Spectral and Energy Efficiencies in Full-Duplex Wireless Information
and Power Transfer
Duong, T. Q., Nguyen, V. D., Tuan, H. D., Shin, O-S., & Poor, H. V. (2017). Spectral and Energy Efficiencies in
Full-Duplex Wireless Information and Power Transfer.
IEEE Transactions on Communications
,
65
(5), 2220-
2233. https://doi.org/10.1109/TCOMM.2017.2665488
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IEEE Transactions on Communications
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Download date:26. Aug. 2022
1
Spectral and Energy Efficiencies in Full-Duplex
Wireless Information and Power Transfer
Van-Dinh Nguyen, Student Member, IEEE, Trung Q. Duong, Senior Member, IEEE, Hoang Duong Tuan,
Oh-Soon Shin, Member, IEEE, and H. Vincent Poor Fellow, IEEE
Abstract—A communication system is considered consisting of
a full-duplex (FD) multiple-antenna base station (BS) and mul-
tiple single-antenna downlink users (DLUs) and single-antenna
uplink users (ULUs), where the latter need to harvest energy
for transmitting information to the BS. The communication is
thus divided into two phases. In the first phase, the BS uses all
available antennas for conveying information to DLUs and wire-
less energy to ULUs via information and energy beamforming,
respectively. In the second phase, ULUs send their independent
information to the BS using their harvested energy while the
BS transmits the information to the DLUs. In the both phases,
the communication is operated at the same time and over the
same frequency band. The aim is to maximize the sum rate and
energy efficiency under ULU achievable information throughput
constraints by jointly designing beamformers and time allocation.
The utility functions of interest are nonconcave and involved
constraints are nonconvex, so these problems are computationally
troublesome. To address them, path-following algorithms are pro-
posed to arrive at least at local optima. The proposed algorithms
iteratively improve the objectives with converge guaranteed.
Simulation results demonstrate that they achieve fast convergence
rate and outperform conventional solutions.
Index Terms—Energy harvesting, full-duplex radios, full-
duplex self-interference, transmit beamforming, wireless infor-
mation and power transfer.
I. INTRODUCTION
Radio-frequency (RF) energy harvesting (EH) communica-
tion has emerged as a promising cost-effective technology for
supplying power to users [1], [2]. Enabling wireless devices
to harvest energy from RF signals, RF-EH communication is
expected to fundamentally reshape the landscape of power
supply in Internet-of-Things (IoT) [3], [4]. Exploring RF
EH communication allows one to transfer information and
energy over the same RF channel [5], [6]. Various cooperative
schemes with/without built-in batteries for energy storage in
This work was supported in part by the Australian Research Councils
Discovery Projects under Project DP130104617, in part by the U.K. Royal
Academy of Engineering Research Fellowship under Grant RF1415/14/22
and U.K. Engineering and Physical Sciences Research Council under Grant
EP/P019374/1, and in part by the U.S. National Science Foundation under
Grants CNS-1456793 and ECCS-1343210.
V.-D. Nguyen and O.-S. Shin are with the School of Electronic Engineering
& Department of ICMC Convergence Technology, Soongsil University, Seoul
06978, Korea (e-mail: {nguyenvandinh, osshin}@ssu.ac.kr).
T. Q. Duong is with the School of Electronics, Electrical Engineering and
Computer Science, Queen’s University Belfast, Belfast BT7 1NN, United
Kingdom (e-mail: trung.q.duong@qub.ac.uk).
H. D. Tuan is with the Faculty of Engineering and Information Technology,
University of Technology Sydney, Broadway, NSW 2007, Australia (email:
Tuan.Hoang@uts.edu.au).
H. V. Poor is with the Department of Electrical Engineering, Princeton
University, Princeton, NJ 08544 USA (e-mail: poor@princeton.edu).
which multiple transceiver pairs communicate with each other
via EH relays were studied in [7]–[10]. In this regard, transmit
beamforming is used to focus information and/or RF energy
at the desired users [11], [12].
An emerging trend is the development of wireless pow-
ered communication networks (WPCNs), which implement
a downlink wireless energy transfer (DWET) followed by
an uplink wireless information transmission (UWIT) [13].
