# MIMO-ISAC: Performance Analysis and Rate Region Characterization

Chongjun Ouyang, Yuanwei Liu, and Hongwen Yang C. Ouyang and H. Yang are with
the School of Information and Communication Engineering, Beijing University of
Posts and Telecommunications, Beijing, 100876, China (e-mail:
{DragonAim,yanghong}@bupt.edu.cn).Y. Liu is with the School of Electronic
Engineering and Computer Science, Queen Mary University of London, London, E1
4NS, U.K. (e-mail: yuanwei.liu@qmul.ac.uk).

###### Abstract

This article analyzes the performance of sensing and communications (S&C)
achieved by a multiple-input multiple-output downlink integrated S&C (ISAC)
system. Three ISAC scenarios are analyzed, including the sensing-centric
design, communications-centric design, and Pareto optimal design. For each
scenario, diversity orders and high signal-to-noise ratio slopes of the
sensing rate (SR) and communication rate (CR) are derived to gain further
insights. Numerical results reveal that i) ISAC achieves the same diversity
order as existing frequency-division S&C (FDSAC) techniques; ii) ISAC achieves
larger high-SNR slopes and a broader SR-CR region than FDSAC.

###### Index Terms:

Integrated sensing and communications (ISAC), performance analysis, sensing-
communication rate region.

##  I Introduction

Integrated sensing and communications (ISAC) is a promising enabler for the
development of the six-generation (6G) and beyond wireless networks [1]. The
main feature of ISAC is to allow communications and sensing to share the same
time-frequency-power-hardware resources. Compared with existing frequency-
division sensing and communications (FDSAC) techniques, in which sensing and
communications (S&C) require isolated frequency bands as well as hardware
infrastructures, ISAC is envisioned to be more spectrum-, energy-, and
hardware-efficient [1, 3, 2]. Due to these attractive characteristics, ISAC
has received considerable attention from both industry and the research
community [1, 3, 4, 5, 7, 6, 2].

From the perspective of information, the S&C performance of ISAC systems is
evaluated by two metrics including the sensing rate (SR) and the communication
rate (CR) [3]. Specifically, the SR measures how much environmental
information can be extracted from the sensing echoes, whereas the CR measures
how much data information can be recovered from the received symbols [3, 4,
5]. These two metrics evaluate the fundamental information-theoretic limits of
the S&C performance achieved by ISAC. With this in mind, the authors in [4]
optimized the weighted sum of the CR and the SR in a downlink multiple-input
multiple-output (MIMO) ISAC system with a single communication user terminal
(UT). This work focused more on the dual-functional S&C (DFSAC) precoding
design and neglected the discussion of basic system insights. As an advance,
the authors in [5] extended the work in [4] to a multiuser case and discussed
the high signal-to-noise ratio (SNR) slopes of the CR and SR as well as an
inner bound of the SR-CR region. Although these two works have made great
progress in understanding the SR and CR in ISAC systems, there are still many
important unsolved problems. For example, the Pareto boundary, i.e., the
upper-right boundary of the rate region that contains all the achievable SR-CR
tuples [8], has not been characterized. For another example, a rigorous
comparison between the SR-CR regions achieved by ISAC and FDSAC is still
missing. It is worth noting that the Pareto boundary and rate region can be
exploited to evaluate the S&C performance limit. Hence, characterizing the
Pareto boundary and rate region is of great theoretical importance in
understanding the superiority of ISAC over FDSAC.

This letter analyzes the S&C performance of a downlink MIMO-ISAC system. The
main contributions are summarized as follows: i) We provide DFSAC precoding
design for three scenarios, including the sensing-centric (S-C) design
(maximizing the SR only), the communications-centric (C-C) design (maximizing
the CR only), and the Pareto optimal design (characterizing the Pareto
boundary of the SR-CR region); ii) For each scenario, we analyze the outage
probability (OP) of the CR and show that ISAC yields the same diversity order
as FDSAC; iii) For each scenario, we derive the high-SNR slopes of the CR and
SR and show that ISAC provides larger high-SNR slopes and thus more degrees of
freedom than FDSAC in terms of both sensing and communications; iv) We provide
a rigorous comparison between the SR-CR regions achieved by ISAC and FDSAC and
prove that ISAC achieves a broader SR-CR region than FDSAC.

Figure 1: Illustration of a downlink MIMO-ISAC system.

##  II System Model

Consider a downlink MIMO-ISAC system as shown in Figure 1, where a DFSAC base
station (BS) is communicating with a set of $M$ UTs, while simultaneously
sensing the targets in its surrounding environment. The BS has $M$ transmit
antennas and $N$ receive antennas and each UT
$m\in{\mathcal{M}}=\left\\{1,\ldots,M\right\\}$ has $K$ receive antennas. The
reason why the number of UTs is set to the same value as that of the BS
transmit antennas will be detailed in Section III-A. Let
$\mathbf{X}=\left[{\mathbf{x}}_{1}\ldots{\mathbf{x}}_{L}\right]\in{\mathbbmss{C}}^{M\times
L}$ be a DFSAC signal matrix, with $L$ being the length of the communication
frame/sensing pulse. From a sensing perspective,
${\mathbf{x}}_{l}\in{\mathbbmss{C}}^{M\times 1}$ for
$l\in{\mathcal{L}}=\left\\{1,\ldots,L\right\\}$ represents the sensing
snapshot transmitted at the $l$th time slot. For communications,
${\mathbf{x}}_{l}$ is the $l$th data symbol vector. Under the framework of
MIMO-ISAC, the signal matrix $\mathbf{X}$ can be written as

$\displaystyle{\mathbf{X}}={\mathbf{W}}{\mathbf{S}}={\mathbf{P}}{\bm{\Xi}}^{1/2}{\mathbf{S}},$
(1)

where ${\mathbf{W}}={\mathbf{P}}{\bm{\Xi}}^{1/2}\in{\mathbbmss{C}}^{M\times
M}$ is the precoding matrix,
${\mathbf{P}}=\left[{\mathbf{p}}_{1}\ldots{\mathbf{p}}_{M}\right]\in{\mathbbmss{C}}^{M\times
M}$ stores the normalized precoders with $\lVert{\mathbf{p}}_{m}\rVert^{2}=1$,
$\forall m$,
${\bm{\Xi}}={\mathsf{diag}}\\{{p_{1}},\ldots,{p_{M}}\\}\succeq{\mathbf{0}}$ is
the power allocation matrix subject to the power budget
$\sum_{m=1}^{M}p_{m}\leq p$, and
$\mathbf{S}=\left[{\mathbf{s}}_{1}\ldots{\mathbf{s}}_{M}\right]^{\mathsf{H}}\in{\mathbbmss{C}}^{M\times
L}$ contains $M$ unit-power data streams intended for the $M$ UTs. Here,
${\mathbf{s}}_{m}^{\mathsf{H}}\in{\mathbbmss{C}}^{1\times L}$ and
$\sqrt{p_{m}}{\mathbf{p}}_{m}\in{\mathbbmss{C}}^{M\times 1}$ denote the data
stream dedicated for UT $m$ and the associated precoding vector, respectively.
The data streams are assumed to be independent with each other so that
$L^{-1}{\mathbf{S}}{\mathbf{S}}^{\mathsf{H}}\approx\mathbf{I}_{M}$ [6].

