# Waveform Optimization and Beam Focusing for Near-field Wireless Power
Transfer with Dynamic Metasurface Antennas and Non-linear Energy Harvesters

Amirhossein Azarbahram, Onel L. A. López, , and Matti Latva-Aho A. Azarbahram,
O. López and M. Latva-Aho are with Centre for Wireless Communications (CWC),
University of Oulu, Finland, (e-mail: {amirhossein.azarbahram,
onel.alcarazlopez, matti.latva-aho}@oulu.fi).This work is partially supported
in Finland by the Finnish Foundation for Technology Promotion, Academy of
Finland (Grants 348515 and 346208 (6G Flagship)), the Finnish-American
Research and Innovation Accelerator, and by the European Commission through
the Horizon Europe/JU SNS project Hexa-X-II (Grant Agreement no. 101095759).

###### Abstract

Radio frequency (RF) wireless power transfer (WPT) is a promising technology
for future wireless systems. However, the low power transfer efficiency (PTE)
is a critical challenge for practical implementations. One of the main
inefficiency sources is the power consumption and loss introduced by key
components such as high-power amplifier (HPA) and rectenna, thus they must be
carefully considered for PTE optimization. Herein, we consider a near-field
RF-WPT system with the emerging dynamic metasurface antenna (DMA) at the
transmitter and non-linear energy harvesters. We provide a mathematical
framework to calculate the power consumption and harvested power from multi-
tone signal transmissions. Then, we propose an approach relying on alternating
optimization and successive convex approximation for waveform optimization and
beam focusing to minimize power consumption while meeting energy harvesting
requirements. Numerical results show that increasing the transmit tones
reduces the power consumption by leveraging the rectifier’s non-linearity.
Moreover, it is demonstrated that increasing the antenna length improves the
performance, while both DMA and fully-digital architectures may be favorable
depending on the setup. Finally, our results verify that the transmitter
generates accurate energy beams pointed to devices located in the near-field,
while energy beams are formed in devices’ direction in the far-field region.

###### Index Terms:

Radio frequency wireless power transfer, waveform design, beamforming, dynamic
metasurface antennas, near-field channels.

##  I Introduction

Future wireless systems will facilitate efficient and eco-friendly
communication across a myriad of low-power devices, fostering a sustainable
society. Achieving this requires uninterrupted connectivity among these
devices and with the underlying infrastructure, all while mitigating
disruptions arising from battery depletion [1, 2, 3]. This is potentially
facilitated by energy harvesting (EH) technologies providing wireless
charging, thus, easing the maintenance of Internet of Things (IoT) devices and
increasing their lifespan. Moreover, EH may lead to improved energy efficiency
and reduced emission footprints across the network [4].

EH receivers may harvest energy from two types of sources: those that exist in
the surrounding environment, and those that are specifically designated for
energy transmission. From a transmission perspective, the latter is supported
by wireless power transfer (WPT) technologies, e.g., based on inductive
coupling, magnetic resonance coupling, laser power beaming, and radio
frequency (RF) radiation. Among them, RF-WPT is promising for charging
multiple users relatively far from the transmitter by exploiting the broadcast
nature of wireless channels. Furthermore, this can be accomplished over the
same infrastructure used for wireless communications. Notably, the most
important challenge toward maturing RF-WPT is related to increasing the
inherently low end-to-end power transfer efficiency (PTE) [4]. Herein, we
focus on RF-WPT, which is referred to as WPT in the following.

###  I-A Preliminaries

Figure 1: Block diagram of a typical WPT system. The power consumption and
loss sources are listed under each block.

The end-to-end PTE depends on the performance of the key building blocks,
namely energy transmitter (ET), wireless channel, and energy receiver (ER) as
illustrated in Fig. 1. At first, a signal is generated and amplified using a
direct current (DC) power source at the ET. Then, it is upconverted to the RF
domain and transmitted over the wireless channel. Finally, the ER converts the
received RF signal to DC for EH purposes. Indeed, the end-to-end PTE
comprises: DC-to-RF, RF-to-RF, and RF-to-DC power conversion efficiency, i.e.,

$e=\underbrace{\frac{P^{t}_{rf}}{P^{t}_{dc}}}_{e_{1}}\times\underbrace{\frac{P^{r}_{rf}}{P^{t}_{rf}}}_{e_{2}}\times\underbrace{\frac{P^{r}_{dc}}{P^{r}_{rf}}}_{e_{3}}=\frac{P^{r}_{dc}}{P^{t}_{dc}}.$
(1)

In WPT, both transmitter and receiver introduce non-linearities to the signals
affecting the amount of harvested power. Indeed, an appropriately designed
transmit signal leveraging these non-linearities may reduce the power
consumption at the transmitter and/or increases the RF-to-DC conversion
efficiency at the receiver [5, 6]. Specifically, using multiple transmit
frequency tones can lead to high peak-to-average power ratio (PAPR) signals at
the receiver, enhancing the rectifier RF-to-DC conversion efficiency [7].
Meanwhile, the RF signal can be focused towards the ER using energy
beamforming (EB), which affects the transmit/receive waveform, to cope with
the channel inefficiencies captured by $e_{2}$, thereby enhancing the amount
of RF power that can be harvested [4]. Notice that the active transmit
components consume power to operate, while the passive elements introduce
power losses, both of which impact $e_{1}$ and must be considered. Therefore,
$e_{1}$, $e_{2}$, and $e_{3}$ are correlated, suggesting that their joint
optimization may lead to significant gains in terms of end-to-end PTE.

The end-to-end PTE is also affected by the system power consumption. One of
the main factors contributing greatly to the power consumption is the
transmitter’s architecture, which also determines the beamforming approach.
For example, in a fully-digital structure, each antenna element necessitates a
dedicated RF chain, with its corresponding high-power amplifier (HPA),
consuming a significant amount of power. An HPA aims to amplify the input
signal to compensate for the path loss and fading in a wireless system.
Moreover, the signal amplification in the HPA requires a DC power source,
which accounts for the majority of the power consumption. The significant
drawback of the fully-digital architecture is the high complexity and cost,
making it impractical for applications requiring massive multiple-input
multiple-output (mMIMO) implementations. Alternatively, analog architectures
using, e.g., passive phase shifters, are less expensive but offer limited
degrees of freedom for EB. Thus, a hybrid architecture implementation
combining both approaches is often more appealing in practice. Hybrid
architectures offer a trade-off between complexity (cost) and beamforming
flexibility [8, 9].

Although hybrid beamforming using phase shifters promotes cost reduction, it
still requires complex analog networks for phase shifting. There are emerging
technologies to provide hybrid beamforming capability with an even lower cost
and complexity, e.g., reflective intelligent surface (RIS)-aided systems [10]
and dynamic metasurface antennas (DMAs) [11]. Notice that RIS is an assisting
node that provides the passive beamforming capability using reflective
elements, while DMA is a transceiver consisting of configurable metamaterial
elements and a limited number of RF chains. DMA avoids analog network
implementation challenges and provides hybrid beamforming capability with low
cost and complexity. Each of these architectures may be favorable based on the
system setup. For instance, employing multiple low-cost RISs helps to cover
the blind spots that are prone to weak signal reception in a large area.
However, reflecting surfaces in RIS-assisted systems lack baseband processing
capability to perform channel estimation and send pilot signals. Thus,
acquiring accurate enough channel state information to attain a suitable
passive beamforming gain might force huge overheads to the system [12]. On the
other hand, DMA is a transceiver and has sufficient baseband processing
capability for channel estimation. However, its implementation requires some
RF chains making it more costly than RIS for large-scale implementation. All
in all, both of these architectures support low-cost transmitter deployments,
while the choice highly depends on the considered system setup. Interestingly,
the authors in [13] utilize a system model comprising both RIS and DMA
structures for uplink MIMO communication, while assuming that the channel is
perfectly known.

###  I-B Prior works

There are many works either focusing on EB, waveform optimization, or joint
waveform and beamforming design for fully-digital WPT systems. The authors in
[14] utilize EB to power multiple devices in a MIMO system consisting of radio
stripes, while the deployment problem of this transmit architecture is
investigated in [15]. Furthermore, a low-complexity beamforming design relying
on the statistics of the channel is proposed in [16] to fairly power a set of
single-antenna devices. In the mentioned works, none of the practical system
non-linearities are considered and the focus is on the received RF power at
the devices. In [17], transmit beamforming and RF and DC combining at the
receiver are leveraged to increase the received DC power in a MIMO WPT system.
Although this work considers the rectifier’s non-linearity, the waveform
design is not investigated. Moreover, a low-complexity waveform design for
single-user setups is proposed in [18], while the large-scale multi-antenna
WPT scenario is addressed in [19]. The authors in [20] leverage beamforming
and a multi-sine waveform in a MIMO WPT system to enhance the harvested power.
Notably, the frameworks in [18, 19, 20] consider the rectifier non-linearity.
Interestingly, the authors in [21] perform waveform and beamforming
optimization while considering both main non-linearity sources (HPA and
rectifier) aiming to maximize the harvested DC power in a WPT system.

Although most of the works on WPT systems in the literature focus on
traditional fully-digital architecture, novel low-cost transmitters have also
attained significant attention recently. For instance, DMA is utilized in [22,
23] for a near-field WPT system, while the authors in [24] propose a minimum-
power beamforming design for meeting quality of service requirements of the
users in a simultaneous wireless information and power transfer (SWIPT)
system. However, none of these studies have considered the rectifier non-
linearity and its impact on the harvested DC power. Notably, the joint
waveform and beamforming design problem in RIS-aided WPT and SWIPT systems is
investigated in [25] and [26], respectively. Furthermore, the two latter works
consider the EH non-linearity at the receiver side, thus, taking into account
its impact on the harvested DC power.