A base station (BS) first transfers energy to the wireless
powered users, who harvest the energy for transmitting their
independent information to the BS. Time allocation for DWET
and UWIT to optimize the sum information rate subject to
per-user achievable information rate thresholds was considered
in [14]. The joint downlink beamforming and uplink power
allocation under fixed time durations to optimize the worst
achievable information user rate was considered in [15] using
alternating optimization. The optimal time allocation is then
searched at grinding points. A similar optimization problem
for the joint energy weight and power allocation with the BS
of massive antenna array was analyzed asymptotically in [16],
while [17] considered energy efficiency in the case of a single
user. The joint time allocation for DWET, time separation
and power allocation in UWIT for the users in optimizing
the WPCN energy efficiency was proposed in [18]. Very
recently, [19] considered a joint downlink (DL) and uplink
(UL) transmission of K-tier heterogeneous cellular networks
with downlink simultaneous wireless information and power
transfer, where outage probability and ergodic capacity of both
DL and UL have been derived.
Meanwhile, full duplex (FD) radio [20] offers enormous
potential to significantly enhance the spectral efficiency com-
pared to its half duplex (HD) counterpart. Recent studies (see,
e.g., [21] and [22]) showed that FD radio may be deployable
in next-generation networks because it can be implemented
at reasonable cost and without complex radio hardware. The
major challenge in FD radio is the residual self-interference
(SI) from the transmit antennas to the receive antennas, which
are co-located and function at the same time and over the same
frequency band. A wide range of SI mitigation techniques were
addressed in [20], [23] and [24]. More recently, FD multiple-
input multiple-output (MIMO) precoding was studied in the
context of multiuser MIMO (MU-MIMO) to improve the over-
all spectral efficiency of downlink and uplink channels [25]–
[28]. A FD single-antenna architecture for energy-recycling
was proposed in [29]. Beamformer design at the FD relay
for harvesting energy and suppressing loop interference was
considered in [30].
2
In this paper, we study the potential of FD radio in WPCNs
to improve both spectral and energy efficiencies. There are
both downlink users (DLUs) and uplink users (ULUs), where
the ULUs need to harvest energy from the BS via DWET by
the BS. The communication operates in two phases in the same
time slot and over the same frequency band. In the first phase,
the BS uses all available antennas to simultaneously transmit
information to DLUs and transfer the energy to ULUs. In the
second phase, the BS operates in FD mode for transmitting
information to DLUs and receiving information from ULUs.
Each ULU uses only the harvested energy in transmitting
information to the BS. We jointly optimize the time allocation
for the phases, the DL information and energy beamforming,
and the UL transmit power allocation subject to the power
budget at the BS and the individual ULU information rate
thresholds. The ULU information rate threshold constraints
are crucial to resolving the so called doubly near-far prob-
lem in WPCNs that discriminates the ULUs by favoring
ones with better channel conditions for both energy transfer
in phase I and information transmission in phase II. The
residual SI and co-channel interference (CCI) from ULUs
to DLUs are taken into account, which potentially offer the
best performance make the optimizations more challenging. In
fact, these problems involve optimization of nonconvex utility
functions subject to nonconvex constraints, for which the
optimal solutions are difficult computationally. Nevertheless,
we propose path-following algorithms to address them. Our
main contributions are summarized as follows:
• We propose a new model for WPCNs to optimize simul-
taneous uplink and downlink information transmission by
exploring FD radio for the BS.
• Assuming perfect channel state information (CSI), we
first develop a path-following algorithm of low complex-
ity for the computational solution of sum rate maximiza-
tion (SRM). The obtained solutions are at least local
optima as they satisfy the Karush-Kuhn-Tucker (KKT)
conditions. Numerical results show fast convergence of
the proposed algorithm and greatly improve the system
performance over the conventional approaches.
• The energy efficiency maximization (EEM) problem is a
difficult nonlinear fraction program since the objective
is not a ratio of a concave and convex function. The
commonly-used Dinkelbach-type algorithms are not ap-
plicable. We develop a novel path-following algorithm
that only invokes one simple convex quadratic program
at each iteration, which again converges at least to a local
optimum.
