# Pattern-Division Multiplexing for Continuous-Aperture MIMO

Zijian Zhang and Linglong Dai  0 Beijing National Research Center for
Information Science and Technology (BNRist)
Department of Electronic Engineering, Tsinghua University, Beijing 100084,
China
Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In recent years, continuous-aperture multiple-input multiple-output (CAP-MIMO)
is reinvestigated to achieve improved communication performance with limited
antenna apertures. Unlike the classical MIMO composed of discrete antennas,
CAP-MIMO has a continuous antenna surface, which is expected to generate any
current distribution (i.e., pattern) and induce controllable spatial
electromagnetic waves. In this way, the information can be modulated on the
electromagnetic waves, which makes it promising to approach the ultimate
capacity of finite apertures. The pattern design for CAP-MIMO is the key
factor to determine the communication performance, but it has not been well
studied in the literature. In this paper, we propose the pattern-division
multiplexing to design the patterns for CAP-MIMO. Specifically, we first
derive the system model of a typical multi-user CAP-MIMO system, which allows
us to formulate the sum-rate maximization problem. Then, we propose a general
pattern-division multiplexing technique to transform the design of continuous
pattern functions to the design of their projection lengths on finite
orthogonal bases. Based on this technique, we further propose a pattern design
scheme to solve the formulated sum-rate maximization problem. Simulation
results show that, the sum-rate achieved by the proposed scheme is about 260%
higher than that achieved by the benchmark scheme.

###### Index Terms:

Continuous-aperture MIMO, electromagnetic information theory, pattern-division
multiplexing.

##  I Introduction

From 3G to 5G, the system performance of wireless communications has been
greatly improved by the use of multiple-input multiple-output (MIMO) [1]. In
recent years, in order to break the performance limit of conventional MIMO
with limited antenna apertures [2], as an ultimate MIMO structure with
infinitely dense antennas, continuous-aperture MIMO (CAP-MIMO), which is also
called as holographic MIMO [3] or large intelligent surface (LIS) [4], is
reinvestigated for wireless communications. Unlike the classical MIMO composed
of multiple discrete antennas, CAP-MIMO takes the form of a spatially-
continuous electromagnetic surface [2], which can generate any current
distribution and induce controllable spatial electromagnetic waves [5]. In
this way, the information for receivers can be directly modulated on the
spatial electromagnetic waves, thus it is promising to achieve the ultimate
capacity of limited apertures.

The patterns, i.e., the current distributions on the continuous CAP-MIMO
aperture, are the key factors determining the performance of CAP-MIMO [6]. To
support multi-stream transmissions, it is necessary for CAP-MIMO to adopt a
series of distinguishable patterns to carry different symbols. To achieve
this, most existing works have directly adopted the patterns generated by the
given special functions [3, 4, 2]. For example, the authors in [3] considered
a near-field line-of-sight scenario with a couple of linear-aperture CAP-MIMO
transceivers. Then, a wavenumber-division multiplexing scheme was proposed to
directly generate orthogonal patterns by Fourier basis functions. In this way,
the transmitted symbols are modulated on different wavenumbers of radiated
electromagnetic waves respectively, thus these symbols become distinguishable,
which is similar to the conventional frequency-division multiplexing.

Despite the existing schemes can improve the communication performance of CAP-
MIMO, most of these schemes are heuristic and can only be applied to some
special scenarios, such as single receiver, near field, and line-of-sight
transmissions [3, 4, 2]. To support CAP-MIMO in general communication
scenarios, it is essential to design the patterns flexibly according to the
real-time channel state information. Unfortunately, according to the best of
our knowledge, the research on such a general pattern design scheme for CAP-
MIMO has not been well studied in the literature.

To fill in this gap, in this paper, we propose the pattern-division
multiplexing technique. Specifically, we first derive the system model
including the capacity and power constraint of CAP-MIMO, which allows us to
formulate the sum-rate maximization problem. Then, we propose the general
pattern-division multiplexing to flexibly design the CAP-MIMO patterns. The
key idea is to use series expansion to project the continuous pattern
functions onto an orthogonal basis space, thus the design of continuous
pattern functions is transformed to the design of their projection lengths on
finite orthogonal bases. Finally, based on the proposed pattern-division
multiplexing, a pattern design scheme is proposed to solve the formulated sum-
rate maximization problem. Simulation results show that, the patterns designed
by the proposed scheme are almost mutually orthogonal, the sum-rate achieved
by the proposed scheme is about 260% higher than that achieved by the existing
wavenumber-division multiplexing scheme [3].

The rest of this paper is organized as follows. Section II introduces the
system model of CAP-MIMO and formulates the sum-rate maximization problem. The
pattern-division multiplexing technique, as well as the pattern design scheme,
are proposed in Section III. Simulation results are provided in Section IV to
validate the effectiveness of the proposed scheme and evaluate the sum-rate
performance of CAP-MIMO. Finally, conclusions are drawn in Section V.

##  II System Model and Problem Formulation

###  II-A System model

Figure 1: (a) illustrates a CAP-MIMO transmitter, which has a continuous
antenna aperture for radiating information-carrying electromagnetic waves [2].
(b) illustrates $K$ electromagnetic-wave receivers, which are able to capture
the electromagnetic waves in space and decode symbols.

As shown in Fig. 1 (a), in this paper, we consider a CAP-MIMO transmitter with
surface ${\cal S}_{\rm T}$ of area $A_{\rm T}=\left|{\cal S}_{\rm T}\right|$.
In the ideal case, CAP-MIMO has a spatially-continuous antenna aperture, which
is able to generate any current distribution on its continuous surface for
wireless communications [2, 3, 4]. Consider that the CAP-MIMO transmitter
simultaneously serves $K$ electromagnetic-wave receivers in the downlink, as
shown in Fig. 1 (b). Let
${{\bf{x}}}\triangleq{\left[{{x_{1}},\cdots,{x_{K}}}\right]^{T}}\in{{\mathbb{C}}^{K}}$
denote $K$ symbols transmitted to $K$ receivers, respectively. Without loss of
generality, we assume that $\mathbb{E}_{\bf
x}\left\\{\mathbf{x}\mathbf{x}^{\rm H}\right\\}=\mathbf{I}_{K}$.

