# Bandwidth-Constrained Capacity Bounds
and Degrees of Freedom for MIMO Antennas

Casimir Ehrenborg, , Mats Gustafsson, ,
and Miloslav Capek Manuscript received . This work was supported by the
Swedish foundation for strategic research (SSF) under the program applied
mathematics and the project Complex analysis and convex optimization for
electromagnetic design and by the Czech Science Foundation under Project
19-06049S.C. Ehrenborg is with the Department of Electrical Engineering, KU
Leuven, Kasteelpark Arenberg 10 postbus 2440, 3001 Leuven, Belgium. (E-mail:
casimir.ehrenborg@kuleuven.be).M. Gustafsson is with the Department of
Electrical and Information Technology, Lund University, Box 118, SE-221 00
Lund, Sweden. (Email: mats.gustafsson@eit.lth.se).M. Capek is with the
Department of Electromagnetic Field, Faculty of Electrical Engineering, Czech
Technical University in Prague, 166 27 Prague, Czech Republic (E-mail:
miloslav.capek@fel.cvut.cz).

###### Abstract

The optimal spectral efficiency and number of independent channels for MIMO
antennas in isotropic multipath channels are investigated when bandwidth
requirements are placed on the antenna. By posing the problem as a convex
optimization problem restricted by the port Q-factor a semi-analytical
expression is formed for its solution. The antennas are simulated by method of
moments and the solution is formulated both for structures fed by discrete
ports, as well as for design regions characterized by an equivalent current.
It is shown that the solution is solely dependent on the eigenvalues of the
so-called energy modes of the antenna. The magnitude of these eigenvalues is
analyzed for a linear dipole array as well as a plate with embedded antenna
regions. The energy modes are also compared to the characteristic modes to
validate characteristic modes as a design strategy for MIMO antennas. The
antenna performance is illustrated through spectral efficiency over the
Q-factor, a quantity that is connected to the capacity. It is proposed that
the number of energy modes below a given Q-factor can be used to estimate the
degrees of freedom for that Q-factor.

###### Index Terms:

MIMO communication, degrees of freedom, computational electromagnetics,
optimization, eigenvalues and eigenfunctions, antenna theory.

##  I Introduction

Modern communication systems often make use of MIMO (MIMO) antennas. They
utilize spatial multiplexing to dramatically increase the transmitted bit-
rate, or capacity, in comparison to single antenna systems [4, 5]. The optimal
performance of MIMO systems is therefore a topic of great interest. Many
different methods have been developed to evaluate maximum capacity under
different conditions and assumptions. For example by considering spherical
regions [6, 7], or treating the problem through an information theoretical
approach [8, 9, 10, 11, 12]. In [13, 14] the authors presented a novel method
for calculating optimal spectral efficiency of arbitrary antenna designs using
convex current optimization.

Design of MIMO antennas is based on effectively exciting discrete
communication channels with low correlation [4]. A proposed strategy for
accomplishing this is to design the antennas such that they effectively excite
modes with orthogonal radiation patterns, such as characteristic modes [15,
16, 17, 18]. A modal analysis method for analyzing MIMO antennas design,
constrained by radiation efficiency, was presented in [14]. This method is
based on analyzing the eigenvalues of a set of modes known as radiation modes
to predict the performance of a MIMO antenna. However, one of the open
questions of this technique is if it can be generalized to other constraining
parameters, such as bandwidth.

Electrically small antennas suffer from a degradation in possible performance
as their size is reduced compared to the wavelength. Some of the parameters
where this is most evident are radiation efficiency, directivity, and
bandwidth [19, 20, 21, 22]. Bandwidth can be estimated, for electrically small
systems, through the Q-factor defined as the quotient of energy stored in a
system over power dissipated by it [23]. This relation accurately estimates
the bandwidth, for single feed, single resonance systems [1], however, it does
not hold for multi-port systems. In this paper, this is addressed by
considering a multi-port system as a superposition of single port systems.
Each port has a well defined Q-factor calculated from the stored energy of the
current induced by that port [24]. These port currents make up the total
current distribution on the antenna when added together. The Q-factor of the
total current is therefore an implicit indication of the Q-factor in each
port. In this way, restricting the Q-factor of the total current density
implicitly imposes a bandwidth requirement on the system.

In this paper optimal spectral efficiency is calculated using the method from
[14] for MIMO antennas transmitting in a uniform multipath environment
restricted by a required Q-factor. The optimization problem is formulated in
the sources of the antenna design problem, either the feeding voltage of a
fixed antenna array, or the current distribution in a design region. These two
cases are studied to gain different insights into how to maximize spectral
efficiency.

The antenna array is optimized by recasting the optimization problem in the
port voltages. This is used to study the placement of canonical dipole
antennas to gain an intuition for maximizing spectral efficiency. An advantage
of this approach is the feasibility of the bound via direct realization of the
port voltage excitation proposed by the optimization routine. The current
optimization problem is solved for a few sub-regions embedded in a plate. The
performance and placement of these sub-regions are compared to optimizing the
current over the entire plate.

The performance is measured in a new quantity, spectral efficiency over
Q-factor, that is related to the capacity of the MIMO system by multiplication
with the center frequency. This quantity illustrates that the Pareto type
bound between the spectral efficiency and Q-factor has an optimal point.

The modal analysis method in [14] is generalized to MIMO antennas restricted
by the Q-factor by considering a different set of modes called energy modes
[3]. It is shown that the energy modes can predict the onset of the solution,
as well as the number of degrees of freedom utilized in the system. The energy
modes are compared to the characteristic modes and it is shown that they have
similar performance, validating previous design techniques. It is also
demonstrated, through simulation, that a uniform Rayleigh channel does not
significantly alter the behavior of the optimal spectral efficiency problem.

The paper is organized in the following way. In Section 1 the full-wave MIMO
system is introduced and the effect of the channel on systems with few ports
is discussed. The problem is stated and solved for port quantities in Section
III and for current density in Section IV. The paper is concluded in Section
V.

##  II Full-wave Model of MIMO Systems

Figure 1: Schematic illustration of a MIMO system, with a channel matrix
${\mathbf{H}}$ modeling wave propagation from transmitter Tx to receiver Rx.

A MIMO system, like the one shown in Fig. 1, is described by

${\mathbf{y}}={\mathbf{H}}{\mathbf{x}}+{\mathbf{n}},$ (1)

where ${\mathbf{y}}$ is an $M\times 1$ vector containing the received signals,
${\mathbf{H}}$ is an $M\times N$ matrix known as the channel matrix,
${\mathbf{x}}$ is an $N\times 1$ vector containing the input signals, and
${\mathbf{n}}$ is an $M\times 1$ vector containing the noise perturbing each
of the receivers [4]. The channel matrix describes the structure of the
antenna and wave propagation between the ports of the transmitter and the
ports of the receiver. There are several ways to model a MIMO system depending
on how we wish to describe the input and output signals. For example, we may
consider the input signals to be the voltage sources in the antenna feeds or
the wave forms sent through the transmission lines connected to the antenna
ports. In this section of the paper we model the MIMO system using a
scattering matrix, and consider the input and output signals in terms of
spherical mode coefficients [30, 6, 7].

An antenna system’s radiating and receiving properties can be described by the
scattering matrix [30],

$\begin{pmatrix}{\mathbf{S}}_{\mathrm{1,1}}&{\mathbf{S}}_{\mathrm{1,2}}\\\
{\mathbf{S}}_{\mathrm{2,1}}&{\mathbf{S}}_{\mathrm{2,2}}\end{pmatrix}\begin{pmatrix}{\boldsymbol{v}}\\\
\boldsymbol{\alpha}\end{pmatrix}=\begin{pmatrix}{\boldsymbol{w}}\\\
\boldsymbol{\beta}\end{pmatrix},$ (2)

where ${\boldsymbol{v}}$ are the signals incoming to the ports of the antenna,
${\boldsymbol{w}}$ are the outgoing signals from the antenna,
$\boldsymbol{\alpha}$ are the incoming spherical waves, $\boldsymbol{\beta}$
are the outgoing spherical waves, ${\mathbf{S}}_{\mathrm{1,1}}$ is the
reflection matrix with dimension $N\times N$, ${\mathbf{S}}_{\mathrm{2,2}}$ is
the matrix representing how the antenna couples incoming and outgoing
spherical waves with dimension $\infty\times\infty$,
${\mathbf{S}}_{\mathrm{2,1}}$ is the $\infty\times N$ matrix coupling the
input signals to the antennas to the radiated spherical waves, and
${\mathbf{S}}_{\mathrm{1,2}}$ is the $N\times\infty$ matrix coupling the
incoming waves to the signals in the ports. The subscripts are used to denote
that these matrices are parts of the scattering matrix. Here, we assume that
our antenna is made of linear, passive, reciprocal materials, such as metals
and dielectrics, making the antenna a reciprocal system. This means that the
antenna behaves equivalently when receiving or transmitting [30].