The rest of this paper is organized as follows. The system
model and problem formulations of SRM and EEM are
described in Section II. We devise the optimal solution to the
SRM and EEM problems in Section III and IV, respectively.
Numerical results are provided in Section V, and Section VI
concludes the paper.
Notation: Bold lower and upper case letters represent vec-
tors and matrices, respectively; X
H
, X
T
, X
∗
, and Trace(X )
are the Hermitian transpose, normal transpose, conjugate, and
trace of a matrix X, respectively. k · k and | · | denote the
h
D
1
h
D
K
D
1
D
K
U
1
U
L
Downlink channels
N + 1
1
2
N
N + 2
N + M
g
U
1
g
U
L
Information transfer
Energy transfer
˜
h
D
1
˜
h
D
K
D
1
D
K
U
1
U
L
Downlink channels
Uplink channels
1
1
2
N
2
M
G
I
BS
h
U
1
h
U
L
Phase I: DL information and energy transfer
Phase II: DL and UL information transfer
BS
g
ℓk
Fig. 1. A WPCN system with WET in the downlink channel and WIT in
both uplink and downlink channels.
Euclidean norm of a matrix or vector and the absolute value
of a complex scalar, respectively. I
N
represents an N × N
identity matrix. x ∼ CN(η, Z) means that x is a random
vector following a complex circular Gaussian distribution with
mean η and covariance matrix Z. E[·] denotes the statistical
expectation. The notation X 0 (X ≻ 0, resp.) means
the matrix X is positive semi-definite (definite, resp.). ℜ{·}
represents real part of the argument. The inner product hX, Yi
is defined as Trace(X
H
Y). ∇
x
f(x) represents the gradient
of f(·) with respect to vector x.
II. SYSTEM MODEL AND OPTIMIZATION PROBLEM
FORMULATIONS
A. Signal Model
We consider a WPCN as illustrated in Fig. 1, which con-
sists of a BS, K DLUs and L ULUs. The BS is equipped
with M receive and N transmit antennas, while all the
users are equipped with a single antenna. Denote by D ,
{D
1
, D
2
, ··· , D
K
} and U , {U
1
, U
2
, ··· , U
L
} the sets of
DLUs and ULUs, respectively. All channels are assumed to
follow independent quasi-static flat fading, i.e., remaining con-
stant during a communication time block, denoted by T , but
change independently from one block to another. Without loss
of generality, the time block T is set as 1. Following [13]–[18],
all ULUs U
ℓ
∈ U are assumed to harvest energy from the RF
signal transmitted by the BS and then transmit information to
BS as illustrated in Fig. 2. During the first fraction 0 < α < 1
of the time block, the BS simultaneously transmits information
to all DLUs D
k
and transfers energy to all ULU U
ℓ
. In the
remaining fraction (1 −α), the BS operates in FD mode, i.e.,
it uses N antennas for transmitting information to the DLUs
and M antennas for receiving information from ULUs.
With all M + N antennas used in phase I, more degrees of
freedom are added to the BS and all ULUs U
ℓ
’s are expected
to harvest more energy from the RF signal. The complex
baseband transmitted signal at the BS in phase I is then
3
αT (1 − α)T
DL information transfer:
BS −→ D
k
BS −→ U
ℓ
DL energy transfer:
DL information transfer:
BS −→ D
k
UL information transfer:
U
ℓ
−→ BS
Fig. 2. The harvest-and-then-transmit protocol.