###  II-A Communication Model

The observation at each UT $m\in{\mathcal{M}}$ can be written as

${\mathbf{Y}}_{{\rm{c}},m}={\mathbf{H}}_{m}(\sqrt{p_{m}}{\mathbf{p}}_{m}{\mathbf{s}}_{m}^{\mathsf{H}}+\sum\nolimits_{m^{\prime}\neq
m}\sqrt{p_{m^{\prime}}}{\mathbf{p}}_{m^{\prime}}{\mathbf{s}}_{m^{\prime}}^{\mathsf{H}})+{\mathbf{N}}_{{\rm{c}},m},$

where ${\mathbf{H}}_{m}\in{\mathbbmss{C}}^{K\times M}$ is the communication
channel matrix and ${\mathbf{N}}_{{\rm{c}},m}\in{\mathbbmss{C}}^{K\times L}$
is the additive white Gaussian noise (AWGN) matrix. It is assumed that all the
communication links shown in Figure 1 suffer Rayleigh fading. In this case,
${\mathbf{N}}_{{\rm{c}},m}$ and ${\mathbf{H}}_{m}$ are mutually independent
complex Gaussian matrices, whose elements are independent and identically
distributed (i.i.d.) with zero mean and unit variance. We consider that each
UT $m$ has access to its effective channel
${\mathbf{P}}_{m}={\mathbf{H}}_{m}{\mathbf{P}}\in{\mathbbmss{C}}^{K\times M}$,
where ${\mathbf{H}}_{m}$ can be estimated by UT $m$ through downlink pilots
and $\mathbf{P}$ can be fed to the UTs from the BS via an error-free link. As
will be shown later, $\mathbf{P}$ is independent of the instantaneous channel
realization $\\{{\mathbf{H}}_{m}\\}_{\forall m}$, which can be fixed for an
extended period of time. Accordingly, the system overhead due to the feedback
of $\mathbf{P}$ could be negligible. We further assume that the channel
estimation error is negligible to characterize the performance bound.
Imperfect channel acquisition can lead to extra interference and thus reduce
the CR, whose influence will be detailed in our future works.

After receiving ${\mathbf{Y}}_{{\rm{c}},m}$, UT $m$ adopts a normalized
equalizer $\mathbf{v}_{m}\in{\mathbbmss{C}}^{K\times 1}$ to eliminate the
inter-user interference (IUI). By assuming $K\geq M$111When $K<M$, the IUI
cannot be thoroughly eliminated, thus yielding a reduced sum CR. Besides, in
this case, maximizing the sum CR requires the joint design of power allocation
and interference cancellation, which is generally NP-hard. This problem is
still open and left for future work., we have
$\mathbf{v}_{m}=\frac{\mathbf{q}_{m}}{\lVert\mathbf{q}_{m}\rVert}$, where
$\mathbf{q}_{m}$ denotes the $m$th column of matrix
${\mathbf{P}}_{m}\left({\mathbf{P}}_{m}^{\mathsf{H}}{\mathbf{P}}_{m}\right)^{-1}$.
It is worth noting that the assumption of $K\geq M$ generally holds for some
low-cost and low-power small cells where the BS has the same number of
antennas as UTs, or even less. Accordingly, the received SNR at UT $m$ is
given as
$\gamma_{m}=p_{m}\lvert{\mathbf{v}}_{m}^{\mathsf{H}}{\mathbf{H}}_{m}{\mathbf{p}}_{m}\rvert^{2}$.
In order to reduce system overhead caused by acquiring channel state
information (CSI) at the BS, it is assumed that the BS does not have the
global CSI $\\{{\mathbf{H}}_{m}\in{\mathbbmss{C}}^{K\times M}\\}_{\forall m}$
and only
$\\{{\mathbf{v}}_{m}^{\mathsf{H}}{\mathbf{H}}_{m}\in{\mathbbmss{C}}^{1\times
M}\\}_{\forall m}$222We comment that the price of the overhead reduction
mainly lies in the calculation of ${\mathbf{v}}_{m}$ at each UT $m$. In fact,
the involved computational complexity is at the same order as that in
designing a zero-forcing-based equalizer, which is acceptable for most
communication terminals. is fed back to or estimated by the BS for power
control. In this case, $\mathbf{P}$ is designed regardless of the global CSI
$\\{{\mathbf{H}}_{m}\\}_{\forall m}$, and ${\bm{\Xi}}$ can be optimized to
improve the CR. After optimizing $\bm{\Xi}$, the BS should feed $p_{m}$ to UT
$m$ for decoding. The sum CR reads
$\mathcal{R}_{\rm{c}}=\sum\nolimits_{m=1}^{M}\log_{2}\left(1+\gamma_{m}\right)$.

###  II-B Sensing Model

By transmitting $\mathbf{X}$ to sense the targets, the BS observes the
following reflected echo signal matrix at its receiver: [6, 7, 5]

$\displaystyle{\mathbf{Y}}_{\rm{s}}={\mathbf{G}}{\mathbf{X}}+{\mathbf{N}}_{\rm{s}},$
(2)

where ${\mathbf{N}}_{\rm{s}}\in{\mathbbmss{C}}^{N\times L}$ is the AWGN matrix
with each entry having zero mean and unit variance, and
$\mathbf{G}=\left[{\mathbf{g}}_{1}\ldots{\mathbf{g}}_{N}\right]^{\mathsf{T}}\in{\mathbbmss{C}}^{N\times
M}$ represents the target response matrix with
${\mathbf{g}}_{n}\in{\mathbbmss{C}}^{M\times 1}$ for
$n\in{\mathcal{N}}=\left\\{1,\ldots,N\right\\}$ representing the target
response from the transmit antenna array to the $n$th receive antenna. The
target response matrix is modeled as [4, 6, 7, 5]

$\displaystyle\mathbf{G}=\sum\nolimits_{t}\beta_{t}{\mathbf{a}}\left(\theta_{t}\right){\mathbf{b}}^{\mathsf{H}}\left(\theta_{t}\right),$
(3)

where $\beta_{t}\sim{\mathcal{CN}}\left(0,\sigma_{t}^{2}\right)$ is the
complex amplitude of the $t$th target with $\sigma_{t}^{2}$ representing the
average strength,
${\mathbf{a}}\left(\theta_{t}\right)\in{\mathbbmss{C}}^{N\times 1}$ and
${\mathbf{b}}\left(\theta_{t}\right)\in{\mathbbmss{C}}^{M\times 1}$ are the
associated receive and transmit array steering vectors, respectively, and
$\theta_{t}$ is its direction of arrival. By considering that the receive
antennas at the BS are widely separated, we have
$\mathbf{g}_{n}\sim{\mathcal{CN}}\left({\mathbf{0}},\mathbf{R}\right)$ for
$n\in{\mathcal{N}}$ and
${\mathbbmss{E}}\left\\{{\mathbf{g}}_{n}{\mathbf{g}}_{n^{\prime}}^{\mathsf{H}}\right\\}={\mathbf{0}}$
for $n\neq n^{\prime}$ [7].

