Energy_Beamforming_for_RF_Wireless_Power_Transfer_With_Dynamic_Metasurface_Antennas (1)

Prévia do material em texto

IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 13, NO. 3, MARCH 2024 781
Energy Beamforming for RF Wireless Power Transfer
With Dynamic Metasurface Antennas
Amirhossein Azarbahram , Graduate Student Member, IEEE, Onel L. A. López , Member, IEEE,
Richard D. Souza , Senior Member, IEEE, Rui Zhang , Fellow, IEEE,
and Matti Latva-Aho , Senior Member, IEEE
Abstract—Radio frequency (RF) wireless power transfer
(WPT) is a promising technology for charging the Internet of
Things. Practical RF-WPT systems usually require energy beam-
forming (EB), which can compensate for the severe propagation
loss by directing beams toward the devices. The EB flexibility
depends on the transmitter architecture, existing a trade-off
between cost/complexity and degrees of freedom. Thus, simpler
architectures such as dynamic metasurface antennas (DMAs) are
gaining attention. Herein, we consider an RF-WPT system with a
transmit DMA for meeting the energy harvesting requirements of
multiple devices and formulate an optimization problem for the
minimum-power design. First, we provide a mathematical model
to capture the frequency-dependant signal propagation effect in
the DMA architecture. Next, we propose a solution based on
semi-definite programming and alternating optimization. Results
show that a DMA-based structure can outperform a fully-digital
implementation and that the required transmit power decreases
with the antenna array size, while it increases and remains almost
constant with frequency in DMA and FD, respectively.
Index Terms—Radio frequency wireless power transfer,
dynamic metasurface antennas, energy beamforming.
I. INTRODUCTION
FUTURE wireless systems must support uninterrupted
operation of massive Internet of Things networks, pro-
viding reliable energy supply for low-power wireless devices.
Radio frequency (RF) wireless power transfer (WPT) is
a promising solution to achieve this goal. The end-to-
end power transfer efficiency of an RF-WPT system is
inherently low, calling for efficient techniques such as
Manuscript received 12 September 2023; revised 1 November 2023;
accepted 13 December 2023. Date of publication 15 December 2023; date
of current version 8 March 2024. This work was supported in part by
the Finnish Foundation for Technology Promotion, Research Council of
Finland (former Academy of Finland), Finland, under Grant 348515 and
Grant 346208 (6G Flagship); in part by the European Commission through
the Horizon Europe/JU SNS Project Hexa-X-II under Grant 101095759;
in part by CNPq, Brazil, under Grant 402378/2021-0, Grant 401730/2022-
0, and Grant 305021/2021-4; and in part by RNP/MCTIC 6G Mobile
Communications Systems under Grant 01245.010604/2020-14. The associate
editor coordinating the review of this article and approving it for publication
was J. Tang. (Corresponding author: Amirhossein Azarbahram.)
Amirhossein Azarbahram, Onel L. A. López, and Matti Latva-Aho are with
the Centre for Wireless Communications, University of Oulu, 90014 Oulu,
Finland (e-mail: amirhossein.azarbahram@oulu.fi; onel.alcarazlopez@oulu.fi;
matti.latva-aho@oulu.fi).
Richard D. Souza is with the Department of Electrical and Electronics
Engineering, Federal University of Santa Catarina, Florianopolis 88034-500,
Brazil (e-mail: richard.demo@ufsc.br).
Rui Zhang is with the School of Science and Engineering, Shenzhen
Research Institute of Big Data, The Chinese University of Hong Kong
(Shenzhen), Shenzhen 518172, Guangdong, China, and also with the
Department of Electrical and Computer Engineering, National University of
Singapore, Singapore (e-mail: elezhang@nus.edu.sg).
Digital Object Identifier 10.1109/LWC.2023.3343563
waveform optimization, energy beamforming (EB), and dis-
tributed antenna systems [1], [2].
EB can focus energy beams toward one or more energy har-
vesting (EH) devices at the same time. The EB flexibility and
potential gains are determined by the transmitter architecture.
Although a fully-digital (FD) structure provides the highest
flexibility, it requires a large number of RF chains, including
power amplifiers (PAs), resulting in high complexity, cost, and
power consumption. The analog architecture alternatives incur
much lower power consumption and cost by reducing the num-
ber of RF chains, and utilizing only analog circuits as phase
shifters. However, they may not offer sufficient flexibility, thus,
hybrid architectures have been introduced to combine digital
and analog approaches and provide beamforming capability
with a trade-off between complexity/cost and flexibility [3].
In hybrid architectures, RF chains and phase shifters can be
connected using, e.g., the fully-connected network, where all
RF chains are connected to all phase shifters. However, since
the complexity and losses of analog circuits affect scalability
and performance, simpler array of subarrays (AoSA)-based
architectures are used [4]. Another technology that avoids
analog circuits is the dynamic metasurface antenna (DMA), a
group of configurable metamaterial elements placed on a set of
waveguides [5]. Unlike reflective surfaces [6], where there is
no correlation between the phase shift and gain introduced by
each reflecting element, the modeling of DMA metamaterial
elements is more involved, making their corresponding design
problems different in general. DMA is utilized in [7] in near-
field WPT to maximize the sum-harvested energy considering
linear RF-to-direct current (DC) power conversion. However,
in practice, there are extensive non-linearities in the EH device.
