# Communication and Localization with Extremely Large Lens Antenna Array

Jie Yang, Yong Zeng, Shi Jin, Chao-Kai Wen, Pingping Xu Jie Yang, Yong Zeng,
Shi Jin, and Pingping Xu are with the National Mobile Communications Research
Laboratory, Southeast University, Nanjing, China (e-mail:
{yangjie;yong_zeng;jinshi;xpp}@seu.edu.cn). Chao-Kai Wen is with the Institute
of Communications Engineering, National Sun Yat-sen University, Kaohsiung,
804, Taiwan (e-mail: chaokai.wen@mail.nsysu.edu.tw).

###### Abstract

Achieving high-rate communication with accurate localization and wireless
environment sensing has emerged as an important trend of beyond-fifth and
sixth generation cellular systems. Extension of the antenna array to an
extremely large scale is a potential technology for achieving such goals.
However, the super massive operating antennas significantly increases the
computational complexity of the system. Motivated by the inherent advantages
of lens antenna arrays in reducing system complexity, we consider
communication and localization problems with an extremely large lens antenna
array, which we call “ExLens”. Since radiative near-field property emerges in
the setting, we derive the closed-form array response of the lens antenna
array with spherical wave, which includes the array response obtained on the
basis of uniform plane wave as a special case. Our derivation result reveals a
window effect for energy focusing property of ExLens, which indicates that
ExLens has great potential in position sensing and multi-user communication.
We also propose an effective method for location and channel parameters
estimation, which is able to achieve the localization performance close to the
Cramér-Rao lower bound. Finally, we examine the multi-user communication
performance of ExLens that serves coexisting near-field and far-field users.
Numerical results demonstrate the effectiveness of the proposed channel
estimation method and show that ExLens with a minimum mean square error
receiver achieves significant spectral efficiency gains and complexity-and-
cost reductions compared with a uniform linear array.

###### Index Terms:

Array response, extremely large lens antenna array, localization, millimeter-
wave communications, spherical wave-front.

##  I Introduction

In comparison with previous generations, the fifth generation (5G) mobile
network is a major breakthrough because of the introduce of massive multiple-
input multiple-output (MIMO), millimeter-wave (mmWave), and ultra-dense
network [1, 2]. However, realizing the full vision of supporting Internet of
Everything services to connect billions of people and machines remains a
challenge for 5G. Thus, research communities worldwide have implemented
initiatives to conceive the next-generation (e.g., the sixth generation (6G))
mobile communication systems [3, 4, 5, 6, 7]. The requirement of various
applications, such as extended reality, autonomous systems, pervasive health
monitoring, and brain computer interactions, are driving the evolution of 6G
towards a more intelligent and software reconfigurable functionality paradigm
that can provide ubiquitous communications and also sense, control, and even
optimize wireless environments.

To fulfill the visions of 6G for high throughput, massive connectivity, ultra-
reliability, and ultra-low latency, on the one hand, mmWave and Tera-Hertz
(THz) frequencies will be exploited further, furthermore, multiple frequency
bands (e.g., microwave/mmWave/THz frequencies) must be integrated to provide
seamless connectivity [8]; on the other hand, the antenna deployment will
evolve towards larger apertures and greater numbers, furthermore, the
extremely large aperture array has been proposed to boost spatial diversity
further [9, 10, 11]. Moreover, intelligent reflecting surfaces or
reconfigurable intelligent surfaces, artificial intelligence, and integrated
terrestrial-aerial-satellite networks are regarded as promising technologies
towards 6G[12, 13, 14, 15, 16]. However, many open problems need to be solved
to reap the full benefits of the aforementioned techniques. In particular,
when the antenna dimension continues to increase, the range of the radiative
near-field of the antenna array expands, and the user equipment (UE) and
significant scatterers are likely to be located in the near-field of the
array. Consequently, the prominent uniform plane wave assumption will no
longer hold for extremely large antenna arrays [17]. Moreover, the use of
thousands or more active antenna elements will generate prohibitive cost in
terms of hardware implementation, energy consumption, and signal processing
complexity [18].

The radiation field of an antenna array is divided into the near-field region
and the far-field region via the Rayleigh distance [19, 20], which is given as
$R={2D^{2}}/{\lambda},$ where $D$ is the maximum dimension of the antenna
array, and $\lambda$ is the wavelength. When the distance between the UE (or
scatterer) and the base station (BS) is smaller than the Rayleigh distance,
the UE (or scatterer) is located in the near-field region, where the spherical
wave-front over the antenna array is observed. For example, a uniform linear
array (ULA) of 1 meter (m) that operates at $30$ GHz corresponds to a Rayleigh
distance of approximately $200\,m$ and nullifies the uniform plane wave-front
model usually assumed in prior research on wireless communications. Few works
have considered the near-field property for modeling and analyzing massive
MIMO channels by proposing the ULA response vector [21] and analyzing the
channel estimation performance [11] under the spherical wave assumption. The
spherical wave-front is also proven to provide an underlying generic
parametric model for estimating the position of UE and scatterers [22, 23].
Several works start to investigate the localization potential with large
advanced antenna arrays to realize the vision of multi-purpose services for 6G
(joint communication, control, localization, and sensing) [24, 25, 26, 27, 28,
29] in addition to communication capabilities. Concentrated and distributed
large antenna arrays are compared in [24] in terms of localization
performance. The theoretical uplink localization and synchronization
performance is analyzed in [25] for ULA illuminated by spherical waves.
Parameter-based localization methods are developed in [26, 27, 28] for lens
antenna arrays in the far-field, [29] considers direct localization by
utilizing the near-field property, and provides a coarse localization
accuracy.

An effective solution to significantly reduce the system complexity and
implementation cost caused by the large number of antennas and UE is to
partition the antenna array into a few disjoint subarrays [10, 30]. In this
work, we propose an alternative solution by using the energy focusing property
of an extremely large lens antenna array denoted as “ExLens”, which can fully
utilize the aperture offered by the large antenna arrays. Recent studies have
confirmed that the signal processing complexity and radio frequency (RF) chain
cost could be significantly reduced without notable performance degradation
for mmWave and massive MIMO systems by utilizing lens antenna arrays [31, 32,
33, 34, 35]. Electromagnetic (EM) lenses can provide variable phase shifting
for EM rays at different points on the lens aperture to achieve angle-
dependent energy focusing property. Therefore, lens antenna arrays can
transform the signal from the antenna space to the beamspace (the latter has
lower dimensions) to reduce the RF chains significantly. In [34, 35], the
array responses of lens antenna arrays have been derived in closed-form as a
“sinc” function of the angle of arrival (AOA)/angle of departure (AOD) of the
impinging/departure signals. However, existing research on the lens antenna
arrays are limited to the far-field assumption. To the best of the authors’
knowledge, the array response of an ExLens for the general spherical wave-
front has not been reported in prior works, let alone conducting a study on
multi-user communication with an ExLens in the coexistence of near-field and
far-field UE.

In this study, we explore the property of ExLens illuminated by spherical
waves, including the capabilities of localization and multi-user
communication, on the basis of the inherent localization information carried
by spherical waves and the great potentials of lens antenna arrays in reducing
system complexity. In summary, we derive a closed-form array response of an
ExLens, based on which we develop an effective method to obtain location
parameters together with channel gains. On the one hand, we can realize
localization with the estimated location parameters. On the other hand, we can
design data transmission with the reconstructed channel. Our main
contributions are presented as follows:

  * •

Array Response: We first derive the closed-form expression for the array
response of ExLens by considering the general spherical wave-front for two
different EM lens designs, and then reveal that the obtained array response
(derived based on the spherical wave assumption) includes the “sinc”- type
array response[34] (derived based on the uniform plane wave assumption) as a
special case. Next, we analyze differences of the energy focusing
characteristics of ExLens illuminated by the spherical and plane wave-fronts.
The window focusing property in the near-field of ExLens shows its great
potential for position sensing and multi-user communication. The approximation
error of the derived closed-form array response is verified ignorable.

  * •

Position Sensing: We analyze the uplink localization ability of an ExLens
equipped at the BS. We first study the theoretical localization performance
from a Fisher information perspective and confirm that the localization
performance improves as the aperture of the lens antenna array increases. By
exploring the energy focusing window of ExLens, we propose an effective
parameterized estimation method to obtain location parameters together with
channel gains. Thus, localization can be performed by directly reusing the
communication signals. Comprehensive simulations show that the localization
performance of the proposed method is close to the Cramér-Rao lower bound
(CRLB) and the channel can also be effectively reconstructed.

  * •

Multi-user Communication: We investigate the multi-user communication
performance of ExLens with coexisting near-field and far-field UE and
scatterers. Power-based antenna selection is applied to ExLens to reduce the
number of RF chains, together with the maximal ratio combining (MRC)- and
minimum mean square error (MMSE)-based combining schemes to maximize the sum-
rate. The multi-user communication performance of the ExLens with perfect and
estimated channel state information (CSIs) are compared. Simulation results
verify the effectiveness of the proposed channel estimation method and show
that the proposed ExLens with an MMSE receiver achieves significant spectral
efficiency gains and complexity-and-cost reductions compared with the
benchmark ULA schemes, when serving coexisting near-field and far-field UE.

The rest of this paper is organized as follows: In Section II, we introduce an
ExLens mmWave system model and derive the closed-form expression of ExLens
array response. The property of ExLens array response is explained in Section
III. In Section IV, we explore the localization capbility of ExLens and
propose an effective method to obtain location parameters together with
channel gains. In Section V, we analyze the multi-user communication
performance of ExLens. Our simulation results are presented in Section VI. We
conclude the paper in Section VII.

Notations—In this paper, upper- and lower-case bold letters denote matrices
and vectors, respectively. For a matrix $\mathbf{A}$, $\mathbf{A}^{-1}$,
$\mathbf{A}^{\text{T}}$, and $\mathbf{A}^{\text{H}}$ represent inverse,
transpose, and Hermitian operators, respectively.
$\mbox{blkdiag}(\mathbf{A}_{1},\ldots,\mathbf{A}_{k})$ denotes a block-
diagonal matrix constructed by $\mathbf{A}_{1},\ldots,\mathbf{A}_{k}$. For a
vector $\mathbf{a}$, the L2-norm is signified by $\|\mathbf{a}\|$. For a
complex value $c$, the module is represented by $|c|$ and the real part is
denoted by $\mathcal{R}\\{c\\}$. For a real number $a$, $\lfloor a\rfloor$
denotes the largest integer that is not greater than $a$. sinc$(\cdot)$ is the
“sinc” function defined as sinc$(x)=\sin(\pi x)/(\pi x)$.
$\mathbb{E}\\{\cdot\\}$ indicates the statistical expectation.

##  II System Model

We consider a BS equipped with an ExLens in the two-dimensional coordinate
system (Fig. 1(a)). The EM lens is placed on the y-axis with physical length
$D_{y}$ and is centered at the origin. The antenna elements are placed on the
focal arc, which is defined as a semi-circle around the center of the EM lens
with radius $F$. As the aperture of an antenna array further increases, UE and
significant scatterers are likely to be located in the near-field of the
array, where the uniform plane wave-front assumption no longer holds.
Therefore, we consider the more general spherical wave-front, which leads to
more novel phase design of the EM lens and has greater energy focus on the
lens antenna array compared with plane wave-front [34].

(a) ExLens

(b) ULA

Figure 1: Different antenna arrays illuminated by spherical wave-front.

We first investigate the receive array response by assuming that ExLens is
illuminated by a spherical wave-front emitted from a UE located at
$\mathbf{u}=[-d\cos\phi,d\sin\phi]$, where $d$ is the distance between the UE
and the center of the EM lens, and $\phi\in(-{\pi}/{2},{\pi}/{2})$ is the
angle of the UE relative to the x-axis (Fig. 1(a)). 111 We assume that the
signal source is in front of the lens antenna array (i.e., it is located at
the opposite side of the EM lens with the array elements). This assumption
practically holds because BSs apply sectorization technique. Each antenna
array serves one sector in practice to cover the range of $60^{\circ}$ to
$120^{\circ}$. Multiple lens antenna arrays can be combined to cover a range
of $360^{\circ}$.  For simplicity, we assume that the UE is equipped with an
omni-directional antenna, and is regarded as a point source. The signal
transmitted by the UE is assumed to be $1$, and the signal arrived at any
point $\mathbf{p}=[0,y]$ on the EM lens aperture is given by [21, 24]

$s(\mathbf{u},\mathbf{p})=\eta(\mathbf{u},\mathbf{p})e^{-jk_{0}\parallel\mathbf{u}-\mathbf{p}\parallel},\vspace{-0.4cm}$
(1)

where $k_{0}={2\pi}/{\lambda}$ is the wave number that corresponds to the
signal wavelength $\lambda$, and
$\eta(\mathbf{u},\mathbf{p})={\lambda}/{(4\pi\\!\parallel\\!\mathbf{u}-\mathbf{p}\\!\parallel})$
corresponds to the free space path loss from point $\mathbf{u}$ to point
$\mathbf{p}$. We define $\theta\in(-{\pi}/{2},{\pi}/{2})$, where $\theta$ is
positive below the x-axis and negative above the x-axis (Fig. 1(a)). The
received signal $r(\theta,d,\phi)$ at any point
$\mathbf{b}=[F\cos\theta,-F\sin\theta]$ at the focal arc 222The case that the
center of the EM lens and the focal arc are not coinciding is left for future
investigation.  can be expressed as

$r(\theta,d,\phi)=\int\limits_{-{{{D_{y}}}\mathord{\left/{\vphantom{{{D_{y}}}2}}\right.\kern-1.2pt}2}}^{{{{D_{y}}}\mathord{\left/{\vphantom{{{D_{y}}}2}}\right.\kern-1.2pt}2}}{s(\mathbf{u},\mathbf{p})\kappa(\mathbf{p},\mathbf{b})e^{-j\varphi(\mathbf{p},\mathbf{b})}}dy=\int\limits_{-{{{D_{y}}}\mathord{\left/{\vphantom{{{D_{y}}}2}}\right.\kern-1.2pt}2}}^{{{{D_{y}}}\mathord{\left/{\vphantom{{{D_{y}}}2}}\right.\kern-1.2pt}2}}{\eta(\mathbf{u},\mathbf{p})e^{-jk_{0}\parallel\mathbf{u}-\mathbf{p}\parallel}\kappa(\mathbf{p},\mathbf{b})e^{-j\varphi(\mathbf{p},\mathbf{b})}}dy.\vspace{-0.2cm}$
(2)

where $\mathbf{p}$ is a function of $y$, $\mathbf{u}$ is a function of
$(d,\phi)$, and $\mathbf{b}$ is a function of $\theta$;
$\kappa(\mathbf{p},\mathbf{b})={\lambda}/{(4\pi\\!\parallel\\!\mathbf{p}-\mathbf{b}\\!\parallel})$
accounts for the free-space path loss from point $\mathbf{p}$ on the EM lens
to point $\mathbf{b}$ on the focal arc;
$\varphi(\mathbf{p},\mathbf{b})=\psi(\mathbf{p})+k_{0}||\mathbf{p}-\mathbf{b}||$,
and $\psi(\mathbf{p})$ is the fixed phase shift determined by the EM lens
design. Therefore, $\varphi(\mathbf{p},\mathbf{b})$ is the total phase shift
of the signal by the EM lens and the propagation delay between EM lens and
focal arc. Eq. (2) follows the principle of linear superposition of signals.