Compared with the instantaneous target response ${\mathbf{G}}$, the
correlation matrix ${\mathbf{R}}\in{\mathbbmss{C}}^{M\times M}$ are rather
fixed for a longer period of time. It is not difficult for the BS to obtain
$\mathbf{R}$ through long-term feedback. In the sequel, we assume that the BS
has access to ${\mathbf{R}}$. This is a commonly used assumption in the
literature, see, e.g., [7]. Besides, we assume that none of the targets are
registered communication UTs in the system. If the targets are also
communication UTs, then we can exploit the sensed results to assist the
communications, which is beyond the scope of this work.

Essentially, the aim of sensing is to extract environmental information
contained in $\mathbf{G}$, e.g., the direction and reflection coefficient of
each target, from the reflected echo signal ${\mathbf{Y}}_{\rm{s}}$ [4, 6, 7,
5]. Particularly, the mutual information (MI) between ${\mathbf{Y}}_{\rm{s}}$
and $\mathbf{G}$ conditioned on the DFSAC signal ${\mathbf{X}}$ characterizes
the information-theoretic limits on how much environmental information can be
extracted, which is also referred to as the sensing MI [7]. On this basis,
from an information-theoretic perspective, we adopt the SR as the performance
metric of sensing, which is defined as the sensing MI per unit time [3, 4, 5].
Assuming that each DFSAC symbol lasts 1 unit time, we write the SR as
${\mathcal{R}}_{\rm{s}}=L^{-1}I\left({\mathbf{Y}}_{\rm{s}};\mathbf{G}|\mathbf{X}\right)$,
where $I\left(X;Y|Z\right)$ denotes the MI between $X$ and $Y$ conditioned on
$Z$.

Given the MIMO-ISAC framework, we intend to analyze its S&C performance by
investigating the CR $\mathcal{R}_{\rm{c}}$ and SR $\mathcal{R}_{\rm{s}}$.
Note that both $\mathcal{R}_{\rm{c}}$ and $\mathcal{R}_{\rm{s}}$ are
influenced by the precoding matrix $\mathbf{W}$. Yet, it is challenging to
find a $\mathbf{W}$ that can maximize $\mathcal{R}_{\rm{s}}$ and
$\mathcal{R}_{\rm{c}}$ simultaneously. As a compromise, we consider three
typical scenarios to unveil further system insights. The first scenario is the
S-C design that aims to maximize the SR, the second scenario is the C-C design
that aims to maximize the CR, and the third scenario is the Pareto optimal
design that aims to find the Pareto boundary of the rate region.

##  III Performance of MIMO-ISAC

###  III-A Sensing-Centric Design

####  III-A1 Performance of Sensing

Under the S-C design, the precoding matrix $\mathbf{W}$ is set to maximize
${\mathcal{R}}_{\rm{s}}$. Particularly, the SR can be calculated as [7]

$\displaystyle{\mathcal{R}}_{\rm{s}}=L^{-1}N\log_{2}\det\left({\mathbf{I}}_{M}+L{\mathbf{W}}^{\mathsf{H}}{\mathbf{R}}{\mathbf{W}}\right).$
(4)

Under the S-C design, the precoding matrix satisfies

$\displaystyle\mathbf{W}_{\rm{s}}=\arg\max\nolimits_{{\mathsf{tr}}\left({\mathbf{W}}{\mathbf{W}}^{\mathsf{H}}\right)\leq
p}{\log_{2}\det\left({\mathbf{I}}_{M}+L{\mathbf{W}}^{\mathsf{H}}{\mathbf{R}}{\mathbf{W}}\right)}.$
(5)

For analytical tractability, we assume that
${\mathbf{R}}\succ{\mathbf{0}}$333When ${\mathbf{R}}\succ{\mathbf{0}}$, the
rank of $\mathbf{G}$ is given by $M$. By [6], to recover the rank-$M$ matrix
$\mathbf{G}$, we should transmit at least $M$ independent signal streams. For
brevity, we assume that the number of data streams or UTs equals $M$.
Actually, the derived results in this letter can be extended to the case with
more than $M$ UTs, which is left as a potential direction for future work..
Then, the following theorem provides an exact expression for the SR as well as
its high-SNR approximation.

######  Theorem 1.

In the S-C design, the maximum SR is given by

$\displaystyle\mathcal{R}_{\rm{s}}^{\rm{s}}={N}{L^{-1}}\sum\nolimits_{m=1}^{M}\log_{2}\left(1+L\lambda_{m}s_{m}^{\star}\right),$
(6)

where $\left\\{\lambda_{m}>0\right\\}_{m=1}^{M}$ are the eigenvalues of matrix
${\mathbf{R}}$ and
$s_{m}^{\star}=\max\left\\{0,\frac{1}{\nu}-\frac{1}{L\lambda_{m}}\right\\}$
with

$\displaystyle\sum_{m=1}^{M}\max\left\\{0,\frac{1}{\nu}-\frac{1}{L\lambda_{m}}\right\\}=p.$

The maximum SR is attained when
${\mathbf{W}}_{\rm{s}}={\mathbf{U}}{\bm{\Delta}}_{\rm{s}}^{1/2}$, where
${\mathbf{U}}{\mathsf{diag}}\left\\{\lambda_{1},\ldots,\lambda_{M}\right\\}{\mathbf{U}}^{\mathsf{H}}$
denotes the eigendecomposition (ED) of ${\mathbf{R}}$ and
${\bm{\Delta}}_{\rm{s}}={\mathsf{diag}}\left\\{s_{1}^{\star},\ldots,s_{M}^{\star}\right\\}$.
As $p\rightarrow\infty$,

$\displaystyle\mathcal{R}_{\rm{s}}^{\rm{s}}\approx\frac{NM}{L}\left(\log_{2}{p}+\frac{1}{M}\sum\nolimits_{m=1}^{M}\log_{2}\left(\frac{L\lambda_{m}}{M}\right)\right).$
(7)

###### Proof:

Similar to the proof of [5, Theorem 3]. ∎

######  Remark 1.

The results in (7) suggest that the high-SNR slope of the SR under the S-C
design is given by $\frac{NM}{L}$.