Following the well-known time-harmonic assumption [7], let ${\bf j}({\bf
s})\in{\mathbb{C}}^{3}$ denote the time-independent current distribution at
location ${\bf s}:=\left(s_{x},s_{y},s_{z}\right)\in{\mathbb{R}}^{3}$. Then,
the symbols to be transmitted are modulated on $K$ different CAP-MIMO patterns
(i.e., current distributions), which aims to make these symbols orthogonal as
much as possible so as to achieve high capacity [2]. Assuming that CAP-MIMO
employs linear pattern combination, the information-carrying current
distribution ${\bf{j}}({\bf{s}})$ on the CAP-MIMO aperture can be modeled as

${\bf{j}}({\bf{s}})=\sum\limits_{k=1}^{K}{{{\bm{\theta}}_{k}}\left({\bf{s}}\right){x_{k}}},~{}~{}{\mathbf{s}\in{{\cal
S}_{\rm T}}},$ (1)

where ${\bm{\theta}}_{k}\left({\bf{s}}\right)\in{\mathbb{C}}^{3}$ is the
pattern that carries symbol $x_{k}$.

To model the radiated information-carrying electromagnetic waves in space, we
define ${\bf e}({\bf r})\in\mathbb{C}^{3}$ as the electric field at point
${\bf r}:=\left(r_{x},r_{y},r_{z}\right)\in\mathbb{R}^{3}$, which is induced
by the current distribution ${\bf j}({\bf s})$ on the CAP-MIMO aperture.
According to Helmholtz wave equation [7], by introducing Green function
$\mathbf{G}(\mathbf{r},\mathbf{s})\in{\mathbb{C}}^{3\times 3}$, the electric
field ${\bf e}({\bf r})$ at point ${\bf r}$ of the receiver can be written as

$\mathbf{e}(\mathbf{r})=\int_{{\cal S}_{\rm
T}}\mathbf{G}(\mathbf{r},\mathbf{s})\mathbf{j}(\mathbf{s})\mathrm{d}\mathbf{s},$
(2)

where Green function $\mathbf{G}(\mathbf{r},\mathbf{s})$ is similar to the
classical definition of wireless channels. Note that,
$\mathbf{G}(\mathbf{r},\mathbf{s})$ is determined by the specific transmission
environment. For example, in ideal unbounded and homogeneous mediums, we have

$\mathbf{G}(\mathbf{r},\mathbf{s})=\frac{\mathrm{j}\kappa
Z_{0}}{4\pi}\frac{e^{\mathrm{j}\kappa\|\mathbf{r}-\mathbf{s}\|}}{\|\mathbf{r}-\mathbf{s}\|}\left(\mathbf{I}_{3}+\frac{\nabla_{\mathbf{r}}\nabla_{\mathbf{r}}^{\mathrm{H}}}{\kappa^{2}}\right),$
(3)

where $Z_{0}$ is the intrinsic impedance of spatial medium.

As shown in Fig. 1 (b), we assume that all $K$ receivers are located in the
far-field region, and each receiver is equipped with an ideal isotropic
antenna with effective aperture area $A_{\rm R}=\frac{\lambda^{2}}{4\pi}$,
which is much smaller than the transmitter aperture area $A_{\rm T}$. In this
case, each receiver can be reasonably approximated by a point in space. Let
${\bf r}_{k}\in{\mathbb{R}}^{3}$ denote the 3-D location of the $k$-th
electromagnetic-wave receiver. In the ideal case, the receiver $k$ is expected
to ideally sense and capture the whole information of the electromagnetic
waves reaching point ${\bf r}_{k}$. According to (1) and (2), the
electromagnetic wave captured by receiver $k$ can be expressed as

$\displaystyle{{\bf{y}}_{k}}={{\bf{e}}_{k}}+{{\bf{n}}_{k}}$ (4)
$\displaystyle=\underbrace{{x_{k}}\\!\\!\int_{{{\cal
S}_{\rm{T}}}}\\!\\!{{{\bf{G}}_{k}}({\bf{s}})}{\bm{\theta}_{k}}\left({\bf{s}}\right){\rm{d}}{\bf{s}}}_{\text{Desired
signal to user $k$}}+\\!\\!\\!\\!\underbrace{\sum\limits_{j=1,j\neq
k}^{K}\\!\\!{{x_{j}}\\!\\!\int_{{{\cal
S}_{\rm{T}}}}\\!\\!{{{\bf{G}}_{k}}({\bf{s}}){\bm{\theta}_{j}}\left({\bf{s}}\right)}{\rm{d}}{\bf{s}}}}_{\text{Interference
from other receivers}}+\underbrace{{{\bf{n}}_{k}}}_{\text{Noise}},$

where ${{\bf{e}}_{k}}:=\mathbf{e}(\mathbf{r}_{k})$, ${{\bf{G}}_{k}}({\bf
s}):=\mathbf{G}(\mathbf{r}_{k},\mathbf{s})$, and ${{\bf{n}}_{k}}$ is the
electromagnetic noise at receiver $k$, which is produced by all incoming
electromagnetic waves that are not generated by the transmitter. Here we
assume that, ${{\bf{n}}_{k}}$ is additive white Gaussian noise (AWGN) with
zero mean and variance $\sigma^{2}{\bf I}_{3}$.

