# ISAC Meets SWIPT: Multi-functional Wireless Systems Integrating Sensing,
Communication, and Powering

Yilong Chen, Haocheng Hua, Jie Xu, and Derrick Wing Kwan Ng
Part of this paper has been presented at the IEEE International Conference on
Communications (ICC), Rome, Italy, 28 May - 1 June, 2023 [1].Y. Chen, H. Hua,
and J. Xu are with the School of Science and Engineering (SSE) and the Future
Network of Intelligence Institute (FNii), The Chinese University of Hong Kong
(Shenzhen), Shenzhen, China (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>xujie@cuhk.edu.cn).D. W. K. Ng is with School of
Electrical Engineering and Telecommunications, the University of New South
Wales, Sydney, Australia (e-mail: w.k.ng@unsw.edu.au).J. Xu is the
corresponding author.

###### Abstract

This paper unifies integrated sensing and communication (ISAC) and
simultaneous wireless information and power transfer (SWIPT), by investigating
a new multi-functional multiple-input multiple-output (MIMO) system that
integrates wireless sensing, communication, and powering. In this system, a
multi-antenna hybrid access point (H-AP) transmits wireless signals to
communicate with a multi-antenna information decoding (ID) receiver,
wirelessly charges a multi-antenna energy harvesting (EH) receiver, and
performs radar target sensing based on the echo signal concurrently. Under
this setup, we aim to reveal the fundamental performance tradeoff limits among
sensing, communication, and powering, in terms of the estimation Cramér-Rao
bound (CRB), achievable communication rate, and harvested energy,
respectively. In particular, we consider two different target models for radar
sensing, namely the point and extended targets, for which we are interested in
estimating the target angle and the complete target response matrix,
respectively. For both models, we define the achievable CRB-rate-energy
(C-R-E) region and characterize its Pareto boundary by maximizing the
achievable rate at the ID receiver, subject to the estimation CRB requirement
for target sensing, the minimum harvested energy requirement at the EH
receiver, and the maximum transmit power constraint at the H-AP. We obtain
partitionable optimal transmit covariance matrix solutions to the two
formulated problems by applying advanced convex optimization techniques. The
numerical results demonstrate the optimal C-R-E region boundary achieved by
our proposed design, as compared to the benchmark schemes based on time
division and eigenmode transmission (EMT).

###### Index Terms:

Integrated sensing and communication (ISAC), simultaneous wireless information
and power transfer (SWIPT), multi-functional wireless systems, optimization.

##  I Introduction

Future sixth-generation (6G) wireless networks are expected to support various
new intelligent Internet-of-Things (IoT) applications such as smart homes,
smart logistics, industrial automation, and smart healthcare [2, 3, 4].
Towards this end, wireless networks need to incorporate billions of low-power
IoT devices and support their localization, sensing, communication,
computation, and control in a sustainable manner [3]. According to the
projection in [4], 6G networks are required to provide peak and user-
experienced data rates of 1 Tbps and 10-100 Gbps, respectively, localization
accuracy of 1 cm indoors and 50 cm outdoors, and battery lifetime of up to 20
years for low-power IoT devices. As a result, innovative wireless techniques
are in high demand to satisfy these stringent key performance indicators
(KPIs).

On the one hand, by integrating the radio-frequency (RF)-based wireless power
transfer (WPT) [5] and wireless communications, simultaneous wireless
information and power transfer (SWIPT) has recently emerged as an efficient
solution for providing ubiquitous energy supplies to massive IoT devices,
enabling their battery-free operation [6]. In SWIPT, the same wireless signals
are reused for simultaneously delivering information and energy to information
decoding (ID) and energy harvesting (EH) receivers, respectively. On the other
hand, integrated sensing and communication (ISAC) has been recognized as an
enabling technique for 6G networks to provide both sensing and communication
functionalities [7], in which the same wireless signals can be exploited for
not only delivering information, but also sensing surrounding targets and
environments based on the echo signals [2]. With the recent advancements in
these areas, we envision that future 6G networks will integrate both SWIPT and
ISAC, evolving towards new multi-functional wireless systems, which can
provide sensing, communication, and powering capabilities at the same time.
These multi-functional wireless systems are expected to significantly enhance
the utilization efficiency of scarce spectrum resources and densely deployed
base station (BS) infrastructures, aiming for supporting emerging IoT
applications, such as smart city and industrial automation. In these
applications, millions of IoT or sensor devices are deployed to monitor the
environment. In particular, these devices require sustainable power supplies
to send their sensed data to the BS via uplink, while the BS needs to provide
control information for these devices via downlink and localize them for
advanced processing [2]. In such scenarios, unifying ISAC and SWIPT can
efficiently facilitate the localization and data collection of massive low-
power devices, thereby reducing the network complexity. On the one hand, by
adopting WPT, IoT nodes can utilize the harvested wireless power to support
their sensing and communication, thus prolonging the lifetime of ISAC
networks. On the other hand, sensing environmental targets or IoT devices can
help to achieve adaptive resource allocation for SWIPT systems.

In the existing literature, there have been extensive prior works
investigating the transmit optimization for SWIPT (e.g., [8, 9, 10, 11, 12,
13]) and ISAC (e.g., [14, 15, 16, 17, 18, 19, 20, 21, 22]) independently. For
instance, the authors in [8] first studied a multiple-input multiple-output
(MIMO) SWIPT system with an ID receiver and an EH receiver, in which the
transmit covariance matrix at the BS was designed to optimally balance the
tradeoff between the communication rate at the ID receiver versus the
harvested energy at the EH receiver. This design was then extended to the
broadcast channel with multiple ID receivers and multiple EH receivers, by
considering the low-complexity linear transmit beamforming [9] and the
capacity-achieving dirty paper coding [10], respectively. Furthermore, other
prior works investigated SWIPT under different setups, e.g., in interference
channels [11] and secrecy communications [12], or in the case with non-linear
EH receivers [13]. On the other hand, for ISAC systems, the works in [14] and
[15] considered the basic setup with a multi-antenna BS, a multi-antenna ID
receiver, and a sensing target, in which the transmit strategies at the BS
were optimized to balance the communication rate versus the estimation Cramér-
Rao bound (CRB) as the performance metric for target sensing. Furthermore, the
authors in [16, 17, 18] studied the ISAC system with multiple communication
users and sensing targets, in which the transmit beamforming design at the BS
was optimized to balance the communication and sensing performances, in terms
of the signal-to-interference-plus-noise ratio (SINR) and radar beampatterns,
respectively. In addition, the work [19] developed a multi-beam transmission
framework for ISAC exploiting analog antenna arrays. Furthermore, advanced
multiple access techniques such as rate-splitting multiple access (RSMA) [20,
21] and non-orthogonal multiple access (NOMA) [22] were adopted to enhance
ISAC performance.

Different from the prior works investigating ISAC and SWIPT independently,
this paper studies a multi-functional MIMO system unifying ISAC and SWIPT, in
which a multi-antenna hybrid access point (H-AP) transmits wireless signals to
simultaneously deliver information to a multi-antenna ID receiver, provide
energy supply to a multi-antenna EH receiver, and estimate a sensing target
based on the echo signals. For such a multi-functional wireless
system,111Practical IoT networks consist of millions of IoT devices, each
having one or more of the three functionalities. As an initial attempt, this
paper mainly considers the separate CRB-rate-energy (C-R-E) setup with an ID
receiver, an EH receiver, and a target. We also investigate the co-located
C-R/C-E (the ID/EH receiver also serves as a target) and R-E (a receiver
performs both ID and EH functionalities) setups as special cases. The
interesting scenario with multiple ID/EH receivers and targets is left for
future work. we aim to optimize the fundamental performance tradeoffs among
sensing, communication, and powering, in terms of the estimation CRB, the
achievable communication rate, and the harvested energy, respectively. To
address this, how to design the transmit strategies at the multi-antenna H-AP
is essential. This problem, however, is particularly challenging, due to the
following reasons. First, MIMO radar sensing, communication, and WPT are
generally designed based on distinct objectives and follow different
principles. In particular, for MIMO radar sensing, it is known that isotropic
transmission, exploiting an identity sample covariance matrix, is beneficial
for leveraging waveform diversity to enhance sensing performance [23, 24]. By
contrast, for point-to-point MIMO communication with perfect channel state
information (CSI) at transceivers, it is well established in [25] that
eigenmode transmission (EMT), which involves implementing singular value
decomposition (SVD) on the MIMO communication channel, along with water-
filling (WF) power allocation over decomposed parallel subchannels, is optimal
for maximizing MIMO channel capacity. As for MIMO WPT with linear radio
frequency (RF)-to-direct current (DC) conversion efficiency, it is shown in
[8] that the strongest EMT based on the MIMO WPT channel is optimal for
maximizing harvested energy. Next, MIMO radar sensing, communication, and WPT
systems handle inter-signal-stream interference in different manners.
Specifically, in MIMO radar sensing, transmitting different independent
probing signal beams is beneficial in reflecting target information from
different perspectives. As in MIMO communication, inter-signal-stream
interference is detrimental, which needs to be mitigated via proper signal
processing techniques. By contrast, in MIMO WPT, inter-signal-stream
interference is beneficial, as it can be harvested by RF-band EH receivers.
Due to these differences in sensing, communication, and powering, finding the
optimal transmission strategy for multi-functional wireless MIMO systems to
balance their performance tradeoffs becomes an important but challenging
problem. These motivate our study in this work.

In this paper, we consider the multi-functional MIMO system by focusing on two
different target models for radar sensing, namely the point and extended
targets, respectively. For the point target case, the H-AP aims to estimate
the target reflection coefficient and the target angle as unknown parameters,
while the target angle estimation CRB is considered as the sensing performance
metric. As for the extended target case, the H-AP aims to estimate the
complete target response matrix and the corresponding matrix estimation CRB is
considered. The main results of this paper are summarized as follows.