####  III-A2 Performance of Communications

Turn now to the communication performance. Particularly, the CR of UT $m$ is
given by
${\overline{\mathcal{R}}}_{{\rm{c}},m}^{\rm{s}}=\log_{2}\left(1+s_{m}^{\star}\rho_{m}\right)$
with $\rho_{m}=\lvert{\mathbf{v}}_{m}^{\mathsf{H}}{\mathbf{h}}_{m}\rvert^{2}$
and ${\mathbf{h}}_{m}\in{\mathbbmss{C}}^{K\times 1}$ being the $m$th column of
matrix ${\mathbf{H}}_{m}{\mathbf{U}}\in{\mathbbmss{C}}^{K\times M}$.
Specifically, we utilize the OP and ergodic CR (ECR) to evaluate the
communication performance. The following theorem provides an exact expression
for the sum ECR
${{\mathcal{R}}}_{{\rm{c}}}^{\rm{s}}={\mathbbmss{E}}\\{{\overline{\mathcal{R}}}_{{\rm{c}}}^{\rm{s}}\\}$
with
${\overline{\mathcal{R}}}_{{\rm{c}}}^{\rm{s}}=\sum_{m=1}^{M}{\overline{\mathcal{R}}}_{{\rm{c}},m}^{\rm{s}}$
as well as its high-SNR approximation.

######  Theorem 2.

In the S-C design, the sum ECR is given by

$\displaystyle{{\mathcal{R}}}_{{\rm{c}}}^{\rm{s}}$
$\displaystyle=\sum\nolimits_{m=1}^{M}\sum\nolimits_{\mu=0}^{K^{\prime}}\frac{(-1/s_{m}^{\star})^{K^{\prime}-\mu}}{(K^{\prime}-\mu)!\ln{2}}\left(-{\rm{e}}^{-\frac{1}{s_{m}^{\star}}}{\rm{Ei}}\left(\frac{1}{s_{m}^{\star}}\right)\right.$
$\displaystyle+\left.\sum\nolimits_{i=1}^{K^{\prime}-\mu}(i-1)!\left(-1/s_{m}^{\star}\right)^{-i}\right),$
(8)

where $K^{\prime}=K-M$ and
${\rm{Ei}}(x)=-\int_{-x}^{\infty}{\rm{e}}^{-t}t^{-1}{\rm{d}}t$ is the
exponential integral function [9, Eq. (8.211.1)]. As $p\rightarrow\infty$,

$\displaystyle{{\mathcal{R}}}_{{\rm{c}}}^{\rm{s}}\approx
M(\log_{2}{p}-\log_{2}{M}+\psi\left(K^{\prime}+1\right)/\ln{2}),$ (9)

where $\psi\left(x\right)=\frac{{\rm d}}{{\rm d}x}\ln{\Gamma\left(x\right)}$
is the Digamma function [9, Eq. (6.461)] and
$\Gamma\left(x\right)=\int_{0}^{\infty}t^{x-1}{\rm e}^{-t}{\rm d}t$ is the
gamma function [9, Eq. (6.1.1)].

###### Proof:

Please refer to Appendix -A for more details. ∎

######  Remark 2.

The results in (9) suggest that the high-SNR slope of the sum ECR under the
S-C design is given by $M$.

It is challenging to derive a closed-form expression for the OP, i.e.,
${{\mathcal{P}}}_{{\rm{c}}}^{\rm{s}}=\Pr({\overline{\mathcal{R}}}_{{\rm{c}}}^{\rm{s}}<\mathcal{R}_{0})$
with $\mathcal{R}_{0}$ denoting the target sum CR. Thus, we focus more on its
high-SNR properties.

######  Theorem 3.

As $p\rightarrow\infty$, the OP of the sum CR achieved by the S-C design
satisfies
${{\mathcal{P}}}_{{\rm{c}}}^{\rm{s}}\simeq{\mathcal{O}}\left(p^{-M(K+M-1)}\right)$.
The notation $f(x)={\mathcal{O}}\left(g(x)\right)$ means that
$\limsup_{x\rightarrow\infty}\frac{\left|f(x)\right|}{g(x)}<\infty$.

###### Proof:

Please refer to Appendix -B for more details. ∎

######  Remark 3.

The above results suggest that a diversity of $M(K-M+1)$ is achievable for the
OP under the S-C design.

###  III-B Communications-Centric Design

Having investigated the S&C performance under the S-C design, we now move to
the C-C design.

####  III-B1 Performance of Communications

It is worth noting that $\mathbf{H}_{m}$ has the same statistical properties
as $\mathbf{H}_{m}{\mathbf{U}}$. In light of this fact as well as the
conclusion drawn in Theorem 1, we design the C-C precoding matrix as
${\mathbf{W}}={\mathbf{P}}{\bm{\Delta}}_{\rm{c}}^{1/2}$ with
${\mathbf{P}}={\mathbf{U}}$ and
${\bm{\Delta}}_{\rm{c}}={\mathsf{diag}}\left\\{c_{1},\ldots,c_{M}\right\\}$,
where $\sum_{m=1}^{M}c_{m}\leq p$ and $c_{m}\geq 0$ for $m\in{\mathcal{M}}$.
The resulting precoding matrix satisfies

$\displaystyle{\mathbf{W}}_{\rm{c}}={\arg\max}_{{\mathbf{W}}={\mathbf{U}}{\bm{\Delta}}_{\rm{c}}^{1/2}}\sum\nolimits_{m=1}^{M}\log_{2}\left(1+c_{m}\rho_{m}\right),$
(10)

where $\log_{2}\left(1+c_{m}\rho_{m}\right)$ calculates the CR of UT $m$.
Particularly, the following lemma provides an expression for the maximum sum
CR achieved by ${\mathbf{W}}_{\rm{c}}$.

######  Lemma 1.

Under the C-C design, the maximum sum CR is

$\displaystyle{\overline{\mathcal{R}}}_{\rm{c}}^{\rm{c}}=\sum\nolimits_{m=1}^{M}\log_{2}\left(1+\rho_{m}c_{m}^{\star}\right),$
(11)

where
$c_{m}^{\star}=\max\left\\{0,\frac{1}{\upsilon}-\frac{1}{\rho_{m}}\right\\}$
with $\sum_{m=1}^{M}c_{m}^{\star}=p$. The maximum sum CR is attained when
$c_{m}=c_{m}^{\star}$ for $m\in{\mathcal{M}}$.