###  I-C Contributions

All in all, WPT systems have received considerable attention for some time.
Still, more effort is needed to reduce the system power consumption, thus
increasing the end-to-end PTE. For this, low-cost multi-antenna transmitters
like DMA are appealing and may pave the way for charging devices efficiently
in massive IoT deployments. Moreover, multiple studies aimed to enhance the
amount of harvested DC power (with rectifier non-linearity) in far-field WPT
systems or received RF power (without rectifier non-linearity) in near-field
WPT. As mentioned before, the receiver does not perform RF-to-DC conversion
linearly and the shape of the waveform may leverage the receiver’s RF-to-DC
conversion efficiency. Thus, it is imperative to take into account the impact
of the rectifier’s non-linearity in the system. To the best of our knowledge,
no work has yet investigated the radiative near-field power transmission and
the power consumption of a multi-antenna WPT system for meeting the EH
requirements of a multi-user setup while considering the receiver non-
linearity, especially when using low-cost transmitters. Herein, we aim
precisely to fill this research gap. Our main contributions are as follows:

1) We formulate a joint waveform optimization and beam focusing problem for a
multi-user multi-antenna WPT system with both a fully-digital and a DMA
architecture. Due to the huge potential of near-field WPT systems for future
practical WPT applications [3], we present our system model relying on a near-
field wireless channel, which can inherently capture far-field conditions as
well. Notice that there are some previous works [25, 21], which focused on
increasing the amount of harvested power in far-field WPT systems while
considering the receiver non-linearity. However, the literature lacks a
minimum-power waveform and beamforming design (even for far-field channels),
which can deal with meeting the EH requirements. This is a critical gap to
fill since such formulation mimics a practical setup where the EH users inform
their DC power demands and the WPT system must serve them with minimum power
consumption, thus maximum end-to-end PTE. Since the HPA is an active element
incurring most of the power consumption at the transmitter side, we model the
power consumption of a class-B HPA as a function of its output power. Notably,
our problem for fully-digital architecture shares some similarities with the
one discussed in [21] as objectives and constraints are interchanged. However,
the main focus of our work here is on the DMA-assisted system, which
introduces much more complexity to the problem due to the coupling between the
variables and their Lorentzian-constrained phase response. Note that the phase
shift introduced by the metamaterial elements is correlated with their
amplitude, which results in a different beamforming problem than other
architectures, e.g., RIS-assisted systems [26] and phase shifter-based hybrid
beamforming [9]. Mathematically, when dealing with those latter architectures,
both RIS passive elements or phase shifters introduce a phase shift to the
signal with constant loss111Note that in most of the works in the literature,
without loss of generality, this phase shifting process by the analog network
or RIS elements is considered to be lossless., while each phase-shifting
configuration in DMA elements leads to a different propagation loss introduced
to the transmit signal. The complexities associated with our specific problem
make the existing optimization frameworks for WPT systems inapplicable to our
system calling for novel approaches.

2) We propose a method relying on alternating optimization and successive
convex approximation (SCA) to efficiently solve the waveform and beamforming
optimization problem in the DMA-assisted WPT system. Specifically, we decouple
the optimization problem to maximize the minimum received DC power by tuning
the frequency response of the metamaterial elements, while minimizing the
consumed power for meeting the EH requirements when optimizing the digital
precoders. Generally, a huge complexity is introduced to the waveform
optimization problems by time sampling since the number of samples should be
relatively large to result in a reliable framework [21, 6, 20]. To cope with
this, we reformulate the received DC power of the users based on the spectrum
of the received waveform, which removes the time dependency in the problem.
Then, the metamaterial elements and the digital precoders are alternatively
optimized using SCA. Motivated by the influence of variable initialization on
the SCA performance, we propose a low-complexity initialization algorithm for
the digital precoders and DMA weights by leveraging the channel
characteristics and dedicating RF signals to the ERs. Furthermore, the
complexity of the proposed optimization framework scales with the number of
users, antenna elements, and frequency tones.

3) We illustrate the convergence of the proposed optimization method
numerically and show that the complexity increases with the antenna length,
number of transmit tones, and number of devices. Furthermore, we provide
evidence that increasing the antenna length and the number of transmit tones
reduces power consumption, while it increases with the number of devices and
user distance. Moreover, our findings evince that DMA performs better in terms
of power consumption when the number of RF chains and transmit tones are
relatively low, while the fully-digital architecture becomes favorable when
the mentioned parameters are sufficiently large. This also depends on the
specific HPAs’ saturation power, number of devices, and user distance.
Additionally, we verify by simulation that the transmitter can accurately
focus the energy on the receiver location in the near-field region, while
energy beams are only formed toward specific directions in the far-field
region.

The remainder of the paper is structured as follows. Section II introduces the
system model, including the transmit architectures, and signal and power
consumption modeling. The optimization problem for a joint waveform and
beamforming design, together with the proposed solving approach, are
elaborated in Section III. Section IV discusses the proposed initialization
algorithm, Section V presents the numerical results, while Section VI
concludes the paper.

Notations: Bold lower-case letters represent column vectors, while non-
boldface characters refer to scalars, ${\mathbf{a}}\odot{\mathbf{b}}$ denotes
the Hadamard product of $\mathbf{a}$ and $\mathbf{b}$, and $\\{x\\}$ is a set
that contains $x$. The $l_{2}$-norm of a vector is denoted as $|\cdot|$. The
mathematical expectation is represented by $\mathbb{E}$ and $(\cdot)^{T}$ and
$(\cdot)^{\star}$ are used to indicate the transpose and conjugate of a matrix
or vector, respectively. Furthermore, the real and the imaginary parts of a
complex number are denoted by $\Re\\{\cdot\\}$ and $\Im\\{\cdot\\}$,
respectively. Additionally, $\lfloor{\cdot}\rfloor$ is the floor operator, and
$\langle{\cdot}\rangle$ denotes the phase of a complex number.

##  II System Model

We consider a multi-antenna WPT system to charge $M$ single-antenna EH
devices. The received RF power at the ER is transferred into the rectifier
input using a matching network. Then, it is converted to DC power by the
rectifier, while $\bar{P}_{m}$ denotes the EH requirement of the $m$th ER.

As previously mentioned, multi-tone waveforms can be exploited to leverage the
rectifier non-linearity and achieve a better end-to-end PTE. Hence, we
consider multi-tone signals with $N_{f}$ tones at frequencies
$f_{1},f_{2},\cdots,f_{N_{f}}$ for power transmission purposes. Without loss
of generality, we set $f_{n}=f_{1}+(n-1)\Delta_{f},\quad n=1,\ldots,N_{f}$,
where $f_{1}$ and $\Delta_{f}$ are the lowest sub-carrier frequency and the
sub-carrier spacing, respectively.

###  II-A Transmit Antenna Architectures

The transmitter is equipped with a uniform planar array (UPA) and $N_{rf}\geq
M$ RF chains. The radiating elements are spaced uniformly, with $N_{h}$ and
$N_{v}$ being the number of elements in the horizontal and vertical direction,
respectively. Thus, the total number of elements is $N=N_{v}\times N_{h}$. Two
types of transmit antenna architectures are considered: 1) Fully-digital
architecture, which requires a dedicated RF chain for each radiating element,
thus $N_{rf}=N$, as shown in Fig. 2a. In a fully-digital architecture, there
is a single-stage beamforming process. Herein, $\omega_{i,n}$ is the complex
weight of the $n$th frequency tone of the $i$th muti-tone waveform. Despite
the high deployment cost and complexity, a fully-digital structure offers the
highest number of degrees of freedom in beamforming.

Figure 2: Transmit antenna architectures. (a) fully-digital architecture
(left) (b) DMA-assisted architecture (right).

2) DMA-assisted architecture, which comprises $N_{v}$ waveguides, each fed by
a dedicated RF chain and composed of $N_{h}$ configurable metamaterial
elements. Therefore, the number of RF chains, and consequently the cost and
complexity, is considerably reduced compared to digital structures, making DMA
suitable for mMIMO applications. Notice that DMA-assisted systems employ a
two-stage beamforming process, i.e., digital beamforming, followed by the
tuning of the amplitude/phase variations introduced by the metamaterial
elements, as illustrated in Fig. 2b. Herein, $q_{i,l}$ is the tunable
frequency response of the $l$th metamaterial element in the $i$th waveguide,
while $h_{i,l}$ is the corresponding waveguide propagation loss, which will be
explained in detail later.

###  II-B Channel Model

In wireless communications, the region where the users are located between the
Fraunhofer and Fresnel distances denoted respectively as $d_{fr}$ and
$d_{fs}$, is the radiative near-field region, which is referred to as the
near-field region in the following. Specifically, a device at distance $r$
from a transmitter experiences near-field conditions
if$\sqrt[3]{\frac{D^{4}}{8\lambda_{1}}}=d_{fs}<d<d_{fr}=\frac{2D^{2}}{\lambda_{1}}$,
where $D$ is the antenna diameter, i.e., the largest size of the antenna
aperture, $\lambda_{1}=\frac{c}{f_{1}}$ is the corresponding wavelength to the
system operating frequency, and $c$ is the speed of light. Notice that both
system frequency and antenna form factor influence the region of operation.
Therefore, by moving toward higher frequencies, e.g., millimeter wave and sub-
THz bands, and/or utilizing larger antenna arrays, the far-field communication
assumption regarding planar wavefronts may not be valid anymore. Instead,
wavefronts impinging a receive node may be strictly spherical, thus, with
advanced capabilities to focus the transmit signals on specific spatial points
rather on spatial directions.

Notice that one of the main applications of WPT systems is in indoor
environments with line-of-sight (LOS) and near-field communication, e.g.,
restaurants, warehouses, and shopping malls. Thus, we employ the near-field
LOS channel model described in [27]. The Cartesian coordinate of the $l$th
radiating element in the $i$th row is
$\mathbf{g}_{i,l}=[x_{i,l},y_{i,l},z_{i,l}]^{T}$. Additionally,
$i=1,2,\ldots,N_{v}$ and $l=1,2,\ldots,N_{h}$. The channel coefficient between
user $m$ and the $l$th element in the $i$th row at the $n$th sub-carrier is
given by

$\gamma_{i,l,m,n}=A_{i,l,m,n}e^{\frac{-j2\pi}{\lambda_{n}}d_{i,l,m}},$ (2)

where ${\frac{2\pi}{\lambda_{n}}d_{i,l,m}}$ is the phase shift caused by the
propagation distance of the $n$th tone, with wavelength $\lambda_{n}$, and
$d_{i,l,m}=|\mathbf{g}_{m}-\mathbf{g}_{i,l}|$ is the Euclidean distance
between the element and the user located at $\mathbf{g}_{m}$. Moreover,

$A_{i,l,m,n}=\sqrt{F(\Theta_{i,l,m})}\frac{\lambda_{n}}{4\pi d_{i,l,m}}$ (3)

is the corresponding channel gain coefficient. Here,
$\Theta_{i,l,m}=(\theta_{i,l,m},\psi_{i,l,m})$ is the elevation-azimuth angle
pair, and $F(\Theta_{i,l,m})$ is the radiation profile of each element. In
addition, we employ the radiation profile as presented in [28], where

$F(\Theta_{i,l,m})=\begin{cases}G_{t}\cos{(\theta_{i,l,m})}^{\frac{G_{t}}{2}-1},&\theta_{i,l,m}\in[0,\pi/2],\\\
0,&\text{otherwise},\end{cases}$ (4)

$G_{t}=2(b+1)$ is the transmit antenna gain, and $b$ denotes the boresight
gain, which depends on the specifications of the antenna elements. Note that
the channel coefficient becomes $A_{m}e^{-j\psi_{i,l,m}}$ for far-field
communication, where $A_{m}$ only depends on the distance of the user $m$ from
the transmitter and $\psi_{i,l,m}$ is solely determined by the user direction
and the relative disposition of the antenna elements within the array.