  * •

We characterize the complete C-R-E regions of the multi-functional MIMO system
for the point and extended target cases, which are defined as the set of all
C-R-E pairs that are simultaneously achievable by sensing, communication, and
WPT. Towards this end, we first identify three vertices on the Pareto boundary
surface of each C-R-E region corresponding to the CRB minimization (C-min),
rate maximization (R-max), and energy maximization (E-max), respectively, as
well as three edges characterizing the optimal C-R, R-E, and C-E tradeoffs,
respectively. Next, for acquiring the remaining points on the two Pareto
boundary surfaces, we formulate two novel MIMO rate maximization problems,
each subject to the corresponding estimation CRB requirement for target
sensing, the harvested energy requirement at the EH receiver, and the maximum
transmit power constraint at the H-AP.

  * •

We derive partitionable optimal solutions to the two CRB-and-energy-
constrained MIMO rate maximization problems using advanced convex optimization
techniques. It is shown that for the point target case, the optimal transmit
covariance matrix solution follows the EMT structure based on a composite
channel matrix, combined with a WF-like power allocation. Specifically, this
solution is generally divided into two parts: one serving the triple roles of
communication, sensing, and powering, and the other dedicated to sensing and
powering only. By contrast, for the extended target case, the optimal transmit
covariance matrix is obtained by applying SVD to the combined ID and EH
channel matrix. In particular, this solution is full rank and consists of two
parts arranged in a block diagonal form: one for balancing the tradeoff among
the three functionalities, and the other solely dedicated to sensing.
Furthermore, we investigate the performance tradeoff of the three
functionalities for special cases with co-located C-R, C-E, and R-E,
respectively.

  * •

We provide numerical results to validate the C-R-E region boundaries achieved
by the optimal solutions for the multi-functional MIMO system. For the point
target case, the resultant C-R-E tradeoff is shown to highly depend on the
correlations among ID, EH, and sensing channels. As for the extended target
case, the resultant C-R-E tradeoff is found to be influenced by system
configuration such as the number of antennas at the H-AP. Furthermore, it is
shown that our proposed optimal designs significantly outperform the benchmark
schemes based on time division, EMT over the ID channel, and EMT over the
combined ID and EH channels.

The remainder of this paper is organized as follows. Section II presents the
investigated multi-functional wireless MIMO system model by considering the
point and extended target cases, respectively. Section III characterizes the
C-R-E regions of the multi-functional MIMO system for the two cases, by first
identifying three vertices and three edges of the Pareto boundary surfaces,
followed by the formulation of two CRB-and-energy-constrained MIMO rate
maximization problems to determine all remaining boundary surface points.
Sections IV and V present the optimal solutions to the two formulated problems
for the point and extended target cases, respectively. Section VI provides
numerical results to validate the performance of our proposed designs.
Finally, Section VII concludes this paper.

Notations: Boldface letters are used for vectors (lower-case) and matrices
(upper-case). For a square matrix $\boldsymbol{A}$,
$\mathrm{tr}(\boldsymbol{A})$, $\det(\boldsymbol{A})$, and
$\boldsymbol{A}^{-1}$ denote its trace, determinant, and inverse,
respectively, while $\boldsymbol{A}\succeq\boldsymbol{0}$ and
$\boldsymbol{A}\succ\boldsymbol{0}$ mean that $\boldsymbol{A}$ is positive
semi-definite and positive definite, respectively. For an arbitrary-size
matrix $\boldsymbol{A}$, $\mathrm{rank}(\boldsymbol{A})$,
$\boldsymbol{A}^{\dagger}$, $\boldsymbol{A}^{T}$, $\boldsymbol{A}^{H}$, and
$\mathcal{R}(\boldsymbol{A})$ denote its rank, conjugate, transpose, conjugate
transpose, and range space, respectively. For a vector $\boldsymbol{a}$,
$\|\boldsymbol{a}\|$ denotes its Euclidean norm. For a complex number $a$,
$|a|$ denotes its magnitude. For a real number $a$,
$(a)^{+}\triangleq\max(a,0)$. $\mathrm{diag}(a_{1},\dots,a_{N})$ denotes a
diagonal matrix whose diagonal entries are $a_{1},\dots,a_{N}$.
$\boldsymbol{I}_{N}$ denotes the identity matrix with dimension $N\times N$.
$\mathbb{C}^{M\times N}$, $\mathbb{S}^{N}$, and $\mathbb{S}_{+}^{N}$ denote
the spaces of $M\times N$ complex matrices, $N\times N$ Hermitian matrices,
and $N\times N$ complex positive semidefinite matrices, respectively.
$\mathrm{span}\\{\cdot\\}$ denotes the linear span. $\mathbb{E}[\cdot]$
denotes the statistic expectation. $\otimes$ denotes the Kronecker product.
$j=\sqrt{-1}$ is the imaginary unit.

##  II System Model

Figure 1: The proposed multi-functional wireless system unifying ISAC and
SWIPT.

This paper considers a multi-functional MIMO system. Fig. 1 shows the separate
C-R-E setup, which consists of an H-AP, an EH receiver, an ID receiver, and a
sensing target to be estimated. The H-AP is equipped with a uniform linear
array (ULA) of $M>1$ transmit antennas and $N_{\text{S}}>M$ receive antennas
for sensing.222For radar estimation tasks, the number of receive antennas
should be greater than that of transmit antennas to prevent any potential
information loss of the sensed targets [26]. The EH and ID receivers are
equipped with $N_{\text{EH}}\geq 1$ and $N_{\text{ID}}\geq 1$ receive
antennas, respectively.

We consider a quasi-static narrowband channel model, in which the wireless
channels remain invariant over the interested transmission block consisting of
$L$ symbols, where $L$ is assumed to be sufficiently large. Let
$\boldsymbol{H}_{\text{EH}}\in\mathbb{C}^{N_{\text{EH}}\times M}$ and
$\boldsymbol{H}_{\text{ID}}\in\mathbb{C}^{N_{\text{ID}}\times M}$ denote the
channel matrices from the H-AP to the EH receiver and the ID receiver,
respectively. Let $\boldsymbol{H}_{\text{S}}\in\mathbb{C}^{N_{\text{S}}\times
M}$ denote the target response matrix from the H-AP transmitter to the sensing
target to the H-AP receiver, which will be specified later for the point and
extended target models, respectively. To reveal the fundamental performance
upper bound limits, it is assumed that the H-AP perfectly knows the CSI of
$\boldsymbol{H}_{\text{EH}}$ and $\boldsymbol{H}_{\text{ID}}$, and the ID
receiver perfectly knows $\boldsymbol{H_{\text{ID}}}$ to facilitate the
transmit optimization, as commonly assumed in the literature, e.g., [8, 9, 10,
11, 12, 13].

At each symbol $l\in\\{1,\dots,L\\}$, let
$\boldsymbol{x}(l)\in\mathbb{C}^{M\times 1}$ denote the transmitted signal at
the H-AP. The transmitted signal over the whole block is expressed as

$\boldsymbol{X}=\big{[}\boldsymbol{x}(1),\dots,\boldsymbol{x}(L)\big{]}\in\mathbb{C}^{M\times
L}.$ (1)

Without loss of optimality, we consider capacity-achieving Gaussian signaling
such that $\boldsymbol{x}(l),\forall l$, are assumed to follow circularly
symmetric complex Gaussian (CSCG) random vectors with zero mean and covariance
matrix
$\boldsymbol{S}=\mathbb{E}\big{[}\boldsymbol{x}(l)\boldsymbol{x}(l)^{H}\big{]}\succeq\boldsymbol{0}$.
As $L$ is sufficiently large, the sample covariance matrix
$\frac{1}{L}\boldsymbol{X}\boldsymbol{X}^{H}$ is approximated as the
statistical covariance matrix $\boldsymbol{S}$, i.e.,
$\frac{1}{L}\boldsymbol{X}\boldsymbol{X}^{H}\approx\boldsymbol{S}$, which is
the optimization variable to be designed.333This approximation and its
accuracy have been verified in the literature of MIMO ISAC systems [26, 15].
Besides, for a maximum transmit power budget $P$ at the H-AP, we have
$\mathbb{E}\big{[}\|\boldsymbol{x}(l)\|^{2}\big{]}=\mathrm{tr}(\boldsymbol{S})\leq
P$.

First, we consider the radar target sensing. The echo signal received by the
H-AP receiver is

$\boldsymbol{Y}_{\text{S}}=\boldsymbol{H}_{\text{S}}\boldsymbol{X}+\boldsymbol{Z}_{\text{S}},$
(2)

where $\boldsymbol{Z}_{\text{S}}\in\mathbb{C}^{N_{\text{S}}\times L}$ denotes
the additive white Gaussian noise (AWGN) at the H-AP receiver that is a CSCG
random matrix with independent and identically distributed (i.i.d.) entries
each with zero mean and variance $\sigma_{\text{S}}^{2}$. The objective of
sensing is to estimate the interested target parameters based on
$\boldsymbol{H}_{\text{S}}$ from the received echo signal
$\boldsymbol{Y}_{\text{S}}$ in (2), in which the transmitted signal
$\boldsymbol{X}$ is known by the H-AP. In particular, we consider two
different target models, namely the point and extended targets, respectively,
as specified in the following.

