# An Information-Theoretic Characterization of MIMO-FAS: Optimization,
Diversity-Multiplexing Tradeoff and $q$-Outage Capacity††thanks: The work of
W. K. New, K. K. Wong and K. F. Tong is supported by the Engineering and
Physical Sciences Research Council (EPSRC) under grant EP/W026813/1.††thanks:
The work of C.-B. Chae is supported by the Institute for Information and
Communication Technology Promotion (IITP) grant funded by the Ministry of
Science and ICT (MSIT), Korea (No. 2021-0-02208, No. 2021-0-00486).††thanks:
The work of H. Xu is supported by the European Union’s Horizon 2020 Research
and Innovation Programme under Marie Sklodowska-Curie Grant No.
101024636.††thanks: W. K. New (email: ${\rm a.new@ucl.ac.uk}$), K. K. Wong
(corresponding author, email: ${\rm kai\text{-}kit.wong@ucl.ac.uk}$), H. Xu
(email: ${\rm hao.xu@ucl.ac.uk}$), and K. F. Tong (email: ${\rm
<EMAIL_ADDRESS>are with the Department of Electronic and Electrical
Engineering, University College London, London, WC1E 6BT, United Kingdom.
C.-B. Chae (email: ${\rm<EMAIL_ADDRESS>is with the School of
Integrated Technology, Yonsei University, Seoul 03722 Korea. K. K. Wong is
also affiliated with Yonsei Frontier Lab., Yonsei University, Seoul 03722,
Korea.

Wee Kiat New, Member, IEEE, Kai-Kit Wong, Fellow, IEEE, Hao Xu, Member, IEEE,
Kin-Fai Tong, Fellow, IEEE, and Chan-Byoung Chae, Fellow, IEEE

###### Abstract

Multiple-input multiple-output (MIMO) system has been the defining mobile
communications technology in recent generations. With the ever-increasing
demands looming towards the sixth generation (6G), we are in need of
additional degrees of freedom that deliver further gains beyond MIMO. To this
goal, fluid antenna system (FAS) has emerged as a new way to obtain spatial
diversity using reconfigurable position-switchable antennas. Considering the
case with more than one ports activated on a 2D fluid antenna surface at both
ends, we take the information-theoretic approach to study the achievable
performance limits of the MIMO-FAS. First of all, we propose a suboptimal
scheme, referred to as QR MIMO-FAS, to maximize the rate at high signal-to-
noise ratio (SNR) via joint port selection, transmit and receive beamforming
and power allocation. We then derive the optimal diversity and multiplexing
tradeoff (DMT) of MIMO-FAS. From the DMT, we highlight that MIMO-FAS
outperforms traditional MIMO antenna systems. Further, we introduce a new
metric, namely $q$-outage capacity, which can jointly consider rate and outage
probability. Through this metric, our results indicate that MIMO-FAS surpasses
traditional MIMO greatly.

###### Index Terms:

6G, Diversity and multiplexing tradeoff, Fluid antenna system, MIMO, Outage
capacity.

##  I Introduction

###  I-A Background

Sixth-generation (6G) mobile communication seeks to push the key performance
indicators (KPIs) way beyond what the current fifth generation (5G) promises
to offer. Such upgrade will require new technologies that can achieve more
from the same amount of bandwidth. Presently, the dominating technology has
been multiple-input multiple-output (MIMO), which also comes in the form of
multiuser MIMO and massive MIMO. In 6G, the desire is to exceed MIMO [1, 2].

To achieve this very ambitious goal, one emerging idea is fluid antenna system
(FAS) [3]. FAS represents any software-controllable fluidic, conductive or
dielectric structure that can adjust its shape and position to reconfigure the
gain, radiation pattern, operating frequency and other radiation
characteristics. This is now feasible, thanks to the recent advances in
utilizing flexible conductive materials such as liquid metals or ionized
solutions [4], switchable pixels [5, 6], and stepper motors for antennas [7,
8]. The concept of fluid antenna includes all forms of movable and non-movable
flexible-position antennas.

Unlike traditional antenna that is placed at a fixed location, fluid antenna
is able to switch its location almost instantly in a limited space. The most
basic single fluid antenna consists of one radio frequency (RF)-chain and $N$
preset locations (also known as ports) that are distributed in a given space
[3]. The radiating element of the fluid antenna can switch its position to
obtain a higher rate, lower outage probability, less interference and other
desirable performance gains depending on the applications. As the ports can be
placed closely to each other, the channels of these ports are strongly
correlated and thus spatial correlation plays a crucial role in FAS.

The main implementation designs for FAS are: i) liquid-based fluid antenna and
ii) RF pixel-based fluid antenna. In the liquid-based fluid antenna, each
liquid droplet can precisely switch its position by controlling the electric
field using the thin conductive lines on top of the dielectric layer while
other technology may use an electronically controlled pump to shift the
position of a fluid radiating element in a tube. On the other hand, in the RF
pixel-based fluid antenna, the RF pixels can be turned on-and-off instantly
regardless of the surface area. One or several pixels when on, form an antenna
port for transmission or reception like a standard antenna. Besides, each
activated port is connected to an RF-chain, operating like a conventional
antenna. In short, the basic principle of FAS is to exploit the dynamic nature
of fluid antenna to achieve ultimate flexibility for diversity and
multiplexing gains.111It is worth pointing out that FAS does not necessarily
use ‘fluid’ materials for antenna and in wireless communications that requires
adaptation in time of milliseconds or less, reconfigurable pixels are more
relevant.

Due to its unprecedented benefits, single-user, single-input single-output
(SISO)-FAS has recently been investigated under different scenarios and
assumptions. Specifically, as the number of ports increases, [9] showed that
the outage probability of FAS could be reduced drastically while [10]
demonstrated that FAS could significantly improve the ergodic capacity.
Motivated by these works, [11] derived the level crossing rate, [12] devised a
port selection algorithm by observing the channels of a few ports, [13]
investigated the performance of FAS over general correlated channels and [14]
analyzed the outage probability of FAS for Terahertz communications while
selection combining and maximum gain combining were further considered.
Optimistic results were obtained in these works but it was illustrated in [15]
that the outage probability of FAS could only be reduced to a floor when a
more accurate spatial correlation model was adopted. The recent work [16]
explained such limitations at an intuitive level and revealed that the
performance of FAS was generally determined by the available space.
Furthermore, only in certain cases might FAS achieve a similar outage
probability as compared to the classical maximal ratio combining (MRC) system.

The unique ability of switching the antenna position finely in FAS can also be
exploited to mitigate interference, which would be impractical in traditional
antenna selection systems. Recently, [17] investigated orthogonal multiple
access to serve multiple users with fluid antennas while [18] used a space
division multiple access approach to minimize the user transmit power.
Nevertheless, an arguably more interesting idea is the fluid antenna multiple
access (FAMA) scheme [19] where the rationale is to exploit the moment of deep
fades in the spatial domain to alleviate inter-user interference. FAMA is
classified into slow FAMA and fast FAMA in which the former switches its port
when the channel changes [20] and the latter switches its port on a symbol-by-
symbol basis [21]. Most recently, the outage probability for two-user FAMA was
revisited in [22].

In summary, FAS has shown promises but much is still not well understood. For
example, the performance of FAS itself can be lifted if more than one ports
are activated. For a point-to-point communication channel, we refer to the
system where both ends are equipped with a multi-port FAS, as MIMO-FAS which
is also known as fluid MIMO or flexible MIMO in [3]. Note that multiple ports
can also be activated in a two dimensional (2D) surface using liquid-based
fluid antenna or RF pixel-based fluid antenna. The schematics of the MIMO-FAS
designs were discussed in [3]. Compared to a traditional MIMO antenna
selection system in which the number of antennas is limited in a given surface
(at least half wavelength separation between the antennas) and the antennas
are fixed in positions, MIMO-FAS is distinct in the sense that the positions
of the radiating elements can be dynamically and finely adjusted and that the
number of preset locations (i.e., ports) within a given surface can be
arbitrarily large, which yields additional gains. Note that FAS has also been
proposed for multiple access recently [20, 21, 22], where the fine resolution
of FAS is absolutely essential and conventional antenna selection would be
unable to cope.

It is anticipated that the capacity and reliability of MIMO-FAS will be
improved over the SISO counterpart. In fact, a related work showed that the
capacity of a movable antenna system (which can be interpreted as MIMO-FAS
with movable fluid antennas) could be improved up to $30.3\%$ as compared to
traditional MIMO systems [23]. In the study, however, spatial correlation due
to rich scattering between the antenna positions was not considered. More
importantly, the optimal diversity and multiplexing tradeoff (DMT) of MIMO-FAS
is unknown.

In information theory, the optimal DMT can be employed as a unified framework
to compare the performance of different multiple-antenna systems [24]. More
concretely, it focuses on the asymptotic high signal-to-noise ratio (SNR)
regime and a scheme is then said to achieve a multiplexing gain of $r$ and a
diversity gain of $d\left(r\right)$ if the rate of the system scales like
$r\log{\rm SNR}$ and its outage probability decays like ${\rm
SNR}^{-d\left(r\right)}$. It is known that $r$ cannot exceed the total degrees
of freedom of the channel and $d\left(0\right)$ is limited by the maximal
diversity gain, i.e., total number of independent channels. In between the two
extremes, a system must tradeoff each type of gains.

###  I-B Contributions

Motivated by the above, this paper analyzes the performance of MIMO-FAS with
the goal of gaining useful insights for designing an efficient MIMO-FAS. To
this end, we first develop a system model of MIMO-FAS while taking into
account of the spatial correlation effect. To characterize the performance
limits of MIMO-FAS, we consider a rich scattering environment since it is well
known that multipath can help to improve the diversity and multiplexing
gains.222Note that rich scattering can help to improve the performance of any
MIMO systems. This includes MIMO-FAS, traditional MIMO and MIMO antenna
selection. Therefore, if the number of scatterers is small, one may further
consider using a reconfigurable intelligent surface to create artificial
scatterers to improve the performance of any MIMO systems. We then propose a
suboptimal scheme that maximizes the rate of MIMO-FAS through joint port
selection, transmit and receive beamforming and power allocation at high SNR.
Based on this scheme, we derive the optimal DMT of MIMO-FAS to reveal the
fundamental limits of MIMO-FAS from an information-theoretic viewpoint. From
the analytical results, we further study the effects of different MIMO-FAS
parameters and reveal the superiority of MIMO-FAS over traditional MIMO and
MIMO antenna selection in terms of DMT. In addition, we introduce a new
metric, referred to as $q$-outage capacity, to showcase the benefits of MIMO-
FAS. Our main contributions are summarized as follows:

  * •

We develop a system model for MIMO-FAS with a 2D fluid antenna surface at both
ends while taking into account of the spatial correlation of the ports. In
particular, we employ a simple yet accurate channel model that considers the
spatial correlation in a three-dimensional (3D) scattering environment. Based
on this channel model, we introduce several system parameters such as active
ports, beamforming matrices and power allocation. The achievable rate of MIMO-
FAS is then derived where its expression resembles the rate of a traditional
MIMO system.

  * •

Also, we formulate a non-convex optimization problem to maximize the rate of
MIMO-FAS via joint port selection, transmit and receive beamforming and power
allocation. We show that the global optimal solution can be obtained using an
exhaustive search, singular value decomposition (SVD) and waterfilling power
allocation, at the expense of a non-polynomial time complexity. To reduce the
time complexity, we propose the QR MIMO-FAS scheme that maximizes the rate of
MIMO-FAS at high SNR via suboptimal port selection, beamforming and power
allocation. It is shown that QR MIMO-FAS has a polynomial time complexity.

  * •

Furthermore, we derive the outer bound of the DMT of MIMO-FAS. By using the
outer bound and QR MIMO-FAS, we obtain the optimal DMT of MIMO-FAS. In this
process, we prove that the spatial correlation matrix $\boldsymbol{J}_{s}$ can
be represented by a finite-size matrix $\boldsymbol{J}_{{\rm red}}^{s}$ even
if the number of ports increases to infinity. Afterwards, we propose methods
to estimate the size of $\boldsymbol{J}_{{\rm red}}^{s}$ and linearly
transform between $\boldsymbol{J}_{s}$ and $\boldsymbol{J}_{{\rm red}}^{s}$
with proof of certificates.

  * •

Extensive results are provided to highlight the effects of several MIMO-FAS
parameters. In the discussions, we provide useful insights for designing an
efficient MIMO-FAS. Although MIMO-FAS provides rate improvements over the
traditional MIMO antenna systems, we highlight that the superiority of MIMO-
FAS actually lies in the diversity gain. Specifically, the diversity gain of
MIMO-FAS for a fixed $r$ is much greater than that of MIMO and MIMO antenna
selection if the total number of active ports or antennas is the same.

  * •

Finally, we introduce a new performance metric, referred to as $q$-outage
capacity, that jointly considers both rate and outage probability. We show
that MIMO-FAS outperforms the traditional MIMO and MIMO antenna selection in
terms of $q$-outage capacity. This result suggests that MIMO-FAS is more
reliable in delivering high data rate transmission than the traditional MIMO
systems.

###  I-C Organization and Notations

The remainder of this paper is organized as follows. Section II introduces the
system model of MIMO-FAS. Section III details the proposed QR MIMO-FAS scheme
that maximizes its rate at high SNR. The optimal DMT of MIMO-FAS is analyzed
in Section IV. Section V presents the numerical results to compare MIMO-FAS
with the traditional MIMO systems and we conclude the paper in Section VI.