###  II-C Transmit & Receive Signals

The signal at the input of the $i$th HPA is given by

$x_{i}(t)=\sum_{n=1}^{N_{f}}{\omega_{i,n}e^{j2\pi f_{n}t}},\quad
i=1,\ldots,N_{rf}.$ (5)

The HPA introduces signal distortion and models such as the Rapp model [29]
capture this non-linearity. It is shown in [21] that when the HPA operates in
the non-linear regime, choosing a single-carrier waveform is preferred to a
multi-carrier one. The reason is that a single-carrier waveform is less
deteriorated by the adverse effect of the signal distortion caused by the HPA
when operating in the non-linear regime. On the other hand, when HPAs operate
in the linear regime, thus, not causing amplitude and phase distortion in the
signal, multi-carrier waveforms are preferred since they leverage the
rectifier’s non-linearity and enhance the harvested power performance.

Note that the non-linear regime of the HPA happens near the saturation
voltage. Thus, in practice, the HPAs can be properly chosen to have a suitable
value of the saturation voltage based on the system setup and avoid operation
in the non-linear regime. Since the aim of this work is to design multi-
carrier waveforms for DMA-assisted WPT, we consider HPAs to operate in the
linear regime. Mathematically, the output signal of the HPA is modeled as
$x^{hpa}_{i}(t)=Gx_{i}(t)$, where $G$ is the HPA gain. The rest of the signal
modeling formulation will be presented separately for different transmit
architectures in the following.

####  II-C1 Fully-Digital Architecture

Herein, $N_{rf}=N$, thus, the real transmit signal at the output of the $l$th
element in the $i$th row of the UPA can be expressed as

$\displaystyle\quad
x^{FD}_{i,l}(t)=\Re\biggl{\\{}x^{hpa}_{u{(i,l)}}(t)\biggr{\\}}=G\sum_{n=1}^{N_{f}}\Re\biggl{\\{}\omega_{{u{(i,l)}},n}e^{j2\pi
f_{n}t}\biggr{\\}},$ (6)

where ${u{(i,l)}}=(i-1)N_{h}+l$. Thereby, the RF signal at the $m$th receiver
when exploiting the fully-digital architecture can be expressed as

$\displaystyle y^{FD}_{m}(t)$
$\displaystyle=\sum_{i=1}^{N_{v}}\sum_{l=1}^{N_{h}}\sum_{n=1}^{N_{f}}\gamma_{i,l,m,n}x^{FD}_{i,l}(t)$
$\displaystyle=G\sum_{i=1}^{N_{v}}\sum_{l=1}^{N_{h}}\sum_{n=1}^{N_{f}}\Re\Bigl{\\{}\gamma_{i,l,m,n}\omega_{{u{(i,l)}},n}e^{j2\pi
f_{n}t}\Bigr{\\}}.$ (7)

Furthermore, by defining
$\mathbf{w}_{n}=[\omega_{1,n},\ldots,\omega_{N,n}]^{T}$ and
$\boldsymbol{\gamma}_{m,n}=[\gamma_{m,n,1,1},\gamma_{m,n,1,2},\ldots,\gamma_{m,n,N_{v},N_{h}}]^{T}$,
(II-C1) can be reformulated as

$y^{FD}_{m}(t)=G\sum_{n=1}^{N_{f}}\Re\Bigl{\\{}\boldsymbol{\gamma}_{m,n}^{T}\mathbf{w}_{n}e^{j2\pi
f_{n}t}\Bigr{\\}}.$ (8)

Figure 3: The Lorentzian constrained (the inner circle) and the ideal weights
(outer circle) in the complex plane. The arrows depict the mapping between the
weights.

####  II-C2 DMA-assisted Architecture

In metasurface antennas, the phase and amplitude that can be configured in the
radiating elements are correlated due to the Lorentzian resonance. Herein, we
capture this correlation by [30]

$q_{i,l}\in\mathcal{Q}=\Big{\\{}{(j+e^{j\phi_{i,l}})}/{2}\Big{|}\phi_{i,l}\in[0,2\pi]\Big{\\}},\quad\forall
i,l.$ (9)

where $\phi_{i,l}$ are the tunable phase of the $l$th metamaterial element in
the $i$th waveguide. As shown in Fig. 3, the ideal phase shifting exhibits a
constant unit amplitude, i.e., no losses, while the amplitude of the
Lorentzian weights depends on the configured phase.

Herein, microstrip lines are used as waveguides, similar to [31, 27]. The
propagation model of the signal within a microstrip is expressed as
$h_{i,l}=e^{-(l-1)d_{l}(\alpha_{i}+j\beta_{i})}$, where $d_{l}$ is the inter-
element distance, $\alpha_{i}$ represents the waveguide attenuation
coefficient, and $\beta_{i}$ is the propagation constant. The mathematical
model of the DMA is represented in Fig. 2b.

Notice that the number of RF chains in the DMA is reduced to $N_{rf}=N_{v}$.
Hence, the real transmit signal radiated from the $l$th element in the $i$th
microstrip can be expressed as

$\displaystyle x_{i,l}^{DMA}(t)$
$\displaystyle\\!=\\!G\Re\left\\{h_{i,l}q_{i,l}x_{i}^{hpa}(t)\right\\}\\!=\\!G\sum_{n=1}^{N_{f}}\Re\left\\{h_{i,l}q_{i,l}{\omega_{i,n}e^{j2\pi
f_{n}t}}\right\\}.$ (10)

Furthermore, the RF signal received at the $m$th user in the DMA-assisted
system is given by

$\displaystyle y^{DMA}_{m}(t)$
$\displaystyle=\sum_{i=1}^{N_{v}}\\!\sum_{l=1}^{N_{h}}\\!\sum_{n=1}^{N_{f}}\gamma_{i,l,m,n}x_{i,l}^{DMA}(t)$
$\displaystyle=G\sum_{i=1}^{N_{v}}\\!\sum_{l=1}^{N_{h}}\\!\sum_{n=1}^{N_{f}}\\!\Re\Bigl{\\{}\gamma_{i,l,m,n}h_{i,l}q_{i,l}\omega_{i,n}e^{j2\pi
f_{n}t}\Bigr{\\}}.$ (11)

Finally, we define

$\displaystyle\bar{\mathbf{w}}_{n}=[\underbrace{\omega_{1,n},\ldots,\omega_{1,n}}_{N_{h}},\ldots,\underbrace{\omega_{N_{v},n},\ldots,\omega_{N_{v},n}}_{N_{h}}]^{T}\in\mathbb{C}^{N\times
1},$

$\mathbf{q}=[q_{1,1},\ldots,q_{N_{v},N_{h}}]^{T}$,
$\mathbf{h}=[h_{1,1},\ldots,h_{N_{v},N_{h}}]^{T}$, and reformulate (II-C2) as

$y^{DMA}_{m}(t)=G\sum_{n=1}^{N_{f}}\Re\Bigl{\\{}(\boldsymbol{\gamma}_{m,n}\odot\mathbf{q}\odot\mathbf{h})^{T}\bar{\mathbf{w}}_{n}e^{j2\pi
f_{n}t}\Bigr{\\}}.$ (12)

###  II-D Rectenna

At the receiver side, the RF signal is converted to DC. This can be modeled by
an antenna equivalent circuit and a single diode rectifier as illustrated in
Fig. 4. The RF signal at the input of the antenna is denoted as $y_{m}(t)$ and
has an average power of $\mathbb{E}\bigl{\\{}{y_{m}(t)}^{2}\bigr{\\}}$. Let us
denote the input impedance of the rectifier and the impedance of the antenna
equivalent circuit by $R_{in}$ and $R_{ant}$, respectively. Thus, assuming
perfect matching $(R_{in}=R_{ant})$, the input voltage at the rectifier of the
$m$th ER is given by $v_{{in},{m}}(t)=y_{m}(t)\sqrt{R_{ant}}$. Furthermore,
the diode current can be formulated as

$i_{d}(t)=i_{s}\bigl{(}e^{\frac{v_{d}(t)}{\hat{n}v_{t}}}-1\bigr{)},$ (13)

where $i_{s}$ is the reverse bias saturation current, $\hat{n}$ is the
ideality factor, $v_{t}$ is the thermal voltage, and
$v_{d}(t)=v_{in}(t)-v_{o}(t)$ is the diode voltage. Moreover, $v_{{o},{m}}(t)$
is the output voltage of the $m$th rectifier, which can be approximated
utilizing the Taylor expansion as [20, 19]

$v_{{o},{m}}=\sum_{i\ even,i\geq
2}^{n_{0}}K_{i}\mathbb{E}\bigl{\\{}y_{m}(t)^{i}\bigr{\\}},$ (14)

where $K_{i}=\frac{R_{ant}^{i/2}}{i!{(\eta_{0}v_{t})}^{i-1}}$. Herein, we
focus on the low-power regime, for which it was demonstrated in [19, 6] that
truncating the Taylor expansion at $n_{0}=4$ is accurate enough. Therefore,
(14) can be written as

$v_{{o},{m}}=K_{2}\mathbb{E}\bigl{\\{}y_{m}(t)^{2}\bigr{\\}}+K_{4}\mathbb{E}\bigl{\\{}y_{m}(t)^{4}\bigr{\\}}$
(15)

and the DC power at the $m$th receiver is given by

$P_{{dc},m}=\frac{{v^{2}_{{o},{m}}}}{R_{L}},$ (16)

where $R_{L}$ is the load impedance of the rectifier, while $y_{m}(t)$ is
equal to $y_{m}^{DMA}(t)$ and $y_{m}^{FD}(t)$ in the DMA and fully-digital.