For the point target, the target response matrix is

$\boldsymbol{H}_{\text{S}}=\alpha\boldsymbol{a}_{r}(\theta)\boldsymbol{a}_{t}^{T}(\theta)\triangleq\alpha\boldsymbol{A}(\theta),$
(3)

where $\alpha\in\mathbb{C}$ denotes the complex-valued reflection coefficient,
$0\leq\theta\leq 2\pi$ denotes the angle of departure/arrival (AoD/AoA), and
$\boldsymbol{a}_{t}(\theta)\in\mathbb{C}^{M\times 1}$ and
$\boldsymbol{a}_{r}(\theta)\in\mathbb{C}^{N_{\text{S}}\times 1}$ denote the
transmit and receive array steering vectors, respectively. By choosing the
center of the ULA antennas as the reference point and assuming half a
wavelength spacing between adjacent antennas, it follows that
$\boldsymbol{a}_{t}(\theta)=\Big{[}e^{-j\frac{M-1}{2}\pi\sin\theta},e^{-j\frac{M-3}{2}\pi\sin\theta},\dots,e^{j\frac{M-1}{2}\pi\sin\theta}\Big{]}^{T}$
and
$\boldsymbol{a}_{r}(\theta)=\Big{[}e^{-j\frac{N_{\text{S}}-1}{2}\pi\sin\theta},e^{-j\frac{N_{\text{S}}-3}{2}\pi\sin\theta},\dots,e^{j\frac{N_{\text{S}}-1}{2}\pi\sin\theta}\Big{]}^{T}$.
In this case, $\theta$ and $\alpha$ are the unknown parameters to be
estimated. Here, the CRB for estimating $\theta$ is adopted as the sensing
performance metric,444As acquiring the embedded target information in $\alpha$
is challenging, in this paper, we focus on the CRB for estimating $\theta$,
similarly as in prior works [26, 27]. Besides, if $\boldsymbol{S}$ is rank-
one, then the CRB in (4) becomes
$\frac{\sigma_{\text{S}}^{2}}{2|\alpha|^{2}L\|\boldsymbol{\dot{a}}_{r}(\theta)\|^{2}\boldsymbol{a}_{t}^{T}(\theta)\boldsymbol{S}\boldsymbol{a}_{t}^{\dagger}(\theta)}$
[23]. which is given by [24]

$\mathrm{CRB}_{1}(\boldsymbol{S})=\frac{\frac{\sigma_{\text{S}}^{2}}{2|\alpha|^{2}L}\mathrm{tr}(\boldsymbol{A}^{H}\boldsymbol{A}\boldsymbol{S})}{\mathrm{tr}(\boldsymbol{\dot{A}}^{H}\boldsymbol{\dot{A}}\boldsymbol{S})\mathrm{tr}(\boldsymbol{A}^{H}\boldsymbol{A}\boldsymbol{S})-\big{|}\mathrm{tr}(\boldsymbol{\dot{A}}^{H}\boldsymbol{A}\boldsymbol{S})\big{|}^{2}},$
(4)

where we define $\boldsymbol{A}\triangleq\boldsymbol{A}(\theta)$ and
$\boldsymbol{\dot{A}}\triangleq\frac{\partial\boldsymbol{A}(\theta)}{\partial\theta}=\boldsymbol{a}_{r}(\theta)\boldsymbol{\dot{a}}_{t}^{T}(\theta)+\boldsymbol{\dot{a}}_{r}(\theta)\boldsymbol{a}_{t}^{T}(\theta)$.
Here, $\boldsymbol{\dot{a}}_{t}(\theta)$ and
$\boldsymbol{\dot{a}}_{r}(\theta)$ denote the derivatives of
$\boldsymbol{a}_{t}(\theta)$ and $\boldsymbol{a}_{r}(\theta)$, respectively,
i.e.,
$\boldsymbol{\dot{a}}_{t}(\theta)=\Big{[}-ja_{t,1}\frac{M-1}{2}\pi\cos\theta,-ja_{t,2}\frac{M-3}{2}\pi\cos\theta,\dots,ja_{t,M}\frac{M-1}{2}\pi\cos\theta\Big{]}^{T}$
and
$\boldsymbol{\dot{a}}_{r}(\theta)=\Big{[}-ja_{r,1}\frac{N_{\text{S}}-1}{2}\pi\cos\theta,-ja_{r,2}\frac{N_{\text{S}}-3}{2}\pi\cos\theta,\dots,$
$ja_{r,N_{\text{S}}}\frac{N_{\text{S}}-1}{2}\pi\cos\theta\Big{]}^{T}$, where
$a_{t,i}$ and $a_{r,i}$ denote the $i$-th entries of
$\boldsymbol{a}_{t}(\theta)$ and $\boldsymbol{a}_{r}(\theta)$, respectively.
According to the symmetry of ULA antennas, we have
$\boldsymbol{a}_{t}^{H}(\theta)\boldsymbol{\dot{a}}_{t}(\theta)=0$ and
$\boldsymbol{a}_{r}^{H}(\theta)\boldsymbol{\dot{a}}_{r}(\theta)=0$.

For the extended target, the target response matrix is[26]

$\boldsymbol{H}_{\text{S}}=\sum_{q=1}^{Q}\alpha_{q}\boldsymbol{a}_{r}(\theta_{q})\boldsymbol{a}_{t}^{T}(\theta_{q}),$
(5)

where $Q$ denotes the number of scatterers at the extended target,
$\alpha_{q}\in\mathbb{C}$ denotes the complex-valued reflection coefficient
associated with the $q$-th scatterer, and $0\leq\theta_{q}\leq 2\pi$ denotes
the AoD/AoA of the $q$-th scatterer. The H-AP aims to estimate the complete
target response matrix $\boldsymbol{H}_{\text{S}}$ as unknown parameters. In
this case, the Fisher information matrix (FIM) for estimating
$\boldsymbol{H}_{\text{S}}$ is given by
$\boldsymbol{J}=\frac{1}{\sigma_{\text{S}}^{2}}\boldsymbol{X}^{\dagger}\boldsymbol{X}^{T}\otimes\boldsymbol{I}_{N_{\text{S}}}=\frac{L}{\sigma_{\text{S}}^{2}}\boldsymbol{S}^{T}\otimes\boldsymbol{I}_{N_{\text{S}}}$,
and the corresponding CRB matrix is given by
$\overline{\mathbf{CRB}}=\boldsymbol{J}^{-1}$ [26]. To facilitate performance
optimization, we adopt the trace of CRB matrix as a scalar performance metric,
i.e.,

$\mathrm{CRB}_{2}(\boldsymbol{S})=\mathrm{tr}(\overline{\mathbf{CRB}})=\frac{\sigma_{\text{S}}^{2}N_{\text{S}}}{L}\mathrm{tr}(\boldsymbol{S}^{-1}).$
(6)

Next, we consider the point-to-point MIMO communication from the H-AP to the
ID receiver. The received signal by the ID receiver at symbol $l$ is given by

$\boldsymbol{y}_{\text{ID}}(l)=\boldsymbol{H}_{\text{ID}}\boldsymbol{x}(l)+\boldsymbol{z}_{\text{ID}}(l),$
(7)

where $\boldsymbol{z}_{\text{ID}}(l)\in\mathbb{C}^{N_{\text{ID}}\times 1}$
denotes the AWGN at the ID receiver that is a CSCG random vector with zero
mean and covariance matrix
$\sigma_{\text{ID}}^{2}\boldsymbol{I}_{N_{\text{ID}}}$ and
$\sigma_{\text{ID}}^{2}$ is the noise power per antenna. With capacity-
achieving Gaussian signaling, the achievable data rate (in bps/Hz) is [25]

${R}(\boldsymbol{S})=\log_{2}\det\Big{(}\boldsymbol{I}_{N_{\text{ID}}}+\frac{1}{\sigma_{\text{ID}}^{2}}\boldsymbol{H}_{\text{ID}}\boldsymbol{S}\boldsymbol{H}_{\text{ID}}^{H}\Big{)}.$
(8)

Then, we consider the WPT from the H-AP to the EH receiver, where the EH
receiver adopts rectifiers to convert the received RF signals into DC signals
for energy harvesting. In general, the harvested DC power is monotonically
non-decreasing with respect to the received RF power. As a result, we exploit
the received RF power (energy over a unit time period, in Watt) at the EH
receiver as the powering performance metric555When the RF-to-DC energy
conversion process of the rectifier is linear, maximizing the received RF
power ${E}(\boldsymbol{S})$ is equivalent to maximizing the harvested DC
power. However, maximizing the DC power becomes an interesting and challenging
problem when the process is non-linear, which will be left for future work.
[6], i.e.,

${E}(\boldsymbol{S})=\mathrm{tr}(\boldsymbol{H}_{\text{EH}}\boldsymbol{S}\boldsymbol{H}_{\text{EH}}^{H}).\vspace{-0.2cm}$
(9)

Our objective is to reveal the fundamental performance tradeoff among the
estimation CRB (i.e., $\mathrm{CRB}_{1}(\boldsymbol{S})$ in (4) and
$\mathrm{CRB}_{2}(\boldsymbol{S})$ in (6) for the point and extended targets,
respectively), the achievable rate ${R}(\boldsymbol{S})$ in (8), and the
received (unit-time) energy ${E}(\boldsymbol{S})$ in (9). Towards this end, we
aim to characterize the C-R-E region, which is defined as the set of all C-R-E
pairs that are simultaneously achievable in the multi-functional wireless MIMO
system. For the point target case ($i=1$) or the extended case ($i=2$), the
C-R-E region is defined as

$\displaystyle\mathcal{C}_{i}\triangleq\big{\\{}(\widehat{\mathrm{CRB}}_{i},\widehat{{R}},\widehat{{E}})\big{|}$
$\displaystyle\widehat{\mathrm{CRB}}_{i}\geq\mathrm{CRB}_{i}(\boldsymbol{S}),\widehat{{R}}\leq{R}(\boldsymbol{S}),$
(10)
$\displaystyle\widehat{{E}}\leq{E}(\boldsymbol{S}),\mathrm{tr}(\boldsymbol{S})\leq
P,\boldsymbol{S}\succeq\boldsymbol{0}\big{\\}}.$

##  III C-R-E Region Characterization

This section characterizes the C-R-E region of the multi-functional MIMO
system $\mathcal{C}_{i}$ in (10), $i=1,2$, by finding its Pareto boundary,
which corresponds to the set of points at which improving one performance
metric will inevitably lead to the degradation of another. For instance, Fig.
2 shows the Pareto boundary of an example C-R-E region. It is observed that
the complete Pareto boundary corresponds to a surface in a three-dimensional
(3D) space, which can be specified by three vertices corresponding to R-max,
E-max, and C-min, respectively, as well as three edges corresponding to the
optimal C-R, R-E, and C-E tradeoffs, respectively. In the following, we first
explain the procedure for acquiring the vertices and edges in Sections III-A
and III-B, respectively, and then formulate the CRB-and-energy-constrained
rate maximization problems to determine the remaining interior points on the
Pareto boundary surface in Section III-C.