###  II-B Sum-rate maximization problem formulation

Based on electromagnetic signal model (4), by calculating the mutual
information [7], the CAP-MIMO channel capacity, i.e., the sum-rate of $K$
receivers, can be written as

${R_{\rm
sum}}=\sum\limits_{k=1}^{K}{{\log_{2}}\det\left|{{{\bf{I}}_{3}}+{{\bm{\alpha}}_{k}}{\bm{\alpha}}_{k}^{\rm{H}}{\bf{J}}_{k}^{-1}}\right|},$
(5)

where ${\bm{\alpha}}_{k}$ and ${\bf{J}}_{k}$ are given by

$\displaystyle{{\bm{\alpha}}_{k}}=$ $\displaystyle\int_{{{\cal
S}_{\rm{T}}}}{{{\bf{G}}_{k}}({\bf{s}}){{\bm{\theta}}_{k}}\left({\bf{s}}\right){\rm
d}{\bf{s}}},$ (6) $\displaystyle{{\bf{J}}_{k}}=$
$\displaystyle\\!\\!\\!\\!\sum\limits_{j=1,j\neq k}^{K}{{\int_{\\!{\cal
S}_{\rm{T}}}\\!\\!\\!{{{\bf{G}}_{k}}({\bf{s}}){{\bm{\theta}}_{j}}\left({\bf{s}}\right){\rm
d}{\bf{s}}}}{{\left({\int_{\\!{\cal
S}_{\rm{T}}}\\!\\!\\!{{{\bf{G}}_{k}}({\bf{s^{\prime}}}){{\bm{\theta}}_{j}}\left({\bf{s^{\prime}}}\right){\rm
d}{\bf{s^{\prime}}}}}\right)}^{\\!\\!\\!\rm{H}}}}\\!\\!+\\!{\sigma^{2}}{{\bf{I}}_{3}}.$

In practical systems, we are interested in investigating the CAP-MIMO channel
capacity under a given power constraint. By integrating the radial component
of the Poynting vector over a sphere with infinite-length radius, the physical
transmit power of CAP-MIMO based communication systems can be bounded by [7]

$\sum\limits_{k=1}^{K}{\int_{{{\cal
S}_{\rm{T}}}}{{{\left\|{{{\bm{\theta}}_{k}}\left({\bf{s}}\right)}\right\|}^{2}}{\rm{d}}{\bf{s}}}}\leq{P_{\rm{T}}},$
(7)

where $P_{\rm T}$ can be viewed as the allowable maximum “transmit power” of
CAP-MIMO, which is implicitly associated with the physical energy and measured
in [${\text{A}}^{2}$].

By combing sum-rate (5) and transmit power constraint (7), the original
problem of CAP-MIMO sum-rate maximization subject to the power constraint can
be formulated as

$\displaystyle\\!\\!\\!\\!{\cal P}_{o}:~{}~{}$
$\displaystyle\mathop{\max}\limits_{{{\bm{\theta}}}\left({\bf
s}\right)}~{}~{}{R_{\rm
sum}}=\sum\limits_{k=1}^{K}{{\log_{2}}\det\left|{{{\bf{I}}_{3}}+{{\bm{\alpha}}_{k}}{\bm{\alpha}}_{k}^{\rm{H}}{\bf{J}}_{k}^{-1}}\right|},$
(8a) $\displaystyle~{}~{}{\rm s.t.}~{}~{}\sum\limits_{k=1}^{K}{\int_{{{\cal
S}_{\rm{T}}}}{{{\left\|{{{\bm{\theta}}_{k}}\left({\bf{s}}\right)}\right\|}^{2}}{\rm{d}}{\bf{s}}}}\leq{P_{\rm{T}}},$
(8b)

where ${\bm{\theta}}({\bf s})$ denotes the set of functions
${\bm{\theta}}_{k}({\bf s})$ for all $k\in\\{1,\cdots,K\\}$. Our goal is to
maximize the sum-rate (8a) by appropriately designing the continuous pattern
functions ${\bm{\theta}}_{k}({\bf s})$ for all $k\in\\{1,\cdots,K\\}$. Note
that, the sum-rate maximization problem ${\cal P}_{o}$ in (8) is difficult to
solve. The reason is that, the coupled continuous functions
${\bm{\theta}}({\bf s})$ within the integral items exist in both optimization
objective and constraint in (8), which is a non-convex density functional
optimization problem. Such kind of continuous functional optimizations are
common in the micro-wave area [2], where they are usually addressed by using
the commercial electromagnetic simulation software such as high frequency
structure simulator (HFSS), which leads to high time and space complexity.

##  III Proposed Pattern-Division Multiplexing Technology

###  III-A A general pattern-division multiplexing technique

To support CAP-MIMO in general communication scenarios, different from the
existing works which adopt the patterns generated by given special functions
[4, 3, 2], we propose the pattern-division multiplexing to flexibly design the
patterns for CAP-MIMO according to the real-time channel state information.
Specifically, the proposed pattern-division multiplexing aims to make the
information-carrying electromagnetic waves reaching different receivers
orthogonal as much as possible. In this way, higher channel capacity is
expected, which is similar to the space-division multiplexing in classical
MIMO systems.

To efficiently optimize the continuous pattern functions
${\bm{\theta}}_{k}({\bf s})$, an intuitive idea is to use series expansion to
project these continuous functions onto an orthogonal space. Since Fourier
series expansion has good generality, in this paper, we use Fourier bases to
expand the continuous functions. For a clear illustration, we first introduce
the following lemma.

######  Lemma 1 (Fourier series expansion)

For any arbitrary continuous function $f(t)\in{\mathbb{C}}$ defined in
$t\in[a,b]\in{\mathbb{R}}$, if $f(t)$ is absolutely integrable, $f(t)$ can be
equivalently rewritten as

$f\left(t\right)=\sum\limits_{n=-\infty}^{\infty}{{F_{n}}{{\Psi_{n}}\left(t\right)}},$
(9)

where the projection length $F_{n}\in{\mathbb{C}}$ and the Fourier basis
${\Psi_{n}}\left(t\right)\in{\mathbb{C}}$ are respectively given by

$\displaystyle{F_{n}}=$
$\displaystyle\frac{1}{{\sqrt{b-a}}}\int_{a}^{b}{f\left(t\right){e^{\frac{-{{\rm{j}}2\pi
n}}{{b-a}}\left({t-\frac{{b-a}}{2}}\right)}}}{\rm{d}}t,$ (10a)
$\displaystyle{\Psi_{n}}\left(t\right)=$
$\displaystyle\frac{1}{{\sqrt{b-a}}}{e^{\frac{{{\rm{j}}2\pi
n}}{{b-a}}\left({t-\frac{{b-a}}{2}}\right)}}.$ (10b)