Figure 4: Antenna equivalent circuit (left) and a single diode rectifier
(right) [20].

###  II-E Power Consumption Model

Lowering the system power consumption is desirable for increasing the end-to-
end PTE and this greatly depends on the power consumption of the HPA. Let us
denote the maximum efficiency and the maximum output power of a class-B HPA as
$\bar{\eta}$ and $P_{max}$, respectively. Hereby, the efficiency of the $i$th
HPA at time $t$ is expressed as
$\eta_{i}(t)={\bar{\eta}}\sqrt{{P_{out,i}(t)}/{P_{max}}}$ [32], where
$P_{out,i}(t)$ is the output power of the HPA. Then, the corresponding power
consumption is

$P_{hpa,i}(t)=\frac{P_{out,i}(t)}{\eta_{i}(t)}=\frac{1}{\bar{\eta}}\sqrt{{P_{max}{P_{out,i}(t)}}}.$
(17)

Notice that each HPA is in charge of supplying the amount of radiating power
by all of the antenna elements, which are fed by its corresponding RF chain.
Thus, $P_{out,i}(t)$ is $\sum_{l=1}^{N_{h}}|x^{DMA}_{i,l}(t)|^{2}$ in DMA-
assisted architecture, and $|x^{FD}_{i,l}(t)|^{2}$ denotes the output power of
the RF chain connected to the $l$th element in the $i$th row.

There are also other power consumption sources in the WPT system. For
instance, digital baseband power consumption, which is required to perform the
digital beamforming, and the RF chain circuit power consumption, including the
mixer, local oscillator, and filter. However, the power consumption of these
sources is usually considered fixed and is negligible compared to the HPA.
Thus, without loss of generality, the total power consumption of the system is
given by

$P_{c}=\sum_{i=1}^{N_{rf}}\mathbb{E}\bigl{\\{}P_{{hpa},i}(t)\bigr{\\}}+P_{in},$
(18)

where $P_{in}=\sum_{i=1}^{N_{rf}}\sum_{n=1}^{N_{f}}|\omega_{i,n}|^{2}$ is the
total input power.

##  III Joint Waveform & Beamforming Optimization

This section formalizes the optimization problem and describes the utilized
approach when employing the aforementioned transmit architectures.

###  III-A Problem Formulation

The goal is to obtain a minimum-power waveform and beamforming design to meet
the EH requirements of the users. Thus, by utilizing (16) and substituting
(17) in (18), the optimization problem can be formulated as

$\displaystyle\operatorname*{minimize}_{\mathcal{V}}\quad$
$\displaystyle\frac{\sqrt{P_{max}}}{\bar{\eta}}\sum_{i=1}^{N_{rf}}\mathbb{E}\bigl{\\{}\sqrt{P_{out,i}}(t)\bigr{\\}}+\sum_{i=1}^{N_{rf}}\sum_{n=1}^{N_{f}}|\omega_{i,n}|^{2}$
(19a) subject to $\displaystyle v_{o,m}^{2}\geq R_{L}\bar{P}_{m},\quad
m=1,\ldots,M,$ (19b)

where $\mathcal{V}$ is the set of optimization variables, which is equal to
$\\{\omega_{i,n},q_{i,l}\\}_{\forall i,l,n}$ and $\\{\omega_{i,n}\\}_{\forall
i,n}$ for DMA-assisted and fully-digital architectures, respectively. Problem
(19) deals with extensive non-linearity since the objective and constraints
are highly non-linear and non-convex functions due to the signal model,
rectenna non-linearity, and the coupling between the digital and analog
beamforming variables in DMA.

###  III-B Optimization Framework for DMA-assisted Architecture

One of the challenges making the problem intractable is the coupling between
the optimization variables. To cope with this, we propose using alternating
optimization by first optimizing the digital precoders when considering fixed
$\\{q_{i,l}\\}_{\forall i,l}$, followed by optimizing the metamaterial
elements’ frequency response for fixed $\\{\omega_{i,n}\\}_{\forall i,n}$.

####  III-B1 Optimization with fixed $\\{q_{i,l}\\}_{\forall i,l}$

Let us proceed by rewriting the optimization problem as

$\displaystyle\operatorname*{minimize}_{\\{\omega_{i,n}\\}_{\forall
i,n}}\quad$
$\displaystyle\frac{\sqrt{P_{max}}}{\bar{\eta}}\sum_{i=1}^{N_{v}}\mathbb{E}\biggl{\\{}\sqrt{\sum_{l=1}^{N_{h}}x^{DMA}_{i,l}(t)^{2}}\biggr{\\}}\\!+\\!\sum_{i=1}^{N_{v}}\sum_{n=1}^{N_{f}}|\omega_{i,n}|^{2}$
(20a) subject to
$\displaystyle\bigl{(}{K_{2}\mathbb{E}\bigl{\\{}y_{m}(t)^{2}\bigr{\\}}+K_{4}\mathbb{E}\bigl{\\{}y_{m}(t)^{4}\bigr{\\}}\bigr{)}}^{2}\geq{R_{L}}\bar{P}_{m},\forall
m.$ (20b)

The problem is still highly non-linear and non-convex. Interestingly, it can
be easily verified that $v_{o,m}$ is convex and increasing with respect to
${y_{m}(t)}^{2}$, and ${y_{m}(t)}$ is affine with respect to
$\bar{\mathbf{w}}$. Therefore, $v_{o,m}$ is a convex function with respect to
$\bar{\mathbf{w}}$ given a fixed configuration for metamaterial elements [21].
Thus, although (20b) is a non-convex constraint, its left-hand side consists
of a convex function. These properties motivate us to adapt the SCA method
[33] to optimize the problem iteratively. Specifically, the objective and
constraints can be approximated by their Taylor expansion.

######  Theorem 1.

By defining
$S_{i}=\sum_{l=1}^{N_{h}}\mathbb{E}\bigl{\\{}x^{DMA}_{i,l}(t)^{2}\bigr{\\}}$,
we can write

$\sum_{i=1}^{N_{rf}}\mathbb{E}\biggl{\\{}\sqrt{\sum_{l=1}^{N_{h}}x^{DMA}_{i,l}(t)^{2}}\biggr{\\}}\leq\sum_{i=1}^{N_{rf}}\tilde{f}\bigl{(}S_{i},S_{i}^{(0)}),$
(21)

where

$\tilde{f}\bigl{(}S_{i},S_{i}^{(0)})=\sqrt{S_{i}^{(0)}}+\frac{1}{2\sqrt{S_{i}^{(0)}}}\bigl{(}S_{i}-S_{i}^{(0)}\bigr{)},\forall
i\vspace{-2mm}$ (22)

is the first order Taylor expansion of ${\sqrt{S_{i}}}$ at point
$S_{i}^{(0)}$.

###### Proof.

The inequality can be proved using Jensen’s inequality [34] and the fact that
${\sqrt{S_{i}}}$ is a concave function with respect to ${S_{i}}$, while the
first-order Taylor expansion of a function is greater than or equal to the
function value at each point. ∎

Then, we rewrite $v_{o,m}$ considering fixed $q_{i,l}$. For this, we define
$\mathbf{a}_{m,n}=\boldsymbol{\gamma}_{m,n}\odot\mathbf{q}\odot\mathbf{h}$ and
leverage the fact that the average power of a signal is equal to the power of
its spectrum. Thus, we have [6]

$\displaystyle\mathbb{E}\bigl{\\{}y_{m}(t)^{2}\bigr{\\}}=g_{m,1}(\\{\bar{\mathbf{w}}_{n}\\}_{\forall
n})=\frac{G^{2}}{2}\sum_{n}|\mathbf{a}^{T}_{m,n}\bar{\mathbf{w}}_{n}|^{2},$
(23)
$\displaystyle\mathbb{E}\bigl{\\{}y_{m}(t)^{4}\bigr{\\}}=g_{m,2}(\\{\bar{\mathbf{w}}_{n}\\}_{\forall
n})=\frac{3G^{4}}{8}\sum_{\begin{subarray}{c}n_{0},n_{1},n_{2},n_{3}\\\
n_{0}+n_{1}=n_{2}+n_{3}\end{subarray}}...$
$\displaystyle(\mathbf{a}^{T}_{m,n_{0}}\bar{\mathbf{w}}_{n_{0}})(\mathbf{a}^{T}_{m,n_{1}}\bar{\mathbf{w}}_{n_{1}})(\mathbf{a}^{T}_{m,n_{2}}\bar{\mathbf{w}}_{n_{2}})^{\star}(\mathbf{a}^{T}_{m,n_{3}}\bar{\mathbf{w}}_{n_{3}})^{\star}.$
(24)