Figure 2: The Pareto boundary of a C-R-E region example.

###  III-A Finding Three Vertices of Pareto Boundary

This subsection finds the three vertices on the Pareto boundary of the C-R-E
region $\mathcal{C}_{i},\forall i\in\\{1,2\\}$, which correspond to R-max,
E-max, and C-min, respectively. First, we find the R-max vertex by optimizing
the transmit covariance matrix $\boldsymbol{S}$ to maximize the communication
rate ${R}(\boldsymbol{S})$, i.e.,
$\max_{\boldsymbol{S}\succeq\boldsymbol{0}}{R}(\boldsymbol{S}),\
\mathrm{s.t.}\ \mathrm{tr}(\boldsymbol{S})\leq P$, where $P$ is the maximum
transmit power budget. This corresponds to a sole MIMO communication system.
It is well-known that the optimal solution to this problem, denoted by
$\boldsymbol{S}_{\text{ID}}$, is obtained by the EMT over the ID channel
(i.e., performing SVD to decompose $\boldsymbol{H}_{\text{ID}}$) combined with
WF power allocation [25]. Accordingly, the maximum rate is
${R}_{\mathrm{max}}={R}(\boldsymbol{S}_{\text{ID}})$ and the corresponding
harvested energy and estimation CRB are
${E}_{\text{ID}}={E}(\boldsymbol{S}_{\text{ID}})$ and
$\mathrm{CRB}_{\text{ID},i}=\mathrm{CRB}_{i}(\boldsymbol{S}_{\text{ID}}),\forall
i\in\\{1,2\\}$, respectively. As a result, the R-max vertex is obtained as
$(\mathrm{CRB}_{\text{ID},i},{R}_{\mathrm{max}},{E}_{\text{ID}})$.666Notice
that for the extended target case with $i=2$, if $\boldsymbol{S}_{\text{ID}}$
is rank-deficient, then we have
$\mathrm{CRB}_{\text{ID},2}=\frac{\sigma_{\text{S}}^{2}N_{\text{S}}}{L}\mathrm{tr}(\boldsymbol{S}_{\text{ID}}^{-1})\to\infty$.

Next, we obtain the E-max vertex by optimizing $\boldsymbol{S}$ to maximize
the harvested energy ${E}(\boldsymbol{S})$, i.e.,
$\max_{\boldsymbol{S}\succeq\boldsymbol{0}}{E}(\boldsymbol{S}),\
\mathrm{s.t.}\ \mathrm{tr}(\boldsymbol{S})\leq P$. This corresponds to a sole
MIMO WPT system. It has been shown in [8] that the optimal rank-one solution
is obtained based on the strongest EMT, denoted by
$\boldsymbol{S}_{\text{EH}}=\sqrt{P}\boldsymbol{v}_{\text{EH},1}\boldsymbol{v}_{\text{EH},1}^{H}$,
with $\boldsymbol{v}_{\text{EH},1}$ being the dominant right singular vector
of $\boldsymbol{H}_{\mathrm{EH}}$. The maximum harvested energy is hence
${E}_{\mathrm{max}}={E}(\boldsymbol{S}_{\text{EH}})$ and the corresponding
communication rate and estimation CRB are
${R}_{\text{EH}}={R}(\boldsymbol{S}_{\text{EH}})$ and
$\mathrm{CRB}_{\text{EH},i}=\mathrm{CRB}_{i}(\boldsymbol{S}_{\text{EH}}),\forall
i\in\\{1,2\\}$, respectively. Thus, the E-max vertex is obtained as
$(\mathrm{CRB}_{\text{EH},i},{R}_{\text{EH}},{E}_{\mathrm{max}})$.777For the
extended target case with $i=2$,
$\mathrm{CRB}_{\text{EH},2}=\frac{\sigma_{\text{S}}^{2}N_{\text{S}}}{L}\mathrm{tr}(\boldsymbol{S}_{\text{EH}}^{-1})\to\infty$
always holds as $\boldsymbol{S}_{\text{EH}}$ is a rank one matrix.

Finally, we consider the C-min vertex by optimizing $\boldsymbol{S}$ to
minimize the estimation CRB $\mathrm{CRB}_{i}(\boldsymbol{S}),\forall
i\in\\{1,2\\}$, i.e.,
$\min_{\boldsymbol{S}\succeq\boldsymbol{0}}\mathrm{CRB}_{i}(\boldsymbol{S}),\
\mathrm{s.t.}\ \mathrm{tr}(\boldsymbol{S})\leq P$. This corresponds to a sole
MIMO radar sensing system. For the point target case with $i=1$, as derived in
[23], the optimal solution is
$\boldsymbol{S}_{\text{S},1}=\frac{P}{M}\boldsymbol{a}_{t}^{\dagger}\boldsymbol{a}_{t}^{T}$
and the minimum CRB is
$\mathrm{CRB}_{\mathrm{min},1}=\mathrm{CRB}_{1}(\boldsymbol{S}_{\text{S},1})=\frac{\sigma_{\text{S}}^{2}}{2|\alpha|^{2}L\|\boldsymbol{\dot{a}}_{r}\|^{2}PM}$.888Since
$\boldsymbol{S}_{\text{S},1}$ is rank-one, the problem for acquiring the C-min
vertex is equivalent to
$\max_{\boldsymbol{S}\succeq\boldsymbol{0}}\boldsymbol{a}_{t}^{T}\boldsymbol{S}\boldsymbol{a}_{t}^{\dagger},\
\mathrm{s.t.}\ \mathrm{tr}(\boldsymbol{S})\leq P$. As for the extended target
case with $i=2$, the optimal solution is
$\boldsymbol{S}_{\text{S},2}=\frac{P}{M}\boldsymbol{I}_{M}$ [15]. In this
case, the minimum CRB is
$\mathrm{CRB}_{\mathrm{min},2}=\mathrm{CRB}_{2}(\boldsymbol{S}_{\text{S},2})=\frac{\sigma_{\text{S}}^{2}N_{\text{S}}M^{2}}{LP}$.
With $\boldsymbol{S}_{\text{S},1}$ and $\boldsymbol{S}_{\text{S},2}$ obtained,
the corresponding communication rate and harvested energy are
${R}_{\text{S},i}={R}(\boldsymbol{S}_{\text{S},i})$ and
${E}_{\text{S},i}={E}(\boldsymbol{S}_{\text{S},i})$, $\forall i\in\\{1,2\\}$,
respectively. Therefore, we obtain the C-min vertice as
$(\mathrm{CRB}_{\mathrm{min},i},{R}_{\text{S},i},{E}_{\text{S},i})$.

###  III-B Finding Three Edges of Pareto Boundary

This subsection focuses on finding the three edges on the Pareto boundary of
the C-R-E region $\mathcal{C}_{i},\forall i\in\\{1,2\\}$, which capture the
optimal C-R, R-E, and C-E tradeoffs, respectively. First, we characterize the
C-R edge connecting the C-min vertex
$(\mathrm{CRB}_{\mathrm{min},i},{R}_{\text{S},i},{E}_{\text{S},i})$ and the
R-max vertex $(\mathrm{CRB}_{\text{ID},i},{R}_{\mathrm{max}},{E}_{\text{ID}})$
by optimizing $\boldsymbol{S}$ to maximize the communication rate
${R}(\boldsymbol{S})$ subject to a maximum CRB constraint, i.e.,
$\max_{\boldsymbol{S}\succeq\boldsymbol{0}}{R}(\boldsymbol{S}),\
\mathrm{s.t.}\
\mathrm{CRB}_{i}(\boldsymbol{S})\leq\mathrm{CRB}_{\text{C-R},i},\
\mathrm{tr}(\boldsymbol{S})\leq P$, where the CRB threshold
$\mathrm{CRB}_{\text{C-R},i}$ ranges from $\mathrm{CRB}_{\mathrm{min},i}$ to
$\mathrm{CRB}_{\text{ID},i}$ to obtain different C-R tradeoff points on the
edge. This corresponds to the CRB-constrained rate maximization problem for a
MIMO ISAC system without WPT, whose optimal solution has been derived in [15]
in a semi-closed-form, denoted by $\boldsymbol{S}_{\text{C-R},i}$. Let
${R}_{\text{C-R},i}(\mathrm{CRB}_{\text{C-R},i})={R}(\boldsymbol{S}_{\text{C-R},i})$
denote the achieved rate and
${E}_{\text{C-R},i}(\mathrm{CRB}_{\text{C-R},i})={E}(\boldsymbol{S}_{\text{C-R},i})$
denote the corresponding harvested energy. Then, the C-R tradeoff edge for
case $i\in\\{1,2\\}$ is
$\Big{\\{}\big{(}\mathrm{CRB}_{\text{C-R},i},{R}_{\text{C-R},i}(\mathrm{CRB}_{\text{C-R},i}),{E}_{\text{C-R},i}(\mathrm{CRB}_{\text{C-R},i})\big{)}\big{|}\mathrm{CRB}_{\mathrm{min},i}\leq\mathrm{CRB}_{\text{C-R},i}\leq\mathrm{CRB}_{\text{ID},i}\Big{\\}}$.