Then, by extending the above Lemma 1 to 3-D scenario, the continuous pattern
functions ${\bm{\theta}}_{k}({\bf s})$ to be designed can be equivalently
rewritten as

${\bm{\theta}}_{k}({\bf
s})=\sum\limits_{\bf{n}}^{\infty}{{{\bf{w}}_{k,{\bf{n}}}}{\Psi_{\bf{n}}}\left({\bf{s}}\right)},~{}~{}{\bf
s}\in{\cal S}_{\rm T},$ (11)

where ${\bf{w}}_{k,{\bf{n}}}\in{\mathbb{C}}^{3}$ is the projection length of
pattern ${\bm{\theta}}_{k}({\bf s})$ on the Fourier basis
${\Psi_{\bf{n}}}\left({\bf{s}}\right)\in{\mathbb{C}}$, and here we define
${\bf{n}}:={\left({n_{x}},{n_{y}},{n_{z}}\right)}$ to distinguish different
expansion items and
$\sum\nolimits_{\bf{n}}^{\infty}{}:=\sum\nolimits_{{n_{x}}=-\infty}^{\infty}{\sum\nolimits_{{n_{y}}=-\infty}^{\infty}{\sum\nolimits_{{n_{z}}=-\infty}^{\infty}{}}}$
for expression simplification. In 3-D scenarios, Fourier basis
${\Psi_{\bf{n}}}\left({\bf{s}}\right)$ becomes

${\Psi_{\bf{n}}}\\!\left({\bf{s}}\right)=\frac{1}{\sqrt{A_{\rm T}}}{e^{{\rm
j}2\pi\left({\frac{{{n_{x}}}}{{{L_{x}}}}{{s_{x}}}+\frac{{{n_{y}}}}{{{L_{y}}}}{{s_{y}}}+\frac{{{n_{z}}}}{{{L_{z}}}}{{s_{z}}}}\right)}}$
(12)

wherein $L_{x}$, $L_{y}$, and $L_{z}$ denote the maximum projection lengths of
CAP-MIMO aperture area $A_{\rm T}$ on the $x$-, $y$-, and $z$-axis of 3-D
coordinate system, respectively. Relying on this transformation, we obtain the
following two corollaries.

######  Corollary 1 (Continuous-discrete transformation for electromagnetic
waves)

By adopting Fourier series expansion, the coupled Green function
${\bf{G}}_{k}({\bf s})$ and pattern function ${\bm{\theta}}_{j}({\bf s})$ in
integral $\int_{{{\cal S}_{\rm T}}}{\rm d}{\bf s}$ (i.e., the electromagnetic
wave), can be equivalently rewritten as

$\displaystyle\int_{{{\cal
S}_{\rm{T}}}}{{{\bf{G}}_{k}}({\bf{s}}){{\bm{\theta}}_{j}}\left({\bf{s}}\right){\rm
d}{\bf{s}}}=\sum\limits_{\bf{n}}^{\infty}{{{\bf{\Omega}}_{k,{\bf{n}}}}{{\bf{w}}_{j,{\bf{n}}}}},$
(13)

where

$\displaystyle{{\bf{\Omega}}_{k,{\bf{n}}}}=\int_{{{\cal
S}_{\rm{T}}}}{{{\bf{G}}_{k}}({\bf{s}}){\Psi_{\bf{n}}}\left({\bf{s}}\right){\rm{d}}{\bf{s}}},$
(14)

and ${{{\bf{\Omega}}_{k,{\bf{n}}}}}\in{\mathbb{C}}^{3\times 3}$ is exactly the
Fourier transform of Green function ${\bf{G}}_{k}({\bf s})$ on the CAP-MIMO
aperture ${\cal S}_{\rm T}$ at the spatial frequency of
$(\frac{{n_{x}}}{L_{x}},\frac{{n_{y}}}{L_{y}},\frac{{n_{z}}}{L_{z}})$.

######  Corollary 2 (Continuous-discrete transformation for power constraint)

According to Parseval’s theorem, the square $L^{2}$ norm of current density
${\bm{\theta}}_{k}({\bf s})$ in integral expressions, i.e., the transmit
power, can be equivalently rewritten as

$\displaystyle\int_{{{\cal
S}_{\rm{T}}}}{{{\left\|{{{\bm{\theta}}_{k}}\left({\bf{s}}\right)}\right\|}^{2}}{\rm{d}}{\bf{s}}}=\sum\limits_{\bf{n}}^{\infty}{{{\left\|{{{\bf{w}}_{k,{\bf{n}}}}}\right\|}^{2}}}.$
(15)

Exploiting Corollary 1 and Corollary 2, the design of the continuous pattern
functions ${\bm{\theta}}_{k}({\bf s})$ becomes the design of the projection
lengths ${\bf{w}}_{k,{\bf{n}}}$, and thus the density functional optimization
can be equivalently transformed as a common digital signal processing problem.
However, since the number of expansion items is infinite, the projection
lengths ${\bf{w}}_{k,{\bf{n}}}$ are still hard to be optimized. Fortunately,
thanks to the mathematical structure of Green function ${\bf{G}}_{k}({\bf
s})$, we notice that, the power of ${\bf{G}}_{k}({\bf s})$ in Fourier space is
mainly distributed in the low-frequency band, while that in the high-frequency
band is very low (see [3, Fig. 4]). In other words, when the number of
expansion items $\bf n$ is large enough, the value of
${\bf{\Omega}}_{k,{\bf{n}}}$ tends to be negligible. This fact inspires us to
approximate the original continuous function with finite Fourier expansion
items, thus we obtain the following proposition.