By leveraging (23) and (III-B1), we can write

${{v}_{o,m}}\geq K_{2}g_{m,1}(\\{\bar{\mathbf{w}}_{n}^{(0)}\\}_{\forall
n})+{K_{4}}g_{m,2}(\\{\bar{\mathbf{w}}_{n}^{(0)}\\}_{\forall n})\\\
+\sum_{n}\tilde{g}_{m,n}(\bar{\mathbf{w}}_{n}^{(0)})\bigl{(}\bar{\mathbf{w}}_{n}-\bar{\mathbf{w}}_{n}^{(0)}\bigr{)},$
(25)

where

$\tilde{g}_{m,n}(\bar{\mathbf{w}}_{n}^{(0)})=G^{2}K_{2}(\mathbf{a}^{T}_{m,n}\bar{\mathbf{w}}_{n}^{(0)})\mathbf{a}^{T}_{m,n}+\\\
\frac{3K_{4}G^{4}}{8}\biggl{[}4{|\mathbf{a}^{T}_{m,n}|}^{4}{|\bar{\mathbf{w}}_{n}^{(0)}|}^{2}{{\bar{\mathbf{w}}{{}_{n}^{(0)}}}}^{T}+\\\
8\sum_{n1}{|\mathbf{a}^{T}_{m,n}|}^{2}{|\mathbf{a}^{T}_{m,n_{1}}|}^{2}{|\bar{\mathbf{w}}_{n_{1}}^{(0)}|}^{2}{{\bar{\mathbf{w}}{{}_{n}^{(0)}}}}^{T}+\\\
\sum_{\begin{subarray}{c}n_{2},n_{3}\\\ n_{2}+n_{3}=2n\\\ n_{2}\neq
n_{3}\end{subarray}}2{(\mathbf{a}^{T}_{m,n_{2}}\bar{\mathbf{w}}_{n_{2}}^{(0)})}^{\star}{(\mathbf{a}^{T}_{m,n_{3}}\bar{\mathbf{w}}_{n_{3}}^{(0)})}^{\star}{(\mathbf{a}^{T}_{m,n}\bar{\mathbf{w}}_{n}^{(0)})}\mathbf{a}^{T}_{m,n}+\\\
2(\mathbf{a}^{T}_{m,n_{2}}\bar{\mathbf{w}}_{n_{2}}^{(0)})(\mathbf{a}^{T}_{m,n_{3}}\bar{\mathbf{w}}_{n_{3}}^{(0)}){{(\mathbf{a}^{T}_{m,n}\bar{\mathbf{w}}_{n}^{(0)})}}^{\star}{\mathbf{a}^{H}_{m,n}}+\\\
\sum_{\begin{subarray}{c}n_{1},n_{2},n_{3}\\\ -n_{1}+n_{2}+n_{3}=n\\\ n\neq
n_{1}\neq n_{2}\neq
n_{3}\end{subarray}}2(\mathbf{a}^{T}_{m,n_{1}}\bar{\mathbf{w}}_{n_{1}}^{(0)}){(\mathbf{a}^{T}_{m,n_{2}}\bar{\mathbf{w}}_{n_{2}}^{(0)})}^{\star}{(\mathbf{a}^{T}_{m,n_{3}}\bar{\mathbf{w}}_{n_{3}}^{(0)})}^{\star}\mathbf{a}^{T}_{m,n}+\\\
2(\mathbf{a}^{T}_{m,n_{1}}\bar{\mathbf{w}}_{n_{1}}^{(0)})(\mathbf{a}^{T}_{m,n_{2}}\bar{\mathbf{w}}_{n_{2}}^{(0)}){(\mathbf{a}^{T}_{m,n_{3}}\bar{\mathbf{w}}_{n_{3}}^{(0)})}^{\star}{\mathbf{a}^{H}_{m,n}}\biggr{]}$
(26)

is the first-order Taylor coefficient of ${{v}_{o,m}}$ at point
$\\{\bar{\mathbf{w}}_{n}^{(0)}\\}_{\forall n}$.

Similarly, it can be verified that

$\mathbb{E}\bigl{\\{}x^{DMA}_{i,l}(t)^{2}\bigr{\\}}=\frac{G^{2}}{2}\sum_{n}|q_{i,l}h_{i,l}w_{i,n}|^{2}.$
(27)

Now, we can reformulate the problem at point
$\\{\bar{\mathbf{w}}_{n}^{(0)},S_{i}^{(0)},{v}_{o,m}^{(0)}\\}_{\forall i,n,m}$
as

$\displaystyle\operatorname*{minimize}_{\begin{subarray}{c}\\{\bar{\mathbf{w}}_{n}\\}_{\forall
n}\\\ \\{{v}_{o,m}\\}_{\forall m}\\\ \\{S_{i}\\}_{\forall
i}\end{subarray}}\quad$
$\displaystyle\frac{\sqrt{P_{max}}}{\bar{\eta}}\sum_{i=1}^{N_{rf}}\tilde{f}\bigl{(}S_{i},S_{i}^{(0)})+\sum_{i=1}^{N_{rf}}\sum_{n=1}^{N_{f}}|\omega_{i,n}|^{2}$
(28a) subject to $\displaystyle\sqrt{{R_{L}}\bar{P}_{m}}\leq
K_{2}g_{m,1}(\\{\bar{\mathbf{w}}_{n}^{(0)}\\}_{\forall n})+$ $\displaystyle
K_{4}g_{m,2}(\\{\bar{\mathbf{w}}_{n}^{(0)}\\}_{\forall n})+$
$\displaystyle\sum_{n}\tilde{g}_{m,n}(\bar{\mathbf{w}}_{n}^{(0)})\bigl{(}\bar{\mathbf{w}}_{n}-\bar{\mathbf{w}}_{n}^{(0)}\bigr{)},\quad\forall
m$ (28b)
$\displaystyle\frac{G^{2}}{2}\sum_{n=1}^{N_{f}}\sum_{l=1}^{N_{h}}\bigl{|}q_{i,l}h_{i,l}\bar{\mathbf{w}}_{n}[i]\bigr{|}^{2}\leq
S_{i},i=1,\ldots,N_{v},$ (28c)
$\displaystyle\bar{\mathbf{w}}_{n}[(i-1)N_{h}+l]=\bar{\mathbf{w}}_{n}[i],\forall
i,l=1,\ldots,N_{h}.$ (28d)

Notice that by utilizing (21) and the fact that (20a) consists of two positive
terms, one can verify that (28a) serves as an upper bound for (20a), thus,
minimizing the latter leads to minimizing the former. Moreover, the inequality
in (25) ensures that the solution to this problem is a feasible solution of
(20) at each point. Interestingly, the problem222Note that for the sake of
notation and facilitating the reader’s understanding, we kept the problem in
the vector form and introduced the constraint (28d) into the problem. However,
the problem can be easily converted to scalar form, which removes the
mentioned constraint. has become convex and can be solved at a given point by
standard convex optimization tools, e.g., CVX [35]. Moreover, the solution can
be iteratively updated using the SCA algorithm [33].

####  III-B2 Optimization with fixed $\\{\omega_{i,n}\\}_{\forall i,n}$

Herein, the non-convex Lorentzian constraint of the metamaterials makes the
problem extremely difficult to solve. To tackle this, we propose decoupling
the problem to first maximize the minimum harvested power when optimizing
$q_{i,l}$. This allows us to leverage the beamforming capability of the
metamaterial elements and provide degrees of freedom to further reduce the
power consumption when optimizing $\omega_{i,n}$ [23]. Hereby, the
optimization problem with fixed $\\{\omega_{i,n}\\}_{\forall i,n}$ can be
reformulated as

$\displaystyle\operatorname*{maximize}_{\\{q_{i,l}\\}_{\forall i,l}}\quad$
$\displaystyle\min_{m}\frac{v_{o,m}^{2}}{R_{L}}$ (29a) subject to
$\displaystyle q_{i,l}\in\mathcal{Q},\forall i,l,$ (29b)

where (29b) is the non-convex Lorentzian constraint. Next, we cope with the
complexity caused by the metamaterial elements.

######  Theorem 2.

Problem (29) is equivalent to

$\displaystyle\operatorname*{maximize}_{\\{q_{i,l}\\}_{\forall i,l}}\quad$
$\displaystyle R$ (30a) subject to
$\displaystyle\Re\bigl{\\{}q_{i,l}\bigr{\\}}^{2}+(\Im\bigl{\\{}q_{i,l}\bigr{\\}}-0.5)^{2}\leq
0.25,\forall i,l,$ (30b) $\displaystyle R\leq
K_{2}{E}\bigl{\\{}y_{m}(t)^{4}\bigr{\\}}+K_{4}{E}\bigl{\\{}y_{m}(t)^{4}\bigr{\\}},\quad\forall
m.$ (30c)

###### Proof.

The proof is provided in Appendix -A. ∎

Notice that (30b) keeps the frequency response of the metamaterials within the
Lorentzian circle, while (30c) ensures that $R$ is below the minimum output
voltage of the devices. Problem (30) is still difficult to solve due to the
non-convex constraint (30c). To cope with this, we define
$\hat{\mathbf{a}}_{m,n}=\boldsymbol{\gamma}_{m,n}\odot\bar{\mathbf{w}}_{n}\odot\mathbf{h}$
and write

$\displaystyle\mathbb{E}\bigl{\\{}y_{m}(t)^{2}\bigr{\\}}={e}_{m,1}({\mathbf{q}})=\frac{G^{2}}{2}\sum_{n}|\hat{\mathbf{a}}^{T}_{m,n}{\mathbf{q}}|^{2},$
(31)
$\displaystyle\mathbb{E}\bigl{\\{}y_{m}(t)^{4}\bigr{\\}}={e}_{m,2}(\mathbf{q})=\frac{3G^{4}}{8}\sum_{\begin{subarray}{c}n_{0},n_{1},n_{2},n_{3}\\\
n_{0}+n_{1}=n_{2}+n_{3}\end{subarray}}...$
$\displaystyle(\hat{\mathbf{a}}^{T}_{m,n_{0}}\mathbf{q})(\hat{\mathbf{a}}^{T}_{m,n_{1}}\mathbf{q})(\hat{\mathbf{a}}^{T}_{m,n_{2}}\mathbf{q})^{\star}(\hat{\mathbf{a}}^{T}_{m,n_{3}}\mathbf{q})^{\star}.$
(32)

Similar to the case of digital precoders, it can be observed that ${v}_{o,m}$
is convex with respect to $\mathbf{q}$, thus, we can write