Table I: The meanings of key notations. Notation |  Meaning
---|---
$\boldsymbol{A}$ |  Activation port matrices at both of the transmitter and receiver sides
$\boldsymbol{A}_{s}$ |  Activation port matrix at side $s$
$C_{{\rm sys}}^{q}$ |  $q$-outage capacity of a system
$d\left(r\right)$ |  Diversity gain for $r$ multiplexing gain
$\boldsymbol{G}$ |  Circularly symmetric complex Gaussian matrix with i.i.d. entries
$\boldsymbol{H}$ |  Complex channels of MIMO-FAS
$\overset{}{\bar{\boldsymbol{H}}}$ |  Complex channels of the activated ports
$\boldsymbol{H}_{a}$ |  Partial channels of the active ports
$\boldsymbol{H}_{i}$ |  Partial channels of the inactive ports
$\boldsymbol{J}_{s}$ |  Spatial correlation matrix at side $s$
$\hat{\boldsymbol{J}}_{s}$ |  Approximated matrix of $\boldsymbol{J}_{s}$
$\boldsymbol{J}_{{\rm red}}^{s}$ |  Full rank spatial correlation matrix at side $s$
$\boldsymbol{K}$ |  Input covariance
$n_{s}$ |  Total number of active ports
$n_{{\rm{max}}}/n_{{\rm{min}}}$ |  Maximum/minimum of $n_{rx}$ and $n_{tx}$
$N_{{\rm{max}}}/N_{{\rm{min}}}$ |  Maximum/minimum of $N_{rx}$ and $N_{tx}$
$N_{s}$ |  Total number of ports at side $s$
$N_{i}^{s}$ |  Number of ports in the $i$-th dimension at side $s$
$N_{s}^{{}^{\prime}}$ |  Rank of $\boldsymbol{J}_{{\rm red}}^{s}$
$N_{\min}^{{}^{\prime}}$ |  Minimum of $N_{rx}^{{}^{\prime}}$ and $N_{tx}^{{}^{\prime}}$
$P_{{\rm sys}}^{{\rm out}}\left({\rm SNR},r\right)$ |  Outage probability of a system in terms of $r\log{\rm SNR}$
$\bar{P}_{{\rm sys}}^{{\rm out}}\left({\rm SNR},q\right)$ |  Outage probability of a system for a fixed $q$-transmission rate
$\boldsymbol{P}$ |  Power allocation matrix
$r$ |  Multiplexing gain
$R_{{\rm sys}}\left({\rm SNR}\right)$ |  Rate of a system for a given ${\rm SNR}$
$s\in\left\\{tx,rx\right\\}$ |  Subscript/superscript to denote the transmit/receiver side
${\rm SNR}$ |  Transmit SNR
$\boldsymbol{v}_{l}^{s}$ |  Certificate between the linear transformation of $\boldsymbol{J}_{s}^{\left(l\right)}$ and $\boldsymbol{J}_{{\rm sub}}^{\left(l-1\right)}$
$W_{s}$ |  Total area of the fluid antenna at side $s$
$W_{i}^{s}$ |  Length of the fluid antenna in the $i$-th dimension at side $s$
$\boldsymbol{W}$ |  Transmit and receive beamforming matrices
$\boldsymbol{W}_{s}$ |  Beamforming matrix at $s$ side
$\lambda$ |  Wavelength of the carrier frequency
$\boldsymbol{\Lambda}_{s}$ |  Matrix whose diagonal entries are the eigenvalues of $\boldsymbol{J}_{s}$
$\boldsymbol{\Sigma}$ |  Matrix whose diagonal entries are singular values of $\boldsymbol{H}$
$\overset{}{\bar{\boldsymbol{\Sigma}}}$ |  Matrix whose diagonal entries are singular values of $\boldsymbol{\bar{H}}$
$\boldsymbol{\Omega}$ |  Special matrix used for port optimization

Throughout this paper, scalar variables are denoted by italic letters (e.g.,
$c$), vectors are denoted by boldface italic small letters (e.g.,
$\boldsymbol{c}$) and matrices are denoted by boldface italic capital letters
(e.g., $\boldsymbol{C}$). Additionally, $\left(\cdot\right)^{T}$ denotes
transpose, $\left(\cdot\right)^{H}$ denotes conjugate transpose while ${\rm
det}\left(\cdot\right)$, ${\rm rank\left(\cdot\right)}$ and ${\rm
trace}\left(\cdot\right)$ represent the determinant, rank and trace of a
matrix, respectively. Moreover, $\left|\cdot\right|$,
$\left\|\cdot\right\|_{2}$ and $\left\|\cdot\right\|_{F}$ denote the absolute,
Euclidean norm and Frobenius norm operations, respectively. Furthermore, we
use $\log(\cdot)$ to denote logarithm with base 2,
$\left[\cdot\right]_{c}^{+}$ outputs the argument that is lower bounded by
$c$, $\min\left\\{\cdot\right\\}$ and $\max\left\\{\cdot\right\\}$ denote the
minimum and maximum value of the argument, respectively. ${\mathbb{E}}[\cdot]$
returns the expected value of the input random quantity, and $\otimes$ denotes
the Kronecker product. Finally, $\boldsymbol{e}_{k}$ represents an all-zero
vector except the $k$-th entry being unity, ${\rm diag}(\cdot)$ denotes a
diagonal matrix whose diagonal entries are the inputs, and
$(\cdot)^{{\dagger}}$ denotes the pseudoinverse of an input matrix. To help
readers follow the mathematical contents, the meanings of the key variables
are listed in Table I.

##  II System Model

As illustrated in Fig. 1, we consider a point-to-point wireless communication
channel in which the transmitter and receiver are equipped with a fluid
antenna. To facilitate our discussions, we use the subscript/superscript $s$
to denote the parameters at the transmitter or receiver as $tx$ or $rx$,
respectively, i.e., $s\in\left\\{tx,rx\right\\}$. We assume that the fluid
antenna takes up a 2D space with an area of $W_{s}$ and has $N_{s}$ ports
spread uniformly over the 2D space. A grid structure is considered where
$N_{i}^{s}$ ports are uniformly distributed along a linear space of length
$\lambda W_{i}^{s}$ for $i\in\left\\{1,2\right\\}$, so that
$N_{s}=N_{1}^{s}\times N_{2}^{s}$ and $W_{s}=\lambda W_{1}^{s}\times\lambda
W_{2}^{s}$, where $\lambda$ is the wavelength of the carrier frequency. In
this MIMO-FAS, the transmitter and receiver can only activate $n_{s}$ out of
$N_{s}$ ports. Note that for SISO-FAS, $n_{s}=1,\forall s$.

Figure 1: A schematic of point-to-point MIMO-FAS.

Considering a 3D environment under rich scattering, the spatial correlation
between the $\left(n_{1}^{s},n_{2}^{s}\right)$-th port and the
$\left(\tilde{n}_{1}^{s},\tilde{n}_{2}^{s}\right)$-th port is given by

$\displaystyle
J_{\left(n_{1}^{s},n_{2}^{s}\right),\left(\tilde{n}_{1}^{s},\tilde{n}_{2}^{s}\right)}^{s}$
$\displaystyle=j_{0}\left(2\pi\sqrt{\left({\textstyle\frac{\left|n_{1}^{s}-\tilde{n}_{1}^{s}\right|}{N_{1}^{s}-1}W_{1}^{s}}\right)^{2}+\left({\textstyle\frac{\left|n_{2}^{s}-\tilde{n}_{2}^{s}\right|}{N_{2}^{s}-1}W_{2}^{s}}\right)^{2}}\right),$
(1)

where $j_{0}\left(\cdot\right)$ is the spherical Bessel function of the first
kind.333Note that (1) can be reduced to a 1D fluid antenna under 2D scattering
environments by setting $N_{1}^{s}=1$ and $\frac{0}{0}\triangleq 0$ and
replacing $j_{0}\left(\cdot\right)$ by $J_{0}\left(\cdot\right)$ where
$J_{0}\left(\cdot\right)$ is the Bessel function of the first kind. A detailed
proof can be found in Appendix I. As seen in (1), it is cumbersome to label a
port in 2D. To simplify our notations, we use the function ${\rm
map}:\mathbb{R}^{2}\rightarrow\mathbb{R}$, e.g., ${\rm
map}\left(n_{1}^{s},n_{2}^{s}\right)=l_{s}$, where
$l_{s}\in\left\\{1,\dots,N_{s}\right\\}$.

Using the mapping function, we can express the spatial correlation matrix
$\boldsymbol{J}_{s}$ as

$\boldsymbol{J}_{s}=\left[\begin{array}[]{cccc}J_{1,1}^{s}&J_{1,2}^{s}&\ldots&J_{1,N_{s}}^{s}\\\
J_{2,1}^{s}&J_{2,2}^{s}&\ldots&J_{2,N_{s}}^{s}\\\
\vdots&\vdots&\ddots&\vdots\\\
J_{N_{s},1}^{s}&J_{N_{s},2}^{s}&\ldots&J_{N_{s},N_{s}}^{s}\end{array}\right],$
(2)

where $J_{k_{s},l_{s}}^{s}$ is the spatial correlation of the $k_{s}$-th and
the $l_{s}$-th port at side $s$, and $k_{s}$ and $l_{s}$ are the labels after
mapping. Since $J_{k_{s},l_{s}}^{s}=J_{l_{s},k_{s}}^{s}$, (2) can be
decomposed into
$\boldsymbol{J}_{s}=\boldsymbol{U}_{s}\boldsymbol{\Lambda}_{s}\boldsymbol{U}_{s}^{H}$
where $\boldsymbol{U}_{s}$ is an $N_{s}\times N_{s}$ matrix whose $l$-th
column (i.e., $\boldsymbol{u}_{l}^{s}$) is the eigenvector of
$\boldsymbol{J}_{s}$ and $\boldsymbol{\Lambda}_{s}={\rm
diag}\left(\lambda_{1}^{s},\ldots,\lambda_{N_{s}}^{s}\right)$ is an
$N_{s}\times N_{s}$ diagonal matrix whose $l$-th diagonal entries are the
corresponding eigenvalues of $\boldsymbol{u}_{l}^{s}$. Without loss of
generality, we assume that the values of the eigenvalues in
$\boldsymbol{\Lambda}_{s}$ are in descending order, i.e.,
$\lambda_{1}^{s}\geq\cdots\geq\lambda_{N_{s}}^{s}$. Note that
$\boldsymbol{U}_{s}\boldsymbol{\Lambda}_{s}\boldsymbol{U}_{s}^{H}$ is computed
independently for each $s\in\left\\{tx,rx\right\\}$.

Given $\boldsymbol{U}_{s}$ and $\boldsymbol{\Lambda}_{s}$ for $\forall s$, the
complex channel of MIMO-FAS can be modelled as

$\boldsymbol{H}=\delta\>\boldsymbol{U}_{rx}\sqrt{\boldsymbol{\Lambda}_{rx}}\boldsymbol{G}\sqrt{\boldsymbol{\Lambda}_{tx}^{H}}\boldsymbol{U}_{tx}^{H},$
(3)

where
$\boldsymbol{G}=\left[\boldsymbol{g}_{1},\ldots,\boldsymbol{g}_{N_{tx}}\right]$,
$\boldsymbol{g}_{l}=\left[g_{1,l},\ldots,g_{N_{rx},l}\right]^{T}$,
$g_{k,l}=x_{k,l}+jy_{k,l}$, and $x_{k,l},y_{k,l}$ are independent Gaussian
random variables with zero mean and variance of $\frac{1}{2}$, $\forall k,l$,
and $\delta^{2}$ is the path loss. Letting $\boldsymbol{h}_{{\rm vec}}={\rm
vec}\left(\boldsymbol{H}\right)$, the covariance of $\boldsymbol{h}_{{\rm
vec}}$ is
$\delta^{2}\left(\boldsymbol{J}_{tx}^{T}\otimes\boldsymbol{J}_{rx}\right)$,
where $\boldsymbol{J}_{tx}^{T}\otimes\boldsymbol{J}_{rx}$ is the spatial
correlation matrix between the transmit and receive ports.444Due to the port
spatial correlation, it can be shown that only a small number of observed
ports/training is required to obtain the full channel state information
regardless of the number of ports of the FAS [25]. Machine learning techniques
have also been proposed to address the channel estimation problem for port
selection in FAS, e.g., [20].

Next, we denote the activation port matrices at the transmitter and receiver,
respectively, as
$\boldsymbol{A}_{tx}=\left[\boldsymbol{a}_{1}^{tx},\ldots,\boldsymbol{a}_{n_{tx}}^{tx}\right]$
and
$\boldsymbol{A}_{rx}=\left[\boldsymbol{a}_{1}^{rx},\ldots,\boldsymbol{a}_{n_{rx}}^{rx}\right]^{T}$,
where $\boldsymbol{a}_{l}^{tx}$ and $\boldsymbol{a}_{l}^{rx}$ are standard
basis vector (i.e.,
$\boldsymbol{a}_{l}^{s}\in\left\\{\boldsymbol{e}_{1},\ldots,\boldsymbol{e}_{N_{s}}\right\\}$).
Since only distinct $n_{s}$ ports can be activated at a time, we have
$\boldsymbol{a}_{k}^{tx}\neq\boldsymbol{a}_{l}^{tx}$ and
$\boldsymbol{a}_{k}^{rx}\neq\boldsymbol{a}_{l}^{rx}$ if $k\neq l$. Let us
further denote $\boldsymbol{W}_{tx}$ and $\boldsymbol{W}_{rx}$ as the transmit
and receive beamforming matrices, with the constraint
$\left\|\boldsymbol{W}_{s}\right\|_{2}=1$. Then the receive signals of MIMO-
FAS can be rewritten as

$\displaystyle\boldsymbol{W}_{rx}\boldsymbol{A}_{rx}\boldsymbol{Y}$
$\displaystyle=\boldsymbol{W}_{rx}\boldsymbol{A}_{rx}\boldsymbol{H}\boldsymbol{A}_{tx}\boldsymbol{W}_{tx}\boldsymbol{x}+\boldsymbol{W}_{rx}\boldsymbol{A}_{rx}\boldsymbol{w}$
(4) $\displaystyle\Rightarrow\tilde{\boldsymbol{Y}}$
$\displaystyle=\tilde{\boldsymbol{H}}\boldsymbol{x}+\tilde{\boldsymbol{w}},$
(5)

where $\boldsymbol{x}$ is the information signal and $\boldsymbol{w}$ is the
additive white Gaussian noise with zero mean and identity covariance. In (5),
we defined
$\tilde{\boldsymbol{Y}}\triangleq\boldsymbol{W}_{rx}\boldsymbol{A}_{rx}\boldsymbol{Y}$,
$\tilde{\boldsymbol{H}}\triangleq\boldsymbol{W}_{rx}\boldsymbol{A}_{rx}\boldsymbol{H}\boldsymbol{A}_{tx}\boldsymbol{W}_{tx}$
and
$\tilde{\boldsymbol{w}}\triangleq\boldsymbol{W}_{rx}\boldsymbol{A}_{rx}\boldsymbol{w}$.
For ease of expositions, we denote
$\boldsymbol{A}=\left[\boldsymbol{A}_{tx},\boldsymbol{A}_{rx}\right]$,
$\boldsymbol{W}=\left[\boldsymbol{W}_{tx},\boldsymbol{W}_{rx}\right]$,
$\bar{\boldsymbol{H}}=\boldsymbol{A}_{rx}\boldsymbol{H}\boldsymbol{A}_{tx}$
and $\boldsymbol{K}=\boldsymbol{W}_{tx}\boldsymbol{P}\boldsymbol{W}_{tx}^{H}$
where $\boldsymbol{K}$ is the input covariance and
$\boldsymbol{P}=\mathbb{E}\left[\boldsymbol{x}\boldsymbol{x}^{H}\right]$ is
the power allocation matrix. Then, the rate of MIMO-FAS is given by

$R\left(\boldsymbol{A},\boldsymbol{W},\boldsymbol{P}\right)=\log{\rm
det}\left(\boldsymbol{I}+\bar{\boldsymbol{H}}\boldsymbol{K}\bar{\boldsymbol{H}}^{H}\right),$
(6)

where ${\rm trace}\left(\boldsymbol{K}\right)\leq{\rm SNR}$ and ${\rm SNR}$ is
the transmit SNR.