There exists many different channel models to describe specific propagation
environments [4, 5, 31]. However, in this paper we aim to consider the most
general scenario to provide a basis from which to construct general
performance bounds for MIMO antennas. Many MIMO devices, e.g., handsets or
small electronics, can be considered to operate in uniform multipath
environments. These environments are characterized by a large number of
independent waves impinging on the antenna from all directions, see Fig. 2.
These waves impinging on the sphere surrounding the antenna system are modeled
by a Rayleigh channel in the spherical modes [32, 7]. The Rayleigh channel
${\mathbf{H}}_{\mathrm{w}}$ is modeled by a random matrix with a complex
Gaussian distribution that is uncorrelated and has zero mean [4, 33, 34].
Because the considered antenna is a reciprocal system it behaves equivalently
when receiving or transceiving. Consequently, we model the transceiving
channel as a Rayleigh channel as well.

The MIMO system in Fig. 1 is therefore simplified to a MIMO antenna in a
spherical Rayleigh channel as in Fig. 2.

Figure 2: Schematic illustration of a MIMO antenna modeled using a scattering
matrix (2). The incoming and outgoing signals are depicted for the
transceiving antenna as ${\boldsymbol{v}}$ and ${\boldsymbol{w}}$, and for the
spherical channel as $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$,
respectively.

This system is governed by the equation

$\boldsymbol{\beta}={\mathbf{H}}_{\mathrm{w}}{\mathbf{S}}_{2,1}{\boldsymbol{v}},$
(3)

when operating in transceiving mode. The two parts of the channel model are
here referred to as: the uniform Rayleigh fading channel,
${\mathbf{H}}_{\mathrm{w}}$, and the deterministic isotropic channel
${\mathbf{S}}_{2,1}$. The instantaneous spectral efficiency of this system can
be calculated by optimizing the time average over the modulation in the input
signals, i.e.,
${\mathbf{A}}=\mathcal{E}\left\\{{\boldsymbol{v}}{\boldsymbol{v}}^{\text{H}}\right\\}/2$,
where $\mathcal{E}\left\\{\cdot\right\\}$ denotes the temporal average. The
matrix ${\mathbf{A}}$ is called the covariance matrix of the input signals.
The optimization problem for instantaneous spectral efficiency is [4]

$\displaystyle\underset{{\mathbf{A}}\succeq{\mathbf{0}}}{\mathrm{maximize}}$
$\displaystyle
C=\log_{2}\det({\mathbf{1}}+\gamma{\mathbf{H}}_{\mathrm{w}}{\mathbf{S}}_{2,1}{\mathbf{A}}{\mathbf{S}}_{2,1}^{\text{H}}{\mathbf{H}}_{\mathrm{w}}^{\text{H}})$
(4) $\displaystyle\mathrm{subject\ to}$
$\displaystyle\mathop{\mathrm{tr}}\nolimits({\mathbf{A}})=1,$

where ${\mathbf{1}}$ is the identity matrix, input power is normalized to $1$,
and $\gamma$ is the SNR (SNR). In many applications the channel
${\mathbf{H}}_{\mathrm{w}}$ changes over time. It is then necessary to
consider the ergodic spectral efficiency, i.e., the spectral efficiency
averaged over channel realizations, $\bar{C}=\mathcal{E}\left\\{C\right\\}$.

It was shown in [7] that the spectral efficiency for systems with high SNR can
be decomposed into a deterministic part describing the antenna and a random
part independent of the antenna, see Appendix A. The physics of maximizing
spectral efficiency are therefore independent of perturbing uniform channels,
such as the Rayleigh channel. This is verified numerically in Section III-A.
We will therefore use the matrix ${\mathbf{S}}_{2,1}$ as the channel matrix
for our calculations (${\mathbf{H}}_{\mathrm{w}}={\mathbf{1}}$).

Modelling a MIMO system using a scattering matrix is useful when studying
channel phenomena. However, in this paper, we focus on how the design of a
MIMO antenna impacts optimal spectral efficiency. Consider the region in Fig.
3a, we want to analyze how to construct the most effective MIMO antenna within
this region. We must therefore switch our modelling from scattering parameters
to quantities that let us incorporate antenna design as a variable.

The ${\mathbf{S}}_{1,1}$ part of the scattering matrix relation (2) can be
rewritten in terms of port quantities

${\mathbf{z}}{\mathbf{i}}={\mathbf{v}},$ (5)

where ${\mathbf{v}}$ are the port voltages, ${\mathbf{i}}$ are the port
currents, and ${\mathbf{z}}$ is the network impedance matrix [35, 36]. Here,
we have moved our perspective to Fig. 3b. In this case, the channel matrix
${\mathbf{S}}_{2,1}$ is written as the connection between the port voltages
and the outgoing spherical waves. We will denote this matrix ${\mathbf{s}}$
for simplicity, defined in Section III.

The network impedance matrix can either be calculated from the (measured)
scattering matrix or from a full-wave EM model. This is useful when studying a
fixed set of antennas, as is done in Section III. We can further rewrite (5)
in terms of meshed structures in order to use full-wave electromagnetic
simulation techniques. In this paper we use MoM (MoM) to simulate our systems,
rewriting (5) as,

${\mathbf{Z}}{\mathbf{I}}={\mathbf{V}},$ (6)

where ${\mathbf{I}}$ is the current column matrix, ${\mathbf{V}}$ is the
excitation column matrix, and ${\mathbf{Z}}$ is the impedance matrix [37], see
Appendix B. This equation models the case in Fig. 3c, where the current can
exist everywhere inside the design region. The current can be split into
regions that are controlled or passively excited by the controlled region
[29]. Similarly to the port case, the channel matrix ${\mathbf{S}}_{2,1}$ is
in this case written as the connection between the currents on the antenna to
the modes in the far field and will be called matrix ${\mathbf{S}}$ throughout
the paper. This formulation is very useful when optimizing the current to
study all possible antenna structures in a region, which is done in Section
IV.

Figure 3: Possible full-wave models of a transmitter Tx from Fig. 2. (a) An
antenna design region to be occupied by a MIMO system. (b) Two dipoles of
fixed geometry are considered within the design region. The feeding voltages,
confined to the blue regions, are controllable variables to be optimized for
maximum capacity. (c) The current density, confined to the green region, is
the controllable quantity. An optimized current is considered to be impressed
in vacuum, the uncontrollable region is made from a given, yet arbitrary,
material.

##  III Array Antenna Excitation

In this section we optimize the excitation of a fixed antenna geometry, see
Fig. 3b, when the antenna is placed in a deterministic isotropic channel. The
maximum capacity given by a set of feeding ports is studied when the antenna’s
bandwidth for each port is prescribed and represented by the Q-factor. These
fixed geometries are used to convey an intuition of how to induce optimal
spectral efficiency.

The current density on an antenna structure is modeled as a linear combination
of currents excited by localized feeders, so-called port modes [38], see Fig.
3b,

${\mathbf{I}}={\mathbf{Z}}^{-1}{\mathbf{C}}{\mathbf{v}}=\sum_{p}v_{p}{\mathbf{I}}_{p},$
(7)

where each port mode ${\mathbf{I}}_{p}$ corresponds to a unitary excitation of
the $p$-th port, ${\mathbf{Z}}$ is the impedance matrix introduced in (6), and
${\mathbf{C}}\in\left\\{0,1\right\\}^{M\times P}$ is an indexation matrix
defined element-wise as

$C_{mp}=\left\\{{\begin{array}[]{*{20}{cl}}1&\textrm{$p$-th port is placed at
$m$-th position,}\\\ 0&\textrm{otherwise}.\end{array}}\right.$ (8)

A remarkable advantage of the port-mode representation is the reduction of the
original $M\times M$ matrices, i.e., of number of degrees of freedom, to
$P\times P$, the number of ports. The construction of the impedance matrix (6)
prior the reduction into port modes (7) is needed since it represents the
full-wave electromagnetic behavior of the systems. The values of physical
quantities defined in MoM-like (6) and port-like (5) representations are
equivalent. To illustrate the transformation let
${\mathbf{M}}={\mathbf{M}}^{\text{H}}\in\mathbb{C}^{M\times M}$ represent a
generic MoM operator, such as the reactance matrix ${\mathbf{X}}$. The
following quadratic forms map this operator to its port equivalent,
${\mathbf{m}}\in\mathbb{C}^{P\times P}$,

$m=\dfrac{1}{2}{\mathbf{I}}^{\text{H}}{\mathbf{M}}{\mathbf{I}}=\dfrac{1}{2}{\mathbf{v}}^{\text{H}}{\mathbf{m}}{\mathbf{v}},$
(9)

where

${\mathbf{m}}={\mathbf{C}}^{\text{H}}{\mathbf{Z}}^{-\mathrm{H}}{\mathbf{M}}{\mathbf{Z}}^{-1}{\mathbf{C}}.$
(10)

Notice that the port-mode representation (10) changes the physical units from
Ohms for operator ${\mathbf{M}}$ to Siemens for port operator ${\mathbf{m}}$.