Thus, a linear power conversion model cannot represent the
amount of harvested power in a practical EH device. Herein,
we also consider a DMA transmitter architecture to charge
multiple single-antenna EH devices but utilize a practical
alternative, which assumes that each device has a specific
DC EH requirement and that the corresponding RF power
requirement can be obtained by the nonlinear EH relationship
of the device. Then, providing that specific RF power required
by the device leads to meeting the EH requirements.
Our main contributions are: i) we optimize the EB aiming
to minimize the transmit power while meeting the EH require-
ments of all receivers; ii) we provide a frequency-dependant
model for the propagation characteristics of the DMA, which
was previously overlooked; iii) we propose an efficient EB
solution relying on semi-definite programming (SDP) and
alternating optimization, which converges in polynomial time;
iv) we show that DMA outperforms the FD structure in terms
of minimum transmit power for meeting the RF power require-
ments, and that the transmit power reduces with increasing
the antenna array size, while it increases with the operating
c© 2023 The Authors. This work is licensed under a Creative Commons Attribution 4.0 License.
For more information, see https://creativecommons.org/licenses/by/4.0/
https://orcid.org/0009-0005-1494-5309
https://orcid.org/0000-0003-1838-5183
https://orcid.org/0000-0002-7389-6245
https://orcid.org/0000-0002-8729-8393
https://orcid.org/0000-0002-6261-0969
782 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 13, NO. 3, MARCH 2024
Fig. 1. DMA-based transmitter architecture.
frequency in the DMA architecture. Next, Section II presents
the system model and the optimization problem. In Section III,
we discuss the proposed solution, while Section IV provides
numerical results. Finally, Section V concludes this letter.
Notations: Bold lowercase (upper-case) letters represent
vectors (matrices), | · | is the l2 norm, (·)T is the transpose and
(·)H is the transpose conjugate. E is the expectation, ⊗ is the
Kronecker product, rank(·) and Tr(·) are the rank and trace
of a matrix, respectively. Vec(·) is the vectorization operation.
II. SYSTEM MODEL & PROBLEM FORMULATION
We consider a multi-antenna RF-WPT system where a
transmitter with a DMA uniform square array charges K
single-antenna devices. As illustrated in Fig. 1, M energy
symbols are the inputs of a digital beamformer, while there
are Nv RF chains atthe output. Each RF chain is connected to
a waveguide with Nh metamaterial elements. Thus, the total
number of elements is N = Nv × Nh .
1) Channel Model: We consider the near-field wireless
channel. The k-th user, located at a distance rk from the
transmitter, lies in the near-field propagation region if dfs <
rk < dfr , where dfs = 3
√
D4
8λ and dfr = 2D2
λ are the
Fresnel and Fraunhofer distance, respectively. Moreover, λ is
the wavelength, D =
√
2L is the antenna diameter, and L is
the antenna length. The location of the lth element in the ith
waveguide is pi ,l = [xi ,l , yi ,l , zi ,l ]
T , i = 1, 2, . . . ,Nv and
l = 1, 2, . . . ,Nh , while pk denotes the kth device location.
The channel coefficient between user k and the lth metama-
terial element in the ith waveguide is given by [8]
γi ,l ,k = Ai ,l ,ke
−j2πdi,l,k/λ, (1)
where 2πdi ,l ,k/λ constitutes the phase shift introduced by the
propagation distance, and di ,l ,k = |pk −pi ,l | is the Euclidean
distance between the element and the user. Additionally, the
channel gain coefficient is
Ai ,l ,k =
√
F
(
θi ,l ,k , φi ,l ,k
) λ
4πdi ,l ,k
, (2)
where θi ,l ,k and φi ,l ,k are the elevation and azimuth angle,
respectively, while F (θi ,l ,k , φi ,l ,k ) denotes the radiation pro-
file of each element. We assume that the latter is given by [9]
F
(
θi ,l ,k , φi ,l ,k
)
=
{
Gtcos
(
θi ,l ,k
)Gt
2
−1
, θi ,l ,k ∈ [
0, π2
]
,
0, otherwise,
(3)
where Gt = 2(b + 1) and b are the transmit antenna gain
and the boresight gain, respectively. Now, let us define γ k =[
γ1,1,k , γ1,2,k , . . . , γNv ,Nh ,k
]T as the N-dimensional channel
coefficients vector between the user k and the elements of the
DMA. Note that under the conventional far-field conditions,
the channel coefficient can be represented as Ake
−jψi,l,k ,
where Ak is only determined by the distance between the user
and the antenna array, and ψi ,l ,k by the spatial direction of
the user and the spacing between the antenna elements.
TABLE I
MICROSTRIP MATERIAL CHARACTERISTICS
2) DMA Model: In practice, microstrip lines are usually
utilized as the DMA waveguides. The propagation coefficient
for the lth element in the ith microstrip is given by [10]
hi ,l = e−(l−1)dl (αi+jβi ), (4)
where dl , αi , and βi are the inter-element distance, waveguide
attenuation coefficient, and propagation constant, respectively.
We assume that all microstrips are of the same type; βi =
β and αi = α. In general, the frequency of the system
affects the signal propagation inside a waveguide, and hence,
the propagation model must capture the frequency-dependent
effects.
