulation results in Sec. IV furthershow that the QP-based solution in Algorithm 1 achieves theupper bound given by the SDR (A.1) described in Appendix A.
B. Iterative Optimization for Min-Max BS Power
Next, we will address the min-max BS power optimizationproblem, which targets minimizing the highest BS radiationpower. The motivation is to limit an undue interferencefrom any BS to the neighboring cell users. Therefore, theinterference (and hence throughput) at individual users isbalanced across the whole network. The min-max BS poweroptimization problem is formulated as follows:
minwk,n∈CM×1,αk,n∈(0,1),∀ k∈K, n∈Nk
maxk∈K
∑n∈Nk
‖wk,n‖2 s.t. (6d), (19a)
|hHk,k,nwk,n|2 ≥ γmink,n ϕk,n(w, αk,n),
∀k ∈ K, n ∈ Nk. (19b)
Here, (6d) requires that the amount of energy harvested byUE (k, n) exceeds some target emin
k,n for useful EH, whereas
(19b) ensures a minimum throughput ln(
1 + γmink,n
)for each
UE (k, n). Similar to the max-min SINR problem (7), thisproblem (19) is nonconvex due to the strong coupling betweenwk,n and αk,n in the harvested energy expression (5).
Given that the SINR constraint (19b) can be expressed as asecond-order cone (SOC) constraint2, we now address problem(19) via second-order cone programming (SOCP) in the vectorvariables wk,n ∈ CM×1. Similar to (8b), we make the variablechange αk,n → α2
k,n in (19) to express (19b) as:
<{hHk,k,nwk,n
}≥√γmink,n
√ϕk,n(w, α2
k,n), ∀k ∈ K, n ∈ Nk,(20)
2This only means the SINR function is quasi-convex. Therefore, the SOCP-based optimization approach cannot be applied to solve problem (7).
Algorithm 2 SOC-based Iterative Optimization to Solve Prob-lem (19)
1: Initialize κ := 0.2: Choose a feasible point (w
(0)k,n, α
(0)k,n), ∀k ∈ K, n ∈ Nk of
(19).3: repeat4: Solve SOCP (24) for w(κ+1)
k,n and α(κ+1)k,n , ∀k ∈ K, n ∈
Nk.5: Set κ := κ+ 1.6: until convergence of the objective (19).
which is equivalent to the following SOC:
<{hHk,k,nwk,n} ≥√γmink,n
∥∥∥∥∥∥∥σa
σctk,n(hHk,k,n
wk,n
)k,n∈K,N\{k,n}
∥∥∥∥∥∥∥2
,
∀k ∈ K, n ∈ N , (21)(tk,n 11 αk,n
)� 0, ∀ k ∈ K, n ∈ N , (22)
where(hHk,k,n
wk,n
)k,n∈K,N\{k,n}
is an (KN−1)×1 column
vector. On the other hand, under the variable change αk,n →α2k,n in (16), the harvested energy expression (5) is inner-
approximated by the following convex constraints:
emink,n
ζk,n(1− α2k,n)− p(κ)
k,n(w) ≤ σ2a, ∀k ∈ K, n ∈ Nk. (23)
As (w(κ), α(κ)) is also feasible to (23), the optimal solution(w(κ+1), α(κ+1)) of the following convex program is a betterpoint to (19) than (w(κ), α(κ))
minwk,n∈CM×1
αk,n∈(0,1), tk,n∀k∈K,n∈Nk
maxk∈K
∑n∈Nk
‖wk,n‖2 s.t. (21), (22), (23).
(24)In Algorithm 2, we propose an SOC-based iterative algo-
rithm to solve problem (19). In order to obtain an initial feasi-ble point to SOCP (24), we cannot use the inner-approximatedEH constraint (23) due to its dependence on the previouslyoptimized beamforming vector w(κ). However, the original EHconstraint (6d) can be implied by the following hard (tighter)constraint in α2
k,n1and a slack variable β2
k,n1:√
emink,n /ζk,n
βk,n−<{hHk,k,nwk,n} ≤ 0,∀k ∈ K, n1 ∈ Nk
(25a)
β2k,n + α2
k,n ≤ 1,∀k ∈ K, n ∈ Nk, (25b)
which is independent of w(κ).3 Hence, the initial point toSOCP (24) can easily be obtained by solving the following
3The satisfaction of the constraint (25) implies the satisfaction of theoriginal EH constraint (6d).
6
SOCP:
minwk,n∈CM×1,
αk,n∈(0,1), βk,n, tk,n,∀k∈K,n∈Nk
maxk∈K
∑n∈Nk ‖wk,n‖2
s.t. (21), (22), (25a), (25b). (26)
Once initialized from a feasible point, Algorithm 2 solves onesimple convex SOCP (24) in each iteration. The solution ofwhich is then used in the next iteration to improve the objectivevalue. Similar to Proposition 1, it can be shown that Algorithm2 generates a sequence {(w(κ),α(κ))} of improved points forproblem (19), which converges to a KKT point. Our simulationresults in Sec. IV further show that the SOC-based solutionin Algorithm 2 achieves the lower bound given by the SDR(A.2a), (A.2b), (A.1e), (A.1f) described in Appendix A.
