# Near- and Far-Field Communications with
Large Intelligent Surfaces

Andrea de Jesus Torres1, Luca Sanguinetti1, Emil Björnson2 L. Sanguinetti was
partially supported by the Italian Ministry of Education and Research (MIUR)
in the framework of the CrossLab project (Departments of Excellence).
1Dipartimento di Ingegneria dell’Informazione, University of Pisa, Italy
<EMAIL_ADDRESS><EMAIL_ADDRESS>2KTH Royal
Institute of Technology and Linköping University, Sweden<EMAIL_ADDRESS>
###### Abstract

This paper studies the uplink spectral efficiency (SE) achieved by two single-
antenna user equipments (UEs) communicating with a Large Intelligent Surface
(LIS), defined as a planar array consisting of $N$ antennas that each has area
$A$. The analysis is carried out with a deterministic line-of-sight
propagation channel model that captures key fundamental aspects of the so-
called geometric near-field of the array. Maximum ratio (MR) and minimum mean
squared error (MMSE) combining schemes are considered. With both schemes, the
signal and interference terms are numerically analyzed as a function of the
position of the transmitting devices when the width/height $L=\sqrt{NA}$ of
the square-shaped array grows large. The results show that an exact near-field
channel model is needed to evaluate the SE whenever the distance of
transmitting UEs is comparable with the LIS’ dimensions. It is shown that, if
$L$ grows, the UEs are eventually in the geometric near-field and the
interference does not vanish. MMSE outperforms MR for an LIS of practically
large size.

###### Index Terms:

Intelligent reflecting surface, reconfigurable intelligent surface,
metasurface, Massive MIMO, MIMO relays, power scaling law, near-field.

##  I Introduction

Large Intelligent Surface (LIS) refers to arrays with a massive number of
antennas in a compact space [1]. In its asymptotic form, it can be thought of
as a spatially-continuous electromagnetic aperture that actively generates
beamformed radio signals or receives them accordingly. Research on this topic
is performed under many different names [2], among them: holographic MIMO [3];
reconfigurable intelligent surface [4]; and software-defined surface (SDS)
[5]. A common practice in multiple antenna communications is to approximate
the received electromagnetic wave with a plane wave. This approximation is
valid when the terminal distance is much larger than the dimensions of the
array and brings to the well-known geometric far-field approximation. This
paper investigates the potential deficiencies of this approximation when an
LIS of size comparable to (or larger than) the distance from the transmitting
devices is considered. For brevity, we consider the deterministic line-of-
sight propagation channel from [6, 7], which allows to study the so-called
geometric near-field of the array by taking into account three fundamental
aspects: 1) the varying distance to the antennas in the LIS; 2) the varying
effective antenna areas; 3) the varying loss from polarization mismatch.
Unlike [6, 7], this channel model is used to evaluate the uplink spectral
efficiency (SE) achieved by two single-antenna user equipments (UEs) when
communicating with an LIS of square geometry, using either maximum ratio (MR)
or minimum mean squared error (MMSE) combining. Comparisons with the far-field
approximation will show that the exact near-field model is unarguably needed
when the LIS has comparable size to UE distances. We will also show that, as
the LIS size grows, the UEs will eventually be in the geometric near-field and
the interference will not vanish, which is different from what is
conventionally considered in the Massive MIMO literature. Moreover, it will be
shown that MMSE provides performance that is superior to MR for an LIS of
practically large size. This makes it the preferred combining scheme.

\begin{overpic}[width=260.17464pt,tics=10]{geometric_setup}
\put(-15.0,10.0){$\mathbf{s}=[x_{s},y_{s},z_{s}]^{\mbox{\tiny$\mathrm{T}$}}$}
\put(-5.0,75.0){Antenna $n$:}
\put(-10.0,67.0){$\mathbf{r}_{n}=[x_{n},y_{n},0]^{\mbox{\tiny$\mathrm{T}$}}$}
\put(-5.0,41.0){Distance}
\put(-42.0,32.0){\small$\sqrt{(x_{n}\\!-\\!x_{s})^{2}\\!+\\!(y_{n}\\!-\\!y_{s})^{2}\\!+\\!z_{s}^{2}}$}
\put(98.0,19.0){$X$} \put(68.0,66.0){$Y$} \put(39.0,0.0){$Z$} \end{overpic}
Figure 1: A source at an arbitrary location $(x_{s},y_{s},z_{s})$ transmits to
the LIS located in the $XY$-plane.