######  Proposition 1 (Finite-item approximation of Fourier series)

Employing a truncation operation on (13) and (15), the electric field and
transmit power can be approximated by

$\displaystyle\int_{{{\cal
S}_{\rm{T}}}}\\!\\!{{{\bf{G}}_{k}}({\bf{s}}){{\bm{\theta}}_{j}}\left({\bf{s}}\right){\rm
d}{\bf{s}}}$ $\displaystyle\approx\sum\limits_{\bf{n}}^{\bf
N}{{{\bf{\Omega}}_{k,{\bf{n}}}}{{\bf{w}}_{j,{\bf{n}}}}},$ (16a)
$\displaystyle\int_{{{\cal
S}_{\rm{T}}}}{{{\left\|{{{\bm{\theta}}_{k}}\left({\bf{s}}\right)}\right\|}^{2}}{\rm{d}}{\bf{s}}}$
$\displaystyle\approx\sum\limits_{\bf{n}}^{\bf
N}{{{\left\|{{{\bf{w}}_{k,{\bf{n}}}}}\right\|}^{2}}},$ (16b)

where we have ${\bf N}:=\left(N_{x},N_{y},N_{z}\right)$ with $N_{x}$, $N_{y}$,
and $N_{z}$ being the numbers of reserved items, and we have
$\sum\nolimits_{\bf{n}}^{\bf{N}}:=\sum\nolimits_{{n_{x}}=\left\lfloor{-{N_{x}}/2}\right\rfloor}^{\left\lfloor{{N_{x}}/2}\right\rfloor}{\sum\nolimits_{{n_{y}}=\left\lfloor{-{N_{y}}/2}\right\rfloor}^{\left\lfloor{{N_{y}}/2}\right\rfloor}{\sum\nolimits_{{n_{z}}=\left\lfloor{-{N_{z}}/2}\right\rfloor}^{\left\lfloor{{N_{z}}/2}\right\rfloor}{}}}$
for clarity.

###  III-B Proposed pattern design scheme for sum-rate maximization

Based on the proposed pattern-division multiplexing technique introduced
above, in this subsection, we propose a pattern design scheme to solve the
sum-rate maximization problem ${\cal P}_{o}$ in (8). Firstly, to decouple
pattern functions ${\bm{\theta}}_{k}({\bf s})$, by adopting an equivalent
transformation [8, Theorem 1] for ${\cal P}_{o}$ in (8), we obtain the
following lemma.

Algorithm 1 Proposed pattern design scheme.

0: Green functions ${{\bm{G}}_{k}\left({\bf s}\right)}$ for all
$k\in\\{1,\cdots,K\\}$.

0: Optimized $R_{{\rm sum}}$, $\bm{\psi}$, and $\bm{\theta}\left({\bf
s}\right)$.

1: Initialize $\bm{\psi}$ and $\bm{\theta}\left({\bf s}\right)$;

2: while No convergence of $R_{{\rm sum}}$ do

3: Update ${\bm{\rho}}$ by (19);

4: Update $\bm{\psi}$ by (20);

5: Update ${\bf w}$ by (26) and (27);

6: Update $\bm{\theta}\left({\bf s}\right)$ by (28);

7: end while

8: return Optimized $R^{\rm opt}_{{\rm sum}}$, $\bm{\psi}^{\rm opt}$, and
$\bm{\theta}^{\rm opt}\left({\bf s}\right)$.

######  Lemma 2 (Equivalent problem for sum-rate maximization)

By introducing an auxiliary variable
${\bm{\rho}}={\left[{{\rho_{k}},\cdots,{\rho_{K}}}\right]^{\rm{T}}}\in{\mathbb{R}}^{K}_{+}$
and the combining vectors
${\bm{\psi}}=\left[{{\bm{\psi}_{1}},\cdots,{\bm{\psi}_{K}}}\right]\in{\mathbb{C}}^{3\times
K}$, the original sum-rate maximization problem ${\cal P}_{o}$ in (8) can be
equivalently reformulated as

$\displaystyle\\!\\!\\!\\!{\cal P}_{1}:$
$\displaystyle\mathop{\max}\limits_{{\bm{\rho}},{\bm{\psi}},{{\bm{\theta}}}\left({\bf
s}\right)}{R_{{\rm{sum}}}^{\prime}}\\!=\\!\sum\limits_{k=1}^{K}{{{\log}_{2}}{\rho_{k}}}\\!-\\!\frac{1}{{\ln
2}}\sum\limits_{k=1}^{K}{{\rho_{k}}{E_{k}}}\\!+\\!\frac{K}{{\ln 2}},$ (17a)
$\displaystyle~{}~{}{\rm s.t.}~{}~{}\sum\limits_{k=1}^{K}{\int_{{{\cal
S}_{\rm{T}}}}{{{\left\|{{{\bm{\theta}}_{k}}\left({\bf{s}}\right)}\right\|}^{2}}{\rm{d}}{\bf{s}}}}\leq{P_{\rm{T}}},$
(17b)

where
$E_{k}:={{\mathbb{E}}_{{\bf{x}},{\bf{n}}}}\left\\{{\left|{{{\hat{x}}_{k}}-{x_{k}}}\right|}^{2}\right\\}$
is the mean-square error (MSE) of the decoded symbol
${\hat{x}}_{k}={\bm{\psi}}_{k}^{\rm H}{\bf y}_{k}$, defined as

$\displaystyle{E_{k}}=$
$\displaystyle{\left|{1-\int_{{S_{\rm{T}}}}{{\bm{\psi}}_{k}^{\rm{H}}{{\bf{G}}_{k}}({\bf{s}})}{{\bm{\theta}}_{k}}\left({\bf{s}}\right){\rm{d}}{\bf{s}}}\right|^{2}}+$
(18) $\displaystyle\sum\limits_{j=1,j\neq
k}^{K}{{{\left|{\int_{{S_{\rm{T}}}}{{\bm{\psi}}_{k}^{\rm{H}}{{\bf{G}}_{k}}({\bf{s}}){{\bm{\theta}}_{j}}\left({\bf{s}}\right)}{\rm{d}}{\bf{s}}}\right|}^{2}}}+{\sigma^{2}}{\left\|{{{\bm{\psi}}_{k}}}\right\|^{2}}.$

To solve the equivalent problem ${\cal P}_{1}$ in (17), a scheme of pattern
design can be established by optimizing variables $\bm{\rho}$, combiners
$\bm{\psi}$, and continuous functions ${\bm{\theta}}({\bf s})$ alternatively
until the convergence of sum-rate $R_{\rm sum}$. For clarity, we summarize the
whole process of this pattern design scheme in Algorithm 1, where the update
steps of $\bm{\rho}$, $\bm{\psi}$, and ${\bm{\theta}}({\bf s})$ will be
introduced in the following three parts, respectively.