##  III QR MIMO-FAS: Suboptimal Port Selection, Beamforming and Power
Allocation

In this section, we aim to maximize the rate of the MIMO-FAS via joint optimal
port selection, beamforming and power allocation. The optimization problem is
formulated as

$\displaystyle\max_{\boldsymbol{A},\boldsymbol{W},\boldsymbol{P}}\leavevmode\nobreak\
\leavevmode\nobreak\ $ $\displaystyle
R\left(\boldsymbol{A},\boldsymbol{W},\boldsymbol{P}\right)$ (7a)
$\displaystyle{\rm s.t.}\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\boldsymbol{a}_{l}^{s}\in\left\\{e_{1},\ldots,e_{N_{s}}\right\\},s\in\left\\{rx,tx\right\\},\forall
l,$ (7b)
$\displaystyle\boldsymbol{a}_{k}^{tx}\neq\boldsymbol{a}_{l}^{tx},\text{\,}\text{{if}}\,k\neq
l,$ (7c)
$\displaystyle\boldsymbol{a}_{k}^{rx}\neq\boldsymbol{a}_{l}^{rx},\,\text{{if}}\,k\neq
l,$ (7d)
$\displaystyle\left\|\boldsymbol{W}_{tx}\right\|_{2}=\left\|\boldsymbol{W}_{rx}\right\|_{2}=1,$
(7e) $\displaystyle{\rm trace}\left(\boldsymbol{K}\right)\leq{\rm SNR}.$ (7f)

Note that (7) is a non-convex optimization problem because i) the optimization
variables are mutually coupled and ii) its domain is non-convex. A systematic
way to solve this problem is to employ an exhaustive search [26], SVD and
waterfilling power allocation [27]. In particular, the maximum rate of MIMO-
FAS can be computed via SVD and waterfilling power allocation over
${\scriptstyle\left(\begin{array}[]{c}N_{tx}\\\
n_{tx}\end{array}\right)}\times{\scriptstyle\left(\begin{array}[]{c}N_{rx}\\\
n_{rx}\end{array}\right)}$ port combinations.555As will be shown later in this
paper, $N_{s},\forall s$ can be represented by a finite constant even in cases
where $N_{s}\rightarrow\infty$. Nevertheless, such a method requires a non-
polynomial time complexity of
$\mathcal{O}\left(N_{tx}^{n_{tx}}N_{rx}^{n_{rx}}\right)$ which is
prohibitively high.

To reduce the time complexity, we propose a suboptimal scheme, namely QR MIMO-
FAS, to maximize the rate of MIMO-FAS in the high SNR regime. This scheme is
useful for analyzing the DMT of MIMO-FAS. In the proposed scheme, we decouple
(7) into two subproblems: i) optimal port selection and ii) optimal
beamforming and power allocation. For optimal port selection, (7) can be
simplified as

$\max_{\boldsymbol{A}}R\left(\boldsymbol{A},\boldsymbol{W},\boldsymbol{P}\right)\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm s.t.}\leavevmode\nobreak\ \leavevmode\nobreak\
(\ref{eq:8}),(\ref{eq:9}),(\ref{eq:10}).$ (8)

However, it is still challenging to solve (8) since the objective function
cannot be directly evaluated. To overcome this problem, we exploit the fact
that $R\left(\boldsymbol{A},\boldsymbol{W},\boldsymbol{P}\right)$ strongly
depends on ${\rm det}(\bar{\boldsymbol{H}}\bar{\boldsymbol{H}}^{H})$ in the
high SNR regime [28]. Thus, (8) can be relaxed as

$\max_{\boldsymbol{A}}{\rm
det}(\bar{\boldsymbol{H}}\bar{\boldsymbol{H}}^{H})\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm s.t.}\leavevmode\nobreak\ \leavevmode\nobreak\
(\ref{eq:8}),(\ref{eq:9}),(\ref{eq:10}),$ (9)

which is unfortunately still an NP-hard problem [29].666The relaxation is done
because
$R\left(\boldsymbol{A},\boldsymbol{W},\boldsymbol{P}\right)\approx\log\det(\bar{\boldsymbol{H}}\boldsymbol{K}\bar{\boldsymbol{H}}^{H})$
at high SNR and the rate of using equal power allocation approaches to that of
waterfilling power allocation as SNR increases [37]. For other SNR regimes,
solving (7) with low complexity remains open. Nevertheless, we can obtain an
efficient solution at low SNR by activating $n_{rx}/n_{tx}$ ports where the
row/column-norm of $\boldsymbol{H}$ are the largest and they are separated by
at least $c_{rx}/c_{tx}$ distance. We refer this scheme as the greedy
selection. However, we can obtain a suboptimal solution by using the strong
rank-revealing QR (RRQR) factorization [30].

Specifically, by applying QR factorization with pivoted column on
$\boldsymbol{H}^{H}$, we have

$\boldsymbol{H}^{H}\boldsymbol{\varPi}=\boldsymbol{Q}\boldsymbol{R},$ (10)

where $\boldsymbol{\varPi}$ is a permutation matrix, $\boldsymbol{Q}$ is an
orthogonal matrix and $\boldsymbol{R}$ is an upper triangular matrix where the
absolute of leading entries in $\boldsymbol{R}$ are decreasing in values. The
upper triangular matrix $\boldsymbol{R}$ can be rewritten as

$\boldsymbol{R}=\left[\boldsymbol{R}_{1}\leavevmode\nobreak\
\boldsymbol{R}_{2}\right]=\left[\begin{array}[]{cc}\boldsymbol{S}_{1}&\boldsymbol{S}_{2}\\\
\boldsymbol{0}&\boldsymbol{S}_{3}\end{array}\right],$ (11)

where $\boldsymbol{R_{1}}=\left[\boldsymbol{S}_{1}\leavevmode\nobreak\
\boldsymbol{0}\right]^{T}$ and
$\boldsymbol{R}_{2}=\left[\boldsymbol{S}_{2}\leavevmode\nobreak\
\boldsymbol{S}_{3}\right]^{T}$. Substituting (11) into (10), we obtain

$\left[\boldsymbol{H}^{H}\boldsymbol{\varPi}_{a}\leavevmode\nobreak\
\boldsymbol{H}^{H}\boldsymbol{\varPi}_{i}\right]=\left[\boldsymbol{H}_{a}^{H}\leavevmode\nobreak\
\boldsymbol{H}_{i}^{H}\right]=\left[\boldsymbol{Q}\boldsymbol{R}_{1}\leavevmode\nobreak\
\boldsymbol{Q}\boldsymbol{R}_{2}\right],$ (12)

in which
$\boldsymbol{\varPi}=\left[\boldsymbol{\varPi}_{a}\leavevmode\nobreak\
\boldsymbol{\varPi}_{i}\right]$. In (12), the left hand side and right hand
side of (10) are separated into two blocks and thus we can interpret
$\boldsymbol{H}_{a}^{H}$ and $\boldsymbol{H}_{i}^{H}$ as the MIMO-FAS channels
of active and inactive ports, respectively. Since the singular values of
$\boldsymbol{H}^{H}$ and $\boldsymbol{R}$ remain the same, it is clear that
(12) provides the following properties

$\displaystyle\prod_{m=1}^{N_{\min}}\sigma_{m}\left(\boldsymbol{H}^{H}\right)$
$\displaystyle=\prod_{m=1}^{n_{s}}\sigma_{m}\left(\boldsymbol{H}_{a}^{H}\right)\prod_{m=n_{s}+1}^{N_{\min}}\sigma_{m}\left(\boldsymbol{H}_{i}^{H}\right)$
$\displaystyle=\prod_{m=1}^{n_{s}}\sigma_{m}\left(\boldsymbol{R}_{1}\right)\prod_{m=n_{s}+1}^{N_{\min}}\sigma_{m}\left(\boldsymbol{R}_{2}\right),$
(13)

where $N_{\min}=\min\left\\{N_{rx},N_{tx}\right\\}$ and
$\sigma_{m}\left(\cdot\right)$ denotes the $m$-th singular value of the matrix
argument.

In alignment with (9), our objective here is to maximize
$\prod_{m=1}^{n_{s}}\sigma_{m}\left(\boldsymbol{H}_{a}^{H}\right)$ in (13) by
permuting the $k$-th column of $\boldsymbol{H}_{a}^{H}$ and the $l$-th column
of $\boldsymbol{H}_{i}^{H}$. To facilitate this objective, we employ the
matrix $\boldsymbol{\Omega}$ where the $(k,l)$-th entry of
$\boldsymbol{\Omega}$ is

$\Omega_{k,l}=\sqrt{\left|\boldsymbol{S}{}_{1}^{\dagger}\boldsymbol{S}_{2}\right|_{k,l}^{2}+\left\|\boldsymbol{s}_{3,l}\right\|_{2}^{2}+\left\|\boldsymbol{s}_{1,k}^{\dagger,T}\right\|_{2}^{2}},$
(14)

where $\left|\boldsymbol{S}\right|_{k,l}$ gives the absolute value of the
$(k,l)$-th entry of $\boldsymbol{S}$, $\boldsymbol{s}_{s,l}$ is the $l$-th
column of $\boldsymbol{S}_{s}$ and $\boldsymbol{s}_{s,k}^{\dagger,T}$ is the
$k$-th row of $\boldsymbol{S}_{s}^{\dagger}$. Furthermore, let us denote
$\boldsymbol{H}_{a,k,l}^{H}$ (or $\boldsymbol{\varPi}_{a,k,l}$) is the new
matrix where the $k$-th column of $\boldsymbol{H}_{a}^{H}$ (or
$\boldsymbol{\varPi}_{a}$) and the $l$-th column of $\boldsymbol{H}_{i}^{H}$
(or $\boldsymbol{\varPi}_{i}$) are permuted.

Conventionally, it is necessary to permute all the $\left(k,l\right)$
combinations and find the maximum
$\prod_{m=1}^{n_{s}}\sigma_{m}\left(\boldsymbol{H}_{a,k,l}^{H}\right)$ in the
presence of spatial correlation. Nevertheless, using (14), we can determine
the increase or decrease of
$\prod_{m=1}^{n_{s}}\sigma_{m}\left(\boldsymbol{H}_{a,k,l}^{H}\right)$ over
$\prod_{m=1}^{n_{s}}\sigma_{m}\left(\boldsymbol{H}_{a}^{H}\right)$ before
permuting them. Hence, we can directly permute the $k$-th column of
$\boldsymbol{H}_{a}^{H}$ (and $\boldsymbol{\varPi}_{a}$) and the $l$-th column
of $\boldsymbol{H}_{i}^{H}$ (and $\boldsymbol{\varPi}_{i}$) and then update
$\boldsymbol{H}^{H}=\boldsymbol{H}_{a,k,l}^{H}$ (and
$\boldsymbol{\varPi}=\boldsymbol{\varPi}{}_{a,k,l}$) if $\Omega_{k,l}>1$. In
this paper, we permute the columns based on the largest $\Omega_{k,l}$ which
yields a suboptimal solution. Furthermore, we perform QR factorization to
obtain the expression in (10) with the existing $\boldsymbol{\varPi}$. These
steps can be repeated until $\Omega_{k,l}\leq 1\leavevmode\nobreak\ \forall
k,l$. Using strong RRQR factorization, we can obtain $\boldsymbol{H}_{a}^{H}$
with the maximum
$\prod_{m=1}^{n_{s}}\sigma_{m}\left(\boldsymbol{H}_{a}^{H}\right)$ where
$\boldsymbol{A}_{rx}^{*}=\boldsymbol{\varPi}_{a}^{T}$. Reapplying strong RRQR
factorization on $\boldsymbol{A}_{rx}^{*}\boldsymbol{H}$, we can also obtain
$\boldsymbol{A}_{tx}^{*}$ where
$\boldsymbol{A}_{tx}^{*}=\boldsymbol{\varPi}_{a}$.