Spectral efficiency is calculated through the covariance of the input signals.
The quadratic forms that typically calculate antenna quantities can be
rewritten in terms of the covariance of the port voltages. Conveniently, the
port voltages is the only part of the antenna expressions that vary in time,
therefore the temporal average can be confined to this variable. For example
the input power can be written as,

$\displaystyle\frac{1}{2}\mathcal{E}\left\\{{\mathbf{v}}^{\text{H}}{\mathbf{r}}{\mathbf{v}}\right\\}$
$\displaystyle=\frac{1}{2}\mathop{\mathrm{tr}}\nolimits\left(\mathcal{E}\left\\{{\mathbf{v}}^{\text{H}}{\mathbf{r}}{\mathbf{v}}\right\\}\right)$
(11)
$\displaystyle=\frac{1}{2}\mathop{\mathrm{tr}}\nolimits\left(\mathcal{E}\left\\{{\mathbf{r}}{\mathbf{v}}{\mathbf{v}}^{\text{H}}\right\\}\right)=\mathop{\mathrm{tr}}\nolimits({\mathbf{r}}{\mathbf{a}})=P_{\mathrm{r}},$

where
${\mathbf{a}}=\mathcal{E}\left\\{{\mathbf{v}}{\mathbf{v}}^{\text{H}}\right\\}/2$
is the covariance of the port voltages, $P_{\mathrm{r}}$ is the radiated
power, and the cyclic properties of the trace have been utilized. We assume
the materials used in the antenna are lossless, therefore the radiated and
input power is the same. This enables the real part of the impedance matrix,
${\mathbf{z}}={\mathbf{r}}+\mathrm{j}{\mathbf{x}}$, to be expanded in the
spherical modes as ${\mathbf{r}}={\mathbf{s}}^{\text{H}}{\mathbf{s}}$. The
matrix ${\mathbf{s}}$ is the same matrix as the deterministic isotropic
channel matrix, which can be calculated through (10),
${\mathbf{s}}={\mathbf{S}}{\mathbf{Z}}^{-1}{\mathbf{C}}$, see Appendix B for
the definition of matrix ${\mathbf{S}}$ [39, 6].

Applying the change of basis (9) and writing the problem in trace formulation,
the spectral efficiency calculation (4) can be written in the port-mode basis
as

$\displaystyle\underset{{\mathbf{a}}}{\mathrm{maximize}}$
$\displaystyle\log_{2}\det\left({\mathbf{1}}+\gamma{\mathbf{s}}{\mathbf{a}}{\mathbf{s}}^{\text{H}}\right)$
(12) $\displaystyle\mathrm{subject\ to}$
$\displaystyle\mathop{\mathrm{tr}}\nolimits({\mathbf{r}}{\mathbf{a}})=1.$

Here, the total radiated power is normalized to unity,
$P_{\mathrm{r}}=1\mathrm{\,W}$. This problem is only restricted by the input
power to each port.

Antennas are most often optimized given certain restrictions to their
performance quantities, the formulation of (12) enables us to add such
restrictions as additional constraints. In this paper we investigate how
bandwidth restrictions affect spectral efficiency optimization.

The Q-factor is an antenna quantity that estimates the fractional bandwidth
[1, 23], see its definition in Appendix C. The connection between the Q-factor
and the bandwidth is explicit for single input systems in free space but there
exists no explicit quantity or expression for calculating the bandwidth for
MIMO systems. However, each of the ports of a MIMO system is a single input
system, if taken in isolation. Therefore, each of these ports have a well
defined Q-factor based on their inputs. We can require a certain Q-factor of
each of the ports in the system as a way of implicitly placing a requirement
on the total systems bandwidth. However, the Q-factor is usually defined in
relation to the system at resonance. For a multi-port system a single
resonance of the total current is not always desired, thus the Q-factor of the
system must be estimated in a different way. Here, we use the average of the
magnetic and electric stored energies over the radiated power to estimate the
bandwidth. This corresponds to adding the condition,

$\dfrac{\mathcal{E}\left\\{{\mathbf{v}}^{\text{H}}{\mathbf{w}}_{\mathrm{x}}{\mathbf{v}}\right\\}}{2P_{\mathrm{r}}}=\frac{\mathop{\mathrm{tr}}\nolimits({\mathbf{w}}_{\mathrm{x}}{\mathbf{a}})}{P_{\mathrm{r}}}\leq
Q,$ (13)

where ${\mathbf{w}}_{\mathrm{x}}$ is the matrix giving the reactive power of
the averaged stored energy, see Appendix C, and $Q$ is the required Q-factor.

The Q-factor constraint (13) is added to (12) to create the optimization
problem that is investigated in this paper,

$\displaystyle\underset{{\mathbf{a}}}{\mathrm{maximize}}$
$\displaystyle\log_{2}\det\left({\mathbf{1}}+\gamma{\mathbf{s}}{\mathbf{a}}{\mathbf{s}}^{\text{H}}\right)$
(14) $\displaystyle\mathrm{subject\ to}$
$\displaystyle\mathop{\mathrm{tr}}\nolimits({\mathbf{r}}{\mathbf{a}})=1,$
$\displaystyle\mathop{\mathrm{tr}}\nolimits({\mathbf{w}}_{\mathrm{x}}{\mathbf{a}})\leq
Q,$ $\displaystyle{\mathbf{a}}\succeq 0.$

A problem of this form can be solved using commercially available software,
such as CVX (CVX) [40], see [13]. However, it can be solved much more
efficiently, and in such a way as to provide greater physical insight, using
the method presented in [14]. The method is based on constructing a convex
dual problem that can be solved semi-analytically. A brief outline of that
procedure is covered here.

Dual problems are constructed from an original optimization problem by forming
linear combinations between the constraints of the original problem. The
solution to the dual problem will therefore always have a value that bounds
the solution of the original problem (14). However, there may exist a duality
gap between the two solutions leading to a “loose” bound on the optimization
problem [25]. The results presented in this paper have been verified
numerically in CVX and no duality gap was observed.

To formulate the dual of the optimization problem (14) we take an affine
combination of the conditions restricting the original problem,

$\displaystyle\min\limits_{\nu}\,\max\limits_{{\mathbf{a}}}$
$\displaystyle\log_{2}\det\left({\mathbf{1}}+\gamma{\mathbf{s}}{\mathbf{a}}{\mathbf{s}}^{\text{H}}\right)$
(15) $\displaystyle\mathrm{subject\ to}$
$\displaystyle\mathop{\mathrm{tr}}\nolimits\left[\left(\nu{\mathbf{r}}+Q^{-1}{\mathbf{w}}_{\mathrm{x}}\right){\mathbf{a}}\right]=(\nu+1),$
$\displaystyle{\mathbf{a}}\succeq 0.$

This problem is convex in the variable $\nu$. The tightest bound on the
solution of the original problem is found when the dual problem is minimized
over $\nu$ [25]. The solution to this problem is found in [14] by
incorporating the matrices of the first condition in (15) into the channel
matrix of the system, see Appendix D. This recasts the system in a form where
it can be solved by water-filling [4, 5].

The water-filling procedure is carried out by taking the singular value
decomposition of the channel matrix. Each singular value represents the loss
associated with feeding power in its corresponding mode. The optimal solution
is found by iteratively allocating energy to the channels associated with the
largest singular values [4, 5, 14]. The algorithm can be expressed as,

$\displaystyle\mathrm{maximize}$
$\displaystyle\sum_{n=1}^{N}\log_{2}\left(1+a_{n}\gamma\sigma_{n}^{2}\right)$
(16) $\displaystyle\mathrm{subject\ to}$ $\displaystyle\sum_{n=1}^{N}a_{n}=1,$

where $a_{n}\geq 0$ is the power allocation fraction in each channel, and
$\sigma_{n}$ is the singular value of the corresponding channel.