The microstrip line comprises a conductor of width υ,
printed on a dielectric substrate with thickness ζ and dielectric
constant εr . Thus, the effective dielectric constant at DC is [11]
ε′e = (εr + 1)/2 + (εr − 1)/2
√
1 + 12ζ/υ. (5)
The effective dielectric constant at frequency f is [11]
εfe = εr −
(
εr − ε′e
)
/
(
1 +G(f )
)
, (6)
where G(f ) = (0.6 + 0.009Z0)f
2/(Z0/8πζ)
2, and
Z0 =
⎧
⎪⎨
⎪⎩
60 ln (8ζ/υ + υ/(4ζ))/
√
εfe , υ ≤ ζ,
120π/
√
εfe
υ/ζ+1.393+0.667 ln (υ/ζ+1.444)
, υ ≥ ζ,
(7)
is the characteristic impedance of the microstrip, while both
cases in (7) are approximately equal for υ/ζ = 1. Thereby,
β = (2π/λ)
√
εfe is the microstrip propagation constant and
the attenuation due to the dielectric loss is given by
αd = πεr
(
εfe − 1
)

/
(
λ
√
ε
f
e(εr − 1)
)
, (8)
where 
 is the loss tangent of the dielectric. Moreover,
αc = Rs/Z0υ is the approximate attenuation due to the
conductor loss and Rs =
√
2πf μ0/2σ is the conductor
surface resistivity with σ and μ0 being the conductivity and
the free space permeability, respectively. Finally, the microstrip
line attenuation coefficient is modeled as α = αd+αc . Table I
lists some microstrip materials with their main characteristics.
Next, H ∈ C
N×N is the microstrip propagation diagonal
matrix with hi ,l being the ((i − 1)Nh + l)th column and
((i − 1)Nh + l)th row element. The Lorentzian-constrained
phase model is utilized to capture the dependency between
the elements amplitude and phase [10], so that the frequency
response of the lth element in the ith microstrip is
qi ,l ∈ Q =
{(
j + ejφi,l
)
/2
∣∣∣φi ,l ∈ [0, 2π]
}
, ∀i , l . (9)
1The thickness of the material is not strictly limited to the mentioned value
and the typical ζ serves as an indicator of the product thickness range.
AZARBAHRAM et al.: EB FOR RF WPT WITH DMAs 783
Additionally, Q ∈ C
N×Nd is the matrix containing the
configurable weights of the metamaterial elements [8], i.e.,
Q(i−1)Nh+l ,n =
{
qi ,l , i = n,
0, i �= n.
(10)
3) Signal Model: Let xm be the mth energy symbol at the
input of the digital precoder where m = 1, . . . ,M and M =
min(Nv ,K ). Also, the energy symbols are independent and
normalized such that E{xHn xr} = 0 and E{xHn xn} = 1, while
wm is the Nv -dimensional precoding vector for xm . Thus, the
transmit signal is s =
∑M
m=1HQwmxm , while the transmit
power is PTx = Ex{sH s} =
∑M
m=1 |HQwm |2. The received
energy signal at the kth EH device is given by yk = γH
k s,
while the corresponding received RF power is
Pk
Rx = Ex
{
yHk yk
}
=
M∑
m=1
|γH
k HQwm |2. (11)
A. Problem Formulation
In practical WPT systems, the transmit power influences
largely the total power consumption. Indeed, the transmit
power determines the power consumption of the PAs, which
are the most power-hungry system components. Motivated by
this, our objective is to minimize the transmit power while
satisfying the users’ RF power requirements. Assuming the
location of the users is known, the optimization problem is
P1: minimize
Q,{wm},∀m
M∑
m=1
|HQwm |2 (12a)
subject to
M∑
m=1
|γH
k HQwm |2 ≥ δk , ∀k , (12b)
(10), qi ,l ∈ Q, ∀i , l . (12c)
where δk is the RF power meeting the EH requirements of the
kth user. The problem is non-convex while the variables are
highly coupled, due to the correlation between the phase and
amplitude of the metamaterial elements through the Lorentzian
constraint, making (12) very difficult to be optimally solved.
III. PROPOSED EB OPTIMIZATION SOLUTION
A. Optimal Digital Precoders With Fixed Q
Let us fix Q and rewrite (11) as
Pk
Rx =
M∑
m=1
((
γH
k HQ
)
wm
)H(
γH
k HQ
)
wm
= Tr
([ M∑
m=1
wmwH
m
]
bkb
H
k
)
= Tr
(
WBk
)
, (13)
which comes from several algebraic transformations and by
defining bk = (γH
k HQ)H ∈ C
Nd×1, W =
∑M
m=1 wmwH
m ,
and Bk = bkb
H
k . Similarly, we reformulate the transmit
power as PTx = Tr(WF), where F = (HQ)HHQ. By
leveraging (13), (12) can be reformulated with a fixed Q as
P2: minimize
W
Tr
(
WF
)
(14a)
subject to Tr
(
WBk
) ≥ δk , ∀k , (14b)
W 	 0, (14c)
which is an SDP that can be solved by standard convex
optimization tools, e.g., CVX [12]. Then, the optimal
precoding vectors {wm}∀m can be obtained as the eigenvec-
tors of W multiplied by the square root of their respective
eigenvalues.