###### Proof:

This lemma can be directly proved by using the water-filling procedure [10]. ∎

We note that deriving a closed-form expression of the sum ECR
${{\mathcal{R}}}_{\rm{c}}^{\rm{c}}={\mathbbmss{E}}\\{{\overline{\mathcal{R}}}_{\rm{c}}^{\rm{c}}\\}$
is a hard task. To unveil more insights, we characterize its high-SNR
behaviour in Theorem 4.

######  Theorem 4.

As $p\rightarrow\infty$, ${{\mathcal{R}}}_{\rm{c}}^{\rm{c}}$ satisfies

$\displaystyle{{\mathcal{R}}}_{{\rm{c}}}^{\rm{c}}\approx
M(\log_{2}{p}-\log_{2}{M}+\psi\left(K^{\prime}+1\right)/\ln{2}),$ (12)

###### Proof:

Please refer to Appendix -C for more details. ∎

######  Remark 4.

The results in (12) suggest that the high-SNR slope of the sum ECR under the
C-C design is given by $M$.

######  Remark 5.

By comparing (9) and (12), we observe that the CR achieved by the C-C design
has the same asymptotic behaviour as that achieved by the S-C design.

Turn to the OP
${{\mathcal{P}}}_{{\rm{c}}}^{\rm{c}}=\Pr({\overline{\mathcal{R}}}_{\rm{c}}^{\rm{c}}<\mathcal{R}_{0})$,
whose asymptotic behaviour is characterized as follows.

######  Theorem 5.

As $p\rightarrow\infty$, the OP of the sum CR achieved by the C-C design
satisfies
${{\mathcal{P}}}_{{\rm{c}}}^{\rm{c}}\simeq{\mathcal{O}}\left(p^{-M(K+M-1)}\right)$.

###### Proof:

Similar to the proof of Theorem 3. ∎

######  Remark 6.

The above results suggest that a diversity order of $M(K-M+1)$ is achievable
for the OP under the C-C design, which is the same as that achieved by the S-C
design.

####  III-B2 Performance of Sensing

For ${\mathbf{W}}={\mathbf{W}}_{\rm{c}}$, the SR reads

$\displaystyle{\overline{\mathcal{R}}}_{\rm{s}}^{\rm{c}}={N}{L^{-1}}\sum\nolimits_{m=1}^{M}\log_{2}\left(1+L\lambda_{m}c_{m}^{\star}\right).$
(13)

Due to the statistics of $c_{m}^{\star}$, we define the average SR as
${{\mathcal{R}}}_{\rm{s}}^{\rm{c}}={\mathbbmss{E}}\\{{\overline{\mathcal{R}}}_{\rm{s}}^{\rm{c}}\\}$,
which can be evaluated numerically. Besides, Theorem 6 describes the high-SNR
behaviour of ${{\mathcal{R}}}_{\rm{s}}^{\rm{c}}$.

######  Theorem 6.

As $p\rightarrow\infty$, ${{\mathcal{R}}}_{\rm{s}}^{\rm{c}}$ satisfies

$\displaystyle{{\mathcal{R}}}_{{\rm{s}}}^{\rm{c}}\approx\frac{NM}{L}\left(\log_{2}{p}+\frac{1}{M}\sum\nolimits_{m=1}^{M}\log_{2}\left(\frac{L\lambda_{m}}{M}\right)\right).$
(14)

###### Proof:

Similar to the proof of Theorem 4. ∎

######  Remark 7.

The SR achieved by the C-C design involves the same asymptotic behaviour as
that achieved by the S-C design.

Taken the conclusions in Remarks 5 and 7 together, we find that the C-C design
degenerates to the S-C design in the high-SNR regime and vice versa.

###  III-C Pareto Optimal Design

In practice, the precoding matrix $\mathbf{W}$ can be designed to satisfy
different qualities of services, which results in a communication-sensing
performance tradeoff. To evaluate this tradeoff, we resort to the Pareto
boundary of the SR-CR region. The Pareto boundary consists of SR-CR tuples at
which it is impossible to improve one of the two rates without simultaneously
decreasing the other [8]. More precisely, let
$(\hat{\mathcal{R}}_{\rm{s}},\hat{\mathcal{R}}_{\rm{c}})$ denote a rate-tuple
on the Pareto boundary, then there is no other rate-tuple
$(\hat{\mathcal{R}}_{\rm{s}}^{\prime},\hat{\mathcal{R}}_{\rm{c}}^{\prime})$
with $\hat{\mathcal{R}}_{\rm{s}}^{\prime}\geq\hat{\mathcal{R}}_{\rm{s}}$,
$\hat{\mathcal{R}}_{\rm{c}}^{\prime}\geq\hat{\mathcal{R}}_{\rm{c}}$, and
$(\hat{\mathcal{R}}_{\rm{s}}^{\prime},\hat{\mathcal{R}}_{\rm{c}}^{\prime})\neq(\hat{\mathcal{R}}_{\rm{s}},\hat{\mathcal{R}}_{\rm{c}})$
[8]. Particularly, we design the precoding matrix as
$\mathbf{W}_{\rm{p}}={\mathbf{P}}{\mathsf{diag}}\left\\{\sqrt{p_{1}},\ldots,\sqrt{p_{M}}\right\\}$
with ${\mathbf{P}}={\mathbf{U}}$, $\sum_{m=1}^{M}p_{m}\leq p$, and $p_{m}\geq
0$, $\forall m$. By [8], any rate-tuple on the Pareto boundary can be obtained
via the rate-profile based method, i.e., solving the following problem:

$\max\nolimits_{{\mathbf{p}},\mathcal{R}}{\mathcal{R}},~{}{\rm{s.t.}}{\mathcal{R}}_{\rm{s}}\geq\alpha{\mathcal{R}},{\mathcal{R}}_{\rm{c}}\geq\bar{\alpha}{\mathcal{R}},{\mathbf{1}}^{\mathsf{T}}{\mathbf{p}}\\!\leq\\!p,p_{m}\\!\geq\\!0,$
(15)

where $\alpha\in[0,1]$ is a particular rate-profile parameter,
$\bar{\alpha}=1-\alpha$, and ${\mathbf{p}}=[p_{1},\ldots,p_{M}]^{\mathsf{T}}$.
We comment that problem (15) is not equivalent to the weighted sum rate
maximization problem (WSRMP) defined in [4]. Besides, solving the WSRMP cannot
guarantee the finding of all Pareto-boundary points [8]. Problem (15) is
convex and can be solved via standard convex problem solvers such as CVX. For
a given $\alpha$, let ${{\mathcal{R}}}_{\rm{c}}^{\alpha}$ and
${{\mathcal{R}}}_{\rm{s}}^{\alpha}$ denote the sum ECR and average SR achieved
by the corresponding optimal precoding matrix, respectively. It follows that
${{\mathcal{R}}}_{\rm{c}}^{\alpha}\in\left[{{\mathcal{R}}}_{\rm{s}}^{\rm{c}},{{\mathcal{R}}}_{\rm{c}}^{\rm{c}}\right]$
and
${{\mathcal{R}}}_{\rm{s}}^{\alpha}\in\left[{{\mathcal{R}}}_{\rm{c}}^{\rm{s}},{{\mathcal{R}}}_{\rm{s}}^{\rm{s}}\right]$
with ${{\mathcal{R}}}_{\rm{s}}^{1}={{\mathcal{R}}}_{\rm{s}}^{\rm{s}}$ and
${{\mathcal{R}}}_{\rm{c}}^{0}={{\mathcal{R}}}_{\rm{c}}^{\rm{c}}$. Accordingly,
we get the following corollaries.