Next, we obtain the R-E edge connecting the R-max vertex
$(\mathrm{CRB}_{\text{ID},i},{R}_{\mathrm{max}},{E}_{\text{ID}})$ and the
E-max vertex $(\mathrm{CRB}_{\text{EH},i},{R}_{\text{EH}},{E}_{\mathrm{max}})$
by optimizing $\boldsymbol{S}$ to maximize the rate ${R}(\boldsymbol{S})$
subject to a minimum energy constraint, i.e.,
$\max_{\boldsymbol{S}\succeq\boldsymbol{0}}{R}(\boldsymbol{S}),\
\mathrm{s.t.}\ {E}(\boldsymbol{S})\geq{E}_{\text{R-E}},\
\mathrm{tr}(\boldsymbol{S})\leq P$, where the energy threshold
${E}_{\text{R-E}}$ ranges from ${E}_{\text{ID}}$ to ${E}_{\mathrm{max}}$ for
searching different R-E tradeoff points on the edge. This corresponds to the
energy-constrained rate maximization problem for a MIMO SWIPT system without
radar sensing, whose optimal transmit covariance has been obtained in [8] as
$\boldsymbol{S}_{\text{R-E}}$ in a semi-closed-form. Accordingly, let
${R}_{\text{R-E}}({E}_{\text{R-E}})={R}(\boldsymbol{S}_{\text{R-E}})$ denote
the achieved rate and
$\mathrm{CRB}_{\text{R-E},i}({E}_{\text{R-E}})=\mathrm{CRB}_{i}(\boldsymbol{S}_{\text{R-E}})$
denote the corresponding CRBs. Thus, the R-E tradeoff edge for case
$i\in\\{1,2\\}$ is given by
$\Big{\\{}\big{(}\mathrm{CRB}_{\text{R-E},i}({E}_{\text{R-E}}),{R}_{\text{R-E}}({E}_{\text{R-E}}),{E}_{\text{R-E}}\big{)}|{E}_{\text{ID}}\leq{E}_{\text{R-E}}\leq{E}_{\mathrm{max}}\Big{\\}}$.

Finally, we consider the C-E edge to connect the C-min vertex
$(\mathrm{CRB}_{\mathrm{min},i},{R}_{\text{S},i},{E}_{\text{S},i})$ and the
E-max vertex $(\mathrm{CRB}_{\text{EH},i},{R}_{\text{EH}},{E}_{\mathrm{max}})$
by optimizing $\boldsymbol{S}$ to maximize the energy ${E}(\boldsymbol{S})$
subject to a maximum CRB constraint, i.e.,
$\max_{\boldsymbol{S}\succeq\boldsymbol{0}}{E}(\boldsymbol{S}),\
\mathrm{s.t.}\
\mathrm{CRB}_{i}(\boldsymbol{S})\leq\mathrm{CRB}_{\text{C-E},i},\
\mathrm{tr}(\boldsymbol{S})\leq P$, where the CRB threshold
$\mathrm{CRB}_{\text{C-E},i}$ ranges from $\mathrm{CRB}_{\mathrm{min},i}$ to
$\mathrm{CRB}_{\text{EH},i}$ to obtain different C-E tradeoff points on the
edge, which corresponds to the case when only sensing and WPT are considered
in our integrated system. For the point target case with $i=1$, the
optimization problem is reformulated into a convex form, as the CRB constraint
$\mathrm{CRB}_{1}(\boldsymbol{S})\leq\mathrm{CRB}_{\text{C-E},1}$ is
equivalent to the convex semi-definite constraint
$\begin{bmatrix}\mathrm{tr}(\boldsymbol{\dot{A}}^{H}\boldsymbol{\dot{A}}\boldsymbol{S})-\frac{\sigma_{\text{S}}^{2}}{2|\alpha|^{2}L\mathrm{CRB}_{\text{C-E},1}}&\mathrm{tr}(\boldsymbol{\dot{A}}^{H}\boldsymbol{A}\boldsymbol{S})^{\dagger}\\\
\mathrm{tr}(\boldsymbol{\dot{A}}^{H}\boldsymbol{A}\boldsymbol{S})&\mathrm{tr}(\boldsymbol{A}^{H}\boldsymbol{A}\boldsymbol{S})\end{bmatrix}\succeq\boldsymbol{0}$
based on the Schur complement [26]. Therefore, the optimal solution
$\boldsymbol{S}_{\text{C-E},1}$ in this case can be found by standard convex
optimization techniques [28].999Note that we can obtain an analytical optimal
solution to this problem by using Lagrangian duality method. However, we omit
the derivation here as it is the subcase of problem (P1) in Section IV, where
we will show that $\boldsymbol{S}_{\text{C-E},1}$ is rank-one and this problem
is equivalent to
$\max_{\boldsymbol{S}\succeq\boldsymbol{0}}E(\boldsymbol{S}),\ \mathrm{s.t.}\
\boldsymbol{a}_{t}^{T}\boldsymbol{S}\boldsymbol{a}_{t}^{\dagger}\geq\frac{\sigma_{\text{S}}^{2}}{2|\alpha|^{2}L\|\boldsymbol{\dot{a}}_{r}\|^{2}\mathrm{CRB}_{\text{C-E},1}},\
\mathrm{tr}(\boldsymbol{S})\leq P$. For the extended target case with $i=2$,
the semi-closed-form optimal solution $\boldsymbol{S}_{\text{C-E},2}$ is
obtained similarly as in [29]. Let
${E}_{\text{C-E},i}(\mathrm{CRB}_{\text{C-E},i})={E}(\boldsymbol{S}_{\text{C-E},i})$
denote the achieved harvested energy and
${R}_{\text{C-E},i}(\mathrm{CRB}_{\text{C-E},i})={R}(\boldsymbol{S}_{\text{C-E},i})$
denote the corresponding communication rate. Accordingly, the C-E tradeoff
edge for case $i\in\\{1,2\\}$ is
$\Big{\\{}\big{(}\mathrm{CRB}_{\text{C-E},i},{R}_{\text{C-E},i}(\mathrm{CRB}_{\text{C-E},i}),{E}_{\text{C-E},i}(\mathrm{CRB}_{\text{C-E},i})\big{)}\big{|}\mathrm{CRB}_{\mathrm{min},i}\leq\mathrm{CRB}_{\text{C-E},i}\leq\mathrm{CRB}_{\text{EH},i}\Big{\\}}$.

###  III-C Characterizing Complete Pareto Boundary Surface

With the vertices and edges obtained, it remains to find the interior points
on the Pareto boundary surface to characterize the complete C-R-E region
$\mathcal{C}_{i}$ for case $i\in\\{1,2\\}$. Towards this end, we optimize the
transmit covariance $\boldsymbol{S}$ to maximize the communication rate
${R}(\boldsymbol{S})$ in (8), subject to a maximum estimation CRB constraint
$\Gamma_{\text{S}}$ and a minimum harvested energy constraint
$\Gamma_{\text{EH}}$. Mathematically, the CRB-and-energy-constrained rate
maximization problem for case $i\in\\{1,2\\}$ is formulated as

$\displaystyle(\text{P}i):\max_{\boldsymbol{S}\succeq\boldsymbol{0}}$
$\displaystyle\
\log_{2}\det\Big{(}\boldsymbol{I}_{N_{\text{ID}}}+\frac{1}{\sigma_{\text{ID}}^{2}}\boldsymbol{H}_{\text{ID}}\boldsymbol{S}\boldsymbol{H}_{\text{ID}}^{H}\Big{)}$
(11a) $\displaystyle\mathrm{s.t.}$ $\displaystyle\
\mathrm{tr}(\boldsymbol{H}_{\text{EH}}\boldsymbol{S}\boldsymbol{H}_{\text{EH}}^{H})\geq\Gamma_{\text{EH}},$
(11b) $\displaystyle\ \mathrm{CRB}_{i}(\boldsymbol{S})\leq\Gamma_{\text{S}},$
(11c) $\displaystyle\ \mathrm{tr}(\boldsymbol{S})\leq P.$ (11d)

By exhausting the possible values of $\Gamma_{\text{EH}}$ and
$\Gamma_{\text{S}}$ enclosed by the projection of the three edges on the C-E
plane, we can obtain all the Pareto boundary surface points [8, 15]. In
particular, let $R_{i}^{\star}(\Gamma_{\text{EH}},\Gamma_{\text{S}})$ denote
the optimal value of problem (P$i$) with given $\Gamma_{\text{EH}}$ and
$\Gamma_{\text{S}}$. Then, we have one Pareto boundary point of the C-R-E
region $\mathcal{C}_{i}$ as
$\big{(}\Gamma_{\text{S}},R_{i}^{\star}(\Gamma_{\text{EH}},\Gamma_{\text{S}}),\Gamma_{\text{EH}}\big{)}$.

In the sequel, we focus on solving problems (P1) and (P2) for the point and
extended target cases, respectively.

