# Wide-Aperture MIMO via
Reflection off a Smooth Surface

Andrea Pizzo, _Member, IEEE_ , Angel Lozano, _Fellow, IEEE_ ,
Sundeep Rangan, _Fellow, IEEE_ , Thomas L. Marzetta, _Life Fellow, IEEE_
Parts of this work were presented at the IEEE Veh. Techn. Conf. (VTC’22
Spring) [1]. A. Pizzo and A. Lozano are with Univ. Pompeu Fabra, 08005
Barcelona, Spain (email: {andrea.pizzo, angel.lozano}@upf.edu). S. Rangan and
T. Marzetta are with New York University, 11201 New York, USA (email:
{s.rangan, tom.marzetta}@nyu.edu). Work supported by the European Research
Council under the H2020 Framework Programme/ERC grant agreement 694974, by the
ICREA Academia program, by the European Union-NextGenerationEU, and by the
Fractus-UPF Chair on Tech Transfer and 6G.

###### Abstract

This paper provides a deterministic channel model for a scenario where
wireless connectivity is established through a reflection off a smooth planar
surface of an infinite extent. The developed model is rigorously built upon
the physics of wave propagation and is as precise as tight are the
unboundedness and smoothness assumptions on the surface. This model allows
establishing how line-of-sight multiantenna communication is altered by a
reflection off an electrically large surface, a situation of high interest for
mmWave and terahertz frequencies.

##  I Introduction

The wealth of unexplored spectrum in the millimeter wave (mmWave) and
terahertz ranges brings an onrush of wireless research seeking its fortune at
higher frequencies [2, 3, 4]. The short range for which these frequencies are
most suitable, in conjunction with the tiny wavelength, enable reasonably
sized arrays to access multiple spatial degrees of freedom (DOF) even in line-
of-sight (LOS) [5]. Precisely, LOS spatial multiplexing is made possible by
the rich pattern of phase variations of the radiated field’s spherical
wavefront, which mimics the diversity richness of multipath propagation at
lower frequencies. This potential has unleashed much research activity on
wide-aperture multiple-input multiple-output (MIMO) communication over LOS
channels [6, 7].

A downside of these high frequencies is blockage and lack of diffraction
around obstacles, which may render LOS MIMO vulnerable to interruptions. This
naturally raises the interest in studying whether wide-aperture MIMO could
also operate through a reflection, capitalizing on the availability in many
environments of interest of surfaces that are electrically (i.e., relative to
the wavelength) large.

This paper seeks to examine MIMO communication via reflection off a smooth
planar surface of infinite extent. To this end, one possibility would be to
apply ray-tracing tools [8], but the accuracy to which the environment should
be characterized to prevent artifacts is not known a priori. Also, ray tracing
does not provide analytical insights into the underlying propagation
mechanisms, which are essential to array optimization. Instead, we derive a
deterministic physics-based scalar channel model that is valid irrespective of
the communication range and embodies other models as particular cases.

###  I-A Contributions

Although an actual reflecting surface is necessarily finite and with some
degree of roughness, at sufficiently high frequencies it may be reasonably
regarded as infinitely large, as the impact of diffraction vanishes.
Oppositely, the roughness is emphasized at high frequencies as irregularities
on the surface become comparable to the small wavelengths. The latter aspect
is not considered in this paper, left for future work.

Motivated by the extensive physics literature on the interaction between a
plane wave and an infinite smooth surface [9, 10], we start by expanding the
3D field generated by an arbitrary source in terms of plane waves [11, 10].
Fundamental principles describing the reflection and transmission phenomena at
the surface can then be applied to each plane wave separately and combined to
obtain the overall field at any point [10]. An LOS channel is seen to be the
cascade of a low-pass filter that cuts off evanescent waves [12], and a
reverse-bowl-shaped filter imposed by the wave equation [13]; a reflection off
a surface adds an additional filtering stage that augments the model in [12,
13] with backward propagation. This paper can also be seen to complement the
zero-mean stochastic model derived in [14], with their conjunction yielding a
Rician fading model.

After discretization through spatial sampling, a deterministic description of
the channel is obtained. This is finally used to numerically evaluate the
eigenvalues, DOF, and spectral efficiency for the purpose of MIMO
communication. Altogether, the contributions are:

  * •

Starting from first principles, a channel model is developed that builds upon
the physics of wave propagation. The analysis is as precise as tight are the
unboundedness and smoothness assumptions on the surface.

  * •

Progress is made, in the wake of [13, 14, 12], towards a comprehensive
physics-based modeling of wireless propagation on which signal processing and
communication theorists can test their algorithms. Propagation is described in
terms of spatial Fourier transforms and linear system theory, notions central
to both communities.

  * •

Classical electromagnetic results such as the image theorem are revisited.
These have fundamental implications on the optimization of antenna spacings as
a function of the signal-to-noise ratio (SNR) and they allow extending results
available for a pure LOS channel [15, 16] to a reflection channel.

###  I-B Outline and Notation

The manuscript is organized as follows. Sec. II revisits the physics behind
plane-wave reflection off a smooth planar surface relying solely on linear
system theory and Fourier transform. In Sec. III, the Fourier spectral
representations of the LOS and reflected transmissions are derived. The
connection with the image theorem is established in Sec. IV, whereas the
channel impulse response follows in Sec. V. After discretization, the channel
response is used in Sec. VI to assess the MIMO performance via reflection. A
comparison with ray-tracing is presented in Sec. VII. Final discussions and
possible extensions are set forth in Sec. VIII.

We use upper (lower) case letters for spatial-frequency (spatial) entities
while $J_{0}(\cdot)$ is the Bessel function of the first kind with order $0$,
$(x)^{+}=\max(x,0)$, and $\delta(\cdot)$ is the Dirac delta function.

##  II Plane-Wave Interaction with Materials

Narrowband propagation is considered at angular frequency $\omega$ in a 3D
medium with an inhomogeneity created by a $z$-oriented planar object of
infinite thickness, dividing the medium into a region 1 ${\\{r_{z}<0\\}}$
(free space) and a region 2 ${\\{r_{z}>0\\}}$ (material). The electromagnetic
properties are constant in each of the two ensuing regions, characterized by
the refractive indexes $n_{1}=1$ and $n_{2}\in\mathbb{C}$ with
$\real(n_{2})\geq 1$ and $\imaginary(n_{2})>0$ modeling the phase variations
and absorption losses occurring inside the material [17, Sec. 4.2]. The
wavenumbers in the two regions are $\kappa_{1}=2\pi/\lambda$ and

$\kappa_{2}=n_{2}\kappa_{1}.$ (1)

###  II-A Dielectric Half-Space

Figure 1: Fresnel reflection coefficient (magnitude) as a function of
${\theta_{\text{i}}}$ for various refractive indices.

We first consider the $xz$-plane containing the direction of propagation and
the surface normal, namely the _plane of incidence_.111This plane can always
be obtained by rotating the Cartesian reference frame opportunely about the
$x$-axis. A point in this plane has coordinates $(r_{x},r_{z})$. An upgoing
_incident_ plane wave

$e_{\text{i}}(r_{x},r_{z})=E_{\text{i}}(\theta_{\text{i}})\,e^{\mathsf{j}\kappa_{1}\left(r_{x}\sin\theta_{\text{i}}+r_{z}\cos\theta_{\text{i}}\right)}$
(2)

with amplitude $E_{\text{i}}(\theta_{\text{i}})$ traveling in region 1 from an
angle $\theta_{\text{i}}$ relative to the surface normal impinges thereon. As
a result of interaction with the surface, this field creates a downgoing
_reflected_ plane wave in region 1,

$e_{\text{r}}(r_{x},r_{z})=E_{\text{r}}(\theta_{\text{r}})\,e^{\mathsf{j}\kappa_{1}\left(r_{x}\sin\theta_{\text{r}}-r_{z}\cos\theta_{\text{r}}\right)},$
(3)

with amplitude $E_{\text{r}}(\theta_{\text{r}})$ and angle $\theta_{\text{r}}$
and another upgoing _transmitted_ plane wave in region 2,

$e_{\text{t}}(r_{x},r_{z})=E_{\text{t}}(\theta_{\text{t}})\,e^{\mathsf{j}\kappa_{2}\left(r_{x}\sin\theta_{\text{t}}+r_{z}\cos\theta_{\text{t}}\right)},$
(4)

with amplitude $E_{\text{t}}(\theta_{\text{t}})$ and angle
$\theta_{\text{t}}$. Derivable from the boundary conditions, Snell’s law
dictates that reflection occurs at the specular angle
$\theta_{\text{r}}=\theta_{\text{i}}$ while transmission is specified by
$\sin(\theta_{\text{t}})=\sin(\theta_{\text{i}})/n_{2}$ [9, Eq. 1.5.6]. The
complex-valued plane-wave amplitudes can be written in terms of the _Fresnel
coefficients_ $R(\theta_{\text{i}})=E_{\text{r}}/E_{\text{i}}$ and
$T(\theta_{\text{i}})=E_{\text{t}}/E_{\text{i}}$, specifying the fraction of
incident field reflected from or transmitted across the surface, for every
incident angle. Their magnitude is always less than unity, and they satisfy
the unitarity relation $T(\theta_{\text{i}})=1+R(\theta_{\text{i}})$ due to
conservation of energy.

