XX Month, XXXX XX Month, XXXX XX Month, XXXX XX Month, XXXX XX Month, XXXX
OJCOMS.2022.1234567

CORRESPONDING AUTHOR: Kenneth M. Mayer (e-mail: kenneth.m.mayer@fau.de). This
work was (partly) funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – SFB 1483 – Project-ID 442419336, EmpkinS.

# Optimal Transmit Antenna Deployment and Power Allocation for Wireless Power
Supply in an Indoor Space

KENNETH M. MAYER (Graduate Student Member  0000-0003-3292-6038 IEEE)
LAURA COTTATELLUCCI (Member  0000-0002-6641-8579 IEEE)  AND ROBERT SCHOBER
(Fellow  0000-0002-6420-4884 IEEE)  Institute for Digital Communications,
Department of Electrical Engineering, Friedrich-Alexander University Erlangen-
Nuremberg, Germany.

###### Abstract

As Internet of Things (IoT) devices proliferate, sustainable methods for
powering them are becoming indispensable. The wireless provision of power
enables battery-free operation and is crucial for complying with weight and
size restrictions. For the energy harvesting (EH) components of these devices
to be small, a high operating frequency is necessary. In conjunction with a
large transmit antenna, the receivers may be located in the radiating near-
field (Fresnel) region, e.g., in indoor scenarios. In this paper, we propose a
wireless power transfer  (WPT) system ensuring reliable supply of power to an
arbitrary number of mobile, low-power, and single-antenna receivers, whose
locations in a three-dimensional cuboid room are unknown. A max-min
optimisation problem is formulated to determine the optimal transmit power
distribution. We rigorously prove that the optimal transmit power
distribution’s support has a lower dimensionality than its domain and thus,
the employment of a continuous aperture antenna, utilised in Holographic MIMO
(HMIMO), is unnecessary in the context of the considered WPT problem. Indeed,
deploying a discrete transmit antenna architecture, i.e., a transmit antenna
array, is sufficient and our proposed solution provides the optimal transmit
antenna deployment and power allocation. Moreover, for a one-dimensional
transmit antenna architecture, a finite number of transmit antennas is proven
to be optimal. The proposed optimal solution is validated through computer
simulations. Our simulation results indicate that the optimal transmit antenna
architecture requires a finite number of transmit antennas and depends on the
geometry of the environment and the dimensionality of the transmit antenna
array. The robustness of the solution, which is obtained under the assumption
that a line-of-sight (LoS) link exists between the transmitter and receiver,
is assessed in an isotropic scattering environment containing a strong LoS
component.

Wireless power transfer, optimisation, radiating near-field region.

##  1 INTRODUCTION

The number of Internet of Things (IoT) devices continues to grow rapidly and
is expected to exceed 43 billion devices globally by 2023 [1]. Among those are
wearable IoT devices, which are primarily intended for collecting data, e.g.,
for patient monitoring, treatment or rehabilitation [2, 3]. Wearable IoT
devices are limited in terms of their weight and size and have to satisfy
aesthetic requirements [4]. For example, these smart devices are not only
designed as wearable accessories, but also as implants or tattoos and they may
be embedded into textiles [2, 3].

Sustainability is a central theme for designing next-generation IoT technology
and includes powering IoT devices, such as wearable systems, health monitoring
systems, and implants, sustainably [1]. Currently, batteries are considered a
hindrance in IoT technology due to the restrictions they impose on devices in
terms of, e.g., size, weight, cost, and the need for replacement [5, 6, 1].
Alternative technologies, such as wireless power transfer (WPT) and energy
harvesting (EH), facilitate the operation of battery-free IoT devices [1, 5,
6]. A reliable supply of power to IoT devices can be ensured through the
combination of available ambient power sources, such as ambient light, and
dedicated power sources [1, 5]. Ambient sources alone can not ensure reliable
powering as they are often uncontrollable. In contrast, dedicated power
sources, such as dedicated electromagnetic (EM) radiation, offer
controllability and thus, can ensure that IoT devices are powered adequately
[5].

The acquisition and use of channel state information (CSI) plays a fundamental
role in wireless information and power transfer. When the acquisition of CSI
is feasible, beamforming-based approaches for directing energy beams towards a
multitude of possibly mobile receiver devices are possible. However, the
resulting WPT system may be highly complex [1], for example, the WPT system
proposed in [7] utilises CSI acquired through uplink training to power devices
in the environment. Often, IoT devices are mobile which makes the acquisition
of reliable CSI at the transmitter challenging. Moreover, certain types of IoT
devices, such as wirelessly powered sensor nodes, may lack the communication
capabilities to broadcast pilot symbols, which makes serving EH nodes with
power a challenge since focusing energy beams towards them is not possible in
a reliable manner due to the missing CSI at the energy transmitter. In this
situation, there is a significant risk that the energy beams are misdirected
and fail to reach the receivers, leading to an insufficient supply of power to
these devices. To ensure the reliable provision of power to an arbitrary
number of devices in an environment, the WPT system must be capable of
supplying the served environment with a minimum guaranteed level of power at
any location without relying on CSI [8, 9].

An essential component for providing a ubiquitous supply of power in an indoor
environment is the transmit antenna architecture. Novel antenna architecture
designs are being discussed as key enabling technologies for future wireless
systems. Advancements beyond massive multiple-input multiple-output (mMIMO)
systems include increasing the physical size of antenna arrays and increasing
the number of transmit antennas while decreasing the inter-element spacing
[10]. In fact, through the incorporation of novel meta-materials, an
approximately infinitesimally small inter-element spacing is achieved,
allowing to pack almost infinitely many radiating elements into an antenna
array, thereby forming a spatially continuous antenna aperture [10]. The
technology revolving around these continuous aperture antennas has been coined
holographic MIMO (HMIMO) and allows for, e.g., holographic beamforming [11,
12]. HMIMO is being considered as an important technology, e.g., for future 6G
wireless communication systems [11]. Moreover, the HMIMO concept is envisioned
to offer a seamless and visually appealing integration of antenna arrays into
the surfaces of an environment including walls, the ceiling, and even windows
[11, 12]. While the benefits of HMIMO have been demonstrated in the context of
communications, the importance of HMIMO for WPT remains unclear.

Indoor wireless systems have been ubiquitous for decades and thus, methods for
accurately modelling the wireless channel have been studied extensively. In
particular, the frequency range of 2.4 GHz to 5 GHz has received significant
attention and thus, deterministic, statistical, and site-specific channel
models have been proposed, which differ in accuracy and computational
efficiency [13]. In [14], the propagation characteristics of indoor wireless
systems operating in the C- and mmWave frequency bands have been investigated
for line-of-sight (LoS) and non-line-of-sight (NLoS) scenarios via a software-
based ray tracing model. Hereby, the increase in attenuation (path loss) upon
increasing the frequency was found to be significantly more severe for the
NLoS scenario. Consequently, in the mmWave frequency band, a strong LoS link
is required for reliable operation of indoor wireless systems, and due to the
severely attenuated scattered components, a LoS channel model is typically
adopted [15, 16]. We note that high-frequency systems are attractive for
short-range applications, such as indoor WPT [17]. The short wavelength allows
the EH antenna of a device to be small [6]. For example, when utilising mmWave
frequency bands, mm-scale antennas can be employed [6]. Consequently, the
overall size of the devices can be reduced, allowing them to be less obtrusive
and easier to integrate, e.g., into fabrics and textiles [5, 1]. Beyond that,
a high carrier frequency in conjunction with large antenna architectures, such
as, the continuous aperture antennas used in HMIMO, leads to a significant
increase in the radiating near-field (Fresnel) region. Although the Fresnel
region could typically be neglected in conventional wireless systems, it has
to be considered in, e.g., indoor scenarios, for high-frequency systems.
Within the Fresnel region, the spherical nature of the EM wavefronts is non-
negligible. Adequately capturing the EM wave characteristics in the Fresnel
region demands modelling the wireless channel according to the spherical
wavefront model (SWM), as the approximation by the planar wavefront model
(PWM), typically utilised in the far-field region, does not hold [18, 19, 20].

Considering the rapidly increasing number of IoT devices [2, 3, 1], the
development of a WPT system, capable of reliably supplying them with power is
an important and timely challenge.  Emerging technologies, such as HMIMO
systems operating in the mmWave frequency band, may play a key role in
addressing this problem. Related works have studied the optimisation of the
deployment of transmit antennas with the goal of supplying devices with power,
e.g., in [21, 22, 23, 24, 25]. In [21, 22, 23, 24], operation in the far-field
regime is assumed, statistical channel models are adopted, and a coverage
area, significantly larger than an indoor space, is considered. For these
systems, statistical metrics, such as, outage probability, are utilised for
optimising or characterising the positions of power beacons. On the other
hand, as discussed previously, a high operating frequency, such as, in the
mmWave frequency band, is affordable and attractive for indoor WPT systems,
where a deterministic LoS channel model appropriately characterises the
propagation of EM waves. As a consequence of the deterministic channel model,
the techniques proposed in [21, 22, 23, 24] are not applicable and an
alternative approach for optimising the deployment of the transmit antennas is
required due to the absence of statistical effects. Furthermore, the authors
of [25] optimise the location of a finite number of radio stripe elements
through geometric programming to maximise the minimum power received by users
in indoor energy hotspots in the radiating near-field region. A continuous
transmit antenna architecture, as in HMIMO, is precluded as a potential
deployment solution when restricting the analysis to a pre-defined number of
transmit antennas. Consequently, this leaves the question unanswered whether
the novel continuous transmit architecture is beneficial in the context of
offering a ubiquitous supply of power in an indoor environment. We aim to
address this open point in this work.

In this paper, we propose a multi-user multiple-input single-output (MU-MISO)
WPT system, which supplies an entire indoor space with a guaranteed minimum
level of power. The  transmit antenna architecture of the WPT system is
realised via a  continuous aperture antenna, which is mounted on the ceiling
of a three-dimensional cuboid room.  The proposed WPT system can serve an
arbitrary number of single-antenna receivers. Assuming free space LoS
propagation of the spherical EM wavefronts, we formulate an optimisation
problem based on the distance information contained in the amplitude
variations of the wireless links between every possible infinitesimally small
radiating element and every possible receiver position to determine the
optimal distribution of power across the continuous aperture antenna of the
transmitter. The objective is to maximise the received power at the worst
possible receiver positions in the indoor space. Thus, the proposed solution
ensures a reliable supply of power to the receivers, which is a relevant
criterion when there is no prior knowledge on the receivers’ locations in the
room.  We establish a connection between the optimal distribution of transmit
power and the optimal transmit antenna architecture to study the relevance of
emerging HMIMO technology to WPT in an indoor environment. Specifically,
employing a continuous aperture antenna would be beneficial if the optimal
transmit power distribution’s support was found to have the same
dimensionality as the distribution’s domain. However, if the optimal transmit
power distribution’s support has a lower dimensionality than its domain, HMIMO
technology was not beneficial.  The adopted continuous  aperture model allows
us to capture the full range of transmit antenna architectures ranging from a
single transmit antenna to a continuous  surface of radiating elements as in
HMIMO, thereby enabling a complete analysis of the problem.  In contrast, the
relevance of HMIMO technology to the considered problem cannot be adequately
studied based on a discrete antenna model. The main contributions of this
paper can be summarised as follows:

  * •

We establish a method for investigating the relevance of continuous aperture
antennas in the context of offering a ubiquitous supply of power in an indoor
environment. To this end, we formulate an optimisation problem for the
distribution of a finite amount of transmit power among an uncountable number
of radiating elements with the objective of maximising the received signal
power at the worst possible receiver position in a given environment.

  * •

We prove that a continuous aperture antenna is unnecessary for the considered
problem since our analysis reveals that the support of the optimal
distribution of transmit power is a set whose closure has empty interior and
Lebesgue measure zero, i.e., the support is shown to have a lower
dimensionality than the domain of the transmit power distribution. The support
of the optimal  distribution corresponds to the optimal positions where
transmit antennas should be deployed and  we conjecture that the optimal
transmit antenna architecture can be realised through a discrete antenna
array. In case the transmit antenna  architecture is constrained to be one-
dimensional, a stronger statement is possible and the optimal number of
radiating elements is shown to be finite analytically.

  * •

Through simulation we verify that the optimal transmit power distribution’s
support is characterised by a finite number of points, and thus, the optimal
deployment consists of a finite number of transmit antennas for both one-
dimensional and two-dimensional transmit antenna  architectures. In the
investigated environments, the optimal number  of transmit antennas is found
to be surprisingly low.

  * •

To validate our findings, the performance of the proposed optimal solution is
evaluated against other power allocation schemes.  Specifically, we
investigate the achievable performance gain of modelling the system in the
Fresnel region and we observe that the performance gap compared to the optimal
distribution for the far-field operating regime decreases when the worst
possible receiver positions in the environment are almost in the far-field. We
also show that the loss in performance is low when removing those transmit
antennas from the optimal transmit antenna array that are allocated a small
amount of transmit power.

  * •

The robustness of the proposed solution is investigated by adding small-scale
fading to the LoS link, which leads to a performance loss of less than $2\%$
in the considered environments.