######  Corollary 1.

For a sufficiently larger SNR,
${{\mathcal{R}}}_{\rm{s}}^{\alpha}\approx\frac{NM}{L}\left(\log_{2}{p}+\frac{1}{M}\sum\nolimits_{m=1}^{M}\log_{2}\left(\frac{L\lambda_{m}}{M}\right)\right)$
and ${{\mathcal{R}}}_{\rm{c}}^{\alpha}\approx
M(\log_{2}{p}-\log_{2}{M}+\psi\left(K^{\prime}+1\right)/\ln{2})$.

###### Proof:

This corollary can be proved by using the results in Theorems 2 and 4 as well
as the Sandwich theorem. ∎

######  Corollary 2.

Let ${\overline{\mathcal{R}}}_{\rm{c}}^{\alpha}$ denote the sum CR for a given
$\alpha$. Then, it has
$\lim_{p\rightarrow\infty}\Pr({\overline{\mathcal{R}}}_{\rm{c}}^{\alpha}<\mathcal{R}_{0})\simeq{\mathcal{O}}\left(p^{-M(K+M-1)}\right)$.

###### Proof:

Similar to the proof of Corollary 1. ∎

######  Remark 8.

Any SR-CR tuple on the Pareto boundary has the same asymptotic behaviour in
the high-SNR regime.

Let ${\mathcal{R}}^{\rm{s}}$ and ${\mathcal{R}}^{\rm{c}}$ denote the
achievable SR and CR, respectively. Then, the rate region achieved by ISAC is
given by

$\displaystyle\mathcal{C}_{\rm{i}}=\left\\{\left({\mathcal{R}}^{\rm{s}},{\mathcal{R}}^{\rm{c}}\right)|{\mathcal{R}}^{\rm{s}}\\!\in\\!\left[0,\mathcal{R}_{\rm{s}}^{\alpha}\right],{\mathcal{R}}^{\rm{c}}\\!\in\\!\left[0,\mathcal{R}_{\rm{c}}^{\alpha}\right],\alpha\\!\in\\!\left[0,\\!1\right]\right\\}.$
(16)

######  Remark 9.

In the above three scenarios, we set ${\mathbf{P}}={\mathbf{U}}$, which is
solely determined by the second order statistics of the target response
$\mathbf{G}$. As stated before, $\mathbf{G}$ or $\mathbf{U}$ can be fixed for
an extended period of time, which is consistent with the statements in Section
II-A.

##  IV Performance of MIMO-FDSAC

We consider FDSAC as a baseline scenario, where the total bandwidth is
partitioned into two sub-bands, one for sensing only and the other for
communications. Besides, the total power is also partitioned into two parts
for sensing and communications, respectively. Specifically, we assume $\kappa$
fraction of the total bandwidth and $\mu$ fraction of the total power is used
for communications. Based on [5], the sum CR and the SR are given by
$\overline{\mathcal{R}}_{\rm{c}}^{\rm{f}}=\max_{\sum_{m=1}^{M}a_{m}\leq\mu
p}\sum_{m=1}^{M}\kappa\log_{2}(1+\frac{a_{m}}{\kappa}\rho_{m})$ and
$\mathcal{R}_{\rm{s}}^{\rm{f}}=\frac{N(1-\kappa)}{L}\max_{\sum_{m=1}^{M}b_{m}\leq(1-\mu)p}\sum\nolimits_{m=1}^{M}\log_{2}\left(1+\frac{L\lambda_{m}}{1-\kappa}b_{m}\right)$,
respectively. Accordingly, we derive the following corollary.

######  Corollary 3.

As $p\rightarrow\infty$, the OP of the sum CR achieved by FDSAC satisfies
$\Pr(\overline{\mathcal{R}}_{\rm{c}}^{\rm{f}}<\mathcal{R}_{0})\simeq{\mathcal{O}}\left(p^{-M(K+M-1)}\right)$.

###### Proof:

Similar to the proof of Corollary 1. ∎

######  Corollary 4.

The high-SNR slopes of
$\mathcal{R}_{\rm{c}}^{\rm{f}}={\mathbbmss{E}}\\{\overline{\mathcal{R}}_{\rm{c}}^{\rm{f}}\\}$
and $\mathcal{R}_{\rm{s}}^{\rm{f}}$ are given by $\kappa M$ and
$(1-\kappa)\frac{NM}{L}$, respectively.

###### Proof:

Similar to the proofs of Theorems 1 and 4. ∎

Moreover, the rate region achieved by FDSAC is given by

$\displaystyle\mathcal{C}_{\rm{f}}=\left\\{\left({\mathcal{R}}^{\rm{s}},{\mathcal{R}}^{\rm{c}}\right)\left|\begin{aligned}
&{\mathcal{R}}^{\rm{s}}\in\left[0,\mathcal{R}_{\rm{s}}^{\rm{f}}\right],{\mathcal{R}}^{\rm{c}}\in\left[0,\mathcal{R}_{\rm{c}}^{\rm{f}}\right],\\\
&\kappa\in\left[0,1\right],\mu\in\left[0,1\right]\end{aligned}\right.\right\\}.$
(17)

After completing all the analyses, we summarize the results related to
diversity order and high-SNR slope in Table I.

######  Remark 10.

The results in Table I suggest that ISAC and FDSAC yield the same diversity
order in terms of the sum CR. Moreover, since $\kappa\in[0,1]$, we note that
ISAC yields larger high-SNR slopes than FDSAC, which means that ISAC provides
more degrees of freedom than FDSAC in terms of both communications and sensing
[10].

We then compare the rate regions $\mathcal{C}_{\rm{i}}$ and
$\mathcal{C}_{\rm{f}}$ as follows.

######  Theorem 7.

The achievable rate regions satisfy
$\mathcal{C}_{\rm{f}}\subseteq\mathcal{C}_{\rm{i}}$.

###### Proof:

Please refer to Appendix -D for more details. ∎

######  Remark 11.

The results in Theorem 7 suggest that the rate region achieved by FDSAC is
entirely covered by that achieved by ISAC. This superiority mainly originates
from ISAC’s integrated utilization of spectrum and power resources.