Multiple reflections that might arise inside an object of finite thickness
would make the interaction with the surface more involved [10, Ch. 2.1.3].
However, these never occur at frequencies high enough such that the material
thickness is much larger than the wavelength, making the reflection phenomenon
highly predictable and suitable for array optimization, as will be seen.

The complex-valued Fresnel reflection coefficient is given by [18, Eq.
7.4.2]222For every angle $\theta_{\text{i}}$ there are two linearly
independent plane waves being the solutions of the two scalar wave equations
characterizing the transverse electric (TE) polarization, where the electric
field is parallel to the surface, and the transverse magnetic (TM)
polarization, where the magnetic field is parallel [10, Ch. 2.1]. We
concentrate on the TE equation as the TM’s is obtainable by invoking the
duality principle.

$R(\theta_{\text{i}})=\frac{\cos(\theta_{\text{i}})-\sqrt{n_{2}^{2}-\sin^{2}(\theta_{\text{i}})}}{\cos(\theta_{\text{i}})+\sqrt{n_{2}^{2}-\sin^{2}(\theta_{\text{i}})}},$
(5)

whose magnitude is plotted in Fig. 1 as a function of ${\theta_{\text{i}}}$
for various dielectric materials [19]. Total reflection is achieved by a
perfect conductor, which behaves as a mirror. Other materials behave as
perfect conductors only at a grazing incidence. In general, denser materials
reflect energy better and, for a given material, close-to-grazing incidences
experience higher reflections than those near the normal.

###  II-B Linear-System-Theoretic Interpretation

We now deviate from physics and provide a different viewpoint on the
interaction mechanism with the surface; this perspective relies only on linear
system theory and Fourier transforms, key results in the toolbox of
communication theorists.

The propagation directions of the incident, reflected, and transmitted plane
waves may alternatively be specified by the wavenumber coordinates

$\displaystyle(\kappa_{x},\pm\kappa_{1z})$
$\displaystyle=(\kappa_{1}\sin\theta_{\text{i}},\pm\kappa_{1}\cos\theta_{\text{i}})$
(6) $\displaystyle(\kappa_{x},\kappa_{2z})$
$\displaystyle=(\kappa_{2}\sin\theta_{\text{t}},\kappa_{2}\cos\theta_{\text{t}})$
(7)

satisfying the dispersion relations
$\kappa_{x}^{2}+\kappa_{iz}^{2}=\kappa_{i}^{2}$ for $i=1,2$. By means of (6),
the plane waves in (2) and (3) can be seen as the 2D Fourier harmonics

$\displaystyle e_{\text{i}}(r_{x},r_{z})$
$\displaystyle=E_{\text{i}}(\kappa_{x})\,e^{\mathsf{j}\left(\kappa_{x}x+\kappa_{1z}z\right)}$
(8) $\displaystyle e_{\text{r}}(r_{x},r_{z})$
$\displaystyle=E_{\text{r}}(\kappa_{x})\,e^{\mathsf{j}\left(\kappa_{x}x-\kappa_{1z}z\right)},$
(9)

which are functions of the spatial-frequency variables
$(\kappa_{x},\kappa_{1z})$. The same holds for (4), expressed as

$e_{\text{t}}(r_{x},r_{z})=E_{\text{t}}(\kappa_{x})\,e^{\mathsf{j}\left(\kappa_{x}x+\kappa_{2z}z\right)}$
(10)

for $(\kappa_{x},\kappa_{2z})$. The connection with Fourier theory that the
above change of variables establishes enables a linear-system-theoretic
interpretation of the reflection and transmission phenomena, with the focus
henceforth being on the reflection.

The response to a harmonic input at spatial frequency
$(\kappa_{x},\kappa_{1z})$ is another harmonic output at the same spatial
frequency—up to a change of sign in $\kappa_{1z}$ due to the reflected wave
traveling in the opposite direction—whose complex amplitude is the product of
the input’s amplitude and the Fresnel spectrum, given by [10, Eq. 2.1.13]

$\displaystyle R(\kappa_{x})$
$\displaystyle=\frac{\kappa_{1z}-\kappa_{2z}}{\kappa_{1z}+\kappa_{2z}}$ (11)

for dielectric materials; this follows from (5) after a change of variables to
wavenumber coordinates while using (1).

Remarkably, a behavior of this sort characterizes a linear and space-invariant
(LSI) system, which is fully described by its wavenumber response
$R(\kappa_{x})$ for any $\kappa_{x}$.

##  III Plane Wave Spectral Representation

Figure 2: Scalar wave propagation in a 3D isotropic and inhomogeneous medium.
View from the plane of incidence.

Consider now every possible vertical plane obtainable by rotating the
$xz$-plane of incidence (i.e., $\phi_{\text{i}}=0$) about the $x$-axis by an
angle ${\phi_{\text{i}}\in[0,2\pi)}$. This brings into play other variables in
the spatial and wavenumber domains, which we embed into the vectors
${\boldsymbol{r}}$ with coordinates $(r_{x},r_{y})$ and
${\boldsymbol{\kappa}}$ with coordinates $(\kappa_{x},\kappa_{y})$.

The field $e_{\text{i}}({\boldsymbol{r}},r_{z})$ radiated by a source of
electric current $j({\boldsymbol{r}},r_{z})$ is described exactly by an
integral superposition of complex harmonics of different amplitudes and
spatial frequencies via the Fourier (plane wave) spectral representation [10,
11]. Precisely, for a source enclosed within a sphere of radius
${0<R_{0}<D_{0}}$ (see Fig. 2),

$\displaystyle e_{\text{i}}({\boldsymbol{r}},r_{z})$
$\displaystyle=\begin{cases}\begin{aligned}
\displaystyle&\iint_{-\infty}^{\infty}E_{\text{i}}^{-}({\boldsymbol{\kappa}})\,e^{\mathsf{j}{\boldsymbol{\kappa}}^{\mbox{\tiny$\mathrm{T}$}}{\boldsymbol{r}}}\frac{d{\boldsymbol{\kappa}}}{(2\pi)^{2}}&\;r_{z}<-R_{0}\\\
\displaystyle&\iint_{-\infty}^{\infty}E_{\text{i}}^{+}({\boldsymbol{\kappa}})\,e^{\mathsf{j}{\boldsymbol{\kappa}}^{\mbox{\tiny$\mathrm{T}$}}{\boldsymbol{r}}}\frac{d{\boldsymbol{\kappa}}}{(2\pi)^{2}}&r_{z}>R_{0}\end{aligned}\end{cases}$
(12)

with complex-valued amplitudes

$E_{\text{i}}^{\pm}({\boldsymbol{\kappa}})=\frac{\kappa_{1}\eta_{1}}{2}\frac{J_{\pm}({\boldsymbol{\kappa}})}{\kappa_{1z}}\,e^{\pm\mathsf{j}\kappa_{1z}r_{z}}$
(13)

specified by the source’s spectrum $J_{\pm}({\boldsymbol{\kappa}})$ obtained
via a 3D Fourier transform of $j({\boldsymbol{r}},r_{z})$ evaluated at
${\kappa_{z}=\pm\kappa_{1z}}$, $\kappa_{iz}$ being defined as

$\kappa_{iz}=\sqrt{\kappa_{i}^{2}-\|{\boldsymbol{\kappa}}\|^{2}},$ (14)

for $i=1,2$. Thus,

$J_{\pm}({\boldsymbol{\kappa}})\\!=\\!\iiint_{-\infty}^{\infty}\\!\\!j({\boldsymbol{s}},s_{z})\,e^{-\mathsf{j}\left({\boldsymbol{\kappa}}^{\mbox{\tiny$\mathrm{T}$}}{\boldsymbol{s}}\pm\kappa_{1z}s_{z}\right)}\,d{\boldsymbol{s}}ds_{z}$
(15)

given ${\eta_{1}\approx 120\pi}$ as the wave impedance of free-space.