B. Suboptimal Q With Fixed Digital Precoders
Now, let us fix {wm}∀m . Then, utilizing the fact that
aTGb = (bT ⊗ aT )Vec(G), we obtain [8]
|γH
k HQwm |2 =
∣∣∣
(
wT
m ⊗ (γH
k H)
)
q
∣∣∣2 = |zHm,kq|2, (15)
where q = Vec(Q) ∈ C
NNv×1 and zm,k = (wT
m ⊗
(γH
k H))H ∈ C
NNv×1. Moreover, we define q̂ ∈ C
N×1 as
the modified version of q without the zero elements, thus,
q̂ = [q1,1, q1,2, . . . , qNv ,Nh
]T . Additionally, ẑm,k ∈ C
N×1 is
obtained by removing the elements with the same index as the
zero elements in q. Then, the RF power received at user k is
Pk
Rx
(a)
=
M∑
m=1
(
zHm,kq
)H (
zm,kq
)
(b)
= q̂H
[ M∑
m=1
ẑm,k ẑ
H
m,k
]
q̂
(c)
=Tr
(
Z̃m,k Q̃
)
, (16)
where (a) comes from using (15) followed by a simple
transformation and (b) from zeroing the terms impacted by the
zero elements of q. Furthermore, (c) comes from defining Q̃ =
q̂q̂H and Z̃m,k =
∑M
m=1 ẑm,k ẑ
H
m,k . As previously mentioned,
the Lorentzian-constrained phase of the elements makes the
problem very complex and difficult to solve. One way to
cope with this complexity is by utilizing approximations torelax (12c) but this may cause the solution to violate the
constraint (12b). Instead, we propose decoupling the problem
to first maximize the minimum RF power at the users given
a fixed W and leveraging only the beamforming capability
of the DMA elements, and then utilizing such a feasible
solution to determine the W that minimizes the transmit
power consumption subject to (12b) and given fixed Q̃. By
conducting this procedure iteratively, a suboptimal solution is
obtained. Notice that such an optimization approach adapts
well to the fact that the gain introduced by the metamaterial
elements is limited and correlated with their phase values,
while the gain in digital precoders can be chosen freely without
any limitation and correlation with the phase values, making
them the dominant factor in determining the transmit power.
Thereby, the optimization problem is formulated as
P3: maximize
Q̃
min
k
Tr
(
Z̃m,k Q̃
)
(17a)
subject to Q̃ 	 0, (17b)
qi ,l ∈ Q, ∀i , l , (17c)
rank
(
Q̃
)
= 1. (17d)
Problem (17) is still difficult to solve due to the Lorentzian
and rank constraints. Therefore, we relax the problem as
P4: maximize
Q̃,t
t (18a)
subject to t ≤ Tr
(
Z̃m,k Q̃
)
, ∀k , (18b)
Tr
(
Q̃
) ≤ N , (18c)
Q̃ 	 0, (18d)
784 IEEE WIRELESS COMMUNICATIONS LETTERS, VOL. 13, NO. 3, MARCH 2024
Fig. 2. Mapping the P4 solution to the Lorentzian constrained weights in
the complex plane.
Algorithm 1 EB Optimization for DMA-Based RF-WPT
1: Input: C, I, {γ k , δk}∀k Output: {w�m}∀m , {q�i ,l}∀i ,l
2: Initialize:
3: Construct Q using random {φi ,l}∀i ,l , (9), and (10)
4: Solve P2 to obtain P�Tx and {w�m}∀m , i ′ = 1, c′ = 0
5: repeat
6: Solve P4 to obtain q′
7: Obtain {qi ,l}∀i ,l using (19) and construct Q
8: Solve P2 to obtain PTx and {wm}∀m ,
9: c′ ← c′ + 1, i ′ ← i ′ + 1
10: if PTx < P�Tx then
11: P�Tx ← PTx , w�m ←wm , q�i ,l ← qi ,l , ∀m, i , l , c′ ← 0
12: end if
13: until c′ = C or i ′ = I
where (18c) prevents the problem from becoming unbounded
and comes from the fact that the maximum value of the
diagonal elements of Q̃ is 1. Hence, the problem becomes an
SDP that can be solved similarly to (14). Then, we transform
the relaxed solution into a feasible solution according to
constraints (17c) and (17d). First, q′ is obtained as the
dominant eigenvector of Q̃ multiplied by the square root of
its eigenvalue. Still, q′ is not Lorentzian constrained, thus, we
map each of the elements in q′ to the nearest point on the
Lorentzian constrained circle as represented in Fig. 2. Hereby,
the configured phase of the lth metamaterial element in the ith
microstrip can be obtained as
φ�i ,l = min
φi,l
∣∣∣
(
j + ejφi,l
)
/2− q ′i ,l
∣∣∣, ∀i , l , (19)
where q ′i ,l is the ((i − 1)Nh + l)th element in q′, while (19)
can be easily solved using a one-dimensional search. Then,
q�i ,l = (j + ejφ
�
i,l )/2, ∀i , l .
C. Overall Algorithm
Algorithm 1 summarizes the overall EB optimization solu-
tion for DMA-assisted RF-WPT system. At first, a random
Lorentzian constrained frequency response is generated for
each metamaterial element, and Q is constructed. Then, a
local optimum solution is obtained for P3 through lines 6-7,
followed by finding the optimal precoders by solving P2.
Moreover, the solution is updated in lines 10-12 if the transmit
power decreases. The above process is repeated iteratively
until there is no improvement in the solution for a maximum
stall counter or a maximum number of iterations is reached.