The paper is organised in the following manner. In Section 2, the system model
is introduced. In Section 3, the proposed optimisation problem is formulated
and the structure of the optimal distribution of power is determined. As a
result, we prove that a finite number of transmit antennas is optimal for the
one-dimensional problem and we conjecture that a finite number of transmit
antennas is also optimal regarding the two-dimensional problem. In Section 4,
we discretise the problem for numerical simulation and, in support of our
conjecture, we verify that the optimal number of transmit antennas remains
unchanged while increasing the granularity of the sampling gird, i.e., for
finer levels of discretisation. In Section 5, we provide a performance and
sensitivity analysis of our approach, and Section 6 concludes this paper.

##  2 SYSTEM MODEL

The WPT system considered in this paper consists of a transmit antenna
architecture employing a continuous aperture antenna and an arbitrary111 In
beamforming-based approaches, such as [7], the number of supported receivers
is limited by the number of transmit antennas. In our work, we do not face
this restriction since we do not rely on beamforming. number of single-antenna
receivers. Throughout this paper, the terms ”transmit antenna” and ”radiating
element” are used synonymously. An illustration of the considered system is
given in Fig. 1, where the transmit antenna architecture is incorporated in
the ceiling of a room.

$x$$z$$y$$L_{z}$$L_{x}$$L_{y}$RXwallscontinuous aperture antennaobstructions

Figure 1: Illustration of a three-dimensional environment with a two-
dimensional transmit antenna architecture consisting of a continuum of
infinitesimally small radiating elements located in a subsection of the
$x$-$z$ plane (orange). The LoS connections of the transmit antenna elements
to one receiver (RX) are indicated in orange.

###  2.1 ENVIRONMENT

The environment considered in this paper is a three-dimensional cuboid room,
where $L_{x}$, $L_{y}$, and $L_{z}$ define the size of the environment (see
Fig. 1). The origin of the coordinate system is located in the middle of the
ceiling of the room.  The receivers, e.g., sensors or wearable IoT devices,
which may be subject to stringent design requirements, have to be supplied
with power wirelessly.  To this end, the receivers are considered to be
equipped with an antenna for harvesting energy.  In this paper, we assume the
transmit antenna architecture, which is realised via a continuous aperture
antenna, is located on the ceiling. The $y$-component of the room (see Fig. 1)
is defined as the following closed interval in the set of real numbers
$\mathbb{R}$

$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\mathcal{Y}}}=\left\\{y\,\big{|}\,0\leq
y\leq L_{y}\,,\;L_{y}>0\right\\}.$ (1)

The transmit antenna architecture is located at $y=0$ and is defined in
Section II-2.2.  At any other height level of the room, i.e., $y>0$, there
exists a plane
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\mathcal{X}}\times\hat{\mathcal{Z}}}$.
A receiver location is defined as the triplet
$(x_{i},y,z_{i})\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\mathcal{X}}\times\hat{\mathcal{Y}}\times\hat{\mathcal{Z}}}$
which corresponds to the $i$-th surface element of
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\mathcal{X}}\times\hat{\mathcal{Z}}}$
at height $y$. The receiver’s antenna, which has the finite aperture $S_{RX}$,
is assumed to be centred at $(x_{i},y,z_{i})$ and to ensure analytical
tractability, we assume that the antennas are centred over a lattice defined
in $\hat{\mathcal{X}}\times\hat{\mathcal{Z}}$. Therefore,
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\mathcal{X}}\times\hat{\mathcal{Z}}}$
is partitioned into disjoint areas of size $S_{RX}$ from which a receiver is
capable of harvesting energy. For better legibility, the index $i$ is dropped
and a receiver location is written as
$(x,y,z)\in{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\mathcal{X}}\times\hat{\mathcal{Y}}\times\hat{\mathcal{Z}}}$.
As a consequence of the position of the origin,
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\mathcal{X}}}$
and
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\hat{\mathcal{Z}}}$
are bounded by $\pm L_{x}/2$ and $\pm L_{z}/2$, respectively.

_Remark 1:_ We note that for the problem considered in Section 3, the
receivers are assumed to lie beyond the reactive near-field of the
transmitter, see also Section II-2.3.

###  2.2 TRANSMIT ANTENNA MODEL

In Section 1, we have established that high-frequency systems are attractive
and affordable for short-range applications, such as indoor WPT [17]. To
leverage this, we consider the WPT system to operate in the mmWave frequency
band in this paper. Primarily, we consider a two-dimensional transmit antenna
architecture. Additionally, we investigate the problem for a one-dimensional
transmit antenna architecture which may be preferable in case of obstructions
in the area of deployment.

The two-dimensional transmit antenna architecture is deployed on the ceiling,
i.e., located in the plane $y=0$, and is defined as the continuous surface
$\mathcal{X}_{a}\times\mathcal{Z}_{a}$ comprising an uncountable infinity of
infinitesimally small radiating elements with

$\displaystyle\mathcal{X}_{a}=\left\\{a_{x}\,\big{|}\,-L_{a_{x}}/2\leq
a_{x}\leq L_{a_{x}}/2\,,\;0<L_{a_{x}}\leq L_{x}\right\\},$ (2)

and

$\displaystyle\mathcal{Z}_{a}=\left\\{a_{z}\,\big{|}\,-L_{a_{z}}/2\leq
a_{z}\leq L_{a_{z}}/2\,,\;0<L_{a_{z}}\leq L_{z}\right\\}.$ (3)

By assuming a continuous transmit antenna architecture, no constraint on the
minimum size of a radiating element is imposed. The location of an
infinitesimally small radiating element is defined by the pair
$(a_{x},a_{z})\in\mathcal{X}_{a}\times\mathcal{Z}_{a}$.

Moreover, we assume that the surface, where the antenna architecture is
located, i.e., $\mathcal{X}_{a}\times\mathcal{Z}_{a}$, is free of
obstructions.  We note that from a practical point of view, obstructions in
the ceiling, such as lights, vents, or sound dampening structures, will
prohibit the deployment of excessively large continuous surfaces in typical
indoor environments, such as, an office or a room in a residential building.
The radiating elements of the transmit antenna architecture are modelled as
identical, omnidirectional antennas. The transmit antenna architecture model
$\mathcal{X}_{a}\times\mathcal{Z}_{a}$ captures any arrangement of radiating
elements ranging from the deployment of a single transmit antenna up to the
distribution of radiating elements over a continuous planar surface222Note
that adopting a discrete transmit antenna model would exclude a continuous
planar surface as a possible solution to Problem (14) discussed in Section 3.
This would preclude the investigation of whether continuous transmit antenna
architectures are optimal for maximising the received signal power at the
worst possible receiver positions in the environment or not..

As a special case of this transmit antenna architecture model, we also
consider a one-dimensional transmit antenna architecture $\mathcal{L}_{a}$
with an arbitrary direction in the plane $y=0$, where the location of a
radiating elements is defined as $a_{l}\in\mathcal{L}_{a}$, and
$\mathcal{L}_{a}$ is assumed to be continuous.

We note that we neglect the impact of mutual coupling among the transmit
radiating elements to ensure analytical tractability of the problem333An
investigation quantifying the impact of mutual coupling and other reactive
near-field effects on system design is an interesting topic for future work.
We point out that, in Section IV-4.3, we show that the optimal deployment only
requires a finite number of transmit antennas as illustrated in Fig. 4. We
also note that mutual coupling effects are typically neglected for discrete
antenna arrays [26, Footnote 4]..

_Remark 2:_ In this paper, we limit our study to transmit antenna
architectures deployed on the ceiling of the room, i.e., at $y=0$.
Nevertheless, our results can be extended to the case where the transmit
antenna architecture is integrated as part of a wall by exchanging the role of
the $y$-component with the $x$-component or the $z$-component.

###  2.3 CHANNEL MODEL

In Section 1, we discussed that the wireless channel is predominantly LoS in
the mmWave frequency band since multipath fading components are weak at such a
high frequency. The problem and the proposed solution presented in Section 3
are developed under LoS conditions. Assuming an LoS channel model allows
establishing an intuitive connection between the maximisation of the received
power and the geometry underlying the problem. The robustness of the optimal
solution is investigated through simulation in Section 5, by assuming
isotropic scattering in the half-space in front of the transmit antenna
architecture in addition to a strong LoS component, based on the Rician fading
channel model.

A suitable channel model must capture the propagation characteristics of the
EM wavefronts from the transmitter to a receiver accurately. The propagation
environment of EM wavefronts can be divided into the near-field and the far-
field region. In conventional wireless systems, the near-field region is
negligibly small, however, when utilising the mmWave frequency band, which
allows the EH antenna of a device to be small, the size of the near-field
region becomes significant. The relationship between the wavelength of the EM
waves, the size of the transmit antenna architecture, and the distance between
the transmitter and the receiver determines the size of the operating regions
[27]. Specifically, a receiver is located within the near-field region if the
distance between the receiver and the transmitter is below the Fraunhofer
distance $d_{\mathrm{Fraunhofer}}=2L^{2}/\lambda$, where $L$ and $\lambda$ are
the antenna’s aperture and the wavelength, respectively [27]. The near-field
region is further subdivided into the reactive near-field and the radiating
near-field region, where the latter is commonly referred to as the Fresnel
region. In this paper, we only consider the region beyond the reactive near-
field for deployment, i.e., we restrict the receivers not to be located in the
reactive near-field of the transmit antenna architecture. The motivation
behind this is twofold. Firstly, the size of the reactive near-field region is
inherently limited since the deployment of overly large surfaces seems
unrealistic (see Section II-2.2). Secondly, in practice, receivers are
unlikely to be located close to the ceiling444 We point out that, in Section
IV-4.3, we show that the optimal deployment only requires a finite number of
transmit antennas, which is realised with a discrete antenna array, and we
illustrate this in Fig. 4. . The set of receiver positions beyond the reactive
near-field is denoted by

$\displaystyle\mathcal{X}\times\mathcal{Y}\times\mathcal{Z}\subset\hat{\mathcal{X}}\times\hat{\mathcal{Y}}\times\hat{\mathcal{Z}},$
(4)

and any receiver position $(x,y,z)$ considered in the following satisfies

$\displaystyle(x,y,z)\in\mathcal{X}\times\mathcal{Y}\times\mathcal{Z}.$ (5)

Therefore, receiver position $(x,y,z)$ lies in the radiating near-field or the
far-field region of the transmitter.

For high carrier frequencies, such as in the mmWave band, the reflection and
scattering losses are severe, causing the LoS component to be dominant in the
wireless channel [18, 28]. Thus, an LoS channel is typically assumed for
wireless systems operating in the near-field region [29, 17, 30]. We note that
interruptions of the LoS due to obstructions cause blockages [17] and we
reduce the risk of channel blockage by placing the transmit antenna
architecture on the ceiling of the room instead of on one of the walls (see
Section II-2.2). For the analytical tractability of the system design, we make
the assumption that an LoS connection between the radiating elements and the
receiver exists and do not consider the impact of channel blockages in this
paper.  Under free-space LoS conditions, the equivalent complex baseband
channel between the radiating element at $(a_{x},a_{z})$ and a receiver at
$(x,y,z)\in\mathcal{X}\times\mathcal{Y}\times\mathcal{Z}$ is given by [27]555
Including effective aperture losses and polarisation losses and studying their
impact on the system performance is an interesting topic for future work.
Here, we aim at determining the maximum achievable system performance and
focus on providing analytical insights, and thus, we neglect these effects.

$\displaystyle
g_{xyz}(a_{x},a_{z})=\frac{\sqrt{c}}{D_{xyz}(a_{x},a_{z})}\,\mathrm{e}^{-\mathrm{j}\frac{2\pi}{\lambda}D_{xyz}(a_{x},a_{z})},$
(6)

where $D_{xyz}(a_{x},a_{z})$ is the distance between the radiating element at
$(a_{x},a_{z})$ and a receiver at $(x,y,z)$, i.e.,
$D_{xyz}(a_{x},a_{z})=\sqrt{(x-a_{x})^{2}+y^{2}+(z-a_{z})^{2}}$ and $c$ is the
channel power gain at the reference distance of 1 meter. By virtue of the
spherical nature of the EM wavefronts, both the amplitude and the phase of
$g_{xyz}(a_{x},a_{z})$ depend on distance $D_{xyz}(a_{x},a_{z})$. In contrast,
when the EM wavefronts are modelled as planar waves (far-field approximation),
the _path loss_ , which is given by the amplitude in (6), between all transmit
radiating elements and a receiver is assumed to be constant.

###  2.4 TRANSMIT SIGNAL MODEL

We denote the transmit power distribution as $P$, which is related to the
transmit power density $p$ as follows

$\displaystyle\mathrm{d}P(a_{x},a_{z})=p(a_{x},a_{z}){\mathrm{d}a_{x}\mathrm{d}a_{z}}.$
(7)

The total transmit power is constrained to be finite, i.e.,

$\displaystyle\iint_{\mathcal{X}_{a}\times\mathcal{Z}_{a}}p(a_{x},a_{z})\,\mathrm{d}a_{x}\,\mathrm{d}a_{z}=\iint_{\mathcal{X}_{a}\times\mathcal{Z}_{a}}\,\mathrm{d}P(a_{x},a_{z})=1,$
(8)

and, without loss of generality, the transmit power is normalised to unity.