The reflected field $e_{\text{r}}({\boldsymbol{r}})$ follows from the
linearity of the spatial filtering operation applied by the surface and the
delay property of the Fourier transform, as the surface is placed at an
arbitrary distance $D_{0}$ from the source, along the $z$-axis; see Fig. 2.
The Fourier spectral representation of $e_{\text{r}}({\boldsymbol{r}})$ is
therefore

$e_{\text{r}}({\boldsymbol{r}},r_{z})=\\!\iint_{-\infty}^{\infty}E_{\text{i}}^{+}({\boldsymbol{\kappa}})R({\boldsymbol{\kappa}})\,e^{-\mathsf{j}\kappa_{1z}(r_{z}-2D_{0})}e^{\mathsf{j}{\boldsymbol{\kappa}}^{\\!\mbox{\tiny$\mathrm{T}$}}\\!{\boldsymbol{r}}}\frac{d{\boldsymbol{\kappa}}}{(2\pi)^{2}}$
(16)

with $R({\boldsymbol{\kappa}})$ the Fresnel spectrum in (11) and $\kappa_{1z}$
as defined in (14). Physically, the reflected field is created by
superimposing the interactions with all possible incident contributions on the
plane of incidence and for all possible vertical planes. With respect to an
incident plane wave, a reflected plane wave exhibits an extra phase shift that
accounts for the round-trip delay accumulated by the incident wave during the
travel to the surface and back, along the $z$-axis. This effect can be
regarded as a _migration_ of the incident field and is directly connected to
the image theorem, as discussed in Sec. IV.

##  IV Image Theorem

$e({\boldsymbol{r}},r_{z})=\begin{cases}\begin{aligned}
\displaystyle&\iint_{-\infty}^{\infty}\left(E_{\text{i}}^{-}({\boldsymbol{\kappa}})e^{-\mathsf{j}\kappa_{1z}r_{z}}+E_{\text{i}}^{+}({\boldsymbol{\kappa}})R({\boldsymbol{\kappa}})e^{-\mathsf{j}\kappa_{1z}(r_{z}-2D_{0})}\right)e^{\mathsf{j}{\boldsymbol{\kappa}}^{\mbox{\tiny$\mathrm{T}$}}{\boldsymbol{r}}}\frac{d{\boldsymbol{\kappa}}}{(2\pi)^{2}}&\quad
r_{z}<-R_{0}\\\
\displaystyle&\iint_{-\infty}^{\infty}E_{\text{i}}^{+}({\boldsymbol{\kappa}})\left(e^{\mathsf{j}\kappa_{1z}r_{z}}+R({\boldsymbol{\kappa}})e^{-\mathsf{j}\kappa_{1z}(r_{z}-2D_{0})}\right)e^{\mathsf{j}{\boldsymbol{\kappa}}^{\mbox{\tiny$\mathrm{T}$}}{\boldsymbol{r}}}\frac{d{\boldsymbol{\kappa}}}{(2\pi)^{2}}&\quad
R_{0}<r_{z}\leq D_{0}\end{aligned}\end{cases}$ (27)

$H({\boldsymbol{\kappa}};r_{z},s_{z})=\begin{cases}\begin{aligned}
\displaystyle&\frac{\kappa_{1}\eta_{1}}{2}\frac{\mathbbm{1}_{\mathcal{D}}({\boldsymbol{\kappa}})}{\kappa_{1z}}\left(e^{-\mathsf{j}\kappa_{1z}(r_{z}-s_{z})}+R({\boldsymbol{\kappa}})e^{-\mathsf{j}\kappa_{1z}(r_{z}+s_{z}-2D_{0})}\right)&\quad
r_{z}<-R_{0}\\\
\displaystyle&\frac{\kappa_{1}\eta_{1}}{2}\frac{\mathbbm{1}_{\mathcal{D}}({\boldsymbol{\kappa}})}{\kappa_{1z}}\left(e^{\mathsf{j}\kappa_{1z}(r_{z}-s_{z})}+R({\boldsymbol{\kappa}})e^{-\mathsf{j}\kappa_{1z}(r_{z}+s_{z}-2D_{0})}\right)&\quad
R_{0}<r_{z}\leq D_{0}\end{aligned}\end{cases}$ (30)

${\boldsymbol{H}}({\boldsymbol{k}},{\boldsymbol{\kappa}})=\frac{\kappa_{1}\eta_{1}}{2}\delta({\boldsymbol{k}}-{\boldsymbol{\kappa}})\frac{\mathbbm{1}_{\mathcal{D}}({\boldsymbol{\kappa}})}{\kappa_{1z}}\cdot\begin{cases}\begin{aligned}
&\begin{pmatrix}0&0\\\
R({\boldsymbol{k}})e^{\mathsf{j}2\kappa_{1z}D_{0}}&1\end{pmatrix}&r_{z}<-R_{0}\\\
&\begin{pmatrix}1&0\\\
R({\boldsymbol{k}})e^{\mathsf{j}2\kappa_{1z}D_{0}}&0\end{pmatrix}&R_{0}<r_{z}\leq
D_{0}\end{aligned}\end{cases}$ (34)

Plugging (13) into (12), the incident field in $\\{r_{z}>R_{0}\\}$ is

$\displaystyle
e_{\text{i}}({\boldsymbol{r}},r_{z})\\!=\\!\frac{\kappa_{1}\eta_{1}}{2}\\!\iint_{-\infty}^{\infty}\\!\\!\frac{J_{+}({\boldsymbol{\kappa}})}{\kappa_{1z}}\,e^{\mathsf{j}\left({\boldsymbol{\kappa}}^{\mbox{\tiny$\mathrm{T}$}}{\boldsymbol{r}}+\kappa_{1z}r_{z}\right)}\frac{d{\boldsymbol{\kappa}}}{(2\pi)^{2}}$
(17)

where $J_{+}({\boldsymbol{\kappa}})$ is given by (15). Similarly, the
reflected field in (16) can be rewritten as

$\displaystyle
e_{\text{r}}({\boldsymbol{r}},r_{z})=\frac{\kappa_{1}\eta_{1}}{2}\\!\iint_{-\infty}^{\infty}\\!\frac{J_{\text{r}}({\boldsymbol{\kappa}})}{\kappa_{1z}}\,e^{\mathsf{j}\left({\boldsymbol{\kappa}}^{\mbox{\tiny$\mathrm{T}$}}{\boldsymbol{r}}-\kappa_{1z}r_{z}\right)}\frac{d{\boldsymbol{\kappa}}}{(2\pi)^{2}}$
(18)

where

$\displaystyle
J_{\text{r}}({\boldsymbol{\kappa}})=J_{+}({\boldsymbol{\kappa}})\,e^{\mathsf{j}\kappa_{1z}2D_{0}}\,R({\boldsymbol{\kappa}}).$
(19)

Notice that (18) and (17) have the same form. Hence,
$J_{\text{r}}({\boldsymbol{\kappa}})$ may be regarded as the Fourier spectrum
of a fictitious source $j_{\text{r}}({\boldsymbol{r}},r_{z})$. For
$R({\boldsymbol{\kappa}})=-1$, the reflected field in (18) may be reproduced
by replicating the source at $r_{z}=2D_{0}$, which accounts for the field
migration to the surface and backward. This is the _image theorem_ , whereby
the reflection elicited by a perfect conductor is equivalent to a mirror image
of the source [20, Sec. 4.7.1]. As an example, for a point source
$j({\boldsymbol{r}},r_{z})=\delta({\boldsymbol{r}})\delta(r_{z})$, i.e., for
$J_{+}({\boldsymbol{\kappa}})=1$, applying Weyl’s identity [10, Eq. 2.2.27]