The number of variables in (14) and (18) is Nv (Nv + 1)/2
and 1 + N(N + 1)/2, respectively. Notice that the rest of the
entries in W and Q̃ are determined according to the Hermitian
Fig. 3. (a) Average transmit power (top) and (b) average gain and the number
of antenna elements (bottom), as a function of frequency for fixed L = 10
cm, K = 1, and K = 2.
structure, and the sizes of these Hermitian matrix sub-spaces
are Nv
2 and N 2, respectively. Additionally, the number of
constraints scales with K in both problems. It is shown that
for a given accuracy, the complexity of SDP problems grows
at most with O(n1/2), where n is the problem size, which is
determined by the number of constraints and variables [13].
Thus, the proposed algorithm, which solves two SDP problems
for I iterations in the worst case, converges in polynomial time
and its complexity increases with K and N 2.
IV. NUMERICAL RESULTS
We consider a 100 m2 indoor area with a transmitter at the
center of the ceiling with a 3 m height. The users are randomly
distributed over 30 realizations and δk = 100 μ W, ∀k , while
the spacing between the DMA metamaterial elements and the
microstrips are λ/5 and λ/2, respectively. Thus, Nv = 
 L
λ/2
�
and Nh = 
 L
λ/5
�. A R©DuPont Pyralux AP-9161 is utilized
to design the microstrip line while its propagation coefficients
are estimated for different frequencies as in Section II-2.
We refer to the proposed method as EB-ASD, while particle
swarm optimization (PSO) [14] with 1000 iterations and 100
particles per iteration is used as a benchmark. The DMA
performance is compared to the FD structure with an inter-
element distance of λ/2. The optimal FD precoders are
obtained by solving P2 with bk = γ k as in [15]. Notably, an
FD setup requires a considerably large number of RF chains
with high power consumption, which is not considered here.
Thus, the performance gap between DMA and FD may be even
larger in practice. Assuming U random locations for a device,
the average gain in the figures is Eu{|γ u |2} and Eu{|γH
u H|2}
with u = 1, . . . ,U for FD and DMA, respectively.
Fig. 3(c) illustrates the simulation results over different
operating frequencies. It is observed that the proposed
approach outperforms PSO. The reason is that the problem
space is large, especially in high-frequency regime, and thus,
PSO needs much more time and particles to perform well.
AZARBAHRAM et al.: EB FOR RF WPT WITH DMAs 785
Fig. 4. (a) Average transmit power (top) and (b) average gain and the number
of antenna elements (bottom), as a function of antenna length for fixed f = 10
GHz, K = 1, and K = 2.
For a multi-user setup, maximum ratio transmission (MRT)-
based [15] precoders can be utilized to derive a lower bound
for the transmit power of the FD structure. Hereby, one may
expect that the received RF power scales with PTx ,k |γ k |2,
where
∑M
m=1 PTx ,m = PTx . Moreover, the transmit power
should be divided among K users and meet all RF power
requirements, thus, the transmit power increases with K, while
it slightly changes over f since |γ k |2 almost remains fixed.
In the DMA, one may expect that the received RF power
by the kth user is influenced by the pattern of |γH
k H|2 since
it includes the majority of the losses. However, there is no
guarantee that the pattern is just influenced by this term and
the structure of Q and its values are also important. Therefore,
although |γH
k H|2 decreases, the increasing pattern of the
transmit power with frequency is not assured and one may
experience other behaviors in different frequency ranges, as
seen in the low-frequency regime in the figure. All in all, the
DMA transmit power pattern may change depending on the
scenario and its parameters, such as the antenna form factor,
distance to the users, and the number of users. Similar to FD,
the transmit power increases with K since the total power
should be split among more users to meet their requirements.
Meanwhile, the number of elements is larger in DMA leading
to more beamforming capability, and additionally, one signal
can feed multiple elements in the DMA. Thus, the total output
power of the RF chains (transmit power) is less compared to
FD, and DMA outperforms FD in the low-frequency regime
in terms of the required transmit power. However, the DMA
losses may become large in higher frequencies depending on
the scenario and antenna form factor, and FD may perform
better.
The influence of the antenna length is illustrated in Fig. 4(c).
The discussions of the previous case are also applicable
here.As expected, the transmit power decreases with antenna
length in both architectures since the average effective gain
increases. Additionally, DMA outperforms FD in this specific
frequency but the situation may change at significantly higher
frequencies, as DMA losses become considerably large. The
proposed approach also outperforms PSO in this configuration.
V. CONCLUSION
We considered a multi-user RF-WPT system with a DMA-
based architecture. Moreover, we proposed an EB design
aiming to minimize the transmit power while meeting the
users’ EH requirements. The solution, using SDP and alter-
nating optimization, outperformed PSO, and our results also
reveal that a DMA transmitter is preferred to the FD alter-
native. It was observed that although increasing the system
frequency does not affect the FD architecture performance, it
increases the required transmit power for the DMA-assisted
system since the effective gain decreases. Meanwhile, increas-
ing the number of elements by utilizing a larger antenna
array may lead to significant performance gains in both
architectures.
REFERENCES
[1] O. L. A. López, H. Alves, R. D. Souza, S. Montejo-Sánchez,
E. M. G. Fernández, and M. Latva-Aho, “Massive wireless energy
transfer: Enabling sustainable IoT toward 6G era,” IEEE Internet Things
J., vol. 8, no. 11, pp. 8816–8835, Jun. 2021.