III. MAX-MIN RATE AND MIN-MAX POWEROPTIMIZATION FOR TRANSMIT TIME-SWITCHING WIPT
SYSTEMS
Unlike the power-switching system model in Sec. II, inthe time-switching (TS) based system, a fraction of time0 < ρ < 1 is used for power transfer while the remainingfraction of time (1 − ρ) for information transfer. Here ρ istermed as the TS ratio. For power transfer, we are to designbeamforming vectors wE
k,n with the achievable harvestedenergy ρEk,n(wE), where
Ek,n(wE) , ζk,n(pk,n(wE) + σ2a),
pk,n(wE) ,∑k∈K
∑n∈Nk
|hHk,k,nwEk,n|
2,
and wE , [wEk,n]k∈K,n∈Nk . For information transfer, we are
to design beamforming vectors wIk,n with the achievable data
rate
(1− ρ) ln
(1 +|hHk,k,nwI
k,n|2
ϕk,n(wI)
)where
ϕk,n(wI) ,∑
n∈Nk\{n}
|hHk,k,nwIk,n|2︸ ︷︷ ︸
intracell interference
+∑
k∈K\{k}
∑n∈Nk
|hHk,k,nwIk,n|
2
︸ ︷︷ ︸intercell interference
+σ2a,
and wI , [wIk,n]k∈K,n∈Nk . Therefore, the individual BS and
total power constraints for the TS-based system are:
ρ∑n∈Nk
‖wEk,n‖2 +(1− ρ)
∑n∈Nk
‖wIk,n‖2 ≤ Pmax
k ,∀k ∈ K
(27a)
ρ∑k∈K
∑n∈Nk
‖wEk,n‖2 + (1− ρ)
∑k∈K
∑n∈Nk
‖wIk,n‖2 ≤ Pmax,
(27b)
respectively. Here, the following constraints must also beimposed:
‖wEk,n‖2 ≤ Pmax, ‖wI
k,n‖2 ≤ Pmaxk , ∀k ∈ K, n ∈ Nk.
(28)
The max-min rate optimization problem for the TS-basedsystem is then formulated as:
max0<ρ<1,
wxk,n∈CM×1, x∈{I,E}
mink∈K,n∈Nk
(1− ρ) ln(1 + fk,n(wI))
(29a)
s.t. ρEk,n(wE) ≥ emink,n , (29b)
(27), (28). (29c)
And the min-max BS power optimization problem for the TS-based system is formulated as:
min0<ρ<1,
wxk,n∈CM×1, x∈{E,I}
maxk∈K
ρ∑n∈Nk
‖wEk,n‖2 + (1− ρ)
∑n∈Nk
‖wIk,n‖2
(30a)
s.t. (1− ρ) ln(1 + fk,n(wI)
)≥ rmin
k,n ,(30b)
(28), (29b) (30c)
where (30b) ensures the minimum rate rmink,n (in nat/sec/Hz) is
achieved.Remark 1: The transmit TS-based WIPT system is different
from the receive TS-based WIPT system [10] which switchesthe received signal yk,n in (1) in the proportion of time 0 <αk,n < 1 for information decoding. Accordingly, the jointdesign of transmit beamformer w and receive TS ratios α ,[αk,n]k∈K,n∈Nk is formulated as:
max0<αk,n<1,
wk,n∈CM×1
mink∈K,n∈Nk
(1− αk,n) ln(1 + fk,n(w))
s.t. (6b), (6c), and αk,nEk,n(w) ≥ emink,n , (31)
and
min0<αk,n<1,
wk,n∈CM×1
maxk∈K
∑n∈Nk
‖wk,n‖2
s.t. αk,nEk,n(w) ≥ emink,n ,
(1− αk,n) ln(1 + fk,n(w)) ≥ rmink,n . (32)
Compared with the receive PS-based optimization problems(7) and (19), the power and EH constraints in (31) and (32)remain the same while the data rate in (31) and (32) is lower.The receive PS-based design thus outperforms the receive TS-based design in general. On the other hand, the transmit TS-based optimizations (29) and (30) exploit the separate designsof wI for ID and wE for EH. For this reason, they outperformthe receive PS-based designs in (7) and (19) as will be shownlater.