##  IV Optimal Solution to Problem (P1) with Point Target

In this section, we present the optimal solution to problem (P1) in the point
target case. Notice that (P1) is not convex due to the estimation CRB
constraint in (11c) for $i=1$. Fortunately, based on the Schur complement,
constraint (11c) for $i=1$ is equivalent to

$\begin{bmatrix}\mathrm{tr}(\boldsymbol{\dot{A}}^{H}\boldsymbol{\dot{A}}\boldsymbol{S})-\frac{1}{\Gamma_{\text{S},1}}&\mathrm{tr}(\boldsymbol{\dot{A}}^{H}\boldsymbol{A}\boldsymbol{S})^{\dagger}\\\
\mathrm{tr}(\boldsymbol{\dot{A}}^{H}\boldsymbol{A}\boldsymbol{S})&\mathrm{tr}(\boldsymbol{A}^{H}\boldsymbol{A}\boldsymbol{S})\end{bmatrix}\succeq\boldsymbol{0},$
(12)

where
$\Gamma_{\text{S},1}\triangleq\frac{2|\alpha|^{2}L}{\sigma_{\text{S}}^{2}}\Gamma_{\text{S}}$.
Therefore, by replacing constraint (11c) with (12), we reformulate (P1) into a
convex form and optimally solve it by adopting the Lagrangian duality method
that paves the way for analyzing the structure of the optimal transmit
covariance matrix.

Let $\lambda\geq 0$, $\nu\geq 0$, and
$\boldsymbol{Z}\triangleq\begin{bmatrix}z_{1}&z_{2}\\\
z_{2}^{\dagger}&z_{3}\end{bmatrix}\succeq\boldsymbol{0}$ denote the dual
variables associated with the constraints in (11b), (11d), and (12),
respectively. The Lagrangian of problem (P1) is given by

$\displaystyle\mathcal{L}(\boldsymbol{S},\lambda,\nu,\boldsymbol{Z})=$
$\displaystyle\log_{2}\det\Big{(}\boldsymbol{I}_{N_{\text{ID}}}+\frac{1}{\sigma_{\text{ID}}^{2}}\boldsymbol{H}_{\text{ID}}\boldsymbol{S}\boldsymbol{H}_{\text{ID}}^{H}\Big{)}-\mathrm{tr}(\boldsymbol{D}\boldsymbol{S})$
(13) $\displaystyle-\lambda\Gamma_{\text{EH}}+\nu
P-\frac{z_{1}}{\Gamma_{\text{S},1}},$

where
$\boldsymbol{D}\triangleq\nu\boldsymbol{I}_{M}-\lambda\boldsymbol{H}_{\text{EH}}^{H}\boldsymbol{H}_{\text{EH}}-z_{1}\boldsymbol{\dot{A}}^{H}\boldsymbol{\dot{A}}-z_{2}\boldsymbol{\dot{A}}^{H}\boldsymbol{A}-z_{2}^{\dagger}\boldsymbol{A}^{H}\boldsymbol{\dot{A}}-z_{3}\boldsymbol{A}^{H}\boldsymbol{A}$.
Note that since
$\boldsymbol{\dot{A}}=\boldsymbol{a}_{r}\boldsymbol{\dot{a}}_{t}^{T}+\boldsymbol{\dot{a}}_{r}\boldsymbol{a}_{t}^{T}$
and
$\boldsymbol{a}_{t}^{H}\boldsymbol{\dot{a}}_{t}=\boldsymbol{a}_{r}^{H}\boldsymbol{\dot{a}}_{r}=0$,
it follows that
$\boldsymbol{D}=\nu\boldsymbol{I}_{M}-\lambda\boldsymbol{H}_{\text{EH}}^{H}\boldsymbol{H}_{\text{EH}}-(z_{1}\boldsymbol{\dot{a}}_{t}^{\dagger}\boldsymbol{\dot{a}}_{t}^{T}+z_{2}\boldsymbol{\dot{a}}_{t}^{\dagger}\boldsymbol{a}_{t}^{T}+z_{2}^{\dagger}\boldsymbol{a}_{t}^{\dagger}\boldsymbol{\dot{a}}_{t}^{T}+z_{3}\boldsymbol{a}_{t}^{\dagger}\boldsymbol{a}_{t}^{T})\|\boldsymbol{a}_{r}\|^{2}-z_{1}\boldsymbol{a}_{t}^{\dagger}\boldsymbol{a}_{t}^{T}\|\boldsymbol{\dot{a}}_{r}\|^{2}$.

Accordingly, the dual function of (P1) is defined as

$g(\lambda,\nu,\boldsymbol{Z})=\max_{\boldsymbol{S}\succeq\boldsymbol{0}}\
\mathcal{L}(\boldsymbol{S},\lambda,\nu,\boldsymbol{Z}).$ (14)

We have the following lemma, which can be verified similarly as in [15, Lemma
1].

###### Lemma 1

If the dual function $g(\lambda,\nu,\boldsymbol{Z})$ is bounded from above,
then it holds that

$\boldsymbol{D}\succeq\boldsymbol{0}~{}{\text{and}}~{}\mathcal{R}(\boldsymbol{H}_{\text{ID}}^{H})\subseteq\mathcal{R}(\boldsymbol{D}).$
(15)

Based on Lemma 1, the dual problem of (P1) is defined as

$(\text{D}1):\min_{\lambda\geq 0,\nu\geq
0,\boldsymbol{Z}\succeq\boldsymbol{0}}\ g(\lambda,\nu,\boldsymbol{Z}),\
\mathrm{s.t.}\ \eqref{D1-constraint}.$ (16)

Since problem (P1) is reformulated into a convex form and satisfies the
Slater’s condition, strong duality holds between (P1) and its dual problem
(D1) [30]. Therefore, we can solve (P1) by equivalently solving (D1). In the
following, we first obtain the dual function $g(\lambda,\nu,\boldsymbol{Z})$
with given dual variables, and then search over $\lambda\geq 0$, $\nu\geq 0$,
and $\boldsymbol{Z}\succeq\boldsymbol{0}$ to minimize
$g(\lambda,\nu,\boldsymbol{Z})$.

###  IV-A Finding Dual Function $g(\lambda,\nu,\boldsymbol{Z})$ of Problem
(P1)

First, we find the dual function $g(\lambda,\nu,\boldsymbol{Z})$ in (14) for
given $\lambda\geq 0$, $\nu\geq 0$, and $\boldsymbol{Z}\succeq\boldsymbol{0}$
satisfying (15). By dropping the constant terms, the problem in (14) is
equivalent to

$\max_{\boldsymbol{S}\succeq\boldsymbol{0}}\
\log_{2}\det\Big{(}\boldsymbol{I}_{N_{\text{ID}}}+\frac{1}{\sigma_{\text{ID}}^{2}}\boldsymbol{H}_{\text{ID}}\boldsymbol{S}\boldsymbol{H}_{\text{ID}}^{H}\Big{)}-\mathrm{tr}(\boldsymbol{D}\boldsymbol{S}).$
(17)

Suppose that $\mathrm{rank}(\boldsymbol{D})=r_{\text{p}}$, the eigenvalue
decomposition (EVD) of $\boldsymbol{D}$ can be expressed as

$\boldsymbol{D}=\begin{bmatrix}\boldsymbol{Q}^{\overline{\mathrm{null}}}\
\boldsymbol{Q}^{\mathrm{null}}\end{bmatrix}\begin{bmatrix}\boldsymbol{\Sigma}^{\overline{\mathrm{null}}}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}\end{bmatrix}\begin{bmatrix}{\boldsymbol{Q}^{\overline{\mathrm{null}}}}^{H}\\\
{\boldsymbol{Q}^{\mathrm{null}}}^{H}\end{bmatrix},$ (18)

where
$\boldsymbol{\Sigma}^{\overline{\mathrm{null}}}=\mathrm{diag}(\sigma_{\text{p},1},\dots,\sigma_{\text{p},r_{\text{p}}})$,
and ${\boldsymbol{Q}^{\overline{\mathrm{null}}}}\in\mathbb{C}^{M\times
r_{\text{p}}}$ and
${\boldsymbol{Q}^{\mathrm{null}}}\in\mathbb{C}^{M\times(M-r_{\text{p}})}$
denote the eigenvectors corresponding to the $r_{\text{p}}$ non-zero
eigenvalues $\sigma_{\text{p},1},\dots,\sigma_{\text{p},r_{\text{p}}}$, and
the remaining $M-r_{\text{p}}$ zero eigenvalues, respectively. Without loss of
generality, we express $\boldsymbol{S}$ as

$\boldsymbol{S}=\begin{bmatrix}\boldsymbol{Q}^{\overline{\mathrm{null}}}\
\boldsymbol{Q}^{\mathrm{null}}\end{bmatrix}\begin{bmatrix}\boldsymbol{S}_{11}&\boldsymbol{C}\\\
\boldsymbol{C}^{H}&\boldsymbol{S}_{00}\end{bmatrix}\begin{bmatrix}{\boldsymbol{Q}^{\overline{\mathrm{null}}}}^{H}\\\
{\boldsymbol{Q}^{\mathrm{null}}}^{H}\end{bmatrix},$ (19)

where $\boldsymbol{S}_{11}\in\mathbb{S}^{r_{\text{p}}}$,
$\boldsymbol{S}_{00}\in\mathbb{S}^{M-r_{\text{p}}}$, and
$\boldsymbol{C}\in\mathbb{C}^{r_{\text{p}}\times(M-r_{\text{p}})}$ are
variables to be optimized. We define
$\mathrm{rank}\big{(}\boldsymbol{H}_{\text{ID}}\boldsymbol{Q}^{\overline{\mathrm{null}}}(\boldsymbol{\Sigma}^{\overline{\mathrm{null}}})^{-\frac{1}{2}}\big{)}=\widetilde{r}_{\text{p}}$
and the truncated SVD
$\boldsymbol{H}_{\text{ID}}\boldsymbol{Q}^{\overline{\mathrm{null}}}(\boldsymbol{\Sigma}^{\overline{\mathrm{null}}})^{-\frac{1}{2}}=\boldsymbol{U}_{\text{p}}\boldsymbol{\Lambda}_{\text{p}}\boldsymbol{V}_{\text{p}}^{H}$,
where
$\boldsymbol{U}_{\text{p}}\in\mathbb{C}^{N_{\text{ID}}\times\widetilde{r}_{\text{p}}}$,
$\boldsymbol{V}_{\text{p}}\in\mathbb{C}^{r_{\text{p}}\times\widetilde{r}_{\text{p}}}$,
and
$\boldsymbol{\Lambda}_{\text{p}}=\mathrm{diag}(\lambda_{\text{p},1},\dots,\lambda_{\text{p},\widetilde{r}_{\text{p}}})$.
We have the following result.