System | Sum CR | SR
---|---|---
$\mathcal{D}$ | $\mathcal{S}$ | $\mathcal{S}$
ISAC | $M(K+M-1)$ | $M$ | ${NM}/{L}$
FDSAC | $M(K+M-1)$ | $\kappa M$ | $(1-\kappa){NM}/{L}$

TABLE I: Diversity Order ($\mathcal{D}$) and High-SNR Slope ($\mathcal{S}$)

##  V Numerical Results

Simulation results will be presented to evaluate the S&C performance of ISAC
systems and also verify the accuracy of the developed analytical results. The
parameters used for simulation are listed as follows: $M=4$, $N=5$, $K=4$,
$L=30$, and the eigenvalues of $\mathbf{R}$ are $\\{1,0.1,0.05,0.01\\}$.

(a) OP of the sum CR.

(b) Sum ECR.

Figure 2: Performance of communications. ${\mathcal{R}}_{0}=2$ bps/Hz.

Figure 2(a) and Figure 2(b) plot the OP and sum ECR versus the SNR $p$ for
$\kappa=\mu=0.5$, respectively. As Figure 2(a) shows, C-C ISAC achieves the
lowest OP while FDSAC achieves the highest OP. In the high-SNR regime, the OP
curves for all the presented cases are parallel to the one representing
$p^{-M(K-M+1)}$, which suggests that the achievable diversity order obtained
in the previous section is tight. This also suggests that ISAC yields the same
diversity order as FDSAC. Let us now turn to Figure 2(b). As expected, C-C
ISAC attains the best ECR performance among the three presented cases.
Besides, the analytical results fit well with the simulations, and the
asymptotic results accurately track the provided simulation results in the
high-SNR regime. Particularly, it can be seen from Figure 2(b) that ISAC
achieves a larger high-SNR slope than FDSAC and the ECRs achieved by C-C ISAC
and S-C ISAC have the same high-SNR asymptotic behaviour, which is consistent
with the results shown in Remarks 5 and 10.

In Figure 3(a), the SR is shown as a function of the SNR $p$. It can be seen
from this graph that S-C ISAC is capable of achieving the best SR performance
among the presented four cases. Moreover, as Figure 3(a) shows, the asymptotic
results track the provided simulation results accurately in the high-SNR
regime. From the data in Figure 3(a), it is apparent that ISAC achieves a
larger high-SNR slope than FDSAC and S-C ISAC and C-C ISAC yield the same
high-SNR slope, which agrees with the conclusions in Remark 7 and Remark 10.

(a) $\kappa=\mu=0.5$.

(b) $p=5$ dB.

Figure 3: (a) Performance of sensing; (b) Rate region.

Figure 3(b) compares the SR-CR regions achieved by ISAC (presented in (16))
and FDSAC (presented in (17)). As shown, the rate region of FDSAC is plotted
by changing the bandwidth allocation factor $\kappa$ and the power allocation
factor $\mu$ from $0$ to $1$. As expected, a larger value of $\kappa$ or $\mu$
yields a higher CR. For ISAC, the point $\mathcal{P}_{\rm{s}}$ and the point
$\mathcal{P}_{\rm{c}}$ are achieved by the S-C design and the C-C design,
respectively. Moreover, the curve segment connecting $\mathcal{P}_{\rm{s}}$
and $\mathcal{P}_{\rm{c}}$ represents the Pareto boundary of ISAC’s rate
region, which is obtained by solving (15) for $\alpha$ changing from 1 to 0.
As Figure 3(b) shows, the rate region achieved by FDSAC is completely included
in that achieved by ISAC, which verifies the correctness of Theorem 7.

##  VI Conclusion

In this letter, we have analyzed the S&C performance of ISAC systems under
three DFSAC precoding designs. The high-SNR slopes and diversity orders
achieved by ISAC have been derived to highlight its superiority. Theoretical
analyses have demonstrated that ISAC can provide more degrees of freedom and
achieve a broader rate region than FDSAC.

###  -A Proof of Theorem 2

Note that $s_{m}^{\star}$ is a constant and the probability density function
(PDF) of
$\rho_{m}=\lvert{\mathbf{v}}_{m}^{\mathsf{H}}{\mathbf{h}}_{m}\rvert^{2}$ is
given by $f_{m}(x)=\frac{{\rm{e}}^{-x}}{K^{\prime}!}x^{K^{\prime}}$ [10, Eq.
(9.80)]. On this basis, we can get (8) by [9, Eq. (4.337.5)]. As stated
before,
$\lim_{p\rightarrow\infty}s_{m}^{\star}\simeq\frac{p}{M}\\!-\\!\frac{1}{L\lambda_{m}}\\!+\\!\sum_{m=1}^{M}\frac{1}{LM\lambda_{m}}$.
This together with the fact of
$\lim_{x\rightarrow\infty}\log_{2}(1\\!+\\!x)\approx\log_{2}{x}$ and [9, Eq.
(4.352.1)] yields (9).

###  -B Proof of Theorem 3

As stated before,
$\lim_{p\rightarrow\infty}s_{m}^{\star}\simeq\frac{p}{M}-\frac{1}{L\lambda_{m}}+\frac{1}{LM}\sum_{m=1}^{M}\frac{1}{\lambda_{m}}$,
which together with the fact of
$\lim_{x\rightarrow\infty}\log_{2}(a+x)\approx\log_{2}{x}$ yields
$\lim_{p\rightarrow\infty}{\overline{\mathcal{R}}}_{{\rm{c}}}^{\rm{s}}\simeq\log_{2}(\prod_{m=1}^{M}\frac{p}{M}\rho_{m})$.
Therefore, the OP satisfies
$\lim_{p\rightarrow\infty}{{\mathcal{P}}}_{{\rm{c}}}^{\rm{s}}\simeq\Pr(\prod_{m=1}^{M}\frac{p}{M}\rho_{m}<2^{{\mathcal{R}}_{0}})$.
On this basis, we can obtain
$\lim_{p\rightarrow\infty}{{\mathcal{P}}}_{{\rm{c}}}^{\rm{s}}\simeq{\mathcal{O}}(p^{-M(K^{\prime}+1)})$
by exploiting the approach in deriving [11, Eq. (39)].

###  -C Proof of Theorem 4

Clearly, as $p\rightarrow\infty$, we have $\upsilon\rightarrow 0$ and thus
$c_{m}^{\star}\simeq\frac{p}{M}-\frac{1}{\rho_{m}}+\frac{1}{M}\sum_{m=1}^{M}\frac{1}{\rho_{m}}$.
It follows from the fact of
$\lim_{x\rightarrow\infty}\log_{2}(a+x)\approx\log_{2}{x}$ that the ECR of UT
$m$ satisfies
$\lim_{p\rightarrow\infty}\mathbbmss{E}\\{\log_{2}(1+\rho_{m}c_{m}^{\star})\\}\simeq\log_{2}{\frac{p}{M}}+{\mathbbmss{E}}\\{\log_{2}{\rho_{m}}\\}$.
Based on [9, Eq. (4.352.1)], the final results follow immediately.