A. Iterative Max-Min Rate Optimization
We will now solve the nonconvex problem (29). First, letus make the following change of variable:
1− ρ = 1/β, (33)
which satisfies the linear constraint
β > 1. (34)
7
Thus, the power constraints (27) become the following convexconstraints:∑
n∈Nk
‖wEk,n‖2 +
1
β
∑n∈Nk
‖wIk,n‖2 ≤
Pmaxk +
1
β
∑n∈Nk
‖wEk,n‖2, ∀k ∈ K
(35a)∑k∈K
∑n∈Nk
‖wEk,n‖2 +
1
β
∑k∈K
∑n∈Nk
‖wIk,n‖2 ≤
Pmax +1
β
∑k∈K
∑n∈Nk
‖wEk,n‖2. (35b)
Similar to (8), problem (29) can now be equivalently expressedby
maxα,β,
wxk,n∈CM×1,x∈{I,E}
mink∈K,n∈Nk
1
βln
(1 +
(<{hHk,k,nwIk,n})2
ϕk,n(wI)
)(36a)
s.t. <{hHk,k,nw
Ik,n
}≥ 0, ∀k ∈ K, n ∈ Nk, (36b)
pk,n(wE) ≥emink,n
ζk,n
(1 +
1
β − 1
)− σ2
a, (36c)
(28), (34), (35). (36d)
Note that unlike (8), the objective function in (36) is quitecomplex to handle due to the additional factor 1/β, while thepower constraint (35) is nonconvex. To deal with this, we firstexploit the fact that function f(x, t) = ln(1+1/x)
t is convex inx > 0, t > 0 which can be seen by examining its Hessian. Thefollowing inequality for all x > 0, x > 0, t > 0 and t > 0then holds true:
ln(1 + 1/x)
t≥ f(x, t) + 〈∇f(x, t), (x, t)− (x, t)〉
= 2ln(1 + 1/x)
t+
1
t(x+ 1)− x
(x+ 1)xt
− ln(1 + 1/x)
t2t. (37)
By replacing 1/x→ x and 1/x→ x in (37), we have:
ln(1 + x)
t≥ a− b
x− ct, (38)
where a = 2 ln(1+x)t + x
t(x+1) > 0, b = x2
t(x+1) > 0, c =ln(1+x)t2 > 0. From that,
1
βln
1 +
(<{hHk,k,nwI
k,n})2
ϕk,n(wI)
≥a(κ) − b(κ) ϕk,n(wI)(
<{hHk,k,nw
Ik,n
})2 − c(κ)β (39)
where
a(κ) = 2ln(1 + d(κ))
β(κ)+
d(κ)
β(κ)(d(κ) + 1)> 0,
b(κ) =(d(κ))2
β(κ)(d(κ) + 1)> 0,
c(κ) =ln(1 + d(κ))
(β(κ))2> 0,
d(κ) =(<{hHk,k,nw
I,(κ)k,n
})2 /ϕk,n(wI,(κ)). (40)
Now, using(<{hHk,k,nw
Ik,n
})2 ≥ 2<{hHk,k,nwI,(κ)k,n }<
{hHk,k,nw
Ik,n
}−(<{hHk,k,nw
I,(κ)k,n
})2
, ψk,n(wIk,n)
together with (39) leads to
1
βln
1 +
(<{hHk,k,nw
Ik,n
})2
ϕk,n(wI)
≥ a(κ) − b(κ) ϕk,n(wI)
ψk,n(wIk,n)
− c(κ)β
, f(κ)k,n(wI , β) (41)
forψk,n(wI
k,n) ≥ 0, ∀k ∈ K, n ∈ Nk. (42)
As the function f(κ)k,n(wI , β) is concave on (42), the follow-
ing convex program provides minorant maximization for thenonconvex program (36) for a given (wE,(κ),wI,(κ), β(κ)):
maxβ,
wxk,n∈CM×1, x∈{I,E}
mink∈K,n∈Nk
f(κ)k,n(wI , β) (43a)
s.t∑n∈Nk
‖wEk,n‖2 +
1
β
∑n∈Nk
‖wIk,n‖2 ≤ Pmax
k
+1
β(κ)
∑n∈Nk
2<{
(wE,(κ)k,n )HwE
k,n
}− β
(β(κ))2
∑n∈Nk
‖wE,(κ)k,n ‖
2, ∀k ∈ K, (43b)
∑k∈K
∑n∈Nk
‖wEk,n‖2 +
1
β
∑k∈K
∑n∈Nk
‖wIk,n‖2 ≤ Pmax
+1
β(κ)
∑k∈K
∑n∈Nk
2<{
(wE,(κ)k,n )HwE
k,n
}− β
(β(κ))2
∑k∈K
∑n∈Nk
‖wE,(κ)k,n ‖
2, (43c)
∑k∈K
∑n∈Nk
[2<{hHk,k,nw
E,(κ)
k,nhHk,k,nw
Ek,n
}−∣∣∣hHk,k,nwE,(κ)
k,n
∣∣∣2] ≥ emink,n
ζk,n
(1 +
1
β − 1
)− σ2
a,
(43d)(28), (34), (36b), (42) (43e)
8
Algorithm 3 Iterative Optimization to Solve Problem (36)1: Initialize κ := 0.2: Choose a feasible point
(wE,(0),wI,(0), β(0)
)of (36).
3: repeat4: Solve convex program (43) for(
wE,(κ+1),wI,(κ+1), β(κ+1)).
5: Set κ := κ+ 1.6: until convergence of the objective in (36).