The singular values of the channel matrix are

$\sigma_{n}^{2}=\dfrac{1+\nu}{w_{n}/Q+\nu},$ (17)

where $w_{n}$ are the eigenvalues of a set of modes, here referred to as
energy modes [3]. The energy modes are calculated through the generalized
eigenvalue problem, defined here over port mode matrices,

${\mathbf{w}}_{\mathrm{x}}{\mathbf{v}}_{n}=w_{n}{\mathbf{r}}{\mathbf{v}}_{n},$
(18)

where ${\mathbf{v}}_{n}$ are the modal port voltages. These modes are similar
to characteristic modes [41] in the sense that they have the property of
orthogonal radiation patterns. In addition, they minimize the total energy
stored by the antenna. This has the effect of implicitly maximizing the
bandwidth of the modes with the lowest eigenvalues.

Figure 4: The singular values (17) as a function of $\nu$ for different energy
mode values. The three insets show water-filling solutions (15) at optimal
$\nu\in\\{22.1,4.5,1.9\\}$ for three different SNR values,
$\gamma\in\\{2,20,200\\}$. For $\gamma\in\\{2,20\\}$ the weakest mode,
$w_{5}$, is not used because the SNR value is insufficient.

The singular values $\sigma^{2}_{n}$ from (17) are depicted in Fig. 4 as a
function of $\nu$ for energy modes with eigenvalues chosen as
$w_{n}/Q=10^{n-3}$, where $n\in\\{1,\dots,5\\}$. The water-filling solutions
(16) of (15) at the optimal $\nu$ are shown in insets for the SNR values
$\gamma\in\\{2,20,200\\}$. We observe that the optimal $\nu$ decreases with
increasing $\gamma$ and that strong channels, corresponding to $w_{n}/Q\leq
1$, are included for all considered $\gamma$, whereas the weaker channels,
$w_{n}/Q>1$, are only included for high $\gamma$. The singular values approach
unity as $\nu\to\infty$ with the explicit equal power allocation solution for
$N$ equal channels

$C=N\log_{2}\left(1+\frac{\gamma}{N}\right).$ (19)

This is also the explicit solution for the case with $\max(w_{n}/Q)\leq 1$,
because the Q-factor constraint in (14) is trivially satisfied. This suggests
that the number of available channels for a given Q-factor can be estimated by
studying the energy mode eigenvalues. Modes with $w_{n}/Q\leq 1$ can be
considered available and will always be used by the system to induce maximum
spectral efficiency. The energy modes not only simplify the optimization
problem for maximum capacity to water-filling solutions (16) but also provide
a physical interpretation of the results. Let us now consider a few examples
to see how the solution to this optimization problem behaves.

###  III-A Example – Two Dipoles of Variable Separation Distance

The first example deals with two thin-strip dipoles of (resonant) length
$L/\lambda=0.49$ and width $L/W=50$. The dipoles are made of perfect electric
conductor (PEC) and separated by the distance $d$ swept in the interval
$d/L\in[0,2]$.

The sole input into the optimization procedure is the set of energy mode
eigenvalues (18). Their dependence on the separation distance $d/L$ is shown
in Fig. 5. There are two types of modes, in-phase and out-of-phase modes,
which become degenerated at $d/L\approx 0.9$ and $d/L\approx 1.9$. The
smallest eigenvalue of the in-phase mode is $w_{1}\approx 3$ for small
distances whereas the out-of-phase mode $w_{2}$ increases rapidly as $d/L\to
0$. This implies an onset around $Q\approx 3$ with a spectral efficiency of a
single channel, $N=1$, used in (19). The out-of-phase mode starts to
contribute for larger Q-factors as $w_{n}/Q$ decreases in the channel singular
values (17), see also Fig. 4 and the insets in Fig. 6 with the water-filling
solutions for $Q=3$ and $Q=10$. At the degeneracy around $d/L\approx 0.9$ the
modes contribute equally following the spectral efficiency (19) from two equal
channels ($N=2$) as $Q$ increases. For larger distances, the two modes have
similar eigenvalues $w_{n}$ as the mutual coupling diminishes.

Figure 5: The eigenvalues of the energy port-modes $w_{n}$ evaluated by (18)
for an array of two dipoles placed in-parallel, separated by the distance $d$
(see the insets). The values of $w_{n}$ are marked for separation distances
$d/L\in\\{0.3,0.6,0.9\\}$. Figure 6: The maximal ergodic spectral efficiency
of two centrally fed dipoles in Rayleigh fading (the blue dashed line) and
deterministic isotropic (the solid red line) channels. The gray area denotes
the one standard deviation limit of the channel realizations. The dipoles have
electrical size $L/\lambda=0.49$, are separated $d/L=0.3$, and the SNR has
been set to $\gamma=20$. The dashed horizontal lines correspond to one and two
ideal equal power allocation channels (19). The two insets show the water-
filling solutions for $Q=3$ and $Q=10$.

The maximum spectral efficiency calculated with (14) is depicted in Fig. 6 for
$\gamma=20$ and spacing $d/L=0.3$ as a function of the Q-factor. The onset is
bounded by the lower bound on the Q-factor, determined from the smallest
eigenvalue $w_{n}$. For two dipoles separated by distance $d/L\approx 0.3$
that bound is just below $Q=3$ as seen in Fig. 5. This supports previous
observations [42, 43] that two mutually coupled dipoles might have lower $Q$
than a single dipole (here $Q\approx 5$). After the onset the spectral
efficiency $C$ increases towards $C=6.9$ which is reached for $Q\approx 31$.
The horizontal dashed lines in Fig. 6 show the spectral efficiency of one and
two ideal equal power allocation channels that do not take into account
antenna geometry or channel effects. These lines serve as an indicator of how
many pathways the system is utilizing [14]. The Q-factor values at which the
two dipoles produce the optimal spectral efficiency of either one or two ideal
channels correspond perfectly to the first and second eigenvalue from Fig. 5.
The two dipoles can be seen to have solutions in between one and two ideal
channels. Indicating that they are utilizing two channel pathways.

The optimization problem (15) can be solved for a MIMO antenna in a complex
propagation channel, by multiplying the matrix ${\mathbf{s}}$ with the channel
matrix. However, the svd (svd) of the channel matrix (17) can no longer be
found efficiently. For each $\nu$ considered in the optimization process, the
svd must be calculated numerically. In Fig. 6, the maximum spectral efficiency
in a uniform Rayleigh fading channel and a deterministic isotropic channel are
compared for the two dipoles. It can be seen that the antenna in the
deterministic channel follows the behaviour of the system in a Rayleigh
channel. This shows that the deterministic channel can be used to estimate the
maximum spectral efficiency characteristics of an electrically small antenna
design problem in a uniform Rayleigh fading multipath environment.

In Fig. 6, the spectral efficiency grows with $Q$. This can be misleading, and
does not mean that higher capacity can be achieved by a narrow-band system.
Spectral efficiency is a measure of the maximum bit rate that can be achieved
on a $1\mathrm{\,Hz}$ bandwidth. By allowing a higher Q-factor in the
optimization problem the solution can concentrate all spectral efficiency to a
specific frequency. The true capacity can be calculated from the spectral
efficiency by multiplying it by the bandwidth. An indication of the capacity
can be found by considering the spectral efficiency divided by the required
Q-factor, $C/Q$. This measure is adopted for the remaining results in the
paper.

Figure 7: The maximal spectral efficiency over Q for two thin-strip dipoles of
separation distance $d$. Dot marks with numbers correspond to particular
spectral efficiencies, evaluated at $C/Q$ maximum. The SNR has been set to
$\gamma=20$.

The spectral efficiency $C$ over $Q$, evaluated via (14), for two dipoles as a
function of their separation distance $d/L$ is shown in Fig. 7. It is seen
that there exists an optimal distance, $d/L\approx 0.3$, for which $C/Q$ is
maximal. This is the distance with the lowest possible eigenvalue in Fig. 5.
All the curves are bounded from the left by the fundamental bound on $Q$ for
two dipoles given by the lowest eigenvalue $w_{n}$. The maximum spectral
efficiency is bounded by (19) with $N=2$, shown by a dashed curve, and is
first reached for the degenerate case around the distance $d/L\approx 0.9$
with $w_{1}=w_{2}\approx 5.1$, see Fig. 5. However, this curve does not
dominate the others, even though it has the highest spectral efficiency. This
is due to higher $Q$ of the dipoles. For distances between $d/L=0.3$ and
$d/L=0.9$, we see a gradual increase from the lowest $Q=w_{1}$ with a $N=1$
spectral efficiency (19) increasing to two channels $N=2$ as $Q$ approaches
$w_{2}$.