####  III-B1 Fix $\bm{\psi}$ and ${\bm{\theta}}({\bf s})$, then optimize
${\bm{\rho}}$

While fixing the combiners $\bm{\psi}$ and patterns ${\bm{\theta}}({\bf s})$,
the optimal solution to ${\bm{\rho}}$ can be obtained by setting
$\frac{{\partial{R^{\prime}_{\rm sum}}}}{{\partial{\rho_{k}}}}$ to zero, given
by

${\rho_{k}^{\rm opt}}=E_{k}^{-1},~{}~{}~{}k\in\\{1,\cdots,K\\}.$ (19)

####  III-B2 Fix ${\bm{\rho}}$ and ${\bm{\theta}}({\bf s})$, then optimize
$\bm{\psi}$

While fixing ${\bm{\rho}}$ and patterns ${\bm{\theta}}({\bf s})$, by defining

$\displaystyle{{\bf{A}}_{k}}\\!=\\!\rho_{k}\\!\\!\sum\limits_{j=1}^{K}\\!\\!{\int_{\\!{{\cal
S}_{\rm{T}}}}\\!\\!{{{\bf{G}}_{k}}({\bf{s}}){{\bm{\theta}}_{j}}\left({\bf{s}}\right)}{\rm{d}}{\bf{s}}{{\left({\int_{\\!{{\cal
S}_{\rm{T}}}}\\!\\!\\!\\!{{{\bf{G}}_{k}}({{\bf{s}}^{\prime}}){{\bm{\theta}}_{j}}\left({{{\bf{s}}^{\prime}}}\right)}{\rm{d}}{{\bf{s}}^{\prime}}}\right)}^{\\!\\!\rm{H}}}}\\!\\!\\!\\!+\\!{\rho_{k}}{\sigma^{2}}{{\bf{I}}_{3}},$
$\displaystyle{{\bm{\alpha}}_{k}}\\!=\\!\rho_{k}\int_{{{\cal
S}_{\rm{T}}}}{{{\bf{G}}_{k}}({\bf{s}}){{\bm{\theta}}_{k}}\left({\bf{s}}\right){\rm
d}{\bf{s}}}.$

and solving $\frac{{\partial{R^{\prime}_{\rm
sum}}}}{{\partial{{\bm{\psi}}_{k}}}}=0$, the optimal solution to
${\bm{\psi}}_{k}$ can be easily calculated as

$\displaystyle{{\bm{\psi}}_{k}^{\rm
opt}}={\bf{A}}_{k}^{-1}{{\bm{\alpha}}_{k}},~{}~{}~{}k\in\\{1,\cdots,K\\}.$
(20)

####  III-B3 Fix ${\bm{\rho}}$ and $\bm{\psi}$, then optimize
${\bm{\theta}}({\bf s})$

Given fixed ${\bm{\rho}}$ and combiners $\bm{\psi}$, after removing the
unrelated components, the subproblem of optimizing the continuous pattern
functions ${\bm{\theta}}({\bf s})$ can be reformulated as

$\displaystyle\\!\\!\\!\\!{\cal P}_{3}:~{}~{}$
$\displaystyle\mathop{\max}\limits_{{\bm{\theta}}\left({\bf
s}\right)}~{}~{}\sum\limits_{k=1}^{K}{\rho_{k}g_{k}\left({\bm{\theta}}\left({\bf
s}\right)\right)},$ (21a) $\displaystyle~{}~{}{\rm
s.t.}~{}~{}\sum\limits_{k=1}^{K}{\int_{{{\cal
S}_{\rm{T}}}}{{{\left\|{{{\bm{\theta}}_{k}}\left({\bf{s}}\right)}\right\|}^{2}}{\rm{d}}{\bf{s}}}}\leq{P_{\rm{T}}},$
(21b)

where function ${g_{k}\left({\bm{\theta}}\left({\bf s}\right)\right)}$ is
defined as

$\displaystyle{g_{k}}\left({{\bm{\theta}}\left({\bf{s}}\right)}\right)=$
$\displaystyle\sum\limits_{j=1}^{K}{{{\left|{\int_{{{\cal
S}_{\rm{T}}}}{{\bm{\psi}}_{k}^{\rm{H}}{{\bf{G}}_{k}}({\bf{s}}){{\bm{\theta}}_{j}}\left({\bf{s}}\right)}{\rm{d}}{\bf{s}}}\right|}^{2}}}-$
(22)
$\displaystyle~{}~{}~{}~{}~{}~{}2\,{\mathop{\Re}\nolimits}\left\\{{\int_{{{\cal
S}_{\rm{T}}}}{{\bm{\psi}}_{k}^{\rm{H}}{{\bf{G}}_{k}}({\bf{s}})}{{\bm{\theta}}_{k}}\left({\bf{s}}\right){\rm{d}}{\bf{s}}}\right\\}.$

Next, to address the challenging issue of density functional optimization as
shown in problem ${\cal P}_{3}$ in (21), we apply the continuous-discrete
transformations in Corollary 1 and Corollary 2, and Proposition 1 to
reformulate problem ${\cal P}_{3}$ in (21), and the reformulated problem can
be written as