expressed as
x
1
=
K
X
k=1
w
1,k
x
k
+ v
e
(1)
where w
1,k
∈ C
(N+M)×1
denotes the k-th information
beamforming vector, x
k
with E{|x
k
|
2
} = 1 is the message
intended for DLU D
k
. The energy beam vector v
e
whose
elements are zero-mean complex Gaussian random variables,
is assumed v
e
∼ CN(0, VV
H
), where V ∈ C
(N+M)×
˜
L
with
˜
L ≤ min
(N + M ), L
. The received signal at DLU D
k
and
ULU U
ℓ
in phase I is, respectively, expressed as
y
D
k
= h
H
D
k
w
1,k
x
k
+
K
X
i=1,i6=k
h
H
D
k
w
1,i
x
i
+ h
H
D
k
v
e
+ n
D
k
, (2)
and
y
U
ℓ
=
K
X
k=1
g
H
U
ℓ
w
1,k
x
k
+ g
H
U
ℓ
v
e
+ n
U
ℓ
(3)
where h
D
k
∈ C
(N+M)×1
and g
U
ℓ
∈ C
(N+M)×1
are the
channel vectors from the BS to DLU D
k
and ULU U
ℓ
,
respectively. They can be explicitly written as
h
D
k
= [
ˆ
h
D
k
,1
, ··· ,
ˆ
h
D
k
,N
|
{z }
˜
h
D
k
,
ˆ
h
D
k
,(N+1)
, ··· ,
ˆ
h
D
k
,(N+M)
]
T
, (4)
g
U
ℓ
= [ˆg
U
ℓ
,1
, ··· , ˆg
U
ℓ
,N
, ˆg
U
ℓ
,(N+1)
, ··· , ˆg
U
ℓ
,(N+M)
|
{z }
˜
g
U
ℓ
]
T
(5)
where
ˆ
h
D
k
,i
∈ C and ˆg
U
ℓ
,i
∈ C, ∀i = 1, ··· , N + M, denote
the baseband channels from the i-th antenna at the BS to
DLU D
k
and ULU U
ℓ
, respectively; n
D
k
∼ CN(0, σ
2
k
) and
n
U
ℓ
∼ CN(0, ˆσ
2
ℓ
) represent the additive white Gaussian noise
(AWGN) at DLU D
k
and ULU U
ℓ
, respectively. The harvested
energy at ULU U
ℓ
is defined by
E
U
ℓ
(w
1
, V, α) = ηαE
|y
U
ℓ
|
2
= ηα
K
X
k=1
|g
H
U
ℓ
w
1,k
|
2
+ kg
H
U
ℓ
Vk
2
(6)
where w
1
, [w
T
1,1
, ··· , w
T
1,K
]
T
∈ C
(N+M)K
, and η denotes
the energy conversion efficiency at the receiver. In (6), the
receive noise can be neglected since it will be negligible
compared to energy transfer from the BS in practice. We can
see that the harvested energy in (6) is also contributed by the
DL information beams. We will show in Section V that the
energy beam v
e
is very beneficial when the ULUs are far from
the DLUs.
From (2), the signal-to-interference-plus-noise ratio (SINR)
at DLU D
k
in phase I can be expressed as
γ
1,k
(w
1
, V) =
|h
H
D
k
w
1,k
|
2
P
K
i=1,i6=k
|h
H
D
k
w
1,i
|
2
+ kh
H
D
k
Vk
2
+ σ
2
k
. (7)
In phase II, the BS uses N antennas for transmitting infor-
mation to the DLUs and M antennas for receiving information
from the ULUs. The received signal at DLU D
k
and the BS
can be, respectively, written as
˜y
D
k
=
˜
h
H
D
k
w
2,k
x
′
k
+
K
X
i=1,i6=k
˜
h
H
D
k
w
2,i
x
′
i
+
L
X
ℓ=1
p
ℓ
g
ℓk
v
ℓ
+ n
D
k
, (8)
and
y
U
=
L
X
ℓ=1
p
ℓ
h
U
ℓ
v
ℓ
+
√
ρ
K
X
k=1
G
H
I
w
2,k
x
′
k
+ n
U
(9)
where
˜
h
D
k
∈ C
N×1
, w
2,k
∈ C
N×1
and x
′
k
with E{|x
′
k
|
2
} = 1
are the transmit channel vector, information beam for DLU D
k
and the message intended for DLU D
k
in phase II. p
ℓ
∈ C,
h
U
ℓ
∈ C
M×1
and v
ℓ
with E{|v
ℓ
|
2
} = 1 are the transmit
power, the receive channel vector, and the message of ULU
U
ℓ
, respectively. n
U
∼ CN(0, ˜σ
2
I) denotes the receive AWGN
at the BS. Since the channel remains unchanged during a
transmission block time,
˜
h
D
k
corresponds to first N elements
of h
D
k
in (4). We also assume the reciprocity for UL and
DL links, i.e., h
U
ℓ
=
˜
g
U
ℓ
, ∀ℓ, where
˜
g
U
ℓ
is defined in (5).