${{v}_{o,m}}\geq
K_{2}{e}_{m,1}({\mathbf{q}}^{(0)})+K_{4}{e}_{m,2}({\mathbf{q}}^{(0)})+\tilde{e}_{m}(\mathbf{q}^{(0)})\bigl{(}{\mathbf{q}}-{\mathbf{q}}^{(0)}\bigr{)},$
(33)

where

$\tilde{e}_{m}(\mathbf{q}^{(0)})=G^{2}K_{2}\sum_{n}(\hat{\mathbf{a}}^{T}_{m,n}{\mathbf{q}}^{(0)})\hat{\mathbf{a}}^{T}_{m,n}+\\\
\frac{3K_{4}G^{4}}{8}\sum_{\begin{subarray}{c}n_{0},n_{1},n_{2},n_{3}\\\
n_{0}+n_{1}=n_{2}+n_{3}\end{subarray}}\biggl{[}(\hat{\mathbf{a}}^{T}_{m,n_{1}}\mathbf{q})(\hat{\mathbf{a}}^{T}_{m,n_{2}}\mathbf{q})^{\star}(\hat{\mathbf{a}}^{T}_{m,n_{3}}\mathbf{q})^{\star}\hat{\mathbf{a}}^{T}_{m,n_{0}}+\\\
(\hat{\mathbf{a}}^{T}_{m,n_{0}}\mathbf{q})(\hat{\mathbf{a}}^{T}_{m,n_{2}}\mathbf{q})^{\star}(\hat{\mathbf{a}}^{T}_{m,n_{3}}\mathbf{q})^{\star}\hat{\mathbf{a}}^{T}_{m,n_{1}}+\\\
(\hat{\mathbf{a}}^{T}_{m,n_{0}}\mathbf{q})(\hat{\mathbf{a}}^{T}_{m,n_{1}}\mathbf{q})(\hat{\mathbf{a}}^{T}_{m,n_{3}}\mathbf{q})^{\star}\hat{\mathbf{a}}_{m,n_{2}}^{H}+\\\
(\hat{\mathbf{a}}^{T}_{m,n_{0}}\mathbf{q})(\hat{\mathbf{a}}^{T}_{m,n_{1}}\mathbf{q})(\hat{\mathbf{a}}^{T}_{m,n_{2}}\mathbf{q})^{\star}\hat{\mathbf{a}}_{m,n_{3}}^{H}\biggr{]}.$
(34)

Hereby, (30) can be reformulated at point $\mathbf{q}^{(0)}$ as

$\displaystyle\operatorname*{maximize}_{\mathbf{q},R}\quad$ $\displaystyle R$
(35a) subject to $\displaystyle R\leq K_{2}\bar{g}_{m,1}({\mathbf{q}}^{(0)})$
$\displaystyle+K_{4}\bar{g}_{m,2}({\mathbf{q}}^{(0)})+\tilde{e}_{m}(\mathbf{q}^{(0)})\bigl{(}{\mathbf{q}}-{\mathbf{q}}^{(0)}\bigr{)},$
(35b) $\displaystyle\Re\bigl{\\{}\mathbf{q}[(i-1)N_{h}+l]\bigr{\\}}^{2}+$
$\displaystyle(\Im\bigl{\\{}\mathbf{q}[(i-1)N_{h}+l]\bigr{\\}}-0.5)^{2}\leq
0.25,\forall i,l,\eqref{probdmaQQconf}$ (35c)

which is convex in the neighborhood of the initial point.

####  III-B3 Alternating Optimization Algorithm

We transformed the original problem into a convex form for both fixed
$\bar{\mathbf{w}}$ and $\mathbf{q}$ in the neighborhood of an initial point.
However, there is still an important challenge, i.e., the initialization of
the variables. Specifically, when using iterative algorithms relying on convex
approximation, e.g., SCA, the starting point must be feasible and the
performance is influenced by it. One can get a feasible point by setting an
extremely large amplitude for the digital weights, but this may lead to poor
performance. To cope with this, we propose a low-complexity initialization
method, which will be explained in the next section. This prevents the initial
consumed power from becoming extremely large, helping the SCA algorithm to
start from a relatively good initial point.

Algorithm 1 illustrates the proposed alternating SCA-based approach for
waveform and beamforming optimization. First, the digital precoders and the
frequency responses of the metamaterial elements are initialized. After that,
digital precoders and metamaterials are optimized in an alternating fashion in
lines 3-17. First, SCA is used to iteratively find a suboptimal solution for
fixed digital precoders. Specifically, $\mathbf{q}$ is updated in each
iteration until convergence in lines 5-8. Then, the obtained $\mathbf{q}$ is
used to run the SCA algorithm for finding suboptimal digital precoders
$\\{\bar{\mathbf{w}}_{n}\\}_{\forall n}$ through line 11-15. These two SCA-
based optimizations are repeated until the alternating optimization converges
to a suboptimal solution.

Algorithm 1 Alternating SCA-based waveform and beamforming design for DMA-
assisted WPT (ASCA-DMA).

1:Input: $\\{\gamma_{i,l,m,n}\\}_{\forall i,l,m,n}$, $\upsilon$ Output:
$P_{c}^{\star}$

2:Initialize: $\mathbf{q}^{(0)}$ and $\bar{\mathbf{w}}^{(0)}_{n},\forall n$,
$P_{c}^{\star}=0$

3:repeat

4: $\xi^{\star}_{1}=0$, $\xi^{\star}_{2}=\infty$, $P_{c}\leftarrow
P_{c}^{\star}$

5: repeat

6: $\xi_{1}\leftarrow\xi^{\star}_{1}$, solve (35) to obtain $\mathbf{q}$ and
$\mathbf{q}^{(0)}\leftarrow\mathbf{q}$

7: $\xi^{\star}_{1}\leftarrow$the objective value in (35a)

8: until $|1-{\xi_{1}^{\star}}/{\xi_{1}}|\leq\upsilon$

9: Compute $v^{(0)}_{out,m},\forall m$ using (15), 23, and III-B1

10: Compute $S_{i}^{(0)},\forall i$ using (27)

11: repeat

12: $\xi_{2}\leftarrow\xi^{\star}_{2}$, solve (28) to obtain $v_{o,m}$ and
$\bar{\mathbf{w}}^{(0)}_{n},\forall n$

13: $v^{(0)}_{out,m}\leftarrow v_{o,m}$,
$\bar{\mathbf{w}}^{(0)}_{n}\leftarrow\bar{\mathbf{w}}^{(0)}_{n},\forall n$,
$S_{i}^{(0)}\leftarrow S_{i},\forall i$

14: $\xi^{\star}_{2}\leftarrow$ the objective value in (28a)

15: until $|1-{\xi_{2}^{\star}}/{\xi_{2}}|\leq\upsilon$

16: Compute $P_{c}^{\star}$ using (19a)

17:until $|1-P_{c}^{\star}/P_{c}|\leq\upsilon$

###  III-C Optimization Framework for Fully-Digital Architecture

Note that the proposed framework for the DMA-assisted system can also be used
for fully-digital architecture with some slight modifications333Note that the
framework can also be straightforwardly adapted for a traditional hybrid
architecture with a fully connected network of phase shifters.. Notably,
alternating optimization is not needed since the variables are just digital
precoders, and adapting SCA to find the suboptimal precoders is sufficient.
Algorithm 2 illustrates the SCA-based optimization for fully-digital
architecture. Notice that problem (28) can be easily modified to match the
fully-digital transmitter. Thus, the expressions are not rewritten to avoid
repetition.

Algorithm 2 SCA-based waveform and beamforming design for fully-digital WPT
(SCA-FD).

1:Input: $\\{\gamma_{i,l,m,n}\\}_{\forall i,l,m,n}$, $\upsilon$ Output:
$P_{c}^{\star}$

2:Initialize: ${\mathbf{w}}^{(0)}_{n},\forall n$, $P_{c}^{\star}=0$

3:repeat

4: $P_{c}\leftarrow P_{c}^{\star}$ , solve (28) to obtain
$\\{v_{o,m},S_{i},{\mathbf{w}}_{n}\\}_{\forall i,m,n}$

5: $v^{(0)}_{out,m}\leftarrow v_{o,m}$,
${\mathbf{w}}^{(0)}_{n}\leftarrow{\mathbf{w}}^{(0)}_{n},\forall n$,
$S_{i}^{(0)}\leftarrow S_{i},\forall i$

6: Compute $P_{c}^{\star}$ using (19a)

7:until $|1-P_{c}^{\star}/P_{c}|\leq\upsilon$

###  III-D Complexity

The proposed ASCA-DMA algorithm consists of a low-complexity initialization,
followed by alternating optimization, while SCA is used to optimize each set
of variables. Each iteration of SCA attempts to solve a quadratic program [33]
((28) or (35)). Moreover, the complexity of quadratic programs scales with a
polynomial function of the problem size, while the degree of the polynomial
mainly depends on the type of the solver. Let us consider a simple solver
based on the Newton method with $\mathcal{O}(n^{3})$ complexity [33], where
$n$ is the problem size. Imagine $U_{1}$ and $U_{2}$ are the number of
required iterations (in the worst-case) for convergence of digital weights and
metamaterial elements’ weights, respectively. Furthermore, $U_{3}$ is
considered the number of iterations required for convergence in alternating
optimization. Hereby, the total complexity of the ASCA-DMA algorithm is
$\mathcal{O}(U_{1}U_{2}U_{3}n^{3})$, where $n$ scales with $M$, $N_{v}$,
$N_{h}$, and $N_{f}$. There is also some additional complexity introduced by
the initialization algorithm, which is negligible since the initialization
procedure is low-complexity. Moreover, the SCA-FD has only a single SCA stage
with a complexity $\mathcal{O}(U_{4}n^{3})$, where $U_{4}$ is the required
number of iterations for convergence of SCA in the worst-case.

##  IV Initialization Algorithm

Algorithm 3 Initialization of digital precoders and the frequency response of
the metamaterial elements.