The transmit signal $t(a_{x},a_{z})$ emitted by the transmit antenna located
at $(a_{x},a_{z})$ is given by

$\displaystyle
t(a_{x},a_{z})=\sqrt{p(a_{x},a_{z})}\,\mathrm{e}^{\mathrm{j}\theta(a_{x},a_{z})},$
(9)

where $p(a_{x},a_{z})\geq 0$ is the non-negative transmit power density at
$(a_{x},a_{z})$ and $\theta(a_{x},a_{z})$ is the phase. Since there is no
information on the location of the receivers, the transmit signal should
illuminate the entire environment. Therefore, the phases are assumed to be
independent and identically distributed (i.i.d.) among the radiating elements.

The received signal in position $(x,y,z)$ is given by

$\displaystyle
r_{xyz}=\iint_{\mathcal{X}_{a}\times\mathcal{Z}_{a}}t(a_{x},a_{z})g_{xyz}(a_{x},a_{z})\,\mathrm{d}a_{x}\,\mathrm{d}a_{z},$
(10)

and the power of the signal collected by the receiver’s antenna with aperture
$S_{RX}$, which is centred at position $(x,y,z)$, is given by

$\displaystyle\gamma_{xyz}$
$\displaystyle=S_{RX}\,\mathrm{E}\left\\{\left|r_{xyz}\right|^{2}\right\\},$
$\displaystyle\overset{\text{i.i.d.}}{=}S_{RX}\,\iint_{\mathcal{X}_{a}\times\mathcal{Z}_{a}}\left|t(a_{x},a_{z})g(a_{x},a_{z})\right|^{2}\,\mathrm{d}a_{x}\,\mathrm{d}a_{z},$
$\displaystyle=c\,S_{RX}\,\iint_{\mathcal{X}_{a}\times\mathcal{Z}_{a}}\frac{p(a_{x},a_{z})}{[D_{xyz}(a_{x},a_{z})]^{2}}\,\mathrm{d}a_{x}\,\mathrm{d}a_{z},$
(11)

where $\mathrm{E}\\{\cdot\\}$ denotes the expectation operator and we exploit
that the phases $\theta(a_{x},a_{z})$ are i.i.d. Therefore, the power received
by the receiver in position $(x,y,z)$ is directly proportional to

$\displaystyle\Phi_{P}\\{f_{xyz}(a_{x},a_{z})\\}=\iint_{\mathcal{X}_{a}\times\mathcal{Z}_{a}}f_{xyz}(a_{x},a_{z})\,\mathrm{d}P(a_{x},a_{z}),$
(12)

with

$\displaystyle
f_{xyz}(a_{x},a_{z})=\left(\frac{1}{D_{xyz}(a_{x},a_{z})}\right)^{2}.$ (13)

Operator $\Phi_{P}\\{\cdot\\}$ represents the integration of function
$f_{xyz}(a_{x},a_{z})$ with respect to (w.r.t.) the transmit power
distribution $P$. Mathematically, this is equivalent to the expectation of a
function $f_{xyz}(a_{x},a_{z})$ w.r.t. a probability distribution. This
equivalence is crucial and it will be exploited in Section III-3.3, where we
will leverage the mathematical equivalence between determining the optimal
transmit power distribution across a continuous aperture antenna, which we
study in this paper, and the problem of finding the capacity-achieving
amplitude-constrained input distribution of a channel, which has been studied
in, e.g., [31, 32, 33]. Note that the impact of additive noise on the received
power is neglected in (2.4), (12). Furthermore, we note that the amount of
power harvested by a device is a non-decreasing function of the received
signal power [34].

##  3 OPTIMAL TRANSMIT POWER DISTRIBUTION

Our goal is to supply power to an arbitrary number of receivers in the
environment without any knowledge of their location. This objective can be
achieved by maximising the received signal power for the worst possible
receiver position(s) in the environment, which leads to a max-min optimisation
problem. In this section, we formulate this optimisation problem to optimise
the transmit power distribution $P$. Hereby, the optimal transmit power
distribution $P^{*}$ characterises the optimal transmit antenna architecture.
Deploying an HMIMO system would be necessary if we found that the optimal
transmit power distribution’s support had the same dimensionality as the
distribution’s domain. However, we will prove in Section III-3.3 that the
optimal transmit power distribution’s support has a lower dimensionality than
the distribution’s domain and thus, a HMIMO system is unnecessary in the
context of the considered WPT problem.

###  3.1 PROBLEM FORMULATION

Maximising the power at the receiver’s antenna centred at point
$(x,y,z)\in\mathcal{X}\times\mathcal{Y}\times\mathcal{Z}$ in the environment
is achieved by maximising $\Phi_{P}\\{f_{xyz}\\}$, defined in (12), via an
optimal distribution of transmit power among the radiating elements, where $P$
is a member of the set of distributions $\Omega$, i.e., $P(a_{x},a_{z})\geq
0,\forall(a_{x},a_{z})\in\mathcal{X}_{a}\times\mathcal{Z}_{a}$, and (8) is
satisfied. Hereby, the non-negativity and finite power constraints, which have
been introduced in Section II-2.2, apply to every element of $\Omega$.
Moreover, the function $P(a_{x},a_{z})$ is restricted to the set
$\mathcal{X}_{a}\times\mathcal{Z}_{a}$, and as a consequence of (8),
$P(a_{x},a_{z})$ is bounded by one and thus, is square-integrable. Therefore,
the set of considered distributions $\Omega$ is a subset of the square-
integrable (finite energy) functions defined over the transmit antenna domain
and denoted as $\mathcal{L}^{2}(\mathcal{X}_{a}\times\mathcal{Z}_{a})$, i.e.,
$P(a_{x},a_{z})\in\Omega\subset\mathcal{L}^{2}(\mathcal{X}_{a}\times\mathcal{Z}_{a})$.
The set $\Omega$ is further characterised in the following lemma.

######  Lemma 1.

The set of distributions
$\Omega\subset\mathcal{L}^{2}(\mathcal{X}_{a}\times\mathcal{Z}_{a})$ is a
weakly compact space in the weak topology and convex.

Proof: The proof is found in Appendix 7.

Ensuring that receivers are reliably supplied with power is achieved by
maximising the received signal power at the worst possible receiver
position(s).  This can be formulated as the following max-min optimisation
problem

$\displaystyle\underset{P\in\Omega}{\text{maximise}}\quad\\!\min_{\begin{subarray}{c}(x,y,z)\\\
\in\mathcal{X}\times\mathcal{Y}\times\mathcal{Z}\end{subarray}}\,\Phi_{P}\\{f_{xyz}\\}.$
(14)

The best worst-case performance, i.e., the optimal objective value of Problem
(14), is denoted by $m$ and is defined as

$\displaystyle m=\min_{\begin{subarray}{c}(x,y,z)\\\
\in\mathcal{X}\times\mathcal{Y}\times\mathcal{Z}\end{subarray}}\,\Phi_{P^{*}}\\{f_{xyz}\\},$
(15)

where $P^{*}$ is the optimal distribution of power. Note that Problem (14) is
linear in $P$. The feasibility of Problem (14) is ensured through the
compactness of $\Omega$ shown in Lemma 1, i.e., $\Omega$ is closed and
bounded.

###  3.2 CRITICAL RECEIVER POSITIONS

By identifying the receiver locations where the received signal power is the
lowest, the complexity of solving Problem (14) can be reduced. These critical
locations are defined w.r.t. the optimal distribution of power $P^{*}$. To
this end, the set containing the critical receiver positions when power is
allocated optimally is denoted by $\mathcal{R_{\mathrm{crit}}}$. The set of
non-critical positions is the complement of $\mathcal{R_{\mathrm{crit}}}$,
i.e., $\mathcal{R_{\mathrm{crit}}^{\text{c}}}$. After the power has been
optimally allocated across the continuous transmit antenna surface, the signal
power received by an antenna centred at a critical receiver position
$(x,y,z)\in\mathcal{R_{\mathrm{crit}}}$ will be below the signal power
received by an antenna centered at any non-critical location
$(x,y,z)\in\mathcal{R_{\mathrm{crit}}^{\text{c}}}$. Next, the set of critical
receiver positions is characterised more precisely.

######  Key Observation 1.

For any $(x,z)\in\mathcal{X}\times\mathcal{Z}$, the worst performance is
attained at $y=L_{y}$. Consequently, all critical positions are located at
$y=L_{y}$, i.e., $\mathcal{Y_{\mathrm{crit}}}=\\{L_{y}\\}$.

Key Observation 1 holds since $f_{xyz}$ is a monotonically decreasing function
of $y$. Thus, the lowest objective value is obtained in the plane with
$y=L_{y}$. As a consequence of Key Observation 1666 Note that the result in
Key Observation 1, i.e., $\mathcal{Y_{\mathrm{crit}}}=\\{L_{y}\\}$, holds
since the radiating elements are assumed to be omnidirectional antennas and
dropping this assumption may change the result., the function $f_{xyz}$ in the
objective function of Problem (14) can be equivalently substituted by
$f_{xz}(a_{x},a_{z}):=f_{xyz}(a_{x},a_{z})|_{y=L_{y}}$.  Problem (14) is
equivalently reformulated in the following lemma.

######  Lemma 2.

Assume that the set of critical positions
$\mathcal{R_{\mathrm{crit}}}\subseteq\mathcal{X}\times\mathcal{Y}_{crit}\times\mathcal{Z}$
is known. Then, solving Problem (14) is equivalent to solving the following
problem

$\displaystyle\underset{o\in\mathbb{R},P\in\Omega,\Lambda}{\text{maximise}}\quad\\!\underbrace{o-\sum_{\mathcal{R_{\mathrm{crit}}}}\left(o-\Phi_{P}\\{f_{xz}\\}\right)\lambda_{xz}}_{J(P)},$
(16)

where $\Lambda$ is the set of positive variables
$\lambda_{xz}>0,\forall(x,y,z)\in\mathcal{R_{\mathrm{crit}}}$. When allocating
power optimally through $P^{*}(a_{x},a_{z})$, the optimal objective value is
$o^{*}=m$ and the following conditions should also hold

$\displaystyle[$ $\displaystyle(\lambda_{xz}>0\implies
m=\Phi_{P^{*}}\\{f_{xz}\\})\,\,\,\mathrm{or}\,\,\,$
$\displaystyle(m<\Phi_{P^{*}}\\{f_{xz}\\}\implies\lambda_{xz}=0)],$
$\displaystyle\forall(x,y,z)\in\mathcal{X}\times\mathcal{Y_{\mathrm{crit}}}\times\mathcal{Z}.$
(17)

Proof: The proof is presented in Appendix 8.

The objective function in Problem (16) is referred to as $J(P)$ in the
following. Note that Problem (14) is a convex optimisation problem, which is
readily apparent when the problem is stated in its epigraph form [35] (see
Appendix 8).

###  3.3 OPTIMAL STRUCTURE OF POWER DISTRIBUTION

In this section, we establish a relation between Problem (16) and the problem
of finding the optimal amplitude-constrained input distribution for reaching
the information capacity of Gaussian channels studied in [31, 32, 33]. In [31]
and [32], the authors determine the optimal input distribution of transmit
symbols, which are bounded in amplitude, for achieving information capacity,
whereas our problem lies in identifying the optimal  distribution of transmit
power, under geometrical constraints regarding the  continuous transmit
antenna architecture. Beyond communication problems [31], similar results on
the optimal input distribution in WPT applications have been established in
[32]. First, we consider the optimisation of a two-dimensional distribution
$P(a_{x},a_{z})$ which is significantly different from the problems involving
one-dimensional distributions such as [31] and [32]. Specifically, the
properties of the optimal two-dimensional distribution’s structure are less
stringent. The analysis of optimisation problems involving input distributions
with a dimensionality greater than one has been studied in [33]. Subsequently,
we consider the optimisation problem with a one-dimensional distribution,
which allows characterising the optimal distribution as in [31] and [32].

In the following, we investigate properties, which apply to the optimal
distribution of power and allow the characterisation of the optimal transmit
antenna architecture. The main results are stated in Propositions 2 and 3 for
the optimal distribution of power over a two-dimensional surface and in
Proposition 4 for the optimal one-dimensional distribution. First, a necessary
and sufficient condition for the optimal distribution of power $P^{*}$ is
established in Proposition 1 and Corollary 1.

######  Proposition 1.

Consider the optimisation problem in (16). There exists an optimal
distribution of power $P^{*}(a_{x},a_{z})$ which attains the maximum $m$ of
Problem (16) if and only if the following condition holds

$\displaystyle\iint_{\mathcal{X}_{a}\times\mathcal{Z}_{a}}\,\bar{f}(a_{x},a_{z})\,\,\mathrm{d}P(a_{x},a_{z})\leq
m,\quad\forall P\in\Omega,$ (18)

where $\bar{f}(a_{x},a_{z})$ is given by

$\displaystyle\bar{f}(a_{x},a_{z})=\frac{\sum_{\mathcal{R}^{\mathrm{crit}}}\,f_{xz}(a_{x},a_{z})\lambda_{xz}}{\sum_{\mathcal{R}^{\mathrm{crit}}}\,\lambda_{xz}},$
(19)

and (18) is satisfied with equality for the optimal distribution of power
$P^{*}(a_{x},a_{z})$.