##  II System model

$\displaystyle\zeta_{\mathbf{s},\mathbf{r}_{n}}=\frac{1}{4\pi}\sum_{x\in\mathcal{X}_{s,n}}\sum_{y\in\mathcal{Y}_{s,n}}$
$\displaystyle\left(\frac{\frac{xy}{z_{s}^{2}}}{3\left(\frac{y^{2}}{z_{s}^{2}}+1\right)\sqrt{\frac{x^{2}}{z_{s}^{2}}+\frac{y^{2}}{z_{s}^{2}}+1}}+\frac{2}{3}\tan^{-1}\left(\frac{\frac{xy}{z_{s}^{2}}}{\sqrt{\frac{x^{2}}{z_{s}^{2}}+\frac{y^{2}}{z_{s}^{2}}+1}}\right)\right)$
(3)

We consider the LIS shown in Fig. 1 consisting of $N$ antennas that each has
area $A$. The antennas have size $\sqrt{A}\times\sqrt{A}$ and are equally
spaced on a $\sqrt{N}\times\sqrt{N}$ grid. The antennas are deployed edge-to-
edge, thus the total area of the LIS is $NA$. The LIS is centered around the
origin in the $XY$-plane. If we number the antennas from left to right, row by
row, according to Fig. 1, the $n$th receive antenna for $n=1,\ldots,N$ is
located at $\mathbf{r}_{n}=[x_{n},y_{n},0]^{\mbox{\tiny$\mathrm{T}$}}$ where

$\displaystyle x_{n}$
$\displaystyle=-\frac{(\sqrt{N}-1)\sqrt{A}}{2}+\sqrt{A}\,\mathrm{mod}(n-1,\sqrt{N})$
(1) $\displaystyle y_{n}$
$\displaystyle=\frac{(\sqrt{N}-1)\sqrt{A}}{2}-\sqrt{A}\left\lfloor\frac{n-1}{\sqrt{N}}\right\rfloor.$
(2)

###  II-A Channel model

The following lemma comes from [6] and extends prior work in [7] to provide a
general way of computing channel gains to each of the $N$ antenna elements of
the LIS.

######  Lemma 1.

Consider a lossless isotropic antenna located at
$\mathbf{s}=[x_{s},y_{s},z_{s}]^{\mbox{\tiny$\mathrm{T}$}}$ that transmits a
signal that has polarization in the $Y$ direction when traveling in the $Z$
direction. The free-space channel gain $\zeta_{\mathbf{s},\mathbf{r}_{n}}$ at
the $n$th receive antenna, located at
$\mathbf{r}_{n}=[x_{n},y_{n},0]^{\mbox{\tiny$\mathrm{T}$}}$, is given by (3)
(at top of next page) where

$\displaystyle\mathcal{X}_{s,n}$
$\displaystyle=\left\\{\sqrt{A}/2+x_{n}-x_{s},\sqrt{A}/2-x_{n}+x_{s}\right\\}$
(4) $\displaystyle\mathcal{Y}_{s,n}$
$\displaystyle=\left\\{\sqrt{A}/2+y_{n}-y_{s},\sqrt{A}/2-y_{n}+y_{s}\right\\}.$
(5)

Lemma 1 is important when quantifying the channel gain in the so-called
geometric near-field of the array [4, 8],111Note that we assume throughout
this paper that $||{\bf s}-{\bf r}_{n}||\gg\lambda$, so the system does not
operate in the reactive near-field of the transmit antenna (even if it is in
the geometric near-field of the array). In fact, this assumption must be made
to derive the expression in Lemma 1; see [7] for details. because it takes
into account the three fundamental properties that makes it different from the
far-field: 1) the distance to the elements varies over the array; 2) the
effective antenna areas vary since the element are seen from different angles;
3) the loss from polarization mismatch varies since the signals are received
from different angles. We will use Lemma 1 in the remainder.