1:Input: $\\{\gamma_{i,l,m,n}\\}_{\forall i,l,m,n}$, $\tau_{s}$, $\varsigma$
Output: $\\{\omega_{i,n},q_{i,l}\\}_{\forall i,l,n}$

2:Initialize: Compute $z_{m},\forall m$ using (36)

3:Allocate one RF chain to each user, $\mathcal{R}_{m}=\\{m\\},\forall m$,
$\bar{\mathcal{R}}=\\{1,\ldots,M\\}$, $w_{m}=0,\forall m$

4:$N^{\prime}_{v}=N_{v}-M$, $N^{\prime}_{rf,m}=\lceil
z_{m}N^{\prime}_{v}\rceil$, $\mathcal{R}^{\prime}=\\{\\}$

5:repeat

6: $m^{\star}=\operatorname*{argmax}_{m\notin\mathcal{R}^{\prime}_{m}}z_{m}$,
$\mathcal{R}^{\prime}\leftarrow\mathcal{R}^{\prime}\cup m^{\star}$,
$i_{c}=M+1$

7: repeat

8: $\mathcal{R}_{m^{\star}}\leftarrow\mathcal{R}_{m^{\star}}\cup i_{c}$,
$i_{c}\leftarrow i_{c}+1$

9: $N^{\prime}_{rf,m}\leftarrow N^{\prime}_{rf,m}-1$,
$N^{\prime}_{v}\leftarrow N^{\prime}_{v}-1$

10: until $N^{\prime}_{rf,m}=0$ or $N^{\prime}_{v}=0$

11:until $N^{\prime}_{v}=0$ or $|\mathcal{R}^{\prime}|=M$

12:for $m=1,\ldots,M$ do

13: Compute $q_{i,l},\forall i\in\mathcal{R}_{m},l$ using (37) and (9),
$w_{m}=\tau_{s}$

14: Solve (40) to obtain $\hat{\omega_{i,n}}^{\star},\forall
i\in\mathcal{R}_{m},n$

15: repeat

16:
$\bar{\omega}_{i,n}=w_{m},\omega_{i,n}=\bar{\omega}_{i,n}e^{j\hat{\omega}_{i,n}}\forall
i\in\mathcal{R}_{m},n$

17: Compute $P_{dc,m}$ using (15), (16), (23), and (III-B1)

18: $w_{m}\leftarrow\varsigma w_{m}$

19: until $P_{dc,m}\geq\bar{P}_{m}$

20:end for

Algorithm 3 illustrates the proposed initialization algorithm. Since the
initialization algorithm has to be adaptable for multi-user scenarios, we
start by proposing a method to allocate the output signal of the RF chains to
the different users444The allocation is only for the initialization process,
and there is no limitation in this regard in the optimization procedure.. For
this, we utilize the channel characteristics by naming
$n_{m}^{\star}=\operatorname*{argmax}_{n}|\boldsymbol{\gamma}_{m,n}|$ as the
strongest sub-carrier channel between user $m$ and the transmitter. Then, a
coefficient $z_{m}$ is assigned to user $m$ based on the gain introduced by
its strongest channel, expressed as

$z_{m}=1-\frac{|\boldsymbol{\gamma}_{m,n_{m}^{\star}}|}{\sum_{\bar{m}=1}^{M}|\boldsymbol{\gamma}_{\bar{m},{n_{\bar{m}}^{\star}}}|}.$
(36)

More precisely, the RF chains are dedicated to the users based on this ratio
such that the users with lower channel gains are served by more signals and
vice versa. The allocation procedure is illustrated in lines 2-11 in Algorithm
3. First, an RF chain is allocated to each user, then, the rest of the RF
chains are divided among users based on their $z_{m}$.

Let us denote $\mathcal{R}_{m}$ as the set of RF chains dedicated to user $m$.
Then, we initialize $q_{i,l},i\in\mathcal{R}_{m}$ to compensate for the phase
shift introduced by both $h_{i,l}$ and $\gamma_{i,l,m,n_{m}^{\star}}$.
Specifically, we need to define $q_{i,l},i\in\mathcal{R}_{m}$ such that

$\phi^{\star}_{i,l}=\operatorname*{argmin}_{\phi_{i,l}}\bigl{\langle}(\frac{j+e^{j\phi_{i,l}}}{2})h_{i,l}\gamma_{i,l,m,n_{m}^{\star}}\bigr{\rangle},\forall
i,l,$ (37)

where $i\in\mathcal{R}_{m}$ and $q_{i,l}$ can be obtained accordingly. Notice
that (37) can be easily solved using a one-dimensional search with negligible
complexity.

The next step is to initialize the amplitude and phase of the digital
precoders. For this, let us proceed by defining the received RF power at the
$m$th user as

$P_{rf,m}=\frac{G^{2}}{2}|\mathbf{a}_{m,n}\bar{\mathbf{w}}_{n}|^{2}\\!=\\!\frac{G^{2}}{2}\sum_{n=1}^{N_{f}}\biggl{|}\sum_{i=1}^{N_{v}}\sum_{l=1}^{N_{h}}\gamma_{i,l,m,n}q_{i,l}h_{i,l}\omega_{i,n}\biggr{|}^{2}.$
(38)

Moreover, the output DC power of the rectifier is an increasing function of
the input RF power when operating in the low-power regime below the breakdown
region of the rectifier circuit’s diode [36]. Motivated by this, we aim to
increase the available RF power at each user during the initialization process
using the phases of the digital precoders with low complexity. One way to
increase the available RF power for each user is to facilitate the coherent
reception of the signals at the receiver. This can be done by reducing the
amount of phase shift introduced in the signal. For this, we assume the
initial digital weights to be
$\omega_{i,n}=\bar{\omega}_{i,n}e^{j\hat{\omega}_{i,n}}$, while the dedicated
signals to each user have the same amplitude $\bar{\omega}_{i,n}=w_{m},\forall
i\in\mathcal{R}_{m},n$. Moreover, the ideal phase initialization for
maximizing the received RF power is obtained by solving

$\operatorname*{argmax}_{\hat{\omega}_{i,n}\in[0,2\pi],\forall
i\in\mathcal{R}_{m},n}\sum_{n=1}^{N_{f}}\biggl{|}\sum_{i=1}^{N_{v}}\sum_{l=1}^{N_{h}}\gamma_{i,l,m,n}q_{i,l}h_{i,l}\bar{\omega}_{i,n}e^{j\hat{\omega}_{i,n}}\biggr{|}^{2}.$
(39)

Meanwhile, solving this problem is not straightforward and introduces much
additional complexity to our framework. Notably, the problem can be decoupled
and solved individually for each sub-carrier without any change in the optimal
solution. Still, there is a coupling between the digital weights of different
RF chains in a similar sub-carrier. For this, we further reduce the complexity
by formulating the problem as

$\operatorname*{argmin}_{\hat{\omega}_{i,n}\in[0,2\pi]}\biggl{|}\big{\langle}\sum_{l=1}^{N_{h}}\gamma_{i,l,m,n}q_{i,l}h_{i,l}e^{j\hat{\omega}_{i,n}}\big{\rangle}\biggr{|},\forall
i\in\mathcal{R}_{m},n.$ (40)

Although the reformulated problem may not have the same solution as (39), it
attempts to reduce the total amount of the phase shift of the received signal.
Therefore, solving (40) can lead to a suboptimal solution to (39) with much
lower complexity, and by using a one-dimensional search. Note that utilizing
such an approach is relevant since the goal is to have a reasonable
initialization for the variables, which leads to feeding a feasible initial
point to the optimization algorithm. The initialization procedure is
illustrated through lines 12-20 in Algorithm 3. For each user, the
metamaterials connected to its dedicated RF chains are initialized. Then, the
phases of the corresponding digital precoders are obtained. Finally, the
amplitudes are iteratively increased until the EH requirement is met and a
feasible solution is found. In the fully-digital architecture, the
initialization algorithm follows the same procedure with one difference, i.e.,
each antenna element has a dedicated RF chain, and thus, a dedicated signal.

##  V Numerical Analysis

In this section, we provide numerical analysis of the system performance. We
consider an indoor office with a transmitter located at the center of the
ceiling. The operating frequency of the system is $f_{1}=5.18$ GHz, which
matches the characteristics of the utilized rectifier model [36]. The spacing
between the elements in the DMA is $\lambda_{1}/5$, while $\lambda_{1}/2$ is
the distance between two consecutive microstrips. Meanwhile, the inter-element
distance is $\lambda_{1}/2$ in the fully-digital architecture. Thus,
$N_{v}=N_{h}=\lfloor\frac{L}{\lambda_{1}/2}\rfloor$ for the fully-digital
system, and
$N_{v}=\lfloor\frac{L}{\lambda_{1}/2}\rfloor,N_{h}=\lfloor\frac{L}{\lambda_{1}/5}\rfloor$
for the DMA-assisted system [27]. Note that $L$ is the array length while the
arrays are considered to be square-shaped. We set the optimization parameters
$\tau_{s}=10^{-3}$, $\varsigma=5$, and $\upsilon=10^{-6}$. Without loss of
generality, we set $G=1$ and $\bar{\eta}=\frac{\pi}{4}$555In practice, the HPA
output power is larger than the input power because $G>1$. However, the
proposed framework applies to all values of $G$.. Finally, the rectifier
parameters are $v_{t}=25$ mV and $\eta_{0}=1.05$ [36, 21, 20].

We utilize the characteristics of the ®Rogers RO4000 series ceramic laminate
to calculate the propagation coefficients of the microstrips. Specifically, we
calculate the attenuation and propagation coefficients of a RO400C LoPro with
a thickness of 20.7 mil (0.5258 mm) using the formulation provided in [23],
which gives $\alpha=0.356$ m-1 and $\beta=202.19$ m-1. Based on the rectifier
circuit design and simulations in [36], the rectifier circuit diode enters the
breakdown region when the received RF power is approximately $100\ \mu$W for a
continuous wave ($N_{f}=1$). Moreover, it has been shown that the maximum RF-
to-DC conversion efficiency is approximately $20\%$ for the mentioned setup.
Hence, we establish a minimum requirement of $\tilde{P}_{dc}=20\ \mu$W for DC
harvested power. In the figures, FD refers to the fully-digital architecture.
Moreover, $d$ represents the distance between the user and the center of the
transmitter.

Fig. 5 provides the convergence performance of the proposed ASCA-DMA approach
by presenting the amount of power consumption at the end of each iteration of
alternating optimization. It is seen that the objective value gradually
decreases with iterations until convergence, while the number of required
iterations for convergence depends on the setup. For instance, it is shown
that increasing $L$, $N_{f}$, or $M$ can increase the complexity of the
problem leading to more iterations. Fig. 6 provides a detailed algorithm
performance by showing all iterations of optimization, including the SCA
algorithm for both optimizing metamaterials and digital precoders. It is seen
that since the minimum harvested power increases when optimizing $\mathbf{q}$,
the power consumption increases, while this facilitates decreasing the amount
of power consumption when optimizing digital precoders. Therefore, the power
consumption decreases gradually at the end of each iteration of the
alternating optimization (red arrows) until convergence.