(a) Design 1: Incident plane wave-front perpendicular to the lens surface
converges at the focal point, where
${\psi}(\mathbf{p})={\phi_{0}}-{k_{0}}\lVert\mathbf{p}-{\mathbf{b}_{0}}\rVert$.

(b) Design 2: Incident spherical wave-front from the left focal point
converges at the right focal point, where
${\psi}(\mathbf{p})={\phi_{0}}-{k_{0}}(\lVert{\mathbf{c}_{0}}-\mathbf{p}\rVert+\lVert\mathbf{p}-\mathbf{b}_{0}\rVert)$.

Figure 2: Two design approaches for the EM lens.

We first review the fundamental principle of operation for EM lenses: the EM
lenses are similar to optical lenses, which can alter the propagation
directions of the EM rays to achieve energy focusing or beam collimation. The
function of EM lens can be effectively realized by appropriately designing
$\psi(\mathbf{p})$ in $\varphi(\mathbf{p},\mathbf{b})$ in (2). In this study,
we consider two different EM lens designs (Fig. 2). In Design 1, where
incident plane wave-front perpendicular to the lens surface converges at the
focal point $\mathbf{b}_{0}=[F,0]$ (Fig. 2(a)), we have
${\phi_{0}}={\psi}(\mathbf{p})+{k_{0}}\lVert\mathbf{p}-{\mathbf{b}_{0}}\rVert$,
where the constant $\phi_{0}$ is the arrived signal phase at the focal point
$\mathbf{b}_{0}$. Hence, we obtain

${\psi}(\mathbf{p})={\phi_{0}}-{k_{0}}\lVert\mathbf{p}-{\mathbf{b}_{0}}\rVert.\vspace{-0.4cm}$
(3)

The total phase shift for a signal from point $\mathbf{p}$ to point
$\mathbf{b}$ is given by

$\begin{array}[]{l}\varphi(\mathbf{p},\mathbf{b})={\psi}(\mathbf{p})+{k_{0}}\lVert\mathbf{p}-\mathbf{b}\rVert={\phi_{0}}+{k_{0}}\left(\lVert\mathbf{p}-\mathbf{b}\rVert-\lVert\mathbf{p}-\mathbf{b}_{0}\rVert\right).\end{array}\vspace{-0.25cm}$
(4)

In Design 2, where the incident spherical wave-front from point
$\mathbf{c}_{0}=[F_{0},0]$ converges at the focal point $\mathbf{b}_{0}=[F,0]$
(Fig. 2(b)), we have
${\phi_{0}}={k_{0}}\lVert{\mathbf{c}_{0}}-\mathbf{p}\rVert+{\psi}(\mathbf{p})+{k_{0}}\lVert\mathbf{p}-\mathbf{b}_{0}\rVert$.
Then, we obtain the following:

${\psi}(\mathbf{p})={\phi_{0}}-{k_{0}}(\lVert{\mathbf{c}_{0}}-\mathbf{p}\rVert+\lVert\mathbf{p}-\mathbf{b}_{0}\rVert).\vspace{-0.5cm}$
(5)

We also obtain the total phase shift as follows:

$\begin{array}[]{l}\varphi(\mathbf{p},\mathbf{b})={\psi}(\mathbf{p})+{k_{0}}\lVert\mathbf{p}-\mathbf{b}\rVert={\phi_{0}}+{k_{0}}(\lVert\mathbf{p}-\mathbf{b}\rVert-\lVert\mathbf{p}-{\mathbf{b}_{0}}\rVert-\lVert{\mathbf{c}_{0}}-\mathbf{p}\rVert).\end{array}\vspace{-0.25cm}$
(6)

Design 1 can be regarded as a special case of Design 2 with
$F_{0}\rightarrow\infty$. The EM lens is designed according to (3) (Design 1)
or (5) (Design 2) and works in the spherical wave-front scenarios (Fig. 1(a)).
Then, we define the response on point $\mathbf{b}=[F\cos\theta,-F\sin\theta]$
at the focal arc as

$a(\theta,d,\phi)={16\pi^{2}Fd}/({{\lambda^{2}{e^{-j{k_{0}d}}}}})\times
r(\theta,d,\phi),\vspace{-0.25cm}$ (7)

the closed-form expression of which can be obtained in the following theorem.

######  Theorem 1.

When illuminated by a spherical wave-front, with the assumption $d,F\gg
D_{y}$, the array response of ExLens on any point
$\mathbf{b}=[F\cos\theta,-F\sin\theta]$ at the focal arc can be approximated
as

$a(\theta,d,\phi)\approx\frac{{\sqrt{\pi}}}{{{\rm{2}}\sqrt{\alpha}}}{e^{-j\left({\frac{{{{{\pi^{2}\beta^{2}}}}}}{{\alpha}}-\frac{5\pi}{4}}\right)}}\left({\mathrm{erf}\left({\frac{{\alpha{D_{y}}+2\pi\beta}}{{2\sqrt{\alpha}}}{e^{j\frac{{3\pi}}{4}}}}\right)+\mathrm{erf}\left({\frac{{\alpha{D_{y}}-2\pi\beta}}{{2\sqrt{\alpha}}}{e^{j\frac{{3\pi}}{4}}}}\right)}\right),\vspace{-0.15cm}$
(8)

where

$\mathrm{erf}\left(x\right)=\frac{2}{{\sqrt{\pi}}}\int\limits_{0}^{x}{{e^{-{t^{2}}}}dt},\vspace{-0.15cm}$
(9)

$\beta=(\sin\theta-\sin\phi)/{\lambda}$ and $\alpha$ for the two different
lens designs is given in Table I.

TABLE I: Parameter $\alpha$ for different lens designs. | Design 1 | Design 2
---|---|---
$\alpha$ | $\dfrac{{\pi{{\sin}^{\rm{2}}}\theta}}{{\lambda F}}-\dfrac{{\pi{{\cos}^{\rm{2}}}\phi}}{{\lambda d}}$ | $\dfrac{{\pi{{\sin}^{\rm{2}}}\theta}}{{\lambda F}}-\dfrac{{\pi{{\cos}^{\rm{2}}}\phi}}{{\lambda d}}+\dfrac{\pi}{{\lambda{F_{0}}}}$

###### Proof.

Please refer to Appendix A. ∎

According to Theorem 1, the parameter $\alpha$ of Design 2 reduces to that of
Design 1 when $F_{0}\to\infty$, as
$\lim\limits_{F_{0}\to\infty}{\pi}/(\lambda{F_{0}})=0$ in Table I. This again
shows that Design 1 can be regarded as a special case of Design 2 with
$F_{0}\rightarrow\infty$. The energy focusing property of ExLens is determined
by the item
${\mathrm{erf}\left({\frac{{\alpha{D_{y}}+2\pi\beta}}{{2\sqrt{\alpha}}}{e^{j\frac{{3\pi}}{4}}}}\right)+\mathrm{erf}\left({\frac{{\alpha{D_{y}}-2\pi\beta}}{{2\sqrt{\alpha}}}{e^{j\frac{{3\pi}}{4}}}}\right)}$
in (8) and is further analyzed in Section III. The parameter $\theta$ in
Theorem 1 is a continuous value, whereas $\theta$ should be sampled for a
particular antenna placement. Here, we assume that
$N_{a}=2\lfloor\tilde{D}_{y}\rfloor+1$ antenna elements are placed on the
focal arc of the EM lens [34], where $\tilde{D}_{y}={D_{y}}/{\lambda}$ denotes
the electrical length of the EM lens. For notational convenience, $N_{a}$ is
assumed to be an odd number. Let $\theta_{n}$ signify the angle of the $n$-th
antenna element relative to the x-axis, where $n\in\\{0,\pm 1,\ldots,\pm N\\}$
and $N={(N_{a}-1)}/{2}$. The deployment of the antenna elements obeys the rule
$\sin\theta_{n}={n}/{N}$. Therefore, the array response of the $n$-th antenna
element located at point $\mathbf{b}_{n}=[F\cos\theta_{n},-F\sin\theta_{n}]$
accroding to (8) can be expressed as

$a_{n}(d,\phi)\approx\frac{{\sqrt{\pi}}}{{{\rm{2}}\sqrt{\alpha}}}e^{-j\left({\frac{{{{{\pi^{2}\beta^{2}}}}}}{{\alpha}}-\frac{5\pi}{4}}\right)}\left({\mathrm{erf}\left({\frac{{\alpha{D_{y}}+2\pi\beta}}{{2\sqrt{\alpha}}}{e^{j\frac{{3\pi}}{4}}}}\right)+\mathrm{erf}\left({\frac{{\alpha{D_{y}}-2\pi\beta}}{{2\sqrt{\alpha}}}{e^{j\frac{{3\pi}}{4}}}}\right)}\right),\vspace{-0.15cm}$
(10)

where $\sin\theta$ in $\alpha$ and $\beta$ is replaced by
$\sin\theta_{n}={n}/{N}$. With the $n$-th element given in (10), the antenna
array response vector $\mathbf{a}(d,\phi)\in\mathbb{C}^{N_{a}\times 1}$ can be
obtained accordingly.

##  III Property of ExLens Array Response

In this section, we analyze the relationship and differences between the array
responses of the lens antenna array illuminated by spherical wave-fronts
(near-field scenarios) and plane wave-fronts (far-field scenarios). Before
entering the in-depth comparison, we review the array response of the lens
antenna array illuminated by plane wave-fronts. The array response of a lens
antenna array for an element located at the focal arc with angle $\theta$ and
illuminated by a uniform plane wave with AOA $\phi$ is given by [34]

$a(\theta,\phi)={D_{y}}\mathrm{sinc}\left({\tilde{D}_{y}\sin\theta-\tilde{D}_{y}\sin\phi}\right).\vspace{-0.45cm}$
(11)

In the far-field scenarios, the array response follows the “sinc” function as
given in (11). For any incident/departure signal from/to a particular
direction $\phi$, only those antennas located near the focal point would
receive/steer significant power. Notably, the focal point reflects the
information of $\phi$, whereas the information of $d$ cannot be reflected from
(11). Furthermore, the angle resolution of the lens antenna array is
determined by the width of the main lobe of the “sinc” function, which is
$2/\tilde{D}_{y}$. When $\tilde{D}_{y}$ increases, the main lobe becomes
narrower such that other multi-paths can be resolved in the spatial domain.

###  III-A Generality Analysis

We reveal the relationship between the array responses of the lens antenna
array illuminated by the spherical wave-front in (8) and plane wave-front in
(11). The following lemma shows that (11) is a special case of the derived
ExLens array response (8).

######  Lemma 1.

When $d$ and $F_{0}$ (for Design 2) increase to infinite and in which the
spherical wave-front reduces to the plane wave-front, the array response given
in (8) converges to (11) as

$\begin{array}[]{ll}\lim\limits_{d,F_{0}\to\infty}\\!\frac{{\sqrt{\pi}}}{{{\rm{2}}\sqrt{\alpha}}}e^{-j\left({\frac{{{{{\pi^{2}\beta^{2}}}}}}{{\alpha}}-\frac{5\pi}{4}}\right)}\\!\\!\left({\mathrm{erf}\left({\frac{{\alpha{D_{y}}+2\pi\beta}}{{2\sqrt{\alpha}}}{e^{j\frac{{3\pi}}{4}}}}\right)\\!\\!+\mathrm{erf}\left({\frac{{\alpha{D_{y}}-2\pi\beta}}{{2\sqrt{\alpha}}}{e^{j\frac{{3\pi}}{4}}}}\right)}\right)\\!=\\!{D_{y}}\mathrm{sinc}\left({\tilde{D}_{y}\sin\theta-\tilde{D}_{y}\sin\phi}\right).\end{array}\vspace{-0.25cm}$
(12)

###### Proof.

Refer to Appendix B. ∎

######  Remark 1.

Lemma 1 reveals that the derived array response of the ExLens illuminated by a
spherical wave-front in (8) is a more general result compared to the result in
[34], which means that the derived array response (8) is applicable to far-
field (plane wave-front) and near-field (spherical wave-front) scenarios.