When the dipoles are separated by a large distance they act as separate
radiators and do not improve their joint Q-factor. This indicates that there
is a trade-off between decoupling the dipole radiators to improve spectral
efficiency and utilizing their mutual coupling to improve their bandwidth [2].

###  III-B Example – Two Dipoles Rotated by An Arbitrary Angle

Here, we study the dependence on the angle between the dipoles. The separation
distance between two dipoles is fixed to the optimal value from the previous
example, i.e., $d/L=0.3$. The dipoles are of the same dimensions, located at
$x=-d/2$ and $x=d/2$, and the left dipole is parallel to the $z$-axis. The
axis of rotation for the right dipole coincides with the $x$-axis, see the
insets of Fig. 9.

Figure 8: The eigenvalues of energy port-modes $w_{n}$ evaluated by (18) for
an array of two dipoles placed in-parallel and rotated by angle $\varphi$.
Figure 9: The maximal spectral efficiency over $Q$ for two thin-strip dipoles
rotated by the angle $\varphi$. The dot marks with numbers correspond to
particular spectral efficiencies, evaluated at $C/Q$ maximum. The SNR has been
set to $\gamma=20$.

The input to the optimization are the stored energy eigenvalues $w_{n}$,
depicted in Fig. 8. The eigenvalues have values $w_{1}=2.8$ and $w_{2}=31$ for
the rotation angle $\varphi=0$. This agrees with the values in Fig. 5 for
$d/L=0.3$ since the arrangements are the same, and a doublet of in-phase and
out-of-phase modes exist. The eigenvalues become degenerate for angle
$\varphi=\pi/2$, i.e., both states can be utilized equally with expected onset
around $Q\approx 5$.

The results of $C/Q$ are depicted in Fig. 9. The curve for $\varphi=0$ is
identical to blue dashed one in Fig. 7 for $d/L$ with the same onset at
$Q\approx 2.8$. Similar trends as in Fig. 7 are observed. The spectral
efficiency over $Q$ decreases with an increasing angle of rotation $\varphi$.
Conversely, the spectral efficiency increases and reaches its maximum for
$\varphi=\pi/2$, where the dipoles are crossed and able to utilize two times
more spherical harmonics, i.e., their radiation is relatively uncoupled, and,
as confirmed by the agreement with the asymptote for $N=2$ (the gray dashed
line), both stored energy modes are exploited. This is confirmed by the
identical value of spectral efficiency, $C\approx 6.9$ for two distant dipoles
in Fig. 7 and two dipoles rotated by angle $\varphi=\pi/2$ in Fig. 9.
Similarly, as in the previous example, the cost for this performance in
capacity is higher required bandwidth (lower $Q$).

###  III-C Example – A Dipole Array of Various Number of Elements

The last example of this section studies the optimal capacity depending on the
increasing number of dipoles in the linear array. Three separation distances,
$d/L\in\\{0.3,0.5,0.7\\}$, are studied, respectively. The dimensions of the
individual radiators are the same as in the previous examples.

Figure 10: The eigenvalues of stored energy modes $w_{n}$ for a dipole array
consisting of a variable number of thin-strip dipole elements (from one to
ten). The spacing between dipoles is $d/L\in\\{0.3,0.5,0.7\\}$.

The stored energy eigenvalues $w_{n}$ are depicted in Fig. 10 in dependence on
number of dipoles and separation distance $d/L$. The eigenvalues monotonically
decrease with adding more dipoles into an array. This effect is emphasized
with increasing separation distance $d/L$, i.e., more modes are closer to the
dominant eigenvalue $w_{1}$ for $d/L=0.7$ than for $d/L=0.3$ and $d/L=0.5$.
This has immediate effect on the results of the optimization.

Figure 11: The optimized $C/Q$ and the spectral efficiency $C$ for the dipole
array. The spacing between the dipoles is $d/L=0.3$, i.e., $kd=0.92n$,
$n\in\\{1,\dots,10\\}$. The dot marks with the numbers correspond to
particular spectral efficiencies, evaluated at $C/Q$ maximum. The SNR has been
set to $\gamma=20$.

The change in spectral efficiency constrained by $Q$ is depicted in Figs.
11–13. It is obvious that the performance is significantly boosted with each
dipole added to the array and, for a given number of radiators, the maximum of
$C/Q$ and the maximum of $C$ coincide. The cost of the higher spectral
efficiency is an increase of electrical size. Adding more dipoles imply more
available states for the optimization. The high-order eigenvalues grow
rapidly, whereas the other ones are accumulated closer to the dominant
eigenvalue $w_{1}$, introducing additional degrees of freedom into the
radiation process, see Fig. 10. This is most apparent for $d/L=0.7$, Fig 13,
where the $C/Q$ for $10$ dipoles approaches the asymptote for $N=10$ (gray
dashed line) even for relatively low values of Q. This is because of large
number of stored energy modes have comparably low values, $w_{n}<5$, when the
number of dipoles reach $10$, see Fig. 10, right pane.

Figure 12: The optimized $C/Q$ and the spectral efficiency $C$ for the dipole
array. The spacing between the dipoles is $d/L=0.5$, i.e., $kd=1.54n$,
$n\in\\{1,\dots,10\\}$. The dot marks with the numbers correspond to
particular spectral efficiencies, evaluated at $C/Q$ maximum. The SNR has been
set to $\gamma=20$. Figure 13: The optimized $C/Q$ and the spectral efficiency
$C$ for the dipole array. The spacing between the dipoles is $d/L=0.7$, i.e.,
$kd=2.16n$, $n\in\\{1,\dots,10\\}$. The dot marks with the numbers correspond
to particular spectral efficiencies, evaluated at $C/Q$ maximum. The SNR has
been set to $\gamma=20$.

##  IV Bound on Antenna Design

In Section III the spectral efficiency is optimized through input parameters
to ports for a fixed geometry. In order to formulate an upper bound on
spectral efficiency that can be gained through antenna design we must make the
antenna structure part of the optimization variable. In this section this is
done through current optimization [3].

An antenna structure, e.g., the structure depicted in Fig. 3b, radiates
electromagnetic fields through the current that the excitation in the ports
induces across it. In MoM this current is modelled through equivalent currents
in free space. The equivalent current is represented through the basis
functions that are connected to the mesh. Consider a MoM mesh covering the
entire design region in Fig. 3a as in Fig. 3c, if this mesh is sufficiently
dense it can support the equivalent current of any antenna structure that
could be designed in the region. The technique of current optimization
optimizes the weights of the basis functions of such a mesh. Because the
equivalent current can represent the radiation properties of any antenna in
the design region, an optimal current bounds those properties. However,
antenna designs have several physical limitations to the current that they can
induce, such as feeding. The equivalent current does not share these
restrictions and therefore can support current distributions that are not
realizable but perform better than any antenna design. This ensures that the
optimal value gained through current optimization is always equal to or
greater than that gained through antenna design in the same region.

Optimizing spectral efficiency through current optimization has been treated
extensively by the authors in [13, 14], and is only briefly reiterated here.

The optimization problem (14) can readily be rewritten in terms of the
equivalent current. Take the covariance of the current,
${\mathbf{P}}=\mathcal{E}\left\\{{\mathbf{I}}{\mathbf{I}}^{\text{H}}\right\\}/2$,
as the optimization variable and replace all port mode matrices by their
equivalent MoM matrices, see (9). The optimization problem is then written as

$\displaystyle\mathrm{maximize}$
$\displaystyle\log_{2}\det\left({\mathbf{1}}+\gamma{\mathbf{S}}{\mathbf{P}}{\mathbf{S}}^{\text{H}}\right)$
(20) $\displaystyle\mathrm{subject\ to}$
$\displaystyle\mathop{\mathrm{tr}}\nolimits({\mathbf{W}}_{\mathrm{x}}{\mathbf{P}})\leq
Q$ $\displaystyle\mathop{\mathrm{tr}}\nolimits({\mathbf{R}}{\mathbf{P}})=1$
$\displaystyle\mathop{\mathrm{rank}}{\mathbf{P}}\leq N$ $\displaystyle\
{\mathbf{P}}\succeq 0,$

where $N$ is the number of degrees of freedom of the system. Here, the port
positions are no longer fixed, only their number is specified. This problem
can be solved in different ways depending on what we wish to investigate.