$\frac{e^{\mathsf{j}\kappa_{1}\|({\boldsymbol{r}},|r_{z}|)\|}}{\|({\boldsymbol{r}},|r_{z}|)\|}=\frac{\mathsf{j}}{2\pi}\iint_{-\infty}^{\infty}\frac{e^{\mathsf{j}({\boldsymbol{\kappa}}^{\mbox{\tiny$\mathrm{T}$}}{\boldsymbol{r}}+\kappa_{1z}|r_{z}|)}}{\kappa_{1z}}\,d{\boldsymbol{\kappa}},$
(20)

from (18) we obtain

$e_{\text{r}}({\boldsymbol{r}},r_{z})=\mathsf{j}\frac{\kappa_{1}\eta_{1}}{4\pi}G({\boldsymbol{r}},r_{z},{\boldsymbol{0}},2D_{0})$
(21)

where

$G({\boldsymbol{r}},r_{z},{\boldsymbol{r}}^{\prime},r_{z}^{\prime})=\frac{e^{\mathsf{j}\kappa_{1}\|({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime},r_{z}-r_{z}^{\prime})\|}}{4\pi\,\|({\boldsymbol{r}}-{\boldsymbol{r}}^{\prime},r_{z}-r_{z}^{\prime})\|}$
(22)

is the Green’s function describing a spherical wave generated at
$({\boldsymbol{r}}^{\prime},r_{z}^{\prime})$ and measured at
$({\boldsymbol{r}},r_{z})$. Hence,
$j_{\text{r}}({\boldsymbol{r}},r_{z})=\delta({\boldsymbol{r}})\delta(r_{z}-2D_{0})$.

For arbitrary materials, $j_{\text{r}}({\boldsymbol{r}},r_{z})$ is obtained
from the spatial convolution

$j_{\text{r}}({\boldsymbol{r}},r_{z})=\iint_{-\infty}^{\infty}j({\boldsymbol{u}},r_{z}-2D_{0})\,r({\boldsymbol{r}}-{\boldsymbol{u}})\,d{\boldsymbol{u}}$
(23)

of the image source and the impulse response of the surface,

$r({\boldsymbol{r}})=\iint_{-\infty}^{\infty}R({\boldsymbol{\kappa}})\,e^{\mathsf{j}{\boldsymbol{\kappa}}^{\mbox{\tiny$\mathrm{T}$}}{\boldsymbol{r}}}\frac{d{\boldsymbol{\kappa}}}{(2\pi)^{2}},$
(24)

which is defined as the 2D inverse Fourier transform of
$R({\boldsymbol{\kappa}})$ in (11). The azimuthal dependance of
$r({\boldsymbol{r}})$ can be eliminated by evaluating (24) at
$(\|{\boldsymbol{r}}\|,0)$, which is possible due to the circular symmetry of
$R({\boldsymbol{\kappa}})$.

From (23), we infer that the spatial filtering applied by the surface creates
a _blurred image_ of the source. This effect vanishes in perfect conductors,
recreating a perfect image. For a point source,
$j_{\text{r}}({\boldsymbol{r}},r_{z})=r({\boldsymbol{r}})\delta(r_{z}-2D_{0})$,
${\boldsymbol{r}}\in\mathbb{R}^{2}$, modeling the impressed currents induced
by the source on the entire surface.

The spatial filtering simplifies when the surface is far enough from the
source that the reflected propagation occurs in the _paraxial regime_. Then,
$R({\boldsymbol{\kappa}})$ is roughly constant for all possible incident
angles and given by the complex material reflectivity [9, Sec. 1.5.3]. Due to
the impulsiveness of the reflection mechanism under the paraxial assumption,
the image source becomes a weakened (and phase-shifted) version of the
original one, which is the premise of ray-tracing algorithms. However, this
need not be the case in wide-aperture MIMO, which rests on the range being
short; this aspect is further expounded in Sec. VII. The implications on the
optimization of antenna spacings in MIMO communication are discussed in Sec.
VI.

##  V Channel Impulse Response

A complete description of what unfolds in region 1 is obtained by combining
all contributions into

$e({\boldsymbol{r}},r_{z})=e_{\text{i}}({\boldsymbol{r}},r_{z})+e_{\text{r}}({\boldsymbol{r}},r_{z})$
(25)

whose expression is given by (27) after substituting (12) and (16). The input-
output relationship between $j({\boldsymbol{s}},s_{z})$ and
$e({\boldsymbol{r}},r_{z})$ is the spatial convolution [14]

$e({\boldsymbol{r}},r_{z})=\iiint_{-\infty}^{\infty}j({\boldsymbol{s}},s_{z})\,h({\boldsymbol{r}},r_{z},{\boldsymbol{s}},s_{z})\,d{\boldsymbol{s}}ds_{z}$
(28)

where $h({\boldsymbol{r}},r_{z},{\boldsymbol{s}},s_{z})$ is the channel
impulse response. Combining (27), (13), and (15), the channel response can be
written as the 2D inverse Fourier transform

$h({\boldsymbol{r}}-{\boldsymbol{s}};r_{z},s_{z})=\iint_{-\infty}^{\infty}H({\boldsymbol{\kappa}};r_{z},s_{z})\,e^{\mathsf{j}{\boldsymbol{\kappa}}^{\mbox{\tiny$\mathrm{T}$}}({\boldsymbol{r}}-{\boldsymbol{s}})}\,\frac{d{\boldsymbol{\kappa}}}{(2\pi)^{2}}$
(29)

of $H({\boldsymbol{\kappa}};r_{z},s_{z})$ in (30). Here, the integration
domain is practically limited to a disk $\mathcal{D}$ of radius
$\kappa_{1}=2\pi/\lambda$, correctly showing the low-pass-filtering behavior
of the wireless propagation [12, 14], which is then converted into a
functional dependence through an indicator function. The reflected channel is
space invariant over any pair of parallel $z$-planes. This extends to any pair
of parallel planes, not necessarily $z$, for an LOS channel.

The space invariance is a direct consequence of the unboundedness and
smoothness of the reflecting surface and enables a linear-system-theoretic
interpretation of the reflection and transmission phenomena. Precisely,
communications between any two different $z$-planes cutting source and
receiver can be regarded as an LSI system with the wavenumber response in
(30). There are three main terms in (30), plus a phase shift due to migration,
that may be interpreted as the cascade of:

  * •

First, $\mathbbm{1}_{\mathcal{D}}({\boldsymbol{\kappa}})$, a low-pass filter
introduced by the migration operation [14, 12].

  * •

Then, $1/\kappa_{1z}$, which confers a reverse-bowl behavior to
$H({\boldsymbol{\kappa}};r_{z},s_{z})$ and is directly attributable to the
wave equation [13, 14].

  * •

Finally, $R({\boldsymbol{\kappa}})$ models the reflection. This depends on
${\boldsymbol{\kappa}}$ via $\kappa_{iz}$ in (14), hence it is circularly
symmetric in the wavenumber domain, which is instrumental to devise an
efficient numerical procedure to generate channel samples (see Appendix).

The space-invariant channel in (29) generated by a specular reflection is
obtainable as a particular instance of the double 2D Fourier transform [14,
Sec. III]

$\displaystyle
h({\boldsymbol{r}},r_{z},{\boldsymbol{s}},s_{z})=\iiiint_{-\infty}^{\infty}H({\boldsymbol{k}},{\boldsymbol{\kappa}};r_{z},s_{z})$
$\displaystyle\hskip 85.35826pt\cdot
e^{\mathsf{j}{\boldsymbol{k}}^{\mbox{\tiny$\mathrm{T}$}}{\boldsymbol{r}}}e^{-\mathsf{j}{\boldsymbol{\kappa}}^{\mbox{\tiny$\mathrm{T}$}}{\boldsymbol{s}}}\,\frac{d{\boldsymbol{k}}}{(2\pi)^{2}}\frac{d{\boldsymbol{\kappa}}}{(2\pi)^{2}}$
(31)

of the wavenumber response

$H({\boldsymbol{k}},{\boldsymbol{\kappa}};r_{z},s_{z})={\boldsymbol{\phi}}^{\mbox{\tiny$\mathrm{H}$}}({\boldsymbol{k}},r_{z}){\boldsymbol{H}}({\boldsymbol{k}},{\boldsymbol{\kappa}}){\boldsymbol{\phi}}({\boldsymbol{\kappa}},s_{z}),$
(32)

given
${\boldsymbol{\phi}}({\boldsymbol{\kappa}},s_{z})=\left(e^{-\mathsf{j}\kappa_{1z}s_{z}},e^{\mathsf{j}\kappa_{1z}s_{z}}\right)^{\mbox{\tiny$\mathrm{T}$}}$.
The above is parametrized by the wavenumber matrix

$\displaystyle{\boldsymbol{H}}({\boldsymbol{k}},{\boldsymbol{\kappa}})=\begin{pmatrix}H_{++}({\boldsymbol{k}},{\boldsymbol{\kappa}})&H_{+-}({\boldsymbol{k}},{\boldsymbol{\kappa}})\\\
H_{-+}({\boldsymbol{k}},{\boldsymbol{\kappa}})&H_{--}({\boldsymbol{k}},{\boldsymbol{\kappa}})\end{pmatrix}$
(33)

that models the coupling between every input spatial frequency
${\boldsymbol{\kappa}}$ and every other output spatial frequency
${\boldsymbol{k}}$. It can also be regarded as an angular response mapping
every incident plane wave traveling along
$({\boldsymbol{\kappa}},\pm\kappa_{1z})$ into every other receive plane wave
from $({\boldsymbol{k}},\pm\kappa_{1z})$. The convention adopted for the
entries of (33) is that the first and second subscripts refer, respectively,
to received and incident plane waves (each one being associated with upgoing
or downgoing waves).