###  III-B Window Effect

We first analyze differences of the energy focusing characteristics of the
lens antenna array illuminated by the spherical wave-front in (8) and plane
wave-front in (11). Specifically, in the near-field scenarios, the array
response (8) has an evident window effect for the energy focusing property,
which does not exist in the far-field scenarios. To better understand the
window effect, we split the array response (8) into three parts as follows:

$a(\theta,d,\phi)=\underbrace{\frac{{\sqrt{\pi}}}{{{\rm{2}}\sqrt{\alpha}}}}_{\left({\rm{a}}\right)}\underbrace{e^{-j\left({\frac{{{{{\pi^{2}\beta^{2}}}}}}{{\alpha}}-\frac{5\pi}{4}}\right)}}_{\left({\rm{b}}\right)}\underbrace{\left({\mathrm{erf}\left({\frac{{\alpha{D_{y}}+2\pi\beta}}{{2\sqrt{\alpha}}}{e^{j\frac{{3\pi}}{4}}}}\right)+\mathrm{erf}\left({\frac{{\alpha{D_{y}}-2\pi\beta}}{{2\sqrt{\alpha}}}{e^{j\frac{{3\pi}}{4}}}}\right)}\right)}_{\left({\rm{c}}\right)},\vspace{-0.2cm}$
(13)

where part $(a)$ is the amplitude, part $(b)$ is the phase, and part $(c)$ is
the window effect for the energy focusing property in the near-field
scenarios. We denote a “window” function as

${w(\theta,d,\phi)}\buildrel\Delta\over{=}{\mathrm{erf}\left({\frac{{\alpha{D_{y}}+2\pi\beta}}{{2\sqrt{\alpha}}}{e^{j\frac{{3\pi}}{4}}}}\right)+\mathrm{erf}\left({\frac{{\alpha{D_{y}}-2\pi\beta}}{{2\sqrt{\alpha}}}{e^{j\frac{{3\pi}}{4}}}}\right)}.\vspace{-0.2cm}$
(14)

######  Lemma 2.

Let $v_{1}$ and $v_{2}$ denote the zero points of $\mathrm{erf}(\xi_{1})$ and
$\mathrm{erf}(\xi_{2})$, respectively, where
$\xi_{1}={\frac{{\alpha{D_{y}}+2\pi\beta}}{{2\sqrt{\alpha}}}{e^{j\frac{{3\pi}}{4}}}}$
and
$\xi_{2}={\frac{{\alpha{D_{y}}-2\pi\beta}}{{2\sqrt{\alpha}}}{e^{j\frac{{3\pi}}{4}}}}$.
We define the center and width of the energy focusing window as
${v_{c}}=({v_{1}}+{v_{2}})/{2}$ and $\Delta v=|v_{1}-v_{2}|$, respectively. In
lens Design 1, the center and width of the energy focusing window are given as
follows:

${v_{c}}=\sin\phi,\ \ \Delta
v=\dfrac{{{D_{y}}{{\cos}^{2}}\phi}}{d}.\vspace{-0.25cm}$ (15)

In lens Design 2, the center and width of the energy focusing window are
obtained as follows:

${v_{c}}=\sin\phi,\ \ \Delta
v={D_{y}}\left|{\dfrac{1}{{{F_{0}}}}-\dfrac{{{{\cos}^{2}}\phi}}{d}}\right|.\vspace{-0.25cm}$
(16)

###### Proof.

Refer to Appendix C. ∎

######  Remark 2.

The similarity of the energy focusing property of the lens antenna array
illuminated by the spherical wave-front in (8) and plane wave-front in (11) is
attributed to the following: the center of the focusing area is approximately
equal to $\sin\phi$, where $\phi$ is the angle of the source point relative to
the x-axis. The differences of the energy focusing property of the “window”
function in (8) (near-field scenarios) and that of the “sinc” function (11)
(far-field scenarios) are as follows: The width of the focusing window
reflects the distance information $d$, according to (15) and (16). This
feature implies that a single ExLens has the positioning capability in the
spherical wave-front scenarios. Therefore, the position of the UE can also be
easily extracted from the information of the energy focusing windows according
to the received communication signals. By contrast, the “sinc” function has a
maximum energy point and cannot reflect the information of $d$, according to
(11).

Next, we explore the changes in the energy focusing properties from the far-
field to the near-field, as aperture size of the lens antenna array increases.
For illustration, we take the lens Design 2 as an example and plot changes in
the energy focusing property by increasing the effective aperture
$\tilde{D}_{y}$ (Fig. 3). Although $\mathrm{erf}(\xi_{1})$ and
$\mathrm{erf}(\xi_{2})$ are complex values, their imaginary parts are close to
zero. Hence we draw the real parts of $\mathrm{erf}(\xi_{1})$ and
$\mathrm{erf}(\xi_{2})$ on the right-hand side of Fig. 3. When $\tilde{D}_{y}$
is small ($\tilde{D}_{y}=5$ and $\tilde{D}_{y}=10$), $d=50\,m$ is much larger
than the Rayleigh distance $R={2D_{y}^{2}}/{\lambda}$, the plane wave-front
assumption holds, and $|\mathrm{erf}(\xi_{1})+\mathrm{erf}(\xi_{2})|$ on the
left-hand side of Fig. 3 presents a representation of the “sinc” function. As
long as the plane wave-front assumption holds, the “sinc” function will become
finer and sharper as $\tilde{D}_{y}$ increases. Thus, the focusing property
and the angle resolution of the lens antenna array improves, as described in
[34]. When $\tilde{D}_{y}$ further increases to a sufficiently large value,
the plane wave-front assumption no longer holds. Furthermore, the “sinc”
function cannot reflect the energy focusing property accurately, say, when
$\tilde{D}_{y}\geqslant 70$. With the further increase in $\tilde{D}_{y}$
($\tilde{D}_{y}=100$ and $\tilde{D}_{y}=200$), the window effect for the
energy focusing property appears. The energy received in the focusing window
is approximatley equal and the energy of the side lobes is extremely small, as
shown in the last two subfigures of Fig. 3. The width of the energy focusing
window increases with $\tilde{D}_{y}$, thereby receiving more energy, but
pronouncing energy diffusion effect. The aforementioned phenomenon is caused
by the relative distance of the lines of $\mathrm{erf}(\xi_{1})$ and
$\mathrm{erf}(\xi_{2})$. The relative distance of the lines of
$\mathrm{erf}(\xi_{1})$ and $\mathrm{erf}(\xi_{2})$ increases with
$\tilde{D}_{y}$, thereby resulting different line shapes of
$|\mathrm{erf}(\xi_{1})+\mathrm{erf}(\xi_{2})|$ (Fig. 3). In summary, the
energy focusing property of the lens antenna array is described by the “sinc”
function in the far-field scenarios and the “window” function in the near-
field scenarios.

######  Remark 3.

The derived array response of the ExLens illuminated by a spherical wave-front
in (8) can describe the transition between the far-field and the near-field
scenarios. In the special case of far-field (i.e., $d,F_{0}\to\infty$), the
angle resolution of the lens antenna array is determined by the width of the
main lobe of the “sinc” function, which is $2/\tilde{D}_{y}$. By contrast, the
width $\Delta v$ determines the angle resolution of the ExLens in the near-
field. The angle resolution makes ExLens illuminated by spherical waves also
suitable for multi-user communication.

Figure 3: Changes in the energy focusing property of the lens antenna array as
the effective aperture $\tilde{D}_{y}$ increases, where $\phi=0$ rad,
$d=50\,m$, $F_{0}=15\,m$, $F=5\,m$, and $\lambda=0.01\,m$.

###  III-C Approximation Tightness and Antenna Deployment

Figure 4: Differences of the energy focusing properties of the ExLens with two
different lens designs, where $\tilde{D}_{y}=100$, $d=7\,m$, $F_{0}=F=5\,m$,
and $\lambda=0.01\,m$.

The approximation error between the derived closed-form array response and the
original one in integral form is compared in Fig. 4. The solid line denotes
the original array response in the integral-form. The dotted line represents
the approximated closed-form array response given in (8). The solid line
matches well with the dashed line when the array power response is above $-20$
dB, thereby indicating that the approximation error is small. Specifically,
the approximation error can be safely ignored when
$\phi\in[-36^{\circ},36^{\circ}]$. When the array power response is below
$-20$ dB, the approximate error slightly increases with $|\phi|$. Moreover,
the two different lens designs also have dissimilar energy focusing
characteristics (Fig. 4). For Design 1, the width of the energy focusing
window $\Delta v={{{D_{y}}{{\cos}^{2}}\phi}}/{d}$ is smaller with larger
$|\phi|$, and this characteristic means that the energy focusing phenomenon is
evident when $|\phi|$ is large. However, for Design 2, we have $\Delta
v={D_{y}}\left|{{1}/{{{F_{0}}}}-{{{{\cos}^{2}}\phi}}/{d}}\right|$ and
$d>{F_{0}}\cos^{2}\phi$, the energy focusing phenomenon is evident when
$|\phi|$ is small. This outcome is expected given that the ExLens of Design 1
has good energy focusing property for the uniform plane incident wave. When we
apply this design into spherical wave-front scenarios, the incident spherical
wave-front becomes closer to the plane wave-front as $|\phi|$ increases;
hence, the energy focusing performance improves. However, the ExLens of Design
2 has good energy focusing property when the source point is near the focal
point $\mathbf{c}_{0}=[-F_{0},0]$. When $|\phi|$ becomes larger, the source
point is farther from the focal point $\mathbf{c}_{0}$; thus, the energy
focusing performance deteriorates. As mentioned before, when $F_{0}\to\infty$,
Design 2 becomes to Design 1. Accordingly, the width of the energy focusing
window of Design 2 equals to that of Design 1, i.e.,
$\lim\limits_{F_{0}\to\infty}{D_{y}}\left|{{1}/{{{F_{0}}}}-{{{{\cos}^{2}}\phi}}/{d}}\right|={{{D_{y}}{{\cos}^{2}}\phi}}/{d}$.
To be more general, we use the lens Design 2 for analysis in the following
sections. We adopt the antenna elements deployment $\sin\theta_{n}={n}/{N}$,
as mentioned in Section II. The right hand-side of Fig. 4 shows that the
energy focusing window is narrower in the center and wider on the edges.
Therefore, placing denser antenna elements in the center of the array can
prevent the non-detection of strong signals. For the far-field scenario, the
antenna array response reduces to the “sinc” function [34], and this kind of
antenna deployment is applicable.

From the analysis in this section, we get such insight that the window effect
for the energy focusing property makes ExLens illuminated by spherical waves
suitable for single-station localization and multi-user communication, which
are analyzed in-depth in the following sections.

##  IV Position Sensing

In this section, we explore the localization capbility of ExLens. The signal
that arrived at different points of the lens aperture has the same incident
angle under the assumption of plane wave-front. However, when the UE is
located in the near-field of ExLens, the signal with a spherical wave-front
arrived at different points of the lens aperture has different incident
angles. Thus, relative to that of the plane wave-front, the received signal
with spherical wave-front contains more abundant angular information that
changes continuously from one edge of the lens aperture to another. According
to the traditional multi-point localization [36], more angular measurements
can ensure more accurate localization. We can infer that a single ExLens has
the localization capbility with the abundant angular information from the
spherical wave-front. Thus, in this section, we analyze the theoretical
localization capbility of ExLens and then propose a parameterized estimation
method to obtain the location parameters.

For ease of expression, we take one UE with single antenna for illustration in
this section. The system model can be easily extended to solve the case with
multiple UEs as long as the pilot signals for different UEs are orthogonal in
time. We consider the narrow band mmWave multi-path channel model. Thus, the
uplink channel is given by

$\textbf{h}=\sum\limits_{l=1}^{L}g_{l}\textbf{a}(d_{l},\phi_{l}),\vspace{-0.25cm}$
(17)

where $g_{l}$ is the complex gain of the $l$-th path,
$\textbf{a}(\cdot)\in\mathbb{C}^{N_{a}\times 1}$ is the array response vector
with elements defined in (10), $l=1$ represents the LOS path,
$(d_{1},\phi_{1})$ is a pair of position parameters of the UE, $l>1$
represents the NLOS path, and $(d_{l},\phi_{l})$ is a pair of position
parameters of the $l$-th scatterer. We only consider the last-jump scatterers.
If all one pilots are used, then, the received signal at the ExLens antenna
array is modelled as

$\textbf{r}=\textbf{h}+\textbf{n}=\sum\limits_{l=1}^{L}g_{l}\textbf{a}(d_{l},\phi_{l})+\textbf{n},\vspace{-0.25cm}$
(18)

where $\textbf{n}\in\mathbb{C}^{N_{a}\times 1}$ represents the circularly
symmetric complex Gaussian noise with zero-mean and covariance matrix
$\sigma^{2}\textbf{I}$. Here, we define the receive signal-to-noise ratio
(SNR) as SNR $=10\lg({\textbf{h}^{\text{H}}\textbf{h}}/{(N_{a}\sigma^{2})})$.
Let $\bm{\eta}_{l}=[g_{l},d_{l},\phi_{l}]$,
$\bm{\eta}=[\bm{\eta}_{1},\ldots,\bm{\eta}_{L}]$, and
$\textbf{h}(\bm{\eta})=\sum\limits_{l=1}^{L}g_{l}\textbf{a}(d_{l},\phi_{l})$.
In the localization, we aim at determining $\bm{\eta}$ based on the received
signal r in (18).

###  IV-A Theoretical Localization Analysis

According to [37], the $3L\times 3L$ positive definite Fisher information
matrix (FIM) of $\bm{\eta}$ is given by

${\bf{F}}\left(\bm{\eta}\right)=\left[\begin{matrix}{\bf{F}}_{11}\left(\bm{\eta}\right)&\ldots&{\bf{F}}_{1L}\left(\bm{\eta}\right)\\\
\vdots&\ddots&\vdots\\\
{\bf{F}}_{L1}\left(\bm{\eta}\right)&\ldots&{\bf{F}}_{LL}\left(\bm{\eta}\right)\end{matrix}\right],$
(19)

where the $3\times 3$ matrix ${\bf{F}}_{ll^{\prime}}\left({\bm{\eta}}\right)$
is defined by

${\bf{F}}_{ll^{\prime}}\left({\bm{\eta}}\right)=\frac{2}{\sigma^{2}}\mathcal{R}\left\\{\dfrac{\partial\textbf{h}^{\text{H}}(\bm{\eta})}{\partial\bm{\eta}_{l}}\dfrac{\partial\textbf{h}(\bm{\eta})}{\partial\bm{\eta}_{l^{\prime}}}\right\\}.\vspace{-0.25cm}$
(20)

The information inequality for the covariance matrix of any unbiased estimate
$\hat{\bm{\eta}}$ reads [37]

$\mathbb{E}\\{(\hat{\bm{\eta}}-\bm{\eta})^{\text{H}}(\hat{\bm{\eta}}-\bm{\eta})\\}\geq{\bf{F}^{-1}}\left(\mathbf{\bm{\eta}}\right).\vspace{-0.3cm}$
(21)

The ${\bf{F}}\left(\bm{\eta}\right)$ is represented in the polar coordinates.
With $x_{l}=-d_{l}\cos\phi_{l}$ and $y_{l}=d_{l}\sin\phi_{l}$, the position of
the UE or scatterer in Cartesian coordinates is given as
$\mathbf{u}_{l}=[x_{l},y_{l}]$, for $l=1,\ldots,L$. Let
$\mathbf{u}=[\mathbf{u}_{1},\ldots,\mathbf{u}_{L}]$,
$\tilde{\mathbf{u}}_{l}=[g_{l},x_{l},y_{l}]$, and
$\tilde{\mathbf{u}}=[\tilde{\mathbf{u}}_{1},\ldots,\tilde{\mathbf{u}}_{L}]$,
thus, transformation to the position domain is achieved as follows: the FIM of
$\tilde{\mathbf{u}}$ is given by
${\bf{F}}\left(\tilde{\mathbf{u}}\right)={\bf{T}}^{\text{T}}{\bf{F}}\left({\bm{\eta}}\right){\bf{T}}$,
where ${\bf{T}}={\rm blkdiag}\\{\mathbf{T}_{1},\ldots,\mathbf{T}_{L}\\}$, and
${\bf{T}}_{l}=[1,0,0;0,x_{l}/d_{l},y_{l}/d_{l};0,y_{l}/d_{l}^{2},-x_{l}/d_{l}^{2}]$
is the Jacobian matrix used to describe the coordinate system transformation,
in which the “;” operator separates rows in a matrix. Then, we define the
position error bound (PEB) from ${\bf{F}}\left(\tilde{\mathbf{u}}\right)$ as

${\text{PEB}}(\mathbf{u})={\sqrt{{\rm
trace}([{\bf{F}^{-1}}\left(\tilde{\mathbf{u}}\right)]_{(\\{2:3,5:6,\ldots,3L-1:3L\\},\\{2:3,5:6,\ldots,3L-1:3L\\})})}}.\vspace{-0.3cm}$
(22)

The root mean-square estimation error (RMSE) of an unbiased estimate of
${\mathbf{u}}$ is lower-bounded by ${\text{PEB}}(\mathbf{u})$. We need
${{\partial\mathbf{a}(d_{l},\phi_{l})}}/{{\partial d_{l}}}$ and
${{\partial\mathbf{a}(d_{l},\phi_{l})}}/{{\partial\phi_{l}}}$, which is
derived in Appendix D, to calculate the PEB given in (22).