Here, convex constraints (43b), (43c) and (43d) are the innerapproximations of nonconvex constraints (35) and (36d) dueto the convexity of function 1
β ‖x‖2, which leads to
‖x‖2
β≥
2<{
(x(κ))Hx}
β(κ)− ‖x
(κ)‖2
(β(κ))2β,
∀x ∈ CN ,x(κ) ∈ CN , β > 0, β(κ) > 0. (44)
The proposed solution for the max-min rate problem (36)(and hence (29)) is summarized in Algorithm 3. Similar toProposition 1, it can be shown that Algorithm 3 generates asequence
{(wE,(κ),wI,(κ), β(κ)
)}of improved points of (36),
which converges to a KKT point. In Algorithm 3, the feasiblepoint
(wE,(0),wI,(0), β(0)
)of (36) is found as follows. We fix
β(0) and solve the following convex problem for fixed rmin >0:
maxwxk,n∈CM×1,x∈{I,E}
mink∈K,n∈Nk
<{hHk,k,nw
Ek,n
}−√emink,n /
(ζk,n
(1− 1/β(0)
)), (45a)
s.t. <{hHk,k,nw
Ik,n
}≥√erminβ(0) − 1
×
∥∥∥∥∥ σa(hHk,k,n
wIk,n
)k,n∈K,N\{k,n}
∥∥∥∥∥2
,
k ∈ K, n ∈ N , (45b)(1− 1/β(0)
) ∑n∈Nk
‖wEk,n‖2 +
(1/β(0)
)×∑n∈Nk
‖wIk,n‖2 ≤ Pmax
k , ∀k ∈ K, (45c)(1− 1/β(0)
)∑k∈K
∑n∈Nk
‖wEk,n‖2 +
(1/β(0)
)×∑k∈K
∑n∈Nk
‖wIk,n‖2 ≤ Pmax, (45d)
(28) (45e)
and then iteratively solve the following convex problem:
maxwxk,n∈CM×1,x∈{I,E}
mink∈K,n∈Nk
∑k∈K
∑n∈Nk
[2<{hHk,k,nw
E,(κ)
k,n
×hHk,k,nwEk,n
}−∣∣∣hHk,k,nwE,(κ)
k,n
∣∣∣2]−emink,n
ζk,n
(1 +
1
β(0) − 1
)− σ2
a
s.t. (28), (45b), (45c), (45d) (46)
Algorithm 4 Iterative Optimization to Solve Problem (47)1: Initialize κ := 0.2: Choose a feasible point
(wE,(0),wI,(0), β(0)
)of (47).
3: repeat4: Solve the convex program (48) for(
wE,(κ+1),wI,(κ+1), β(κ+1)).
5: Set κ := κ+ 1.6: until convergence of the objective (48).
where the initial point wE,(0)k,n for (46) is obtained from the
solution of (45). Problem (46) is solved for κ = 0, 1, 2, . . .until a positive optimal value is attained. If problem (45)or (46) is infeasible with β(0) or solving (46) fails to givea positive optimal value, we repeat the above process fora different value of β(0) in order to find a feasible point(wE,(0),wI,(0), β(0)
).4
B. Iterative Min-Max Power Optimization
We now turn our attention to the min-max BS poweroptimization problem (30), which is equivalently expressedas:
minβ>0,
wxk,n∈CM×1, x∈{E,I}
maxk∈K
[ ∑n∈Nk
‖wEk,n‖2 +
1
β
∑n∈Nk
‖wIk,n‖2
− 1
β
∑n∈Nk
‖wEk,n‖2
](47a)
s.t. (28), (34), (36d), (47b)1
βln(1 + fk,n(wI)
)≥ rmin
k,n . (47c)
From (41) and (44), the following convex program providesmajorant minimization for the nonconvex program (47) for agiven (w
E,(κ)k,n ,w
I,(κ)k,n , β(κ)):
minβ>0,
wxk,n∈CM×1, x∈{E,I}
maxk∈K
[ ∑n∈Nk
‖wEk,n‖2 +
1
β
∑n∈Nk
‖wIk,n‖2
− 1
β(κ)
∑n∈Nk
2<{
(wE,(κ)k,n )HwE
k,n
}+
β
(β(κ))2
∑n∈Nk
‖wE,(κ)k,n ‖
2
](48a)
s.t. (28), (34), (36b), (42), (43d), (48b)
f(κ)k,n
(wI , β
)≥ rmin
k,n , (48c)
where f (κ)k,n(wI , β) is defined in (41).
The proposed solution for the min-max BS power opti-mization problem (47) (and hence (30)) is summarized inAlgorithm 4. Similar to Proposition 1, it can be shown thatAlgorithm 4 generates a sequence
{(wE,(κ),wI,(κ), β(κ)
)}of
improved points of (48), which converges to a KKT point. In
4Simulation results in Sec. IV show that in almost all of the scenariosconsidered, problems (45) or (46) are feasible and a positive optimal value of(46) is obtained in one single iteration for the first tried value β(0) = 1.11.