We next find the parameterization of
${\boldsymbol{H}}({\boldsymbol{k}},{\boldsymbol{\kappa}})$ that models the
scenario in Sec. III. By inspection, comparing (29)–(30) against (31), yields
(34). The entries of the angular matrix are impulsive because incident and
received plane waves are in one-to-one correspondence: each incident wave
turns into a received wave with specular direction, as specified by Snell’s
law.

Generally, the surface of a material object may appear as either smooth or
rough depending on the frequency. A rough surface at microscopic level
reflects every impinging plane wave off multiple directions creating a diffuse
reflection spectrum, typically centered around the specular direction. These
surface irregularities are accounted by a non-impulsive
${\boldsymbol{H}}({\boldsymbol{k}},{\boldsymbol{\kappa}})$ in (32), whose
computation is left for future work.

##  VI Application to MIMO Communication

Figure 3: ULAs separated by $D$ and equipped with
$N_{\text{t}}=N_{\text{r}}=8$ antennas with spacing ${\sf d}$. Arrays have
arbitrary orientations $\vartheta_{\text{t}}$ and $\vartheta_{\text{r}}$ with
respect to the $x$-axis. The clear and solid circles at source and receiver
indicate the antennas and their projections onto the $x$-axis, respectively.
Antennas are connected either via a LOS or a reflected channel off a
$z$-oriented surface. We denote by $\theta_{0}$ the angle formed by the
surface normal and the geometrical path connecting the centroids of the image
source and receiver.

Let us now apply the developed model to evaluate the channel eigenvalues, DOF,
and spectral efficiency.

With $N_{\text{t}}$ transmit and $N_{\text{r}}$ receive antennas, the channel
matrix ${\boldsymbol{H}}\in\mathbb{C}^{N_{\text{r}}\times N_{\text{t}}}$ is
obtained by sampling the impulse response at the antenna locations,
${[{\boldsymbol{H}}]_{m,n}=h({\boldsymbol{r}}_{m},{\boldsymbol{s}}_{n})}$ for
$m=0,\dots,N_{\text{r}}-1$ and $n=0,\dots,N_{\text{t}}-1$. The transmit array
is centered at the origin whereas the centroid of the receive array is at
${\boldsymbol{r}}_{0}=(r_{0x},r_{0y},r_{0z})$. An efficient numerical
generation procedure for ${\boldsymbol{H}}$ is provided in the Appendix.

Let $N_{\min}=\min(N_{\text{r}},N_{\text{t}})$ and
$N_{\max}=\max(N_{\text{r}},N_{\text{t}})$. We consider uniform linear arrays
(ULAs) at $57.5$ GHz (see Fig. 3) under the proviso that those ULAs are
substantially shorter than their separation range, the so-called _paraxial
approximation_ , so we can leverage results available for LOS channels [15,
16]. The transmitting and receiving ULAs have arbitrary orientations
$\vartheta_{\text{t}}$ and $\vartheta_{\text{r}}$ with respect to the
$x$-axis.

We hasten to emphasize that the reliance on the paraxial approximation is
confined to the production of benchmark results for LOS MIMO, with our channel
model being valid regardless. The frequency, in turn, is motivated by mmWave
applications [2] and by availability of refractive indices for most common
materials [19].

###  VI-A Parallel Arrays Optimized for LOS Transmission

Consider parallel ULAs aligned with the $x$-axis, with
${N_{\text{t}}=N_{\text{r}}=8}$ and antenna spacing ${\sf d}$. The range is
${D=10}$ m whereas the surface is at ${D_{0}=15}$ m. First, we validate the
model in LOS, for which the closed-form solution in (22) is available. The
channel matrix obtained by sampling (22) is compared to the LOS component in
our model, derivable after an inverse Fourier transform of the first term in
(30), the LOS term, according to (29), followed by spatial sampling. Setting

${\sf d}(D)=\sqrt{\lambda D/N_{\max}},$ (35)

renders ${\boldsymbol{H}}$ a Fourier matrix and is optimum at high SNR [5,
15]. The normalized eigenvalues of
${\boldsymbol{H}}{\boldsymbol{H}}^{\mbox{\tiny$\mathrm{H}$}}$,
$\lambda_{n}({\boldsymbol{H}})$, are plotted in Fig. 4. The perfect match
validates the numerical procedure in the Appendix for this LOS setting.

Then, we validate the model under perfect reflection. To this end, the channel
obtained by imaging the source is compared against the one associated with the
perfect reflection in our model; the latter is obtained by plugging the second
term in (30) with $R({\boldsymbol{\kappa}})=-1$ into (29) and sampling.

The eigenvalues of the reflected channel matrix are further shown in Fig. 4
for different materials. These undergo two effects relative to their LOS
brethren:

  * •

_Power loss_ caused by the longer range and by the reflection of only a share
of the incident power, with dense materials and shallow angles reflecting
better.

  * •

_Spatial selectivity_ due the antenna spacing in (35) being suboptimally small
for the longer range of the reflected channel.

Figure 4: Normalized channel eigenvalues for various materials. Parallel ULAs
separated by $D=10$ m with spacing ${\sf d}(D)$ in (35). Figure 5: Spectral
efficiency as a function of SNR for various materials. Parallel ULAs separated
by $D=10$ m with spacing ${\sf d}(D,\mathsf{SNR})$ in (38).

We now gauge the capacity with channel-state information at the transmitter,
which equals [21, 22]

$\displaystyle C({\boldsymbol{H}},{\sf
SNR})=\sum_{n=1}^{N_{\min}}\log_{2}\\!\left(1+\left(\nu-\frac{1}{\lambda_{n}({\boldsymbol{H}})}\right)^{\\!\\!+}\lambda_{n}({\boldsymbol{H}})\\!\right)$
(36)

where $\nu$ is such that
${\sum_{n=1}^{N_{\min}}\left(\nu-{1}/{\lambda_{n}({\boldsymbol{H}})}\right)^{+}={\sf
SNR}}$ while
${\sum_{n=1}^{N_{\min}}\lambda_{n}({\boldsymbol{H}})=N_{\text{r}}N_{\text{t}}}$.
At any given SNR, $C({\sf SNR})=\max_{{\boldsymbol{H}}}C({\boldsymbol{H}},{\sf
SNR})$ satisfies [15, 16]

$C({\sf
SNR})\leq\max_{\rho\in\\{1,2,\ldots,N_{\min}\\}}\rho\log_{2}\\!\left(1+\frac{{\sf
SNR}}{\rho}\frac{N_{\text{r}}N_{\text{t}}}{\rho}\right),$ (37)

with the upper bound corresponding to $\rho$ nonzero eigenvalues equal to
$N_{\text{r}}N_{\text{t}}/\rho$ and to the SNR-dependent antenna spacing

${\sf d}(D,{\sf SNR})=\sqrt{\eta({\sf SNR})\lambda D/N_{\max}},$ (38)

for a fraction $\eta({\sf SNR})=\rho({\sf SNR})/N_{\min}$ of the $N_{\min}$
potential DOF. Thus, $\eta\in[0,1]$ with $\eta=1$ at high enough SNR. The
capacity $C({\boldsymbol{H}},{\sf SNR})$ is reported in Fig. 5 for the antenna
spacing, ${\sf d}(D,{\sf SNR})$, that is optimum for the LOS channel at every
SNR. With respect to the LOS case, the capacity of the reflected channel
experiences an offset (power loss, due to the longer range) and a reduced
slope (DOF loss, due to the spatial selectivity).