Fig. 5 shows PEBs as functions of $d$, $\phi$, $D_{y}$, and $F_{0}$ ($F$ shows
a similar property as $F_{0}$). Given that some approximations are made to
derive the closed-form array response (10), we denote the obtained PEB with
the approximated closed-form array response as APEB. We also calculate the PEB
with the original received signal in the integral form denoted as OPEB. We can
evaluate the accuracy of the approximation by comparing APEB with OPEB. Given
the minimal approximation error shown in Fig. 5 (only when $|\phi|>1.2$ rad or
${D}_{y}>3\,m$, the APEB slightly deviates from the OPEB), the theoretical
localization analysis based on (10) is accurate.

######  Remark 4.

The localization performance of ExLens degrades with the increase in $d$ and
$|\phi|$. This finding is expected given that the UE transits from near-field
to far-field because $d$ increases, thereby demonstrating that a single BS
loses its localization capability. The localization performance improves with
the increase in ${D}_{y}$. By contrast, the value of $F_{0}$ has a little
effect on PEBs. Given that the increase in ${D}_{y}$ can bring rich angle
measurements, which is similar to adding additional BSs in the multi-point
localization. Under the given configuration in Fig. 5, an ExLens with the
electrical aperture $\tilde{D}_{y}=100$ and SNR $=20$ dB can theoretically
provide around centimeter-level localization accuracy for a UE located with
$d<50\,m$ and $\phi<1$ rad to the BS.

Figure 5:  (a) PEB as a function of $d$ and $\phi$ with $D_{y}=1\,m$,
$F_{0}=F=5\,m$, $\lambda=0.01\,m$, $N_{a}=201$, $L=1$, and SNR $=20$ dB. (b)
PEB as a function of ${D}_{y}$ and $F_{0}$ with $d=18\,m$, $\phi=0$ rad,
$F=5\,m$, $\lambda=0.01\,m$, $N_{a}=201$, $L=1$, and SNR $=20$ dB.

###  IV-B Location Parameter Estimation Method

In this subsection, we propose a location parameters estimation method to
determine $(d_{l},\phi_{l})$, and the gain $g_{l}$ for $l=1,\ldots,L$. The
maximum likelihood (ML) estimator is given by

$(\bm{\hat{d}},\bm{\hat{\phi}},\bm{\hat{g}})=\mathop{\arg\min}\limits_{\bm{d}\in\mathbb{R}^{L},\bm{\phi}\in(-\frac{\pi}{2},\frac{\pi}{2})^{L},\bm{g}\in\mathbb{C}^{L}}\left\Arrowvert\textbf{r}-\sum\limits_{l=1}^{L}g_{l}\textbf{a}(d_{l},\phi_{l})\right\Arrowvert^{2},\vspace{-0.25cm}$
(23)

where ${\bm{d}}=[d_{1},\ldots,d_{L}]$,
${\bm{\phi}}=[\phi_{1},\ldots,\phi_{L}]$, and ${\bm{g}}=[g_{1},\ldots,g_{L}]$.
333 We assume that only $M_{RF}$ RF chains are available in the ExLens system,
where $M_{RF}<N_{a}$. Thus, the low-complexity power-based antenna selection
method is applied. We let $\textbf{r}\in\mathbb{C}^{N_{a}\times 1}$ denote the
received signal after the antenna selection, which has $M_{RF}$ non-zero
elements. The brute-force search for the optimal estimate of
$(\bm{d},\bm{\phi},\bm{g})$ in the whole continuous domain
($\bm{d}\in\mathbb{R}^{L}$, $\bm{\phi}\in(-{\pi}/{2},{\pi}/{2})^{L}$, and
$\bm{g}\in\mathbb{C}^{L}$) is infeasible. Hence, we propose an effective
localization method, which contains three stages: (1) Initialization stage,
where we propose a window-based coarse localization algorithm to determine the
grid search region. (2) detection stage, in which we find a relatively
accurate estimate of $(d_{l},\phi_{l})$, for $l=1,\ldots,L$, from discrete
grids by discrete OMP (DOMP) algorithm. (3) refinement stage, where we
iteratively refine the location parameters $(d_{l},\phi_{l})$ and gain $g_{l}$
for $l=1,\ldots,L$ by Newton algorithm [38, 39].

####  IV-B1 Initialization stage

We utilize the window effect for energy focusing property of ExLens to narrow
down the search region. Lemma 2 is developed for single-path scenarios. For
multi-path scenarios, we let $v_{1l}$ and $v_{2l}$ denote the window edges for
the $l$-th path, which are affected by the position parameters
$(d_{l},\phi_{l})$, where $l=1,\ldots,L$. In Appendix C, we derive the
relationships between the window edges and the location parameters. After
parameters $v_{1l}$ and $v_{2l}$ are measured, we can obtain a coarse
estimation of $(d_{l},\phi_{l})$ for $l=1,\ldots,L$. We apply power detection
to the received signal by each antenna, and for a given threshold, we can
obtain $\hat{v}_{1l}$ and $\hat{v}_{2l}$ for $l=1,\ldots,L$. According to (67)
and (69), we have the following expression for the $l$-th path:

$\mathbf{g}_{l}=\mathbf{G}\mathbf{q}_{l}+\mathbf{e}_{l},\vspace{-0.5cm}$ (24)

where

$\mathbf{g}_{l}=\\!\left(\\!\left(\\!\hat{v}_{1l}\\!+\\!\frac{F}{D_{y}}\right)^{2}\\!\\!\\!-\\!\left(\\!\frac{F}{D_{y}}\\!\right)^{2}\\!\\!\\!+\\!\frac{F}{F_{0}},\\!\left(\\!\hat{v}_{2l}\\!-\\!\frac{F}{D_{y}}\\!\right)^{2}\\!\\!\\!-\\!\left(\frac{F}{D_{y}}\right)^{2}\\!\\!\\!+\\!\frac{F}{F_{0}}\right)^{\text{T}}\\!\\!\\!,\
\ \mathbf{G}=\begin{pmatrix}\frac{2F}{D_{y}}&F\\\
\frac{-2F}{D_{y}}&F\end{pmatrix},\ \
\mathbf{q}_{l}=\begin{pmatrix}\sin\phi_{l}\\\
\frac{\cos^{2}\phi_{l}}{d_{l}}\end{pmatrix},\vspace{-0.1cm}$ (25)

and $\mathbf{e}_{l}$ is the noise vector caused by measurement error. Then,
the least squares estimator is given by

$\mathbf{\hat{q}}_{l}=(\mathbf{G}^{\text{H}}\mathbf{G})^{-1}\mathbf{G}^{\text{H}}\mathbf{g}_{l},\vspace{-0.3cm}$
(26)

with $\mathbf{\hat{q}}_{l}=[\hat{q}_{1l},\hat{q}_{2l}]$. Thus, the position
parameters $(d_{l},\phi_{l})$ can be recovered by

$\hat{\phi}_{l}=\arcsin\hat{q}_{1l},\ \
\hat{d}_{l}=({1-\hat{q}_{1l}^{2}})/{\hat{q}_{2l}}.\vspace{-0.3cm}$ (27)

We denote the sets $\mathbb{S}_{d}=\cup\mathbb{S}_{d}^{l}$ and
$\mathbb{S}_{\phi}=\cup\mathbb{S}_{\phi}^{l}$, for $l=1,\ldots,L$, where
$\mathbb{S}_{d}^{l}=\\{\hat{d}_{l}-\Delta d\leq d\leq\hat{d}_{l}+\Delta d\\}$
and
$\mathbb{S}_{\phi}^{l}=\\{\hat{\phi}_{l}-\Delta\phi\leq\phi\leq\hat{\phi}_{l}+\Delta\phi\\}$.
We generate finite discrete sets by taking $N_{d}$ and $N_{\phi}$ grids on the
obtained sets $\mathbb{S}_{d}$ and $\mathbb{S}_{\phi}$ as
$\mathbb{\bar{S}}_{d}$ and $\mathbb{\bar{S}}_{\phi}$, respectively. The total
search region is initialized as $\mathbb{\bar{S}}_{d}$ and
$\mathbb{\bar{S}}_{\phi}$.

####  IV-B2 Detection stage

We apply DOMP algorithm to detect $d_{l}$ and $\phi_{l}$ from the discrete
sets $\mathbb{\bar{S}}_{d}$ and $\mathbb{\bar{S}}_{\phi}$, respectively, for
$l=1,\ldots,L$. We take the detection of the $l^{\prime}$-th path as an
example for illustration. Let $(\hat{d}_{l},\hat{\phi}_{l},\hat{g}_{l})$, for
$l=1,\ldots,l^{\prime}-1$, denote the estimates of the first $l^{\prime}-1$
paths. Then, the residual measurement is given by

$\textbf{r}_{r}=\textbf{r}-\sum\limits_{l=1}^{l^{\prime}-1}\hat{g}_{l}\textbf{a}(\hat{d}_{l},\hat{\phi}_{l}).\vspace{-0.25cm}$
(28)

We apply the ML estimates by minimizing the residual power
$\left\Arrowvert\textbf{r}_{r}-g\textbf{a}(d,\phi)\right\Arrowvert^{2}$, or
equivalently, by maximizing $S(d,\phi,g)$, where

$S(d,\phi,g)=2\mathcal{R}\left\\{\textbf{r}_{r}^{\text{H}}g\textbf{a}(d,\phi)\right\\}-\left\Arrowvert
g\textbf{a}(d,\phi)\right\Arrowvert^{2}.\vspace{-0.25cm}$ (29)

The generalized likelihood ratio test estimate of
$(d_{l^{\prime}},\phi_{l^{\prime}})$ of the $l^{\prime}$-th path is the
solution of the following optimization problem

$(\hat{d}_{l^{\prime}},\hat{\phi}_{l^{\prime}})=\mathop{\arg\max}\limits_{d\in\mathbb{\bar{S}}_{d},\phi\in\mathbb{\bar{S}}_{\phi}}|\textbf{a}(d,\phi)^{\text{H}}\textbf{r}_{r}|^{2}/\left\Arrowvert\textbf{a}(d,\phi)\right\Arrowvert^{2}.\vspace{-0.25cm}$
(30)

The corresponding gain of the $l^{\prime}$-th path that maximizes
$S(d,\phi,g)$ is given by

$\hat{g}_{l^{\prime}}=\left(\textbf{a}(\hat{d}_{l^{\prime}},\hat{\phi}_{l^{\prime}})^{\text{H}}\textbf{r}_{r}\right)/\left\Arrowvert\textbf{a}(\hat{d}_{l^{\prime}},\hat{\phi}_{l^{\prime}})\right\Arrowvert^{2}.\vspace{-0.25cm}$
(31)

####  IV-B3 Refinement stage

Given that $d_{l^{\prime}}$ and $\phi_{l^{\prime}}$ can take any value in
$\mathbb{R}$ and $(-{\pi}/{2},{\pi}/{2})$, respectively, we add a refinement
stage by utilizing Newton algorithm to reduce the off-grid effect and enhance
the estimation accuracy. Let
$(\hat{d}_{l^{\prime}},\hat{\phi}_{l^{\prime}},\hat{g}_{l^{\prime}})$ denote
the current estimates. The Newton refinement is given by

$\begin{pmatrix}\hat{\hat{d}}_{l^{\prime}}\\\
\hat{\hat{\phi}}_{l^{\prime}}\end{pmatrix}=\begin{pmatrix}\hat{{d}}_{l^{\prime}}\\\
\hat{{\phi}}_{l^{\prime}}\end{pmatrix}-\begin{pmatrix}\frac{\partial^{2}S}{\partial
d^{2}}&\frac{\partial^{2}S}{\partial d\partial\phi}\\\
\frac{\partial^{2}S}{\partial\phi\partial
d}&\frac{\partial^{2}S}{\partial\phi^{2}}\end{pmatrix}^{-1}\begin{pmatrix}\frac{\partial
S}{\partial d}\\\ \frac{\partial
S}{\partial\phi}\end{pmatrix}\vspace{-0.25cm}$ (32)

where the first-order partial derivatives of $S(d,\phi,g)$ is given by

$\frac{\partial S}{\partial
x}=\mathcal{R}\left\\{(\textbf{r}_{r}-g\textbf{a}(d,\phi))^{\text{H}}g\frac{\partial\textbf{a}(d,\phi)}{\partial
x}\right\\},\vspace{-0.25cm}$ (33)

where $x$ can be $d$ or $\phi$. The second-order partial derivatives of
$S(d,\phi,g)$ is given by

$\frac{\partial^{2}S}{\partial x\partial
y}=\mathcal{R}\left\\{\left(\textbf{r}_{r}-g\textbf{a}(d,\phi)\right)^{\text{H}}g\frac{\partial^{2}\textbf{a}(d,\phi)}{\partial
x\partial y}-|g|^{2}\frac{\partial\textbf{a}^{\text{H}}(d,\phi)}{\partial
x}\frac{\partial\textbf{a}(d,\phi)}{\partial y}\right\\},\vspace{-0.25cm}$
(34)

where $x$ and $y$ can be $d$ or $\phi$. Refer to (74)-(81) and some tedious
calculations, (33) and (34) can be obtained. The gain is then updated to

$\hat{\hat{g}}_{l^{\prime}}=\left(\textbf{a}(\hat{\hat{d}}_{l^{\prime}},\hat{\hat{\phi}}_{l^{\prime}})^{\text{H}}\textbf{r}_{r}\right)/\left\Arrowvert\textbf{a}(\hat{\hat{d}}_{l^{\prime}},\hat{\hat{\phi}}_{l^{\prime}})\right\Arrowvert^{2}.\vspace{-0.25cm}$
(35)

We accept a refinement only if the new residual power
$\left\Arrowvert\textbf{r}_{r}-\hat{\hat{g}}_{l^{\prime}}\textbf{a}(\hat{\hat{d}}_{l^{\prime}},\hat{\hat{\phi}}_{l^{\prime}})\right\Arrowvert^{2}$
is smaller than the old residual power
$\left\Arrowvert\textbf{r}_{r}-\hat{g}_{l^{\prime}}\textbf{a}(\hat{d}_{l^{\prime}},\hat{\phi}_{l^{\prime}})\right\Arrowvert^{2}$.