###  II-B Signal model

We consider two single-antenna UEs that communicate with the LIS in Fig. 1
under the following assumption, shown in Fig. 2. This setup is sufficient to
demonstrate a few key results.

######  Assumption 1.

UE $k$ for $k=1,2$ is located in the $XZ$-plane at distance $d_{k}$ from the
center of the array with angle $\theta_{k}\in[-\pi/2,\pi/2]$. Both UEs send a
signal that has polarization in the $Y$ direction when traveling in the $Z$
direction.

We denote by
$\mathbf{h}_{k}=[h_{k1},\ldots,h_{kN}]^{\mbox{\tiny$\mathrm{T}$}}\in\mathbb{C}^{N}$
for $k=1,2$ the channel between UE $k$ and the LIS. Particularly,
$h_{kn}=|h_{kn}|e^{-j\phi_{kn}}$ is the channel from the source to the $n$th
receive antenna with $|h_{kn}|^{2}\in[0,1]$ being the channel gain and
$\phi_{kn}\in[0,2\pi)$ the phase shift. Following the geometry stated in
Assumption 1, the two UEs are located at (see Fig. 2)

$\displaystyle\mathbf{s}_{1}$
$\displaystyle=[x_{s_{1}},y_{s_{1}},z_{s_{1}}]^{\mbox{\tiny$\mathrm{T}$}}=[d_{1}\sin(\theta_{1}),0,d_{1}\cos(\theta_{1})]^{\mbox{\tiny$\mathrm{T}$}}$
(6) $\displaystyle\mathbf{s}_{2}$
$\displaystyle=[x_{s_{2}},y_{s_{2}},z_{s_{2}}]^{\mbox{\tiny$\mathrm{T}$}}=[d_{2}\sin(\theta_{2}),0,d_{2}\cos(\theta_{2})]^{\mbox{\tiny$\mathrm{T}$}}.$
(7)

From Lemma 1, the following corollaries are found.

######  Corollary 1 (Exact model).

Under Assumption 1, the channel $h_{kn}=|h_{kn}|e^{-j\phi_{kn}}$ to the $n$th
receive antenna is obtained as

$|h_{kn}|^{2}=\zeta_{{\bf s}_{k},{\bf r}_{n}},\hskip
5.69046pt\phi_{kn}=2\pi\,\mathrm{mod}\Bigg{(}\frac{{||\mathbf{s}_{k}-\mathbf{r}_{n}||}}{\lambda},1\Bigg{)}.$
(8)

######  Corollary 2 (Far-field approximation).

Under Assumption 1, if UE $k$ is in the geometric far-field of the array, in
the sense that $d_{k}\cos(\theta_{k})\gg\sqrt{NA}$, then
$h_{kn}=|h_{kn}|e^{-j\phi_{kn}}$ is obtained as

$|h_{kn}|^{2}\\!=\\!A\frac{\cos(\theta_{k})}{4\pi d_{k}^{2}},\hskip
2.84544pt\phi_{kn}\\!=\\!2\pi\,\mathrm{mod}\Big{(}\frac{d_{k}\\!-\\!x_{n}\sin(\theta_{k})}{\lambda},1\\!\Big{)}.$
(9)

The received signal ${\bf y}\in\mathbb{C}^{M\times 1}$ at the LIS is
$\mathbf{y}=s_{1}\mathbf{h}_{1}+s_{2}\mathbf{h}_{2}+\mathbf{n}$ where
$s_{i}\sim\mathcal{N}_{\mathbb{C}}({0},p_{i})$ is the data signal from UE $i$
and $\mathbf{n}\in\mathbb{C}^{M\times N}$ is thermal noise with i.i.d.
elements distributed as $\mathcal{N}_{\mathbb{C}}(0,\sigma^{2})$. We define
the average received signal-to-noise ratio (SNR) of UE $i$ as
${\mathsf{SNR}}_{i}={p_{i}}/{\sigma^{2}}$. The channels are deterministic and
thus can be estimated arbitrarily well from pilot signals. Hence, perfect
channel state information is assumed. The impact of imperfect knowledge of the
interfering UE channel will be investigated in Section III-D.