Proof: This proposition is proven in Appendix 9.

Intuitively, function $\bar{f}$ can be interpreted as the weighted average of
$f_{xz}$ over the receiver positions $\mathcal{X}\times\mathcal{Z}$ at
$y=L_{y}$. The weights are given by the variables
$\lambda_{xz}/\sum_{\mathcal{R}^{\mathrm{crit}}}\,\lambda_{xz}$ and thus, only
critical receiver positions contribute to $\bar{f}$. Note that the optimal
distribution $P^{*}$ is not necessarily unique since $J(P)$ is linear in $P$
and not strictly concave as in the problems considered in [31, 32].
Additionally, Proposition 1 provides a condition for the existence of $P^{*}$.
Obtaining $P^{*}$ requires determining the variables $\lambda_{xz}$ satisfying
(18) and (19).

Next, the optimality condition for $P^{*}$, given in Proposition 1, is
restated in a more intuitive manner. This requires defining the support of
$P^{*}$, which is analogous to the definition in [33]. A radiating element’s
position $(a_{x},a_{z})$ is called a _point of increase_ of the optimal
distribution $P^{*}$ if for any open subset $O\subset\mathbb{R}^{2}$
containing $(a_{x},a_{z})$, $P^{*}(O)>0$777Here, the argument of the optimal
transmit power distribution is a set instead of a tuple of points, i.e.,
$P^{*}(O)$ corresponds to $\iint_{O}\mathrm{d}P^{*}(a_{x},a_{z})$. holds [33].
The set of points of increase of $P^{*}$ is defined as
$\mathcal{E}(P^{*})\subseteq\mathbb{R}^{2}$. Note that
$P^{*}(\mathcal{E}(P^{*}))=1$ [33]. Moreover, $\mathcal{E}(P^{*})$ is the
smallest closed subset of $\mathbb{R}^{2}$ such that
$P^{*}(\mathcal{E}(P^{*}))=1$ [33]. Note that $P\in\Omega$ is restricted to
$\mathcal{X}_{a}\times\mathcal{Z}_{a}\subset\mathbb{R}^{2}$ and thus,
$\mathcal{E}(P^{*})\subseteq\mathcal{X}_{a}\times\mathcal{Z}_{a}$. We define
$\mathcal{E}(P^{*})$ as the support of $P^{*}$.

######  Corollary 1.

Consider the optimisation problem in (16). There exists an optimal
distribution of power $P^{*}$ for Problem (16) if and only if
$\bar{f}(a_{x},a_{z})$ satisfies the following equations

$\displaystyle\bar{f}(a_{x},a_{z})$ $\displaystyle\leq
m,\quad\forall(a_{x},a_{z})\in\mathcal{X}_{a}\times\mathcal{Z}_{a},$ (20)
$\displaystyle\bar{f}(a_{x},a_{z})$
$\displaystyle=m,\quad\forall(a_{x},a_{z})\in\mathcal{E}(P^{*}).$ (21)

Proof: The proof is provided in Appendix 10.

Thus, Corollary 1 describes the relationship between the optimal distribution
of transmit power $P^{*}$ for attaining the optimal objective value $m$ of
Problem (16) and the optimal set of positions regarding the radiating elements
$\mathcal{E}(P^{*})$, i.e., the support of $P^{*}$.

Based on the optimality conditions on $P^{*}$ defined in Proposition 1 and
Corollary 1, the set of optimal positions regarding the radiating elements is
characterised more precisely. First, we complete our analysis of the two-
dimensional distribution $P^{*}(a_{x},a_{z})$. Subsequently, we characterise
the set of optimal positions under the constraint of a  continuous one-
dimensional deployment.  The characterisation requires the following
definition in the two-dimensional scenario.

######  Definition 1 (see [33]).

A subset $S_{1}\subset S_{0}$ is _dense_ in $S_{0}$ if every element $s\in
S_{0}$ is either an element of $S_{1}$ or an accumulation point of $S_{1}$. A
subset $S_{1}\subset S_{0}$ is _nowhere dense_ if $S_{2}\cap S_{1}$ is not
_dense_ in $S_{0}$, where $S_{2}$ is any non-empty open set $S_{2}\subset
S_{0}$.

######  Proposition 2.

The support $\mathcal{E}(P^{*})\subset\mathcal{X}_{a}\times\mathcal{Z}_{a}$ of
the optimal two-dimensional distribution of power $P^{*}$ is a nowhere dense
set of $\mathcal{X}_{a}\times\mathcal{Z}_{a}$.

Proof: The proof is provided in Appendix 11.

######  Proposition 3.

The support $\mathcal{E}(P^{*})\subset\mathcal{X}_{a}\times\mathcal{Z}_{a}$ of
the optimal two-dimensional distribution of power $P^{*}$ is of Lebesgue
measure zero.

Proof: The proof is provided in Appendix 12.

Thus, $\mathcal{E}(P^{*})$ is a set of pairs $(a_{x},a_{z})$, which define the
optimal  positions of radiating elements, that is nowhere dense and has
Lebesgue measure zero, i.e., the set $\mathcal{E}(P^{*})$ does not occupy any
measurable surface in $\mathcal{X}_{a}\times\mathcal{Z}_{a}$. Intuitively, the
set $\mathcal{E}(P^{*})$ could consist of isolated points scattered throughout
$\mathcal{X}_{a}\times\mathcal{Z}_{a}$ since both properties hold in this
case. On the other hand, Propositions 2 and 3 do not preclude the possibility
of $\mathcal{E}(P^{*})$ containing more complex structures, such as continuous
one-dimensional structures, as long as $\mathcal{E}(P^{*})$ is nowhere dense
and of measure zero. This is further investigated in Section 4.  We note that
based on Propositions 2 and 3, a continuous planar transmit antenna
architecture, which is utilised in HMIMO, is unnecessary for WPT. Moreover,
observe that, by virtue of the max-min problem, the optimal distribution must
be symmetrical w.r.t. the $x$-axis and the $z$-axis, i.e.,
$P^{*}(a_{x},a_{z})=P^{*}(-a_{x},a_{z})=P^{*}(-a_{x},-a_{z})=P^{*}(a_{x},-a_{z})$.
Violating this symmetry condition would lead to a decrease in performance at
the worst possible receiver positions, i.e., a decrease of the objective
value.

The results presented in this section, which allowed drawing conclusions
regarding the optimal deployment of transmit radiating elements
$\mathcal{E}(P^{*})$, pertained to the analysis of a two-dimensional
distribution of power. In the following, we restrict the continuous transmit
antenna architecture to lie on the one-dimensional set $\mathcal{L}_{a}$,
which results in a problem analogous to [31, 32] (see Section 2.2).
Consequently, the optimal distribution of power $\tilde{P}^{*}(a_{l})$ along
$\mathcal{L}_{a}$ is one-dimensional. This problem is investigated in the
following, whereby the optimal set of positions of the radiating elements
$\tilde{\mathcal{E}}(\tilde{P}^{*})\subset\mathcal{L}_{a}$ is shown to be
finite.

######  Proposition 4.

The support $\tilde{\mathcal{E}}(\tilde{P}^{*})\subset\mathcal{L}_{a}$ of the
optimal one-dimensional distribution is a finite set of points.

Proof: The proof is provided in Appendix 13.

We note that a continuous transmit antenna architecture is unnecessary for
one-dimensional transmit antenna architectures since Proposition 4 guarantees
that a finite number of radiating elements is sufficient.

##  4 SYSTEM DISCRETISATION

In the following, to shed light on our analytical results in Section 3, the
optimal  transmit antenna architecture is studied via numerical analysis. In
order to solve Problem (14) numerically, the  continuous sets
$\mathcal{X}_{a}\times\mathcal{Z}_{a}$ and $\mathcal{L}_{a}$ have to be
discretised. For a one-dimensional transmit antenna architecture, we have
proven in Proposition 4 that a finite number of transmit antennas placed in
$\tilde{\mathcal{E}}(\tilde{P}^{*})\subset\mathcal{L}_{a}$ is optimal. For a
two-dimensional transmit antenna architecture, Proposition 3 guarantees that
the optimal deployment of transmit antennas is on a set
$\mathcal{E}(P^{*})\subset\mathcal{X}_{a}\times\mathcal{Z}_{a}$ with Lebesgue
measure zero. We obtain approximations of the optimal solution to Problem (14)
through discretising the problem, and subsequently, solving it numerically. We
verify numerically that the solution of the optimisation problem remains
unchanged for finer levels of discretisation (LoD), i.e., for an increasing
granularity of the sampling grid. Further, since Propositions 2 and 3 ensure
that the optimal set of transmit antennas
$\mathcal{E}(P^{*})\subset\mathcal{X}_{a}\times\mathcal{Z}_{a}$ is nowhere
dense and has Lebesgue measure zero, which are properties satisfied by a set
of isolated points scattered within $\mathcal{X}_{a}\times\mathcal{Z}_{a}$, we
conjecture that such a set of isolated points is optimal. Thereby, to support
this conjecture by simulation results, it is necessary to investigate whether
the number of transmit antennas increases monotonically for finer LoD or if
the number of transmit antennas remains constant. Intuitively, if the latter
is true, there is an optimal, finite number of transmit antennas, which
depends on the geometrical properties of the environment, i.e., $L_{x}$,
$L_{y}$, and $L_{z}$, at least from a practical point of view. The three-
dimensional environments, which are investigated in this paper, are listed in
Table 1. The difference between the environments is the ratio of height to
width since the width and depth of the room are chosen to be equal. The
environments are denoted by their height-to-width ratio.

Table 1: Definition of the four environments analysed in Sections 4 and 5. Height $L_{y}$ in [m] | Width $L_{x}$ in [m] | Depth $L_{z}$ in [m] | $\frac{\mathrm{Height}}{\mathrm{Width}}$
---|---|---|---
2 | 2 | 2 | 1/1
2 | 6 | 6 | 1/3
2 | 8 | 8 | 1/4
2 | 10 | 10 | 1/5

We note that the dimensions of the environments listed in Table 1 are
realistic, e.g., for an office or a room in a residential building. In
environments spanning a significantly larger area, such as, a factory
building, mmWave operating frequencies become unaffordable due to the large
path loss, which would result in a very low received power when receivers are
located far away from a transmitter, and thus, operation at a lower frequency
would be preferred. However, NLoS components become increasingly important for
decreasing operating frequency, and thus, the wireless channel may not be
accurately described by a deterministic LoS channel. In this case, it is
preferable to consider alternative wireless systems and models, such as those
in [21, 22, 23, 24], for which optimal transmit antenna deployment and power
allocation differ significantly in methodology and solutions.

###  4.1 DISCRETISED OPTIMISATION PROBLEM

In the following, the set of transmit antenna positions, i.e.,
$\mathcal{X}_{a}\times\mathcal{Z}_{a}$ for the two-dimensional problem and
$\mathcal{L}_{a}$ for the one-dimensional problem, is discretised. In this
paper, we only investigate systems where the discretised transmit antenna
architecture, which we also refer to as a (discrete) antenna array, covers the
entire ceiling888 In Section II-2.2, the deployment of excessively large
continuous surfaces in typical indoor environments was deemed unsuitable due
to obstructions in the ceiling. This problem is substantially less severe when
deploying discrete transmit antennas within
$\mathcal{X}_{a}\times\mathcal{Z}_{a}$ or $\mathcal{L}_{a}$ as considered in
this section. , i.e., $L_{a_{x}}=L_{x}$ and $L_{a_{z}}=L_{z}$. The one-
dimensional antenna array is also placed on the ceiling along the $x$-axis,
i.e., $\mathcal{L}_{a}=\mathcal{Z}_{a}$. Moreover, we set $L_{x}=L_{z}$. The
LoD is defined as the number $N+1$ of equally-sized intervals into which the
sets $\mathcal{X}_{a}$, $\mathcal{Z}_{a}$ and $\mathcal{L}_{a}$ are
partitioned, respectively. Consequently, we have a grid of antenna positions
with $(N+1)^{2}$ elements in $\mathcal{X}_{a}\times\mathcal{Z}_{a}$ and $N+1$
elements in $\mathcal{L}_{a}$, respectively. The position of transmit antenna
$i$, which corresponds to one element of the grid, is given as
$(a_{x},a_{z})_{i}$, with $i=1,\dots,(N+1)^{2}$, in two dimensions and
$(a_{l})_{i}$, with $i=1,\dots,N+1$, in one dimension. In the following, the
coordinates of position $(a_{x},a_{z})_{i}$ are denoted by $a_{xi}$ and
$a_{zi}$. Similarly, the coordinate of position $(a_{l})_{i}$ is given by
$a_{li}$. By utilising the definitions in Section II-2.4, Problem (14) is
discretised as follows

$\displaystyle\underset{o\in\mathbb{R},p(a_{xi},a_{zi})}{\text{maximise}}$
$\displaystyle o$ (22a) subject to $\displaystyle
o\leq\sum_{(a_{xi},a_{zi})}f_{xz}(a_{xi},a_{zi})p(a_{xi},a_{zi}),\quad$
$\displaystyle\forall(x,L_{y},z)\in\mathcal{X}\times\mathcal{Y_{\mathrm{crit}}}\times\mathcal{Z},$
(22b) $\displaystyle\sum_{(a_{xi},a_{zi})}p(a_{xi},a_{zi})=1,$ (22c)
$\displaystyle p(a_{xi},a_{zi})\geq 0,$ (22d)

where the problem has been rewritten in epigraph form [35]. After
discretisation, the convex optimisation problem in (22) can be solved
efficiently, for example, by applying the interior-point method [35]. The
problem is solved using standard numerical tools for convex optimisation, such
as CVXPY [36], [37]. For all results shown, any value below $10^{-6}$ in the
optimal distribution of power was deemed to a  representation and computation
error in computer arithmetics and was set to zero. In the worst case, values
below $10^{-6}$ accounted for less than $10^{-5}\%$ of the total power
distributed among the transmit antennas.