Note that, on the one hand, we can realize localization with the estimated
location parameters; on the other hand, we can design data transmission with
the reconstructed the channel between the BS and the UE by using (17). We can
reconstruct the channel between the BS and all different UEs by orthogonal
pilot signals. The spectral efficiency based on the reconstructed channel is
also analyzed in Section VI.

##  V Multi-user Communication

Figure 6: Multi-user communication with ExLens in spherical-wave scenarios.

The method proposed in Section IV can obtain the location parameters together
with the channel gains. Multiple UEs can be simultaneously served by the
ExLens system with the channels of all UEs reconstructed. In this section, we
analyze the multi-user communication performance of ExLens with limited RF
chains in the coexistence of near-field and far-field UE.

We assume that $K$ single-antenna UEs are served by a BS, which is equipped
with an ExLens with $N_{a}$ antenna elements (Fig. 6). We consider the multi-
path environment because of the presence of scatterers. For illustration
purposes, we only draw two paths for each UE (one line-of-sight (LoS) path and
one non-line-of-sight (NLoS) path) in Fig. 6 and omit the UE in the far-field.
Each path corresponds to an energy focusing window (FW) at the focal arc, and
the overlap of the FWs of different UEs introduces inter-user-interference
(IUI). We consider the narrow band mmWave multi-path channel model [11].
According to (17), the channel between the BS and the $k$-th UE can be
expressed as

$\textbf{h}_{k}=\sum\limits_{l=1}^{L_{k}}g_{kl}\textbf{a}(d_{kl},\phi_{kl}),\vspace{-0.25cm}$
(36)

where $L_{k}$ is the number of paths of the $k$-th UE, in which $l=1$
corresponds to the LoS path and $1<l\leq L_{k}$ corresponds to the NLoS path,
$g_{kl}$ is the complex path gain for the $l$-th path of the $k$-th UE,
$(d_{k1},\phi_{k1})$ are the position parameters of the $k$-th UE,
$(d_{kl},\phi_{kl})$ are the position parameters of the $l$-th scatterer of
the $k$-th UE with ($1<l\leq L_{k}$), and $\textbf{a}(d_{kl},\phi_{kl})$ is
the array response vector with elements given in (10). Let
$x_{k}=\sqrt{p_{k}}s_{k}$ represent the transmitted signal by the $k$-th UE,
where $\sqrt{p_{k}}$ denotes the transmitted power, and $s_{k}$ denotes the
independent information-bearing symbol with $\mathbb{E}\\{|s_{k}|^{2}\\}=1$.
The signal received at the BS is given as

$\tilde{\textbf{r}}=\sum\limits_{k=1}^{K}\mathbf{h}_{k}{x_{k}}+\textbf{n}=\sum\limits_{k=1}^{K}\sum\limits_{l=1}^{L_{k}}g_{kl}\textbf{a}(d_{kl},\phi_{kl}){x_{k}}+\textbf{n},\vspace{-0.25cm}$
(37)

where $\textbf{n}\in\mathbb{C}^{N_{a}\times 1}$ represents the Gaussian noise
with zero-mean and covariance matrix ${\sigma^{2}}\textbf{I}$.

We can reduce the number of RF chains for systems equipped with such array by
exploiting the energy focusing property of the ExLens in near-field and far-
field. We assume that only $M_{RF}$ RF chains are available, where
$M_{RF}<N_{a}$. Thus, antenna selection (e.g., by the low-complexity power-
based antenna selection method) must be applied. Let
$\textbf{W}_{\text{RF}}\in\mathbb{R}^{N_{a}\times M_{RF}}$ denote the power-
based antenna selection matrix, where the elements of $\textbf{W}_{\text{RF}}$
are $0$ or $1$. To avoid the scenario in which antenna selection favors nearby
UE over distant ones, we assume that the channel-inversion based power control
is applied during the antenna selection phase. Thus, the received signals at
the BS from different UEs have comparable strength. The signal received by the
selected antennas can be expressed as

$\textbf{W}_{\text{RF}}^{\text{H}}\tilde{\textbf{r}}=\sum\limits_{k=1}^{K}\textbf{W}_{\text{RF}}^{\text{H}}\mathbf{h}_{k}{x_{k}}+\textbf{W}_{\text{RF}}^{\text{H}}\textbf{n}.\vspace{-0.25cm}$
(38)

Given
$\tilde{\textbf{r}}_{s}\buildrel\Delta\over{=}\textbf{W}_{\text{RF}}^{\text{H}}\tilde{\textbf{r}}$,
$\mathbf{h}_{k,s}\buildrel\Delta\over{=}\textbf{W}_{\text{RF}}^{\text{H}}\mathbf{h}_{k}$,
and
$\textbf{n}_{s}\buildrel\Delta\over{=}\textbf{W}_{\text{RF}}^{\text{H}}\textbf{n}$,
we have

$\tilde{\textbf{r}}_{s}=\sum\limits_{k=1}^{K}\mathbf{h}_{k,s}{x_{k}}+\textbf{n}_{s},\vspace{-0.25cm}$
(39)

which can be rewritten as

$\tilde{\textbf{r}}_{s}=\mathbf{h}_{k,s}{x_{k}}+\sum\limits_{k^{\prime}\neq
k}^{K}\mathbf{h}_{k^{\prime},s}{x_{k^{\prime}}}+\textbf{n}_{s},\vspace{-0.25cm}$
(40)

where the term $\sum\limits_{k^{\prime}\neq
k}^{K}\mathbf{h}_{k^{\prime},s}{x_{k^{\prime}}}$ is the IUI for the $k$-th UE,
and $\textbf{n}_{s}\in\mathbb{C}^{M_{RF}\times 1}$ represents the Gaussian
noise at the selected antennas with zero-mean and covariance matrixc
${\sigma^{2}}\textbf{I}$. Let $\textbf{u}_{k}\in\mathbb{C}^{M_{RF}\times 1}$
represent the baseband combining vector for the $k$-th UE, where
$||\textbf{u}_{k}||=1$. The bandwidth-normalized achievable rate for the
$k$-th UE is given by

$R_{k}=\log_{2}\left(1+\dfrac{p_{k}|\textbf{u}_{k}^{\text{H}}\mathbf{h}_{k,s}|^{2}}{\sum\limits_{k^{\prime}\neq
k}^{K}{p_{k^{\prime}}}|\textbf{u}_{k}^{\text{H}}\mathbf{h}_{k^{\prime},s}|^{2}+\sigma^{2}}\right),\vspace{-0.25cm}$
(41)

and for all the $K$ UE, we obtain the sum-rate as

$R=\sum\limits_{k=1}^{K}R_{k}=\sum\limits_{k=1}^{K}\log_{2}\left(1+\dfrac{p_{k}|\textbf{u}_{k}^{\text{H}}\mathbf{h}_{k,s}|^{2}}{\sum\limits_{k^{\prime}\neq
k}^{K}{p_{k^{\prime}}}|\textbf{u}_{k}^{\text{H}}\mathbf{h}_{k^{\prime},s}|^{2}+\sigma^{2}}\right).\vspace{-0.25cm}$
(42)

In near-field and far-field scenarios, the ExLens has the energy focusing
ability. When the incident angles of different UEs are sufficiently separated,
the ExLens can resolve various UEs. For general systems where the BS cannot
resolve all the UEs perfectly, we apply the linear receivers described in the
following subsection to detect the signals from different UEs.

###  V-A Linear Receivers

The combining vector $\textbf{u}_{k}$ is applied to (40) to detect $s_{k}$.
First, we consider the MRC scheme, which disregards the IUI term in (40). In
this case, $\textbf{u}_{k}$ is designed to simply maximize the desired signal
power of the $k$-th UE, as given by

$\textbf{u}_{k}^{*}=\mathop{\arg\max}_{\parallel\textbf{u}_{k}\parallel=1}\mid\textbf{u}_{k}^{\text{H}}\mathbf{h}_{k,s}\mid^{2}.\vspace{-0.25cm}$
(43)

The optimal solution to (43) is

$\textbf{u}_{k}^{*}=\dfrac{\mathbf{h}_{k,s}}{\parallel\mathbf{h}_{k,s}\parallel}.\vspace{-0.25cm}$
(44)

The combining vector $\textbf{u}_{k}$ designed by the MRC scheme in (44) is
sub-optimal in general because it ignores the IUI.

To further mitigate the IUI, we apply the MMSE-based combining scheme. The
MMSE considers the interference and finds the $\textbf{u}_{k}$, which
minimizes the mean square error of the combined received and the desired
signals, given as

$\textbf{u}_{k}^{*}=\mathop{\arg\min}_{\parallel\textbf{u}_{k}\parallel=1}\mathbb{E}\\{\mid\textbf{u}_{k}^{\text{H}}\tilde{\mathbf{r}}_{s}-s_{k}\mid^{2}\\}.\vspace{-0.25cm}$
(45)

The optimal MMSE solution is

$\textbf{u}_{k}^{*}=\dfrac{\mathbf{R}_{rr}^{-1}\mathbf{R}_{rs}}{\parallel\mathbf{R}_{rr}^{-1}\mathbf{R}_{rs}\parallel},\vspace{-0.25cm}$
(46)

where
$\mathbf{R}_{rr}=\sum\limits_{k=1}^{K}p_{k}\mathbf{h}_{k,s}\mathbf{h}_{k,s}^{\text{H}}+\sigma^{2}\mathbf{I}$
denotes the autocorrelation matrix of received signal $\tilde{\mathbf{r}}_{s}$
and $\mathbf{R}_{rs}=\sqrt{p_{k}}\mathbf{h}_{k,s}$ denotes the cross-
correlation of the received signal $\tilde{\mathbf{r}}_{s}$ and the desired
signal $s_{k}$.

###  V-B Benchmark Schemes

We compare the multi-user communication performance of ExLens with a
conventional ULA in which both are illuminated by spherical wave-fronts. We
assume that both types of arrays have the same electrical aperture
$\tilde{D}_{y}$ and number of antenna elements
($N_{a}=2\lfloor\tilde{D}_{y}\rfloor+1$). Let ULA be placed along the y-axis
centered at the origin, and the space between two adjacent antenna elements is
$\Delta d={\lambda}/{2}$ (Fig. 1(b)). Take one UE for example, whose position
parameters are $(d,\phi)$, with $d$ denoting the distance between the UE and
the original point, and $\phi\in(-\pi/2,\pi/2)$ as the angle of the UE
relative to the x-axis. According to [22], the array response of the ULA
illuminated by the spherical wave-front is given by

$\textbf{a}(d,\phi)=\left(\dfrac{d}{d_{{}_{-N}}}e^{-jk_{0}(d_{{}_{-N}}-d)},\ldots,\dfrac{d}{d_{{}_{-1}}}e^{-jk_{0}(d_{{}_{-1}}-d)},1,\dfrac{d}{d_{{}_{1}}}e^{-jk_{0}(d_{{}_{1}}-d)},\ldots,\dfrac{d}{d_{{}_{N}}}e^{-jk_{0}(d_{{}_{N}}-d)}\right),\vspace{-0.15cm}$
(47)

where $d_{n}=\sqrt{d^{2}+n^{2}\Delta d^{2}-2nd\Delta d\sin\phi}$ and
$n\in\\{0,\pm 1,\ldots,\pm N\\}$. When $d\to\infty$, ${d}/{d_{n}}\to 1$ and
$d_{n}-d\to-n\Delta d\sin\phi$ are clearly observed. Then, the array response
illuminated by the spherical wave-front given in (47) reduces to that
illuminated by plane wave-front. Thus, the array response given in (47) for
ULA can be used for near-field and far-field scenarios. We obtain the signal
received by the ULA with a spherical wave-front by substituting (47) into (37)
with $K$ UE. We assume that the ULA is equipped with $M_{RF}$ RF chains. Thus,
antenna selection is necessary. However, given that the optimal antenna
selection scheme for the ULA system illuminated by the spherical wave-front
with multi-users is unknown in general, we apply two analog combining matrix
design schemes as benchmarks for a remarkable comparison. In the first
benchmark, we adopt the power-based antenna selection because of its
simplicity. Then, we apply the MRC and MMSE-based digital combining vector
design schemes to the ULA system. However, when $M_{RF}$ is small, the
performance for ULA is limited because of the limited array gain with the
small number of antennas selected. Thus, we also consider the second benchmark
by applying the approximate Gram Schmidt-based hybrid precoding scheme to
design the analog combining matrix for ULA [40]. To mitigate the IUI, the
MMSE-based digital combining vector design scheme is then applied.