\begin{overpic}[width=303.53267pt,tics=10]{figure_from_above}
\put(42.0,50.0){LIS} \put(30.0,20.0){$d_{1}$} \put(20.0,32.0){$d_{2}$}
\put(32.0,28.0){$\theta_{2}$} \put(40.0,21.0){$\theta_{1}$}
\put(48.0,1.0){$Z$} \put(95.0,40.0){$X$} \put(18.0,0.0){UE 1}
\put(2.0,15.0){UE 2} \end{overpic} Figure 2: The two UEs are located in the
$XZ$-plane at distances $d_{k}$ for $k=1,2$ and have angles $\theta_{k}$ for
$k=1,2$.

$\displaystyle||\mathbf{h}_{k}||^{2}=\xi_{d_{k},\theta_{k},N}=$
$\displaystyle\sum_{i=1}^{2}\Bigg{(}\frac{B_{k}+(-1)^{i}\sqrt{B_{k}}\tan(\theta_{k})}{6\pi(B_{k}+1)\sqrt{2B_{k}+\tan^{2}(\theta_{k})+1+2(-1)^{i}\sqrt{B_{k}}\tan(\theta_{k})}}$
$\displaystyle+\frac{1}{3\pi}\tan^{-1}\Bigg{(}\frac{B_{k}+(-1)^{i}\sqrt{B_{k}}\tan(\theta_{k})}{\sqrt{2B_{k}+\tan^{2}(\theta_{k})+1+2(-1)^{i}\sqrt{B_{k}}\tan(\theta_{k})}}\Bigg{)}\Bigg{)}$
(15)

To detect $s_{1}$ from $\mathbf{y}$, the LIS uses the combining vector
$\mathbf{v}_{1}\in\mathbb{C}^{M}$, multiplied by the vector $\mathbf{y}$. By
treating the interference as noise, the SE for UE $1$ is
$\mathsf{SE}_{1}=\log_{2}\left(1+\gamma_{1}\right)$ where

$\displaystyle\gamma_{1}=\frac{\mathsf{SNR}_{1}|\mathbf{v}_{1}^{\mbox{\tiny$\mathrm{H}$}}{\mathbf{h}}_{1}|^{2}}{\mathsf{SNR}_{2}|\mathbf{v}_{1}^{\mbox{\tiny$\mathrm{H}$}}{\mathbf{h}}_{2}|^{2}+||{\mathbf{v}}_{1}||^{2}}$
(10)

is the signal-to-interference-and-noise ratio (SINR). We begin by considering
MR combining, defined as $\mathbf{v}_{1}=\mathbf{h}_{1}/||\mathbf{h}_{1}||$,
leading to

$\displaystyle\gamma_{1}^{\rm{MR}}\\!=\\!\\!\frac{\mathsf{SNR}_{1}||\mathbf{h}_{1}||^{2}}{\mathsf{SNR}_{2}\frac{{|\mathbf{h}_{1}^{\mbox{\tiny$\mathrm{H}$}}{\mathbf{h}}_{2}|^{2}}}{||\mathbf{h}_{1}||^{2}}+1}\\!\\!=\\!\mathsf{SNR}_{1}||\mathbf{h}_{1}||^{2}\left(1-\alpha^{\rm{MR}}\right)$
(11)

with
$\alpha^{\rm{MR}}=\frac{\mathsf{SNR}_{2}\frac{|\mathbf{h}_{1}^{\mbox{\tiny$\mathrm{H}$}}{\mathbf{h}}_{2}|^{2}}{||\mathbf{h}_{1}||^{2}}}{1+\mathsf{SNR}_{2}\frac{{|\mathbf{h}_{1}^{\mbox{\tiny$\mathrm{H}$}}{\mathbf{h}}_{2}|^{2}}}{||\mathbf{h}_{1}||^{2}}}\vspace{-.1cm}$.
The term