###  4.2 LEVEL OF DISCRETISATION

There is an inherent trade-off between the computational complexity of solving
a problem and the accuracy of its solution. The necessary LoD is determined
based on a desired accuracy and a criterion to assess whether the proposed
solution is sufficiently accurate. To this end, Problem (22) was solved
repeatedly for increasing granularity of the sampling grid. Subsequently, the
differences between the objective values of consecutive iterations were
evaluated.

In Fig. 2, the absolute value of the relative difference between the objective
values, i.e., the absolute value of the difference between the current and
previous objective value divided by the current objective value, was
calculated for every $N$. As the LoD increases, the accuracy increases for all
investigated environments since the objective value stabilises.  From Fig. 2,
at LoD $N+1=81$, the relative difference lies below $0.05\%$ and beyond that,
the improvements were deemed negligible. Consequently, unless specified
otherwise, all simulations in the following are performed at LoD $N+1=81$.

3141516171819110111100.050.10.15$N+1$$|$ Relative difference in % $|$1/1
(1D)1/3 (1D)1/4 (1D)1/5 (1D)

31415161718191101111$0$$1\cdot 10^{-1}$$0.2$$0.3$$N+1$$|$ Relative difference
in % $|$1/1 (2D)1/3 (2D)1/4 (2D)1/5 (2D)

Figure 2: Absolute value of relative difference between objective values in
percent for increasing $N$ for the height-to-width ratios defined in Table 1.

###  4.3 FINITENESS OF OPTIMAL POWER DISTRIBUTION

In this section, the optimal number of transmit antennas999The optimal number
of transmit antennas is defined as the number of non-zero values in the
solution $p^{*}(a_{ix},a_{iz})$ of Problem (22). $N_{t}^{*}$ is investigated
for increasing LoD. To this end, the optimal number of transmit antennas for
each considered environment is displayed in Fig. 3.

21416181101121141$0$$2$$4$$6$$8$$10$$N+1$$N_{t}^{*}$1/1 (1D)1/3 (1D)1/4
(1D)1/5 (1D)

21416181101121141$0$$20$$40$$60$$N+1$$N_{t}^{*}$1/1 (2D)1/3 (2D)1/4 (2D)1/5
(2D)

Figure 3: Optimal number of transmit antennas, i.e., number of positions where
non-zero powers are allocated by the optimal distribution, for increasing $N$
for the height-to-width ratios given in Table 1.

.

In Fig. 3, one can observe that the optimal number of transmit antennas
$N_{t}^{*}$ does not increase monotonically for different LoDs, which suggests
that a finite number of antennas is indeed optimal, even for a two-dimensional
array. The optimal number of antennas to be deployed depends on the geometry
of the environment and the dimensionality of the transmit antenna
architecture.  We note that in Section 3, the optimal solution of the problem
is characterised in the context of a continuous architecture and a slight
mismatch between the optimal positions regarding the continuous architecture
and the optimal positions regarding the discretised architecture may occur
depending on how well the optimal solution can be embedded in the grid at a
certain granularity (see Section V-5.1). Consequently, slight fluctuations
regarding the optimal number of transmit antennas are expected, which are
visible in Fig. 3. However, note that these fluctuations are negligible
compared to the difference in the number of grid cells among different LoDs,
especially for the two-dimensional array with $(N+1)^{2}$ cells.

In general, the optimal number of transmit antennas is larger when employing a
two-dimensional antenna array compared to the deployment over a one-
dimensional transmit antenna array. Moreover, for a decreasing height-to-width
ratio, the optimal number of transmit antennas increases, especially when
employing a two-dimensional antenna array.

##  5 PERFORMANCE AND SENSITIVITY ANALYSIS

We assume a LoD of $N+1=81$, a transmit power of $10$ W,
$S_{RX}=\lambda^{2}/4$, which is on the scale of the antenna aperture proposed
in [38], and a wavelength of $\lambda=12.5$ mm for the considered
system101010A wavelength of $\lambda=12.5$ mm corresponds to an operating
frequency of $24$ GHz, which is at the low end of the mmWave frequency band
[6]. The electromagnetic field (EMF) radiation exposure limit for the proposed
system operating at $24$ GHz is around $9.9$ W/m2 [39]. Considering the worst-
scenario, i.e., when only a single transmit antenna is deployed which radiates
a power of $10$ W, the minimum distance $d_{\mathrm{min}}$ from the single
antenna to be below the exposure limit is around
$d_{\mathrm{min}}=\sqrt{10\;\text{W}/(4\pi\,9.9\;\text{W/m}^{2})}\approx 0.28$
m. The exposure to EMF radiation decreases for an increasing distance between
the transmit antenna array and the receiver [39]. When the transmit power is
spread across multiple antennas, which is optimal for the majority of
considered environments as shown in Fig. 4, the required distance from the
ceiling to be below the exposure limit reduces. In our system, the transmit
antenna array is positioned on the ceiling of the environment and thus, from a
practical point of view, the EMF exposure to humans in the environment will
typically be below the required limit since they will have a certain distance
to the ceiling..

Next, we evaluate the boundaries of the near-field region in the context of
the discretised antenna array. The room is contained within
$d_{\mathrm{Fraunhofer}}$ since the antenna architecture covers the entire
ceiling, which has a side length of at least the height of the room. Reactive
near-field effects are typically limited to a short distance around the
individual radiating elements in the context of discrete antenna architectures
[40, 26]. Specifically, reactive near-field effects are negligible beyond a
distance of
$d_{\mathrm{Fresnel}}=\sqrt[3]{\frac{L_{\mathrm{element}}^{4}}{8\lambda}}$
from an antenna element with aperture $L_{\mathrm{element}}$, which is
typically on the order of the wavelength. Thus, $d_{\mathrm{Fresnel}}$ is
typically limited to a few wavelengths from the transmit antenna [40, 27].

###  5.1 HEATMAPS OF OPTIMAL DISTRIBUTIONS

The solution $p^{*}(a_{xi},a_{zi})$ of Problem (22) is visualised in Fig. 4
using heatmaps for the height-to-width ratios considered in Table 1. The value
zero is depicted using white colour, while the non-zeros are represented by a
colour corresponding to the percentage of power allocated to the position.
Thus, transmit antennas are represented by a colour different from white.

1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40 non-zeros)1/1 (1D)
(1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5 (1D) (4 non-
zeros)1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40 non-
zeros)1/1 (1D) (1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5
(1D) (4 non-zeros)1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40
non-zeros)1/1 (1D) (1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5
(1D) (4 non-zeros)1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40
non-zeros)1/1 (1D) (1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5
(1D) (4 non-zeros)1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40
non-zeros)1/1 (1D) (1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5
(1D) (4 non-zeros)1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40
non-zeros)1/1 (1D) (1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5
(1D) (4 non-zeros)1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40
non-zeros)1/1 (1D) (1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5
(1D) (4 non-zeros)1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40
non-zeros)1/1 (1D) (1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5
(1D) (4 non-zeros)1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40
non-zeros)1/1 (1D) (1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5
(1D) (4 non-zeros)1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40
non-zeros)1/1 (1D) (1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5
(1D) (4 non-zeros)1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40
non-zeros)1/1 (1D) (1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5
(1D) (4 non-zeros)1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40
non-zeros)1/1 (1D) (1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5
(1D) (4 non-zeros)1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40
non-zeros)1/1 (1D) (1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5
(1D) (4 non-zeros)1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40
non-zeros)1/1 (1D) (1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5
(1D) (4 non-zeros)1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40
non-zeros)1/1 (1D) (1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5
(1D) (4 non-zeros)1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40
non-zeros)1/1 (1D) (1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5
(1D) (4 non-zeros)1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40
non-zeros)1/1 (1D) (1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5
(1D) (4 non-zeros)1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40
non-zeros)1/1 (1D) (1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5
(1D) (4 non-zeros)1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40
non-zeros)1/1 (1D) (1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5
(1D) (4 non-zeros)1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40
non-zeros)1/1 (1D) (1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5
(1D) (4 non-zeros)1/1 (1 non-zero)1/3 (12 non-zeros)1/4 (13 non-zeros)1/5 (40
non-zeros)1/1 (1D) (1 non-zero)1/3 (1D) (4 non-zeros)1/4 (1D) (4 non-zeros)1/5
(1D) (4 non-zeros)

Figure 4: Heatmaps of optimal power distributions for the height-to-width
ratios listed in Table 1 with $N+1=81$. Each heading indicates the height-to-
width ratio from Table 1, the dimensionality of the transmit antenna array,
and the number of non-zeros of $p^{*}(a_{ix},a_{iz})$ in brackets, i.e, the
optimal number of transmit antennas.  The relative amount of power allocated
to an antenna is colour-coded according to the colour bar at the bottom. The
antennas are partly magnified to aid visualising the solution. The colours of
the unmagnified antennas follow from the symmetry of the optimal solution
discussed in Section 3.

This depiction supports our conjecture that the optimal transmit antenna
positions are isolated points.  Except for a height-to-width ratio of one, the
room is contained within the near-field of the optimal antenna deployment
since the largest distance between individual transmit antennas defines the
aperture of the array generated by the antennas [40]. From a practical point
of view, the proposed optimal deployment may be realised via a distributed
antenna system.

For a height-to-width ratio equal to one, it is optimal to place a single
antenna in the centre of the ceiling.  In this case, the proposed system does
not have to be optimised for operating in the Fresnel region since the
critical receiver positions, which are in the corners of the room at
$y=L_{y}$, are in the far-field in this scenario.

###  5.2 MINIMUM RECEIVED POWER AND COMPARISON TO ALTERNATIVE DEPLOYMENTS

In this section, the performance of the proposed optimal two-dimensional
distribution of power is measured in terms of the minimum received power in
the environment. This value corresponds to the guaranteed level of received
power, which is available at any single receiver position $(x,y,z)$ beyond the
reactive near-field region of the room. Thus, we display the performance of
the worst-case scenario, which applies to a subset of receiver positions at
the floor $y=L_{y}$ of the room since the antennas are deployed on the
ceiling.

The proposed optimal method is denoted by M-OPT. Moreover, the performance is
compared to other power allocation methods, which are listed below:

$\frac{1}{1}$$\frac{1}{2}$$\frac{1}{3}$$\frac{1}{4}$$\frac{1}{5}$0.11$\frac{\mathrm{Height}}{\mathrm{Width}}$Minimum
Received Power $\gamma_{xyz}$ [µW]M-OPTM-FFM-UNIM-S75

$\frac{1}{1}$$\frac{1}{2}$$\frac{1}{3}$$\frac{1}{4}$$\frac{1}{5}$$0.4$$0.6$$0.8$$1$$\frac{\mathrm{Height}}{\mathrm{Width}}$LossM-
FFM-UNIM-S75

Figure 5: Minimum received power at the worst location in the environment and
loss of the benchmark schemes and suboptimal schemes compared to proposed
optimal deployment and power allocation. The height is fixed at $2$ m.

.

  * •

The achievable performance gain of modelling the system in the Fresnel region
is highlighted by comparing the proposed approach to the optimal power
distribution designed under the far-field approximation. By virtue of the max-
min problem, under the far-field assumption, the ideal deployment location
lies in the centre of the ceiling since the path losses corresponding to
different transmit antennas are assumed to be identical. As a consequence of
the constant path loss, a single transmit antenna is deployed to which all the
available power is allocated. This benchmark scheme is referred to as M-FF.

  * •

A uniform power distribution across the antenna array, where an antenna is
located at every grid cell and the power is split equally among the antennas.
This benchmark scheme is referred to as M-UNI.

  * •

We derive a suboptimal scheme from the proposed optimal approach by removing
optimally deployed transmit antennas operating with low transmit power.
Thereby, small non-zero values are set to zero in the distribution, and
subsequently, the allocated powers are re-normalised for a fair comparison at
equal total transmit power. More specifically, the x-th percentile of the
allocated power levels was calculated and transmit antennas operating at a
power level lower than the one corresponding to the x-th percentile were
removed, followed by the re-normalisation of power. We investigated the 75th,
percentile of the non-zero values. The scheme is referred to as M-S75.