9
0 50 100 1500
20
40
60
80
100
120
140
160
x−coordinate (m)
y−co
ordi
nate
(m
)
Base StationUser
Fig. 2. Topology of the multicell network used in the numerical examples
Algorithm 4, the feasible{(
wE,(0),wI,(0), β(0))}
of (47) canbe found by first fixing β(0) and solving the convex feasibilityproblem with the following constraints:√
emink,n /
(ζk,n
(1− 1/β(0)
))−<{hHk,k,nwE
k,n} ≤ 0,
k ∈ K, n ∈ N , (49a)
<{hHk,k,nw
Ik,n
}≥√er
mink,n β
(0) − 1
×
∥∥∥∥∥ σa(hHk,k,n
wIk,n
)k,n∈K,N\{k,n}
∥∥∥∥∥2
,
k ∈ K, n ∈ N , (49b)(28), (36b). (49c)
IV. NUMERICAL EXAMPLES AND COMPLEXITY ANALYSIS
In the numerical examples, a 3-cell network model with 4UEs per cell shown in Fig. 2 is used. The cell radius is set as 40m and the BS-to-UE distance as 20 m to enable practical WIPT[7], [8]. For large-scale propagation loss, a pathloss exponentequal to 4 is assumed. For small-scale fading, a Rician fadingchannel is generated according to
hk,k,n =
√KR
1 +KRhLOSk,k,n +
√1
1 +KRhNLOSk,k,n, ∀k, k, n
(50)
where KR = 10 dB is the Rician factor; hLOSk,k,n
∈ CM×1 isthe line-of-sight (LOS) deterministic component; and hNLOS
k,k,n∼
CN (0, 1) is the circularly-symmetric complex Gaussian ran-dom variable that models the Rayleigh fading component.Here, the far-field uniform linear antenna array model is usedwith
hLOSk,k,n =
[1, ejθk,k,n , ej2θk,k,n , . . . , ej(M−1)θk,k,n
]Tfor θk,k,n = 2πd sin(φk,k,n)/λ, where d = λ/2 is theantenna spacing, λ is the carrier wavelength and φk,k,n is
the direction of UE (k, n) to BS k [21]. In the simulations,φk,k,n is generated as a random angle between 0o and 360o.For simplicity and without loss of generality, we assume thatγmink,n = γ in (19b), rmin
k,n = r in (30b), and ζk,n = ζ,emink,n = e, ∀k, n. Unless specified otherwise, we set the target
minimum EH threshold as e = −20 dBm. In all simulations,we also set ζ = 0.5, σ2
a = −90 dBm and σ2c = −90 dBm.
The error tolerance used in the stopping condition is set as10−3 for all algorithms. All simulations are conducted usingMATLAB 2015b and CVX 2.1 [35].
A. Results for Max-Min Rate Problems (7) and (29)
Algorithm 1, the nonsmooth optimization algorithm of [27]and the SDR approach are used to solve the PS-based problem(7), whereas Algorithm 3 is to solve the TS-based problem(29). Assuming that Pmax = 22 dBW, Figs. 3 and 4 plotthe maximized minimum UE rate for different values of BStransmit power Pmax
k and BS transmit antenna number M .Figs. 3 and 4 show that the minimum UE rate improves byincreasing the power budget Pmax
k and the number of BSantennas M , respectively, due to an increase in the availableradio resources. In Fig. 4, we also evaluate the performanceof the algorithms for different values of the target minimumEH threshold, e = {−20,−10} dBm. Fig. 4 shows that byincreasing the target EH threshold from e = −20 dBm toe = −10 dBm, the achievable information rate is reduced sincemore time (in the TS-based system) or power (in the PS-basedsystem) is required to meet the increased EH requirement.However, it is important to mention that the percentage ofdecrease in the information rate for the TS-based system isquite less than that for the PS-based system.
In addition, we can see from Figs. 3 and 4 that theperformance of Algorithm 1 coincides with the upper boundobtained by the SDR approach in all the considered simulationsetups. Although the proposed algorithm of [27] also achieves
10
8 9 10 11 12 13 14 15 16
Pmax
k(dBW)
1.6
1.8
2
2.2
2.4
2.6
2.8
3
3.2
min
rate
(bits/sec/Hz)
SDR (upper bound for PS)Algorithm [26]Proposed Algorithm 1 (PS)Proposed Algorithm 3 (TS)
Fig. 3. Maximized minimum UE rate for M = 4 and Pmax = 22 dBW.
4 5 6
Number of BS transmit antenna, M
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
min
rate
(bits/sec/Hz)
SDR (upper bound for PS)Algorithm [26]Proposed Algorithm 1 (PS)Proposed Algorithm 3 (TS)
e = −20 dBm
e = −10 dBm
Fig. 4. Maximized minimum UE rate for Pmax = 22 dBW, Pmaxk = 16
dBW and e = {−20,−10} dBm.
-20 -18 -16 -14 -12 -10
Target minimum EH threshold, e (dBm)
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
TSratio(ρ)
Proposed Algorithm 3, M=4Proposed Algorithm 3, M=5Proposed Algorithm 3, M=6
Fig. 5. Optimzed TS ratio ρ by Algorithm 3 for different numbers of BSantennas M and Pmax
k = 16 dBW.
0 2 4 6 8 10 12 14 16 18iterations
0
1
2
3
4
5
6
7
8
9
min
rate
(bits/sec/Hz)
SDR (upper bound for PS)Proposed Algorithm 1 (PS)Proposed Algorithm 3 (TS)
Fig. 6. Convergence of proposed Algorithms 1 and 3 for M = 4 andPmaxk = 16 dBW.
this bound, it requires much higher computational complexitythan Algorithm 1 as will be analyzed shortly. It should benoted that Algorithm 1 does not perform any bisection searchas is the case for both the SDR approach and the algorithm of[27]. Note further that the SDR approach only provides rank-one matrices W?
k,n in no more than 61.7% of the time [27]. Incontrast, the nonsmooth optimization algorithm of [27] alwaysreturns rank-one matrix solutions, and Algorithm 1 of coursedirectly gives the optimal vectors w?
k,n because no matrixoptimization is involved. Figs. 3 and 4 also show that transmitTS-based WIPT system with Algorithm 3 considerably out-performs the receive PS-based counterpart. Such throughputenhancement is generally not possible with the receive TS-based WIPT system as has been reported in the literature. Withits high performance and easy implementation, the transmitTS-based solution could be an attractive candidate for practicalWIPT systems.