$\displaystyle\\!\\!\\!\\!{\cal P}_{4}:~{}~{}$
$\displaystyle\mathop{\max}\limits_{\bf
w}~{}~{}\sum\limits_{k=1}^{K}{\rho_{k}{{\hat{g}}_{k}}\left({\bf{w}}\right)},$
(23a) $\displaystyle~{}~{}{\rm
s.t.}~{}~{}\sum\limits_{k=1}^{K}{\sum\limits_{\bf{n}}^{\bf
N}{{{\left\|{{{\bf{w}}_{k,{\bf{n}}}}}\right\|}^{2}}}}\leq{P_{\rm{T}}},$ (23b)

where we have defined ${\bf{w}}$ as the set of ${\bf{w}}_{k,{\bf n}}$ and

$\displaystyle{{\hat{g}}_{k}}\left({\bf{w}}\right)\\!=\\!\sum\limits_{j=1}^{K}{{{\left|{\sum\limits_{\bf{n}}^{\bf
N}{{\bf{h}}_{k,{\bf{n}}}^{\rm{H}}{{\bf{w}}_{j,{\bf{n}}}}}}\right|}^{2}}}-2{\mathop{\Re}\nolimits}\left\\{{\sum\limits_{\bf{n}}^{\bf
N}{{\bf{h}}_{k,{\bf{n}}}^{\rm{H}}{{\bf{w}}_{k,{\bf{n}}}}}}\right\\},$ (24)

in which
${{\bf{h}}_{k,{\bf{n}}}}:={\bf{\Omega}}_{k,{\bf{n}}}^{\rm{H}}{{\bm{\psi}}_{k}}$.

To further simplify the expression, we define ${\bf h}_{k}$ and ${\bf w}_{k}$
as the vectorized sets of ${\bf h}_{k,{\bf n}}$ and ${\bf w}_{k,{\bf n}}$ for
all ${\bf
n}=[n_{x},n_{y},n_{z}]\in\left\\{\\{-N_{x},\cdots,N_{x}\\},\\{-N_{y},\cdots,N_{y}\\},\\{-N_{z},\cdots,N_{z}\\}\right\\}$.
Thus problem ${\cal P}_{4}$ in (23) can be equivalently reorganized as

$\displaystyle\\!\\!\\!\\!{\cal P}_{5}:~{}~{}$
$\displaystyle\mathop{\max}\limits_{{\bf
w}}~{}~{}\sum\limits_{k=1}^{K}\rho_{k}{\left({\sum\limits_{j=1}^{K}{{{\left|{{\bf{h}}_{k}^{\rm{H}}{{\bf{w}}_{j}}}\right|}^{2}}}\\!-2{\mathop{\Re}\nolimits}\left\\{{{\bf{h}}_{k}^{\rm{H}}{{\bf{w}}_{k}}}\right\\}}\right)},$
(25a) $\displaystyle~{}~{}{\rm
s.t.}~{}~{}\sum\limits_{k=1}^{K}{{{\left\|{{{\bf{w}}_{k}}}\right\|}^{2}}}\leq{P_{\rm{T}}},$
(25b)

which is a standard quadratically constrained quadratic program (QCQP). By
adopting Lagrange multiplier method, the optimal solution to problem ${\cal
P}_{5}$ in (25) is given by

$\displaystyle{{\bf{w}}_{k}^{\rm
opt}}={\rho_{k}}{\left({{\rho_{k}}\sum\limits_{j=1}^{K}{{{\bf{h}}_{j}}{\bf{h}}_{j}^{\rm{H}}+\zeta{{\bf{I}}_{3{N_{F}}}}}}\right)^{\\!\\!-1}}{{\bf{h}}_{k}},$
(26)

wherein $N_{F}:=(N_{x}+1)(N_{y}+1)(N_{z}+1)$ is the total number of Fourier
expansion items. Note that $\zeta$ is the Lagrange multiplier, which should be
chosen such that the complementarity slackness condition of power constraint
(25b) is satisfied, i.e.,

${\zeta^{\rm opt}}=\min\left\\{{{\zeta}\geq
0:\sum\limits_{k=1}^{K}{{{\left\|{{{\bf{w}}_{k}}}\right\|}^{2}}}\leq{P_{\rm{T}}}}\right\\}.$
(27)

One-dimensional binary search can be an efficient way to solve (27) and obtain
the optimal multiplier $\zeta^{\rm opt}$.

After calculating the optimal projection lengths ${{\bf{w}}_{k}^{\rm opt}}$,
according to (11), the final solution to the CAP-MIMO patterns can be obtained
by

${\bm{\theta}}_{k}^{\rm opt}({\bf s})=\sum\limits_{\bf{n}}^{\bf
N}{{{\bf{w}}^{\rm
opt}_{k,{\bf{n}}}}{\Psi_{\bf{n}}}\left({\bf{s}}\right)},~{}~{}{\bf s}\in{\cal
S}_{\rm T},$ (28)

which completes the proposed pattern design scheme.

##  IV Simulation Results

###  IV-A Simulation setup

We consider a 3-D scenario for simulation, where one CAP-MIMO transmitter
simultaneously serves $K=8$ receivers. We assume that the CAP-MIMO transmitter
is deployed on the $xy$-plane with its center located at $\left(0,0,0\right)$,
i.e.,

${{\cal
S}_{T}}:=\left\\{{{\left({{s_{x}},{s_{y}},{s_{z}}}\right)}{\Big{|}}\left|{{s_{x}}}\right|\leq\frac{{{L_{x}}}}{2},\left|{{s_{y}}}\right|\leq\frac{{{L_{y}}}}{2},{s_{z}}=0}\right\\},$
(29)

where the CAP-MIMO aperture has a square shape with the area of $A_{\rm
T}=0.25\,{\rm m}^{2}$, i.e., $L_{x}=L_{y}=0.5\,{\rm m}$. All receivers are
located within a square region, where four electromagnetic-wave receivers are
located at $\left(\pm 1\,{\rm m},\pm 1\,{\rm m},30\,{\rm m}\right)$, and the
other four receivers are located at $\left(\pm 5\,{\rm m},\pm 5\,{\rm
m},30\,{\rm m}\right)$, respectively. The frequency is set to $f=2.4$ ${\rm
GHz}$, and the intrinsic impedance is set to $Z_{0}=376.73$ $\Omega$ [7].
Unless specially specified, the maximum transmit power is set to $P_{\rm
T}=10^{-4}$ ${\rm A}^{2}$, and the noise power is set to $\sigma^{2}=5.6\times
10^{-3}$ ${\rm V}^{2}/{\rm m}^{2}$. The Green functions ${\bf G}_{k}\left({\bf
s}\right)$ are generated by the free-space model (3). The sampling number of
the integral operation $\int_{{{\cal S}_{\rm T}}}{\rm d}{\bf s}$ is set to
$I_{s}=1024$. The numbers of the reserved Fourier expansion items are set to
$N_{x}=5$, $N_{y}=5$, and $N_{z}=0$, respectively, i.e., $N_{F}=36$. All
pattern functions ${\bm{\theta}}_{k}({\bf s})$ and receiver combiners
${\bm{\psi}}_{k}$ are randomly initialized.