We present the performance results in Fig. 5. Hereby, we investigate the
performance by evaluating the minimum received power111111 Typical EH circuits
are designed for receive powers in the µW range [41]. The constantly available
and guaranteed received power is in the range of, e.g., the continuous standby
power required by the nanowatt sensors presented in [42]. Moreover, receivers
capable of receiving energy at multiple positions, e.g., through multiple
receiving antennas can increase the amount of received power. , i.e., the
power at the critical receiver positions, for M-OPT, M-FF, M-UNI, and M-S75
for environments with different height-to-width ratios. Moreover, we determine
the performance loss incurred by utilising M-FF, M-UNI, and M-S75 in
comparison to the proposed optimal scheme M-OPT. To this end, we calculate the
loss compared to the proposed optimal solution by dividing the objective value
of a benchmark scheme, i.e., M-FF or M-UNI, or the suboptimal scheme, i.e.,
M-S75, by the objective value obtained through the optimal deployment and
power allocation, i.e., the solution of Problem (22), i.e., M-OPT.

For all schemes, we observe that the minimum received power decreases for a
decreasing height-to-width ratio. This effect occurs due to the increasing
environment size for decreasing height-to-width ratio while keeping the total
transmit power constant. Note that the received power can be increased by
adding more receive antennas or increasing the receiver’s antenna size
$S_{RX}$.

We observe that the proposed optimal power distribution outperforms or matches
the performance of the other schemes.  For a height-to-width ratio of one, we
see from Fig. 4 that the proposed optimal solution coincides with M-FF and we
find no performance loss between M-OPT and the other schemes. On the other
hand, for decreasing height-to-width ratio the performance gap between M-OPT
and the alternative methods grows. We observe the largest performance gap
between M-OPT and M-FF which indicates the relevance of accurately modelling
the wireless channel by taking into account the spherical EM wavefronts when
operating in the near-field region. Moreover, a uniform distribution of power,
i.e., in the case of M-UNI, is also found to be highly suboptimal, especially
for small height-to-width ratios. On the other hand, the loss in performance
compared to the optimal distribution is negligible for M-S75 with a slight
loss observable for a height-to-width ratio of $1/5$. This is because antennas
radiating a relatively low amount of power contribute little to the minimum
received power. Therefore, removing these transmit antennas comes at a small
loss in received power for the investigated environments.

###  5.3 SENSITIVITY ANALYSIS

While the assumption of a strong LoS channel, which was made throughout this
paper, is valid for very high carrier frequencies, a low-power small-scale
fading component may still be present. This requires including an NLoS
component in the wireless channel model [43]. Assuming isotropic scattering,
the NLoS component can be modelled as spatially correlated Rayleigh fading
[44], [45], [46]. The superposition of the NLoS and LoS components is modelled
by a Rician fading channel.

In this section, we assess the robustness of the proposed optimal two-
dimensional power distribution in the presence of scattering objects in the
environment using a Rician fading channel model. The Rician fading channel
between transmit antenna $i$ and the receiver position $(x,L_{y},z)$ is
defined as

$\displaystyle\tilde{g}_{xz}(a_{xi},a_{zi})=\sqrt{\frac{\kappa}{\kappa+1}}\,g_{xz}(a_{xi},a_{zi})+\sqrt{\frac{1}{\kappa+1}}\,h(a_{xi},a_{zi}),$
(23)

where constant $\kappa$ denotes the Rician $K$-factor and the LoS channel is
given in (6). The NLoS channel vector
$\boldsymbol{h}=[h(a_{x1},a_{z1}),\dots,h(a_{x(N+1)^{2}},a_{z(N+1)^{2}})]^{T}\in\mathbb{C}^{{(N+1)^{2}}\times
1}$ is a realisation of the circularly invariant complex Gaussian distribution
$\boldsymbol{h}\sim\mathcal{N}_{\mathbb{C}}\left(\boldsymbol{0},\,\frac{{c}}{\tilde{D}^{2}}A\boldsymbol{R}\right)$,
where $\tilde{D}$ is the average distance between transmitter and receiver
[44]. $A$ is the area of a transmit antenna and the entries of the spatial
correlation matrix
$\boldsymbol{R}\in\mathbb{C}^{{(N+1)^{2}}\times{(N+1)^{2}}}$ are given by

$\displaystyle
R_{ij}=\mathrm{sinc}\left(\frac{2}{\lambda}\sqrt{(a_{xi}-a_{xj})^{2}+(a_{zi}-a_{zj})^{2}}\right),$
$\displaystyle\forall i,j=1,\dots,(N+1)^{2},$ (24)

with $\mathrm{sinc}(x)=\mathrm{sin}(\pi x)/(\pi x)$ [44]. The propagation
environment is no longer deterministic in case of small-scale fading. For the
simulations, the coefficient of variation (CoV) is provided to indicate the
variability of the simulation values in relation to the mean. This
dimensionless measure is comparable across the simulation results of different
environments. For the simulations, we consider $\kappa=10$, and
$A=(\lambda/2)^{2}$.

####  5.3.1 Discreteness of the Optimal Solution

We investigate the sparsity of the transmit antenna deployment at LoD $N+1=81$
by calculating the number of antennas relative to the number of available grid
cells $(N+1)^{2}$. The optimal distribution of transmit power for an
individual channel realisation is determined by solving (22), whereby we
utilise the squared magnitude of the Rician channel (23), after normalisation
by the reference channel gain $c$, instead of the normalised squared magnitude
of the LoS channel, i.e., $f_{xz}(a_{xi},a_{zi})$, in the constraints (22b).
This yields the optimal number of transmit antennas for an individual channel
realisation. Subsequently, we average the optimal number of transmit antennas
over the different channel realisations. The results are given in Table 2 and
we list the percentage of non-zeros of the proposed deployment developed under
LoS conditions for comparison.

Table 2:  Average number of transmit antennas relative to number of grid cells under fading conditions compared with the percentage of non-zeros in the proposed distribution developed under LoS conditions at LoD $N+1=81$. $\frac{\mathrm{Height}}{\mathrm{Width}}$ | LoS [%] | Rician Mean [%] | Rician CoV [%]
---|---|---|---
1/1 | 0.152 | 0.155 | 77.10
1/3 | 0.183 | 0.65 | 16.81
1/4 | 0.198 | 1.14 | 10.87
1/5 | 0.61 | 1.751 | 10.45

Table 2 reveals that the distribution of power, when optimised under fading
conditions, is less sparse across all environments compared to the
distribution of power under perfect LoS conditions, i.e., a larger number of
transmit antennas is required. Moreover, the CoV is similar across different
environments, except for the environment with a height-to-width ratio of one.
This system is especially sensitive to channel fluctuations since under LoS
conditions, the optimal deployment requires only a single transmit antenna.

####  5.3.2 Performance Comparison

Next, the performance of the proposed optimal deployment and power allocation
is investigated in the small-scale fading environment. For each channel
realisation, we determine the minimum received power which is achieved when
utilising the proposed deployment and power allocation, which was optimised
for LoS conditions. As a benchmark, we optimise the deployment and power
allocation for each channel realisation and then determine the received power
at the critical receiver positions. We average the minimum received power of
the optimal LoS solution and the benchmark scheme across the channel
realisations, respectively. The loss in performance due to small-scale fading
is measured by the ratio of the averaged objective value of the proposed
scheme developed under LoS conditions divided by the averaged objective value
of the benchmark scheme. Simulations are performed for environments with
different height-to-width ratios and the results are displayed in Table 3.

Table 3: Performance loss due to small-scale fading. $\frac{\mathrm{Height}}{\mathrm{Width}}$ |  Average Relative Performance [%] | CoV [%]
---|---|---
1/1 | 98.752 | 0.529
1/3 | 98.77 | 0.153
1/4 | 98.79 | 0.126
1/5 | 98.8 | 0.131

The performance loss caused by small-scale fading was found to be below 2% for
all considered environments.

##  6 CONCLUSION

In this paper, we formulated an optimisation problem for determining the
optimal distribution of transmit power across a continuous aperture antenna.
The proposed solution maximises the received powers at the worst possible
receiver positions in an indoor space. This system is especially relevant for
environments with a multitude of mobile low-power devices, which require a
reliable supply of power, such as wearables. In this scenario, focusing energy
beams to every mobile device may not be possible in a reliable manner.
Mathematically, the formulated problem bears a resemblance to that of finding
the optimal amplitude-constrained input distribution for achieving the
information capacity. Consequently, analogous theoretical results are
obtained, which guarantee that the optimal transmit antenna positions are
spread out within their area of deployment such that they do not form any
regions of concentration and do not occupy any measurable area.  Specifically,
a transmit antenna architecture with a spatially continuous aperture, which is
utilised in HMIMO systems, was proven to be unnecessary for the considered WPT
problem. Furthermore, the optimal number of transmit antennas is shown to be
finite analytically for a one-dimensional transmit antenna array. Having
excluded a continuous aperture antenna as a suitable transmit antenna
architecture, the problem was discretised and the resulting solution yielded
the optimal transmit antenna deployment including the optimal allocation of
transmit power. In the environments investigated through simulation, deploying
a finite number of antennas was found to be optimal in all cases. Based on our
results, the optimal deployment may be achieved by a finite, and surprisingly
small, number of transmit antennas radiating a fixed amount of power.
Moreover, our proposed method yields a considerable gain compared to benchmark
schemes. Finally, the robustness of the approach has been validated by
determining the optimal solution in a fading environment with a strong LoS
component. The loss in performance under fading conditions was found to be
around 2% in all considered environments.

Possible extensions of the proposed approach offer several opportunities for
future work. These include the extension of the method to more geometrically
challenging environments and more flexible transmit antenna architecture
placements, such as positioning transmitters on multiple walls as well as the
ceiling of a room. Furthermore, a small-scale multipath fading component for
modelling the wireless channel may be directly incorporated into the analysis
of the problem to improve the robustness of the solution. Moreover, an
investigation of the impact of the reactive near-field region when the
deployment of large continuous antenna arrays is affordable presents another
interesting topic for future work.  While we characterise the solution to our
problem through tools from topology theory and functional analysis,
determining the optimal distribution of power and the transmit antenna
deployment through different mathematical techniques may provide additional
insights. Furthermore, a possible extension to our problem is the joint
optimisation of the amplitudes and the phases of the transmit signals of the
individual radiating elements.

## 7

By definition, any $P\in\Omega$ is non-negative and bounded by one (see (8)).
Observe that given any two distributions $P_{1},P_{2}\in\Omega$, their convex
combination $\lambda P_{1}+(1-\lambda)P_{2}\in\Omega$ with coefficient
$\lambda\in[0,1]$ is a distribution as well. It follows that $\Omega$ is a
convex set [35].

In order to further characterise the set $\Omega$, the weak topology on a
Hilbert space is defined. A linear functional in $\Omega$ is a linear map from
the Hilbert space to the underlying field. The weak topology on a Hilbert
space is defined to be the weakest topology such that all continuous linear
functionals on the Hilbert space are continuous [47].

Observe that $\mathcal{L}^{2}(\mathcal{X}_{a}\times\mathcal{Z}_{a})$ is an
infinite-dimensional vector space equipped with the following inner product
[48]

$\displaystyle\iint_{\mathcal{X}_{a}\times\mathcal{Z}_{a}}w(a_{x},a_{z})\,v(a_{x},a_{z})\,\mathrm{d}a_{x}\,\mathrm{d}a_{z},$
$\displaystyle\forall
w(a_{x},a_{z}),v(a_{x},a_{z})\in\mathcal{L}^{2}(\mathcal{X}_{a}\times\mathcal{Z}_{a}),$
(25)

where $w(a_{x},a_{z})$ and $v(a_{x},a_{z})$ are arbitrary elements of
$\mathcal{L}^{2}(\mathcal{X}_{a}\times\mathcal{Z}_{a})$. This inner product
space is complete since completeness is ensured by Lebesgue measurability and
Lebesgue integration [48]. Thus,
$\mathcal{L}^{2}(\mathcal{X}_{a}\times\mathcal{Z}_{a})$ is a Hilbert space,
which is separable since it has a countable orthonormal basis [48]. Note that
the set $\Omega$ is a subset of
$\mathcal{L}^{2}(\mathcal{X}_{a}\times\mathcal{Z}_{a})$ with bounded elements
in $\Omega$ (see (8)). In a separable Hilbert space, every bounded sequence
has a weakly convergent subsequence in the weak topology [47] and therefore,
$\Omega$ is sequentially compact [49]. The concepts of sequential compactness
and compactness are equivalent for a separable Hilbert space [47]. Thus,
$\Omega$ is a compact space.