Applying Algorithm 3 for the max-min rate problem (29),Fig. 5 plots the optimized value of the transmit TS ratio ρ fordifferent values of the target EH threshold e and the numberof BS antennas M = {4, 5, 6}. As can be seen, by increasingthe target EH threshold e the optimized TS ratio ρ increasessince more time duration is required to fulfill the increased EHrequirement. It is further observed that the optimized value ofthe TS ratio ρ is smaller in the presence of a larger numberof BS antennas.
Fig. 6 illustrates the fast convergence of Algorithms 1 and 3which terminate in as few as 8 and 4 iterations, respectively.Here, each iteration corresponds to solving one simple QP(17) in Algorithm 1, one convex problem (43) in Algorithm3, and one SDP (A.1a)–(A.1f) in the SDR approach. Note thatinitializing the proposed Algorithms 1 and 3 only requires asingle iteration.
The computational complexities of Algorithm 1, the nons-
11
2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4
Target minimum rate, r (bits/sec/Hz)
2
4
6
8
10
12
14
16m
ax B
S tra
nsm
it p
ow
er
(dB
W)
SDR (lower bound for PS)Proposed Algorithm 2 (PS)Proposed Algorithm 4 (TS)
Fig. 7. Minimized maximum BS transmit power for M = 5.
4 5 6
Number of BS transmit antenna, M
2
4
6
8
10
12
14
16
18
20
22
max B
S tra
nsm
it p
ow
er
(dB
W)
SDR (lower bound for PS)Proposed Algorithm 2 (PS)Proposed Algorithm 4 (TS)
e = −10 dBm
e = −20 dBm
Fig. 8. Minimized maximum BS transmit power for r = 2.31 bits/sec/Hz(i.e., γ = 6 dB) and e = {−20,−10} dBm.
TABLE ICOMPLEXITY ANALYSIS FOR ALG. 1, SDR APPROACH AND [27] (TO SOLVE PROBLEM (7)), AND ALG. 3 (TO SOLVE PROBLEM (29))
Algorithms avg. # iter scal var lin cons quad cons SD consAlg. 1 (PS) 11 60 24 16 0
Algorithm of [27] (PS) 26.5 132 40 0 36SDR approach (PS) 17 132 40 24 12
Alg. 3 (TS) 6.8 97 25 20 0
mooth optimization algorithm of [27], the SDR method andAlgorithm 3 are O
(iA1(M + 1)3K3N3(3KN +K + 1)
),
O(i[26]
((M2 +M + 2)KN/2
)3(6KN +K + 1)),
O(iSDR
((M2 +M + 2)KN/2
)3(6KN +K + 1)
)and
O(iA3 (2KNM + 1)3(3KN +2K + 3)
), respectively [36].
Here, iA1 = 11 is the average number of times that QP (17)is solved by Algorithm 1; i[26] = 26.5 is the average numberof times that an SDP is solved by [27]; iSDR = 17 is theaverage number of times that the feasibility (convex) SDR(A.1b)–(A.1f) is solved; and iA3 = 6.8 is the average numberof times that QP (43) is solved by Algorithm 3. Note thatthe initialization (46) for Algorithm 3 requires 1.1 iterationson average. For the particular case of M = 4, N = 4,K = 3and Pmax
k = 16 dBW, Table I shows the average number ofiterations required (‘avg. # iter.’) as well as the numbersof scalar variables (‘scal var’), linear constraints (‘lincons’), quadratic constraints (‘quad cons’) and semidefiniteconstraints (‘SD cons’) of the concerned algorithms. Clearly,Algorithms 1 and 3 are the most computationally efficient asthey involve the smallest numbers of iterations, variables andconstraints.
B. Results for Min-Max BS Power Optimization Problems (19)and (30)
Algorithm 2 and the SDR approach are used to solveproblem (19) whereas Algorithm 4 is to solve problem (30).
Figs. 7 and 8 plot the minimized maximum BS transmitpower for different values of the minimum rate r and BStransmit antenna number M . As expected, the BS transmitpower requirement increases by setting higher target rates anddecreases by using more BS antennas, respectively. In Fig. 8,we also evaluate the performance of the proposed algorithmsfor different values of the target minimum EH thresholde = {−20,−10} dBm. By increasing the target EH thresholdfrom e = −20 dBm to e = −10 dBm, the required BS transmitpower increases to meet the increased EH requirement. As canbe observed, Algorithm 2 achieves the lower bound given bySDR under all the network settings considered. Furthermore,the transmit TS-based WIPT system by Algorithm 4 clearlyoutperforms the receive PS-based WIPT system by at least 3.5dB in power.