$\displaystyle\\!\\!\\!\mathsf{SNR}_{2}\frac{{|\mathbf{h}_{1}^{\mbox{\tiny$\mathrm{H}$}}{\mathbf{h}}_{2}|^{2}}}{||\mathbf{h}_{1}||^{2}}=\mathsf{SNR}_{2}\frac{\left|\sum_{n=1}^{N}|h_{1n}||h_{2n}|e^{j(\phi_{1n}-\phi_{2n})}\right|^{2}}{||\mathbf{h}_{1}||^{2}}$
(12)

accounts for the interference generated by UE 2 whereas
$\mathsf{SNR}_{1}||\mathbf{h}_{1}||^{2}$ in (11) represents the received SNR
in the absence of interference. Since MR does not do anything against the
interference, the term $\alpha^{\rm{MR}}$ in (11) must be interpreted as the
performance loss due to the presence of UE 2.

Instead of using the suboptimal MR combining, we note that $\gamma_{1}$ in
(10) is a generalized Rayleigh quotient with respect to ${\bf v}_{1}$ and thus
is maximized by MMSE combining:

$\displaystyle{\bf
v}_{1}=\Bigg{(}\sum\limits_{i=1}^{2}\mathsf{SNR}_{i}{\mathbf{h}}_{i}{\mathbf{h}}_{i}^{\mbox{\tiny$\mathrm{H}$}}+\mathbf{I}_{M}\Bigg{)}^{\\!-1}\\!\\!{\mathbf{h}}_{1}$
(13)

leading to

$\displaystyle\gamma_{1}^{\rm{MMSE}}=\mathsf{SNR}_{1}||\mathbf{h}_{1}||^{2}\big{(}1-\alpha^{\rm{MMSE}}\big{)}$
(14)

with
$\alpha^{\rm{MMSE}}=\frac{\mathsf{SNR}_{2}\frac{|\mathbf{h}_{1}^{\mbox{\tiny$\mathrm{H}$}}{\mathbf{h}}_{2}|^{2}}{||\mathbf{h}_{1}||^{2}}}{1+\mathsf{SNR}_{2}||\mathbf{h}_{2}||^{2}}$.

The SINR expression in (14) contains some of the same terms as in (11), but
has a different structure which makes it behave differently. Unlike
$\alpha^{\rm{MR}}$ in (11), the term $\alpha^{\rm{MMSE}}$ in (14) must be
interpreted as the performance loss encountered with MMSE when cancelling the
interference from UE 2.

##  III Near- and Far-Field Analysis

We now analyze the system in the near- and far-field cases. We begin by
reviewing the behavior of the channel gain [6], and then we will look into the
interference gain and spectral efficiency. From [6, Prop. 1], we have the
following result.

######  Proposition 1.

Under Assumption 1, we have that
$||\mathbf{h}_{k}||^{2}=\xi_{d_{k},\theta_{k},N}$ where
$\xi_{d_{k},\theta_{k},N}$ is given by (15) with

$\displaystyle B_{k}=\frac{NA}{4d_{k}^{2}\cos^{2}(\theta_{k})}.$ (16)

In the far-field case, it becomes

$\displaystyle||\mathbf{h}_{k}||^{2}=NA\frac{\cos(\theta_{k})}{4\pi
d_{k}^{2}}.$ (17)

\begin{overpic}[width=476.98463pt,tics=10]{Fig3_ChannelGain_vs_size.eps}
\put(15.0,32.0){\footnotesize{Upper limit $1/3$}} \end{overpic} Figure 3:
Behavior of the channel gain $||\mathbf{h}_{1}||^{2}$ using either the exact
model or the far-field approximation. The desired UE 1 has
$\theta_{1}=2^{\circ}$ and is at a distance $d_{1}=25\lambda=2.5$ m.