Figure 5: The convergence performance of ASCA-DMA for (a) $M=1$, $L=15$ cm,
$N_{f}=1$ (left), (b) $M=1$, $L=25$ cm, $N_{f}=1$ (middle), and (c) $M=2$,
$L=25$ cm, $N_{f}=4$ (right) over iterations with $d=2.5$ m and $P_{max}=100$
W. Figure 6: The convergence performance of ASCA-DMA for $M=1$, $L=25$ cm,
$d=2.5$ m, $N_{f}=1$, and $P_{max}=1$ W over all iterations (alternating and
SCA). The red and green arrows indicate the starting points of optimization
for the metamaterial elements and digital precoders, respectively.

Fig. 7 showcases the power consumption of the system as a function of the
antenna length. Note that increasing the $L$ reduces the power consumption
since the number of elements and the array aperture increases. This leads to
more capability in beam focusing, thus, delivering more power to the devices
given the same transmit power. However, the performance comparison between DMA
and FD highly depends on the system setup. It is seen that DMA outperforms FD
when $L$ and $P_{max}$ are relatively low. Note that the output of each RF
chain in DMA has to feed all the corresponding elements, while the the amount
of power consumption scales with the saturation power of the HPA. Thus, when
$P_{max}$ is low, the output power of each RF chain in DMA is multiplied by a
small value, leading to potential performance gains compared to FD. However,
this is only the case when $L$ is low and the number of RF chains is small. On
the other hand, in FD, $N_{rf}$ increases with $L$ at a higher rate compared
to DMA. Thus, it is easier to reduce the amount of output power of the HPAs by
distributing the required output power among them, leading to lower power
consumption compared to DMA. However, the value of $L$ that shifts the
favorable architecture from DMA to FD or vice versa depends on $P_{max}$ and
the transmit power required for meeting the EH requirements. For instance,
when $d=2.5$ m and $P_{max}=1$ W, the required transmit power and the power
consumption multiplier are both low, leading to a shift in performance with
lower $L$. Meanwhile, when $d=6.5$ m and $P_{max}=100$ W, DMA outperforms FD
up to a higher value of $L$ compared to the previous case. Meanwhile, when $d$
is small and $P_{max}$ is sufficiently large, FD outperforms DMA over all $L$
values with the performance gap increasing with $L$. In contrast, when $d$ is
relatively large and $P_{max}$ is small, DMA becomes the favorable choice for
sufficiently large $L$.

Figure 7: The power consumption as a function of $L$ for $N_{f}=4$,
$P_{max}\in\\{1,100\\}$ W, and $d\in\\{2.5,6.5\\}$ m.

Our results in Fig. 8 corroborate that increasing $N_{f}$ reduces $P_{c}$. As
discussed in Section II-C, this is because the HPAs operate in the linear
regime, and higher $N_{f}$ can leverage the rectifier non-linearity and
deliver more DC power to the ERs via waveform optimization. However, as
previously mentioned, the preference for DMA or FD highly depends on the
system parameters. For instance, when $L=10$ cm and $P_{max}=1$ W, both
$N_{rf}$ and saturation power have small values, leading to a lower power
consumption for DMA compared to FD. The reason is that the multiplier of the
HPA output power is low, thus, a lower value is multiplied by the output power
of each RF chain in DMA. Combining this with a small $L$, thus a small
$N_{rf}$ for FD, leads to DMA outperforming FD. On the other hand, different
parameters and system setups may lead to different performances. For example,
it is seen that when $L$ and saturation power are both sufficiently large, the
favorable architecture shifts from DMA to FD. The reason is that the number of
RF chains is much higher for FD in this case, leading to lower output power
for each. Then, combining this with a large value of saturation voltage leads
to smaller HPA outputs in FD, thus, lower power consumption. Meanwhile,
increasing $N_{f}$ affects the performance gap between FD and DMA, and
increasing $N_{f}$ leads to more performance gains in FD compared to DMA since
the number of signals is relatively larger in FD. Thus, when DMA outperforms
FD for a given $L$ and $P_{max}$, FD may start outperforming DMA by
sufficiently increasing $N_{f}$. On the other hand, when FD performs better,
increasing $N_{f}$ can increase the performance gap.

Figure 8: The power consumption as a function of $N_{f}$ for $L=25$ cm,
$P_{max}\in\\{1,100\\}$ W, $L\in\\{10,20\\}$ cm, and $d=2.5$ m.

Fig. 9 illustrates the impact of user distance on performance. It is obvious
that the power consumption increases with distance since the path loss becomes
larger and more transmit power is required to overcome that and meet the EH
requirements. Interestingly, the performance shift between FD and DMA is also
shown here. It is shown that when both $P_{max}$ and $L$ are relatively low,
DMA outperforms FD, especially over large distances. On the other hand, when
$L$ increases, DMA starts performing better after a certain $d$ since the
number of HPAs is much larger in FD, and increasing their output power affects
the power consumption considerably. For instance, when $L=20$ cm and
$P_{max}=100$ W, FD starts with better performance than DMA, but the
performance gap decreases as $d$ becomes larger.

Figure 9: The power consumption as a function of user distance for $N_{f}=4$,
$P_{max}\in\\{1,100\\}$ W, and $L\in\\{10,20\\}$ cm.

The impact of the number of users on system performance has been illustrated
in Fig. 10 for different system parameters. As expected, the power consumption
increases with $M$ since more EH requirements must be met. As previously
shown, both DMA or FD may be the favorable choice depending on the system
setup and the number of devices. For example, when $P_{max}$ and $L$ are both
sufficiently large, DMA outperforms FD for a small $M$, but FD becomes
favorable when $M$ is relatively large. The reason is that increasing $M$
leads to more required transmit power to satisfy the requirements and as
earlier mentioned, a large $L$ leads to relatively smaller output powers
compared in FD to DMA.

Some discussions on the complexity of implementing a large-scale FD setup are
in order. Notice that 36 RF chains are needed in a 400 cm2 FD array and just 6
for a DMA with the same size at $f_{1}=5.18$ GHz. Thus, although FD may
outperform DMA in many setups, as seen earlier, DMA may still be favorable
since it can achieve relatively good performance with a reduced $N_{rf}$ and
complexity.

Figure 10: The power consumption over $M$ for (a) $P_{max}=1$ W (top) and (b)
$P_{max}=100$ W (bottom), while $N_{f}=4$, and $L\in\\{15,25\\}$ cm. The users
are located at $d=4.4$ m.

Fig. 11 provides some insights regarding the beam focusing capability in the
near-field WPT by illustrating the normalized received RF power in each
spatial point of the area. Note that the received signal at each point is
normalized by its path loss to remove the impact of the distance. In Fig. 11a,
it is seen that when the device is located in the near-field region, the beam
is focused around the device location, while the beam trace fades increasingly
past the device. Such phenomena can have a huge benefit in reducing the RF
emission footprint in the environment, which facilitates the implementation of
environmentally friendly WPT systems. Meanwhile, the beam pattern for a
located device in the far-field is formed in the user’s direction, as
illustrated in Fig. 11b. This may be highly disadvantaged in interference-
sensitive applications since the generated beam may cause difficult-to-handle
interference in the signals conveying information, e.g., in SWIPT.

Figure 11: The normalized received RF power (W) in the area when the energy
receiver is located at (a) the near-field region (top) and (b) the far-field
region (bottom) in the DMA-assisted system with $L=30$ cm and $N_{f}=1$.

The received RF waveform in DMA-assisted and fully-digital systems is
presented in Fig.12a and Fig.12b, respectively. As previously mentioned, high
PAPR waveforms are beneficial for enhancing the performance in terms of DC
harvested power [7]. Moreover, our simulations verify this by showing that the
received signal experiences high peak amplitudes at specific intervals. Recall
that when HPAs are operating in the linear regime, as in our case, the HPA
does not introduce distortion to the signal. Thus, it is beneficial to utilize
multiple tones to leverage the rectifier’s non-linearity. Note that the peak-
to-peak time depends on the characteristics of the EH circuit, mainly the
capacitor.

Figure 12: The received signal at the device using (a) DMA (top) and (b) FD
(bottom) for $M=1$, $L=25$ cm, and $N_{f}=8$.

##  VI Conclusion and Future Work

In this paper, we investigated a multi-antenna near-field WPT system with a
DMA as the transmitter to charge multiple non-linear EH devices. Furthermore,
we proposed an optimization framework relying on alternating optimization and
SCA for the joint waveform optimization and beam focusing to minimize the
system power consumption while meeting the EH requirements. Numerical results
showed that both DMA and fully-digital architecture may be the favorable
choice in terms of power consumption depending on the system setups and
parameters such as antenna length, saturation power of the HPAs, number of
users, and user distance. Moreover, we showed that increasing the antenna
length or the number of tones can enhance the performance. Finally, we
verified that the transmitter can focus the energy beams on the spatial points
in the near-field region, while energy beams are formed toward the devices’
direction in the far-field.

As a prospect for future research, we may delve deeper into the signal
generation aspect by analyzing the power consumption based on the number of
tones. Another research direction is to utilize optimization approaches with
lower complexity, e.g., relying on machine learning, to learn online the
input-output relation of the system’s non-linear components while optimizing
the transmit waveform accordingly.

###  -A Proof of Theorem 2

We relax (29b) by limiting the values of $q_{i,l}$ to lie within the
Lorentzian circle in the complex plane. By utilizing the fact that (29a) is a
positive and increasing function of the rectifier’s output voltage and using
the epigraph form, the relaxed problem can be written as (30). Note that each
configuration of a metamaterial corresponds to a point on the Lorentzian-
constrained circle. Imagine $\vec{e}$ is the vector that represents the
direction and gain of a point on the Lorentzian circle, while this direction
and gain impacts the transmit signal. Meanwhile, the goal of (30) is to
increase the minimum output voltage of the ERs. Therefore, when shaping the
transmit signal toward different ERs, it is obvious that $\vec{e}$ should be
chosen with the most possible gain in the required direction to improve the
signal strength at the receiver. Furthermore, the most possible introduced
gain by a metamaterial element along a specified direction happens when the
point is exactly on the Lorentzian-constrained circle in that direction.
Hence, although (30b) is a relaxed version of the constraint (29b),
considering the final solution of the metamaterials as $a\vec{e},0\leq a\leq
1$, the only solution that leads to the most gain for the desired direction is
$a=1$. Thus, the solution of (30) is the same as (29) leading to a
configuration positioned on the Lorentzian circle and the equivalence between
the problems.

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