##  VI Numerical Results

###  VI-A Localization Performance

In this subsection, we discuss the performance of the proposed localization
method. We define
$\mbox{CRLB}(\mathbf{d})=\sqrt{\sum_{k=1}^{L}{\bf{F}}^{-1}(\bm{\eta})_{(2k-1,2k-1)}}$
for comparison, where ${\bf{F}}^{-1}(\bm{\eta})_{(2k-1,2k-1)}$ denote the
$(2k-1,2k-1)$-th element of the matrix ${\bf{F}}^{-1}(\bm{\eta})$. Similarly,
we define
$\mbox{CRLB}({\bm{\phi}})=\sqrt{\sum_{k=1}^{L}{\bf{F}}^{-1}(\bm{\eta})_{(2k,2k)}}$.
The PEB($\mathbf{u}$) is calculated according to (22). Let $\mathbf{x}$
represent $\mathbf{d}$, ${\bm{\phi}}$, or $\mathbf{u}$, and
$\hat{\mathbf{x}}_{i}$ denote the estimate of $\mathbf{x}$ at the $i$-th Monte
Carlo simulation. The RMSE is defined as
$\mbox{RMSE}(\mathbf{x})=\sqrt{\sum_{i=1}^{T}||\hat{\mathbf{x}}_{i}-\mathbf{x}||^{2}/T}$.
We define the normalized mean square error (NMSE) as
$\sum_{i=1}^{T}||\mathbf{h}-\hat{\mathbf{h}}_{i}||^{2}/||\mathbf{h}||^{2}/T$
for channel estimation, where $\hat{\mathbf{h}}_{i}$ is the estimate of
$\mathbf{h}$ at the $i$-th Monte Carlo simulation. All numerical results
provided in this subsection are obtained from $T=1000$ independent Monte Carlo
simulations. The position parameters $(d,\phi)$ of a UE is generated with
$d\sim\mathcal{U}[7,30]\,m$ and $\phi\sim\mathcal{U}[-\pi/5,\pi/5]$ rad, where
$\mathcal{U}$ denotes the uniform distribution. The settings for the ExLens
are fixed to $D_{y}=1\,m$, $F_{0}=F=5\,m$, $\lambda=0.01\,m$, and $N_{a}=201$.

Fig. 7 demonstrates that (a) RMSE versus CRLB for the estimate of
$\mathbf{d}$; (b) RMSE versus CRLB for the estimate of ${\bm{\phi}}$; (c) RMSE
versus PEB for the estimate of $\mathbf{u}$; and (d) NMSE for the channel
estimation. First, we observe an improvement in the estimation accuracy of
proposed method (denoted as NOMP) with respect to DOMP, where DOMP reaches
performance floors with the increase in SNR. The performance plateau shows a
fundamental algorithmic limitation of DOMP, and highlights the critical role
of cyclic Newton refinements in NOMP, as explained in [38]. Second, NOMP does
not achieve the CRLB in the estimate of $\mathbf{d}$ (Fig. 7(a)), but it
closely follows the bound for all SNRs. In the estimate of ${\bm{\phi}}$ (Fig.
7(b)), NOMP can achieve the CRLB. The array response of the ExLens is more
sensitive to the changes in $\phi$ than that in $d$ in spherical scenarios.
Accordingly, NOMP performs better in ${\bm{\phi}}$ estimation than
$\mathbf{d}$ estimation. Third, the performance of $\mathbf{u}$ estimates
(Fig. 7(c)) shows similar trends to that of $\mathbf{d}$ (Fig. 7(a)). The
estimation error of position $\mathbf{u}$ is mainly determined by the
estimation error of $\mathbf{d}$ because the estimate of ${\bm{\phi}}$
achieves CRLB. In the simulation settings, the proposed localization method
can achieve meter-, decimeter-, and centimeter-level accuracies when SNR $>0$,
$>20$, and $>40$ dB, respectively. Moreover, Fig. 7(d) shows that the proposed
method performs well in channel estimation, thereby demonstrating that the
channel and location parameter estimation can be simultaneously performed in
spherical-wave scenarios by directly reusing the communication signals.

Figure 7:  (a) RMSE versus CRLB for the estimate of $\mathbf{d}$. (b) RMSE
versus CRLB for the estimate of ${\bm{\phi}}$. (c) RMSE versus PEB for the
estimate of $\mathbf{u}$. (d) NMSE for the estimate of $\mathbf{h}$. The
settings for the ExLens are fixed to $D_{y}=1\,m$, $F_{0}=F=5\,m$,
$\lambda=0.01\,m$, and $N_{a}=201$. In the case $L=1$, $d=16.8837\,m$ and
$\phi=0.0693$ rad. In the case $L=2$, $d_{1}=12.8657\,m$, $\phi_{1}=-0.1935$
rad, $d_{2}=14.4962\,m$, and $\phi_{2}=0.1897$ rad.

###  VI-B Multi-user Communication Performance

In this subsection, we compare the multi-user communication performance of
ExLens with that of the conventional ULA antenna array. In the following
simulations, the ExLens and ULA systems serve near-field and far-field UE
simultaneously. We assume that ExLens and ULA have the same electrical
aperture $\tilde{D}_{y}$ and number of antenna elements
$N_{a}=2\lfloor\tilde{D}_{y}\rfloor+1$. For the approximate Gram Schmidt-based
hybrid precoding scheme applied to the benchmark ULA system, the size of the
beamsteering codebook $N_{cb}$ is set to $1024$, and the resolution of the
phase shifters in the analog combining network is assumed to be $10$ bits. All
numerical results provided in this section are obtained from Monte Carlo
simulations with $1000$ independent channel realizations. The position
parameters $(d,\phi)$ of a UE is generated with $d\sim\mathcal{U}[20,320]\,m$
and $\phi\sim\mathcal{U}[-\pi/5,\pi/5]$ rad. The low-complexity power-based
antenna selection method is applied to “LENS MMSE”, “LENS MRC”, “ULA MMSE”,
and “ULA MRC”, and the Gram Schmidt-based analog combining method is applied
to “ULA GS MMSE”.

Figure 8: Comparison of the spectral efficiencies for single-user with
different SNRs, where $K=1$, $M_{RF}=5$, $L=2$, $F=5\,m$, $F_{0}=15\,m$,
$\tilde{D}_{y}=100$, $\lambda=0.01\,m$, and $N_{a}=201$.

Figure 9: Comparison of the spectral efficiencies for multi-user scenarios
with different SNRs, where $K=5$, $M_{RF}=25$, $L_{k}=2$, $F=5\,m$,
$F_{0}=15\,m$, $\tilde{D}_{y}=100$, $\lambda=0.01\,m$, and $N_{a}=201$.

Figs. 9 and 9 compare the spectral efficiencies of different schemes with
varying SNRs. The single-user (Fig. 9 with $K=1$, $L_{k}=2$, $M_{RF}=5$) and
multi-user (Fig. 9 with $K=5$, $L_{k}=2$, $M_{RF}=25$) scenarios are
considered. The other parameters for ExLens are fixed to $F=5\,m$,
$F_{0}=15\,m$, $\tilde{D}_{y}=100$, $\lambda=0.01\,m$, and $N_{a}=201$. In the
benchmark ULA system, we assume that perfect CSI is available at the BS. In
the ExLens system, we consider both cases with perfect and estimated CSIs. In
the ExLens system with a limited number of RF chains, we apply the power based
antenna selection before channel estimation. The channel is then estimated by
the method proposed in Section IV-B with the received signal from the selected
antennas. First, the spectral efficiency of the ExLens systems outperforms the
ULA systems because most energy of the received signal is concentrated on the
selected antennas for the ExLens systems with the energy focusing property. By
contrast, the energy in the ULA system is almost evenly spread across each
antenna. The simple power-based antenna selection method causes significant
energy loss for the ULA systems, thereby resulting in poor performance in
terms of spectral efficiency. Second, in single-user scenarios (Fig. 9), the
“ULA GS MMSE” scheme outperforms the simple power-based antenna selection
schemes “ULA MMSE” and “ULA MRC”. This phenomenon is expected because the
approximate Gram Schmidt-based hybrid precoding method considers the channel
characteristic when it is applied to the “ULA GS MMSE” scheme. However, the
performance of the “ULA GS MMSE” scheme with much higher computational
complexity is still worse than that of “LENS MMSE” and “LENS MRC” schemes.
Given the absence of the IUI for single-user scenarios, the MRC and MMSE
schemes have the same performance. In multi-user scenarios (Fig. 9), the
advantages of the ExLens systems are more pronounced over the ULA systems. The
MRC schemes perform worse than the MMSE schemes, especially for high SNRs, due
to the presence of the IUI. Lastly, the performance of the “LENS MMSE” and
“LENS MRC” schemes with the estimated CSI is close to that based on perfect
CSI, thereby showing the effectiveness of the proposed channel estimation
method.

Figure 10: Comparison of the spectral efficiencies for single-user and multi-
user scenarios with different number of RF chains, where $L_{k}=2$ for the
$k$-th UE, $F=5\,m$, $F_{0}=15\,m$, $\tilde{D}_{y}=100$, SNR$=10$ dB,
$\lambda=0.01\,m$, and $N_{a}=201$.

Figure 11: Comparison of the spectral efficiencies for multi-user scenarios
with different number of UE, where $L_{k}=2$ for the $k$-th UE, $M_{RF}=20$,
$F=5\,m$, $F_{0}=15\,m$, $\tilde{D}_{y}=100$, SNR$=10$ dB, $\lambda=0.01\,m$,
and $N_{a}=201$.

Then, we compare the spectral efficiencies of different schemes by increasing
the number of RF chains for both single-user ($K=1$) and multi-user ($K=5$)
scenarios (Fig. 11). As it benefits from the energy focusing property of
ExLens, the “LENS MMSE” scheme always outperforms the ULA schemes for
different numbers of RF chains. For single-user scenarios, as $M_{RF}$
increases, the spectral efficiencies of different schemes improve. When
$M_{RF}=15$, the performance of the ExLens schemes almost reaches maximum. The
ULA schemes require much more RF chains to achieve a similar performance.
Therefore, the energy focusing property of ExLens is beneficial for reducing
the number of RF chains, and this outcome helps to significantly reduce the
signal processing complexity and hardware cost without notable performance
degradation. For the ULA schemes, “ULA GS MMSE” shows advantages over other
schemes when $M_{RF}$ is small, but with further increase in $M_{RF}$, the
performance of “ULA GS MMSE” saturates, a situation which is also explained in
[40]. For multi-user scenarios, the number of RF chains required by the “LENS
MMSE” to achieve the optimal performance increases to around $45$ when $K$
increases to $5$. Thus, more RF chains are needed to distinguish more UEs.
However, compared with the total number of antenna elements, the number of RF
chains needed by the ExLens system remains low ($45<201$). The advantage of
the “LENS MMSE” scheme is more evident than the “ULA MMSE” scheme with a
smaller number of RF chains ($M_{RF}<45$).

The performance of spectral efficiency versus the number of served UE for
different schemes is shown in Fig. 11, by fixing the number of the RF chains
to $20$, the value of SNR to $10$ dB, and parameters for ExLens to $F=5\,m$,
$F_{0}=15\,m$, and $\tilde{D}_{y}=100$. The “LENS MMSE” scheme always has the
highest spectral efficiency among all others. Therefore, the UE resolution of
the ExLens system is greater than that of the ULA system with limited RF
chains. Since the IUI becomes larger as $K$ increases, the performance of the
MRC schemes become worse than that of the MMSE schemes. The beamsteering
codebook for the approximate Gram Schmidt scheme is designed for the single-
user systems of the ULA, where the analog combiner designed by the approximate
Gram Schmidt scheme exhibits larger deviation from the real channel as $K$
increases. Such higher deviation causes the worse performance of the “ULA GS
MMSE” than the “ULA MMSE” when $K>5$. The spectral efficiency of the “LENS
MMSE” scheme initially increases with $K$, then decreases when $K>17$. This
trend is because the ability of the ExLens system to serve UE becomes limited
given a number of RF chains.

Figure 12: Comparison of the spectral efficiencies for multi-user scenarios
with different antenna array aperture sizes, where $K=5$, $L_{k}=2$ for the
$k$-th UE, $M_{RF}=5$, $F=5\,m$, $F_{0}=15\,m$, SNR$=10$ dB, and
$\lambda=0.01\,m$.

Finally, we evaluate the spectral efficiencies of different schemes by
increasing the electrical aperture $\tilde{D}_{y}$ of the lens antenna array.
The results are shown in Fig. 12, with $K=5$, $L_{k}=2$ for the $k$-th UE,
$F=5\,m$, $F_{0}=15\,m$, $M_{RF}=5$, and SNR$=10$ dB. Channel inversion-based
power control during the antenna selection phase is applied for all
simulations. With unchanged total received power of all systems, as
$\tilde{D}_{y}$ increases, the total number of antenna elements increases for
ULA systems, and the energy at each antenna element decreases. Thus, with a
fixed number of RF chains, the total received energy decreases accordingly,
thereby leading to lower spectral efficiency for the ULA schemes. However,
ExLens systems present an interesting phenomenon, i.e., the spectral
efficiency of the ExLens schemes shows a trend of first increasing and then
decreasing. According to the change of the energy focusing effects from far-
field to near-field (Fig. 3), this phenomenon is easily understood. When
$\tilde{D}_{y}$ is very small, the “sinc” function holds, and the width of the
main lobe is $2/\tilde{D}_{y}$ and determines the system resolution to the UE.
As $\tilde{D}_{y}$ increases, the system resolution rises. Hence, the spectral
efficiency of the lens systems increases with $\tilde{D}_{y}$ initially. As
$\tilde{D}_{y}$ further increases, the “sinc” function no longer holds, and
the near-field effect becomes obvious. Then, the width of the focusing window
determines the system resolution to the UE. As $\tilde{D}_{y}$ further
increases, the system resolution decreases. Thus, the spectral efficiency of
the lens systems decreases with the further increase of $\tilde{D}_{y}$.
During this process, the ExLens system resolution to the UE will reach the
maximum at some value of $\tilde{D}_{y}$. Moreover, the optimal size of
$\tilde{D}_{y}$ is $60$ under the simulation configuration given in Fig. 12.
These observations are instructive for the design of the electrical aperture
size of the ExLens.

##  VII Conclusion

We considered the communication and localization problems with an ExLens.
First, we derived the closed-form antenna array response of ExLens by
considering the spherical wave-front for two different EM lens designs. The
relationship between the antenna array response of ExLens in the near-field
and far-field revealed that the derived near-field array response includes the
existing “sinc” function response as a special case. We further analyzed the
changes in the energy focusing properties from the far-field to the near-field
and the difference of the energy focusing properties of the two EM lens
designs. The window focusing property in the near-field also revealed the
great potential of ExLens for position sensing and multi-user communication.
The theoretical uplink localization ability of an ExLens was analyzed through
the Fisher information. To utilize the window focusing property for position
sensing, an effective location parameter estimation method was next proposed.
The results showed that the localization performance is close to the CRLB and
can be enhanced as the aperture of ExLens increase. In addition, the channel
can be effectively reconstructed by the proposed estimation method. Finally,
the multi-user communication performance of ExLens that serves UE in near-
field and far-field was investigated with perfect and estimated CSIs.
Simulation results verified the effectiveness of the proposed channel
estimation method and showed that the proposed ExLens with MMSE receiver
achieves significant spectral efficiency gains and complexity-and-cost
reductions compared with the ULA systems.