Applying Algorithm 4 for the min-max BS power opti-mization problem (30), Fig. 9 plots the optimized value ofthe transmit TS ratio ρ for different values of the target EHthreshold e and the number of BS antennas M = {4, 5, 6}.Similar trends for the optimized TS ratio for Algorithm 3in Fig. 5 can now be observed for Algorithm 4 in Fig. 9.Finally, Fig. 10 shows that Algorithm 2 quickly convergeswithin 3 iterations to the theoretical lower bound obtainedafter solving the relaxed SDR (A.2a), (A.2b), (A.1e), (A.1f)[see AppendixA]. In this algorithm, each iteration correspondsto solving one SOCP (24). On the other hand, Algorithm 4requires about 6 iterations to converge where each iteration
12
-20 -18 -16 -14 -12 -10
Target minimum EH threshold, e (dBm)
0
0.05
0.1
0.15
0.2
0.25TSratio(ρ)
Proposed Algorithm 4, M=4Proposed Algorithm 4, M=5Proposed Algorithm 4, M=6
Fig. 9. Optimized TS ratio ρ by Algorithm 4 for different numbers of BSantennas M and r = 2.31 bits/sec/Hz.
0 1 2 3 4 5 6 7iterations
3
4
5
6
7
8
9
10
11
12
max B
S tra
nsm
it p
ow
er
(dB
W)
SDR (lower bound for PS)Proposed Algorithm 2 (PS)Proposed Algorithm 4 (TS)
Fig. 10. Convergence of Algorithms 2 and 4 for M = 5 and r = 2.31bits/sec/Hz.
solves one QP (48).The computational complexities of Algorithm 2, the
SDR method and Algorithm 4 are O(iA2(M + 2)3K3
N34KN), O
(((M2 +M + 2)KN/2
)36KN
)and
O(iA4(2KNM + 1)3(4KN +K + 2)
), respectively [36].
Here iA2 = 3 and iA4 = 6.99 are the average number ofiterations required for Algorithms 2 and 4 to converge.For the particular case of M = 4, N = 4,K = 3 andr = 2.316 bit/sec/Hz, Table II shows the required numberof variables and constraints, where ‘SOC cons’ denotes therequired number of second-order cone constraints. AlthoughAlgorithm 4 for the transmit TS-based WIPT system requiresmore computational effort than Algorithm 2 for the receivePS-based WIPT system, the former system outperforms thelatter system as previously shown in Figs. 7 and 8.
V. CONCLUSIONS
In this paper, we have jointly designed the BS transmitbeamformers with either the receive PS ratios or the transmitTS ratio for wireless energy harvesting multicell network.The design objectives include maximization of the minimumdata rate among all UEs and minimization of the maximumBS transmit power. To solve the highly nonconvex problemformulations, we have proposed new iterative optimizationalgorithms of low computational complexity that are based onquadratic programming and second-order cone programming.Simulation results with practical parameters show that thealgorithms converge quickly and that the transmit TS-basedWIPT system outperforms the receive PS-based WIPT system.In the case of PS-based designs, the proposed algorithmstightly approach the theoretical bound in the considered nu-merical examples.
APPENDIX ASDR-BASED APPROACH TO SOLVE PROBLEMS (7) AND (19)
In the SDR-based approach, problem (7) in the beamform-ing vectors wk,n is recast as the following problem in their
outer products Wk,n , wk,nwHk,n < 0:
maxWk,n∈CM×M ,αk,n∈(0,1), γ,∀ k∈K, n∈Nk
γ (A.1a)
s.t.1
γTr{Hk,k,nWk,n} −
∑n∈Nk\{n}
Tr{Hk,k,nWk,n}
−∑
k∈K\{k}
∑n∈Nk
Tr{Hk,k,nWk,n}
≥ σ2a +
σ2c
αk,n, ∀k ∈ K, n ∈ Nk
(A.1b)∑n∈Nk
Tr{Wk,n} ≤ Pmaxk , ∀k ∈ K (A.1c)∑
k∈K
∑n∈Nk
Tr{Wk,n} ≤ Pmax (A.1d)∑k∈K
∑n∈Nk
Tr{Hk,k,nWk,n}
≥emink,n
ζk,n(1− αk,n)− σ2
a, ∀k ∈ K, n ∈ Nk(A.1e)
Wk,n < 0, ∀k ∈ K, n ∈ Nk (A.1f)rank(Wk,n) = 1, ∀k ∈ K, n ∈ Nk. (A.1g)
Let us denote W , [Wk,n]k∈K,n∈Nk . By fixing γ andfurther ignoring the difficult rank-one constraint (A.1g), (A.1)is relaxed to the feasibility SDP (A.1b)–(A.1f). Because (A.1b)is the only constraint that involves γ and it is monotonic inγ, the optimal value of γ can be found via a bisection searchin an outer loop. The optimization process is repeated until(W,α, γ) converges to (W?,α?, γ?), ∀k ∈ K, n ∈ Nk,in which case (A.1a)–(A.1f) is solved. The obtained solutionby SDR approach is not guaranteed to be of rank one,i.e., rank(W?
k,n) > 1 is mostly observed. Thus, SDR-based
13
TABLE IICOMPLEXITY ANALYSIS FOR ALG. 2 AND SDR APPROACH (TO SOLVE PROBLEM (19)), AND ALG. 4 (TO SOLVE PROBLEM (30))
Algorithms avg. # iter scal var lin cons quad cons SD cons SOC consAlg. 2 (PS) 3 72 12 24 0 12
SDR approach (PS) 1 132 36 24 12 0Alg. 4 (TS) 6.99 97 25 28 0 0
solution can serve as an upper bound for max-min rate problem(7) .