###  -D Proof of Theorem 7

To proceed, we construct two auxiliary regions as follows:

$\displaystyle\mathcal{C}_{1}=\\{\left({\mathcal{R}}^{\rm{s}},{\mathcal{R}}^{\rm{c}}\right)|{\mathcal{R}}^{\rm{s}}\\!\in\\!\left[0,\mathcal{R}_{{\rm{s}},1}^{\epsilon}\right],{\mathcal{R}}^{\rm{c}}\\!\in\\![0,\mathbbmss{E}\\{\overline{\mathcal{R}}_{{\rm{c}},1}^{\epsilon}\\}],\epsilon\\!\in\\!\left[0,\\!1\right]\\},$
$\displaystyle\mathcal{C}_{2}=\\{\left({\mathcal{R}}^{\rm{s}},{\mathcal{R}}^{\rm{c}}\right)|{\mathcal{R}}^{\rm{s}}\\!\in\\!\left[0,\mathcal{R}_{{\rm{s}},2}^{\epsilon}\right],{\mathcal{R}}^{\rm{c}}\\!\in\\!\left[0,\mathcal{R}_{{\rm{c}},2}^{\epsilon}\right],\epsilon\\!\in\\!\left[0,\\!1\right]\\}.$

Here,

$\displaystyle\mathcal{R}_{{\rm{s}},1}^{\epsilon}\\!=\\!\frac{N}{L}\\!\max\nolimits_{\sum_{m=1}^{M}\\!k_{m}\leq(1-\epsilon)p}\\!\sum\nolimits_{m=1}^{M}\\!\log_{2}\left(1\\!+\\!L\lambda_{m}k_{m}\right)$
(18)

and

$\displaystyle\overline{\mathcal{R}}_{{\rm{c}},1}^{\epsilon}=\max\nolimits_{\sum_{m=1}^{M}u_{m}\leq\epsilon
p}\sum\nolimits_{m=1}^{M}\\!\log_{2}(1\\!+\\!u_{m}\rho_{m}).$ (19)

Let $\\{k_{m}^{\epsilon}\\}_{m=1}^{M}$ and $\\{u_{m}^{\epsilon}\\}_{m=1}^{M}$
denote the optimal solutions to $\\{k_{m}\\}_{m=1}^{M}$ and
$\\{u_{m}\\}_{m=1}^{M}$ for a given $\epsilon$, respectively. Besides,
$\mathcal{R}_{{\rm{s}},2}^{\epsilon}=\mathbbmss{E}\\{\overline{\mathcal{R}}_{{\rm{s}},2}^{\epsilon}\\}$
and
$\mathcal{R}_{{\rm{c}},2}^{\epsilon}=\mathbbmss{E}\\{\overline{\mathcal{R}}_{{\rm{c}},2}^{\epsilon}\\}$,
where $\overline{\mathcal{R}}_{{\rm{s}},2}^{\epsilon}$ and
$\overline{\mathcal{R}}_{{\rm{c}},2}^{\epsilon}$ denote the SR and sum CR
achieved by the precoding matrix
$\mathbf{W}={\mathbf{U}}{\bm{\Delta}}_{\epsilon}^{1/2}$ with
${\bm{\Delta}}_{\epsilon}={\mathsf{diag}}\\{k_{1}^{\epsilon}+u_{1}^{\epsilon},\ldots,k_{M}^{\epsilon}+u_{M}^{\epsilon}\\}$,
respectively. Clearly, we have
$\mathcal{C}_{2}\subseteq{\mathcal{C}}_{\rm{i}}$ and
${\mathcal{C}}_{\rm{f}}\subseteq\mathcal{C}_{1}$. Furthermore,
$\\{(\mathcal{R}_{{\rm{s}},1}^{\epsilon},\mathbbmss{E}\\{\overline{\mathcal{R}}_{{\rm{c}},1}^{\epsilon}\\})|\epsilon\\!\in\\!\left[0,\\!1\right]\\}$
serves as the boundary of $\mathcal{C}_{1}$. Given $\epsilon_{1}\in[0,1]$,
when
$\overline{\mathcal{R}}_{{\rm{c}},1}^{\epsilon_{1}}\in[{{\mathcal{R}}}_{\rm{s}}^{\rm{c}},{\overline{\mathcal{R}}}_{\rm{c}}^{\rm{c}}]$,
there exists an $\epsilon_{2}\in[0,1]$ with
$\overline{\mathcal{R}}_{{\rm{c}},2}^{\epsilon_{2}}=\overline{\mathcal{R}}_{{\rm{c}},1}^{\epsilon_{1}}$.
Using the monotonicity of function $\log_{2}(1+ax)$ ($a>0$) with respect to
$x\geq 0$, we obtain that $\epsilon_{2}\leq\epsilon_{1}$. By continuously
using the monotonicity of $\log_{2}(1+ax)$, we can get
$\overline{\mathcal{R}}_{{\rm{s}},2}^{\epsilon_{2}}\geq{\mathcal{R}}_{{\rm{s}},1}^{\epsilon_{1}}$.
When
$\overline{\mathcal{R}}_{{\rm{c}},1}^{\epsilon_{1}}\in[0,{{\mathcal{R}}}_{\rm{s}}^{\rm{c}}]$,
we have
$\overline{\mathcal{R}}_{{\rm{c}},1}^{\epsilon_{1}}\leq{{\mathcal{R}}}_{\rm{s}}^{\rm{c}}=\overline{\mathcal{R}}_{{\rm{c}},2}^{0}$
and
$\overline{\mathcal{R}}_{{\rm{s}},1}^{\epsilon_{1}}\leq{{\mathcal{R}}}_{\rm{s}}^{\rm{s}}=\overline{\mathcal{R}}_{{\rm{s}},2}^{0}$.
The above arguments imply that any rate-tuple on the boundary of
${\mathcal{C}}_{1}$ falls within ${\mathcal{C}}_{2}$ and thus
${\mathcal{C}}_{1}\subseteq\mathcal{C}_{2}$ holds. As stated before,
$\mathcal{C}_{2}\subseteq{\mathcal{C}}_{\rm{i}}$ and
${\mathcal{C}}_{\rm{f}}\subseteq\mathcal{C}_{1}$. Taken together, we obtain
$\mathcal{C}_{\rm{f}}\subseteq{\mathcal{C}}_{\rm{i}}$.

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