###  VI-B Parallel Arrays Optimized for the Reflected Transmission

Figure 6: Normalized channel eigenvalues for various materials. Parallel ULAs
separated by $D=10$ m with spacing ${\sf d}(D)$ for the LOS channel and ${\sf
d}(D_{\text{e}})$ for the reflected channel. Figure 7: Spectral efficiency as
a function of SNR for various materials. Parallel ULAs separated by $D=10$ m
with spacing ${\sf d}(D,{\sf SNR})$ for the LOS channel and ${\sf
d}(D_{\text{e}},\mathsf{SNR})$ for the reflected channel.

While the power loss is inevitable, because of the longer range, the spatial
selectivity can be corrected by tailoring the antenna spacing to the
equivalent LOS transmission from the image source. To this end, recall from
the image theorem that the reflected channel can be regarded as an LOS channel
with augmented distance $D_{\text{e}}>D$; in the setting of Figs. 4 and 5,
${D_{\text{e}}=2D_{0}-D}$. For a perfect conductor, this alone justifies the
choice of an antenna spacing equal to ${\sf d}(D_{\text{e}})$. The argument is
somewhat more involved for arbitrary materials, due to the distortion
introduced by reflection, but it ultimately leads to the same observation as
illustrated in Fig. 6. Numerically, this is supported by the invariance of the
curves for the materials in Fig. 4. Physically, it is explained by the
paraxial approximation, whereby the field has an approximately constant
wavenumber response in magnitude. Hence, the reflection has an approximately
multiplicative effect on the channel impulse response in (30) and the whole
interaction phenomenon with the surface is described by the reflectivity
coefficient, $R(\theta_{0})$, which is derivable from (5) after setting
${\theta_{\text{i}}=\theta_{0}}$ with $\theta_{0}$ as per Fig. 3.

Similarly, the eigenvalues of the reflected MIMO channel
${\boldsymbol{H}}{\boldsymbol{H}}^{\mbox{\tiny$\mathrm{H}$}}$ are obtained by
scaling the LOS eigenvalues uniformly by $|R(\theta_{0})|^{2}$. From (5), for
the chosen materials, setting ${\theta_{\text{i}}=\theta_{0}}$ yields a
scaling of $7.19$ dB (concrete), $9.63$ dB (floorboard), and $13.98$ dB
(plaster board). These values describe the gap in Fig. 6 between the
eigenvalues of the reflected channel for various materials and those of a
perfect conductor. The additional gap to the LOS channel is due to the
enhanced range, a loss of $6.02$ dB in our setting.

For completeness, Fig. 7 shows the spectral efficiency corresponding to the
eigenvalue distributions in Fig. 6. With respect to Fig. 5, the antenna
spacing is ${\sf d}(D,{\sf SNR})$ for the LOS channel and ${\sf
d}(D_{\text{e}},{\sf SNR})$ for the reflected channel, which lead to the same
DOF.

###  VI-C Power Loss and Spatial Selectivity for Parallel Arrays

Figure 8: Pathloss as a function of $D$ for different materials at normal
incidence.

We have seen that the power loss is determined by the additional range and by
the share of incident power not reflected by the surface. This is constant
over the arrays themselves as amplitude variations thereon are negligible with
the proviso that propagation occurs in the paraxial regime. From the image
theorem,

$\beta=|R(0)|^{2}\left(\frac{\lambda}{4\pi D_{\text{e}}}\right)^{2}$ (39)

where $D_{\text{e}}=2D_{0}-D$ and $R(0)=(1-n_{2})/(1+n_{2})$. In Fig. 8,
$\beta$ is plotted as a function of $(D_{0}-D)$ for different materials. The
interface is at $D_{0}=15$ m from the source, while the range between receiver
and surface varies accordingly to $(D_{0}-D)$.

Figure 9: Number of DOF as a function of $(D_{0}-D)$ when the material is
concrete. Parallel ULAs.

Receiver motion away from the surface, if unaccounted for, leads to a
decreasing stepwise function of $D-D_{0}\in[0,D_{0}]$; this is shown in Fig.
9, where the DOF equal the number of eigenvalues that are at most $40$ dB
below the maximum. Correcting the antenna spacing as a function of $D$
prevents this decrease.

###  VI-D Non-Parallel Arrays

Figure 10: Pathloss as a function of $\theta_{0}$ for various materials.
Oblique incidence with the receiver at $r_{0x}\in[0,100]$ m and $r_{0z}=10$ m.

Non-parallel ULA configurations arise either when the receiver is shifted
along the $x$-axis, creating an oblique incidence ($\theta_{0}>0$), or when
arrays are oriented differently in elevation
($\vartheta_{\text{t}}\neq\vartheta_{\text{r}}$); see Fig. 3. The relative
azimuth angle is set to zero, as it is immaterial to ULAs [16]. With the focus
on oblique incidence and its impact on power loss and spatial selectivity, the
ULAs are aligned with the $x$-axis
($\vartheta_{\text{t}}=\vartheta_{\text{r}}=0$).

First, let us consider the power loss. Due to rotational symmetry about the
$x$-axis, the $xz$-plane can be selected without loss of generality. The
pathloss in (39) generalizes to arbitrary receive positions when using

$D_{\text{e}}(\theta_{0})=\frac{2D_{0}-r_{0z}}{\cos(\theta_{0})},$ (40)

and $R(\theta_{0})$, which are parametrized by the incident angle

$\theta_{0}=\arccos\\!\left(\frac{2D_{0}-r_{0z}}{\sqrt{D^{2}+4D_{0}(D_{0}-r_{0z})}}\right).$
(41)

Fig. 10 depicts $\beta$ for various materials. The receiver is shifted along
the $x$-axis on the interval $r_{0x}\in[0,100]$ m with $r_{0z}=10$ m such that
$D=(r_{0x}^{2}+r_{0z}^{2})^{1/2}$.

(a) Projected view on the $yz$-plane (top view of Fig. 3).

(b) Projected view on the $xz$-plane (side view of Fig. 3).

(c) Modeling of oblique incidence as a relative orientation.

Figure 11: Non-parallel ULA configuration arising from an oblique incidence
with ULAs oriented as the $x$-axis. Figure 12: Normalized channel eigenvalues
for various materials. Oblique incidence with the receiving ULA at
${\boldsymbol{r}}_{0}=(1,4,10)$ m (hence, $\vartheta=5.3^{\circ}$ and
$\vartheta_{\text{e}}=2.8^{\circ}$). The antenna spacings are ${\sf d}(D,{\sf
SNR},\vartheta)$ for the LOS channel and ${\sf d}(D_{\text{e}},{\sf
SNR},\vartheta_{\text{e}})$ for the reflected channel. Figure 13: Spectral
efficiency as a function of SNR for different materials. Non-parallel ULAs
with spacing ${\sf d}(D,{\sf SNR},\vartheta)$ for the LOS channel and ${\sf
d}(D_{\text{e}},{\sf SNR},\vartheta_{\text{e}})$ for the reflected channel.