Given $\boldsymbol{A}^{*}$, (7) reduces to the optimal beamforming and power
allocation problem which can be formulated as

$\max_{\boldsymbol{W},\boldsymbol{P}}R\left(\boldsymbol{W},\boldsymbol{P}|\boldsymbol{A}^{*}\right)\leavevmode\nobreak\
\leavevmode\nobreak\ {\rm s.t.}\leavevmode\nobreak\ \leavevmode\nobreak\
(\ref{eq:11}),(\ref{eq:12}).$ (15)

Interestingly, (15) can be easily solved via SVD and waterfilling power
allocation [27]. In particular, the solution to (15) is
$\boldsymbol{W}_{r}^{*}=\bar{\boldsymbol{M}}^{H}$ and
$\boldsymbol{W}_{t}^{*}=\bar{\boldsymbol{N}}$ where
$\bar{\boldsymbol{H}}=\bar{\boldsymbol{M}}\bar{\boldsymbol{\Sigma}}\bar{\boldsymbol{N}}^{H}$,
$\bar{\boldsymbol{M}}$ is the left singular matrix of $\bar{\boldsymbol{H}}$,
$\bar{\boldsymbol{N}}$ is the right singular matrix of $\bar{\boldsymbol{H}}$,
$\bar{\boldsymbol{\Sigma}}={\rm
diag}\left(\bar{\Sigma}_{1},\dots,\bar{\Sigma}_{n_{\min}}\right)$ denotes the
diagonal matrix whose $l$-th entry is the $l$-th singular value of
$\bar{\boldsymbol{H}}$,
$\bar{\Sigma}_{1}\geq\cdots\geq\bar{\Sigma}_{n_{\min}}$ and
$n_{\min}=\min\left\\{n_{tx},n_{rx}\right\\}$. In addition,
$\boldsymbol{P}^{*}={\rm diag}\left(p_{1}^{*},\dots,p_{n_{\min}}^{*}\right)$,
$p_{l}^{*}=\left[\mu-\frac{1}{\overset{}{\bar{\Sigma}_{l}^{2}}}\right]_{0}^{+}$,
and ${\rm SNR}=\sum p_{l}^{*}$.777Note that it is possible to employ equal
power allocation at high SNR. In fact, our analysis leverages this assumption
for tractability. Nevertheless, we will consider waterfilling power allocation
here since it is optimal regardless of the SNR. In addition, it would be
useful later to make a fair comparison between different benchmarking schemes.
Thus, the optimal input covariance is
$\boldsymbol{K}^{*}=\boldsymbol{W}_{tx}^{*}\boldsymbol{P}^{*}\boldsymbol{W}_{tx}^{*H}$.
Using the above methods, the rate of the QR MIMO-FAS for a given ${\rm SNR}$
can be expressed as

$R_{{\rm QR}}\left({\rm
SNR}\right)=\sum_{l=1}^{n_{\min}}\log\left(1+p_{l}^{*}\bar{\Sigma}_{l}^{2}\right),$
(16)

where (16) helps analyze the optimal DMT of MIMO-FAS.

1: Compute (10) via QR factorization with pivoted column

2: Compute $\boldsymbol{\Omega}$ using (14)

3: While $\Omega_{k,l}>1$

4: Permute the $k$-th column of
$\boldsymbol{H}_{a}^{H}/\boldsymbol{\varPi}_{a}$ and the $l$-th column of
$\boldsymbol{H}_{i}^{H}/\boldsymbol{\varPi}_{i}$ with the largest
$\Omega_{k,l}>1$

5: Update $\boldsymbol{H}^{H}=\boldsymbol{H}_{a,k,l}^{H}$ and
$\boldsymbol{\varPi}=\boldsymbol{\varPi}{}_{a,k,l}$

6: Perform QR factorization using $\boldsymbol{\varPi}$ and return to Step 2

7: end

8: Set $\boldsymbol{A}_{r}^{*}=\boldsymbol{\varPi}_{a}^{T}$

9: Repeat Steps 1–7 by replacing $\boldsymbol{H}^{H}$ with
$\boldsymbol{A}_{r}^{*}\boldsymbol{H}$

10: Set $\boldsymbol{A}_{r}^{*}=\boldsymbol{\varPi}_{a}^{T}$

11: Compute SVD on
$\boldsymbol{A}_{r}^{*}\boldsymbol{H}\boldsymbol{A}_{t}^{*}$ to obtain
$\boldsymbol{W}^{*}$

12: Use bisection to obtain $\boldsymbol{P}^{*}$

Algorithm 1 Pseudocode of QR MIMO-FAS

The pseudocode of the proposed scheme is presented in Algorithm 1. Let us
denote $N_{\max}=\max\left\\{N_{tx},N_{rx}\right\\}$ and
$n_{\max}=\max\left\\{n_{tx},n_{rx}\right\\}$. The worst computational cost of
Step 1 is
$\frac{4}{3}n_{\max}^{3}-4N_{\max}n_{\max}^{2}+4N_{\max}^{2}n_{\max}$ flops
[30]. Steps 2–7 require
$\left(1+t_{i}\right)\left[\left(\frac{2}{3}n_{\max}^{3}N_{\max}^{2}+2n_{\max}^{2}N_{\max}^{3}+2N_{\max}^{3}\right)\right]$
flops [31], where $t_{i}$ is a finite number of permutations and it is usually
small due to Step 1 [28]. Step 9 has the same total computational cost as
Steps 1–7, which is also finite. In addition, Steps 11 and 12 require
$21n_{\max}^{3}$ and
$\log\left(\frac{\mu_{\max}}{\epsilon_{0}}\right)n_{\max}$ flops,
respectively, where $\mu_{\max}$ is the interval for searching $\mu$ and
$\epsilon_{0}$ is the tolerance for bisection method. Summing up the
computational costs, the proposed scheme has a polynomial time complexity of
$\mathcal{{O}}\left(n_{\max}^{3}N_{\max}^{3}\right)$. Compared to the global
optimal solution, the proposed scheme significantly reduces the time
complexity.

##  IV Optimal DMT

In this section, we analyze the optimal DMT of MIMO-FAS. As defined in [24], a
MIMO scheme is said to achieve a multiplexing gain of $r$ and a diversity gain
of $d$ if

$\lim_{{\rm SNR}\rightarrow\infty}\frac{R_{{\rm sys}}\left({\rm
SNR}\right)}{\log{\rm SNR}}=r,$ (17)

and the outage probability satisfies888Here, we use the fact that the error
probability can be arbitrarily close to the outage probability [32, 33].

$\lim_{{\rm SNR}\rightarrow\infty}\frac{\log\left(P_{{\rm sys}}^{{\rm
out}}\left({\rm SNR},r\right)\right)}{\log{\rm SNR}}=-d\left(r\right),$ (18)

in which $R_{{\rm sys}}\left({\rm SNR}\right)$ and $P_{{\rm sys}}^{{\rm
out}}\left({\rm SNR},r\right)$ are, respectively, the rate and outage
probability of the system. Similar to [24], we use the symbol $\dot{=}$ to
denote exponential equality. In particular, $f\left({\rm
SNR}\right)\dot{=}\leavevmode\nobreak\ {\rm SNR}^{-d}$ if

$\lim_{{\rm SNR}\rightarrow\infty}\frac{\log\left(f\left({\rm
SNR}\right)\right)}{\log{\rm SNR}}=-d.$ (19)

To obtain the optimal DMT of MIMO-FAS, we present the following lemmas.

######  Lemma 1.

Given a 2D space with an area of $W_{s}$ where both $W_{1}^{s}\gg 0$ and
$W_{2}^{s}\gg 0$, $\boldsymbol{J}_{s}$ in (2) can be represented by
$\boldsymbol{J}_{{\rm red}}^{s}$, i.e., a full rank symmetric
$N_{s}^{{}^{\prime}}\times N_{s}^{{}^{\prime}}$ finite-size matrix even if
$N_{s}\rightarrow\infty$.

###### Proof:

For a positive $W_{s}$, consider (2) where $N_{s}\rightarrow\infty$. Without
loss of generality, we focus on two neighboring ports: the
$\left(n_{1}^{s},n_{2}^{s}\right)$-th and
$\left(\tilde{n}_{1}^{s},\tilde{n}_{2}^{s}\right)$-th port. In cases where
$N_{1}^{s}\rightarrow\infty$, we analyze
$\left(\tilde{n}_{1}^{s},\tilde{n}_{2}^{s}\right)=\left(n_{1}^{s}\pm
1,n_{2}^{s}\right)$. The spatial correlation between the
$\left(n_{1}^{s},n_{2}^{s}\right)$-th and
$\left(\tilde{n}_{1}^{s},n_{2}^{s}\right)$-th port is given by

$J_{\left(n_{1}^{s},n_{2}^{s}\right),\left(\tilde{n}_{1}^{s},n_{2}^{s}\right)}^{s}=\underset{N_{1}^{s}\rightarrow\infty}{\lim}j_{0}\left(\frac{2\pi}{N_{1}^{s}-1}W_{1}^{s}\right)=1,$
(20)

since $\underset{N_{1}^{s}\rightarrow\infty}{\lim}\frac{1}{N_{1}^{s}-1}=0$. In
other words, the spatial correlation of the $\tilde{n}_{1}^{s}$-th port is
identical to that of the $n_{1}^{s}$-th port in the limit. For ease of
exposition, let us refer to the spatial correlation of a port as an entry.

For $n_{2}^{s}=\left\\{1,\ldots,N_{2}^{s}\right\\}$, we can remove the
identical entries (e.g., $\tilde{n}_{1}^{s}$ for $\exists n_{1}^{s}$) and
obtain $\bar{N}_{1}^{s}$ distinct entries. Next, let us denote the
$\left(\dot{n}_{1}^{s},n_{2}^{s}\right)$-th port as the farthest port away
from the $\left(n_{1}^{s},n_{2}^{s}\right)$-th port. Since $W_{1}^{s}\gg 0$,
it is clear that
$J_{\left(n_{1}^{s},n_{2}^{s}\right),\left(\dot{n}_{1}^{s},n_{2}^{s}\right)}^{s}=j_{0}\left(2\pi\dot{c}\right)$
where $\dot{c}\gg 0$ is the distance between the
$\left(\dot{n}_{1}^{s},n_{2}^{s}\right)$-th and
$\left(n_{1}^{s},n_{2}^{s}\right)$-th ports. By intermediate value theorem, we
can conclude there is an $\varepsilon$ such that

$\varepsilon={\rm
inf}\left\\{e\left|J_{\left(n_{1}^{s},n_{2}^{s}\right),\left(n_{1}^{s}\pm
e,n_{2}^{s}\right)}^{s}\right.\neq 1,\begin{array}[]{l}e\in\mathbb{N},\\\
0<e\leq N_{1}^{s}-1\end{array}\right\\}.$ (21)

Note that one cannot make any assumption on $e$ and $\varepsilon$ except their
existence. Let us define
$c_{1}^{s}\triangleq\frac{\varepsilon}{N_{1}^{s}-1}W_{1}^{s}$ as the minimal
distance required for the spatial correlation between the
$\left(n_{1}^{s},n_{2}^{s}\right)$-th and $\left(n_{1}^{s}\pm
e,n_{2}^{s}\right)$-th ports to be completely distinct. Then we can verify
that $\bar{N}_{1}^{s}$ is finite since
$W_{1}^{s}\geq\bar{N}_{1}^{s}c_{1}^{s}>0$. Let us write the $\bar{N}_{1}^{s}$
distinct entries as a vector $\boldsymbol{v}_{n_{2}^{s}}$ for each
$n_{2}^{s}\in\left\\{1,\ldots,N_{2}^{s}\right\\}$. If these vectors are
linearly dependent, then we can similarly remove the dependent vectors and
obtain $\bar{N}_{2}$ independent vectors. As a result, we can rewrite
$\boldsymbol{J}_{s}$ as a symmetric $\bar{N}_{s}\times\bar{N}_{s}$ finite-size
matrix with distinct entries where
$\bar{N}_{s}=\bar{N}_{1}^{s}\bar{N}_{2}^{s}$.

Using a similar argument, we see that $\boldsymbol{J}_{s}$ can be represented
by an $\bar{N}_{s}\times\bar{N}_{s}$ finite-size matrix if
$N_{2}^{s}\rightarrow\infty$. Combining the two cases, it is straightforward
to see that $\boldsymbol{J}_{s}$ can be rewritten as a symmetric
$\bar{N}_{s}\times\bar{N}_{s}$ matrix as $N_{1}^{s}\rightarrow\infty$ and
$N_{2}^{s}\rightarrow\infty$ since both $W_{1}^{s}\gg 0$ and $W_{2}^{s}\gg 0$,
and we have (21) and

$\varepsilon_{2}={\rm
inf}\left\\{e\left|J_{\left(n_{1}^{s},n_{2}^{s}\right),\left(n_{1}^{s},n_{2}^{s}\pm
e\right)}^{s}\right.\neq 1,\begin{array}[]{l}e\in\mathbb{N},\\\ 0<e\leq
N_{2}^{s}-1\end{array}\right\\}.$ (22)

From the above, it is clear that we can use the same argument to show that
$\boldsymbol{J}_{s}$ can be rewritten as a symmetric
$\bar{N}_{s}\times\bar{N}_{s}$ matrix if $N_{1}^{s}/N_{2}^{s}$ is finite since
we can always remove the entries where their vertical/horizontal distances
between the adjacent ports are less than $c_{1}^{s}/c_{2}^{s}$.

Let us denote $N_{s}^{{}^{\prime}}$ as the full rank of the symmetric
$\bar{N}_{s}\times\bar{N}_{s}$ matrix where
$N_{s}^{{}^{\prime}}\leq\bar{N}_{s}$. Then we can further reduce the symmetric
$\bar{N}_{s}\times\bar{N}_{s}$ matrix to a full rank symmetric
$N_{s}^{{}^{\prime}}\times N_{s}^{{}^{\prime}}$ submatrix
$\boldsymbol{J}_{{\rm red}}^{s}$ by removing the
$(\bar{N}_{s}-N_{s}^{{}^{\prime}})$ dependent rows and columns. Thus,
$\boldsymbol{J}_{s}$ can always be represented by $\boldsymbol{J}_{{\rm
red}}^{s}$, i.e., a full rank symmetric $N_{s}^{{}^{\prime}}\times
N_{s}^{{}^{\prime}}$ finite-size matrix. In other words, it suffices to
consider $\boldsymbol{J}_{{\rm red}}^{s}$ instead of $\boldsymbol{J}_{s}$
since some rows/columns of $\boldsymbol{J_{s}}$ are identical or a linear
combination of the others. A more general result is given in Appendix II. ∎

######  Lemma 2.

If $\boldsymbol{J}_{tx}$ and $\boldsymbol{J}_{rx}$ are full rank, the optimal
DMT of any MIMO system with channel $\boldsymbol{H}$ is the same as that of a
system with channel $\boldsymbol{G}$.