Note that the term
√
ρ
P
K
k=1
G
H
I
w
2,k
x
k
in (9) represents
the FD SI left over from the so called analog-circuit domain
SI cancellation [24], where G
I
is a fading loop channel and
0 ≤ ρ ≤ 1 is used for modeling the degree of SI propagation
[20]. The CCI from ULU U
ℓ
to DLU D
k
is denoted by g
ℓk
.
From (8), the SINR at DLU D
k
in phase II is
γ
2,k
(w
2
, p) =
|
˜
h
H
D
k
w
2,k
|
2
P
K
i=1,i6=k
|
˜
h
H
D
k
w
2,i
|
2
+
P
L
ℓ=1
p
2
ℓ
|g
ℓk
|
2
+ σ
2
k
(10)
where w
2
, [w
T
2,1
, w
T
2,2
, ··· , w
T
2,K
]
T
∈ C
NK
and p ,
[p
1
, p
2
, ··· , p
L
]
T
. The achieved SR of DL transmission is thus
R
D
w
1
, w
2
, V, p, α
= α
K
X
k=1
ln
1 + γ
1,k
w
1
, V
+
1 − α
K
X
k=1
ln
1 + γ
2,k
w
2
, p
. (11)
For simplicity, we adopt the minimum mean square error and
successive interference cancellation (MMSE-SIC) receiver at
the BS to maximize the received SINR of U
ℓ
in (9). Assuming
that the decoding order follows the ULU index order, the
resultant SINR in decoding U
ℓ
’s information is [31]
γ
ℓ
(w
2
, p) =p
2
ℓ
h
H
U
ℓ
L
X
j>ℓ
p
2
j
h
U
j
h
H
U
j
+ ρ
K
X
k=1
G
H
I
w
2,k
w
H
2,k
G
I
+ ˜σ
2
I
!
−1
h
U
ℓ
. (12)
Then, the achieved SR of UL transmission is
R
U
(w
2
, p, α) =
1 − α
L
X
ℓ=1
ln
1 + γ
ℓ
w
2
, p
. (13)
4
B. Optimization Problem Formulations
Our main goal is to maximize both the total SR and EE of
the system by jointly deriving the time allocation (for two
phases), the DL beamformers, and the UL transmit power
allocation under the ULU rate thresholds.
1) SRM Problem Formulation: The SRM problem of jointly
designing w
1
, w
2
, V, p, and α can be expressed as
maximize
w
1
,w
2
,V,p,α
R
D
(w
1
, w
2
, V, p, α) + R
U
(w
2
, p, α) (14a)
s.t. (1 − α) ln
1 + γ
ℓ
(w
2
, p)
≥ ¯r
u
, ∀ℓ = 1, ··· , L, (14b)
p
2
ℓ
≤ p
eh
U
ℓ
(w
1
, V, α), ∀ℓ = 1, ··· , L, (14c)
p
ℓ
≥ 0, ∀ℓ = 1, ··· , L, (14d)
α
kw
1
k
2
+ kVk
2
+ (1 − α)kw
2
k
2
≤ P
BS
, (14e)
0 < α < 1 (14f)
where without loss of generality the same rate threshold ¯r
u
for all ULUs is set, P
BS
is the maximum transmit power at
the BS, and
p
eh
U
ℓ
(w
1
, V, α) =
E
U
ℓ
(w
1
, V, α)
(1 − α)
=
ηα
1 − α
K
X
k=1
|g
H
U
ℓ
w
1,k
|
2
+ kg
H
U
ℓ
Vk
2
. (15)
The optimization problem in (14) is known as the spectral
efficiency maximization problem. The constraints in (14b)
impose a quality-of-service (QoS) requirement for ULU U
ℓ
,
i.e., the achievable information decoding rate should be not
less than a given threshold ¯r
u
to prevent the FD to maximize
R
D
only in maximizing the objective value in (14), leading
to an extremely low QoS for ULUs. More importantly, they
cognitively rule out the doubly near-far occurrence of favoring
ULUs with better channel conditions. According to (6) and
(11), the ULUs with better channel conditions are advanta-
geous in both harvesting energy in phase I and transmitting
information in phase II. The constraints (14c) merely mean
that each ULU U
ℓ
utilizes its energy harvested from phase I
to transmit information to the BS, as illustrated in Fig. 2. The
constraint in (14e) is the power constraint at the BS, which
differs from the following one that was studied previously
[15]:
kw
1
k
2
+ kVk
2
+ kw
2
k
2
≤ P
BS
. (16)
However, in contrast to the left-hand side (LHS) of (14e),
which is the total transmit power at the BS, the LHS of (16)
is a sum of power rates so the constraint (16) is meaningless.