[2] B. Clerckx, R. Zhang, R. Schober, D. W. K. Ng, D. I. Kim, and
H. V. Poor, “Fundamentals of wireless information and power transfer:
From RF energy harvester models to signal and system designs,” IEEE
J. Sel. Areas Commun., vol. 37, no. 1, pp. 4–33, Jan. 2019.
[3] I. Ahmed et al., “A survey on hybrid beamforming techniques in 5G:
Architecture and system model perspectives,” IEEE Commun. Surveys
Tuts., vol. 20, no. 4, pp. 3060–3097, 4th Quart., 2018.
[4] L. Yan, C. Han, and J. Yuan, “A dynamic array-of-subarrays architecture
and hybrid precoding algorithms for terahertz wireless communica-
tions,” IEEE J. Sel. Areas Commun., vol. 38, no. 9, pp. 2041–2056,
Sep. 2020.
[5] N. Shlezinger, G. C. Alexandropoulos, M. F. Imani, Y. C. Eldar, and
D. R. Smith, “Dynamic metasurface antennas for 6G extreme massive
MIMO communications,” IEEE Wireless Commun., vol. 28, no. 2,
pp. 106–113, Apr. 2021.
[6] Q. Wu, X. Guan, and R. Zhang, “Intelligent reflecting surface-aided
wireless energy and information transmission: An overview,” Proc.
IEEE, vol. 110, no. 1, pp. 150–170, Jan. 2022.
[7] H. Zhang, N. Shlezinger, F. Guidi, D. Dardari, M. F. Imani, and
Y. C. Eldar, “Near-field wireless power transfer with dynamic metasur-
face antennas,” in Proc. IEEE (SPAWC), 2022, pp. 1–5.
[8] H. Zhang, N. Shlezinger, F. Guidi, D. Dardari, M. F. Imani, and
Y. C. Eldar, “Beam focusing for near-field multiuser MIMO communi-
cations,” IEEE Trans. Wireless Commun., vol. 21, no. 9, pp. 7476–7490,
Sep. 2022.
[9] S. W. Ellingson, “Path loss in reconfigurable intelligent surface-enabled
channels,” in Proc. IEEE (PIMRC), 2021, pp. 829–835.
[10] D. R. Smith, O. Yurduseven, L. P. Mancera, P. Bowen, and
N. B. Kundtz, “Analysis of a waveguide-fed metasurface antenna,” Phys.
Rev. Appl., vol. 8, Nov. 2017, Art. no. 54048. [Online]. Available:
https://link.aps.org/doi/10.1103/PhysRevApplied.8.054048
[11] D. M. Pozar, Microwave Engineering. Hoboken, NJ, USA: Wiley, 2012.
[12] M. Grant and S. Boyd, “CVX: MATLAB software for disciplined
convex programming (version 2.1).” Cvxr.com. Accessed: Jan. 28, 2020.
[Online]. Available: http://cvxr.com/cvx, Mar. 2014.
[13] L. Vandenberghe and S. Boyd, “Semidefinite programming,” SIAM
Rev., vol. 38, no. 1, pp. 49–95, 1996. [Online]. Available:
https://doi.org/10.1137/1038003
[14] J. Kennedy and R. Eberhart, “Particle swarm optimization,” in Proc. Int.
Conf. Neural Netw., 1995, pp. 1942–1948.
[15] O. L. A. López, D. Kumar, R. D. Souza, P. Popovski, A. Tölli,
and M. Latva-Aho “Massive MIMO with radio stripes for indoor
wireless energy transfer,” IEEE Trans. Wireless Commun., vol. 21, no. 9,
pp. 7088–7104, Sep. 2022.
<<
 /ASCII85EncodePages false
 /AllowTransparency false
 /AutoPositionEPSFiles false
 /AutoRotatePages /None
 /Binding /Left
 /CalGrayProfile (Gray Gamma 2.2)
 /CalRGBProfile (sRGB IEC61966-2.1)
 /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2)
 /sRGBProfile (sRGB IEC61966-2.1)
 /CannotEmbedFontPolicy /Warning
 /CompatibilityLevel 1.4
 /CompressObjects /Off
 /CompressPages true
 /ConvertImagesToIndexed true
 /PassThroughJPEGImages true
 /CreateJobTicket false
 /DefaultRenderingIntent /Default
 /DetectBlends true
 /DetectCurves 0.0000
 /ColorConversionStrategy /LeaveColorUnchanged
 /DoThumbnails false
 /EmbedAllFonts true
 /EmbedOpenType false
 /ParseICCProfilesInComments true
 /EmbedJobOptions true
 /DSCReportingLevel 0
 /EmitDSCWarnings false
 /EndPage -1
 /ImageMemory 1048576
 /LockDistillerParams true
 /MaxSubsetPct 100
 /Optimize true
 /OPM 0
 /ParseDSCComments false
 /ParseDSCCommentsForDocInfo false
 /PreserveCopyPage true
 /PreserveDICMYKValues true
 /PreserveEPSInfo false
 /PreserveFlatness true
 /PreserveHalftoneInfo true
 /PreserveOPIComments false
 /PreserveOverprintSettings true
 /StartPage 1
 /SubsetFonts false
 /TransferFunctionInfo /Remove
 /UCRandBGInfo /Preserve
 /UsePrologue false
 /ColorSettingsFile ()
 /AlwaysEmbed [ true
 /Arial-Black
 /Arial-BoldItalicMT
 /Arial-BoldMT
 /Arial-ItalicMT
 /ArialMT
 /ArialNarrow