Note that both (15) and (17) depend on the total LIS area, $NA$. Hence, the
channel gain is independent of the wavelength. Since practical elements are
sub-wavelength-sized, the number of elements $N$ that is needed to achieve a
given channel gain is inversely proportional to $\lambda^{2}$ .

\begin{overpic}[width=476.98463pt,tics=10]{Fig4_Interferece_vs_size.eps}
\end{overpic} Figure 4: Behavior of the interference gain
$\frac{{|\mathbf{h}_{1}^{\mbox{\tiny$\mathrm{H}$}}{\mathbf{h}}_{2}|^{2}}}{||\mathbf{h}_{1}||^{2}}$
using either the exact model or the far-field approximation. The desired UE 1
is at $\theta_{1}=2^{\circ}$ while the interfering UE 2 is at
$\theta_{2}=-2^{\circ}$. Both UEs are at a distance $d_{1}=d_{2}=2.5$ m. Note
that the valleys not always reaching their correct value, $-\infty$ dB, due to
the limited numerical precision.

Fig. 3 shows the channel gain $||\mathbf{h}_{1}||^{2}$ as a function of the
size of the LIS, i.e., $L=\sqrt{NA}$, using either the exact expression in
(15) or the far-field approximation in (17). We consider a setup with
$\lambda=0.1$ m in which UE 1 has $\theta_{1}=2^{\circ}$ and
$d_{1}=25\lambda=2.5$ m. The results of Fig. 3 show that an LIS larger than
$L=1$ m is already enough to notice the approximation gap, whereas $L\geq 10$
m is needed to approach the upper limit of $1/3$ (this limit comes from
considering the polarization mismatch loss [6] and differs from the $1/2$
limit in [1]). This shows the importance of properly modeling the near-field.

###  III-A Interference gain

We now analyze the normalized interference gain
$\frac{{|\mathbf{h}_{1}^{\mbox{\tiny$\mathrm{H}$}}{\mathbf{h}}_{2}|^{2}}}{||\mathbf{h}_{1}||^{2}}$.
We notice that the computation of a closed-form expression with the exact
channel model is challenging while it takes the simple form with the far-field
approximation [9, Eq. (12)]

$\frac{{|\mathbf{h}_{1}^{\mbox{\tiny$\mathrm{H}$}}{\mathbf{h}}_{2}|^{2}}}{||\mathbf{h}_{1}||^{2}}=\frac{A\cos(\theta_{2})}{4\pi
d_{2}^{2}}\left|\frac{\sin(\pi
L\Omega/\lambda)}{\sin(\pi\sqrt{A}\Omega/\lambda)}\right|^{2}$ (18)

with $L=\sqrt{NA}$ and $\Omega=\sin(\theta_{2})-\sin(\theta_{1})$. Fig. 4
plots the the normalized interference gain
$\frac{{|\mathbf{h}_{1}^{\mbox{\tiny$\mathrm{H}$}}{\mathbf{h}}_{2}|^{2}}}{||\mathbf{h}_{1}||^{2}}$
as a function of $L$, using either the exact model and the far-field
approximation from Corollaries 1 and 2, respectively. We consider a setup with
$\lambda=0.1$ m in which the two UEs have different angles
$\theta_{1}=2^{\circ}$ and $\theta_{2}=-2^{\circ}$, but have the same distance
from the LIS, given by $d_{1}=d_{2}=2.5$ m. In line with the results of Fig.
3, the gap between the exact model and the far-field approximation is
noticeable when $L\geq 1$ m. We notice that $L>100$ m is needed with the exact
model to approach its upper limit. This is more than $3$ dB higher than the
upper limit with the far-field approximation.