Note that (16) is much stricter than (14e), i.e., by using
(16), the BS does not use all allowable power and thus
the corresponding performance is not optimal. This will be
elaborated in the next sections.
2) EEM Problem Formulation: Another performance met-
ric of interest is to maximize EE of the system. Energy
consumption in green wireless networks has attracted much
attention of both academia and industry recently [17], [32],
[33]. In this paper, the power consumed by the BS and ULUs
is taken into account.
Power consumption model: A linear power model [34] is
adopted in this paper, where the total power consumption at
the BS is modeled as
P
DL
=
1
ǫ
α(kw
1
k
2
+ kVk
2
) + (1 − α)kw
2
k
2
+ αM P
dyn
BS
+ N P
dyn
BS
+ P
sta
BS
(17)
where ǫ ∈ (0, 1] is the power amplifier efficiency, P
dyn
BS
is the
dynamic power consumption associated to the power radiation
of all circuit blocks in each active radio frequency chain, and
P
sta
BS
is the static power consumed by cooling system, power
supply, etc. Similarly, the total power consumption of all users
in the UL channel is given by
P
UL
= (1 − α)
L
X
ℓ=1
P
dyn
U
ℓ
+
L
X
ℓ=1
P
sta
U
ℓ
. (18)
Note that the formulation in (18) does not include a power
consumed by sending data by ULUs as it is already incor-
porated in (17). For notational simplicity, we denote P
0
,
NP
dyn
BS
+P
sta
BS
+
P
L
ℓ=1
P
sta
U
ℓ
as the circuit power of the system,
which is independent from the optimization variables. The
EEM problem is thus
maximize
w
1
,w
2
,V,p,α
R
D
(w
1
, w
2
, V, p, α) + R
U
(w
2
, p, α)
χ(w
1
, w
2
, V, α) + P
0
(19a)
s.t. (14b), (14c), (14d), (14e), (14f) (19b)
where χ(w
1
, w
2
, V, α) ,
1
ǫ
α
kw
1
k
2
+ kVk
2
+ (1 −
α)kw
2
k
2
+ αM P
dyn
BS
+ (1 − α)
P
L
ℓ=1
P
dyn
U
ℓ
.
Naturally, the reciprocity for UL and DL links in time
duplex division (TDD) mode is adopted for small cell systems
as those considered in this paper, under which CSI is easily
obtained by requesting all DLUs and ULUs to send their pilots
to the BS and thus can be assumed perfectly available. The
performance under perfect CSI also serves as a benchmark for
the achievement of the FD systems. Moreover, our proposed
algorithms can be further adjusted to robust optimization
problems in dealing with the worst case of imperfect CSI.
III. SUM RATE MAXIMIZATION
Finding an optimal solution to the SRM problem (14) is
challenging due to the non-concavity of its objective function
and nonconvexity of its feasible set. In this section, we propose
a path-following computation procedure to obtain a local
optimum.
The following inequalities, whose proofs are given in the
Appendix A, will be frequently used in the paper:
ln
1 +
|x|
2
y
≥ ln
1 +
|¯x|
2
¯y
−
|¯x|
2
¯y
+ 2
ℜ{¯x
∗
x}
¯y
−
|¯x|
2
(|x|
2
+ y)
¯y(¯y + |¯x|
2
)
, (20)
|x|
2
y
≥ 2
¯x
∗
x
¯y
−
|¯x|
2
¯y
2
y, (21)
∀x ∈ C, ¯x ∈ C, y > 0, ¯y > 0.