 /ArialNarrow-Bold
 /ArialNarrow-BoldItalic
 /ArialNarrow-Italic
 /ArialUnicodeMS
 /BookAntiqua
 /BookAntiqua-Bold
 /BookAntiqua-BoldItalic
 /BookAntiqua-Italic
 /BookmanOldStyle
 /BookmanOldStyle-Bold
 /BookmanOldStyle-BoldItalic
 /BookmanOldStyle-Italic
 /BookshelfSymbolSeven
 /Century
 /CenturyGothic
 /CenturyGothic-Bold
 /CenturyGothic-BoldItalic
 /CenturyGothic-Italic
 /CenturySchoolbook
 /CenturySchoolbook-Bold
 /CenturySchoolbook-BoldItalic
 /CenturySchoolbook-Italic
 /ComicSansMS
 /ComicSansMS-Bold
 /CourierNewPS-BoldItalicMT
 /CourierNewPS-BoldMT
 /CourierNewPS-ItalicMT
 /CourierNewPSMT
 /EstrangeloEdessa
 /FranklinGothic-Medium
 /FranklinGothic-MediumItalic
 /Garamond
 /Garamond-Bold
 /Garamond-Italic
 /Gautami
 /Georgia
 /Georgia-Bold
 /Georgia-BoldItalic
 /Georgia-Italic
 /Haettenschweiler
 /Helvetica
 /Helvetica-Bold
 /HelveticaBolditalic-BoldOblique
 /Helvetica-BoldOblique
 /Helvetica-Condensed-Bold
 /Helvetica-LightOblique
 /HelveticaNeue-Bold
 /HelveticaNeue-BoldItalic
 /HelveticaNeue-Condensed
 /HelveticaNeue-CondensedObl
 /HelveticaNeue-Italic
 /HelveticaNeueLightcon-LightCond
 /HelveticaNeue-MediumCond
 /HelveticaNeue-MediumCondObl
 /HelveticaNeue-Roman
 /HelveticaNeue-ThinCond
 /Helvetica-Oblique
 /HelvetisADF-Bold
 /HelvetisADF-BoldItalic
 /HelvetisADFCd-Bold
 /HelvetisADFCd-BoldItalic
 /HelvetisADFCd-Italic
 /HelvetisADFCd-Regular
 /HelvetisADFEx-Bold
 /HelvetisADFEx-BoldItalic
 /HelvetisADFEx-Italic
 /HelvetisADFEx-Regular
 /HelvetisADF-Italic
 /HelvetisADF-Regular
 /Impact
 /Kartika
 /Latha
 /LetterGothicMT
 /LetterGothicMT-Bold
 /LetterGothicMT-BoldOblique
 /LetterGothicMT-Oblique
 /LucidaConsole
 /LucidaSans
 /LucidaSans-Demi
 /LucidaSans-DemiItalic
 /LucidaSans-Italic
 /LucidaSansUnicode
 /Mangal-Regular
 /MicrosoftSansSerif
 /MonotypeCorsiva
 /MSReferenceSansSerif
 /MSReferenceSpecialty
 /MVBoli
 /PalatinoLinotype-Bold
 /PalatinoLinotype-BoldItalic
 /PalatinoLinotype-Italic
 /PalatinoLinotype-Roman
 /Raavi
 /Shruti
 /Sylfaen
 /SymbolMT
 /Tahoma
 /Tahoma-Bold
 /Times-Bold
 /Times-BoldItalic
 /Times-Italic
 /TimesNewRomanMT-ExtraBold
 /TimesNewRomanPS-BoldItalicMT
 /TimesNewRomanPS-BoldMT
 /TimesNewRomanPS-ItalicMT
 /TimesNewRomanPSMT
 /Times-Roman
 /Trebuchet-BoldItalic
 /TrebuchetMS
 /TrebuchetMS-Bold
 /TrebuchetMS-Italic/Tunga-Regular
 /Verdana
 /Verdana-Bold
 /Verdana-BoldItalic
 /Verdana-Italic
 /Vrinda
 /Webdings
 /Wingdings2
 /Wingdings3
 /Wingdings-Regular
 /ZapfChanceryITCbyBT-MediumItal
 /ZWAdobeF
 ]
 /NeverEmbed [ true
 ]
 /AntiAliasColorImages false
 /CropColorImages true
 /ColorImageMinResolution 200
 /ColorImageMinResolutionPolicy /OK
 /DownsampleColorImages false
 /ColorImageDownsampleType /Average
 /ColorImageResolution 300
 /ColorImageDepth -1
 /ColorImageMinDownsampleDepth 1
 /ColorImageDownsampleThreshold 1.50000
 /EncodeColorImages true
 /ColorImageFilter /DCTEncode
 /AutoFilterColorImages false
 /ColorImageAutoFilterStrategy /JPEG
 /ColorACSImageDict <<
 /QFactor 0.76
 /HSamples [2 1 1 2] /VSamples [2 1 1 2]
 >>
 /ColorImageDict <<
 /QFactor 0.76
 /HSamples [2 1 1 2] /VSamples [2 1 1 2]
 >>
 /JPEG2000ColorACSImageDict <<
 /TileWidth 256
 /TileHeight 256
 /Quality 15
 >>
 /JPEG2000ColorImageDict <<
 /TileWidth 256
 /TileHeight 256
 /Quality 15
 >>
 /AntiAliasGrayImages false
 /CropGrayImages true
 /GrayImageMinResolution 200
 /GrayImageMinResolutionPolicy /OK
 /DownsampleGrayImages false
 /GrayImageDownsampleType /Average
 /GrayImageResolution 300
 /GrayImageDepth -1
 /GrayImageMinDownsampleDepth 2
 /GrayImageDownsampleThreshold 1.50000
 /EncodeGrayImages true
 /GrayImageFilter /DCTEncode
 /AutoFilterGrayImages false
 /GrayImageAutoFilterStrategy /JPEG
 /GrayACSImageDict <<
 /QFactor 0.76
 /HSamples [2 1 1 2] /VSamples [2 1 1 2]
 >>
 /GrayImageDict <<
 /QFactor 0.76
 /HSamples [2 1 1 2] /VSamples [2 1 1 2]
 >>
 /JPEG2000GrayACSImageDict <<
 /TileWidth 256
 /TileHeight 256
 /Quality 15
 >>
 /JPEG2000GrayImageDict <<
 /TileWidth 256
 /TileHeight 256
 /Quality 15
 >>
 /AntiAliasMonoImages false