As for the benchmark scheme, we consider the existing wavenumber-division
multiplexing recently proposed in [3] for comparison, of which the key idea is
to directly adopt multiple orthogonal Fourier basis functions to generate the
patterns for different receivers respectively, as discussed in Section I. To
compare the interference elimination ability of different schemes, we consider
the interference-free sum-rate as the upper bound for comparison, which can be
realized by assuming that all inter-receiver interference can be ideally
eliminated and then employing the proposed pattern design.

###  IV-B Sum-rate against aperture area $A_{\rm T}$

Figure 2: Sum-rate against the aperture area $A_{\rm T}$ of CAP-MIMO.

We plot the sum-rate against the aperture area $A_{\rm T}$ in Fig. 2, where
the aperture shape of CAP-MIMO always remains a square, i.e., $L_{x}=L_{y}$.
From this figure, we can observe that, for all schemes, the sum-rate quickly
improves as the aperture area $A_{\rm T}$ increases. However, compared with
the existing wavenumber-division multiplexing, the proposed pattern-division
multiplexing achieves a higher sum-rate. For example, when $A_{\rm T}=1\,{\rm
m}^{2}$, the sum-rate achieved by the proposed scheme is 16.46 bps/Hz, which
is about 260% higher than 4.53 bps/Hz achieved by the existing wavenumber-
division multiplexing. The reason is that, the wavenumber-division
multiplexing directly uses $K$ different orthogonal Fourier bases to generate
$K$ different patterns. Despite this scheme can make the symbols orthogonal at
the transmitter, their orthogonality cannot be guaranteed at the receivers.
After passing through the electromagnetic channels, the orthogonality among
different symbols may be destroyed, which will result in high inter-receiver
interference or even make the radiated electromagnetic wave unable to reach
the receivers accurately. By contrast, the proposed pattern-division
multiplexing jointly designs the patterns for $K$ receivers according to the
specific channel state information. During the process of pattern design, the
proposed scheme actually makes a trade-off between the amplification of
desired signals and the elimination of inter-receiver interference, and it can
also achieve a power allocation among different users. In this way, the
electromagnetic waves carrying symbols can be focused on the receivers
accurately with stronger orthogonality, and thus higher sum-rate can be
achieved.

###  IV-C Patterns ${\bm{\theta}}({\bf s})$ of CAP-MIMO

(a) $k=1$ (b) $k=2$

(c) $k=3$ (d) $k=4$

Figure 3:  The normalized amplitude of the x-component of the optimized
pattern ${\bm{\theta}}_{k}({\bf s})$.

To show the pattern functions optimized by the proposed pattern-division
multiplexing, we present the normalized amplitude of the x-component of the
optimized pattern functions ${\bm{\theta}}_{k}({\bf s})$ for the former four
receivers 1-4 in Fig. 3, and their phase in Fig. 4, respectively. From these
two figures, it is interesting to observe that, after designing the patterns
(i.e., the current distributions) via the proposed pattern-division
multiplexing scheme, the patterns ${\bm{\theta}}_{k}({\bf s})$ for different
receivers are nearly orthogonal. In particular, as shown in Fig. 3, the power
of the patterns that carry different symbols are distributed in non-
overlapping regions, in this way, the inter-receiver interference at each
electromagnetic-wave receiver can be well eliminated. Besides, from Fig. 4 one
can notice that, the phase of the patterns carrying different symbols are
symmetrically distributed, which means that the electromagnetic-waves for the
four receivers are radiated towards four different spatial directions and
focused on the four receivers respectively. This interesting phenomenon is
similar to the design result of MIMO beamforming, which aims to generate
orthogonal beams towards multiple receivers. It also intuitively shows the
reason why the proposed pattern design scheme can improve the sum-rate. We can
conclude that, both of these two figures have provided intuitive explanations
for the sum-rate improvement, which have demonstrated the effectiveness of the
proposed pattern-division multiplexing technique111More simulation results and
insights can be found in the journal version of this paper [9]..

(a) $k=1$ (b) $k=2$

(c) $k=3$ (d) $k=4$

Figure 4:  The phase of the x-component of the optimized pattern
${\bm{\theta}}_{k}({\bf s})$.

##  V Conclusions

In this paper, we proposed the pattern-division multiplexing to fill in the
gap of a flexible pattern design scheme for CAP-MIMO. Specifically, we first
derived the system model of CAP-MIMO, which allowed us to formulate the sum-
rate maximization problem. Then, we proposed a general pattern-division
multiplexing technique to flexibly design the patterns of CAP-MIMO. By
applying series expansion to project the continuous functions of the patterns
onto an orthogonal basis space, the design of continuous pattern functions
were transformed to the design of their projection lengths on finite
orthogonal bases. Based on this technique, we proposed a pattern design scheme
to solve the formulated sum-rate maximization problem. Simulation results
showed that, the patterns designed by the proposed scheme for multiple
receivers were almost mutually orthogonal. Particularly, the sum-rate achieved
by the proposed scheme was about 260% higher than that achieved by the
existing wavenumber-division multiplexing scheme, which demonstrated the
effectiveness of the proposed scheme.

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  *[MIMO]: multiple-input multiple-output
  *[CAP-MIMO]: continuous-aperture MIMO
  *[LIS]: large intelligent surface
  *[AWGN]: additive white Gaussian noise
  *[QCQP]: quadratically constrained quadratic program