## Appendix A

In this section, we derive the array response of ExLens illuminated by a
spherical wave-front. For Design 1, by bringing (4) into (2), together with
$||\mathbf{u}-\mathbf{p}||=\sqrt{{d^{2}}+{y^{2}}-2dy\sin\phi}$,
$||\mathbf{p}-\mathbf{b}_{0}||=\sqrt{{F^{2}}+{y^{2}}}$,
$||\mathbf{p}-\mathbf{b}||=\sqrt{{F^{2}}+{y^{2}}+2yF\sin\theta}$,
$\eta(\mathbf{u},\mathbf{p})={\lambda/{\vphantom{\lambda{({2\pi\sqrt{{d^{2}}+{y^{2}}-2dy\sin\phi}})}}}{({4\pi\sqrt{{d^{2}}+{y^{2}}-2dy\sin\phi}})}}$,
and
$\kappa(\mathbf{p},\mathbf{b})={\lambda}/{(4\pi\\!\sqrt{{F^{2}}+{y^{2}}+2yF\sin\theta})}$,
we get

$r(\theta,d,\phi)=\int\limits_{-{{{D_{y}}}\mathord{\left/{\vphantom{{{D_{y}}}2}}\right.\kern-1.2pt}2}}^{{{{D_{y}}}\mathord{\left/{\vphantom{{{D_{y}}}2}}\right.\kern-1.2pt}2}}{\frac{\lambda^{2}{e^{-j{k_{0}}\sqrt{{d^{2}}+{y^{2}}-2dy\sin\phi}}}{e^{-j\left({{\rm{}}{\phi_{0}}+{k_{0}}\left({\sqrt{{F^{2}}+{y^{2}}+2yF\sin\theta}{\rm{-}}\sqrt{{F^{2}}+{y^{2}}}}\right)}\right)}}}{{16\pi^{2}\sqrt{{d^{2}}+{y^{2}}-2dy\sin\phi}\
\sqrt{{F^{2}}+{y^{2}}+2yF\sin\theta}}}}dy.\vspace{-0.25cm}$ (48)

To derive the closed form of (48), we have to make the following assumptions:
(A1) $d\gg y$ and (A2) $F\gg y$, where $y\in[-D_{y}/2,D_{y}/2]$. Given (A1),
we utilize Taylor series approximation and have

$\sqrt{{d^{2}}+{y^{2}}-2dy\sin\phi}\approx
d-y\sin\phi+{y^{2}}\frac{{{{\left({\cos\phi}\right)}^{2}}}}{{2d}}.\vspace{-0.25cm}$
(49)

Moreover, for the same reason, we have

$\frac{1}{{\sqrt{{d^{2}}+{y^{2}}-2dy\sin\phi}}}=\frac{1}{d}{\left({1+\frac{{{y^{2}}}}{{{d^{2}}}}-\frac{{2y}}{d}\sin\phi}\right)^{-\frac{1}{2}}}\approx\frac{1}{d}.\vspace{-0.25cm}$
(50)

Then, given (A2), we also utilize Taylor series approximation and obtain

$\sqrt{{F^{2}}+{y^{2}}+2yF\sin\theta}{\rm{-}}\sqrt{{F^{2}}+{y^{2}}}\approx\sqrt{{F^{2}}+{y^{2}}}\times\left[{\frac{{yF\sin\theta}}{{{F^{2}}+{y^{2}}}}{\rm{-}}\frac{{\rm{1}}}{{\rm{2}}}{{\left({\frac{{yF\sin\theta}}{{{F^{2}}+{y^{2}}}}}\right)}^{\rm{2}}}}\right],\vspace{-0.25cm}$
(51)

for further simplification, we obtain

$\sqrt{{F^{2}}+{y^{2}}+2yF\sin\theta}{\rm{-}}\sqrt{{F^{2}}+{y^{2}}}\approx
y\sin\theta-\frac{{{{\left({y\sin\theta}\right)}^{2}}}}{{2F}}.\vspace{-0.25cm}$
(52)

Moreover, for the same reason, we have

$\frac{1}{\sqrt{{F^{2}}+{y^{2}}+2yF\sin\theta}}=\frac{1}{F}{\left({1+\frac{{{y^{2}}}}{{{F^{2}}}}-\frac{{2y}}{F}\sin\theta}\right)^{-\frac{1}{2}}}\approx\frac{1}{F}.\vspace{-0.25cm}$
(53)

Substituting (49)-(53) into (48), we have

$r(\theta,d,\phi)\approx\int\limits_{-{{{D_{y}}}\mathord{\left/{\vphantom{{{D_{y}}}2}}\right.\kern-1.2pt}2}}^{{{{D_{y}}}\mathord{\left/{\vphantom{{{D_{y}}}2}}\right.\kern-1.2pt}2}}\frac{\lambda^{2}}{16\pi^{2}dF}{e^{-j{k_{0}}\left({d-y\sin\phi+{y^{2}}\frac{\cos^{2}\phi}{2d}}\right)}}e^{-j\left(\phi_{0}+k_{0}y\sin\theta-\frac{k_{0}y^{2}\sin^{2}\theta}{2F}\right)}dy.\vspace{-0.25cm}$
(54)

Rewritten (54), we get

$r(\theta,d,\phi)\approx\frac{\lambda^{2}{e^{-j\left({{k_{0}}d+{\phi_{0}}}\right)}}}{16\pi^{2}dF}\int\limits_{-{{{D_{y}}}\mathord{\left/{\vphantom{{{D_{y}}}2}}\right.\kern-1.2pt}2}}^{{{{D_{y}}}\mathord{\left/{\vphantom{{{D_{y}}}2}}\right.\kern-1.2pt}2}}{{e^{j{y^{2}}{k_{0}}\left({\frac{{{{\sin}^{\rm{2}}}\theta}}{{{\rm{2}}F}}-\frac{{{{\cos}^{2}}\phi}}{{2d}}}\right)}}{e^{-jy{k_{0}}\left({\sin\theta-\sin\phi}\right)}}}dy.\vspace{-0.25cm}$
(55)

Without loss of generality, we assume $\phi_{0}=2\pi$ for the first lens
design. Since $\phi_{0}$ is common for all antenna elements, the phase term
$e^{-j\phi_{0}}$ can be ignored. Denote
$\alpha=\frac{{\pi\sin^{2}\theta}}{{\lambda
F}}-\frac{{\pi\cos^{2}\phi}}{{\lambda d}}$, and
$\beta=({{\sin\theta-\sin\phi}})/{\lambda}$, we obtain

$r(\theta,d,\phi)\approx\frac{\lambda^{2}{e^{-j{{k_{0}}d}}}}{16\pi^{2}dF}\int\limits_{-{{{D_{y}}}\mathord{\left/{\vphantom{{{D_{y}}}2}}\right.\kern-1.2pt}2}}^{{{{D_{y}}}\mathord{\left/{\vphantom{{{D_{y}}}2}}\right.\kern-1.2pt}2}}{{e^{j\alpha{y^{2}}}}{e^{-j2\pi\beta
y}}}dy.\vspace{-0.25cm}$ (56)

For Design 2, with
$\lVert{\mathbf{c}_{0}}-\mathbf{p}\rVert=\sqrt{{F_{0}}^{2}+{y^{2}}}$, we have

$r(\theta,d,\phi)=\int\limits_{-{{{D_{y}}}\mathord{\left/{\vphantom{{{D_{y}}}2}}\right.\kern-1.2pt}2}}^{{{{D_{y}}}\mathord{\left/{\vphantom{{{D_{y}}}2}}\right.\kern-1.2pt}2}}{\frac{\lambda^{2}{e^{-j{k_{0}}(\sqrt{{d^{2}}+{y^{2}}-2dy\sin\phi}-\sqrt{{F_{0}}^{2}+{y^{2}}})}}{e^{-j\left({{\rm{}}{\phi_{0}}+{k_{0}}\left({\sqrt{{F^{2}}+{y^{2}}+2yF\sin\theta}{\rm{-}}\sqrt{{F^{2}}+{y^{2}}}}\right)}\right)}}}{{16\pi^{2}\sqrt{{d^{2}}+{y^{2}}-2dy\sin\phi}\
\sqrt{{F^{2}}+{y^{2}}+2yF\sin\theta}}}}dy.\vspace{-0.25cm}$ (57)

Similarly, without loss of generality, we assume $\phi_{0}-k_{0}F_{0}=2\pi$
for the second lens design. Since $\phi_{0}-k_{0}F_{0}$ is common for all
antenna elements, the phase term $e^{-j(\phi_{0}-k_{0}F_{0})}$ can be ignored.
The received signal can have the same approximate expression as (56) with
$\alpha=\frac{{\pi{{\sin}^{\rm{2}}}\theta}}{{\lambda
F}}-\frac{{\pi{{\cos}^{\rm{2}}}\phi}}{{\lambda
d}}+\frac{\pi}{{\lambda{F_{0}}}}$, as summarized in Table I. Let
${J_{a}}=\int\limits_{-{{{D_{y}}}\mathord{\left/{\vphantom{{{D_{y}}}2}}\right.\kern-1.2pt}2}}^{{{{D_{y}}}\mathord{\left/{\vphantom{{{D_{y}}}2}}\right.\kern-1.2pt}2}}{{e^{j\alpha{y^{2}}}}{e^{-j2\pi\beta
y}}dy}$, we have

${J_{a}}=\frac{{\sqrt{\pi}}}{{{\rm{2}}\sqrt{\alpha}}}{e^{-j\left({\frac{{{{\left({2\pi\beta}\right)}^{2}}}}{{4\alpha}}-\frac{5\pi}{4}}\right)}}\left({\mathrm{erf}\left({\frac{{\alpha{D_{y}}+2\pi\beta}}{{2\sqrt{\alpha}}}{e^{j\frac{{3\pi}}{4}}}}\right)+\mathrm{erf}\left({\frac{{\alpha{D_{y}}-2\pi\beta}}{{2\sqrt{\alpha}}}{e^{j\frac{{3\pi}}{4}}}}\right)}\right).\vspace{-0.25cm}$
(58)

Hence, we obtain

$r(\theta,d,\phi)\approx\frac{{\lambda^{2}{e^{-j{k_{0}d}}}}}{{16\pi^{2}dF}}J_{a}.\vspace{-0.25cm}$
(59)

Then, we define that the effective lens antenna array response on point
$\mathbf{b}=[F\cos\theta,-F\sin\theta]$ at the focal arc as
$a(\theta,d,\phi)=r(\theta,d,\phi)\times{16\pi^{2}dF}/({{\lambda^{2}{e^{-j{k_{0}d}}}}})$.
It then follows from (59) that we have

$a(\theta,d,\phi)\approx\frac{{\sqrt{\pi}}}{{{\rm{2}}\sqrt{\alpha}}}{e^{-j\left({\frac{{{{\left({2\pi\beta}\right)}^{2}}}}{{4\alpha}}-\frac{5\pi}{4}}\right)}}\left({\mathrm{erf}\left({\frac{{\alpha{D_{y}}+2\pi\beta}}{{2\sqrt{\alpha}}}{e^{j\frac{{3\pi}}{4}}}}\right)+\mathrm{erf}\left({\frac{{\alpha{D_{y}}-2\pi\beta}}{{2\sqrt{\alpha}}}{e^{j\frac{{3\pi}}{4}}}}\right)}\right),\vspace{-0.25cm}$
(60)

where $\beta={{(\sin\theta-\sin\phi)}}/{\lambda}$ and $\alpha$ is given in
Table I for different lens designs.

## Appendix B

In this section, we give the proof of Lemma 1. By substituting the definition
of $\mathrm{erf}(x)$ given in (9) into (8), and after some manipulations, we
have

$a(\theta,d,\phi)=-\left.{{{e^{j\frac{\pi}{4}}}\left({\int\limits_{0}^{\left({\frac{{{D_{y}}\sqrt{\alpha}}}{2}+\frac{{\pi\beta}}{{\sqrt{\alpha}}}}\right){e^{j\frac{{3\pi}}{4}}}}{{e^{-{t^{2}}}}dt}+\int\limits_{0}^{\left({\frac{{{D_{y}}\sqrt{\alpha}}}{2}-\frac{{\pi\beta}}{{\sqrt{\alpha}}}}\right){e^{j\frac{{3\pi}}{4}}}}{{e^{-{t^{2}}}}dt}}\right)}}\middle/{{(\sqrt{\alpha}{e^{j\frac{{{{\left({\pi\beta}\right)}^{2}}}}{\alpha}}})}}\right..\vspace{-0.25cm}$
(61)

The assumption of plane wave-front holds when $d\to\infty$ and
$F_{0}\to\infty$ (for the second lens design), and also with the assumption
that $F\gg y$, we can assume that in the far-field $\alpha\rightarrow 0$. Let
$x\triangleq\sqrt{\alpha}$, we have

$\lim\limits_{d,F_{0}\to\infty}a(\theta,d,\phi)=\mathop{\lim}\limits_{x\to
0}{\rm{-}}\left.{{{e^{j\frac{\pi}{4}}}\left({\int\limits_{0}^{\left({\frac{{{D_{y}}x}}{2}+\frac{{\pi\beta}}{x}}\right){e^{j\frac{{3\pi}}{4}}}}{{e^{-{t^{2}}}}dt}+\int\limits_{0}^{\left({\frac{{{D_{y}}x}}{2}-\frac{{\pi\beta}}{x}}\right){e^{j\frac{{3\pi}}{4}}}}{{e^{-{t^{2}}}}dt}}\right)}}\middle/{{x{e^{j\frac{{{{\left({\pi\beta}\right)}^{2}}}}{{{x^{2}}}}}}}}\right..\vspace{-0.25cm}$
(62)