The optimization problem (20) can be made convex by dropping the rank
constraint [14]. The problem can then be solved directly using freely
available convex optimization software such as CVX [40, 13]. However, this
method can become computationally cumbersome for bigger problems.

The problem (20) can instead be solved using the method detailed in [14], see
Appendix D. The solution is equivalent to replacing the singular values in
(16), with those calculated through (17) using the $N$ lowest energy mode
eigenvalues $W_{n}$ calculated by the generalized eigenvalue problem,

${\mathbf{W}}_{\mathrm{x}}{\mathbf{I}}_{n}=W_{n}{\mathbf{R}}{\mathbf{I}}_{n}.$
(21)

These modes have the same properties as their port equivalents (18), i.e.,
orthogonal radiation patterns and minimization of the total stored energy.

###  IV-A Modal Analysis for a rectangular plate

Figure 14: Energy mode eigenvalues $W_{n}$ (21) for a rectangular region with
side lengths $\ell\times\ell/2$, the first three eigencurrents at $ka=1$ are
shown as well.

Fig. 14 depicts the eigenvalues of energy modes for a rectangular plate with
side lengths $\ell\times\ell/2$ as a function of electrical size $0.2\leq
ka\leq 2$. The modal behaviour of two electric dipoles and one magnetic dipole
dominate for electrically small structures, where the scaling
$W_{n}\sim(ka)^{-3}$ is found [44, 45]. Higher-order modes have very large
eigenvalues $W_{n}$ and require similarly high $Q$ to contribute. The energy-
mode eigenvalues $W_{n}$ decrease with increasing electrical size and several
modes are available for larger $ka$. For $ka=1$, we have the first eigenvalue
$W_{1}\approx 4$, proposing an onset around $Q=4$, and two additional
eigenvalues at $9$ and $15$ indicate three independent channels below $Q=15$,
see Fig. 15.

When optimizing over the current density the number of channels in (4) can no
longer be considered small. Therefore, we must analyze the impact of the
Rayleigh channel on the optimization results. Consider (20) with the objective
function,

$C_{{\mathbf{H}}_{\mathrm{w}}}=\log_{2}\det\left({\mathbf{1}}+\gamma{\mathbf{H}}_{\mathrm{w}}{\mathbf{S}}{\mathbf{P}}{\mathbf{S}}^{\text{H}}{\mathbf{H}}_{\mathrm{w}}^{\text{H}}\right).$
(22)

This can be done by numerically calculating the SVD of the channel matrix for
each $\nu$.

Figure 15: Maximum ergodic spectral efficiency of a $ka=1$, $\ell\times\ell/2$
plate in a uniform Rayleigh fading channel (blue dashed line) and a
deterministic isotropic channel (solid red line). The gray area denotes one
standard deviation from the average of the Rayleigh channel case. The dashed
horizontal lines correspond to one, two, and three ideal equal power
allocation channels [14].

In Fig. 15 the current on a $ka=1$ plate has been optimized for maximal
spectral efficiency with and without the inclusion of a uniform Rayleigh
channel. The average of the uniform Rayleigh fading channel realizations has
the same performance as the plate in the deterministic isotropic spherical
channel. This supports the claims in [13, 14] that the Rayleigh channel can be
neglected when studying the fundamentals of antenna design for optimal
spectral efficiency in uniform multipath environments.

The standard deviation of the Rayleigh channel realizations for the plate in
Fig. 15 is significantly smaller than what can be seen for two dipoles in Fig.
6. This is due to the added degrees of freedom in optimizing an entire plate.
The plate can support higher order modes at lower Q-values and its current can
be tailored to specific channel realizations.

###  IV-B Sub-region Energy modes

Antennas inside communication devices are, in general, much smaller than the
total device size [46]. This means that only a sub-region of the device is
dedicated to antenna design. The antennas therein excite currents across the
entire device that contribute to communication. From an optimization
perspective this can be seen as only controlling the current within the
antenna sub-region. The optimization of arrays through their port quantities
in Section III is similar to this concept. The ports control the current on
the dipole on a very small sub-region of the antenna. In this section this
concept is generalized to controlling the current in larger sub-regions, see
Fig. 3c. Optimizing the current on a sub-region for maximal spectral
efficiency calculates a bound on the spectral efficiency available from
designing an antenna in a sub-region of a device. This is interesting to
investigate since bandwidth and Q-factor are usually harshly restricted by
reducing the antenna size [20, 47].

Figure 16: The eigenvalues $W_{n}$ of a spherical shell, a rectangular plate
with aspect ratio $2:1$, and several of its sub-region arrangements with size
$0.1\ell\times 0.05\ell$ for the cases A-D and width $0.05\ell$ for the case
E, where the current is controlled inside the blue regions. The electrical
size is $ka=1$, where $k$ is the wave number, and $a$ is the radius of the
smallest sphere circumscribing the entire structure. The dashed lines
correspond to the characteristic mode currents evaluated in the Rayleigh
quotient (23) [18].

Consider the placement of sub-regions for antenna design within a device.
Which orientation of sub-region has the greatest potential spectral efficiency
can be evaluated by comparing their energy mode eigenvalues. In Fig. 16, five
different sub-region eigenvalues, as well as the plate and circumscribing
spherical shell eigenvalues, are shown. By studying how close each of the sub-
region cases come to the eigenvalues of the full plate we can evaluate how
well they induce the full performance of the plate. In Fig. 16, we see that
there is a gap between the eigenvalue of the full plate compared to small
segregated sub-regions, whereas the case E, the loop region, has eigenvalues
close to the full plate. We see that the two diagonally placed elements in
case B outperform the two elements in case C for the first and third mode.
This is similar to the case in [14] due to the first and second order modes
being induced diagonally across the plate. Therefore, the two diagonally
situated sub-regions do not effectively induce the second mode across the
opposite diagonal. The eigenvalues of the circumscribing spherical shell are
significantly lower than those of the plate. These modes are grouped in sets
of three because the spherical shell supports three degenerate dipole modes.

The dashed lines in Fig. 16 correspond to the characteristic mode currents
evaluated in the Rayleigh quotient of the eigenvalue problem (21), i.e.,

$W_{\mathrm{c},n}=\frac{{\mathbf{I}}_{\mathrm{c},n}^{\text{H}}{\mathbf{W}}_{\mathrm{x}}{\mathbf{I}}_{\mathrm{c},n}}{{\mathbf{I}}_{\mathrm{c},n}^{\text{H}}{\mathbf{R}}{\mathbf{I}}_{\mathrm{c},n}},$
(23)

where ${\mathbf{I}}_{\mathrm{c},n}$ are the characteristic mode currents [41].
These values indicate how well the characteristic mode currents perform in the
metric of energy modes. In Fig. 16, we see that the characteristic modes
basically overlap the energy mode eigenvalues for all considered cases. This
indicates that the characteristic modes are a good tool for MIMO antenna
design, validating previous design methodologies [17, 16, 15].

###  IV-C Sub-region Optimization

In Fig. 17, the spectral efficiency over $Q$ is depicted for the electrical
size $ka=1$ as a function of the required Q-factor using the energy modes
depicted in Fig. 16. This Pareto-type curve delimits the feasible region of
the problem and reveals that the capacity and the Q-factor are strictly
conflicting parameters. It can be seen that there is a preferred Q-factor for
optimal spectral efficiency for different sizes and geometries found at the
maximum of the curves. This maximum is found close to the minimum feasible
Q-factor found in Fig. 17. However, the performance in terms of $C/Q$
deteriorates slowly with increasing Q-factor. This indicates that sub-region
MIMO antennas use less bandwidth but more spatial diversity to maintain high
performance.

The dashed gray lines in Fig. 17 depict the equal power allocation channels
(19). When the solution of the different sub-region cases passes one of these
lines it is utilizing at least that many different channels to induce optimal
spectral efficiency. This number is the structure’s number of effective modes
for a certain allowed Q-factor. The number of effective modes can be estimated
from Fig. 16 by counting the number of eigenvalues below $W_{n}=Q$. This is
the limit for what is considered an effective mode, see Fig. 4. When the
number of ports is restricted by a certain number, in (20), the spectral
efficiency over $Q$ maximally reaches the dashed gray curve associated with
that number of ideal channels.