###### Proposition 1

The optimal solution to problem (17) is

$\boldsymbol{S}^{*}=\begin{bmatrix}\boldsymbol{Q}^{\overline{\mathrm{null}}}\
\boldsymbol{Q}^{\mathrm{null}}\end{bmatrix}\begin{bmatrix}\boldsymbol{S}_{11}^{*}&\boldsymbol{C}^{*}\\\
{\boldsymbol{C}^{*}}^{H}&\boldsymbol{S}_{00}^{*}\end{bmatrix}\begin{bmatrix}{\boldsymbol{Q}^{\overline{\mathrm{null}}}}^{H}\\\
{\boldsymbol{Q}^{\mathrm{null}}}^{H}\end{bmatrix},$ (20)

where

$\boldsymbol{S}_{11}^{*}=(\boldsymbol{\Sigma}^{\overline{\mathrm{null}}})^{-\frac{1}{2}}\boldsymbol{V}_{\text{p}}\mathrm{diag}(\widetilde{p}_{1},\dots,\widetilde{p}_{\widetilde{r}_{\text{p}}})\boldsymbol{V}_{\text{p}}^{H}(\boldsymbol{\Sigma}^{\overline{\mathrm{null}}})^{-\frac{1}{2}},$
(21)

with

$\widetilde{p}_{k}=\Big{(}\frac{1}{\ln
2}-\frac{\sigma_{\text{ID}}^{2}}{\lambda_{\text{p},k}^{2}}\Big{)}^{+},\forall
k\in\\{1,\dots,\widetilde{r}_{\text{p}}\\},$ (22)

and $\boldsymbol{C}^{*}$ and $\boldsymbol{S}_{00}^{*}\succeq\boldsymbol{0}$
can be chosen arbitrarily such that
$\boldsymbol{S}^{*}\succeq\boldsymbol{0}$.101010Note that $\boldsymbol{C}^{*}$
and $\boldsymbol{S}_{00}^{*}$ are generally non-unique here. As a result, an
additional step is needed to determine them for solving the primal problem
(P1) later. Here, we can simply choose $\boldsymbol{C}^{*}=\boldsymbol{0}$ and
$\boldsymbol{S}_{00}^{*}=\boldsymbol{0}$ for obtaining the dual function
$g(\lambda,\nu,\boldsymbol{Z})$ only.

Proof: See Appendix A. $\square$

###  IV-B Solving Dual Problem (D1)

Next, we solve the dual problem (D1), which is convex but not necessarily
differentiable. Therefore, we can solve (D1) by applying subgradient-based
methods such as the ellipsoid method [30]. First, for the objective function
$g(\lambda,\nu,\boldsymbol{Z})$, the subgradient at
$(\lambda,\nu,z_{1},z_{2},z_{3})$ is
$\big{[}\mathrm{tr}(\boldsymbol{H}_{\text{EH}}^{H}\boldsymbol{H}_{\text{EH}}\boldsymbol{S}^{*})-\Gamma_{\text{EH}},P-\mathrm{tr}(\boldsymbol{S}^{*}),\mathrm{tr}(\boldsymbol{\dot{A}}^{H}\boldsymbol{\dot{A}}\boldsymbol{S}^{*})-\frac{1}{\Gamma_{\text{S},1}},\mathrm{tr}(\boldsymbol{\dot{A}}^{H}\boldsymbol{A}\boldsymbol{S}^{*}+\boldsymbol{A}^{H}\boldsymbol{\dot{A}}\boldsymbol{S}^{*})+j\mathrm{tr}(\boldsymbol{\dot{A}}^{H}\boldsymbol{A}\boldsymbol{S}^{*}-\boldsymbol{A}^{H}\boldsymbol{\dot{A}}\boldsymbol{S}^{*}),\mathrm{tr}(\boldsymbol{A}^{H}\boldsymbol{A}\boldsymbol{S}^{*})\big{]}^{T}$.
Then, let $\boldsymbol{q}_{1}$ denote the eigenvector corresponding to the
minimum eigenvalue of $\boldsymbol{D}$. Since constraint
$\boldsymbol{D}\succeq\boldsymbol{0}$ is equivalent to
$\boldsymbol{q}_{1}^{H}\boldsymbol{D}\boldsymbol{q}_{1}\geq 0$, the
subgradient of constraint $\boldsymbol{D}\succeq\boldsymbol{0}$ at
$(\lambda,\nu,z_{1},z_{2},z_{3})$ is
$\big{[}\boldsymbol{q}_{1}^{H}\boldsymbol{H}_{\text{EH}}^{H}\boldsymbol{H}_{\text{EH}}\boldsymbol{q}_{1},-1,\boldsymbol{q}_{1}^{H}\boldsymbol{\dot{A}}^{H}\boldsymbol{\dot{A}}\boldsymbol{q}_{1},\boldsymbol{q}_{1}^{H}(\boldsymbol{\dot{A}}^{H}\boldsymbol{A}+\boldsymbol{A}^{H}\boldsymbol{\dot{A}})\boldsymbol{q}_{1}+j\boldsymbol{q}_{1}^{H}(\boldsymbol{\dot{A}}^{H}\boldsymbol{A}-\boldsymbol{A}^{H}\boldsymbol{\dot{A}})\boldsymbol{q}_{1},\boldsymbol{q}_{1}^{H}\boldsymbol{A}^{H}\boldsymbol{A}\boldsymbol{q}_{1}\big{]}^{T}$.
Furthermore, let $\boldsymbol{q}_{2}=[q_{2,1},q_{2,2}]^{T}$ denote the
eigenvector corresponding to the minimum eigenvalue of $\boldsymbol{Z}$. The
subgradient of constraint $\boldsymbol{Z}\succeq\boldsymbol{0}$ at
$(\lambda,\nu,z_{1},z_{2},z_{3})$ is
$\big{[}0,0,-|q_{2,1}|^{2},-(q_{2,1}^{\dagger}q_{2,2}+q_{2,2}^{\dagger}q_{2,1})-j(q_{2,1}^{\dagger}q_{2,2}-q_{2,2}^{\dagger}q_{2,1}),-|q_{2,2}|^{2}\big{]}^{T}$.
With these derived subgradients, the ellipsoid method can be implemented
efficiently, based on which we can obtain the optimal dual solution to (D1) as
$\lambda^{*}$, $\nu^{*}$, and $\boldsymbol{Z}^{*}$.

###  IV-C Optimal Solution to Primal Problem (P1)

Now, we present the optimal solution to the primal problem (P1). With the
optimal dual variables $\lambda^{*}$, $\nu^{*}$, and $\boldsymbol{Z}^{*}$ at
hand, the corresponding unique optimal solution $\boldsymbol{S}_{11}^{*}$ to
problem (17) is directly used for constructing the optimal primal solution to
(P1), denoted by $\boldsymbol{S}_{11}^{\mathrm{opt}}$. However, as indicated
in Section IV-A, the optimal solutions of $\boldsymbol{C}^{*}$ and
$\boldsymbol{S}_{00}^{*}$ to (17) are not unique. As a result, with given
$\boldsymbol{S}_{11}^{\mathrm{opt}}$, we need to obtain the optimal solutions
of $\boldsymbol{C}$ and $\boldsymbol{S}_{00}$, denoted by
$\boldsymbol{C}^{\mathrm{opt}}$ and $\boldsymbol{S}_{00}^{\mathrm{opt}}$,
respectively, by exploiting the following proposition.

###### Proposition 2

The optimal solution to problem (P1) is

$\boldsymbol{S}^{\mathrm{opt},1}=\begin{bmatrix}\boldsymbol{Q}^{\overline{\mathrm{null}}}\
\boldsymbol{Q}^{\mathrm{null}}\end{bmatrix}\begin{bmatrix}\boldsymbol{S}_{11}^{\mathrm{opt}}&\boldsymbol{C}^{\mathrm{opt}}\\\
{\boldsymbol{C}^{\mathrm{opt}}}^{H}&\boldsymbol{S}_{00}^{\mathrm{opt}}\end{bmatrix}\begin{bmatrix}{\boldsymbol{Q}^{\overline{\mathrm{null}}}}^{H}\\\
{\boldsymbol{Q}^{\mathrm{null}}}^{H}\end{bmatrix},$ (23)

where $\boldsymbol{S}_{11}^{\mathrm{opt}}$ is given by (21) based on
$\lambda^{*}$, $\nu^{*}$, and $\boldsymbol{Z}^{*}$, while
$\boldsymbol{C}^{\mathrm{opt}}$ and $\boldsymbol{S}_{00}^{\mathrm{opt}}$ are
obtained by solving the following feasibility problem.