###  III-B Spectral efficiency analysis

\begin{overpic}[width=476.98463pt,tics=10]{Fig5_SE_vs_size.eps}
\put(30.0,16.0){\vector(1,-1){3.0}} \put(25.0,16.5){\footnotesize{MMSE}}
\put(42.5,12.0){\footnotesize{MR}} \put(42.0,13.0){\vector(-1,0){3.0}}
\end{overpic} (a) Spectral efficiency versus $L=\sqrt{NA}$

\begin{overpic}[width=476.98463pt,tics=10]{Fig6_SE_vs_dx.eps}
\put(61.0,17.0){\vector(-1,-1){3.0}} \put(60.0,18.0){\footnotesize{MMSE}}
\put(47.5,12.0){\footnotesize{MR}} \put(52.0,13.0){\vector(2,1){3.0}}
\end{overpic} (b) Spectral efficiency versus $z=z_{s_{1}}=z_{s_{2}}$ when
$L=6$ m

Figure 5: SE behavior with MR and MMSE. The SE in the ideal interference-free
case is reported as reference.

Fig. 5(a) plots the SE achieved by UE 1 with MR and MMSE as a function of $L$,
using either the exact model or the far-field approximation from Corollaries 1
and 2. We consider the same setup of Fig. 4; that is, $\lambda=0.1$ m,
$\theta_{1}=2^{\circ}$, $\theta_{2}=-2^{\circ}$ and $d_{1}=d_{2}=2.5$ m. We
set $p_{1}=p_{2}=30$ dBm and $\sigma^{2}=0$ dBm. The SE
$\log_{2}(1+\mathsf{SNR}_{1}||\mathbf{h}_{1}||^{2})$ computed with the exact
model in the ideal interference-free case is also reported as a reference.
Fig. 5(a) shows that the SE saturates with both schemes as $L$ increases.
However, while MMSE quickly converges to the interference-free case, the
performance gap with MR is substantial with the exact model, while it
asymptotically vanishes with the far-field model as in [9, Sec. III-D].

\begin{overpic}[width=390.25534pt,tics=10]{Fig_Focusing_SE_MMSE.eps}
\put(50.0,35.0){\vector(-1,1){4.0}} \put(49.0,32.0){\footnotesize{Point-of-
interest}} \end{overpic} (a) SE with MMSE

\begin{overpic}[width=390.25534pt,tics=10]{Fig_Focusing_SE_MR.eps}
\put(50.0,35.0){\vector(-1,1){4.0}} \put(49.0,32.0){\footnotesize{Point-of-
interest}} \end{overpic} (b) SE with MR

Figure 6: SE in bit/s/Hz achieved by UE $1$ when the interfering UE $2$ is
transmitting at different locations over the $XZ-$plane. Both MMSE and MR are
considered.

Fig. 5(b) plots the SE achieved by UE 1 with MR and MMSE as a function of
$z=z_{s_{1}}=z_{s_{2}}$, using either the exact model or the far-field
approximation. We consider an LIS of size $L=6$ m, and set $p_{1}=p_{2}=30$
dBm and $\sigma^{2}=0$ dBm. MMSE provides the same SE as in the interference-
free case for $z\leq 10$ m, whereas it is lower for larger values. The SE gap
between MMSE and MR is relatively large for the values of $z$ of practical
interest, i.e., smaller than tens of meters. MMSE converges to MR only when
$z\geq 40$ m.

To gain further insights into the large performance gap between MMSE and MR in
the near-field, Fig. 6 shows the SE in bit/s/Hz when the desired UE $1$ is
fixed at $\theta_{1}=2^{\circ}$ $d_{1}=2.5$ m and the interfering UE $2$ is
transmitting from different locations over the $XZ-$plane, whose distance from
the point-of-interest is measured in wavelengths. We assume $\lambda=0.1$ m
and use an LIS of size $L=6$ m. Fig. 6(a) reveals that the SE with MMSE is low
only in an elliptic region around the point-of-interest, whose semi-major axis
(along $z$ direction) is roughly half-a-wavelength. This means that MMSE can
efficiently reject any interfering signal that comes from a location that is
at least half-a-wavelength away. On the contrary, Fig. 6(b) shows that a low
SE is achieved with MR for most of the locations the interfering UE is
transmitting from. This is because MR does not do anything against the
interference. The region where the SE achieves its minimum value is still an
ellipse but with a larger area. We see that the SE with both MMSE and MR
ranges from $\approx 7$ to $1$ bit/s/Hz (a reduction of $86\%$), but while in
the MMSE case the highest values are located in most of the observed area,
that is not the case when using MR.