Utilizing the L’Hopital’s rule, (62) can be simplified to

$\begin{array}[]{ll}&\mathop{\lim}\limits_{x\to
0}-\frac{{{e^{j\pi}}\left({\left({\frac{{{D_{y}}}}{2}-\frac{{\pi\beta}}{{{x^{2}}}}}\right){e^{j\left({\frac{{{D_{y}}^{2}{x^{2}}}}{4}+\frac{{{{\left({\pi\beta}\right)}^{2}}}}{{{x^{2}}}}+{D_{y}}\pi\beta}\right)}}+\left({\frac{{{D_{y}}}}{2}+\frac{{\pi\beta}}{{{x^{2}}}}}\right){e^{j\left({\frac{{{D_{y}}^{2}{x^{2}}}}{4}+\frac{{{{\left({\pi\beta}\right)}^{2}}}}{{{x^{2}}}}-{D_{y}}\pi\beta}\right)}}}\right)}}{{{e^{j\frac{{{{\left({\pi\beta}\right)}^{2}}}}{{{x^{2}}}}}}-2j\frac{{{{\left({\pi\beta}\right)}^{2}}}}{{{x^{2}}}}{e^{j\frac{{{{\left({\pi\beta}\right)}^{2}}}}{{{x^{2}}}}}}}}\\\
=&\mathop{\lim}\limits_{x\to
0}-\frac{{{e^{j\pi}}\left({\left({\frac{{{D_{y}}}}{2}{x^{2}}-\pi\beta}\right){e^{j\left({\frac{{{D_{y}}^{2}{x^{2}}}}{4}+{D_{y}}\pi\beta}\right)}}+\left({\frac{{{D_{y}}}}{2}{x^{2}}+\pi\beta}\right){e^{j\left({\frac{{{D_{y}}^{2}{x^{2}}}}{4}-{D_{y}}\pi\beta}\right)}}}\right)}}{{{x^{2}}-2j{{\left({\pi\beta}\right)}^{2}}}},\end{array}\vspace{-0.25cm}$
(63)

and with ${x\to 0}$, we have

$\mathop{\lim}\limits_{x\to
0}-\frac{{{e^{j\pi}}\left({\left({\frac{{{D_{y}}}}{2}{x^{2}}-\pi\beta}\right){e^{j\left({\frac{{{D_{y}}^{2}{x^{2}}}}{4}+{D_{y}}\pi\beta}\right)}}+\left({\frac{{{D_{y}}}}{2}{x^{2}}+\pi\beta}\right){e^{j\left({\frac{{{D_{y}}^{2}{x^{2}}}}{4}-{D_{y}}\pi\beta}\right)}}}\right)}}{{{x^{2}}-2j{{\left({\pi\beta}\right)}^{2}}}}=\frac{{{e^{j{D_{y}}\pi\beta}}-{e^{-j{D_{y}}\pi\beta}}}}{{2j\pi\beta}},\vspace{-0.25cm}$
(64)

since
$({{{e^{j{D_{y}}\pi\beta}}-{e^{-j{D_{y}}\pi\beta}}}})/({{2j\pi\beta}})={D_{y}}\mathrm{sinc}\left({{{{\tilde{D}_{y}}}}\sin\theta-{{{\tilde{D}_{y}}}}\sin\phi}\right)$,
obviously, we have

$\lim\limits_{d,F_{0}\to\infty}a(\theta,d,\phi)={D_{y}}\mathrm{sinc}\left({{{{\tilde{D}_{y}}}}\sin\theta-{{{\tilde{D}_{y}}}}\sin\phi}\right).\vspace{-0.25cm}$
(65)

## Appendix C

In this section, we give the proof of Lemma 2. To analyze the property of
${w(\theta,d,\phi)}$ in (14), we treat $\alpha$ and $\beta$ as a continuous
function of $\theta$. We firstly review the property of the $\mathrm{erf}(x)$
defined in (9), where $\mathrm{erf}(0)=0$, $\mathrm{erf}(\infty)=1$, and
$\mathrm{erf}(-x)=-\mathrm{erf}(x)$. According to (14), ${w(\theta,d,\phi)}$
is the sum of two $\mathrm{erf}$ functions, and the amplitude of
${w(\theta,d,\phi)}$ is shown in Fig. 3, which is similar to a rectangular
window function in the near-field. The edges of the window are determined by
zero points of two $\mathrm{erf}$ functions, namely $v_{1}$ and $v_{2}$. Take
the second lens design as an example, a given received waveform is shown in
the last two subfigures of Fig. 3, where $v_{1}$ is obtained when $\xi_{1}=0$,
and similarly $v_{2}$ is obtained when $\xi_{2}=0$. Note that
$\xi_{1}={\frac{{\alpha{D_{y}}+2\pi\beta}}{{2\sqrt{\alpha}}}{e^{j\frac{{3\pi}}{4}}}}$
and
$\xi_{2}={\frac{{\alpha{D_{y}}-2\pi\beta}}{{2\sqrt{\alpha}}}{e^{j\frac{{3\pi}}{4}}}}$.
Denote $v\buildrel\Delta\over{=}\sin\theta$, where $v\in(-1,1)$. Let
$\xi_{1}=0$, we have

${v^{2}}+\frac{{2F}}{{{D_{y}}}}v+\left({\frac{F}{{{F_{0}}}}-\frac{{2F\sin\phi}}{{{D_{y}}}}-\frac{{F{{\cos}^{2}}\phi}}{d}}\right)=0.\vspace{-0.25cm}$
(66)

Since $v\in(-1,1)$, the only solution of (66) is

${v_{1}}=-{{\frac{{F}}{{{D_{y}}}}+\sqrt{{{\left({\frac{{F}}{{{D_{y}}}}}\right)}^{2}}-\left({\frac{F}{{{F_{0}}}}-\frac{{2F\sin\phi}}{{{D_{y}}}}-\frac{{F{{\cos}^{2}}\phi}}{d}}\right)}}}.\vspace{-0.25cm}$
(67)

Similarly, let $\xi_{2}=0$, we have

${v^{2}}-\frac{{2F}}{{{D_{y}}}}v+\left({\frac{F}{{{F_{0}}}}+\frac{{2F\sin\phi}}{{{D_{y}}}}-\frac{{F{{\cos}^{2}}\phi}}{d}}\right)=0.\vspace{-0.25cm}$
(68)

Since $v\in(-1,1)$, the only solution of (68) is

${v_{\rm{2}}}={{\frac{{F}}{{{D_{y}}}}{\rm{-}}\sqrt{{{\left({\frac{{F}}{{{D_{y}}}}}\right)}^{2}}-\left({\frac{F}{{{F_{0}}}}{\rm{+}}\frac{{2F\sin\phi}}{{{D_{y}}}}-\frac{{F{{\cos}^{2}}\phi}}{d}}\right)}}}.\vspace{-0.25cm}$
(69)

According to (67) and (69), we can further obtain the center and width of the
focusing window. Let $v_{c}$ denote the center of the focusing window, we have

${v_{c}}=\frac{{{v_{1}}+{v_{2}}}}{2}=\frac{{\frac{{16F\sin\phi}}{{{D_{y}}}}}}{{\frac{{8F}}{{{D_{y}}}}\left({\sqrt{1-\left({\frac{{{D_{y}}^{2}}}{{F{F_{0}}}}+\frac{{2{D_{y}}\sin\phi}}{F}-\frac{{{D_{y}}^{2}{{\cos}^{2}}\phi}}{{Fd}}}\right)}+\sqrt{1-\left({\frac{{{D_{y}}^{2}}}{{F{F_{0}}}}-\frac{{2{D_{y}}\sin\phi}}{F}-\frac{{{D_{y}}^{2}{{\cos}^{2}}\phi}}{{Fd}}}\right)}}\right)}},\vspace{-0.25cm}$
(70)

since $F,F_{0},d\gg D_{y}$, we obtain

${v_{c}}\approx\left.\left({{\frac{{16F\sin\phi}}{{{D_{y}}}}}}\right)\middle/\left({{\frac{{16F}}{{{D_{y}}}}}}\right)\right.=\sin\phi.\vspace{-0.25cm}$
(71)

Let $\Delta v$ denote the width of the focusing window, we have

$\begin{array}[]{ll}\Delta
v\\!\\!\\!\\!\\!&=\left|{{v_{\rm{1}}}-{v_{\rm{2}}}}\right|\\\
&={{\left|{\frac{{F}}{{{D_{y}}}}\left({{\rm{1}}\\!-\\!\sqrt{{\rm{1}}-\left({\frac{{{D_{y}}^{2}}}{{F{F_{0}}}}+\frac{{{\rm{2}}{D_{y}}\sin\phi}}{F}-\frac{{{D_{y}}^{2}{{\cos}^{2}}\phi}}{{Fd}}}\right)}}\right){\rm{+}}\frac{{F}}{{{D_{y}}}}\left({{\rm{1}}\\!-\\!\sqrt{{\rm{1}}-\left({\frac{{{D_{y}}^{2}}}{{F{F_{0}}}}\\!-\\!\frac{{{\rm{2}}{D_{y}}\sin\phi}}{F}-\frac{{{D_{y}}^{2}{{\cos}^{2}}\phi}}{{Fd}}}\right)}}\right)}\right|}},\end{array}\vspace{-0.25cm}$
(72)

similarly, since $F,F_{0},d\gg D_{y}$, we get

$\begin{array}[]{ll}\Delta
v&\approx\dfrac{{{\left|{\frac{F}{{{D_{y}}}}\left({\frac{{{D_{y}}^{2}}}{{F{F_{0}}}}+\frac{{{\rm{2}}{D_{y}}\sin\phi}}{F}-\frac{{{D_{y}}^{2}{{\cos}^{2}}\phi}}{{Fd}}}\right){\rm{+}}\frac{F}{{{D_{y}}}}\left({\frac{{{D_{y}}^{2}}}{{F{F_{0}}}}-\frac{{{\rm{2}}{D_{y}}\sin\phi}}{F}-\frac{{{D_{y}}^{2}{{\cos}^{2}}\phi}}{{Fd}}}\right)}\right|}}}{2}\\\
&={D_{y}}\left|{\dfrac{1}{{{F_{0}}}}-\dfrac{{{{\cos}^{2}}\phi}}{d}}\right|.\end{array}\vspace{-0.25cm}$
(73)

The same procedure can be applied to obtain the approximate center and width
of the focusing window for the first lens design.

## Appendix D

For the $n$-th element in $\mathbf{a}(d_{l},\phi_{l})$, we have

$a_{n}(d_{l},\phi_{l})=am_{n}(d_{l},\phi_{l})\times
ph_{n}(d_{l},\phi_{l})\times w_{n}(d_{l},\phi_{l}),\vspace{-0.25cm}$ (74)

where $am_{n}(d_{l},\phi_{l})=\frac{{\sqrt{\pi}}}{{{\rm{2}}\sqrt{\alpha}}}$,
$ph_{n}(d_{l},\phi_{l})=e^{-j\left({\frac{{{{{\pi^{2}\beta^{2}}}}}}{{\alpha}}-\frac{5\pi}{4}}\right)}$,
and $w_{n}(d_{l},\phi_{l})$ is the discrete “window” function derived from
(14) by replacing $\sin\theta$ with $\sin\theta_{n}={n}/{N}$ in $\alpha$ and
$\beta$ for $n\in\\{0,\pm 1,\ldots,\pm N\\}$. We simplify
$am_{n}(d_{l},\phi_{l})$, $ph_{n}(d_{l},\phi_{l})$ and $w_{n}(d_{l},\phi_{l})$
as $am_{n}$, $ph_{n}$ and $w_{n}$. Then, we have

$\dfrac{{\partial a_{n}(d_{l},\phi_{l})}}{{\partial d_{l}}}=\dfrac{{\partial
am_{n}}}{{\partial d_{l}}}\times ph_{n}\times
w_{n}+am_{n}\times\dfrac{{\partial ph_{n}}}{{\partial d_{l}}}\times
w_{n}+am_{n}\times ph_{n}\times\dfrac{{\partial w_{n}}}{{\partial
d_{l}}},\vspace{-0.25cm}$ (75)

where

$\dfrac{{\partial am_{n}}}{{\partial
d_{l}}}=\dfrac{\pi\sqrt{\pi}\cos^{2}\phi_{l}}{4\lambda
d_{l}^{2}\alpha\sqrt{\alpha}},\vspace{-0.25cm}$ (76) $\dfrac{{\partial
ph_{n}}}{{\partial
d_{l}}}=e^{-j\left({\frac{{{{{\pi^{2}\beta^{2}}}}}}{{\alpha}}-\frac{5\pi}{4}}\right)}\dfrac{{{e^{j\frac{\pi}{2}}}{\pi^{3}}{\beta^{2}}{{\cos}^{2}}\phi_{l}}}{{\lambda{d_{l}^{2}}{\alpha^{2}}}},\vspace{-0.25cm}$
(77)

and

$\dfrac{{\partial w_{n}}}{{\partial
d_{l}}}=\dfrac{{\sqrt{\pi}{e^{j\frac{{3\pi}}{4}}}{{\cos}^{2}}\phi_{l}}}{{\lambda{d_{l}^{2}}\alpha}}\left({\zeta_{1}{e^{j\zeta^{2}_{2}}}+\zeta_{2}{e^{j\zeta^{2}_{1}}}}\right),\vspace{-0.4cm}$
(78)

where $\zeta_{1}=\frac{{\alpha{D_{y}}+2\pi\beta}}{{2\sqrt{\alpha}}}$ and
$\zeta_{2}=\frac{{\alpha{D_{y}}-2\pi\beta}}{{2\sqrt{\alpha}}}$. Similarly,
$\dfrac{{\partial a_{n}(d_{l},\phi_{l})}}{{\partial\phi_{l}}}$ can be obtained
with

$\dfrac{{\partial
am_{n}}}{{\partial\phi_{l}}}=-\dfrac{\pi\sqrt{\pi}\sin(2\phi_{l})}{4\lambda
d_{l}\alpha\sqrt{\alpha}},\vspace{-0.25cm}$ (79) $\dfrac{{\partial
ph_{n}}}{{\partial\phi_{l}}}=e^{-j\left({\frac{{{{{\pi^{2}\beta^{2}}}}}}{{\alpha}}-\frac{5\pi}{4}}\right)}\left(\dfrac{2\pi^{2}{e^{j\frac{\pi}{2}}}\beta\cos\phi}{\lambda\alpha}+\dfrac{\pi^{3}{e^{j\frac{\pi}{2}}}\beta^{2}\sin(2\phi)}{\lambda
d\alpha^{2}}\right),\vspace{-0.4cm}$ (80)

and

$\dfrac{{\partial
w_{n}}}{{\partial\phi_{l}}}=\dfrac{{\sqrt{\pi}{e^{j\frac{{3\pi}}{4}}}{{\sin}}(2\phi_{l})}}{{\lambda{d_{l}}\alpha}}\left({\zeta_{1}{e^{j\zeta^{2}_{2}}}+\zeta_{2}{e^{j\zeta^{2}_{1}}}}\right)+\dfrac{{2\sqrt{\pi}{e^{j\frac{{3\pi}}{4}}}{{\cos}}\phi_{l}}}{{\lambda\sqrt{\alpha}}}\left({{e^{j\zeta_{2}^{2}}}-{e^{j\zeta_{1}^{2}}}}\right).\vspace{-0.25cm}$
(81)

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