###### Proof:

See [34]. ∎

######  Lemma 3.

The optimal DMT of using only $n_{rx}\times n_{tx}$ channels from the MIMO
channel $\boldsymbol{G}$, where $n_{tx}\leq N_{tx}$ and $n_{rx}\leq N_{rx}$,
is a piecewise linear function connecting the points $\left(n_{\min},0\right)$
and

$\left\\{r,\left(N_{rx}-r\right)\left(N_{tx}-r\right)\right\\},\quad
r=0,\ldots,N,$ (23)

where

$N=\arg\min_{{\eta\in\mathbb{Z}\atop 0\leq\eta\leq
n_{\min}-1}}\frac{\left(N_{rx}-\eta\right)\left(N_{tx}-\eta\right)}{n_{\min}-\eta}.$
(24)

###### Proof:

See [35]. ∎

######  Corollary 4.

If the antennas are placed based on a grid structure with at least half a
wavelength apart and the transmit/receive spatial correlation matrices are
full rank, then the optimal DMT of $n_{rx}\times n_{tx}$ MIMO antenna
selection is a piecewise linear function connecting the points
$\left(n_{\min},0\right)$ and

$\left\\{r,\left(w_{rx}-r\right)\left(w_{tx}-r\right)\right\\},\quad
r=0,\ldots,N_{\text{as}},$ (25)

where

$N_{\text{as}}=\arg\min_{{\eta\in\mathbb{Z}\atop 0\leq\eta\leq
n_{\min}-1}}\frac{\left(w_{rx}-\eta\right)\left(w_{tx}-\eta\right)}{n_{\min}-\eta}.$
(26)

###### Proof:

Given a fixed $W_{s}^{1}$ and $W_{s}^{2}$, there can be at most
$w_{s}=\left(\left\lfloor\frac{W_{s}^{1}}{0.5}\right\rfloor+1\right)$$\left(\left\lfloor\frac{W_{s}^{2}}{0.5}\right\rfloor+1\right)$
antennas at side $s$. Using Lemma 2 and Lemma 3, we obtain (25) and (26). ∎

Using the above lemmas, we can now obtain the outer bound of the DMT of MIMO-
FAS.

######  Theorem 5.

For finite $W_{rx}$ and $W_{tx}$, the outer bound of the DMT of MIMO-FAS is a
piecewise linear function connecting the points $\left(n_{\min},0\right)$ and

$\left\\{r,\left(N_{rx}^{{}^{\prime}}-r\right)\left(N_{tx}^{{}^{\prime}}-r\right)\right\\},\quad
r=0,\dots,N^{\prime},$ (27)

where

$N^{\prime}=\arg\min_{{\eta\in\mathbb{Z}\atop 0\leq\eta\leq
n_{\min}-1}}\frac{\left(N_{rx}^{{}^{\prime}}-\eta\right)\left(N_{tx}^{{}^{\prime}}-\eta\right)}{n_{\min}-\eta}.$
(28)

###### Proof:

By using SVD, we can decompose
$\boldsymbol{H}=\boldsymbol{M}\boldsymbol{\Sigma}\boldsymbol{N}$ where
$\boldsymbol{M}$ is the left singular matrix of $\boldsymbol{H}$,
$\boldsymbol{N}$ is the right singular matrix of $\boldsymbol{H}$,
$\boldsymbol{\Sigma}={\rm
diag}\left(\Sigma_{1},\dots,\Sigma_{N_{\min}}\right)$ is a diagonal matrix
whose $l$-th entry is the singular value of $\boldsymbol{H}$ and
$\Sigma_{1}\geq\cdots\geq\Sigma_{N_{\min}}$. According to the Cauchy’s
Interlacing theorem [36], it follows that
$\Sigma_{1}\geq\bar{\Sigma}_{1}\geq\cdots\geq\bar{\Sigma}_{n_{\min}}\geq\Sigma_{n_{\min}}\geq\cdots\geq\Sigma_{N_{\min}}$.
Therefore, the rate of MIMO-FAS can be upper bounded by

$R\left({\rm
SNR}\right)=\sum_{l=1}^{n_{\min}}\log\left(1+\tilde{p}_{l}^{*}\Sigma_{l}^{2}\right),$
(29)

where $\tilde{p}_{l}^{*}=\left[\mu-\frac{1}{\Sigma_{l}^{2}}\right]_{0}^{+}$
and ${\rm SNR}=\sum\tilde{p}_{l}^{*}$. At high SNR, (29) can be simplified as

$R\left({\rm SNR}\right)=\sum_{l=1}^{n_{\min}}\log\left(1+\frac{{\rm
SNR}}{n_{\min}}\Sigma_{l}^{2}\right),$ (30)

since the rate of using equal power allocation approaches to that of
waterfilling power allocation as ${\rm SNR}$ increases [37]. Consequently, the
outage probability of MIMO-FAS can be lower bounded by

$P_{{\rm out}}\left({\rm SNR},r\right)=\mathbb{{P}}\left\\{R\left({\rm
SNR}\right)<r\log{\rm SNR}\right\\}.$ (31)

At high SNR, the outage probability is rewritten as [24]

$P_{{\rm out}}\left({\rm
SNR},r\right)=\mathbb{{P}}\left\\{\sum_{l=1}^{n_{\min}}\log\left(1+{\rm
SNR}\Sigma_{l}^{2}\right)<r\log{\rm SNR}\right\\},$ (32)

since

$\displaystyle\lim_{{\rm
SNR}\rightarrow\infty}\frac{\log\left(\mathbb{{P}}\left\\{\sum_{l=1}^{n_{\min}}\log\left(1+\frac{{\rm
SNR}}{n_{\min}}\Sigma_{l}^{2}\right)<r\log{\rm SNR}\right\\}\right)}{\log{\rm
SNR}}$ $\displaystyle=$ $\displaystyle\lim_{{\rm
SNR}\rightarrow\infty}\frac{\log\left(\mathbb{{P}}\left\\{\sum_{l=1}^{n_{\min}}\log\left(1+\frac{{\rm
SNR}}{n_{\min}}\Sigma_{l}^{2}\right)<r\log{\rm
SNR}\right\\}\right)}{\log\frac{{\rm SNR}}{n_{\min}}}$ $\displaystyle=$
$\displaystyle\lim_{{\rm
SNR}\rightarrow\infty}\frac{\log\left(\mathbb{{P}}\left\\{\sum_{l=1}^{n_{\min}}\log\left(1+{\rm
SNR}\Sigma_{l}^{2}\right)<r\log{\rm SNR}\right\\}\right)}{\log{\rm SNR}}.$
(33)

Using (19), we can obtain the outer bound on the diversity gain for a fixed
multiplexing gain $r$ as

$P_{{\rm out}}\left({\rm SNR},r\right)\dot{=}\leavevmode\nobreak\
\mathbb{{P}}\left\\{\sum_{l=1}^{n_{\min}}\log\left(1+{\rm
SNR}\Sigma_{l}^{2}\right)<r\log{\rm SNR}\right\\}.$ (34)

According to [38], the joint probability density function (PDF) of
$\boldsymbol{\Sigma}^{2}$ is given by

$\displaystyle f\left(\boldsymbol{\Sigma}^{2}\right)$
$\displaystyle=\sum_{\boldsymbol{q}}\frac{\left(-1\right)^{\frac{N_{\min}\left(N_{\min}-1\right)}{2}}\mathcal{A}}{N_{\min}!\triangle\left(\boldsymbol{q}\right)}\triangle\left(\boldsymbol{\Sigma}^{2}\right)\times$
$\displaystyle\qquad\qquad{\rm
det}\left(\boldsymbol{\Sigma}_{k}^{2\left(q_{l}+N_{\max}-N_{\min}\right)}\right),$
(35)

where $\boldsymbol{q}=\left[q_{1},\dots,q_{N_{\max}}\right]^{T}$,
$N_{\min}=\min\left\\{N_{tx},N_{rx}\right\\}$,

$\displaystyle\mathcal{{A}}$
$\displaystyle=\frac{\prod_{k=1}^{N_{\min}}\lambda_{k}^{s}\prod_{l=1}^{N_{\max}}\lambda_{l}^{s^{\prime}}{\rm
det}\left(\left(-\lambda_{k}^{s}\right)^{q_{l}}\right)}{\triangle\left(\boldsymbol{\lambda}^{s}\right)\triangle(\boldsymbol{\lambda}^{s^{\prime}})}\times$
$\displaystyle\frac{{\rm
det}\left(\left.\left(\lambda_{k}^{s^{\prime}}\right)^{q_{l}+N_{\max}-N_{\min}}\right|_{k=1}^{N_{\min}},\left.\left(\lambda_{i}^{s^{\prime}}\right)^{N_{\max}-k}\right|_{k=N_{\min}+1}^{N_{\max}}\right)}{\prod_{l=1}^{N_{\min}}\left(q_{l}+N_{\max}-N_{\min}\right)!},$
(36)

where

$\left\\{\begin{aligned}
s&=\left\\{s|N_{s}=\min\left\\{N_{tx},N_{rx}\right\\}\right\\},\\\
s^{\prime}&=\left\\{s^{\prime}|N_{s^{\prime}}=\max\left\\{N_{tx},N_{rx}\right\\}\right\\},\\\
\boldsymbol{\lambda}^{s}&=\left[\lambda_{1}^{s},\dots,\lambda_{N_{s}}^{s}\right]^{T},\end{aligned}\right.$
(37)

$\triangle\left(\boldsymbol{\lambda}^{s}\right)$ denotes the Vandermonde
determinant of vector $\boldsymbol{\lambda}^{s}$, ${\rm
det}\left(f\left(k,l\right)\right)$ is the determinant of a matrix with the
$\left(k,l\right)$-th entry given by the function $f\left(k,l\right)$ and
$q_{l}=b_{l}+N_{\max}-l$. In addition,
$\boldsymbol{b}=\left[b_{1},\dots,b_{N_{\max}}\right]^{T}$ is the irreducible
representation of unitary group such that $b_{1}\geq\cdots\geq
b_{N_{\max}}\geq 0$ are integers.

Conventionally, we can analyze the outer bound by simplifying the joint PDF of
$\boldsymbol{\Sigma}^{2}$ and then analyzing the exponents of $\Sigma_{l}$.
Nevertheless, it is found that no simplification can be made to keep the
exponents of $\Sigma_{l}$ tractable when the rows and columns of
$\boldsymbol{H}$ are fully correlated [39]. To alleviate this problem, we
employ Lemma 1, Lemma 2, and Lemma 3.

Specifically, according to [40], the PDF of $\boldsymbol{H}$ can be obtained
by removing the dependent entries and the PDF of the singular values of
$\boldsymbol{H}$ can be obtained via coordinate changes [41]. Using Lemma 1,
we know that $\boldsymbol{J}_{s}$ can be represented by a full rank symmetric
finite-size matrix, i.e, $\boldsymbol{J}_{{\rm
red}}^{s}\in\mathbb{{R}}^{N_{s}^{{}^{\prime}}\times N_{s}^{{}^{\prime}}}$ and
${\rm rank}\left(\boldsymbol{J}_{{\rm red}}^{s}\right)=N_{s}^{{}^{\prime}}$.
It follows that $\boldsymbol{H}$ and $\boldsymbol{G}$ in (3) can be rewritten
as $N_{rx}^{{}^{\prime}}\times N_{tx}^{{}^{\prime}}$ matrices with the same
PDFs. From Lemma 2, it is known that the DMT of any MIMO system with channel
$\boldsymbol{H}\in\mathbb{C}^{N_{rx}^{{}^{\prime}}\times
N_{tx}^{{}^{\prime}}}$ is the same as that of a system with channel
$\boldsymbol{G}\in\mathbb{C}^{N_{rx}^{{}^{\prime}}\times
N_{tx}^{{}^{\prime}}}$. Since $\boldsymbol{G}$ is an independent and
identically distributed (i.i.d.) circularly symmetric complex Gaussian matrix,
it follows that ${\rm
rank}\left(\boldsymbol{G}\right)=N_{\min}^{{}^{\prime}}=\min\\{N_{rx}^{{}^{\prime}},N_{tx}^{{}^{\prime}}\\}$
with probability one. Using Lemma 3, we can conclude that the outer bound of
the DMT of MIMO-FAS is a piecewise linear function connecting the points as
given in (27) and (28). ∎

Using Theorem 5, we can now derive the optimal DMT of MIMO-FAS by considering
its outer bound and inner bound. Specifically, the DMT of QR MIMO-FAS can be
regarded as the inner bound of the DMT of MIMO-FAS.

######  Theorem 6.

The DMT of QR MIMO-FAS is equivalent to the outer bound of the DMT of MIMO-
FAS, and thus it is also the optimal DMT of MIMO-FAS.