## 8

First, Problem (14) is rewritten in its epigraph form [35] which yields the
following constrained optimisation problem

$\displaystyle\underset{o\in\mathbb{R},P\in\Omega}{\text{maximise}}$
$\displaystyle o$ (26a) subject to $\displaystyle
o\leq\Phi_{P}\\{f_{xz}\\},\quad\forall(x,L_{y},z)\in\mathcal{X}\times\mathcal{Y_{\mathrm{crit}}}\times\mathcal{Z},$
(26b)

where $o$ is bounded by $m=\underset{\begin{subarray}{c}(x,L_{y},z)\\\
\in\mathcal{X}\times\mathcal{Y_{\mathrm{crit}}}\times\mathcal{Z}\end{subarray}}{\min}\,\Phi_{P^{*}}\\{f_{xz}\\}$.
Observe that the objective function and the feasible set of Problem (26) are
both affine. Then, if the linear programming problem in (26) admits a
solution, i.e., if $o^{*}=m$ is finite, duality holds. Furthermore, there
exist some $\lambda_{xz}\geq
0,\forall(x,L_{y},z)\in\mathcal{X}\times\mathcal{Y_{\mathrm{crit}}}\times\mathcal{Z}$
such that the augmented problem

$\displaystyle\underset{o\in\mathbb{R},P\in\Omega,\Lambda}{\text{maximise}}\quad\\!o-\sum_{\mathcal{X}\times\mathcal{Y_{\mathrm{crit}}}\times\mathcal{Z}}\left(o-\Phi_{P}\\{f_{xz}\\}\right)\lambda_{xz},$
(27)

and Problem (26) are equivalent if the solution $(o^{*},P^{*})$ is feasible,
i.e., the problem constraints are satisfied in $(o^{*},P^{*})$, and the
following complementary slackness conditions hold:

$\displaystyle\lambda_{xz}\left(m-\Phi_{P^{*}}\\{f_{xz}\\}\right)$
$\displaystyle\overset{!}{=}0,\quad\forall(x,L_{y},z)\in\mathcal{X}\times\mathcal{Y_{\mathrm{crit}}}\times\mathcal{Z}.$
(28)

Since $f_{xz}$ is bounded, the optimal objective value $o^{*}=m$ is finite and
it is attained in $P^{*}$. The conditions in (28) imply

$\displaystyle[$ $\displaystyle(\lambda_{xz}>0\implies
m=\Phi_{P^{*}}\\{f_{xz}\\})\,\,\,\mathrm{or}\,\,\,$
$\displaystyle(m<\Phi_{P^{*}}\\{f_{xz}\\}\implies\lambda_{xz}=0)],\,$
$\displaystyle\forall(x,L_{y},z)\in\mathcal{X}\times\mathcal{Y_{\mathrm{crit}}}\times\mathcal{Z}$
(29)

From (8), if $\lambda_{xz}>0$, then $o^{*}=m=\Phi_{P^{*}}\\{f_{xz}\\}$ and
position $(x,L_{y},z)$ is critical. If $o^{*}=m<\Phi_{P^{*}}\\{f_{xz}\\}$,
then $\lambda_{xz}=0$ and position $(x,L_{y},z)$ is non-critical. Thus, the
dual variables $\lambda_{xz}$ are only non-zero for critical receiver
locations and it is sufficient to sum over $\mathcal{R}_{\mathrm{crit}}$ in
(27). Thus, under the assumption that $\mathcal{R}_{\mathrm{crit}}$ is known,
the modified augmented problem in (16) along with (8) is equivalent to the
augmented problem in (27) together with (8) and thus, to Problem (14).

## 9

The continuity of $\Phi_{P}\\{f_{xz}\\}$ in $P$ is established in the
following lemma.

######  Lemma 9.1.

$\Phi_{P}\\{f_{xz}\\}$ is weakly continuous in the weak topology on a Hilbert
space if $f_{xz}$ is a continuous and bounded function.

###### Proof.

Based on Theorem 1 in [33], the linear functional $\Phi_{P}\\{f_{xz}\\}$ is
weakly continuous on $\Omega$ if $f_{xz}$ is a continuous and bounded
function. From Key Observation 1, $f_{xz}$ is a continuous function and
bounded since $y=L_{y}$. ∎

Recall the mathematical equivalence between the WPT problem considered in this
paper, where we optimise the transmit power distribution across a continuous
aperture antenna and the problem of determining the amplitude-constrained
capacity-achieving input distribution of a channel studied in, e.g., [31, 32,
33]. For the proof of Proposition 1 we apply Theorem 9 in [33]. The convexity
and compactness of $\Omega$ is established in Lemma 1. The continuity of
$J(P)$ is shown in Lemma 9.1. The concavity of $J(P)$ follows from the
linearity of $J(P)$ in $P$. Formulating the optimality condition for $P^{*}$
requires defining the Gâteaux derivative of $J(P)$.

######  Definition 9.1.

The Gâteaux derivative of the function $J:\Omega\rightarrow\mathbb{R}$ at $P$
in the direction of $Q$ with $P,Q\in\Omega$ is defined as follows [33]

$\displaystyle\Delta_{Q}J(P)=\underset{\theta\rightarrow
0}{\mathrm{lim}}\,\frac{J((1-\theta)P+\theta Q)-J(P)}{\theta}.$ (30)

Moreover, the Gâteaux derivative of $J(P)$ in (30) can be expressed as [33]

$\displaystyle\Delta_{Q}J(P)=J(Q)-J(P),$ (31)

since it is a linear functional of $P$. Theorem 9 in [33] establishes that if
$\Omega$ is a convex set and $J(P)$ is continuous, concave, and Gâteaux
differentiable, then $J(P)$ attains the maximum value at $P^{*}\in\Omega$ if
and only if $\Delta_{P}J(P^{*})\leq 0,\forall P\in\Omega$. Applying (31) and
cancelling the terms that are independent of the distributions $P$ and
$P^{*}$, we obtain

$\displaystyle\Delta_{P}J(P^{*})=\sum_{\mathcal{R}^{\mathrm{crit}}}\,\lambda_{xz}\Phi_{P}\\{f_{xz}\\}-\sum_{\mathcal{R}^{\mathrm{crit}}}\,\lambda_{xz}\Phi_{P^{*}}\\{f_{xz}\\}.$
(32)

We can further enforce in (32) the constraints on $\lambda_{xz}$ and
$\Phi_{P}\\{f_{xz}\\}$ for the optimal solution $P^{*}$ in (2). Then,
$\lambda_{xz}$ is only non-zero for critical receiver locations w.r.t. the
optimal distribution $P^{*}$. Furthermore, the objective value for the
critical receiver positions is the constant $\Phi_{P^{*}}\\{f_{xz}\\}=m$. By
expanding the expectation operator in (32), the optimality condition
$\Delta_{P}J(P^{*})\leq 0$ can be expressed as

$\displaystyle\iint_{\mathcal{X}_{a}\times\mathcal{Z}_{a}}\,\frac{\sum_{\mathcal{R}^{\mathrm{crit}}}\,f_{xz}(a_{x},a_{z})\lambda_{xz}}{\sum_{\mathcal{R}^{\mathrm{crit}}}\,\lambda_{xz}}\,\,\mathrm{d}P(a_{x},a_{z})$
$\displaystyle=$
$\displaystyle\iint_{\mathcal{X}_{a}\times\mathcal{Z}_{a}}\,\bar{f}(a_{x},a_{z})\,\,\mathrm{d}P(a_{x},a_{z})\leq
m,$ (33)

which is satisfied with equality for the optimal distribution of power
$P^{*}$.

## 10

If the conditions (20) and (21) are satisfied, then conditions (18) and (19)
in Proposition 1 are satisfied. In the following, we show that the converse
holds as well, i.e., if Proposition 1 holds, then Corollary 1 holds as well.
We proceed by contradiction and assume that Proposition 1 holds, however, (20)
does not hold. In this case,
$\exists(a_{x1},a_{z1})\in\mathcal{X}_{a}\times\mathcal{Z}_{a}$, such that
$\bar{f}(a_{x1},a_{z1})>m$. Suppose, $(a_{x1},a_{z1})$ is the only transmit
antenna, i.e., $(a_{x1},a_{z1})$ is allocated all the power. Then (18) yields

$\displaystyle\iint_{\mathcal{X}_{a}\times\mathcal{Z}_{a}}\,\bar{f}(a_{x},a_{z})\,\,\mathrm{d}P(a_{x},a_{z})=\bar{f}(a_{x1},a_{z1})>m,$
(34)

which contradicts (18) in Proposition 1. Consequently, if (18) holds, then
(20) holds as well.
Next, suppose Proposition 1 holds, however, (21) does not. Define a subset
$\mathcal{E}_{0}(P^{*})\subset\mathcal{E}(P^{*})$ such that
$P^{*}(\mathcal{E}_{0}(P^{*}))=0$. Observe that $\bar{f}(a_{x},a_{z})$ is
continuous on $\mathbb{R}^{2}$ since the function’s numerator is a weighted
sum of continuous functions and the denominator is always a positive constant.
Assume there exists a point of increase
$(a_{x1},a_{z1})\in\mathcal{E}(P^{*})\setminus\mathcal{E}_{0}(P^{*})$ such
that $\bar{f}(a_{x1},a_{z1})>m$. For any $\delta>0$ such that
$\bar{f}(a_{x1},a_{z1})-\delta>m$, there exists an open ball $\mathcal{B}$
centred at $(a_{x1},a_{z1})$ such that
$\bar{f}(a_{x},a_{z})>\bar{f}(a_{x1},a_{z1})-\delta>m,\forall(a_{x},a_{z})\in\mathcal{B}$,
which follows from the continuity of $\bar{f}(a_{x},a_{z})$. Since the open
ball $\mathcal{B}$ contains the point of increase
$(a_{x1},a_{z1})\in\mathcal{E}(P^{*})\setminus\mathcal{E}_{0}(P^{*})$, it
follows that $P^{*}(\mathcal{B})>0$. This contradicts Proposition 1, i.e.,
$m=\iint_{\mathcal{X}_{a}\times\mathcal{Z}_{a}}\,\bar{f}(a_{x},a_{z})\,\,\mathrm{d}P^{*}(a_{x},a_{z})$,
since

$\displaystyle m=$
$\displaystyle\underbrace{\iint_{(\mathcal{E}(P^{*})\setminus\mathcal{E}_{0}(P^{*}))\cap\mathcal{B}}\,\bar{f}(a_{x},a_{z})\,\,\mathrm{d}P^{*}(a_{x},a_{z})}_{>\,mP^{*}(\mathcal{B})}$
$\displaystyle+\underbrace{\iint_{(\mathcal{E}(P^{*})\setminus\mathcal{E}_{0}(P^{*}))\setminus\mathcal{B}}\,\bar{f}(a_{x},a_{z})\,\,\mathrm{d}P^{*}(a_{x},a_{z})}_{=\,m(1-P^{*}(\mathcal{B}))}$
$\displaystyle+\underbrace{\iint_{\mathcal{E}_{0}(P^{*})}\,\bar{f}(a_{x},a_{z})\,\,\mathrm{d}P^{*}(a_{x},a_{z})}_{=0}>m.$
(35)

From this contradiction we have shown that
$\bar{f}(a_{x},a_{z})=m,\forall(a_{x},a_{z})\in\mathcal{E}(P^{*})\setminus\mathcal{E}_{0}(P^{*})$.
Furthermore, since $P^{*}(\mathcal{E}_{0}(P^{*}))=0$, $\mathcal{E}_{0}(P^{*})$
has no interior points. Any open set of $\mathbb{R}^{2}$ containing a point in
$\mathcal{E}_{0}(P^{*})$ intersects points in
$\mathcal{E}(P^{*})\setminus\mathcal{E}_{0}(P^{*})$ and therefore, every point
in $\mathcal{E}_{0}(P^{*})$ is in the closure of
$\mathcal{E}(P^{*})\setminus\mathcal{E}_{0}(P^{*})$. Moreover, since
$\bar{f}(a_{x},a_{z})$ is continuous and bounded, it must also be constant on
the closure of $\mathcal{E}(P^{*})\setminus\mathcal{E}_{0}(P^{*})$. This
implies $\bar{f}(a_{x},a_{z})=m,\forall(a_{x},a_{z})\in\mathcal{E}(P^{*})$.
Consequently, if (18) holds, then (21) holds as well. Thus, (20) and (21) are
necessary and sufficient conditions for $P^{*}(a_{x},a_{z})$.

## 11

The proof is similar to Theorem 12 in [33].
The characterisation of the support $\mathcal{E}$ requires the Identity
Theorem for real-analytic functions, which is stated, e.g., in [50]. The
Identity Theorem states that if two real-analytic functions $f_{1}$ and
$f_{2}$, which are defined on an open set $U\subseteq\mathbb{R}^{n}$, take the
same value on an open subset $W\subseteq U$, i.e., $f_{1}(w)=f_{2}(w),\forall
w\in W$, then $f_{1}(u)=f_{2}(u),\forall u\in U$.

Next, we show that $\bar{f}(a_{x},a_{z})$ is not constant in
$\mathcal{X}_{a}\times\mathcal{Z}_{a}$. Thereby, it suffices to show that the
partial derivative of $\bar{f}(a_{x},a_{z})$ w.r.t. $a_{x}$, with $a_{z}=0$,
is not zero $\forall(a_{x},a_{z})\in\mathcal{X}_{a}\times\mathcal{Z}_{a}$.
Note that

$\displaystyle\frac{\partial\bar{f}(a_{x},a_{z})}{\partial
a_{x}}=\frac{\sum_{\mathcal{R}_{\mathrm{crit}}}\,\dfrac{2\left(x-a_{x}\right)}{\left(\left(z-a_{z}\right)^{2}+\left(x-a_{x}\right)^{2}+L_{y}^{2}\right)^{2}}\lambda_{xz}}{\sum_{\mathcal{R}_{\mathrm{crit}}}\,\lambda_{xz}},$
(36)

is only zero for $x=a_{x}$ and non-zero otherwise. Consequently,
$\bar{f}(a_{x},a_{z})$ is not constant in
$\mathcal{X}_{a}\times\mathcal{Z}_{a}$.