\begin{overpic}[width=476.98463pt,tics=10]{Fig7_SE_Polarization.eps}
\put(28.0,22.0){\vector(1,-1){4.0}} \put(23.0,23.0){\footnotesize{MMSE}}
\put(43.0,13.0){\vector(-1,1){4.0}} \put(42.0,10.0){\footnotesize{MR}}
\end{overpic} Figure 7: Impact of polarization on the SE and interference gain
of UE $1$ for the same setup in Fig. 5(a).

###  III-C Impact of polarization mismatch

Unlike [1], the analysis and results above take into account the polarization
mismatch, which varies the received power of each antenna element since the
signals are received from different angles [6]. This has a double impact on
the system performance. Firstly, the global channel gain is reduced and
converges to $1/3$ instead of $1/2$ as $L$ grows large; see Fig. 3. Secondly,
the mutual interference between UEs changes due to the varying power footprint
induced by the polarization mismatch. To quantify the joint effect of these
two facts, Fig. 7 considers the same setup as in Fig. 5(a) and compares the SE
obtained with the exact model when ignoring or considering the polarization
mismatch loss. The two effects have a non-negligible impact on both MMSE and
MR when $L$ is large. For example, for $L=100$ the SE reduction is $6.5\%$
with MMSE, whereas it is $15\%$ with MR. However, for values of $L$ of
practical interest, i.e., in the range between $L=1$ and $L=10$ meters, the
polarization effects loss is limited to $5\%$.

###  III-D Imperfect knowledge of interfering UE channel

The SE analysis above reveals that MMSE performs much better than MR in both
the near- and far- fields. However, the analysis relies on perfect knowledge
of the channel vectors $\\{{\bf h}_{1},{\bf h}_{2}\\}$. We now evaluate the
robustness of MMSE against imperfect knowledge of ${\bf h}_{2}$ in (13). This
is done by assuming that the location of UE $2$ for the computation of ${\bf
h}_{2}$ in (13) is imperfectly known. Particularly, we assume that the
estimated position differs from the true one by an error that is uniformly
distributed in a circle of radius $r$, centered at the true position. Fig. 8
shows the average SE as a function of $r$, expressed in wavelengths with
$\lambda=0.1$ m, in the same setup of Fig. 5(a) with $L=6$ m. The average is
taken with respect to the randomly generated errors. We see that MR is
unaffected by the imperfect knowledge of the interfering UE position since MR
does not rely on the knowledge of ${\bf h}_{2}$. On the other hand, the SE
with MMSE decreases very fast. In line with Fig. 6(a), an estimation error of
half-a-wavelength is enough for a $65\%$ reduction. This calls for estimation
schemes with centimeter accuracy (having $\lambda=10$ cm) which is far beyond
what we can achieve with state-of-the-art solutions in wireless applications.

Figure 8: SE with the exact model when the position of the interfering UE is
uncertain within a radius $r$, measured in wavelenghts. We assume
$\lambda=0.1$, $L=6$ m and an inter-UE distance is $17.5$ cm$=1.75\,\lambda$.

##  IV Summary

This paper showed that a realistic assessment of the uplink SE achievable by
two single-antenna UEs communicating with an LIS requires the use of an exact
near-field channel model, whenever the distance of transmitting UEs is
comparable with the LIS size, $L$. It also showed that increasing $L$
unboundedly does not guarantee the suppression of interference with MR
combining, especially when the UEs are closely located in space. MMSE
combining is still needed to efficiently suppress the interference.
Particularly, the SE with MMSE quickly converges to the interference-free case
when the UEs are transmitting from a distance of few meters and values of $L$
of practical interest are considered, i.e., in the range $1\leq L\leq 10$
meters. However, the SE with MMSE deteriorates fast in the presence of channel
estimation errors. Lastly, we showed that the polarization mismatch should not
be ignored since it has a non-negligible impact on SE, especially when $L$
grows.

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