It is evident that the small segregated sub-regions from the case A to D do
not effectively induce the available spectral efficiency of the full plate, as
predicted from Fig. 16. However, the loop region reaches $C/Q$ values much
closer to the full plate. In contrast to the case studied in [14], that was
restricted by efficiency, this gap between the sub-regions and the full plate
remains the same when the size of the plate is reduced. Instead of narrowing
the gap between the full plate and its segregated sub-regions, as in [14], a
reduction of size makes the sub-region solution unfeasible, due to the lower
Q-factor bound [48]. Therefore, we can conclude that the optimal performance
of the embedded MIMO antennas is more restricted by the limited bandwidth than
the requirements on radiation efficiency.

Figure 17: The optimal spectral efficiency over $Q$ for the $ka=1$ spherical
shell, plate with aspect ratio $2:1$, and its sub-regions presented in Fig.
16, for different Q-factor restrictions. The dashed gray lines correspond to
increasing numbers of ideal equal power allocation channels. The SNR has been
set to $\gamma=20$.

##  V Conclusions

It has been shown that a set of modes known as energy modes, as well as
characteristic modes, serves as a useful design tool when analyzing
performance of MIMO antennas. The conclusion that a set of properly weighted
characteristic modes approximately form an optimal current for capacity can be
utilized in practice, specifically since simultaneous excitation of
characteristic modes is well explored. However, due to the harsh penalties on
stored energy when reducing the design region, it has been concluded that it
is not possible to reach the full potential of the plate while only feeding it
with a set of small segregated sub-regions. However, the use of a connected
loop region was shown to be much more efficient.

Spectral efficiency bounds of MIMO antennas in uniform Rayleigh fading
channels and a deterministic isotropic channel have been considered when
restricted by the Q-factor. It was shown that the uniform Rayleigh fading
channel does not affect the characteristics of the spectral efficiency bound.
The Pareto-type bound has been illustrated in spectral efficiency over $Q$,
where a certain Q-factor was shown to be Pareto optimal.

The modal analysis illustrated in this paper could also be carried out on
designed antennas to evaluate their adherence to the principals suggested
here. Generalizing this method to include several different design parameters
remains an interesting future prospect.

##  Appendix A Rayleigh Channel Contribution

To study the dependence of the system on the uniform Rayleigh fading channel,
consider a MIMO system with a finite and small number of ports $N$. Since the
number of ports is small, the SNR can be considered to be large in each port,
i.e.,
$\gamma{\mathbf{H}}_{\mathrm{w}}{\mathbf{S}}_{\mathrm{2,1}}{\mathbf{A}}{\mathbf{S}}_{\mathrm{2,1}}^{\text{H}}{\mathbf{H}}_{\mathrm{w}}^{\text{H}}\succ\alpha{\mathbf{1}}$,
$\alpha\gg 1$. The ergodic spectral efficiency is then gained by averaging (4)
over channel realizations,

$\begin{split}\bar{C}=\,&\mathcal{E}\left\\{\log_{2}\det(\gamma{\mathbf{H}}_{\mathrm{w}}{\mathbf{S}}_{\mathrm{2,1}}{\mathbf{A}}{\mathbf{S}}_{\mathrm{2,1}}^{\text{H}}{\mathbf{H}}_{\mathrm{w}}^{\text{H}})\right\\}\\\
=\,&\log_{2}\det(\gamma)+\mathcal{E}\left\\{\log_{2}\det({\mathbf{H}}_{\mathrm{w}})\right\\}\\\
&+\mathcal{E}\left\\{\log_{2}\det({\mathbf{H}}_{\mathrm{w}}^{\text{H}})\right\\}+\log_{2}\det({\mathbf{S}}_{\mathrm{2,1}}{\mathbf{A}}{\mathbf{S}}_{\mathrm{2,1}}^{\text{H}}).\end{split}$
(24)

The effect of the channel in such a system is only an additive term in the
spectral efficiency evaluation. Consider the correlation loss, defined as

$\Delta\bar{C}=\bar{C}-\bar{C}_{{\mathbf{1}}},$ (25)

where $\bar{C}_{{\mathbf{1}}}$ is the equal power allocation solution. This
quantity is independent of the additive channel contribution in (24). This
implies that the physics of maximizing ergodic spectral efficiency is
independent of the channel in uniform multipath environment, as long as the
number of ports in the system is low [7]. Note that (25) only contains a
temporal average over the input signals, significantly simplifying its
evaluation.

##  Appendix B Electric Field Integral Equation

All antenna structures in this paper are modeled as perfectly conducting
bodies with the electric field integral equation (EFIE), [37], which relates
tangential component of incident and scattered electric fields as

$\hat{\boldsymbol{n}}\times\boldsymbol{E}_{\mathrm{i}}\left(\boldsymbol{r}\right)=-\hat{\boldsymbol{n}}\times\boldsymbol{E}_{\mathrm{s}}\left(\boldsymbol{r}\right),$
(26)

where $\boldsymbol{E}_{\mathrm{i}}$ is the incident field,
$\boldsymbol{E}_{\mathrm{s}}$ is the scattered field

$\boldsymbol{E}_{\mathrm{s}}\left(\boldsymbol{r}\right)=-\mathrm{j}kZ_{0}\int\limits_{\varOmega}\mathbf{G}\left(\boldsymbol{r},\boldsymbol{r}^{\prime}\right)\cdot\boldsymbol{J}\left(\boldsymbol{r}^{\prime}\right)\,\mathrm{d}S^{\prime},$
(27)

$Z_{0}$ is the impedance of vacuum, $k$ is wavenumber, and $\mathbf{G}$
denotes the free-space dyadic Green’s function defined as

$\mathbf{G}\left(\boldsymbol{r},\boldsymbol{r}^{\prime}\right)=\left(\mathbf{1}+\frac{\nabla\nabla}{k^{2}}\right)\frac{\mathrm{e}^{-\mathrm{j}k|\mathbf{r}-\mathbf{r}^{\prime}|}}{4\pi\left|\mathbf{r}-\mathbf{r}^{\prime}\right|}$
(28)

with $\mathbf{1}$ being the identity dyadic.

By applying a suitable set of basis functions

$\boldsymbol{J}(\boldsymbol{r})\approx\sum_{n=1}^{N}I_{n}\boldsymbol{\psi}_{n}(\boldsymbol{r})$
(29)

and the same set as testing functions (Galerkin method), the relation (26)
transforms into algebraic form

$\mathbf{Z}\mathbf{I}=\mathbf{V},$ (30)

where the impedance matrix $\mathbf{Z}=[Z_{nm}]\in\mathbb{C}^{N\times N}$ and
excitation vector $\mathbf{V}=[V_{n}]\in\mathbb{C}^{N\times 1}$ are defined
element-wise as

$Z_{nm}=-\mathrm{j}kZ_{0}\int\limits_{\varOmega}\int\limits_{\varOmega}\boldsymbol{\psi}_{n}\left(\boldsymbol{r}\right)\cdot\mathbf{G}\left(\boldsymbol{r},\boldsymbol{r}^{\prime}\right)\cdot\boldsymbol{\psi}_{m}\left(\boldsymbol{r}^{\prime}\right)\,\mathrm{d}S\,\mathrm{d}S^{\prime},$
(31)

and

$V_{n}=\int\limits_{\varOmega}\boldsymbol{\psi}_{n}\left(\boldsymbol{r}\right)\cdot\boldsymbol{E}_{\mathrm{i}}\left(\boldsymbol{r}\right)\,\mathrm{d}S,$
(32)

respectively.

The real part ${\mathbf{R}}$ of the impedance matrix
${\mathbf{Z}}={\mathbf{R}}+\mathrm{j}{\mathbf{X}}$ can be factorized as

${\mathbf{R}}={\mathbf{S}}^{\mathrm{T}}{\mathbf{S}},$ (33)

where the matrix ${\mathbf{S}}$ is defined element-wise as

$S_{\alpha
n}=k\sqrt{Z_{0}}\int\limits_{\varOmega}\boldsymbol{\psi}_{n}\cdot{\mathbf{u}}^{(1)}_{\alpha}(k\boldsymbol{r})\,\mathrm{d}S$
(34)

with ${\mathbf{u}}^{(1)}_{\alpha}$ being the regular spherical waves, see [49]
for details.