###### Proof:

At high SNR, (16) can be simplified as

$R_{{\rm QR}}\left({\rm
SNR}\right)=\sum_{l=1}^{n_{\min}}\log\left(1+\frac{{\rm
SNR}}{n_{tx}}\bar{\Sigma}_{l}^{2}\right)$ (38)

since the rate of using equal power allocation approaches to that of
waterfilling power allocation as ${\rm SNR}$ increases [37]. Consequently, the
outage probability is characterized by

$P_{{\rm QR}}^{{\rm out}}\left({\rm SNR},r\right)=\mathbb{{P}}\left\\{R_{{\rm
QR}}\left({\rm SNR}\right)<r\log{\rm SNR}\right\\}.$ (39)

At high SNR, the difference between (30) and (38) is

$R\left({\rm SNR}\right)-R_{{\rm QR}}\left({\rm
SNR}\right)\approx\sum_{l=1}^{n_{\min}}\log\left(\frac{n_{tx}\Sigma_{l}^{2}}{n_{\min}\bar{\Sigma}_{l}^{2}}\right)=c_{0},$
(40)

which is a constant.999Note that a constant gap amounts to a finite scaling of
SNR. This implies that $R\left({\rm SNR}\right)=R_{{\rm QR}}\left({\rm
SNR}+\gamma\right)$ for some $\gamma>0$. As
$\lim_{c\rightarrow\infty}\log\left(\frac{1+c}{c}\right)=0$, the approximation
of (40) is tight as ${\rm SNR}\rightarrow\infty$. As such, we have

$\displaystyle\lim_{{\rm SNR}\rightarrow\infty}\frac{R_{{\rm QR}}\left({\rm
SNR}\right)}{\log{\rm SNR}}$ $\displaystyle=\lim_{{\rm
SNR}\rightarrow\infty}\frac{R\left({\rm SNR}\right)-c_{0}}{\log{\rm SNR}}$
$\displaystyle=\lim_{{\rm SNR}\rightarrow\infty}\frac{R\left({\rm
SNR}\right)}{\log{\rm SNR}},$ (41)

and

$\displaystyle\lim_{{\rm SNR}\rightarrow\infty}\frac{\log\left(P_{{\rm
QR}}^{{\rm out}}\left({\rm SNR},r\right)\right)}{\log{\rm SNR}}$
$\displaystyle=$ $\displaystyle\lim_{{\rm
SNR}\rightarrow\infty}\frac{\log\left(\mathbb{P}\left\\{R\left({\rm
SNR}\right)<r\log{\rm SNR}+c_{0}\right\\}\right)}{\log{\rm SNR}}$
$\displaystyle=$ $\displaystyle\lim_{{\rm
SNR}\rightarrow\infty}\frac{\log\left(\mathbb{P}\left\\{R\left({\rm
SNR}\right)<r\log{\rm SNR}\right\\}\right)}{\log{\rm SNR}}$ $\displaystyle=$
$\displaystyle\lim_{{\rm SNR}\rightarrow\infty}\frac{\log\left(P_{{\rm
out}}\left({\rm SNR},r\right)\right)}{\log{\rm SNR}}.$ (42)

Thus, the DMT of QR MIMO-FAS is equivalent to the outer bound, and also the
optimal DMT of MIMO-FAS. ∎

It is challenging to explicitly obtain $N_{s}^{{}^{\prime}}$ because
$\boldsymbol{J}_{s}$ might be near to being singular. It is also unclear how
$\boldsymbol{J}_{s}$ can be reduced to/reconstructed from
$\boldsymbol{J}_{{\rm red}}^{s}$. To address these issues, we propose methods
to reliably estimate $N_{s}^{{}^{\prime}}$, and to linearly transform
$\boldsymbol{J}_{s}$ to $\boldsymbol{J}_{{\rm red}}^{s}$ and vice versa with a
proof of certificates. These methods are based on the following theorems.

######  Theorem 7.

Since $\boldsymbol{J}_{s}$ can be well-approximated by
$\hat{\boldsymbol{J}}_{s}$ where
$\hat{\boldsymbol{J}}_{s}=\boldsymbol{U}_{s}\hat{\boldsymbol{\Lambda}}\boldsymbol{U}_{s}^{H}$
and $\hat{\boldsymbol{\Lambda}}={\rm
diag}(\lambda_{1}^{s},\dots,\lambda_{N_{s}^{{}^{\prime}}}^{s},0,\dots,0)$, the
rank of $\boldsymbol{J}_{s}$ can be estimated as $N_{s}^{{}^{\prime}}$.

###### Proof:

Define an arbitrarily small $\xi>0$ as the threshold where the numerical
values of the eigenvalues are negligible. According to [42],
$\boldsymbol{J}_{s}$ can be regularized by $\hat{\boldsymbol{J}}_{s}$, where
$\hat{\boldsymbol{J}}_{s}=\boldsymbol{U}_{s}\hat{\boldsymbol{\Lambda}}\boldsymbol{U}_{s}^{H}$,
$\hat{\boldsymbol{\Lambda}}={\rm
diag}(\lambda_{1}^{s},\dots,\lambda_{N_{s}^{{}^{\prime}}}^{s},0,\dots,0)$ and
$\lambda_{l}^{s}<\xi$, $l\in\\{N_{s}^{{}^{\prime}}+1,\dots,N_{s}\\}$. The
Frobenius norm between $\boldsymbol{J}_{s}$ and $\hat{\boldsymbol{J}}_{s}$ is
given by

$E=\|\boldsymbol{J}_{s}-\hat{\boldsymbol{J}}_{s}\|_{F}=\sum_{l=N_{s}^{{}^{\prime}}+1}^{N_{s}}\lambda_{l}^{s},$
(43)

which can be upper bounded by $(N_{s}-N_{s}^{{}^{\prime}})\xi$. In practice,
we have
$\lambda_{1}^{s}\gg\lambda_{N_{s}^{{}^{\prime}}+1}^{s}>\ldots>\lambda_{N_{s}}^{s}$
and thus (43) is usually very small. Therefore, the rank of
$\boldsymbol{J}_{s}$ can be estimated as $N_{s}^{{}^{\prime}}$. ∎

######  Theorem 8.

For arbitrary $N_{s}=N_{1}^{s}\times N_{2}^{s}$ and a finite
$W_{s}=W_{1}^{s}\times W_{2}^{s}$, $\boldsymbol{J}_{s}$ in (2) can be reduced
to $\boldsymbol{J}_{{\rm red}}^{s}$ using the set
$\mathcal{V}_{s}=\\{\boldsymbol{v}_{l}^{s}|\boldsymbol{J}_{s}^{\left(l\right)}\tilde{\boldsymbol{v}}_{l}^{s}=\boldsymbol{0},l=N_{s}^{{}^{\prime}}+1,\ldots,N_{s}\\}$
where
$\tilde{\boldsymbol{v}}_{l}^{s}=[\boldsymbol{v}_{l}^{s}\leavevmode\nobreak\
-1]^{T}$. Conversely, $\boldsymbol{J}_{s}$ in (2) can be reconstructed from
$\boldsymbol{J}_{{\rm red}}^{s}$ with the set $\mathcal{V}_{s}$.

###### Proof:

Let us introduce a vector
$\tilde{\boldsymbol{v}}_{l}^{s}=[\boldsymbol{v}_{l}^{s}\leavevmode\nobreak\
-1]^{T}$ where $\boldsymbol{v}_{l}^{s}\in\mathbb{R}^{l}$ and
$\boldsymbol{v}_{l}^{s}\neq\boldsymbol{0}$. In addition, let us denote

$\boldsymbol{J}_{s}^{\left(l\right)}=\left[\begin{array}[]{cc}\boldsymbol{J}_{{\rm
sub}}^{\left(l-1\right)}&\boldsymbol{j}_{l}\\\
\boldsymbol{j}_{l}^{T}&j_{l,l}\end{array}\right].$ (44)

Note that $\boldsymbol{J}_{s}=\boldsymbol{J}_{s}^{\left(N_{s}\right)}$ and
$\boldsymbol{J}_{{\rm red}}^{s}=\boldsymbol{J}_{s}^{(N_{s}^{{}^{\prime}})}$.
If $N_{s}^{{}^{\prime}}=N_{s}$, then $\mathcal{V}_{s}$ is an empty set and
$\boldsymbol{J}_{s}=\boldsymbol{J}_{{\rm red}}^{s}$. Thus, we can focus on the
case where $N_{s}^{{}^{\prime}}<N_{s}$. Without loss of generality, we may
assume that the last column of $\boldsymbol{J}_{s}^{\left(l\right)}$ is a
linear combination of the first $\left(l-1\right)$ columns of
$\boldsymbol{J}_{s}^{\left(l\right)}$. This implies that
$\boldsymbol{J}_{s}^{\left(l\right)}\tilde{\boldsymbol{v}}_{l}^{s}=\boldsymbol{0}$,
i.e., $\tilde{\boldsymbol{v}}_{l}^{s}$ is in the null space of
$\boldsymbol{J}_{s}^{\left(l\right)}$ and $\tilde{\boldsymbol{v}}_{l}^{s}\neq
0$. If
$\boldsymbol{J}_{s}^{\left(l\right)}\tilde{\boldsymbol{v}}_{l}^{s}=\boldsymbol{0}$
and $l\neq N_{s}^{{}^{\prime}}$, we can reduce
$\boldsymbol{J}_{s}^{\left(l\right)}$ to $\boldsymbol{J}_{{\rm
sub}}^{\left(l-1\right)}$ by setting
$\boldsymbol{v}_{l}^{s}\in\mathcal{V}_{s}$. Otherwise, we have
$\boldsymbol{J}_{s}^{\left(l\right)}=\boldsymbol{J}_{{\rm red}}^{s}$ since
$\boldsymbol{J}_{s}^{\left(l\right)}$ must be a full rank matrix. Conversely,
$\boldsymbol{J}_{s}^{\left(l\right)}$ can be reconstructed from
$\boldsymbol{J}_{{\rm sub}}^{\left(l-1\right)}$ if $\boldsymbol{v}_{l}^{s}$ is
given. Specifically, we can define
$\boldsymbol{j}_{l}\triangleq\boldsymbol{J}_{{\rm
sub}}^{\left(l-1\right)}\boldsymbol{v}_{l}^{s}$ and
$j_{l,l}\triangleq\boldsymbol{j}_{l}^{T}\boldsymbol{v}_{l}^{s}$ and they are
sufficient to reconstruct $\boldsymbol{J}_{s}^{\left(l\right)}$. Hence,
$\boldsymbol{v}_{l}^{s}$ can be interpreted as a certificate that
$\boldsymbol{J}_{s}^{\left(l\right)}$ can be reduced to/reconstructed from
$\boldsymbol{J}_{{\rm sub}}^{\left(l-1\right)}$ for
$l=\\{N_{s}^{{}^{\prime}}+1,\dots,N_{s}\\}$. ∎

Table II:  Estimation of $N_{s}^{{}^{\prime}}$ for different $W_{s}$ in terms
of $\lambda^{2}$, where $\xi=10^{-3}$, $N_{s}=100$, $N_{1}^{s}=N_{2}^{s}$, and
$W_{1}^{s}=W_{2}^{s}$.

$W_{s}$ | $0.5\times 0.5$ | $1\times 1$ | $1.5\times 1.5$ | $2\times 2$ | $2.5\times 2.5$ | $3\times 3$
---|---|---|---|---|---|---
$N_{s}^{{}^{\prime}}$ | $13$ | $23$ | $34$ | $48$ | $60$ | $73$
$E$ | $0.001$ | $0.002$ | $0.003$ | $0.002$ | $0.005$ | $0.005$

[‡] Note that the figures regarding the diversity order of MIMO-FAS described
in [25] were based on an earlier version of this paper considering the spatial
correlation in 2D environments only. The correct diversity orders for MIMO-FAS
in 3D environments should be referred to this table. For example, for MIMO-FAS
with $0.5\lambda\times 0.5\lambda$ FAS at both ends, the diversity order is
$13\times 13=169$ which is much higher than originally reported.

The proposed methods enable us to estimate $N_{s}^{{}^{\prime}}$ for given
$N_{1}^{s}$, $N_{2}^{s}$, $W_{1}^{s}$, and $W_{2}^{s}$. Furthermore, they
allow us to verify that $\boldsymbol{J}_{s}$ indeed can be reduced to (or
reconstructed from) $\boldsymbol{J}_{{\rm red}}^{s}$ with proof of
certificates. An example of the estimations of $N_{s}^{{}^{\prime}}$ is given
in Table II. This table can help us better understand the performance of MIMO-
FAS. For example, by substituting the estimation of $N_{s}^{{}^{\prime}}$ into
Theorem 5, we observe that MIMO-FAS yields massive diversity gains if
$r<n_{\min}$. Thus, it is worth investigating how we can leverage MIMO-FAS
effectively. To answer this question, we introduce the $q$-outage capacity.

######  Definition 9.

The $q$-outage capacity of a system is defined as

$C_{{\rm sys}}^{q}=q\left(1-\bar{P}_{{\rm sys}}^{{\rm out}}\left({\rm
SNR},q\right)\right),$ (45)

where

$\bar{P}_{{\rm sys}}^{{\rm out}}\left({\rm
SNR},q\right)=\mathbb{{P}}\left\\{R_{{\rm sys}}\left({\rm
SNR}\right)<q\right\\},$ (46)

where $\bar{P}_{{\rm sys}}^{{\rm out}}\left(q\right)$ is the outage
probability of a system for a fixed target rate or transmission rate $q$
independent of the SNR. It is worth noting that (45) differs from the
$\epsilon$-outage capacity as it is not measuring the largest $q$ such that
$\bar{P}_{{\rm sys}}^{{\rm out}}\left({\rm SNR},q\right)\leq\epsilon$.
Instead, (45) is interpreted as the average rate that a system can reliably
transmit over period of time such that the statistics of the fading do not
change. Furthermore, both $q$ and $\bar{P}_{{\rm sys}}^{{\rm out}}\left({\rm
SNR},q\right)$ play important roles in (45). For example, if $q\approx 0$, we
usually have $\bar{P}_{{\rm sys}}^{{\rm out}}\left({\rm SNR},q\right)\approx
0$. If $q$ is large, then we typically have $\bar{P}_{{\rm sys}}^{{\rm
out}}\left({\rm SNR},q\right)\approx 1$. Nevertheless, both cases yield
$C_{{\rm sys}}^{q}\approx 0$. In between the two extremes, we have $C_{{\rm
sys}}^{q}\gg 0$ and there is an optimal $q$ for a system such that (45) is
maximized. In addition, the $q$-outage capacity gain of MIMO-FAS over a
traditional antenna system can be characterized as

$C_{{\rm MIMO\text{-}FAS}}^{q}-C_{{\rm sys}}^{q}\\\ =q\left[\bar{P}_{{\rm{\rm
sys}}}^{{\rm out}}\left({\rm SNR},q\right)-\bar{P}_{{\rm
MIMO\text{-}FAS}}^{{\rm out}}\left({\rm SNR},q\right)\right].$ (47)

Using (47), we can easily see the benefits that can be harnessed by MIMO-FAS
over a traditional antenna system.

##  V Results and Discussions

Here, we present the analytical and Monte Carlo simulation results to evaluate
the performance of MIMO-FAS. For brevity, we focus on a symmetric MIMO-FAS
design where $N_{1}^{s}=N_{2}^{s}$, $W_{1}^{s}=W_{2}^{s}$ and $W_{rx}=W_{tx}$.
Unless stated otherwise, we assume that $N_{s}=100$, $n_{s}=4$,
$W_{s}=1\lambda^{2}$, $\delta_{s}^{2}=1$ and ${\rm SNR}=30{\rm dB}$. We also
consider multiple schemes based on different combinations of techniques to
highlight the respective gains and effects. These benchmarking schemes
are:101010Note that [23] has proposed a solution where the antennas can move
to locally optimal positions. We can interpret these positions as activating
some ports as $N_{s}\rightarrow\infty$. However, the solution cannot be
employed here due to two impediments. Firstly, the spatial correlation of
these positions cannot be obtained. Secondly, we are considering the cases
where the positions might be discrete. Therefore, the solution is not
considered in this paper.