Similarly, problem (19) in beamforming vectors wk,n isrecast as the following rank-one constrained SDP in the outerproducts Wk,n , wk,nw
Hk,n < 0, ∀k ∈ K, n ∈ Nk:
minWk,n∈CM×M ,αk,n∈(0,1),∀ k∈K, n∈Nk
maxk∈K
∑n∈Nk
Tr{Wk,n} (A.2a)
s.t. Tr{Hk,k,nWk,n} ≥ γmink,n
∑n∈Nk\{n}
Tr{Hk,k,nWk,n}
+∑
k∈K\{k}
∑n∈Nk
Tr{Hk,k,nWk,n}+ σ2a +
σ2c
αk,n
,
∀k ∈ K, n ∈ Nk(A.2b)
(A.1e), (A.1f), (A.1g). (A.2c)
By ignoring the rank-one constraint (A.1g), the optimal solu-tion
∑k∈K
∑n∈Nk Tr{W?
k,n} of the SDR formed by (A.2a),(A.2b), (A.1e), (A.1f) provides a lower bound of the actualoptimal value of problem (19).
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Ali Arshad Nasir (S09-M13) 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.
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 Centre for Health Technologies, Universityof Technology Sydney. He has been involved in research with the areasof optimization, control, signal processing, wireless communication, andbiomedical engineering for more than 20 years.
Duy Trong Ngo (S’08-M’15) received the B.Eng.(with First-class Honours and the University Medal)degree in telecommunication engineering from TheUniversity of New South Wales, Australia in 2007,the M.Sc. degree in electrical engineering (com-munication) from University of Alberta, Canada in2009, and the Ph.D. degree in electrical engineeringfrom McGill University, Canada in 2013.
Since 2013, he has been a Lecturer with theSchool of Electrical Engineering and Computing,The University of Newcastle, Australia, where he
currently leads the research effort in design and optimization for 5G wirelesscommunications networks.
In 2013, Dr. Ngo was awarded two prestigious Postdoctoral Fellowshipsfrom the Natural Sciences and Engineering Research Council of Canada andthe Fonds de recherche du Quebec – Nature et technologies. At The Universityof Newcastle, he received the 2015 Vice-Chancellor’s Award for Research andInnovation Excellence and the 2015 Pro Vice-Chancellor’s Award for ResearchExcellence in the Faculty of Engineering and Built Environment.
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 UniversityBelfast, UK as a Lecturer (Assistant Professor). Hiscurrent research interests include physical layer se-curity, energy-harvesting communications, cognitiverelay networks. He is the author or co-author of morethan 220 technical papers published in scientificjournals (110 articles) and presented at internationalconferences (110 papers).
Dr. Duong currently serves as an Editor for the IEEE TRANSACTIONSON WIRELESS COMMUNICATIONS, IEEE TRANSACTIONS ON COMMUNI-CATIONS, IEEE COMMUNICATIONS LETTERS, IET COMMUNICATIONS. Hewas a Editor of WILEY TRANSACTIONS ON EMERGING TELECOMMUNI-CATIONS TECHNOLOGIES, ELECTRONICS LETTERS and has also servedas the Guest Editor of the special issue on some major journals includ-ing IEEE JOURNAL IN SELECTED AREAS ON COMMUNICATIONS, IETCOMMUNICATIONS, IEEE ACCESS, IEEE WIRELESS COMMUNICATIONSMAGAZINE, IEEE COMMUNICATIONS MAGAZINE, EURASIP JOURNALON WIRELESS COMMUNICATIONS AND NETWORKING, EURASIP JOUR-NAL ON ADVANCES SIGNAL PROCESSING. He was awarded the Best PaperAward at the IEEE Vehicular Technology Conference (VTC-Spring) in 2013,IEEE International Conference on Communications (ICC) 2014. He is therecipient of prestigious Royal Academy of Engineering Research Fellowship(2016-2021).
Vincent Poor (S72, M77, SM82, F87) received thePh.D. degree in EECS from Princeton University in1977. 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 the Michael Henry Strater UniversityProfessor of Electrical Engineering. From 2006 till2016, he served as Dean of Princetons School ofEngineering and Applied Science. Dr. Poors researchinterests are in the areas of statistical signal pro-cessing, stochastic analysis and information theory,
and their applications in wireless networks and related fields. Among hispublications in these areas is the recent book Mechanisms and Games forDynamic Spectrum Allocation (Cambridge University Press, 2014).
Dr. Poor is a member of the National Academy of Engineering and theNational Academy of Sciences, and a foreign member of the Royal Society.He is also a Fellow of the American Academy of Arts and Sciences andthe National Academy of Inventors, and of other national and internationalacademies. He received the Technical Achievement and Society Awards ofthe IEEE Signal Processing Society in 2007 and 2011, respectively. Recentrecognition of his work includes the 2014 URSI Booker Gold Medal, the2015 EURASIP Athanasios Papoulis Award, the 2016 John Fritz Medal, andhonorary doctorates from Aalborg University, Aalto University, HKUST andthe University of Edinburgh.