Second, we turn to spatial selectivity. Consider oblique incidence on a
vertical plane, not necessarily the $xz$-plane. Its projected views on the
$yz$-plane and on the $xz$-plane are illustrated in Figs. 11(a) and 11(b). For
the side view in Fig. 11(b), we define

$\displaystyle\widehat{D}$
$\displaystyle=D/\sqrt{1+\left(\frac{r_{0y}}{r_{0z}}\right)^{\\!2}}$ (42)
$\displaystyle\widehat{D}_{\text{e}}$
$\displaystyle=D_{\text{e}}/\sqrt{1+\left(\frac{r_{0y}}{2D_{0}-r_{0z}}\right)^{\\!2}},$
(43)

which are obtained by projecting their counterparts $D$ and $D_{\text{e}}$
onto the $xz$-plane; see Fig. 11(a). As sketched in Fig. 11(c), shifting the
receiver along the $x$-axis is equivalent to rotating the transmitting and
receiving ULAs with respect to the $x$-axis by an angle

$\vartheta=\arccos\\!\left(r_{0z}/\widehat{D}\right)$ (44)

for the LOS channel, and by another angle

$\vartheta_{\text{e}}=\arccos\\!\left(\frac{2D_{0}-r_{0z}}{\widehat{D}_{\text{e}}}\right)$
(45)

for the reflected channel. Unlike the power loss, spatial selectivity can be
corrected by tailoring the ULA spacing opportunely [16]. To this end, for the
LOS channel,

${\sf d}(D,{\sf SNR},\vartheta)=\frac{{\sf d}(D,{\sf SNR})}{\cos(\vartheta)},$
(46)

with ${\sf d}(D,{\sf SNR})$ the optimal antenna spacing for parallel ULAs in
(38) whereas, for the reflected channel, ${\sf d}(D_{\text{e}},{\sf
SNR},\vartheta_{\text{e}})$ with $D_{\text{e}}$ in (40) and
$\vartheta_{\text{e}}$ in (45). Compared to parallel ULAs, non-parallel ULAs
have antennas that are spaced further apart due to the division by
$\cos(\cdot)$ in (46). The potential DOF thereby shrink for ULAs tilted
sideways. At high SNR, since $\eta({\sf SNR})=1$ in (38), ${\sf d}(D,{\sf
SNR})$ reduces to ${\sf d}(D)$ in (35) leading to full-rank channel matrices
for the LOS and reflected channels. This property is validated in Fig. 12 for
an $x$-oriented ULA located at ${\boldsymbol{r}}_{0}=(1,4,10)$ m, i.e., for
$\vartheta=5.3^{\circ}$ and $\vartheta_{\text{e}}=2.8^{\circ}$. Finally, the
spectral efficiency with ULA spacings optimized at every SNR for the LOS and
reflected transmissions is also shown in Fig. 13 for different materials.

##  VII Implications for Ray Tracing Algorithms

Figure 14: Spectral efficiency as a function of the array apertures over
equivalent distance ratio at ${\sf SNR}=0$ dB when the material is concrete.
The antenna spacing is optimized for the reflected transmission.

NLOS connectivity is typically established via multiple reflections involving
possibly distinct materials and orientations. Analysis becomes unwieldy in
such general settings and the recourse are numerical algorithm such as ray
tracing [17]. Our setup provides insights into the mechanisms involved at each
stage of reflection.

Our exact channel model describes the reflected propagation as an LSI
filtering, whereas the ray-tracing model regards the convolving response as an
impulse weighted by the reflectivity coefficient in (5) at
$\theta_{\text{i}}=\theta_{0}$. To appreciate the difference between the exact
and the approximated method (ray-tracing) one should increase the array
apertures $L_{\text{r}}$ and $L_{\text{t}}$ for a given communication range
$D_{\text{e}}$, thus violating the sufficient condition for the reflected
transmission to be paraxial. To this end, Fig. 14 depicts the spectral
efficiency of the reflected channel between two ULAs of apertures
$L_{\text{r}}=L_{\text{t}}=L$ as a function of $L/D_{\text{e}}$ at ${\sf
SNR}=0$ dB. The ULAs are separated by ${D=2}$ m and the surface distance is
${D_{0}=3}$ m so that $D_{\text{e}}=4$ m. In turn, the antenna spacing is
optimized for the reflected transmission, which implies array apertures
linearly increase with the number of antennas.

The ray-tracing curve yields a tight match with the exact one, except for the
regime where the two arrays have an aperture $L$ comparable to the range
$D_{\text{e}}$ of the reflected transmission. Hence, ray tracing algorithms
leveraging the paraxial approximation offer a good fit to reality, as also
supported by the robustness of the underlying approximation against changes in
the propagation geometry.

##  VIII Conclusion

$h(\delta_{\rho};r_{z},s_{z})=\begin{cases}\begin{aligned}
\displaystyle&\frac{\kappa_{1}\eta_{1}}{4\pi}\int_{0}^{\kappa_{1}}\kappa_{\rho}\frac{J_{0}(\kappa_{\rho}\delta_{\rho})}{\kappa_{1z}}\left(e^{-\mathsf{j}\kappa_{1z}(r_{z}-s_{z})}+R(\kappa_{\rho})e^{-\mathsf{j}\kappa_{1z}(r_{z}+s_{z}-2D_{0})}\right)d\kappa_{\rho}&r_{z}<-R_{0}\\\
\displaystyle&\frac{\kappa_{1}\eta_{1}}{4\pi}\int_{0}^{\kappa_{1}}\kappa_{\rho}\frac{J_{0}(\kappa_{\rho}\delta_{\rho})}{\kappa_{1z}}\left(e^{\mathsf{j}\kappa_{1z}(r_{z}-s_{z})}+R(\kappa_{\rho})e^{-\mathsf{j}\kappa_{1z}(r_{z}+s_{z}-2D_{0})}\right)d\kappa_{\rho}&R_{0}<r_{z}\leq
D_{0}\end{aligned}\end{cases}$ (47)

Through a physics-based formulation, we have confirmed that reflection off a
large and smooth planar surface, say a wall or ceiling, can serve as
alternatives to LOS for wide-aperture MIMO communication. With respect to an
LOS link, a reflected counterpart exhibits:

  * •

A power loss determined by the additional range and by the share of incident
power not reflected by the surface.

  * •

A reduction in the number of DOF because of the antenna spacing tailored to
the LOS link being smaller than the one that the reflected link would require
at the same SNR.

If the arrays are outright configured for the reflected transmission, then the
second effect is corrected. The above observations bode well for flexible LOS
MIMO communication aided by reflections, with further work required to
determine the impact of surface finiteness and roughness. This paper ignores
mutual coupling effects among antenna elements, which are most impactful at
sub-wavelength spacings [23]. This ought not to be the case for wide-aperture
MIMO that envisions electrically large antenna spacings, with follow-up
studies needed to confirm this hypothesis.

Connection with the image theorem that underlies ray tracing showed that, with
non-planar wavefronts, the image of the transmitter is blurred by the
convolution with a response modeling the not perfect reflectivity of the
surface. Ray tracing ignores this blurring, which is to say it regards the
convolving response as an impulse. However, our findings show that only for
very large arrays does the response depart from an impulse, justifying the use
of ray tracing algorithms [8] in most situation.

## Generation of the MIMO channel matrix

For the sake of compactness, let us define the space-lag variable
${{\boldsymbol{\delta}}={\boldsymbol{r}}-{\boldsymbol{s}}}$ with coordinates
${\delta_{x}=r_{x}-s_{x}}$ and ${\delta_{y}=r_{y}-s_{y}}$, indicating the
displacement between source and receiver on the $z$-plane. Due to circular
symmetry of $H({\boldsymbol{\kappa}};r_{z},s_{z})$ in (30), we eliminate the
azimuthal dependance of the channel impulse response by evaluating (29) at
$(\delta_{\rho},0)$, $\delta_{\rho}=\|{\boldsymbol{\delta}}\|$. The result is
reported in (47) for any given $s_{z}$ and $r_{z}$ where we introduced
$\kappa_{\rho}=\|{\boldsymbol{\kappa}}\|\in[0,\kappa_{1}]$ within
${\boldsymbol{\kappa}}\in\mathcal{D}$. Hence, the impulse response is
invariant under any affine transformation that preserves the distance between
source and receiver on the $z$-plane. Eq. (47) is a Sommerfeld-type integral
[10, Eq. 2.2.30]. This describes the received field as integral superpositions
of cylindrical waves times an upgoing or downgoing plane wave in the
$z$-direction. Analytical solutions of (47) are hardly available and problem-
specific [10, Ch. 2.7.3]. Hence, we resort to a numerical integration
procedure that accounts for the singularities on the complex
$\kappa_{\rho}$-plane.333The error of a numerical integration routine is
proportional to the derivatives of the integrand and are unbounded near a
singularity [24].