Next, we show that $\bar{f}(a_{x},a_{z})$ is real-analytic in $\mathbb{R}^{2}$
and thus, in $\mathcal{X}_{a}\times\mathcal{Z}_{a}$. Let
$\bar{f}(a_{x},a_{z})=kn(a_{x},a_{z})$, where
$k=\left(\sum_{\mathcal{R}_{\mathrm{crit}}}\,\lambda_{xz}\right)^{-1}$ is a
positive constant for all $(a_{x},a_{z})$ and
$n(a_{x},a_{z})=\sum_{\mathcal{R}_{\mathrm{crit}}}\,f_{xz}(a_{x},a_{z})\lambda_{xz}$.
The function $\bar{f}(a_{x},a_{z})$ is real-analytic in $(a_{x},a_{z})$ if
$n(a_{x},a_{z})$ is an analytic function. The function $n(a_{x},a_{z})$ is
real-analytic in $(a_{x},a_{z})$ if $f_{xz}(a_{x},a_{z})\lambda_{xz}$ is real-
analytic for fixed $(x,z)$. Thereby, it suffices to show that
$f_{xz}(a_{x},a_{z})$ is real-analytic since $\lambda_{xz}$ is some non-
negative constant scaling $f_{xz}(a_{x},a_{z})$.

Note that the composition of real-analytic functions is real-analytic [50]. By
rewriting $f_{xz}(a_{x},a_{z})$ as
$f(w_{xz}(a_{x},a_{z}))=1/(L_{y}^{2}+w_{xz}(a_{x},a_{z}))$, where
$w_{xz}(a_{x},a_{z})=(x-a_{x})^{2}+(z-a_{z})^{2}\geq 0$, it is sufficient to
show that both $w_{xz}(a_{x},a_{z})$ and $f(w)=1/({w+L_{y}^{2}})>0$ are real-
analytic. By definition, polynomials are real-analytic functions [50], thus
$w_{xz}(a_{x},a_{z})$ is real-analytic. Next, the real function $f(w)$ is
shown to be analytic by proving that it’s complex extension
$f(r_{1}+\mathrm{j}r_{2})=u(r_{1},r_{2})+\mathrm{j}v(r_{1},r_{2})=(r_{1}+L_{y}^{2})/((r_{1}+L_{y})^{2}+r_{2}^{2})-\mathrm{j}r_{2}/((r_{1}+L_{y})^{2}+r_{2}^{2})$
is holomorphic. Function $f(w)$ is holomorphic and thus, analytic, if it
satisfies the Cauchy-Riemann (CR) equations [51]. The CR equations are given
by [51]

$\displaystyle\frac{\partial u(r_{1},r_{2})}{\partial r_{1}}=\frac{\partial
v(r_{1},r_{2})}{\partial r_{2}},$ (37) $\displaystyle\frac{\partial
u(r_{1},r_{2})}{\partial r_{2}}=-\frac{\partial v(r_{1},r_{2})}{\partial
r_{1}},$ (38)

Clearly, the CR equations are satisfied for the complex extension
$f(r_{1}+\mathrm{j}r_{2})$ since

$\displaystyle\frac{\partial u(r_{1},r_{2})}{\partial r_{1}}=\frac{\partial
v(r_{1},r_{2})}{\partial
r_{2}}=\frac{r_{2}^{2}-(L_{y}^{2}+r_{1})^{2}}{((L_{y}^{2}+r_{1})^{2}+r_{2}^{2})^{2}},$
(39) $\displaystyle\frac{\partial u(r_{1},r_{2})}{\partial
r_{2}}=-\frac{\partial v(r_{1},r_{2})}{\partial
r_{1}}=\frac{-2r_{2}(L_{y}^{2}+r_{1})}{((L_{y}^{2}+r_{1})^{2}+r_{2}^{2})^{2}},$
(40)

which proves that the complex extension $f(r_{1}+\mathrm{j}r_{2})$ is
analytic. Consequently, $\bar{f}(a_{x},a_{z})$ is real-analytic in
$\forall(a_{x},a_{z})\in\mathbb{R}^{2}$ and thus, in
$\forall(a_{x},a_{z})\in\mathcal{X}_{a}\times\mathcal{Z}_{a}$.

Suppose that $\mathcal{E}(P^{*})$ is not a nowhere dense set in
$\mathcal{X}_{a}\times\mathcal{Z}_{a}$. Consequently, there exists some non-
empty open set $\mathcal{E}_{2}\subset\mathcal{X}_{a}\times\mathcal{Z}_{a}$
such that $\mathcal{E}_{2}\cap\mathcal{E}(P^{*})$ is dense in
$\mathcal{X}_{a}\times\mathcal{Z}_{a}$. From (21), $\bar{f}(a_{x},a_{z})=m$ is
constant on $\mathcal{E}(P^{*})$ and thus, constant on
$\mathcal{E}_{2}\cap\mathcal{E}(P^{*})$. Since $\bar{f}(a_{x},a_{z})$ is real-
analytic, the Identity Theorem of real-analytic functions applies and thus,
$\bar{f}(a_{x},a_{z})$ must be constant on
$\mathcal{X}_{a}\times\mathcal{Z}_{a}$. However,
$\bar{f}(a_{x},a_{z})\neq\text{const.},\forall(a_{x},a_{z})\in\mathcal{X}_{a}\times\mathcal{Z}_{a}$.
From this contradiction,
$\mathcal{E}(P^{*})\subset\mathcal{X}_{a}\times\mathcal{Z}_{a}$ must be a
nowhere dense set of $\mathcal{X}_{a}\times\mathcal{Z}_{a}$.

## 12

Suppose $\mathcal{E}(P^{*})$ is of positive Lebesgue measure. For any open
subset $\mathcal{V}\subset\mathcal{E}(P^{*})$, the function
$\bar{f}(a_{x},a_{z})$ is constant on $\mathcal{V}$ since it is constant on
$\mathcal{E}(P^{*})$. Since $\bar{f}(a_{x},a_{z})$ is real-analytic as shown
in Appendix 11, the Identity Theorem, e.g., [50] of real-analytic functions
applies and thus, $\bar{f}(a_{x},a_{z})$ must be constant on
$\mathcal{X}_{a}\times\mathcal{Z}_{a}$. However, as shown in Appendix 11,
$\bar{f}(a_{x},a_{z})\neq\text{const.},\forall(a_{x},a_{z})\in\mathcal{X}_{a}\times\mathcal{Z}_{a}$.
From this contradiction,
$\mathcal{E}(P^{*})\subset\mathcal{X}_{a}\times\mathcal{Z}_{a}$ must have
Lebesgue measure zero.

## 13

The proof is also analogous to [31, 32].
This proof requires the application of the Bolzano-Weierstrass theorem which
states that in the set of real numbers any infinite bounded sequence of real
numbers has a convergent subsequence and the limit of that subsequence is an
accumulation point of the original sequence [52]. In the following
$\bar{f}(a_{l})$ is a function defined over a one-dimensional set, i.e.,
$\bar{f}:\mathcal{L}_{a}\rightarrow\mathbb{R}$, with
$\mathcal{L}_{a}\subset\mathcal{X}_{a}\times\mathcal{Z}_{a}$ and
$\tilde{\mathcal{E}}(\tilde{P}^{*})$ is the optimal set of transmit antenna
positions, which is bounded, i.e.,
$\tilde{\mathcal{E}}(\tilde{P}^{*})\subset\mathcal{L}_{a}$. Suppose the set
$\tilde{\mathcal{E}}(\tilde{P}^{*})$ is not finite. From the Bolzano-
Weierstrass theorem, it follows that $\tilde{\mathcal{E}}(\tilde{P}^{*})$ has
an accumulation point $q$. Based on (21), define the function
$s(a_{l})=m-\bar{f}(a_{l})$, which is equal to zero $\forall
a_{l}\in\tilde{\mathcal{E}}$. Moreover, $s(a_{l})$ is analytic since it is the
sum of analytic functions [50]. By assuming
$\tilde{\mathcal{E}}(\tilde{P}^{*})$ is not finite, $s$ has infinitely many
zeros. Moreover, the accumulation point $q$ is also a zero of function
$s(a_{l})$, i.e., $s(q)=0$, since $s(a_{l})$ is a continuous function, which
is implied by the analycity of $s(a_{l})$. Then, from the Identity Theorem,
e.g. [50], $s(a_{l})$ coincides with zero in the entire set $\mathcal{L}_{a}$
or equivalently

$\displaystyle\bar{f}(a_{l})=m,\forall a_{l}\in\mathcal{L}_{a},$ (41)

which implies that $\bar{f}$ is constant on $\mathcal{L}_{a}$. However,
$\bar{f}$ is not constant on $\mathcal{L}_{a}$ as shown in Appendix 11.
Consequently, $\tilde{\mathcal{E}}(\tilde{P}^{*})$ must be a finite set.

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[]KENNETH M. MAYER (Graduate Student Member, IEEE) received the B.Sc. degree
in Medical Engineering (2019) and the M.Sc. degree in Advanced Signal
Processing and Communications Engineering (2021) from Friedrich-Alexander
University (FAU) Erlangen-Nuremberg, Germany. He is currently pursuing the
Ph.D. degree at the Institute for Digital Communications at FAU focusing on
wireless communications and wireless power transfer. []LAURA COTTATELLUCCI
(Member, IEEE) received the M.Sc. degree in electrical engineering from La
Sapienza University of Rome, Italy, the Ph.D. degree from Technical University
of Vienna, Austria (2006), and the Habilitation degree from University of
Nice-Sophia Antipolis, France. Since December 2017 and September 2021, she has
been a professor for digital communications at Friedrich Alexander Universität
(FAU) Erlangen-Nuremberg, Germany, and an Adjunct Professor at EURECOM,
France, respectively. She worked for 5 years (1995-2000) at Telecom Italia in
the design of telecommunications networks and as Senior Researcher at
Forschungszentrum Telekommunikation Wien, Austria (2000-2005). She was a
Research Fellow at INRIA, France, in the quarter October-December 2005, and at
the University of South Australia in 2006. From December 2006 to November
2017, she was an Assistant Professor at EURECOM, France, where she was also an
Adjunct Professor (2018-2019). She was an Elected Member of the IEEE Technical
Committee on Signal Processing for Communications and Networking (2017-22),
served as an Associate Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS
(2015-2020) and the IEEE TRANSACTIONS ON SIGNAL PROCESSING (2016- 2020). Since
September 2022 she has been senior member of the editorial board of IEEE
Signal Processing Magazine. Her research interests are in the field of
communication and information theory and signal processing for wireless
communications, satellite, and complex networks.

[]ROBERT SCHOBER (Fellow, IEEE) received the Diplom (Univ.) and the Ph.D.
degrees in electrical engineering from Friedrich-Alexander University of
Erlangen-Nuremberg (FAU), Germany, in 1997 and 2000, respectively. From 2002
to 2011, he was a Professor and Canada Research Chair at the University of
British Columbia (UBC), Vancouver, Canada. Since January 2012 he is an
Alexander von Humboldt Professor and the Chair for Digital Communication at
FAU. His research interests fall into the broad areas of Communication Theory,
Wireless and Molecular Communications, and Statistical Signal Processing.

Robert received several awards for his work including the 2002 Heinz Maier
Leibnitz Award of the German Science Foundation (DFG), the 2004 Innovations
Award of the Vodafone Foundation for Research in Mobile Communications, a 2006
UBC Killam Research Prize, a 2007 Wilhelm Friedrich Bessel Research Award of
the Alexander von Humboldt Foundation, the 2008 Charles McDowell Award for
Excellence in Research from UBC, a 2011 Alexander von Humboldt Professorship,
a 2012 NSERC E.W.R. Stacie Fellowship, a 2017 Wireless Communications
Recognition Award by the IEEE Wireless Communications Technical Committee, and
the 2022 IEEE Vehicular Technology Society Stuart F. Meyer Memorial Award.
Furthermore, he received numerous Best Paper Awards for his work including the
2022 ComSoc Stephen O. Rice Prize and the 2023 ComSoc Leonard G. Abraham
Prize. Since 2017, he has been listed as a Highly Cited Researcher by the Web
of Science. Robert is a Fellow of the Canadian Academy of Engineering, a
Fellow of the Engineering Institute of Canada, and a Member of the German
National Academy of Science and Engineering.

He served as Editor-in-Chief of the IEEE TRANSACTIONS ON COMMUNICATIONS, VP
Publications of the IEEE Communication Society (ComSoc), ComSoc Member at
Large, and ComSoc Treasurer. Currently, he serves as Senior Editor of the
PROCEEDINGS OF THE IEEE and as ComSoc President.