  * •

Optimal MIMO-FAS: It considers the MIMO-FAS setup that utilizes an exhaustive
search, SVD and waterfilling for port selection, beamforming and power
allocation, respectively.

  * •

QR MIMO-FAS: This is the proposed MIMO-FAS that employs strong RRQR
factorization, SVD and waterfilling power allocation for a suboptimal
solution.

  * •

Greedy MIMO-FAS: This is the MIMO-FAS that employs greedy selection, SVD and
waterfilling power allocation for an efficient solution in the low SNR regime.

  * •

Random MIMO-FAS: It randomly activates the ports and uses SVD as well as
waterfilling power allocation.

  * •

MIMO: This refers to the traditional MIMO that employs SVD and waterfilling
power allocation. Unless otherwise stated, we assume that the number of
antennas is $n_{s}^{{\rm mimo}}=4$ and the antennas are spatially correlated
based on (1).

  * •

MIMO-AS: This refers to the traditional MIMO antenna selection system that
employs strong RRQR factorization, SVD and waterfilling power allocation.
Unless stated otherwise, the number of active antennas is $n_{s}^{{\rm mimo-
as}}=4$. Also, a maximum number of antennas is considered in the given surface
where the antennas are placed based on a grid structure with at least half a
wavelength apart, and they are spatially correlated due to (1).

Figure 2: Outage probability of FAS versus SNR for different number of ports
and dimensional space, with $q=7{\rm bps/Hz}$.

(a)  (b)  (c)

Figure 3: Average rates of the benchmarking schemes for different values of
$n_{s}$: a) ${\rm{SNR}=30}$dB; b) ${\rm{SNR}=10}$dB; c) ${\rm{SNR}=-10}$dB.

To highlight the superiority of 2D space, we first consider a simplified
scenario where there is only a single fluid antenna at the receiver. Fig. 2
shows the outage probability of FAS versus SNR for various number of ports and
dimensional space. Here, the outage probability is obtained using (46) by
replacing $R_{{\rm sys}}\left({\rm SNR}\right)$ with $R_{{\rm QR}}\left({\rm
SNR}\right)$. Given the same number of ports, the ports that are distributed
in 2D space can achieve a much lower outage probability as compared to the
ones that are distributed in 1D space. This improvement can be explained from
the fact that a 2D space has an additional dimension for the fluid antenna to
move around. Hence, it contains more spatial diversity and yields a lower
outage probability. This suggests that the ports in MIMO-FAS should be
designed using the entire 2D space for better performance.

Next in Fig. 3, we study the average rates of the benchmarking schemes for
different $n_{s}$ at different SNR. Since the optimal MIMO-FAS is computed
using an exhaustive search, we set $N_{s}=12$ where $N_{1}^{s}=3$ and
$N_{2}^{s}=4$. Fig. 3(a) illustrates that QR MIMO-FAS achieves a similar
average rate as compared to the optimal MIMO-FAS at high SNR. Furthermore, the
average rate of QR MIMO-FAS scales like $n_{s}\log{\rm SNR}$ when $n_{s}$
ranges from $1$ to $6$. Nevertheless, it suffers from a diminishing rate gain
when $n_{s}$ ranges from $7$ to $12$. Thus, MIMO-FAS is most effective when
each active port has the freedom of being at least half a wavelength apart
from each other. In addition, the average rate of QR MIMO-FAS outperforms
traditional MIMO when the number of active ports or antennas is the same
(i.e., $n_{s}=4$). This is because QR MIMO-FAS activates the optimal ports in
each realization, which reduces the spatial correlation effect. In Fig. 3(b),
we further observe that QR MIMO-FAS provides a higher sum-rate in the medium
SNR regime when $n_{s}$ is small while greedy MIMO-FAS yields a better
performance when $n_{s}$ is large. Nevertheless, as shown in Fig. 3(c), greedy
MIMO-FAS is generally more efficient than QR MIMO-FAS in the low SNR regime.
These results suggest that an efficient scheme with low complexity is still
required to maximize the rate of MIMO-FAS in the medium and low SNR regimes.
Besides, QR MIMO-FAS yields a similar or higher rate than MIMO-AS when the
number of active ports or antennas is the same.

(a)  (b)

Figure 4: The performance of the benchmarking schemes for different values of
$N_{s}$: (a) average rate; (b) outage probability, with $q=39{\rm bps/Hz}$;
(solid) without mutual coupling; (dotted) liquid-based fluid antenna with
mutual coupling; (dashed) RF pixel-based fluid antenna with mutual coupling.

Fig. 4 presents the average rates and outage probabilities of the benchmarking
schemes for different values of $N_{s}$. In these results, we omit the optimal
MIMO-FAS because it is difficult to perform exhaustive search online for large
$N_{s}$. On the other hand, since the activated ports can be placed very close
to each other, it would be useful to consider the mutual coupling effect and
investigate its effect on the performance of MIMO-FAS. In particular, we
consider two designs: liquid-based and RF pixel-based fluid antennas. To make
a fair comparison between the cases with and without mutual coupling, we
assume that $N_{1}^{s}=\left\lfloor\frac{W_{1}^{s}}{0.5}\right\rfloor+1$ and
vary $N_{2}^{s}$ accordingly. This means the resolution in one direction is
fixed while we change the resolution of FAS in another direction to examine
the impact of mutual coupling. The details of the mutual coupling model are
given in Appendix III.

As seen in Fig. 4, generally speaking, the performance of QR MIMO-FAS and
random MIMO-FAS with mutual coupling are not vastly different from the case
without mutual coupling regardless of whether liquid-based or RF pixel-based
fluid antenna is considered. This suggests that the active ports can be placed
close to each other, typically much less than half a wavelength, and still
yield a similar performance. By contrast, the rate performance of greedy MIMO-
FAS appears to suffer more from mutual coupling as $N_{s}$ increases. However,
greedy MIMO-FAS is not supposed to work well here because the setting is under
the high SNR regime.

It is worth pointing out that for RF pixel-based fluid antenna, mutual
coupling can exist regardless of whether the pixels are on or off. It is
therefore essential to improve the S-matrix via antenna design in order to
achieve a good performance. However, the advantage of RF pixel-based fluid
antenna is that the mutual coupling matrix is deterministic given $N_{1}^{s}$
and $N_{2}^{s}$ regardless of which pixels are the active ones. Thus, one can
directly obtain the optimal port selection, beamforming and power allocation
while taking into account of the mutual coupling effect. Moreover, matching
networks can be employed directly to further improve the performance of MIMO-
FAS but this technique is not considered in our results.

Based on the above observations, we hence focus on the performance of MIMO-FAS
without mutual coupling effect. In Fig. 4(a), we see that the average rates of
random MIMO-FAS and greedy MIMO-FAS generally decrease as $N_{s}$ increases.
This suggests that efficient port selection in MIMO-FAS is essential.
Furthermore, the average rate of QR MIMO-FAS is $1{\rm bps/Hz}$ higher than
that of MIMO-AS and $6{\rm bps/Hz}$ higher than that of MIMO when $N_{s}$ is
large. In Fig. 4(b), the outage probabilities of random MIMO-FAS, Greedy MIMO-
FAS and MIMO are near one (i.e., $0.99$) while the outage probability of MIMO-
AS is in the order of $10^{-2}$. In contrast, the outage probability of QR
MIMO-FAS is much lower (i.e., the order of $10^{-3}$). Nevertheless, the
outage probability of QR MIMO-FAS decreases to a floor as $N_{s}$ continues to
increase. This limitation is due to the fact that there are approximately
$N_{s}^{{}^{\prime}}$ diversity in $\boldsymbol{J}_{s}$ for a fixed $W_{s}$.
Therefore, the outage probability of QR MIMO-FAS is limited by
$N_{s}^{{}^{\prime}}$ for a fixed $W_{s}$ (see, Table II).

(a)  (b)

Figure 5: The performance of QR MIMO-FAS, MIMO and MIMO-AS for different
values of $W_{s}$: (a) average rate; (b) outage probability, with
$q=42.5$bps/Hz.

To verify this explanation, we further investigate the outage probabilities of
QR MIMO-FAS, MIMO and MIMO-AS for different values of $W_{s}$. For brevity, we
omit random MIMO-FAS and greedy MIMO-FAS as we now know that these schemes do
not provide an effective performance for large $N_{s}$. As seen in Fig. 5(a),
the average rates of QR MIMO-FAS, MIMO and MIMO-AS increase and then plateau.
Nevertheless, as shown in Fig. 5(b), the outage probabilities of QR MIMO-FAS
decrease without bound as $W_{s}$ increases if $N_{s}$ is sufficiently large.
By analyzing Table II, we can observe that although $N_{s}$ remains fixed,
$N_{s}^{{}^{\prime}}$ generally increases if $W_{s}$ is increased. In
contrast, the outage probability of MIMO is always near to being one since the
diversity gain of MIMO remains the same even if $W_{s}$ is increased. On the
other hand, the outage probability of MIMO-AS decreases at a slower rate due
to the lower resolution (i.e., the number of antennas is less than the number
of ports in the given space). Overall, Figs. 4 and 5 suggest that the value of
$N_{s}$ be determined by $W_{s}$.

From the above results, one might be amazed with the rate improvement of QR
MIMO-FAS. Nevertheless, we highlight that the superiority of QR MIMO-FAS lies
in the diversity gain. In particular, QR MIMO-FAS can reduce its outage
probability to a much lower value than MIMO and MIMO-AS if $q$ is low (e.g.,
$q<n_{\min}\log{\rm SNR}$). To examine this phenomenon more closely, Fig. 6
illustrates the outage probabilities of QR MIMO-FAS, MIMO and MIMO-AS for
different $q$. Within $6{\rm bits/Hz}$, we see that the outage probability of
QR MIMO-FAS reduces at a steeper rate (e.g., from the order of $10^{-1}$ to
$10^{-5}$) while the outage probability of MIMO-AS reduces at a slower rate
(e.g., from the order of $10^{-1}$ to $10^{-2}$) and the outage probability of
MIMO remains roughly the same.

Figure 6: The outage probabilities of QR MIMO-FAS, MIMO and MIMO-AS for
different values of $q$. Figure 7: The optimal DMT of MIMO-FAS, MIMO and MIMO-
AS.

To understand this at a more fundamental level, we present the DMT of QR MIMO-
FAS in Fig. 7. Note that the DMT of QR MIMO-FAS is also the optimal DMT of
MIMO-FAS. As shown in Table II, the value of $N_{s}^{{}^{\prime}}$ depends on
$W_{s}$ as long as $N_{s}\geq N_{s}^{{}^{\prime}}$. In Fig. 7, it can be seen
that the diversity gain of MIMO-FAS is much superior than that of an
$n_{rx}\times n_{tx}$ MIMO system for a fixed $r$. For example, the maximum
diversity of $4\times 4$ MIMO is $16$ as $r\rightarrow 0$. This is because the
optimal DMT of a traditional $n_{rx}\times n_{tx}$ MIMO system is a piecewise
linear function connecting the point
$\left(r,\left(n_{rx}-r\right)\left(n_{tx}-r\right)\right)$[24]. Meanwhile,
the maximum diversity of MIMO-AS is limited by $w_{rx}w_{tx}$. For instance,
if $W_{s}=1\lambda^{2}$, the maximum diversity of MIMO-AS is $81$. In
contrast, the maximum diversity gain of MIMO-FAS is approximately $23\times
23=529\gg\left\\{81,16\right\\}$ if $W_{rx}=W_{tx}=1\lambda^{2}$. Hence, the
diversity gain of MIMO-FAS is massive because outage only occurs when all the
ports experience deep fades. To obtain the same diversity gain at multiplexing
gain $r$ from $0$ to $N^{\prime}$, a traditional $N_{rx}^{{}^{\prime}}\times
N_{tx}^{{}^{\prime}}$ MIMO would have been required. However, the downside is
that it can only have at most $n_{\min}$ multiplexing gain.

Figure 8: The $q$-outage capacity of MIMO-FAS, MIMO and MIMO-AS.

Finally, we investigate the $q$-outage capacity to showcase the benefits of
MIMO-FAS. Fig. 8 presents the $q$-outage capacities of QR MIMO-FAS, MIMO and
MIMO-AS as well as the $q$-outage capacity gain of QR MIMO-FAS over MIMO and
MIMO-AS. For ease of exposition, the optimal $q^{*}$ of QR MIMO-FAS, MIMO and
MIMO-AS are denoted as $q_{{\rm MIMO\text{-}FAS}}^{*}$, $q_{{\rm MIMO}}^{*}$,
$q_{{\rm MIMO\text{-}AS}}^{*}$, respectively. As it is seen, the outage
capacities of the schemes increase up to $q^{*}$ and decrease thereafter. To
the left side of $q^{*}$, the capacity is limited by $q$ (i.e., rate) since
$\bar{P}_{{\rm sys}}^{{\rm out}}\left({\rm SNR},q\right)$ is small. To the
right side of $q^{*}$, the capacity is limited by the outage probability
because $\bar{P}_{{\rm sys}}^{{\rm out}}\left({\rm SNR},q\right)$ is large.
Since MIMO provides limited diversity gain and MIMO-AS has a limited number of
antennas within a given space, both schemes fail to achieve certain $q$-outage
capacity achievable by QR MIMO-FAS. This suggests that MIMO-FAS can reliably
deliver a much higher rate than traditional MIMO and MIMO-AS systems.