Assuming the analyticity of the integrand, we can invoke Cauchy’s integral
theorem and deform the contour integration path to avoid singularities. The
integral value is unchanged along this new integration path. This should lie
in the fourth orthant due to ${\real(\kappa_{\rho})\geq 0}$ and Sommerfeld
radiation conditions (i.e., ${\imaginary(\kappa_{1z})\geq 0}$ and
${\real(\kappa_{1z})\geq 0}$) that ensures convergence of the improper
integral in (47) [10, Ch. 2.2.3].444The half-planes
${\imaginary(\kappa_{1z})=0}$ and ${\real(\kappa_{1z})=0}$ map to the
hyperbola [10, Eq. 2.2.33] in the complex $\kappa_{\rho}$ plane; see [10, Fig.
2.2.8]. We follow [25] and choose a semi-elliptical integration path
$\mathcal{C}$ that goes around the pole singularities with semi-axes of the
ellipse chosen as [25]

$\displaystyle\kappa_{\rho}^{\text{maj}}=(\kappa_{1}+\kappa_{2})/2$
$\displaystyle\qquad\kappa_{\rho}^{\text{min}}=\kappa_{\rho}^{\text{maj}}/10^{3}$
(48)

so that the contour of $\mathcal{C}$ is sufficiently away from the singularity
but $\kappa_{\rho}$ is small enough for the argument of the Bessel function in
(47) to ensure controlled oscillations. For complex integration, we
parametrize the curve as $\kappa_{\rho}(\theta):[\pi,2\pi)\to\mathcal{C}$
where
$\kappa_{\rho}(\theta)=\kappa_{\rho}^{\prime}(\theta)+\mathsf{j}\kappa_{\rho}^{\operatorname*{\prime\prime}}(\theta)$
with

$\displaystyle\kappa_{\rho}^{\prime}(\theta)=\frac{\kappa_{\rho}^{\text{maj}}}{2}(1+\cos(\theta))$
$\displaystyle\;\;\;\kappa_{\rho}^{\operatorname*{\prime\prime}}(\theta)=\frac{\kappa_{\rho}^{\text{min}}}{2}\sin(\theta),$
(49)

leading to [24, Ch. 10.5]

$\int_{\mathcal{C}}f(\kappa_{\rho})\,d\kappa_{\rho}=\int_{0}^{\pi}f(\kappa_{\rho}(\theta))\left|\frac{\partial\kappa_{\rho}(\theta)}{\partial\theta}\right|\,d\theta$
(50)

where $f(\kappa_{\rho})$ is the integrand of (47) and the Jacobian is

$\frac{\partial\kappa_{\rho}(\theta)}{\partial\theta}=\frac{1}{2}\left(-\kappa_{\rho}^{\text{maj}}\sin(\theta)+\mathsf{j}\kappa_{\rho}^{\text{min}}\cos(\theta)\right).$
(51)

The presented numerical generation procedure performs superbly as long as the
transverse distance $\rho$ is not too large compared to the wavelength
$\lambda$. Numerical simulations show no issue for $\rho<18$ m at $60$ GHz,
i.e., $\delta_{\rho}/\lambda<3600$. For larger $\delta_{\rho}$, the integrand
in (47) becomes a rapidly oscillating function of $\kappa_{\rho}$, due to the
large variations into the Bessel function, and $\mathcal{C}$ must be chosen
according to the steepest descent path [10, Ch. 2.7.3].

## References

  * [1] A. Pizzo, A. Lozano, S. Rangan, and T. L. Marzetta,  “Line-of-sight MIMO via reflection from a smooth surface,”  in IEEE Veh. Technol. Conf. (VTC), 2022.
  * [2] S. Rangan, T. S. Rappaport, and E. Erkip,  “Millimeter-wave cellular wireless networks: Potentials and challenges,”  Proc. IEEE, vol. 102, no. 3, pp. 366–385, 2014.
  * [3] T. S. Rappaport, G. R. MacCartney, M. K. Samimi, and S. Sun,  “Wideband millimeter-wave propagation measurements and channel models for future wireless communication system design,”  IEEE Trans. Commun., vol. 63, no. 9, pp. 3029–3056, 2015.
  * [4] T. S. Rappaport, Y. Xing, O. Kanhere, S. Ju, A. Madanayake, S. Mandal, A. Alkhateeb, and G. C. Trichopoulos,  “Wireless communications and applications above 100 GHz: Opportunities and challenges for 6G and beyond,”  IEEE Access, vol. 7, pp. 78729–78757, 2019.
  * [5] E. Torkildson, U. Madhow, and M. Rodwell,  “Indoor millimeter wave MIMO: Feasibility and performance,”  IEEE Trans. Wireless Commun., vol. 10, no. 12, pp. 4150–4160, 2011\.
  * [6] Frode Bohagen, Pal Orten, and Geir E. Oien,  “On spherical vs. plane wave modeling of line-of-sight MIMO channels,”  IEEE Trans. Commun., vol. 57, no. 3, pp. 841–849, 2009.
  * [7] H. Do, S. Cho, J. Park, H.-J. Song, N. Lee, and A. Lozano,  “Terahertz line-of-sight MIMO communication: Theory and practical challenges,”  IEEE Commun. Magazine, vol. 59, no. 3, pp. 104–109, 2021.
  * [8] Yaqi Hu, Mingsheng Yin, Sundeep Rangan, and Marco Mezzavilla,  “Parametrization of high-rank line-of-sight MIMO channels with reflected paths,”  in 2022 IEEE 23rd Int. Workshop Signal Process. Advances Wireless Commun. (SPAWC), 2022, pp. 1–5.
  * [9] M. Born and E. Wolf,  Principles of Optics,  Pergamon Press, 6 edition, 1980.
  * [10] W. C. Chew,  Waves and Fields in Inhomogenous Media,  Wiley-IEEE Press, 1995.
  * [11] T. B. Hansen and A. D. Yaghjian,  Plane-Wave Theory of Time-Domain Fields,  Wiley-IEEE Press, New York, 1999.
  * [12] A. Pizzo, A de J. Torres, L. Sanguinetti, and T. L. Marzetta,  “Nyquist sampling and degrees of freedom of electromagnetic fields,”  IEEE Trans. on Signal Process., pp. 1–12, 2022, doi=10.1109/TSP.2022.3183445.
  * [13] A. Pizzo, T. L. Marzetta, and L. Sanguinetti,  “Spatially-stationary model for Holographic MIMO small-scale fading,”  IEEE J. Sel. Areas Commun., vol. 38, no. 9, pp. 1964–1979, 2020\.
  * [14] A. Pizzo, L. Sanguinetti, and T. L. Marzetta,  “Spatial characterization of electromagnetic random channels,”  IEEE Open J. Commun. Soc., vol. 3, pp. 847–866, 2022.
  * [15] H. Do, N. Lee, and A. Lozano,  “Capacity of line-of-sight MIMO channels,”  in IEEE Int. Symp. Inf. Theory (ISIT), 2020, pp. 2044–2048.
  * [16] H. Do, N. Lee, and A. Lozano,  “Reconfigurable ULAs for line-of-sight MIMO transmission,”  IEEE Trans. Wireless Commun., vol. 20, no. 5, pp. 2933–2947, 2021\.
  * [17] A. F. Molisch,  Wireless Communications,  Wiley-IEEE Press, 2 edition, 2011.
  * [18] S. J. Orfanidis,  Electromagnetic Waves and Antennas,  2016\. [Online] Available: https://www.ece.rutgers.edu/~orfanidi/ewa/.
  * [19] K. Sato, H. Kozima, H. Masuzawa, T. Manabe, T. Ihara, Y. Kasashima, and K. Yamaki,  “Measurements of reflection characteristics and refractive indices of interior construction materials in millimeter-wave bands,”  in IEEE Veh. Technol. Conf. (VTC), 1995.
  * [20] C. A Balanis,  Antenna Theory: Analysis and Design,  Wiley-Interscience, 4 edition, 2005.
  * [21] A Tulino, A. Lozano, and S. Verdu,  “MIMO capacity with channel state information at the transmitter,”  in IEEE Int’l Symp. Spread Spectrum Techniques and Applications (ISSSTA), 2004.
  * [22] R. W. Heath Jr. and A. Lozano,  Foundations of MIMO Communication,  Cambridge University Press, 2018.
  * [23] Michel T. Ivrlač and Josef A. Nossek,  “Toward a circuit theory of communication,”  IEEE Trans. Circuits Syst., vol. 57, no. 7, pp. 1663–1683, 2010\.
  * [24] F. B. Hildebrand,  Advanced Calculus for Applications,  Prentice Hall, 1962.
  * [25] M. Paulus, P. Gay-Balmaz, and O. J. F. Martin,  “Accurate and efficient computation of the Green’s tensor for stratified media,”  Phys. Rev. E, vol. 62, pp. 5797–5807, Oct 2000.