 /CropMonoImages true
 /MonoImageMinResolution 400
 /MonoImageMinResolutionPolicy /OK
 /DownsampleMonoImages false
 /MonoImageDownsampleType /Bicubic
 /MonoImageResolution 600
 /MonoImageDepth -1
 /MonoImageDownsampleThreshold 1.50000
 /EncodeMonoImages true
 /MonoImageFilter /CCITTFaxEncode
 /MonoImageDict <<
 /K -1
 >>
 /AllowPSXObjects false
 /CheckCompliance [
 /None
 ]
 /PDFX1aCheck false
 /PDFX3Check false
 /PDFXCompliantPDFOnly false
 /PDFXNoTrimBoxError true
 /PDFXTrimBoxToMediaBoxOffset [
 0.00000
 0.00000
 0.00000
 0.00000
 ]
 /PDFXSetBleedBoxToMediaBox true
 /PDFXBleedBoxToTrimBoxOffset [
 0.00000
 0.00000
 0.00000
 0.00000
 ]
 /PDFXOutputIntentProfile (None)
 /PDFXOutputConditionIdentifier ()
 /PDFXOutputCondition ()
 /PDFXRegistryName ()
 /PDFXTrapped /False
 /CreateJDFFile false
 /Description <<
 /CHS <FEFF4f7f75288fd94e9b8bbe5b9a521b5efa7684002000410064006f006200650020005000440046002065876863900275284e8e55464e1a65876863768467e5770b548c62535370300260a853ef4ee54f7f75280020004100630072006f0062006100740020548c002000410064006f00620065002000520065006100640065007200200035002e003000204ee553ca66f49ad87248672c676562535f00521b5efa768400200050004400460020658768633002>
 /CHT <FEFF4f7f752890194e9b8a2d7f6e5efa7acb7684002000410064006f006200650020005000440046002065874ef69069752865bc666e901a554652d965874ef6768467e5770b548c52175370300260a853ef4ee54f7f75280020004100630072006f0062006100740020548c002000410064006f00620065002000520065006100640065007200200035002e003000204ee553ca66f49ad87248672c4f86958b555f5df25efa7acb76840020005000440046002065874ef63002>
 /DAN <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>
 /DEU <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>
 /ESP <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>
 /FRA <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>
 /ITA (Utilizzare queste impostazioni per creare documenti Adobe PDF adatti per visualizzare e stampare documenti aziendali in modo affidabile. I documenti PDF creati possono essere aperti con Acrobat e Adobe Reader 5.0 e versioni successive.)
 /JPN <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>
 /KOR <FEFFc7740020c124c815c7440020c0acc6a9d558c5ec0020be44c988b2c8c2a40020bb38c11cb97c0020c548c815c801c73cb85c0020bcf4ace00020c778c1c4d558b2940020b3700020ac00c7a50020c801d569d55c002000410064006f0062006500200050004400460020bb38c11cb97c0020c791c131d569b2c8b2e4002e0020c774b807ac8c0020c791c131b41c00200050004400460020bb38c11cb2940020004100630072006f0062006100740020bc0f002000410064006f00620065002000520065006100640065007200200035002e00300020c774c0c1c5d0c11c0020c5f40020c2180020c788c2b5b2c8b2e4002e>/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken waarmee zakelijke documenten betrouwbaar kunnen worden weergegeven en afgedrukt. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
 /NOR <FEFF004200720075006b00200064006900730073006500200069006e006e007300740069006c006c0069006e00670065006e0065002000740069006c002000e50020006f0070007000720065007400740065002000410064006f006200650020005000440046002d0064006f006b0075006d0065006e00740065007200200073006f006d002000650072002000650067006e0065007400200066006f00720020007000e5006c006900740065006c006900670020007600690073006e0069006e00670020006f00670020007500740073006b007200690066007400200061007600200066006f0072007200650074006e0069006e006700730064006f006b0075006d0065006e007400650072002e0020005000440046002d0064006f006b0075006d0065006e00740065006e00650020006b0061006e002000e50070006e00650073002000690020004100630072006f00620061007400200065006c006c00650072002000410064006f00620065002000520065006100640065007200200035002e003000200065006c006c00650072002e>
 /PTB <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>
 /SUO <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>
 /SVE <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>
 /ENU (Use these settings to create PDFs that match the "Recommended" settings for PDF Specification 4.01)
 >>
>> setdistillerparams
<<
 /HWResolution [600 600]
 /PageSize [612.000 792.000]
>> setpagedevice