##  VI Conclusions

In this paper, we analyzed the performance limits of MIMO-FAS. To this end, we
developed a system model for MIMO-FAS where a 2D fluid antenna surface was
used at both ends, while taking into account of the spatial correlation of the
ports. We then proposed a suboptimal scheme to maximize the rate of MIMO-FAS
at high SNR via joint port selection, beamforming and power allocation, namely
QR MIMO-FAS. One key contribution was the derivation of the outer bound of the
DMT for MIMO-FAS. Through the outer bound and the proposed scheme, we then
obtained the optimal DMT of MIMO-FAS which revealed the fundamental limits of
MIMO-FAS. Extensive results were presented, illustrating that QR MIMO-FAS
achieved a similar rate as compared to the optimal MIMO-FAS in the high SNR
regime. By fixing other MIMO-FAS parameters, we found that the average rate
and outage probability of QR MIMO-FAS approached to a limit as $N_{s}$
increased. Likewise, the average rate of QR MIMO-FAS improved up to a certain
level as $W_{s}$ increased. Nevertheless, the outage probability of QR MIMO-
FAS decreased without bound as $W_{s}$ increased. For the same multiplexing
gain, we also showed that MIMO-FAS achieved massive diversity gain as compared
to the traditional MIMO and MIMO-AS systems. Motivated by this, we further
illustrated that MIMO-FAS could reliably deliver a much higher rate than the
traditional MIMO and MIMO-AS systems in terms of $q$-outage capacity that
jointly considered both rate and outage probability.

## Appendix I: Spatial Correlation of 2D Fluid Antenna Surface over 3D
Scattering Environment

Without loss of generality, let us refer to the position of the
($n_{1},n_{2}$)-port as
$\boldsymbol{n}_{l_{s}}=\left[0,\frac{n_{2}^{s}-1}{N_{2}^{s}-1}W_{2}^{s},\frac{n_{1}^{s}-1}{N_{1}^{s}-1}W_{1}^{s}\right]{}^{T}$
where ${\rm{map}}\left(n_{1},n_{2}\right)=l_{s}$ and $\lambda$ is the
wavelength. Suppose a plane wave impinges on the fluid antenna surface from
azimuth angle $\varphi$ and elevation angle $\theta$. Then, the array response
vector can be expressed as [43]

$\boldsymbol{a}\left(\varphi,\theta\right)=\left[e^{j\frac{2\pi}{\lambda}\boldsymbol{k}\left(\varphi,\theta\right)^{T}\boldsymbol{n}_{1}\lambda},\dots,e^{j\frac{2\pi}{\lambda}\boldsymbol{k}\left(\varphi,\theta\right)^{T}\boldsymbol{n}_{N_{s}}\lambda}\right]^{T},$
(48)

where

$\boldsymbol{k}\left(\varphi,\theta\right)=\left[\begin{array}[]{ccc}\cos\left(\theta\right)\cos\left(\varphi\right)&\cos\left(\theta\right)\sin\left(\varphi\right)&\sin\left(\theta\right)\end{array}\right]^{T},$
(49)

is the normalized wave vector.

Let us denote the spatial correlation matrix as
$\boldsymbol{J}_{s}=\mathbb{{E}}\left\\{\boldsymbol{a}\left(\varphi,\theta\right)\boldsymbol{a}\left(\varphi,\theta\right)^{H}\right\\}$.
From (48), we know that the $\left(k_{s},l_{s}\right)$-th entry of
$\boldsymbol{J}_{s}$ can be expressed as

$\displaystyle\left[\boldsymbol{J}_{s}\right]_{k_{s},l_{s}}$
$\displaystyle=\mathbb{{E}}\left\\{e^{j\frac{2\pi}{\lambda}\boldsymbol{k}\left(\varphi,\theta\right)^{T}\boldsymbol{n}_{k_{s}}\lambda}e^{-j\frac{2\pi}{\lambda}\boldsymbol{k}\left(\varphi,\theta\right)^{T}\boldsymbol{n}_{l_{s}}\lambda}\right\\}$
$\displaystyle=\mathbb{{E}}\left\\{e^{j2\pi\boldsymbol{k}\left(\varphi,\theta\right)^{T}\left(\boldsymbol{n}_{k_{s}}-\boldsymbol{n}_{l_{s}}\right)}\right\\},$
(50)

where ${\rm{map}}\left(\tilde{n}_{1}^{s},\tilde{n}_{2}^{s}\right)=k_{s}$.
Using the Jacobi-Anger plane wave expansion [44], (50) can be rewritten as

$\left[\boldsymbol{J}_{s}\right]_{k_{s},l_{s}}=4\pi\sum_{m=0}^{\infty}\sum_{n=-m}^{m}\left(-i\right)^{m}\times\\\
\alpha_{m}^{n}Y_{m}^{n}\left(\frac{\boldsymbol{n}_{k_{s}}-\boldsymbol{n}_{l_{s}}}{\left\|\boldsymbol{n}_{k_{s}}-\boldsymbol{n}_{l_{s}}\right\|}\right)\times\\\
j_{m}\left(2\pi\left\|\boldsymbol{n}_{k_{s}}-\boldsymbol{n}_{l_{s}}\right\|\right),$
(51)

where $j_{m}\left(.\right)$ is the spherical Bessel function of the first
kind, $Y_{m}^{n}\left(\cdot\right)$ is the spherical harmonics, and

$\displaystyle\alpha_{m}^{n}$
$\displaystyle=\int_{\Omega}f_{\boldsymbol{k}\left(\varphi,\theta\right)}\left(\boldsymbol{k}\right)Y_{m}^{n}\left(\boldsymbol{k}\right)d\Omega\left(\boldsymbol{k}\right),$
(52)

where $\Omega\left(\boldsymbol{k}\right)$ is a surface element of a unit
sphere $\Omega$. For a 3D isotropic scattering environment, we have
$f_{\boldsymbol{k}\left(\varphi,\theta\right)}\left(\boldsymbol{k}\right)=\frac{1}{4\pi}$
and thus (52) reduces to

$\alpha_{m}^{n}=\frac{1}{4\pi}\int_{\Omega}Y_{m}^{n}\left(\boldsymbol{k}\right)d\Omega\left(\boldsymbol{k}\right).$
(53)

Using the fact that

$Y_{m}^{n}\left(\frac{\boldsymbol{n}_{k_{s}}-\boldsymbol{n}_{l_{s}}}{\left\|\boldsymbol{n}_{k_{s}}-\boldsymbol{n}_{l_{s}}\right\|}\right)=\frac{1}{\sqrt{4\pi}}\quad{\rm{if}}\leavevmode\nobreak\
n=m=0,$ (54)

and

$\int_{\Omega}Y_{m}^{n}\left(\tilde{\boldsymbol{k}}\right)d\Omega\left(\tilde{\boldsymbol{k}}\right)=\begin{cases}\sqrt{4\pi}&{\rm{if}}\leavevmode\nobreak\
n=m=0,\\\ 0&{\rm{otherwise}},\end{cases}$ (55)

(51) can be rewritten as

$\left[\boldsymbol{J}_{s}\right]_{k_{s},l_{s}}=j_{0}\left(2\pi\left\|\boldsymbol{n}_{k_{s}}-\boldsymbol{n}_{l_{s}}\right\|\right),$
(56)

which gives (1). Note that (1) conforms with [45, 46, 47] since
$j_{0}\left(c\right)=\frac{\sin c}{c}.$ In addition, (1) can be reduced to a
1D fluid antenna with 2D scattering environment by setting $N_{1}^{s}=1$ and
$\frac{0}{0}\triangleq 0$ and replacing $j_{0}\left(\cdot\right)$ by
$J_{0}\left(\cdot\right)$ where $J_{0}\left(\cdot\right)$ is the Bessel
function of the first kind [9].

## Appendix II: Generalization to Other Spatial Correlation Models

Without loss of generality, let us consider a 1D fluid antenna since a similar
argument can be made for a 2D fluid antenna surface. To begin with, let us
denote $\boldsymbol{J}$ as the $N\times N$ spatial correlation matrix. Suppose
that $W\gg 0$, and the spatial correlation between the $k$-th port and the
$l$-th port is $J_{k,l}=f\left(k,l,N\right)$ where the spatial correlation
function $f$ satisfies two conditions: i)
$\lim_{N\rightarrow\infty}f\left(k,k\pm 1,N\right)=1$ and ii) there are some
$\exists l\neq k$ such that $f\left(k,l,N\right)\neq 1$. The first condition
implies that the $(k\pm 1)$-th row/column of $\boldsymbol{J}$ can be removed
from $\boldsymbol{J}$ since the $(k\pm 1)$-th row/column of $\boldsymbol{J}$
is always identical to the $k$-th row/column of $\boldsymbol{J}$ in the limit.
The second condition implies that there exist some $l$-th port whose $l$-th
row/column must be retained in $\boldsymbol{J}$ since its spatial correlation
is completely distinct from the $k$-th port. If $f$ satisfies these two
conditions, then there must exist a minimal spacing $c$ between the $k$-th and
$l$-th ports such that their spatial correlation is distinct. Using the fact
that $W$ is finite, it is clear that there are at most $\bar{N}$ non-identical
ports since conditions (i) and (ii) hold. In particular, $\bar{N}$ must be
finite because $W\geq\bar{N}c>0$. As a result, $\boldsymbol{J}$ can be
rewritten as a symmetric $\bar{N}\times\bar{N}$ finite size matrix. Let us
denote $N^{{}^{\prime}}$ as the full rank of the symmetric
$\bar{N}\times\bar{N}$ matrix where $N^{{}^{\prime}}\leq\bar{N}$. Then, we can
further reduce the symmetric $\bar{N}\times\bar{N}$ matrix to a full rank
symmetric $N^{{}^{\prime}}\times N^{{}^{\prime}}$ submatrix
$\boldsymbol{J}_{{\rm red}}$ by removing the $(\bar{N}-N^{{}^{\prime}})$
dependent rows and columns. Consequently, $\boldsymbol{J}$ can be represented
by $\boldsymbol{J}_{{\rm red}}$ which is a full rank symmetric
$N^{{}^{\prime}}\times N^{{}^{\prime}}$ finite-size matrix.

## Appendix III: The Effect of Mutual Coupling

In liquid-based fluid antenna, mutual coupling only occurs between the active
ports. Thus, the MIMO-FAS channel with mutual coupling effect can be modeled
as [48]

$\boldsymbol{\bar{H}}_{mc}=\boldsymbol{Z}_{mc}^{rx,l}\boldsymbol{\bar{H}}\boldsymbol{Z}_{mc}^{tx,l},$
(57)

where $\boldsymbol{Z}_{mc}^{rx,l}$ and $\boldsymbol{Z}_{mc}^{tx,l}$ are the
mutual coupling matrices which can be pre-computed offline given the antenna
technologies used. The mutual coupling matrix is given as

$\boldsymbol{Z}_{mc}^{s,l}=\left(Z_{A}+Z_{L}\right)\left(\boldsymbol{Z}_{s}^{l}+Z_{L}\boldsymbol{I}\right)^{-1},s\in\left\\{rx,tx\right\\},$
(58)

where $Z_{A}$, $Z_{L}$ and $\boldsymbol{Z}_{s}^{l}$ are the antenna impedance,
load impedance and mutual impedance matrix of the active ports, respectively.
To compute $\boldsymbol{Z}_{s}^{l}$, we assume that each active port is a
dipole element with a length of $0.5\lambda$ and a width of $0.001\lambda$.
Consequently, $Z_{A}=73.08+42.21j$ and $Z_{L}=Z_{A}^{*}$. Due to the dipole’s
physical constraint, we fix
$N_{1}^{s}=\left\lfloor\frac{W_{1}^{s}}{0.5}\right\rfloor+1$ and vary
$N_{2}^{s}$ accordingly. If all the active ports are far from each other, we
have $\boldsymbol{Z}_{mc}^{s}\approx\boldsymbol{I}$. Given (57), SVD and
waterfilling power allocation are then performed. It is worth highlighting
that this model is a conservative one because the mutual coupling effect is
not considered when optimizing $A_{tx}$ and $A_{rx}$. In fact, the performance
of MIMO-FAS can be further improved when considering the mutual coupling
effect and allowing the optimal active ports to be freely located within the
given surface.

In RF pixel-based fluid antenna, mutual coupling can exist regardless of
whether the pixels are on or off. Thus, in practice, it is important to
improve the S-matrix via antenna design. To a coarse approximation, the
S-matrix is modeled as

$\boldsymbol{S}_{mc}^{s}=\left[\begin{array}[]{cccc}\alpha_{\text{rl}}S_{1,1}^{s}&\alpha_{\text{iso}}S_{1,2}^{s}&\cdots&\alpha_{\text{iso}}S_{1,N_{s}}^{s}\\\
\alpha_{\text{iso}}S_{2,1}^{s}&\alpha_{\text{rl}}S_{2,2}^{s}&&\vdots\\\
\vdots&&\ddots\\\
\alpha_{\text{iso}}S_{N_{s},1}^{s}&\cdots&&\alpha_{\text{rl}}S_{N_{s},N_{s}}^{s}\end{array}\right],$
(59)

where $\alpha_{\text{rl}}$ and $\alpha_{\text{iso}}$ determine the improvement
level of the return loss and isolation, respectively, while
$S_{k_{s},l_{s}}^{s}$ is the S-parameter between the $k_{s}$-th and $l_{s}$-th
ports. In this paper, we assume that the return loss and isolation levels are
$-15$dB and $30$dB, respectively, which are typical values that can be
achieved using state-of-the-art technologies [49]. Given (59), the mutual
impedance matrix of the RF pixel-based fluid antenna can be computed as

$\boldsymbol{Z}_{s}^{p}=z_{0}\left(\boldsymbol{I}+\boldsymbol{S}_{mc}^{s}\right)\left(\boldsymbol{I}-\boldsymbol{S}_{mc}^{s}\right)^{-1},$
(60)

where $z_{0}=50\Omega$ is the reference impedance. Similar to (58), the mutual
coupling matrix is given as

$\boldsymbol{Z}_{mc}^{s,p}=\left(Z_{A}+Z_{L}\right)\left(\boldsymbol{Z}_{s}^{p}+Z_{L}\boldsymbol{I}\right)^{-1},s\in\left\\{rx,tx\right\\}.$
(61)

Similar to (57), the MIMO-FAS channel with mutual coupling effect is modeled
as

$\boldsymbol{H}_{mc}=\boldsymbol{Z}_{mc}^{rx,p}\boldsymbol{H}\boldsymbol{Z}_{mc}^{tx,p}.$
(62)

In contrast to (57) and (58), it is worth noting that (61) and (62) are
$N_{s}\times N_{s}$ matrices. Given (62), the port selections, beamforming and
power allocation can be performed by different schemes. Note that matching
networks can be employed to further improve the performance of MIMO-FAS [50].

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