##  Appendix C Definition of Q-factor

The radiation Q-factor is defined as

$Q_{\mathrm{rad}}=\dfrac{2\omega\max\left\\{W_{\mathrm{m}},W_{\mathrm{e}}\right\\}}{P_{\mathrm{r}}},$
(35)

where the stored magnetic and electric energies are

$\displaystyle\omega W_{\mathrm{m}}$
$\displaystyle=\dfrac{1}{8}{\mathbf{I}}^{\text{H}}\left(\omega\dfrac{\partial{\mathbf{X}}}{\partial\omega}+{\mathbf{X}}\right){\mathbf{I}},$
(36a) $\displaystyle\omega W_{\mathrm{e}}$
$\displaystyle=\dfrac{1}{8}{\mathbf{I}}^{\text{H}}\left(\omega\dfrac{\partial{\mathbf{X}}}{\partial\omega}-{\mathbf{X}}\right){\mathbf{I}},$
(36b)

the radiated power is defined as

$P_{\mathrm{r}}=\dfrac{1}{2}{\mathbf{I}}^{\text{H}}{\mathbf{R}}{\mathbf{I}},$
(37)

and ${\mathbf{Z}}={\mathbf{R}}+\mathrm{j}{\mathbf{X}}$ is the impedance
matrix, see Appendix B. For purposes of this paper, the stored energy matrix
is introduced as

${\mathbf{W}}_{\mathrm{x}}=\dfrac{1}{2}\omega\dfrac{\partial{\mathbf{X}}}{\partial\omega},$
(38)

and the radiation Q-factor is rewritten as

$Q_{\mathrm{rad}}=Q+Q_{\mathrm{t}}=\dfrac{\omega\left(W_{\mathrm{m}}+W_{\mathrm{e}}\right)}{P_{\mathrm{r}}}+\dfrac{\omega|W_{\mathrm{m}}-W_{\mathrm{e}}|}{P_{\mathrm{r}}},$
(39)

where the first part ($Q$) belongs to the antenna itself and the second part
($Q_{\mathrm{t}}$) belongs to a lumped element tuning an antenna to the
resonance [1]. Generally, it holds that $Q_{\mathrm{rad}}\geq Q$, and
$Q_{\mathrm{rad}}=Q$ only at the self-resonance of an antenna. In this paper,
we use only the first part of (39), $Q$, since it is a low-bound estimate of
an achievable Q-factor irrespective of how a multi-port antenna is tuned.

##  Appendix D SVD of channel matrix

The optimal solution of (15) is found by writing the system on the form (16).
This is achieved through a change of variables. Define the matrix

${\mathbf{r}}_{\nu}=\frac{1}{(\nu+1)}\left(\nu{\mathbf{r}}+Q^{-1}{\mathbf{w}}_{\mathrm{x}}\right).$
(40)

This matrix is positive semi-definite for sufficiently large antenna sizes,
and appropriate values of $\nu$. Positive semi-definite matrices can be
decomposed with the Cholesky decomposition,
${\mathbf{r}}_{\nu}={\mathbf{b}}^{\text{H}}{\mathbf{b}}$. With this
decomposition the condition in (16) can be rewritten as,

$\mathcal{E}\left\\{{\mathbf{v}}^{\text{H}}{\mathbf{r}}_{\nu}{\mathbf{v}}\right\\}=\mathop{\mathrm{tr}}\nolimits({\mathbf{r}}_{\nu}{\mathbf{a}})=\mathop{\mathrm{tr}}\nolimits({\mathbf{b}}{\mathbf{a}}{\mathbf{b}}^{\text{H}})=\mathop{\mathrm{tr}}\nolimits(\widetilde{{\mathbf{a}}}),$
(41)

where the cyclic permutation of the trace has been utilized and
$\widetilde{{\mathbf{a}}}={\mathbf{b}}{\mathbf{a}}{\mathbf{b}}^{\text{H}}$ is
a change of variables. With this change of variables (15) is written as

$\displaystyle\min\limits_{\nu}\,\max\limits_{\widetilde{{\mathbf{a}}}}$
$\displaystyle\log_{2}\det({\mathbf{1}}+\gamma{\mathbf{s}}{\mathbf{b}}^{-1}\widetilde{{\mathbf{a}}}{\mathbf{b}}^{-\text{H}}{\mathbf{s}}^{\text{H}})$
(42) $\displaystyle\mathrm{subject\ to}$
$\displaystyle\mathop{\mathrm{tr}}\nolimits(\widetilde{{\mathbf{a}}})=1.$

This is a problem that can be solved by water-filling [4, 5], as in (16), with
the channel matrix, ${\mathbf{h}}={\mathbf{s}}{\mathbf{b}}^{-1}$. All that is
required is the singular value decomposition of ${\mathbf{h}}$. The singular
values can be calculated from the eigenvalues of the matrix times itself,

$\mathop{\mathrm{svd}}\nolimits({\mathbf{h}})=(\mathop{\mathrm{eig}}\nolimits({\mathbf{h}}{\mathbf{h}}^{\text{H}}))^{1/2}=(\mathop{\mathrm{eig}}\nolimits({\mathbf{s}}{\mathbf{r}}_{\nu}^{-1}{\mathbf{s}}^{\text{H}}))^{1/2}.$
(43)

The eigenvalues can be determined by utilizing the decomposition of the
radiation matrix, ${\mathbf{r}}={\mathbf{s}}^{\text{H}}{\mathbf{s}}$. Putting
this into (43),

$\mathop{\mathrm{eig}}\nolimits({\mathbf{s}}{\mathbf{r}}_{\nu}^{-1}{\mathbf{s}}^{\text{H}})=\mathop{\mathrm{eig}}\nolimits((\nu+1)\left(\nu+Q^{-1}{\mathbf{s}}^{-\text{H}}{\mathbf{w}}_{\mathrm{x}}{\mathbf{s}}^{-1}\right)^{-1})\\\
=(\nu+1)\left(\nu+Q^{-1}\mathop{\mathrm{eig}}\nolimits({\mathbf{s}}^{-\text{H}}{\mathbf{w}}_{\mathrm{x}}{\mathbf{s}}^{-1})\right)^{-1}.$
(44)

The eigenvalues
$\mathop{\mathrm{eig}}\nolimits({\mathbf{s}}^{-\text{H}}{\mathbf{w}}_{\mathrm{x}}{\mathbf{s}}^{-1})$
are determined from the generalized eigenvalue problem (18). This gives the
singular values (17),

$\sigma_{n}^{2}=\dfrac{(1+\nu)}{w_{n}/Q+\nu}.$ (45)

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|  Casimir Ehrenborg (S’15) received his M.Sc. degree in engineering physics
in 2014, and his Ph.D. degree in electrical engineering in 2019, from Lund
University, Sweden. He is currently a postdoctoral fellow in the META research
group at KU Leuven. In 2015, he participated in and won the IEEE Antennas and
Propagation Society Student Design Contest for his body area network antenna
design. In 2019, he was awarded the IEEE AP-S Uslenghi Letters Prize for best
paper published in IEEE Antennas and Propagation Letters during 2018. His
research interests include Spacetime metamaterials, MIMO antennas, physical
bounds, and stored energy.
---|---
|  Mats Gustafsson received the M.Sc. degree in Engineering Physics 1994, the
Ph.D. degree in Electromagnetic Theory 2000, was appointed Docent 2005, and
Professor of Electromagnetic Theory 2011, all from Lund University, Sweden. He
co-founded the company Phase holographic imaging AB in 2004. His research
interests are in scattering and antenna theory and inverse scattering and
imaging. He has written over 100 peer reviewed journal papers and over 100
conference papers. Prof. Gustafsson received the IEEE Schelkunoff Transactions
Prize Paper Award 2010, the IEEE Uslenghi Letters Prize Paper Award 2019, and
best paper awards at EuCAP 2007 and 2013. He served as an IEEE AP-S
Distinguished Lecturer for 2013-15.
---|---
|  Miloslav Capek (M’14, SM’17) received the M.Sc. degree in Electrical
Engineering 2009, the Ph.D. degree in 2014, and was appointed Associate
Professor in 2017, all from the Czech Technical University in Prague, Czech
Republic. He leads the development of the AToM (Antenna Toolbox for Matlab)
package. His research interests are in the area of electromagnetic theory,
electrically small antennas, numerical techniques, fractal geometry, and
optimization. He authored or co-authored over 100 journal and conference
papers. Dr. Capek is member of Radioengineering Society, regional delegate of
EurAAP, and Associate Editor of IET Microwaves, Antennas & Propagation.
---|---
  *[MIMO]:
  *[SNR]:
  *[MoM]:
  *[CVX]:
  *[svd]: