#  A generalization of the achievable rate of a MISO system using Bode-Fano
wideband matching theory

Nitish Deshpande, Student Member, IEEE, Miguel R. Castellanos, Member, IEEE,
Saeed R. Khosravirad, Member, IEEE, Jinfeng Du, Member, IEEE, Harish
Viswanathan, Fellow, IEEE, and Robert W. Heath Jr., Fellow, IEEE
Nitish Deshpande, Miguel R. Castellanos, and Robert W. Heath Jr. are with the
Department of Electrical and Computer Engineering, North Carolina State
University, Raleigh, NC 27606 USA (email: {nvdeshpa, mrcastel,
rwheathjr}@ncsu.edu). Saeed R. Khosravirad, Jinfeng Du, and Harish Viswanathan
are with Nokia Bell Laboratories, Murray Hill, NJ 07974, USA (email:
{saeed.khosravirad, jinfeng.du, harish.viswanathan}@nokia-bell-labs.com). This
project is funded in part by Nokia Bell Laboratories, Murray Hill, NJ 07974,
USA. This material is also based upon work supported in part by the National
Science Foundation under Grant No. NSF-CNS-2147955 and NSF-ECCS-2153698 and by
funds from federal agencies and industry partners as specified in the
Resilient & Intelligent NextG Systems (RINGS) program.

###### Abstract

Impedance-matching networks affect power transfer from the radio frequency
(RF) chains to the antennas. Their design impacts the signal to noise ratio
(SNR) and the achievable rate. In this paper, we maximize the information-
theoretic achievable rate of a multiple-input-single-output (MISO) system with
wideband matching constraints. Using a multiport circuit theory approach with
frequency-selective scattering parameters, we propose a general framework for
optimizing the MISO achievable rate that incorporates Bode-Fano wideband
matching theory. We express the solution to the achievable rate optimization
problem in terms of the optimized transmission coefficient and the Lagrangian
parameters corresponding to the Bode-Fano inequality constraints. We apply
this framework to a single electric Chu’s antenna and an array of two electric
Chu’s antennas. We compare the optimized achievable rate obtained numerically
with other benchmarks like the ideal achievable rate computed by disregarding
matching constraints and the achievable rate obtained by using sub-optimal
matching strategies like conjugate matching and frequency-flat transmission.
We also propose a practical methodology to approximate the achievable rate
bound by using the optimal transmission coefficient to derive a physically
realizable matching network through the ADS software.

###### Index Terms:

Bode-Fano matching theory, achievable rate maximization, scattering
parameters, matching network design

##  I Introduction

As wireless systems exploit higher bandwidths, it is crucial to design
matching networks that support the desired power transfer in the band of
interest to achieve the target data rate[1, 2, 3, 4]. For narrowband systems,
matching networks are optimized for power transfer between source and load at
a single frequency. For wideband arrays, it is challenging to design matching
networks because the load depends on the frequency-selectivity of the array
including mutual coupling between antennas [4]. The Bode-Fano theory captures
these practical matching constraints with a frequency-selective circuit theory
approach based on scattering parameters[2, 5]. In this paper, we incorporate
these constraints in achievable rate analysis unlike conventional wideband
MIMO literature which does not treat matching networks as a part of the
analysis[6, 7].

The problem of matching a source impedance to a load impedance in conventional
RF literature is mostly based on power transfer based metrics[5, 8, 9, 10,
11]. In general, the transmit matching network is designed to maximize power
transfer efficiency while the receive matching network to minimize the noise
figure. For a narrowband system, the conjugate matching network is designed
such that the effective load impedance equals complex conjugate of the source
impedance. For broadband matching, the constant quality factor circle
technique can be used [5]. Designing matching networks for systems operating
at higher fractional bandwidths is challenging because of the frequency-
selectivity of the load. Recently, a globally optimal approach to designing
wideband matching networks defined a unique trajectory connecting source and
load on Smith chart using the power transfer efficiency metric [10]. Although
power transfer efficiency is important, it only quantifies the power transfer
from the RF chain to the antennas within a transmitter. From a communication
theoretic perspective, the most relevant metric is the end-to-end achievable
data rate. The communication rate depends on factors like the bandwidth,
wireless propagation channel, beamforming response at both receiver and
transmitter, mutual coupling between antenna elements, and their radiation
patterns. The achievable rate metric captures all factors. Hence, wireless
devices should optimize the matching network to maximize the rate rather than
the power transfer efficiency.

For analysis of wideband systems, it is essential to understand the
fundamental design tradeoffs between gain and bandwidth[12]. In large phased-
arrays operating at higher fractional bandwidths, there is a phase mismatch
between the frequency-flat phase-shifter and frequency-selective array
response[13]. The frequency-selectivity of antennas and matching networks was
considered for analysis of dense array wideband massive MIMO[14]. The results
in [14, 15] showed that for systems which use matching networks based on the
conjugate matching strategy, the SNR drops drastically for frequencies away
from the center frequency. A matching network based on a narrowband assumption
is sub-optimal in a wideband setting. Hence, it is necessary to optimize
physically realizable matching networks over the bandwidth of interest. These
examples show that the shift from frequency-flat to frequency-selective models
is necessary as wireless systems transition from narrowband to wideband
operation[16, 13, 12, 14, 15].

The circuit theory approach to modeling wireless communication systems enables
incorporating physically consistent frequency-selective models for the
antennas, arrays, wireless channel, and the RF chain components in the
analysis[17]. This approach captures effects like mutual coupling in the form
of impedance or scattering matrices thus making the system analysis more
realistic and tractable[4, 18, 19, 17, 15, 20, 21, 22, 23, 24]. Although
circuit theoretic abstractions have been used for decades for the design of
individual RF components like antennas[25, 26], matching networks[5], and
amplifiers[27], the application of circuit models for MIMO communication
systems is more recent. Phenomena like super-directivity[19] and super-
bandwidth[23] that occur in tightly coupled arrays can be explained with the
circuit theory approach. Hardware effects like amplifier current
constraints[14] and matching network limitations[15] can also be incorporated
through circuit models. Hence, the circuit theory approach to communication is
useful to design matching networks for optimizing achievable rate.

Prior work has studied achievable rate maximization through impedance-matching
only for specific matching network topologies[4, 21, 22, 28, 29]. In [21], an
upper bound on the MIMO-OFDM capacity was proposed by optimizing the receiver
matching network parameters based on a T-network topology. In [4], the
achievable rate of a MISO and SIMO system was optimized in terms of the
inductances and transformer turns ratio of a single port matching network.
Although [4] and [21] used a communication theoretic objective, the methods
used for optimizing the matching network parameters are specific to a given
topology and do not guarantee optimality over a general family of passive and
linear matching networks. From a circuit theory perspective, there exists a
fundamental limit on the wideband performance of a passive matching network,
popularly known as the Bode-Fano limits[5, 8, 9]. Recent work derived an upper
bound on the single-input-single-output (SISO) achievable rate by applying the
Bode-Fano wideband matching constraints at the receiver[22] and
transmitter[30]. Recently, a multiport extension of the Bode-Fano matching
theory proposed new bounds applicable to a system with multiple transmit
antennas driven by multiple sources[2, 31]. The application of the improved
Bode-Fano matching limits to a MIMO system from an achievable rate perspective
is not investigated in prior work.

In this paper, we analyze a MISO system from a joint circuit and communication
theoretic perspective. We answer two fundamental questions. The first question
is “What is the upper bound on the achievable rate of a MISO system over all
physically realizable linear and passive matching networks that satisfy the
Bode-Fano wideband matching constraints?” We demonstrate how ignoring the
Bode-Fano constraints leads to an over-estimation of the rate for wideband
systems. The second question is “How to design impedance-matching networks
that achieve rate close to the proposed upper bound?” In contrast to prior
work, we design realizable matching networks that maximize achievable rate.
The main contributions of this paper are as follows.

  * •

We derive a frequency-selective circuit theoretic model of a MISO system with
a single RF chain at the transmitter that supplies power to the antenna array
through an impedance-matching network and an analog beamforming network. The
system model formulation is in terms of the scattering parameters, which
enables a direct application to any RF system whose scattering parameters can
be measured.

  * •

We propose a general framework for optimizing the achievable rate of wideband
MISO systems as a function of the impedance-matching network. The constraints
are based on a generalized version of the Bode-Fano wideband matching theory
recently proposed in [2, 31]. For deriving the constraints, we use a rational
and passive approximation of the equivalent load comprising of the analog
beamforming network and the transmit antennas. The maximum achievable rate is
expressed in terms of the optimized transmission coefficient and Lagrangian
parameters associated with the Bode-Fano inequalities. The transmission
coefficient depends not only on the antenna and array parameters but also on
the wireless propagation channel and the analog beamformer.

  * •

We propose a three step procedure to design circuits that approximate the
desired optimal response obtained through the achievable rate optimization
solution. Our simulation results show that the matching network designed using
this procedure achieves rates close to the maximum achievable rate bound.

  * •

We demonstrate this three step procedure for two specific models: a single
Chu’s electric antenna and an array of two Chu’s antennas. We use a practical
LC ladder matching network topology whose components are numerically optimized
in ADS to fit the corresponding optimal transmission coefficient.

  * •

We compare our proposed bound and the performance of the designed matching
network with the ideal Shannon’s bound, frequency-flat transmission, conjugate
matching at center frequency, and the no matching case. We also analyze the
achievable rate trend with bandwidth. We show the existence of an optimal
bandwidth for the achievable rate bound obtained with Bode-Fano constraints
and the corresponding circuit simulations.

This paper is organized as follows: In Section II, we discuss the choice of
the modeling framework in comparison with other frameworks used in literature.
Choosing a circuit theoretic modeling methodology, we formulate a frequency-
selective model for a MISO system where the linear network parameters are
described using the scattering parameter notation. In Section III, we discuss
the general form of the Bode-Fano matching constraints followed by the
achievable rate optimization problem formulation and derivation of the optimal
transmission coefficient. In Section IV, we propose a methodology to design
matching networks based on the derived transmission coefficient supported by
circuit illustrations using ADS software. In Section V, we present numerical
results for SNR and achievable rate using the derived theoretical bounds,
circuit simulations, and comparison with conventional matching benchmarks. In
Section VI, we summarize the key takeaways and discuss future research
directions. The simulation code for generating achievable rate optimization
results and the corresponding circuit ADS files are made publicly available to
facilitate
reproducibility111https://github.com/nvdeshpa/AchievableRateWidebandMatching.

Notation: A bold lowercase letter $\bm{{\mathsf{a}}}$ denotes a column vector,
a bold uppercase letter $\bm{\mathsf{A}}$ denotes a matrix, $(\cdot)^{\ast}$
denotes conjugate, $(\cdot)^{T}$ denotes transpose, $|\cdot|$ indicates
absolute value, ${\mathbf{I}}_{N}$ represents the identity matrix of size $N$,
$\mathbf{0}_{N}$ represents an all zero matrix of size $N$, $\mathcal{R}(z)$
denotes the real part of a complex number $z$, $\\{i\\}_{1}^{N}$ is shorthand
for $i=\\{1,2,\dots,N\\}$, $[z]^{+}=\mbox{max}(0,z)$.

##  II System model

###  II-A Modeling frameworks for wireless communication system

(a) A general two-port network model where $\mathsf{v}_{\mathsf{T}}(f)$ is the
voltage, $\mathsf{i}_{\mathsf{T}}(f)$ is the current at the input, and
$\mathsf{v}_{\mathsf{N}}(f)$ is noise source at output. Alternate
representation using root power waves $\mathsf{a}_{\mathsf{T}}(f)$,
$\mathsf{b}_{\mathsf{T}}(f)$, and $\mathsf{b}_{\mathsf{R}}(f)$.

(b) Circuit model of a MISO system where $\mathsf{a}_{\mathsf{F}}(f)$ is the
incident root power wave and $\mathsf{b}_{\mathsf{F}}(f)$ is the reflected
root power wave from the analog beamforming network,
$\bm{{\mathsf{a}}}_{\mathsf{A}}(f)\in{\mathbb{C}}^{N\times 1}$ is the incident
root power wave vector and
$\bm{{\mathsf{b}}}_{\mathsf{A}}(f)\in{\mathbb{C}}^{N\times 1}$ is the
reflected root power wave vector from the transmit antenna array. The linear
blocks are described using scattering parameter matrices
$\bm{\mathsf{S}}_{\mathsf{M}}(f)$, $\bm{\mathsf{S}}_{\mathsf{F}}(f)$, and
$\bm{\mathsf{S}}_{\mathsf{H}}(f)$.

Figure 1: In (a), a communication system with one RF chain at both receiver
and transmitter is modeled as a two-port network. In (b), the two-port network
model is specified for a MISO system with one transmit RF chain connected to
$N$ antennas through impedance-matching network and analog beamforming
network.

With every wireless generation, there has been evolution in the communication
system modeling approach. The analysis of a wireless system is impacted by the
choice of model. It is important to choose a model that encompasses the proper
assumptions and ensures the validity and applicability of the insights to the
target wireless application.

We overview different frameworks that are used for modeling physical layer
wireless communication. In terms of frequency dependence, the system model is
classified as frequency-flat or frequency-selective. For frequency-flat
models, the network parameters are evaluated at a specific center frequency
and assumed to be fixed over the narrow bandwidth of interest. For narrowband
wireless applications like sensor networks, RFID[32], and narrowband Internet
of Things[33], it suffices to use a frequency-flat modeling framework. For
wideband wireless applications like satellite communication, Wi-Fi, and
cellular[34], frequency-selective models are required to guarantee that the
analytical or numerical results from the model are useful for the desired
frequency range.

In terms of the modeling methodology, the models can be classified as
dimensionless (non-circuit theoretic models) and physically consistent
(circuit theoretic models). Non-circuit theoretic models have been useful for
analyzing achievable rate, interference analysis, power allocation[7], and
beamforming optimization. However, the definition of power with these models
is based on a single complex dimensionless variable. The power definition
using circuit models is based on a pair of variables like the current and
voltage or incident and reflected root power waves[17],[5]. For problems like
impedance-matching network design which relate to power flow[4], or analysis
of new array architectures like dynamic metasurface antennas[35], it is
essential to use a circuit model to capture the relevant hardware and
electromagnetic effects like mutual coupling[20] and polarization[36].

The circuit theoretic MIMO models can be further classified based on
impedance/admittance versus scattering parameters. Although impedance and
scattering parameters can be converted to each other through algebraic
transformations, the scattering parameters are more applicable because they
can be easily measured for any general load and directly relate to the flow of
power[5]. Scattering parameters are widely recognized in the RF community for
design of individual RF components like antennas[26], matching networks[5],
and amplifiers[27]. The use of scattering parameters for the analysis of
wireless communication systems allows us to leverage several results on
matching network analysis developed in the microwave and antenna community[2,
31]. It also makes our work generalizable to any practical RF system. As we
target the achievable rate analysis and matching network design problem for
wideband systems, we use the circuit theoretic frequency-selective model with
scattering parameters.

###  II-B A two-port linear network model of communication system

In Fig. 1(a), we represent a communication system with a single source and
single load using a two-port network model[22]. On the transmitter side, the
source generates the transmit signal obtained from the output of the transmit
RF chain, i.e., the signal obtained after up-conversion and amplification.
This signal is input to a cascade of linear networks effectively modeled as a
linear two-port network. The two-port network is used to model several linear
blocks of a communication system like the impedance-matching network, analog
beamforming network, transmit antenna array network, wireless propagation
channel, and receive antenna as shown in Fig. 1(b) and described in Section
II-C. We model the noise contribution from background radiation at the receive
antenna by a voltage source at the output of the linear two-port network[17].
The receiver RF chain is modeled as a load. For simplifying the analysis, we
do not model the low noise amplifier, receive matching network, and intrinsic
noise source.

We use a frequency domain representation for describing the signal flow
through the two-port network. The subscript $``\mathsf{T}"$ indicates
transmitted signal while subscript $``\mathsf{R}"$ indicates received signal.
Let the voltage of the real-valued passband transmit signal in time domain at
the input of the two-port network be $v_{\mathsf{T}}(t)$. Let the current
entering the two-port network be $i_{\mathsf{T}}(t)$. We assume that the
stochastic signals $v_{\mathsf{T}}(t)$ and $i_{\mathsf{T}}(t)$ are Gaussian
wide-sense stationary random processes so that these signals are completely
described by their mean and second-order moments[37]. As these signals do not
have finite energy, a windowed Fourier transform with interval $T_{0}$ is used
for defining the spectrum[38]. For frequency domain representation, we use
frequency $f$ in Hertz. Let
$\mathsf{v}_{\mathsf{T}}(f)=\int_{-\frac{T_{0}}{2}}^{\frac{T_{0}}{2}}v_{\mathsf{T}}(t)e^{-{\mathsf{j}}2\pi
ft}\mathrm{d}t\big{[}\frac{\text{V}}{\text{Hz}}\big{]}$ and
$\mathsf{i}_{\mathsf{T}}(f)=\int_{-\frac{T_{0}}{2}}^{\frac{T_{0}}{2}}i_{\mathsf{T}}(t)e^{-{\mathsf{j}}2\pi
ft}\mathrm{d}t\big{[}\frac{\text{A}}{\text{Hz}}\big{]}$. The main purpose of
using the frequency domain representation is to simplify the analysis.

For further simplification, we use a root power wave representation of the
signals which directly relates to the flow of power[5]. The root power waves
at different ports are related by the scattering parameters, which can be
easily measured compared to impedance or admittance parameters. The root power
wave is a stochastic process which can be expressed as a linear combination of
the current and voltage stochastic processes. In terms of the voltage and
current, assuming characteristic impedance of the transmit circuit as
$\mathsf{Z}_{0}$, the root power wave incident on the two-port network is
defined as
$\mathsf{a}_{\mathsf{T}}(f)=\frac{\mathsf{v}_{\mathsf{T}}(f)+\mathsf{Z}_{0}\mathsf{i}_{\mathsf{T}}(f)}{2\sqrt{\mathcal{R}\\{\mathsf{Z}_{0}\\}}}\big{[}\frac{\sqrt{\text{W}}}{\text{Hz}}\big{]}$
[5, Eq 2.118]. The root power wave reflected back from the two-port network on
the transmit side is defined as
$\mathsf{b}_{\mathsf{T}}(f)=\frac{\mathsf{v}_{\mathsf{T}}(f)-\mathsf{Z}_{0}^{\ast}\mathsf{i}_{\mathsf{T}}(f)}{2\sqrt{\mathcal{R}\\{\mathsf{Z}_{0}\\}}}\big{[}\frac{\sqrt{\text{W}}}{\text{Hz}}\big{]}$
[5, Eq 2.118]. The power spectral density of the power incident (or available)
on the transmitter side of the two-port network is [5, 22]

$P_{\mathsf{T}}(f)=\underset{T_{0}\rightarrow\infty}{\mbox{lim}}\frac{1}{T_{0}}{\mathbb{E}}[|\mathsf{a}_{\mathsf{T}}(f)|^{2}]\left[\frac{\text{W}}{\text{Hz}}\right].$
(1)

Although currents and voltage signals can be used for formulating the
communication system model, the root power wave notation is a good
mathematical tool for simplifying the impedance-matching problem in terms of
metrics like power loss ratio and transmission coefficient[39].

On the receiver side, we assume an ideal load termination $\mathsf{Z}_{0}$ to
avoid reflected root power wave from the load and simplify the analysis. This
load termination requires the use of a receive matching network that transfers
all incident power to the receive RF chain. The modeling of a practical
receive matching network was done in [22] while we focus only on the practical
transmit matching network analysis and design. The root power wave at the
output of the linear two-port network represents the received signal component
$\mathsf{b}_{\mathsf{RS}}(f)$. The voltage source at the receiver port models
the noise from background radiation. The noise voltage source is denoted as
$\mathsf{v}_{\mathsf{N}}(f)$. The resulting root power wave is
$\mathsf{b}_{\mathsf{RN}}(f)=\frac{\mathsf{v}_{\mathsf{N}}(f)}{\sqrt{\mathcal{R}\\{\mathsf{Z}_{0}\\}}}$[20].
Adding the signal and noise root power waves, the resulting root power wave
incident on the load is denoted as $\mathsf{b}_{\mathsf{R}}(f)$[20]. By
replacing $\mathsf{a}_{\mathsf{T}}(f)$ with $\mathsf{b}_{\mathsf{RS}}(f)$ in
(1), we obtain the received signal power spectral density
$P_{\mathsf{RS}}(f)$. Similarly, by replacing $\mathsf{a}_{\mathsf{T}}(f)$
with $\mathsf{b}_{\mathsf{RN}}(f)$ in (1), we obtain the received noise
spectral density $P_{\mathsf{RN}}(f)$. Let $k_{B}$ be the Boltzmann constant
in J/K and $T$ be temperature in K. We set
$P_{\mathsf{RN}}(f)=\mathsf{N}_{0}=k_{B}T\left[\frac{\text{W}}{\text{Hz}}\right]$.
At the receiver, we define
$\mathsf{SNR}(f)=\frac{P_{\mathsf{RS}}(f)}{\mathsf{N}_{0}}$. The
$\mathsf{SNR}(f)$ is non-zero in the band for which $P_{\mathsf{T}}(f)$ is
non-zero.

Assuming Gaussian channel noise, the mutual information per unit time (bits/s)
between the transmit and received Gaussian random process is
$\int_{-\infty}^{\infty}\frac{1}{2}\log_{2}(1+\mathsf{SNR}(f))\mathrm{d}f$.
For a real-valued passband signal, $\mathsf{SNR}(f)$ is symmetric about $f=0$.
This simplifies the definition to an integral over positive frequencies as
$\int_{0}^{\infty}\log_{2}(1+\mathsf{SNR}(f))\mathrm{d}f$[22]. Although
integration upper limit is unbounded, SNR is positive only for a specific
frequency range corresponding to the signal bandwidth which results in a
finite integral value. In Section II-C, we describe the model of a MISO
communication system and define $\mathsf{SNR}(f)$ in terms of
$P_{\mathsf{T}}(f)$ and the scattering parameters of the individual linear
sub-networks.

###  II-C Circuit model of a MISO communication system in terms of scattering
parameters

We analyze the achievable rate of a MISO wireless system consisting of a
transmit array with $N$ antennas and a single receive antenna. The transmitter
hardware consists of a single RF chain connected to $N$ antennas through an
impedance-matching network and an analog beamforming network. The proposed
model is applicable to any general type of antenna, array, analog beamformer,
and matching network. In this paper, we focus on the design of transmit
matching network.

The circuit theoretic model of the MISO communication system is shown in Fig.
1(b). The transmit array with $N$ antennas and the single receive antenna form
an $(N+1)$ port network. Let
$\bm{\mathsf{S}}_{\mathsf{T}}(f)\in{\mathbb{C}}^{N\times N}$ be the scattering
parameter matrix for the transmit array and
$\mathsf{S}_{\mathsf{R}}(f)\in{\mathbb{C}}$ be the scattering parameter of the
receive antenna. The wireless propagation channel scattering parameter is
$\bm{{\mathsf{s}}}_{\mathsf{RT}}^{T}(f)\in{\mathbb{C}}^{1\times N}$, which
accounts for the antenna gains and frequency-selective fading between receiver
and transmitter. Similarly,
$\bm{{\mathsf{s}}}_{\mathsf{TR}}(f)\in{\mathbb{C}}^{N\times 1}$[17]. We assume
that the transmit array is sufficiently far from the receive antenna such that
the signal attenuation between them is large[17, 20]. Hence, we can use the
unilateral approximation by setting
$\bm{{\mathsf{s}}}_{\mathsf{TR}}(f)=\mathbf{0}_{N\times 1}$, i.e., we assume
that the transmitter is unaffected by the electromagnetic fields at the
receiver. The $(N+1)$ port network block scattering parameter matrix is
defined and simplified as

$\bm{\mathsf{S}}_{\mathsf{H}}(f)=\begin{bmatrix}\bm{\mathsf{S}}_{\mathsf{T}}(f)&\bm{{\mathsf{s}}}_{\mathsf{TR}}(f)\\\
\bm{{\mathsf{s}}}_{\mathsf{RT}}^{T}(f)&\mathsf{S}_{\mathsf{R}}(f)\\\
\end{bmatrix}=\begin{bmatrix}\bm{\mathsf{S}}_{\mathsf{T}}(f)&\mathbf{0}_{N\times
1}\\\ \bm{{\mathsf{s}}}_{\mathsf{RT}}^{T}(f)&\mathsf{S}_{\mathsf{R}}(f)\\\
\end{bmatrix}.$ (2)

The incident root power wave vector on the transmit antenna array is denoted
as $\bm{{\mathsf{a}}}_{\mathsf{A}}(f)\in{\mathbb{C}}^{N\times 1}$ and the
reflected root power wave vector is
$\bm{{\mathsf{b}}}_{\mathsf{A}}(f)\in{\mathbb{C}}^{N\times 1}$, as shown in
Fig. 1(b). The root power wave vectors at the receiver and transmitter ports
are related using $\bm{\mathsf{S}}_{\mathsf{H}}(f)$ as

$[\bm{{\mathsf{b}}}^{T}_{\mathsf{A}}(f),\mathsf{b}_{\mathsf{RS}}(f)]^{T}=\bm{\mathsf{S}}_{\mathsf{H}}(f)[\bm{{\mathsf{a}}}^{T}_{\mathsf{A}}(f),0]^{T}.$
(3)

With the unilateral approximation, we can isolate the transmitter circuit
model by writing
$\bm{{\mathsf{b}}}_{\mathsf{A}}(f)=\bm{\mathsf{S}}_{\mathsf{T}}(f)\bm{{\mathsf{a}}}_{\mathsf{A}}(f)$.
Using (2) and (3), the received signal root power wave is

$\mathsf{b}_{\mathsf{RS}}(f)=\bm{{\mathsf{s}}}_{\mathsf{RT}}^{T}(f)\bm{{\mathsf{a}}}_{\mathsf{A}}(f).$
(4)

The unilateral approximation enables simplification of the communication
system analysis and design. It is also reasonable from a practical perspective
because the signal attenuates heavily from transmitter to the receiver [17].

The transmitter network is characterized through the transmit impedance-
matching network and analog beamforming network. The scattering parameter
matrix of the analog beamforming network is denoted as the $(N+1)\times(N+1)$
complex matrix
$\bm{\mathsf{S}}_{\mathsf{F}}(f)=\begin{bmatrix}\mathsf{S}_{{\mathsf{F}},11}(f)&\bm{{\mathsf{s}}}_{{\mathsf{F}},12}^{T}(f)\\\
\bm{{\mathsf{s}}}_{{\mathsf{F}},21}(f)&\bm{\mathsf{S}}_{{\mathsf{F}},22}(f)\\\
\end{bmatrix}$, where $\mathsf{S}_{{\mathsf{F}},11}(f)\in{\mathbb{C}}$,
$\bm{{\mathsf{s}}}_{{\mathsf{F}},12}^{T}(f)\in{\mathbb{C}}^{1\times N}$,
$\bm{{\mathsf{s}}}_{{\mathsf{F}},21}(f)\in{\mathbb{C}}^{N\times 1}$, and
$\bm{\mathsf{S}}_{{\mathsf{F}},22}(f)\in{\mathbb{C}}^{N\times N}$. The
combination of the antenna array and the analog beamforming network can be
treated as an equivalent load with scattering parameter denoted as
$\mathsf{S}_{\mathsf{eq}}(f)$. We express $\mathsf{S}_{\mathsf{eq}}(f)$ in
terms of the scattering parameter matrix elements of the array and the analog
network as [2]

$\displaystyle\mathsf{S}_{\mathsf{eq}}(f)$
$\displaystyle=\mathsf{S}_{{\mathsf{F}},11}(f)+\bm{{\mathsf{s}}}_{{\mathsf{F}},12}^{T}(f)\bm{\mathsf{S}}_{\mathsf{T}}(f)$
(5)
$\displaystyle\times({\mathbf{I}}-\bm{\mathsf{S}}_{{\mathsf{F}},22}(f)\bm{\mathsf{S}}_{\mathsf{T}}(f))^{-1}\bm{{\mathsf{s}}}_{{\mathsf{F}},21}(f).$

The scattering parameter matrix of the transmit impedance-matching network,
which connects the transmit source to the equivalent load, is denoted as
$\bm{\mathsf{S}}_{\mathsf{M}}(f)=\begin{bmatrix}\mathsf{S}_{{\mathsf{M}},11}(f)&\mathsf{S}_{{\mathsf{M}},12}(f)\\\
\mathsf{S}_{{\mathsf{M}},21}(f)&\mathsf{S}_{{\mathsf{M}},22}(f)\\\
\end{bmatrix}$. For the single antenna case, the transmitter network only
consists of the matching network. As there is no analog beamformer for the
single antenna, the scattering parameter of the equivalent load is the
scattering parameter of the antenna, i.e.,
$\mathsf{S}_{\mathsf{eq}}(f)=\mathsf{S}_{\mathsf{T}}(f)$.

To establish a linear relationship between the received signal root power wave
$\mathsf{b}_{\mathsf{RS}}(f)$ and the transmit signal root power wave
$\mathsf{a}_{\mathsf{T}}(f)$, we apply the scattering parameter definition for
each block shown in Fig. 1(b). Let the incident root power wave on the
combined load of antennas and analog beamforming network be denoted as
$\mathsf{a}_{\mathsf{F}}(f)$. The reflected root power wave from the combined
load is
$\mathsf{b}_{\mathsf{F}}(f)=\mathsf{S}_{\mathsf{eq}}(f)\mathsf{a}_{\mathsf{F}}(f)$.
We express $\bm{{\mathsf{a}}}_{\mathsf{A}}(f)$ in terms of
$\mathsf{a}_{\mathsf{F}}(f)$ using the scattering parameter matrix elements as

$\bm{{\mathsf{a}}}_{\mathsf{A}}(f)=({\mathbf{I}}-\bm{\mathsf{S}}_{{\mathsf{F}},22}(f)\bm{\mathsf{S}}_{\mathsf{T}}(f))^{-1}\bm{{\mathsf{s}}}_{{\mathsf{F}},21}(f)\mathsf{a}_{\mathsf{F}}(f).$
(6)

Finally, $\mathsf{a}_{\mathsf{F}}(f)$ is expressed in terms of the incident
root power wave on the transmit impedance-matching network

$\mathsf{a}_{\mathsf{F}}(f)=\frac{\mathsf{S}_{{\mathsf{M}},21}(f)}{1-\mathsf{S}_{{\mathsf{M}},22}(f)\mathsf{S}_{\mathsf{eq}}(f)}\mathsf{a}_{\mathsf{T}}(f).$
(7)

Using (4), (5), (6), (7), we relate $\mathsf{b}_{\mathsf{RS}}(f)$ to
$\mathsf{a}_{\mathsf{T}}(f)$ using a channel coefficient corresponding to an
equivalent SISO channel

$H_{\mathsf{SISO}}(f)=\frac{\bm{{\mathsf{s}}}_{\mathsf{RT}}^{T}(f)({\mathbf{I}}-\bm{\mathsf{S}}_{{\mathsf{F}},22}(f)\bm{\mathsf{S}}_{\mathsf{T}}(f))^{-1}\bm{{\mathsf{s}}}_{{\mathsf{F}},21}(f)\mathsf{S}_{{\mathsf{M}},21}(f)}{1-\mathsf{S}_{{\mathsf{M}},22}(f)\mathsf{S}_{\mathsf{eq}}(f)}.$
(8)

Hence, we have
$\mathsf{b}_{\mathsf{RS}}(f)=H_{\mathsf{SISO}}(f)\mathsf{a}_{\mathsf{T}}(f)$.
The equivalent channel in (8) depends not only on the wireless propagation
channel but also captures the frequency-selectivity effect of antennas and
matching network.

The equivalent channel expression is used for relating the received signal
power spectral density to that available from the transmitter side. Using (8)
and the definition of the power spectral density in (1), we relate
$P_{\mathsf{RS}}(f)$ to $P_{\mathsf{T}}(f)$ as

$P_{\mathsf{RS}}(f)=|H_{\mathsf{SISO}}(f)|^{2}P_{\mathsf{T}}(f)\left[\frac{\text{W}}{\text{Hz}}\right].$
(9)

The $\mathsf{SNR}(f)$ in terms of $P_{\mathsf{T}}(f)$ and the equivalent
channel is

$\mathsf{SNR}(f)=|H_{\mathsf{SISO}}(f)|^{2}\frac{P_{\mathsf{T}}(f)}{\mathsf{N}_{0}}.$
(10)

The mutual information per unit time is
$\int_{0}^{\infty}\log_{2}\left(1+|H_{\mathsf{SISO}}(f)|^{2}\frac{P_{\mathsf{T}}(f)}{\mathsf{N}_{0}}\right)\mathrm{d}f$.
It depends on the design of the matching network and the transmit power
allocation at each frequency. We assume a bandlimited source that supplies a
maximum power per frequency
$\mathsf{E}_{\mathsf{s}}\left[\frac{\text{W}}{\text{Hz}}\right]$ for
$f\in[f_{\mathsf{min}},f_{\mathsf{max}}]$[22]. For the bandwidth
$B=f_{\mathsf{max}}-f_{\mathsf{min}}$, we assumed a fixed total supplied power
$B\mathsf{E}_{\mathsf{s}}\text{W}$. Assuming that the source supplies the
maximum available power at each frequency, we define the achievable rate in
bits/s as

$\mathsf{R}=\int_{f_{\mathsf{min}}}^{f_{\mathsf{max}}}\log_{2}\left(1+|H_{\mathsf{SISO}}(f)|^{2}\frac{\mathsf{E}_{\mathsf{s}}}{\mathsf{N}_{0}}\right)\mathrm{d}f.$
(11)

In Section III, we optimize the achievable rate $\mathsf{R}$ by optimally
designing the matching network.

##  III Optimizing achievable rate under Bode-Fano matching constraints

In this section, we propose a general framework for optimizing the achievable
rate as a function of the matching network. The rate depends on the matching
network through the term $H_{\mathsf{SISO}}(f)$ as shown in (8) which depends
on the matching network scattering parameter matrix elements
$\mathsf{S}_{{\mathsf{M}},21}(f)$ and $\mathsf{S}_{{\mathsf{M}},22}(f)$. The
values for these elements at each frequency are hardware specific, i.e.,
dependent on the actual physical elements like the inductors and capacitors
used for building the matching network circuit. One approach of optimizing
rate is to define the rate objective in terms of the physical component values
for a fixed structure[4]. A limitation of this approach is that it does not
guarantee that the specific circuit gives better performance theoretically
than any other physically realizable matching network. In this paper, instead
of optimizing the rate in terms of the physical components of a specific
matching circuit, we formulate and solve a general problem that applies to any
matching network made of passive and linear elements. In Section III-A, we
describe the constraints associated with any general passive matching network.

###  III-A General form of the Bode-Fano matching constraint

The first systematic approach to study the bandwidth limitation of a matching
network was proposed by Bode for a special type of reactive load[8]. The work
by Bode was generalized by Fano for arbitrary reactive loads[9]. These results
on the bandwidth limitation of matching networks are popularly known as Bode-
Fano limits[5]. Recently, a generalization of the Bode-Fano matching limits
was proposed for arbitrary loads like an antenna array with an analog
beamforming network[2, 31]. We use these results from [2, 31] to formulate the
matching network constraints in the achievable rate optimization problem.

The Bode-Fano constraints place a bound on the power loss ratio metric [2].
The constraints are expressed in terms of the power loss ratio, which
indicates the ratio of expected power lost (due to reflection and dissipation)
to the expected input power where the expectation is over the input random
signal. Mathematically, it is given as

$\displaystyle r^{2}(f)$
$\displaystyle=1-\frac{{\mathbb{E}}[|{\mathsf{a}}_{\mathsf{F}}(f)|^{2}-|{\mathsf{b}}_{\mathsf{F}}(f)|^{2}]}{{\mathbb{E}}[|{\mathsf{a}}_{\mathsf{T}}(f)|^{2}]}$
(12)
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}1-\frac{|\mathsf{S}_{{\mathsf{M}},21}(f)|^{2}}{|1-\mathsf{S}_{{\mathsf{M}},22}(f)\mathsf{S}_{\mathsf{eq}}(f)|^{2}}(1-|\mathsf{S}_{\mathsf{eq}}(f)|^{2}),$

where equality $(a)$ follows from the definition of
$\mathsf{b}_{\mathsf{F}}(f)$ and (7). We also define the transmission
coefficient as

$\mathcal{T}(f)=1-r^{2}(f).$ (13)

A lower value of $r^{2}(f)$ or a higher value of $\mathcal{T}(f)$ for a
specified bandwidth indicates a better power transfer to the equivalent load
in the desired band.

Cauchy’s integral relations in complex variable calculus can be applied to any
linear circuit model for deriving Bode-Fano constraints[9]. Mathematically, it
is convenient to analyze the circuit model as a function of the complex
frequency $s=\sigma+{\mathsf{j}}2\pi f$[9][31]. Similar to the original Bode-
Fano constraints, we assume that the load should be realizable by means of
finite passive lumped elements[9]. Therefore, we use a rational approximation
of $\mathsf{S}_{\mathsf{eq}}(f)$ defined in the whole complex plane and
denoted as $\hat{\mathsf{S}}_{\mathsf{eq}}(s)$[31]. We assume that
$\hat{\mathsf{S}}_{\mathsf{eq}}(s)$ should be in the rational form and satisfy
the passivity condition[31]. The guidelines for obtaining
$\hat{\mathsf{S}}_{\mathsf{eq}}(s)$ from $\mathsf{S}_{\mathsf{eq}}(f)$ are
discussed in detail in [31]. Note that $\hat{\mathsf{S}}_{\mathsf{eq}}(s)$ is
not unique and depends on the technique used to approximate
$\mathsf{S}_{\mathsf{eq}}(f)$. We briefly summarize the approximation
techniques from [31] in Appendix A.

The Bode-Fano theory provides a set of constraints on the power loss ratio
$r^{2}(f)$ for any passive and linear impedance-matching network terminated
with a passive load realized using lumped elements. These constraints are
expressed as bounds on the integral of logarithm of the power loss ratio [2,
15, 9]. For a simple load of resistor $R$ and capacitor $C$ in parallel, there
is only one Bode-Fano constraint expressed as
$\int_{0}^{\infty}\log\left(\frac{1}{r^{2}(f)}\right)\mathrm{d}f\leq\frac{1}{RC}$
[5]. For an arbitrary load, the number of necessary constraints for the
physical realizability of $r^{2}(f)$ is determined using a Darlington
equivalent representation of the load. From Darlington’s theory, any
physically realizable impedance is equivalent to the input impedance of a
reactive two-port network terminated with a $1\Omega$ resistor[40]. The number
of such necessary constraints on $r^{2}(f)$ equals the number of independent
parameters used to define the Darlington equivalent network of the load[9].
For example, for a load of resistor, inductor, and capacitor in series, the
Darlington equivalent is specified using the quality factor value and the
resonant frequency which results in two Bode-Fano bounds[15, Eq 11].

For an equivalent load with rational approximation
$\hat{\mathsf{S}}_{\mathsf{eq}}(s)$, we assume there are $N_{\mathsf{BF}}$
number of distinct constraints for describing the physical realizability of
the power loss ratio. For the $i$th constraint where
$i\in\\{1,2,\dots,N_{\mathsf{BF}}\\}$, we define two positive terms
$\xi_{\mathsf{BF},i}(f)$ and $B_{\mathsf{BF},i}$. The term
$\xi_{\mathsf{BF},i}(f)$ is a prelog term in the integrand which is multiplied
by the logarithm of the power loss ratio. The term $B_{\mathsf{BF},i}$ is an
upper bound on the Bode-Fano integral. For a load whose scattering parameter
after rational approximation is $\hat{\mathsf{S}}_{\mathsf{eq}}(s)$, the
$N_{\mathsf{BF}}$ distinct constraints required for the realizability of power
loss ratio $r^{2}(f)$ are[2]

$\int_{0}^{\infty}\\!\\!\\!\xi_{\mathsf{BF},i}(f)\log\left(\frac{1}{r^{2}(f)}\right)\mathrm{d}f\leq
B_{\mathsf{BF},i},\text{for }\\{i\\}_{1}^{N_{\mathsf{BF}}},$ (14)

where $\xi_{\mathsf{BF},i}(f)$ and $B_{\mathsf{BF},i}$ are positive terms
evaluated using $\hat{\mathsf{S}}_{\mathsf{eq}}(s)$ as discussed in Appendix
B. The detailed proof of the general form of the Bode-Fano constraint is given
in [31].

In Section III-B, we will use the constraints defined in (14) for formulating
the achievable rate optimization problem as a function of the transmission
coefficient $\mathcal{T}(f)$ defined in (13). Before proceeding to the problem
formulation, we first rewrite the achievable rate in terms of the transmission
coefficient so that both constraints and objective in the optimization can be
expressed as function of the variable $\mathcal{T}(f)$.

###  III-B Achievable rate optimization problem formulation

The achievable rate metric depends on the matching network through the
scattering parameters $\mathsf{S}_{{\mathsf{M}},21}(f)$ and
$\mathsf{S}_{{\mathsf{M}},22}(f)$ as shown through (8) and (11). Using (8),
(10), (12), and (13), we express $\mathsf{SNR}(f)$ in terms of
$\mathcal{T}(f)$ as

$\displaystyle\mathsf{SNR}(f)$
$\displaystyle=|\bm{{\mathsf{s}}}_{\mathsf{RT}}^{T}(f)({\mathbf{I}}-\bm{\mathsf{S}}_{{\mathsf{F}},22}(f)\bm{\mathsf{S}}_{\mathsf{T}}(f))^{-1}\bm{{\mathsf{s}}}_{{\mathsf{F}},21}(f)|^{2}$
(15)
$\displaystyle\times\frac{P_{\mathsf{T}}(f)\mathcal{T}(f)}{(1-|\mathsf{S}_{\mathsf{eq}}(f)|^{2})\mathsf{N}_{0}}.$

From (15), the achievable rate expression is

$\displaystyle\mathsf{R}=$
$\displaystyle\int_{f_{\mathsf{min}}}^{f_{\mathsf{max}}}\\!\\!\\!\\!\log_{2}\bigg{(}1+\frac{|\bm{{\mathsf{s}}}_{\mathsf{RT}}^{T}(f)({\mathbf{I}}-\bm{\mathsf{S}}_{{\mathsf{F}},22}(f)\bm{\mathsf{S}}_{\mathsf{T}}(f))^{-1}\bm{{\mathsf{s}}}_{{\mathsf{F}},21}(f)|^{2}}{(1-|\mathsf{S}_{\mathsf{eq}}(f)|^{2})}$
(16)
$\displaystyle\times\frac{\mathsf{E}_{\mathsf{s}}}{\mathsf{N}_{0}}\mathcal{T}(f)\bigg{)}\mathrm{d}f.$

In the ideal matching network case, i.e., no power loss due to reflection or
dissipation, $\mathcal{T}(f)=1$, and the ideal lossless SNR is expressed as

$\mathsf{SNR}_{\text{ideal}}(f)=\frac{|\bm{{\mathsf{s}}}_{\mathsf{RT}}^{T}(f)({\mathbf{I}}-\bm{\mathsf{S}}_{{\mathsf{F}},22}(f)\bm{\mathsf{S}}_{\mathsf{T}}(f))^{-1}\bm{{\mathsf{s}}}_{{\mathsf{F}},21}(f)|^{2}}{(1-|\mathsf{S}_{\mathsf{eq}}(f)|^{2})}\frac{\mathsf{E}_{\mathsf{s}}}{\mathsf{N}_{0}}.$
(17)

The achievable rate for the ideal case is
$\mathsf{R}_{\text{ideal}}=\int_{f_{\mathsf{min}}}^{f_{\mathsf{max}}}\log_{2}\left(1+\mathsf{SNR}_{\text{ideal}}(f)\right)\mathrm{d}f$.
The ideal SNR depends on the wireless propagation channel, the scattering
parameters of the antenna array, and the analog beamforming network but does
not depend on the matching network. For a physically realizable matching
network, $\mathcal{T}(f)\leq 1$. Therefore,
$\mathsf{R}\leq\mathsf{R}_{\text{ideal}}$ meaning that the achievable rate is
over-estimated when Bode-Fano constraints are disregarded.

We formulate the achievable rate optimization problem to optimally design the
transmission coefficient $\mathcal{T}(f)$. In (16), we defined the
optimization objective in terms of $\mathcal{T}(f)$. Similarly, the Bode-Fano
inequalities from (14) can be expressed in terms of $\mathcal{T}(f)$ using
(13). The achievable rate optimization problem in terms of $\mathcal{T}(f)$
using (14), (16), and (17) is

$\displaystyle\mathsf{R}_{\text{max}}=\underset{\mathcal{T}(f)}{\mbox{ max
}}\int_{f_{\mathsf{min}}}^{f_{\mathsf{max}}}\log_{2}(1+\mathsf{SNR}_{\text{ideal}}(f)\mathcal{T}(f))\mathrm{d}f,$
(18a) $\displaystyle\text{s.t.
}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!\xi_{\mathsf{BF},i}(f)\log\left(\frac{1}{1-\mathcal{T}(f)}\right)\mathrm{d}f\leq
B_{\mathsf{BF},i},\text{for }\\{i\\}_{1}^{N_{\mathsf{BF}}}$ (18b)
$\displaystyle 0\leq\mathcal{T}(f)\leq 1.$ (18c)

The constraint in (18c) follows from the definition of the transmission
coefficient in (13). Comparing this formulation to [22], the key difference is
that $\mathcal{T}(f)$ was defined on the receiver side between a single
antenna and the low-noise amplifier in [22]. This led to $\mathcal{T}(f)$
appearing in both signal power and the extrinsic noise power in [22, Eq 21].
The problem formulation in our work expressed in (18) uses the
$\mathcal{T}(f)$ defined between the transmit RF chain and the equivalent load
of multiple transmit antennas and analog beamforming network. So
$\mathcal{T}(f)$ appears only in the signal power leading to an optimal
solution expression different from [22, Eq 25].

###  III-C Optimal transmission coefficient

We use the Lagrangian to solve the optimization problem in (18). The total
number of constraints in (18b) and (18c) is $N_{\mathsf{BF}}+2$. For the $i$th
constraint, we denote the Lagrangian parameter as $\mu_{i}$. The Lagrangian is
[41]

$\displaystyle\mathcal{L}\left(\mathcal{T}(f),\mu_{i}|_{i=1}^{N_{\mathsf{BF}}+2}\right)=-\int_{f_{\mathsf{min}}}^{f_{\mathsf{max}}}\log_{2}(1+\mathsf{SNR}_{\text{ideal}}(f)\mathcal{T}(f))\mathrm{d}f$
$\displaystyle+\sum_{i=1}^{N_{\mathsf{BF}}}\mu_{i}\left(\int_{0}^{\infty}\xi_{\mathsf{BF},i}(f)\log\left(\frac{1}{1-\mathcal{T}(f)}\right)\mathrm{d}f-B_{\mathsf{BF},i}\right)$
$\displaystyle-\mu_{N_{\mathsf{BF}}+1}\mathcal{T}(f)+\mu_{N_{\mathsf{BF}}+2}(\mathcal{T}(f)-1).$
(19)

The solution to (18) is obtained after applying Karush-Kuhn-Tucker (KKT)
conditions in Appendix C[22]. The maximum rate is defined in terms of the
optimal transmission coefficient $\mathcal{T}^{\star}(f)$ as

$\mathsf{R}_{\text{max}}=\int_{f_{\mathsf{min}}}^{f_{\mathsf{max}}}\log_{2}(1+\mathsf{SNR}_{\text{ideal}}(f)\mathcal{T}^{\star}(f))\mathrm{d}f.$
(20)

The expression for $\mathcal{T}^{\star}(f)$ is in terms of the optimal
Lagrangian parameters $\mu^{\star}_{i}|_{i=1}^{N_{\mathsf{BF}}}$ described as
follows.

###### Theorem 1

The relationship between the variables $\mathcal{T}^{\star}(f)$ and
$\mu^{\star}_{i}|_{i=1}^{N_{\mathsf{BF}}}$ corresponding to the optimal
solution of the optimization problem in (18) is as follows.

$\displaystyle\mathcal{T}^{\star}(f)=\left[\frac{1-\ln
2\frac{\sum_{i=1}^{{N_{\mathsf{BF}}}}\mu^{\star}_{i}\xi_{\mathsf{BF},i}(f)}{\mathsf{SNR}_{\text{ideal}}(f)}}{1+\ln
2\sum_{i=1}^{N_{\mathsf{BF}}}\mu^{\star}_{i}\xi_{\mathsf{BF},i}(f)}\right]^{+},\quad\mu^{\star}_{i}|_{i=1}^{N_{\mathsf{BF}}}\geq
0,$ (21a)
$\displaystyle\mu_{i}^{\star}\\!\left(\int_{0}^{\infty}\\!\\!\\!\xi_{\mathsf{BF},i}(f)\log\left(\frac{1}{1-\mathcal{T}^{\star}(f)}\right)\mathrm{d}f-B_{\mathsf{BF},i}\right)=0,\\{i\\}_{1}^{N_{\mathsf{BF}}},$
(21b)
$\displaystyle\left(\int_{0}^{\infty}\\!\\!\\!\xi_{\mathsf{BF},i}(f)\log\left(\frac{1}{1-\mathcal{T}^{\star}(f)}\right)\mathrm{d}f-B_{\mathsf{BF},i}\right)\leq
0,\\{i\\}_{1}^{N_{\mathsf{BF}}}.$ (21c)

###### Proof:

Refer to Appendix C for proof. ∎

The expression for the optimal transmission coefficient computed using (21)
can be interpreted as waterfilling in the frequency domain which is a
fundamental result by Shannon in information theory[42]. From (21a), we
observe that $\mathcal{T}^{\star}(f)$ is higher for frequencies with better
$\mathsf{SNR}_{\text{ideal}}(f)$. As $\mathsf{SNR}_{\text{ideal}}(f)$ is
inversely proportional to bandwidth, the peak gain in $\mathcal{T}^{\star}(f)$
for higher bandwidths is lower and vice versa. This shows that the fundamental
gain-bandwidth tradeoff of matching networks is captured in the expression of
$\mathcal{T}^{\star}(f)$ in (21a).

The variables $\mathcal{T}^{\star}(f)$ and
$\mu^{\star}_{i}|_{i=1}^{N_{\mathsf{BF}}}$ corresponding to the optimal
solution of the optimization problem in (18) are tightly coupled in the
equations (21a), (21b), and (21c). We use a numerical approach to compute a
sub-optimal solution. We set all but one Lagrangian parameters to 0, apply a
bisection search on the non-zero parameter till (21b) is satisfied within a
threshold, and repeat this process for all parameters to obtain the values of
$\mu^{\star}_{i}|_{i=1}^{N_{\mathsf{BF}}}$ that maximize the rate. This low-
complexity numerical approach ensures that (21b) is satisfied for
$N_{\mathsf{BF}}-1$ parameters and within a specific tolerance for one
parameter. The approximate solution for $\mathcal{T}^{\star}(f)$ is obtained
by substituting the optimized values of
$\mu^{\star}_{i}|_{i=1}^{N_{\mathsf{BF}}}$ in (21a).

All passive and linear matching networks will provide an achievable rate less
than the value of $\mathsf{R}_{\text{max}}$. This maximum achievable rate
based on Bode-Fano bounds is more accurate than $\mathsf{R}_{\text{ideal}}$
which disregards the matching theory. This rate $\mathsf{R}_{\text{max}}$ is a
new benchmark for designing matching networks instead of a metric like power
transfer efficiency which does not capture the effect of wireless propagation
channel.

##  IV Matching network circuit design methodology and illustrations

From a system design perspective, it is crucial to provide a practical
methodology to approximate the theoretical achievable rate bound from Section
III. In this section, we address the second question, “How to design
impedance-matching networks to approximate this achievable rate bound?” We
propose a practical matching network design approach using
$\mathcal{T}^{\star}(f)$ from (21a).

###  IV-A General methodology to design matching network

We provide a three step procedure to design a matching network based on the
achievable rate upper bound as follows.

  1. 1.

Evaluation of Bode-Fano constraints

    1. (a)

For a given scattering matrix of an antenna $\bm{\mathsf{S}}_{\mathsf{T}}(f)$
and analog beamforming network $\bm{\mathsf{S}}_{\mathsf{F}}(f)$, obtain a
passive rational approximation as a function of the complex frequency to
evaluate the rational function of the scalar equivalent load
$\hat{\mathsf{S}}_{\mathsf{eq}}(s)$.

    2. (b)

Evaluate $N_{\mathsf{BF}}$ Bode-Fano constraints using the expression of
$\hat{\mathsf{S}}_{\mathsf{eq}}(s)$ based on Table I in Appendix B.

  2. 2.

Optimal transmission coefficient

    1. (a)

Solve the optimization problem (18) for the $N_{\mathsf{BF}}$ Bode-Fano
constraints by numerically solving the system of equations and inequalities
given by (21).

    2. (b)

Compute $\mathcal{T}^{\star}(f)$ for the optimized
$\mu^{\star}_{i}|_{i=1}^{N_{\mathsf{BF}}}$ using (21a).

  3. 3.

Approximating $\mathcal{T}^{\star}(f)$ with a practical matching network
topology

    1. (a)

Choose a general reactive ladder circuit with a fixed order.

    2. (b)

Optimize the component values of the matching network circuit topology to fit
the desired frequency response of the optimal transmission coefficient
$\mathcal{T}^{\star}(f)$.

In the first step, overfitting when approximating
$\mathsf{S}_{\mathsf{eq}}(f)$ with $\hat{\mathsf{S}}_{\mathsf{eq}}(s)$ can
result in loose Bode-Fano bounds. Sometimes, there exists poles and zeros in
the rational approximation which are close to each other. As observed from
expression of $B_{\mathsf{BF}}$ from Table I in Appendix B, overfitting may
result in higher bounding values of $B_{\mathsf{BF}}$[31]. Overfitting is an
issue if the computed value of $\mathcal{T}^{\star}(f)$ is close to one even
for higher bandwidths. The gain-bandwidth tradeoff will not be captured due to
overfitting. This issue can be avoided by reducing the model order.

Figure 2:  Circuit model of the 4th order LC ladder in ADS software. This
simple topology can be used to approximate the transmission coefficient
response obtained from the optimization problem.

###  IV-B Application of matching network design methodology to a single Chu’s
antenna

We present numerical illustrations for the matching network design
methodology. For simulations, we use Chu’s antenna at receiver and
transmitter, similar to [22], to provide generic insights without using a
specific antenna design. Although Bode-Fano bounds depend on the antenna type,
the achievable rate optimization methodology is general and can be applied to
any antenna design.

Chu’s antenna model: Let the Chu’s antenna structure be enclosed in a
spherical volume with radius $a$. Let the speed of light be denoted as $c$ and
characteristic resistance be denoted as $R$. The Chu’s electric antenna is
modeled with an equivalent circuit consisting of a capacitor $C=\frac{a}{cR}$
in series with a parallel combination of an inductor $L=\frac{aR}{c}$ and
resistor $R$ [22, Fig. 2]. The input impedance is
$\mathsf{Z}_{\mathsf{T}}(f)=\mathsf{Z}_{\mathsf{R}}(f)=\frac{R}{{\mathsf{j}}2\pi
f\frac{a}{c}}+\frac{R}{1+({{\mathsf{j}}2\pi f\frac{a}{c}})^{-1}}$. The
scattering parameter in rational form is
$\hat{\mathsf{S}}_{\mathsf{T}}(s)=({2s^{2}\frac{a^{2}}{c^{2}}+2s\frac{a}{c}+1})^{-1}.$
For a single antenna, there is no analog beamforming network, hence,
$\hat{\mathsf{S}}_{\mathsf{eq}}(s)=\hat{\mathsf{S}}_{\mathsf{T}}(s)$.

Bode-Fano bounds for a single Chu’s antenna: Substituting
$\hat{\mathsf{S}}_{\mathsf{eq}}(s)$ in (26), we obtain $s^{4}=0$. For repeated
roots with multiplicity 4, we apply [31, Eq. 22] to derive the bounds

$\displaystyle\int_{0}^{\infty}\frac{1}{2\pi^{2}f^{2}}\log\left(\frac{1}{1-\mathcal{T}(f)}\right)\mathrm{d}f\leq\frac{2a}{c},$
(22a)
$\displaystyle\int_{0}^{\infty}\frac{1}{8\pi^{4}f^{4}}\log\left(\frac{1}{1-\mathcal{T}(f)}\right)\mathrm{d}f\leq\frac{4a^{3}}{3c^{3}}.$
(22b)

The bounds are in the form specified in (18b) with $N_{\mathsf{BF}}=2$.

(a) Bandwidth of 4.2 GHz.

(b) Bandwidth of 0.7 GHz.

Figure 3:  For (a), the optimal transmission curve is higher than the
frequency-flat transmission curve for frequencies greater than 6.1 GHz. For
(b), the optimal transmission coincides with the frequency-flat transmission
curves. Conjugate matching response is same for both bandwidths whereas the
proposed optimal transmission and frequency-flat transmission responses change
with the bandwidth.

Parameter setup: Let $f_{\mathsf{c}}=7$ GHz. The corresponding wavelength
$\lambda_{\mathsf{c}}=4.29$ cm. Let $a=\frac{\lambda_{\mathsf{c}}}{10}=4.29$
mm, bandwidth $B\in\\{0.1f_{\mathsf{c}},0.6f_{\mathsf{c}}\\}=\\{0.7,4.2\\}$
GHz,
$\mathsf{E}_{\mathsf{s}}=\frac{0.25}{B}\left[\frac{\text{W}}{\text{Hz}}\right]$,
and $R=50\Omega$. Let the distance between receiver and transmitter be
$d_{\mathsf{t}x-rx}=500$m and the antenna gain $G=1.5$. The wireless channel
is known at the transmitter and defined as
$\mathsf{S}_{\mathsf{RT}}(f)=\frac{1-\mathsf{S}_{\mathsf{T}}(f)}{\mathsf{Z}_{0}+Z_{\mathsf{R}}(f)}\frac{c}{2\pi
fd_{\mathsf{t}x-rx}}G\mathcal{R}(Z_{\mathsf{T}}(f))$[20]. Using
$k_{B}=1.38\times 10^{-23}$ J/K and $T=290K$, we get $\mathsf{N}_{0}=4\times
10^{-21}\left[\frac{\text{W}}{\text{Hz}}\right]$.

Optimal transmission coefficient approximation with an LC ladder: We compute
the optimal transmission coefficient by solving the achievable rate
optimization problem using two Bode-Fano bounds in (22). To approximate this
transmission coefficient, it suffices to use a simple 4th order LC ladder
shown in Fig. 2. We use Keysight ADS which is a circuit simulation software
for characterizing and optimizing RF systems. In ADS, we define 8 design
variables, $L_{i}$ and $C_{i}$ for $\\{i\\}_{1}^{4}$, and set the optimization
goal based on the optimal transmission coefficient. The output is the
optimized values of $L_{i}$ and $C_{i}$. The transmission coefficient
corresponding to the optimized circuit is used as a comparison benchmark.

Frequency-flat transmission coefficient approximation with an LC ladder: For
comparison with the box-car matching approach in [15], we assume frequency-
flat transmission coefficient in a band spanning $f_{\mathsf{min}}$ to
$f_{\mathsf{max}}$, i.e., $\mathcal{T}(f)=\mathcal{T}_{\mathsf{ff}}$ for
$f\in[f_{\mathsf{min}},f_{\mathsf{max}}]$. The value of
$\mathcal{T}_{\mathsf{ff}}$ should satisfy both (22a) and (22b). We define
$r_{1}=\exp\bigg{(}\frac{-2a/c}{\int_{f_{\mathsf{min}}}^{f_{\mathsf{max}}}{1}/({2\pi^{2}f^{2}})\mathrm{d}f}\bigg{)}$
and
$r_{2}=\exp\left(\frac{-{4a^{3}}/({3c^{3}})}{\int_{f_{\mathsf{min}}}^{f_{\mathsf{max}}}{1}/({8\pi^{4}f^{4}})\mathrm{d}f}\right)$.
The value of $\mathcal{T}_{\mathsf{ff}}$ satisfying both constraints is
$\mathcal{T}_{\mathsf{ff}}=1-\mbox{Max}\\{r_{1},r_{2}\\}$. The frequency-flat
transmission coefficient is approximated similarly using a 4th order LC ladder
in ADS.

In Fig. 3, we plot the transmission coefficient versus frequency for the
theoretical case and the 4th order LC ladder circuit in Fig. 2 optimized in
ADS. We see that the circuit implemented in ADS provides a good approximation
of the desired transmission coefficient in the bandwidth of interest. This
shows that with a simple matching network topology, it is possible to
approximate the transmission coefficient. We also show the conjugate matching
transmission coefficient benchmark in Fig. 3 which remains the same
irrespective of the bandwidth. The proposed optimal transmission coefficient
curves are dependent on the bandwidth. In Fig. 3(a), although conjugate
matching transmission has higher $\mathcal{T}(f)$ than the optimal
transmission for frequencies 6.5 GHz to 7.4 GHz, this is because the optimal
transmission is optimized for a larger band from 4.9 GHz to 9.1 GHz. In Fig.
3(b), the optimal transmission is optimized for 6.65 GHz to 7.35 GHz and is
higher than fixed conjugate matching. We use these transmission coefficients
in Section V to compute the SNR, achievable rate, and its comparison with
other benchmarks.

###  IV-C Application of matching network design methodology to an array of
two Chu’s antennas

We present numerical illustrations for the matching network design methodology
applied to an array of two Chu’s antennas.

Chu’s antenna array model: We assume an array of two parallel Chu’s antennas,
each enclosed in a spherical volume of radius $a$ and separated by a distance
$d$. The self impedance for each antenna is
$\mathsf{Z}_{\mathsf{T11}}(f)=\mathsf{Z}_{\mathsf{T22}}(f)=\frac{R}{{\mathsf{j}}2\pi
f\frac{a}{c}}+\frac{R}{1+\frac{1}{{\mathsf{j}}2\pi f\frac{a}{c}}}$. The mutual
impedance between two antennas is [23]

$\displaystyle\mathsf{Z}_{\mathsf{T12}}(f)=\mathsf{Z}_{\mathsf{T21}}(f)=-1.5\sqrt{\mathcal{R}(\mathsf{Z}_{\mathsf{T11}}(f))\mathcal{R}(\mathsf{Z}_{\mathsf{T22}}(f))}$
(23) $\displaystyle\times\left(\\!\frac{1}{{\mathsf{j}}2\pi
f\frac{d}{c}}-\frac{1}{(2\pi f\frac{d}{c})^{2}}+\frac{j}{(2\pi
f\frac{d}{c})^{3}}\\!\right)\\!e^{-{\mathsf{j}}2\pi f\frac{d}{c}}.$

The array impedance matrix is defined as
$\bm{\mathsf{Z}}_{\mathsf{T}}(s)=\begin{bmatrix}\mathsf{Z}_{\mathsf{T11}}(s)&\mathsf{Z}_{\mathsf{T12}}(s)\\\
\mathsf{Z}_{\mathsf{T21}}(s)&\mathsf{Z}_{\mathsf{T22}}(s)\end{bmatrix}.$ The
scattering matrix is
$\bm{\mathsf{S}}_{\mathsf{T}}(s)=(\bm{\mathsf{Z}}_{\mathsf{T}}(s)+R{\mathbf{I}}_{2})^{-1}(\bm{\mathsf{Z}}_{\mathsf{T}}(s)-R{\mathbf{I}}_{2})$.

Analog beamforming network model: For the scattering matrix of the analog
beamforming network, we assume $\mathsf{S}_{{\mathsf{F}},11}(f)=0$,
$\bm{\mathsf{S}}_{{\mathsf{F}},22}(f)=\mathbf{0}_{2}$, and
$\bm{{\mathsf{s}}}_{{\mathsf{F}},12}(f)=\bm{{\mathsf{s}}}_{{\mathsf{F}},21}^{T}(f)$
represents the beamforming vector corresponding to an ideal frequency-flat
phased array. We simulate two beamforming modes similar to [31]. The even mode
corresponds to
$\bm{{\mathsf{s}}}_{{\mathsf{F}},21}(f)=\frac{-{\mathsf{j}}}{\sqrt{2}}[1,1]^{T}$
and the odd mode corresponds to
$\bm{{\mathsf{s}}}_{{\mathsf{F}},21}(f)=\frac{-{\mathsf{j}}}{\sqrt{2}}[1,-1]^{T}$.
We also assume no insertion loss.

(a) Odd mode beamforming for $\theta=\frac{\pi}{2}$

(b) Even mode beamforming for $\theta=0$

Figure 4: For an array of two Chu’s antennas with even and odd mode analog
beamforming, the optimal transmission curves are higher than the frequency-
flat transmission curves for a major portion of the bandwidth. As matching
network order is fixed for both beamforming modes, the gap between theoretical
and circuit response is different.

Parameter setup: For $f_{\mathsf{c}}=7$ GHz, let
$d=\frac{\lambda_{\mathsf{c}}}{2}$ and $a=\frac{\lambda_{\mathsf{c}}}{15}$,
bandwidth $B=0.6f_{\mathsf{c}}=4.2$ GHz, and
$\mathsf{E}_{\mathsf{s}}=\frac{0.25}{B}\left[\frac{\text{W}}{\text{Hz}}\right]$.
For receiver at angle $\theta$ from broadside, the wireless channel is [20]

$\bm{{\mathsf{s}}}_{\mathsf{RT}}(f)=\frac{cG\mathcal{R}(\mathsf{Z}_{\mathsf{T}}(f))({\mathbf{I}}_{2}-\bm{\mathsf{S}}_{\mathsf{T}}(f))[1,\exp({\mathsf{j}}2\pi
f\frac{d}{c}\sin(\theta))]^{T}}{2\pi
fd_{\mathsf{t}x-rx}(\mathsf{Z}_{0}+\mathsf{Z}_{\mathsf{R}}(f))}.$ (24)

In (24), the mutual coupling effect between antennas is captured through the
dependence on $\bm{\mathsf{S}}_{\mathsf{T}}(f)$.

Bode-Fano bounds for Chu’s antenna array: For both even and odd beamforming
modes, we compute $\hat{\mathsf{S}}_{\mathsf{eq}}(s)$ and substitute it in
(26) to solve for $s$. We obtain two unique roots with $\mathcal{R}(s_{0})>0$
which correspond to two Bode-Fano inequalities computed using Table I in
Appendix B.

Optimal transmission coefficient approximation with an LC ladder: We compute
the optimal transmission coefficient by solving the achievable rate
optimization problem using two Bode-Fano constraints. For approximating
$\mathcal{T}^{\star}(f)$, it suffices to choose a 7th order LC ladder. In ADS,
we define 14 design variables: $L_{i}$ and $C_{i}$ for $\\{i\\}_{1}^{7}$ and
set the optimization goal based on $\mathcal{T}^{\star}(f)$. Note that the
choice of model order can be changed depending on other design requirements.

Frequency-flat transmission coefficient approximation with an LC ladder: For
comparison with the box-car matching approach in [15], let
$\mathcal{T}_{\mathsf{ff}}$ satisfy both Bode-Fano constraints. Using Table I
in Appendix B, for $\\{i\\}_{1}^{2}$, we define

$r_{i}=\exp\left(\\!\\!\frac{\log\left(\left|\hat{\mathsf{S}}_{\mathsf{eq}}(s_{i})\frac{\prod_{\ell=1}^{N_{\mathsf{z}}}(s_{i}+z_{\mathsf{eq},\ell})}{\prod_{\ell=1}^{N_{\mathsf{z}}}(s_{i}-z_{\mathsf{eq},\ell})}\right|\right)}{\int_{f_{\mathsf{min}}}^{f_{\mathsf{max}}}\mathcal{R}\\{(s_{i}-{\mathsf{j}}2\pi
f)^{-1}+(s_{i}+{\mathsf{j}}2\pi f)^{-1}\\}\mathrm{d}f}\\!\\!\right)\\!\\!.$
(25)

The value of $\mathcal{T}_{\mathsf{ff}}$ satisfying both constraints is
$\mathcal{T}_{\mathsf{ff}}=1-\mbox{Max}\\{r_{1},r_{2}\\}$. The frequency-flat
transmission coefficient can be approximated similarly using a 7th order LC
ladder in ADS.

In Fig. 4, we plot the transmission coefficient versus frequency for the
theoretical case and the 7th order LC ladder circuit optimized in ADS. For
$\theta=\frac{\pi}{2}$, i.e., endfire incidence, we use the odd mode
beamforming as shown in Fig. 4(a), whereas even mode beamforming is used for
the broadside incidence as shown in Fig. 4(b). We observe that for a major
portion of the 4.2 GHz bandwidth for both beamforming modes, optimal
transmission response is higher than frequency-flat transmission response.
This leads to a higher SNR and achievable rate.

##  V SNR and achievable rate simulations

In this section, we present the results for SNR and achievable rate
corresponding to the following six cases.

(a) Bandwidth of 4.2 GHz.

(b) Bandwidth of 0.7 GHz.

Figure 5: For a single Chu’s antenna - In (a), the SNR curve for the optimal
transmission ADS circuit is higher than the frequency-flat transmission curve
for frequencies above 6.1 GHz. In (b), the SNR curves for the optimal and
frequency-flat transmission coincide.

Upper bound (Shannon): This bound corresponds to the case when the Bode-Fano
constraints are ignored, i.e., $\mathcal{T}(f)=1\forall f$. The ideal SNR
denoted as $\mathsf{SNR}_{\text{ideal}}(f)$ is defined in (17).

Proposed upper bound (Shannon and Bode/Fano): This upper bound is obtained as
a solution to problem (18) which maximizes the achievable rate over all
physically realizable matching networks at transmitter.

Conjugate matching at center frequency: The matching network is designed such
that the load impedance gets transformed to the complex conjugate of the
source impedance.

Proposed optimal transmission (circuit in ADS): This case corresponds to the
optimized matching network design obtained through the three step procedure in
Section IV-A.

Benchmark of frequency-flat transmission (circuit in ADS): The matching
network is designed to approximate the frequency-flat coefficient
$\mathcal{T}_{\mathsf{ff}}$ satisfying Bode-Fano constraints.

Benchmark of no matching: Finally, we also compare with the case when matching
network is absent, i.e., the source is directly connected to the antenna
array. The resulting SNR is defined as $\mathsf{SNR}_{\text{No-
match}}(f)={|\bm{{\mathsf{s}}}_{\mathsf{RT}}^{T}(f)({\mathbf{I}}-\bm{\mathsf{S}}_{{\mathsf{F}},22}(f)\bm{\mathsf{S}}_{\mathsf{T}}(f))^{-1}\bm{{\mathsf{s}}}_{{\mathsf{F}},21}(f)|^{2}}\frac{\mathsf{E}_{\mathsf{s}}}{\mathsf{N}_{0}}$.

###  V-A Simulation results for a single Chu’s antenna

In Fig. 5(a), we use the parameter setup and the transmission coefficient
based on circuit design in Section IV-B for computing the SNR versus frequency
for the six cases. We observe that the SNR solely based on Shannon’s upper
bound is higher than that of the bound which incorporates Bode-Fano wideband
matching theory. The Shannon upper bound technique overestimates the SNR. This
upper bound cannot be attained by any practical matching network. The bound
proposed by combining Shannon’s theory and Bode-Fano theory is more realistic
as it incorporates the gain-bandwidth tradeoff in matching networks. We show
that this bound can be approximated using a practical matching network
topology optimized using ADS as discussed in Section IV-B.

Figure 6:  Achievable rate versus bandwidth for a single Chu’s antenna with
center frequency 7 GHz. The circuit based on optimal transmission
significantly outperforms the frequency-flat transmission circuit for larger
bandwidths. Also, the optimal bandwidth for getting the highest achievable
rate is 2.8 GHz beyond which rate decreases.

For a major portion of the 4.2 GHz bandwidth, the SNR for the ADS circuit
corresponding to the optimal transmission is greater than that of the
frequency-flat transmission circuit. This leads to a higher achievable rate as
well for the optimal transmission based circuit. The SNR corresponding to the
conjugate matching network is higher than the optimal transmission for
frequencies 6.5 GHz to 7.4 GHz but drastically decreases outside this range.
As rate depends on the SNR for the whole band from 4.9 GHz to 9.1 GHz, the
achievable rate for conjugate matching is less than the rate for optimal
transmission and frequency-flat transmission. Even for a bandwidth of 0.7 GHz
as shown in Fig. 5(b), conjugate matching is still worse compared to the
proposed approach because the optimal transmission solution in (21a) depends
on bandwidth unlike conjugate matching as shown in Fig. 3.

(a) Odd mode beamforming for $\theta=\frac{\pi}{2}$

(b) Even mode beamforming for $\theta=0$

Figure 7: For an array of two Chu’s antennas with frequency-flat even and odd
mode analog beamforming. SNR degradation for $\theta=\frac{\pi}{2}$ compared
to $\theta=0$ because of beam squint effect.

In Fig. 6, we plot the achievable rate as a function of bandwidth. The
achievable rate plot based on the Shannon upper bound continuously increases
with the bandwidth. A realistic trend is observed for the upper bound obtained
after incorporating Bode-Fano theory, i.e., we observe that the rate decreases
beyond a certain bandwidth. The results highlight how the matching network
limits the bandwidth and achievable rate of the system. We show the existence
of an optimal bandwidth which gives the highest possible achievable rate
because of the gain-bandwidth tradeoff of matching networks. From the proposed
upper bound plot and the corresponding ADS circuit simulation, the optimal
bandwidth for highest achievable rate is 2.8 GHz. For conjugate matching, this
optimal bandwidth is 2.1 GHz. It is lower than the optimal bandwidth of the
proposed ADS circuit because conjugate matching response is invariant of the
bandwidth. The proposed matching approach accounts for the bandwidth
dependence and enables use of higher bandwidth for maximizing rate.

(a) Odd mode beamforming for $\theta=\frac{\pi}{2}$

(b) Even mode beamforming for $\theta=0$

Figure 8: Achievable rate versus bandwidth for an array of two Chu’s antennas
with even and odd mode analog beamforming. From the circuit based plots for
both beamforming modes, the achievable rate peak occurs at a bandwidth of 1.4
GHz. At larger bandwidths, odd mode achievable rates lower than even mode
because of beam squint.

###  V-B Simulation results for an array of two Chu’s antennas

In Fig. 7, we use the parameter setup and the transmission coefficient based
on circuit design in Section IV-C for computing the SNR versus frequency for
the six cases. Most of the SNR comparison and trends are similar to that
discussed in Section V-A. At broadside incidence, i.e., $\theta=0$, there is
no relative phase difference between two antennas. Hence, even mode
beamforming works well. For endfire incidence, i.e., $\theta=\frac{\pi}{2}$,
the phase-difference between two antennas varies as a function of frequency as
shown in (24). Using a frequency-flat beamforming for endfire at higher
bandwidths results in a phase mismatch and subsequent SNR reduction. This
effect is commonly known as beam squint. So, we observe SNR degradation for
endfire compared to broadside in Fig. 7.

In Fig. 8, for the odd mode, we observe that the upper bound obtained after
incorporating Bode-Fano theory significantly deviates from the Shannon bound
beyond 1.4 GHz bandwidth. This trend is also consistent with the rate obtained
from the circuit simulations in ADS. For the even mode, the achievable rate
increases faster with bandwidth compared to odd mode because there is no phase
mismatch with even mode. Frequency-selective true time delay (TTD) beamforming
can be used to mitigate the beam squint effect. The achievable rate variation
with bandwidth for TTD systems is a future direction.

##  VI Conclusion

In this paper, we generalized the achievable rate analysis for a MISO system
by incorporating constraints from Bode-Fano wideband matching theory. We
proposed a general optimization framework which maximizes the achievable rate
over all physically realizable linear and passive matching networks. The
proposed upper bound based on the combination of Shannon’s theory and Bode-
Fano theory is more realistic because it captures the gain-bandwidth tradeoff
of matching networks. We also proposed a simple three step procedure to design
matching networks that approximate this bound. We demonstrated this procedure
for a single Chu’s antenna and an array with two Chu’s antennas. From the
derived theoretical bound and the ADS circuit simulations, an optimal
bandwidth behavior is observed in the achievable rate analysis as function of
the bandwidth. In future work, we propose the application of this methodology
to other antenna types like dipoles or patch antenna. The main challenge is to
numerically compute the upper bound because the number of Bode-Fano
constraints increases for complicated antenna geometries. In future work, we
plan to extend this work to MIMO systems with multiple RF chains and study the
bandwidth-multiplexing tradeoff[31]. The main challenge in extension to MIMO
is the joint optimization of multiport transmit and receive impedance-matching
networks.

TABLE I: Evaluation of $\xi_{\mathsf{BF},i}(f)$ and $B_{\mathsf{BF},i}$ (Based on [2, Table 1])  Location in WCP | $\xi_{\mathsf{BF},i}(f)$ | $B_{\mathsf{BF},i}$
---|---|---
$s_{i}={\mathsf{j}}2\pi f_{i}$ | $\frac{1}{4\pi^{2}}\left[\frac{1}{(f_{i}-f)^{2}}+\frac{1}{(f_{i}+f)^{2}}\right]$ | \- $\left[\sum_{\ell=1}^{N_{\mathsf{p}}}(p_{\mathsf{eq},\ell}-{\mathsf{j}}2\pi f_{i})^{-1}+\sum_{m=1}^{N_{\mathsf{z}}}(z_{\mathsf{eq},m}+{\mathsf{j}}2\pi f_{i})^{-1}\right]$
$\mathcal{R}\\{{s_{i}}\\}>0$ | $\mathcal{R}\\{(s_{i}-{\mathsf{j}}2\pi f_{i})^{-1}+(s_{i}+{\mathsf{j}}2\pi f_{i})^{-1}\\}$ | $-\log\left(\left|\hat{\mathsf{S}}_{\mathsf{eq}}(s_{i})\frac{\prod_{m=1}^{N_{\mathsf{z}}}(s_{i}+z_{\mathsf{eq},m})}{\prod_{m=1}^{N_{\mathsf{z}}}(s_{i}-z_{\mathsf{eq},m})}\right|\right)$
$s_{i}=\infty$ | 1 | $\frac{-1}{2}\left[\sum_{\ell=1}^{N_{\mathsf{p}}}p_{\mathsf{eq},\ell}+\sum_{m=1}^{N_{\mathsf{z}}}z_{\mathsf{eq},m}\right]$

##  Appendix A Computing the scattering parameter
$\hat{\mathsf{S}}_{\mathsf{eq}}(s)$ in rational form [31]

Case 1: The impedance parameter of the load is analytically known in the
rational form in the whole complex plane and denoted as
$\hat{\mathsf{Z}}_{\mathsf{eq}}(s)$. The corresponding scattering parameter of
the load in the rational form is
$\hat{\mathsf{S}}_{\mathsf{eq}}(s)=\frac{\hat{\mathsf{Z}}_{\mathsf{eq}}(s)-\mathsf{Z}_{0}}{\hat{\mathsf{Z}}_{\mathsf{eq}}(s)+\mathsf{Z}_{0}}$.

Case 2: The measured value of the scattering parameter of the load
${\mathsf{S}}_{\mathsf{eq}}(f)$ is available for the frequency $f$ in the
range of interest $[f_{1},f_{2}]$. A passive and rational approximation
$\hat{\mathsf{S}}_{\mathsf{eq}}(s)$ is obtained such that
$\hat{\mathsf{S}}_{\mathsf{eq}}({\mathsf{j}}2\pi f)$ is close to
${\mathsf{S}}_{\mathsf{eq}}(f)$ for $f\in[f_{1},f_{2}]$ within a specified
error tolerance. This can be done numerically using the rationalfit function
in MATLAB[43].

##  Appendix B Computing $\xi_{\mathsf{BF},i}(f)$ and $B_{\mathsf{BF},i}$ for
$\\{i\\}_{1}^{N_{\mathsf{BF}}}$ from $\hat{\mathsf{S}}_{\mathsf{eq}}(s)$

Using closed-form expression of $\hat{\mathsf{S}}_{\mathsf{eq}}(s)$, we first
solve the following for $s$[2].

$\hat{\mathsf{S}}_{\mathsf{eq}}(-s)\hat{\mathsf{S}}_{\mathsf{eq}}(s)-1=0.$
(26)

Let $s_{i}$ be a distinct root of (26). The value of $s_{i}$ in (26) can be
obtained analytically or numerically by using vpasolve function in MATLAB. Let
$\\{z_{\mathsf{eq},1},\dots,z_{\mathsf{eq},m},\dots,z_{\mathsf{eq},N_{\mathsf{z}}}\\}$
be the zeros and
$\\{p_{\mathsf{eq},1},\dots,p_{\mathsf{eq},\ell},\dots,p_{\mathsf{eq},N_{\mathsf{p}}}\\}$
be the poles of the rational equivalent load
$\hat{\mathsf{S}}_{\mathsf{eq}}(s)$. For each $s_{i}$, there is a
corresponding $\xi_{\mathsf{BF},i}(f)$ and $B_{\mathsf{BF},i}$ depending on
the location of $s_{i}$ in the whole complex plane (WCP) categorized in Table
I. For the case of multiplicity of $s_{i}$ more than one, i.e. there are
repeated roots, $\xi_{\mathsf{BF},i}(f)$ and $B_{\mathsf{BF},i}$ can be
computed using [31, Eq 21- Eq 23] for each repeated root.

##  Appendix C Proof of Theorem 2

The variables $\mathcal{T}^{\star}(f)$ and
$\mu^{\star}_{i}|_{i=1}^{N_{\mathsf{BF}}+2}$ satisfy the KKT conditions[22,
41] applied to (18).

Primal feasibility:

$\displaystyle\int_{0}^{\infty}\\!\\!\\!\xi_{\mathsf{BF},i}(f)\log\left(\frac{1}{1-\mathcal{T}^{\star}(f)}\right)\mathrm{d}f-B_{\mathsf{BF},i}\leq
0,\\{i\\}_{1}^{N_{\mathsf{BF}}},$ (27a)
$\displaystyle\mathcal{T}^{\star}(f)-1\leq 0,$ (27b)
$\displaystyle-\mathcal{T}^{\star}(f)\leq 0.$ (27c)

Dual feasibility: $\mu_{i}^{\star}\geq 0.$

Complementary slackness:

$\displaystyle\mu_{i}^{\star}\\!\\!\left(\int_{0}^{\infty}\\!\\!\\!\xi_{\mathsf{BF},i}(f)\log\left(\frac{1}{1-\mathcal{T}^{\star}(f)}\right)\mathrm{d}f-B_{\mathsf{BF},i}\\!\\!\right)\\!=\\!0,\\{i\\}_{1}^{N_{\mathsf{BF}}}\\!\\!\\!\\!\\!,$
(28a)
$\displaystyle\mu_{N_{\mathsf{BF}}+1}^{\star}\mathcal{T}^{\star}(f)\\!=0,$
(28b)
$\displaystyle\mu_{N_{\mathsf{BF}}+2}^{\star}(\mathcal{T}^{\star}(f)-1)=0.$
(28c)

Stationarity: Let $\chi(f)$ be an arbitrary shaped function and $\epsilon$
represents the magnitude of variation [22]. Using variational calculus and the
stationarity condition, we set
$\frac{\mathrm{d}}{\mathrm{d}\epsilon}\left[\mathcal{L}\left(\mathcal{T}^{\star}(f)+\epsilon\chi(f)\right)\right]\bigg{|}_{\epsilon=0}\\!\\!\\!\\!\\!\\!=0,$
simplified using chain rule as

$\displaystyle=\frac{\mathrm{d}\left(\mathcal{T}^{\star}(f)+\epsilon\chi(f)\right)}{\mathrm{d}\epsilon}\frac{\mathrm{d}\left[\mathcal{L}\left(\mathcal{T}^{\star}(f)+\epsilon\chi(f)\right)\right]\bigg{|}_{\epsilon=0}}{\mathrm{d}\left(\mathcal{T}^{\star}(f)+\epsilon\chi(f)\right)}$
$\displaystyle=\frac{\mathrm{d}\left[\mathcal{L}\left(\mathcal{T}^{\star}(f)+\epsilon\chi(f)\right)\right]}{\mathrm{d}\left(\mathcal{T}^{\star}(f)+\epsilon\chi(f)\right)}\chi(f)\bigg{|}_{\epsilon=0}$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\int_{0}^{\infty}\frac{\chi(f)}{\ln
2}\bigg{[}\left(\frac{-\mathsf{SNR}_{\text{ideal}}(f)}{1+\mathsf{SNR}_{\text{ideal}}(f)\mathcal{T}^{\star}(f)}\right)+$
(29) $\displaystyle\sum_{i=1}^{N_{\mathsf{BF}}}\mu_{i}^{\star}\frac{\ln
2\xi_{\mathsf{BF},i}(f)}{(1-\mathcal{T}^{\star}(f))}\bigg{]}\mathrm{d}f-\mu_{N_{\mathsf{BF}}+1}^{\star}\chi(f)+\mu_{N_{\mathsf{BF}}+2}^{\star}\chi(f),$

where $(a)$ follows from (III-C). Let $\mu_{N_{\mathsf{BF}}+1}^{\star}=0$ and
$\mu_{N_{\mathsf{BF}}+2}^{\star}=0$ to satisfy (28b) and (28c) respectively.
Equating (C) to 0 is equivalent to setting the integrand inside the integral
to 0. As $\chi(f)$ is an arbitrary function, the non-trivial condition is

$\left[\left(\frac{-\mathsf{SNR}_{\text{ideal}}(f)}{1+\mathsf{SNR}_{\text{ideal}}(f)\mathcal{T}^{\star}(f)}\right)+\sum_{i=1}^{N_{\mathsf{BF}}}\mu_{i}^{\star}\frac{\ln
2\xi_{\mathsf{BF},i}(f)}{(1-\mathcal{T}^{\star}(f))}\right]=0.$ (30)

Simplifying (30), we obtain

$\mathcal{T}^{\star}(f)\left(1+\ln
2\sum_{i=1}^{N_{\mathsf{BF}}}\mu^{\star}_{i}\xi_{\mathsf{BF},i}(f)\right)=1-\frac{\ln
2\sum_{i=1}^{N_{\mathsf{BF}}}\mu^{\star}_{i}\xi_{\mathsf{BF},i}(f)}{\mathsf{SNR}_{\text{ideal}}(f)}.$
(31)

For (27a) to be satisfied, $\mathcal{T}^{\star}(f)-1\leq 0$ which requires
atleast one $\mu_{i}^{\star}$ to be strictly positive based on the expression
of $\mathcal{T}^{\star}(f)$ from (31). Combining (27c) and (31), we get (21a).

## References

  * [1] Buon Kiong Lau, J. Andersen, G. Kristensson, and A. Molisch, “Impact of matching network on bandwidth of compact antenna arrays,” _IEEE Trans. Antennas Propag._ , vol. 54, no. 11, pp. 3225–3238, Nov. 2006.
  * [2] D. Nie and B. M. Hochwald, “Bandwidth analysis of multiport radio frequency systems—part I,” _IEEE Trans. Antennas Propag._ , vol. 65, no. 3, pp. 1081–1092, Mar. 2017.
  * [3] P. S. Taluja and B. L. Hughes, “Bandwidth limitations and broadband matching for coupled multi-antenna systems,” in _Proc. IEEE Globecom_. Houston, TX, USA: IEEE, Dec. 2011, pp. 1–6.
  * [4] S. Saab, A. Mezghani, and R. W. Heath, “Optimizing the mutual information of frequency-selective multi-port antenna arrays in the presence of mutual coupling,” _IEEE Trans. Commun._ , vol. 70, no. 3, pp. 2072–2084, Mar. 2022.
  * [5] M. Steer, _Microwave and RF design (Third Edition, 2019)_. NC State University, 2019.
  * [6] S. Park, A. Alkhateeb, and R. W. Heath, “Dynamic subarrays for hybrid precoding in wideband mmWave MIMO systems,” _IEEE Trans. Wirel. Commun._ , vol. 16, no. 5, pp. 2907–2920, May 2017.
  * [7] R. W. Heath Jr and A. Lozano, _Foundations of MIMO Communication_ , 1st ed. Cambridge University Press, Dec. 2018.
  * [8] H. W. Bode _et al._ , “Network analysis and feedback amplifier design,” 1945\.
  * [9] R. M. Fano, “Theoretical limitations on the broadband matching of arbitrary impedances,” _Journal of the Franklin Institute_ , vol. 249, no. 1, pp. 57–83, 1950.
  * [10] C. R. Chappidi and K. Sengupta, “Globally optimal matching networks with lossy passives and efficiency bounds,” _IEEE Trans. Circuits Syst. I: Regul. Pap._ , vol. 65, no. 1, pp. 257–269, 2018.
  * [11] D. Nie, B. M. Hochwald, and E. Stauffer, “Systematic design of large-scale multiport decoupling networks,” _IEEE Trans. Circuits Syst. I: Regul. Pap._ , vol. 61, no. 7, pp. 2172–2181, Jul. 2014.
  * [12] N. Deshpande, M. R. Castellanos, S. R. Khosravirad, J. Du, H. Viswanathan, and R. W. Heath, “A wideband generalization of the near-field region for extremely large phased-arrays,” _IEEE Wireless Commun. Lett._ , vol. 12, no. 3, pp. 515–519, 2023.
  * [13] M. Cai, K. Gao, D. Nie, B. Hochwald, J. N. Laneman, H. Huang, and K. Liu, “Effect of wideband beam squint on codebook design in phased-array wireless systems,” in _Proc. IEEE Globecom_ , 2016, pp. 1–6.
  * [14] N. V. Deshpande, S. R. Khosravirad, J. Du, H. Viswanathan, M. R. Castellanos, and R. W. Heath, “Analysis of dense array massive MIMO with practical constraints,” in _Proc. 26th Int. ITG Workshop Smart Antennas and 13th Conference on Systems, Communications, and Coding_. VDE, 2023, pp. 1–6.
  * [15] P. S. Taluja and B. L. Hughes, “Diversity limits of compact broadband multi-antenna systems,” _IEEE J. Sel. Areas Commun._ , vol. 31, no. 2, p. 12, 2013.
  * [16] B. Wang, F. Gao, S. Jin, H. Lin, and G. Y. Li, “Spatial- and frequency-wideband effects in millimeter-wave massive MIMO systems,” _IEEE Trans. Signal Process._ , vol. 66, no. 13, pp. 3393–3406, 2018.
  * [17] M. T. Ivrlac and J. A. Nossek, “Toward a circuit theory of communication,” _IEEE Trans. Circuits Syst. I: Regul. Pap._ , vol. 57, no. 7, pp. 1663–1683, Jul. 2010.
  * [18] T. A. de Vasconcelos, A. L. F. de Almeida, and J. A. Nossek, “Matching strategies for multiantenna arrays,” in _Proc. 24th Int. ITG Workshop Smart Antennas (WSA)_ , 2020, pp. 1–6.
  * [19] T. L. Marzetta, “Super-directive antenna arrays: Fundamentals and new perspectives,” in _Proc. 53rd Asilomar Conf. Signals, Syst., Comput._ , 2019, pp. 1–4.
  * [20] J. Wallace and M. Jensen, “Mutual coupling in MIMO wireless systems: A rigorous network theory analysis,” _IEEE Trans. Wirel. Commun._ , vol. 3, no. 4, pp. 1317–1325, Jul. 2004.
  * [21] L. Kundu and B. L. Hughes, “Enhancing capacity in compact MIMO-OFDM systems with frequency-selective matching,” _IEEE Trans. Commun._ , vol. 65, no. 11, pp. 4694–4703, Nov. 2017.
  * [22] V. Shyianov, M. Akrout, F. Bellili, A. Mezghani, and R. W. Heath, “Achievable rate with antenna size constraint: Shannon meets Chu and Bode,” _IEEE Trans. Commun._ , vol. 70, no. 3, pp. 2010–2024, Mar. 2022.
  * [23] M. Akrout, V. Shyianov, F. Bellili, A. Mezghani, and R. W. Heath, “Super-wideband massive mimo,” _arXiv preprint arXiv:2208.01556_ , 2022\.
  * [24] M. Akrout, F. Bellili, A. Mezghani, and J. A. Nossek, “Physically consistent models for intelligent reflective surface-assisted communications under mutual coupling and element size constraint,” _arXiv preprint arXiv:2302.11130_ , 2023.
  * [25] L. J. Chu, “Physical limitations of omni-directional antennas,” _Journal of applied physics_ , vol. 19, no. 12, pp. 1163–1175, 1948.
  * [26] C. A. Balanis, _Antenna theory: analysis and design_. John wiley & sons, 2015.
  * [27] P. R. Gray, P. J. Hurst, S. H. Lewis, and R. G. Meyer, _Analysis and design of analog integrated circuits_. John Wiley & Sons, 2009.
  * [28] S. Saab, A. Mezghani, and R. W. Heath, “Capacity based optimization of compact wideband antennas,” in _Proc. IEEE-APS Topical Conf. Antennas Propag. Wireless Commun. (APWC)_. Granada, Spain: IEEE, Sep. 2019, pp. 322–325.
  * [29] ——, “Capacity based analysis of a wideband SIMO system in the presence of mutual coupling,” in _Proc. IEEE Globecom_. Waikoloa, HI, USA: IEEE, Dec. 2019, pp. 1–6.
  * [30] N. Deshpande, M. R. Castellanos, S. R. Khosravirad, J. Du, H. Viswanathan, and R. W. Heath, “Achievable rate of a SISO system under wideband matching network constraints,” in _Proc. IEEE Globecom_. Kuala Lumpur, Malaysia: IEEE, Dec. 2023.
  * [31] D. Nie and B. M. Hochwald, “Bandwidth analysis of multiport radio-frequency systems—part II,” _IEEE Trans. Antennas Propag._ , vol. 65, no. 3, pp. 1093–1107, Mar. 2017.
  * [32] N. Decarli, “On phase-based localization with narrowband backscatter signals,” _EURASIP J. Adv. Signal Process_ , vol. 2018, no. 1, pp. 1–12, 2018.
  * [33] K. Kousias, G. Caso, O. Alay, A. Brunstrom, L. D. Nardis, M.-G. D. Benedetto, and M. Neri, “Coverage and deployment analysis of narrowband internet of things in the wild,” _IEEE Commun. Mag._ , vol. 58, no. 9, pp. 39–45, 2020\.
  * [34] C. Han, A. O. Bicen, and I. F. Akyildiz, “Multi-ray channel modeling and wideband characterization for wireless communications in the Terahertz band,” _IEEE Trans. Wirel. Commun._ , vol. 14, no. 5, pp. 2402–2412, 2015\.
  * [35] R. J. Williams, P. Ramírez-Espinosa, J. Yuan, and E. de Carvalho, “Electromagnetic based communication model for dynamic metasurface antennas,” _IEEE Trans. Wirel. Commun._ , vol. 21, no. 10, pp. 8616–8630, 2022.
  * [36] M. R. Castellanos and R. W. Heath, “MIMO communication with polarization reconfigurable antennas,” in _Proc. 55th Asilomar Conf. Signals, Syst., Comput._ , 2021, pp. 427–431.
  * [37] W. B. Davenport, W. L. Root _et al._ , _An introduction to the theory of random signals and noise_. McGraw-Hill New York, 1958, vol. 159.
  * [38] J. A. Russer and P. Russer, “Modeling of noisy EM field propagation using correlation information,” _IEEE Trans. Microw. Theory Tech._ , vol. 63, no. 1, pp. 76–89, 2014.
  * [39] D. Williams, “Traveling waves and power waves: Building a solid foundation for microwave circuit theory,” _IEEE Microw. Mag._ , vol. 14, no. 7, pp. 38–45, 2013.
  * [40] S. Darlington, “Synthesis of reactance 4-poles which produce prescribed insertion loss characteristics: including special applications to filter design,” _Journal of Mathematics and Physics_ , vol. 18, no. 1-4, pp. 257–353, 1939.
  * [41] S. Boyd, S. P. Boyd, and L. Vandenberghe, _Convex optimization_. Cambridge university press, 2004.
  * [42] C. Shannon, “Communication in the presence of noise,” _Proceedings of the IRE_ , vol. 37, no. 1, pp. 10–21, 1949.
  * [43] B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” _IEEE Trans. Power Deliv._ , vol. 14, no. 3, pp. 1052–1061, 1999.