$\displaystyle\mathrm{find}\ \boldsymbol{C},\boldsymbol{S}_{00},$
$\displaystyle\mathrm{s.t.}\ \eqref{Ra},\ \eqref{Rc},\ \eqref{Rb-
reformulation},$ (24) $\displaystyle\ \ \
\boldsymbol{S}=\begin{bmatrix}\boldsymbol{Q}^{\overline{\mathrm{null}}}\
\boldsymbol{Q}^{\mathrm{null}}\end{bmatrix}\begin{bmatrix}\boldsymbol{S}_{11}^{\mathrm{opt}}&\boldsymbol{C}\\\
\boldsymbol{C}^{H}&\boldsymbol{S}_{00}\end{bmatrix}\begin{bmatrix}{\boldsymbol{Q}^{\overline{\mathrm{null}}}}^{H}\\\
{\boldsymbol{Q}^{\mathrm{null}}}^{H}\end{bmatrix}.$

###### Remark 1

Based on Proposition 2, we have the following interesting observations on the
optimal transmit covariance matrix $\boldsymbol{S}^{\mathrm{opt},1}$ for the
point target case. First, it is observed that
$\boldsymbol{S}^{\mathrm{opt},1}$ can be divided into two parts, i.e.,
$\boldsymbol{S}_{11}^{\mathrm{opt}}$ for the triple roles of communication,
sensing, and powering; and $\boldsymbol{C}^{\mathrm{opt}}$ and
$\boldsymbol{S}_{00}^{\mathrm{opt}}$ for sensing and WPT only. Next, due to
the competition of the three functionalities,111111From (20) and (21),
$\boldsymbol{S}^{\mathrm{opt,1}}$ can be decomposed as
$\boldsymbol{S}^{\mathrm{opt,1}}=\boldsymbol{F}\boldsymbol{\Xi}\boldsymbol{F}^{H}$.
Here, $\boldsymbol{\Xi}$ is a positive semidefinite matrix, while
$\boldsymbol{F}$ is a semi-unitary matrix whose columns are a set of
orthogonal uniform basis of
$\mathrm{span}\big{\\{}\boldsymbol{a}_{t}^{\dagger},\boldsymbol{\dot{a}}_{t}^{\dagger},\\{\boldsymbol{v}_{\text{ID},i}\\},\\{\boldsymbol{v}_{\text{EH},i}\\}\big{\\}}$.
Here, $\boldsymbol{v}_{\text{ID},i}$’s and $\boldsymbol{v}_{\text{EH},i}$’s
are the right singular vectors corresponding to the non-zero singular values
of $\boldsymbol{H}_{\text{ID}}$ and $\boldsymbol{H}_{\text{EH}}$,
respectively. $\boldsymbol{S}_{11}^{\mathrm{opt}}$ follows the EMT structure
based on the ID channel $\boldsymbol{H}_{\text{ID}}$ and the composite channel
matrix $\boldsymbol{D}$ consisting of sensing and EH channels (cf. (21)),
together with a WF-like power allocation (cf. (22)). Here, the matrix
$\boldsymbol{D}$ properly balances the tradeoff between sensing and WPT, and
the EMT structure of $\boldsymbol{S}_{11}^{\mathrm{opt}}$ highlights the
tradeoff between the communication versus the combined sensing and WPT.
Therefore, the obtained solution provides an efficient tradeoff between ISAC
and SWIPT. Notice that based on extensive simulations, we find that with
randomly generated channels, it follows that
$\boldsymbol{D}\succ\boldsymbol{0}$ and
$\boldsymbol{S}^{\mathrm{opt},1}=\boldsymbol{Q}^{\overline{\mathrm{null}}}\boldsymbol{S}_{11}^{\mathrm{opt}}{\boldsymbol{Q}^{\overline{\mathrm{null}}}}^{H}$,
i.e., only $\boldsymbol{S}_{11}^{\mathrm{opt}}$ is necessary; by contrast, in
some special cases (e.g., $\boldsymbol{H}_{\text{ID}}$ and
$\boldsymbol{H}_{\text{EH}}$ being orthogonal to $\boldsymbol{A}$), it could
happen that $\boldsymbol{D}$ is rank-deficient, and thus
$\boldsymbol{C}^{\mathrm{opt}}$ and $\boldsymbol{S}_{00}^{\mathrm{opt}}$ are
also required.

###  IV-D Special Cases with Co-located C-R and C-E

####  IV-D1 Co-located C-R setup

In this setup, we perform SWIPT for separate ID and EH receivers, while
locating the ID receiver concurrently. Accordingly, the LoS communication
channel is
$\boldsymbol{H}_{\text{ID}}=\alpha_{\text{ID}}\boldsymbol{a}_{\text{ID}}(\theta)\boldsymbol{a}_{t}^{T}(\theta)$,
where $\alpha_{\text{ID}}$ is the path loss and
$\boldsymbol{a}_{\text{ID}}(\theta)=\Big{[}e^{-j\frac{N_{\text{ID}}-1}{2}\pi\sin\theta},e^{-j\frac{N_{\text{ID}}-3}{2}\pi\sin\theta},\dots,e^{j\frac{N_{\text{ID}}-1}{2}\pi\sin\theta}\Big{]}^{T}$
is the steering vector at the ID receiver. Since
$\boldsymbol{a}_{\text{ID}}^{H}\boldsymbol{a}_{\text{ID}}=N_{\text{ID}}$ and
the communication rate is
$R(\boldsymbol{S})=\log_{2}\Big{(}1+\frac{\alpha_{\text{ID}}^{2}N_{\text{ID}}}{\sigma_{\text{ID}}^{2}}\boldsymbol{a}_{t}^{T}\boldsymbol{S}\boldsymbol{a}_{t}^{\dagger}\Big{)}$,
we can treat $\alpha_{\text{ID}}\sqrt{N_{\text{ID}}}\boldsymbol{a}_{t}^{T}$ as
the equivalent ID channel. According to Proposition 1, the optimal solution to
(P1) in this case is rank-one, and thus (P1) is equivalent to

$\max_{\boldsymbol{S}\succeq\boldsymbol{0}}\
\boldsymbol{a}_{t}^{T}\boldsymbol{S}\boldsymbol{a}_{t}^{\dagger},\
\mathrm{s.t.}\
\mathrm{tr}(\boldsymbol{H}_{\text{EH}}\boldsymbol{S}\boldsymbol{H}_{\text{EH}}^{H})\geq\Gamma_{\text{EH}},\
\mathrm{tr}(\boldsymbol{S})\leq P,$ (25)

whose feasibility requires that
$\boldsymbol{a}_{t}^{T}\boldsymbol{S}\boldsymbol{a}_{t}^{\dagger}\geq\frac{1}{\|\boldsymbol{\dot{a}}_{r}\|^{2}\Gamma_{\text{S,1}}}$
holds. On the Pareto boundary, the C-min and R-max vertices coincide, as both
are found by maximizing
$\boldsymbol{a}_{t}^{T}\boldsymbol{S}\boldsymbol{a}_{t}^{\dagger}$. Also, the
C-E and R-E tradeoff edges coincide, since they can be acquired by solving
(25).

####  IV-D2 Co-located C-E setup

In this setup, we perform SWIPT and locate the EH receiver concurrently.
Accordingly, the LoS WPT channel is
$\boldsymbol{H}_{\text{EH}}=\alpha_{\text{EH}}\boldsymbol{a}_{\text{EH}}(\theta)\boldsymbol{a}_{t}^{T}(\theta)$,
where $\alpha_{\text{EH}}$ is the path loss and
$\boldsymbol{a}_{\text{EH}}(\theta)=\Big{[}e^{-j\frac{N_{\text{EH}}-1}{2}\pi\sin\theta},e^{-j\frac{N_{\text{EH}}-3}{2}\pi\sin\theta},\dots,e^{j\frac{N_{\text{EH}}-1}{2}\pi\sin\theta}\Big{]}^{T}$
is the steering vector at the EH receiver. Since
$\boldsymbol{a}_{\text{EH}}^{H}\boldsymbol{a}_{\text{EH}}=N_{\text{EH}}$, the
harvested energy is given by
$E(\boldsymbol{S})=\alpha_{\text{EH}}^{2}N_{\text{EH}}\boldsymbol{a}_{t}^{T}\boldsymbol{S}\boldsymbol{a}_{t}^{\dagger}$.
We consider a special case with a single antenna equipped at the ID receiver,
i.e., $N_{\text{ID}}=1$, and denote the ID channel vector as
$\boldsymbol{h}_{\text{ID}}\in\mathbb{C}^{M\times 1}$. In this case, similar
to (25), (P1) is simplified to

$\displaystyle\max_{\boldsymbol{S}\succeq\boldsymbol{0}}$ $\displaystyle\
\boldsymbol{h}_{\text{ID}}^{H}\boldsymbol{S}\boldsymbol{h}_{\text{ID}}$ (26)
$\displaystyle\mathrm{s.t.}$ $\displaystyle\
\boldsymbol{a}_{t}^{T}\boldsymbol{S}\boldsymbol{a}_{t}^{\dagger}\geq\max\Big{(}\frac{\Gamma_{\text{EH}}}{\alpha_{\text{EH}}^{2}N_{\text{EH}}},\frac{1}{\|\boldsymbol{\dot{a}}_{r}\|^{2}\Gamma_{\text{S,1}}}\Big{)},\
\mathrm{tr}(\boldsymbol{S})\leq P.$

On the Pareto boundary, the C-min and E-max vertices coincide, and so do the
C-R and R-E tradeoff edges, as they can be obtained by solving (26).

(a) C-R-E view. (b) C-R view. (c) C-E view.

Figure 3: The Pareto boundaries of the C-R-E regions for the separate setup
and co-located C-R and C-E setups.

The following simulation verifies the observations above. For the simulations
in this paper, the H-AP is equipped with a ULA of $M=6$ and $N_{\text{S}}=16$
antennas with half a wavelength spacing between adjacent antennas. The target
reflection coefficient is set as $\alpha=10^{-6}$, accounting for a round-trip
path loss of $120$ dB. The transmit power at the H-AP is $30$ dBm. The noise
powers at the sensing receiver and the ID receiver over the transmitted signal
bandwidth of $100$ KHz are set to be
$\sigma_{\text{S}}^{2}=\sigma_{\text{ID}}^{2}=-90$ dBm. For the EH receiver,
the energy conversion efficiency is $\zeta=50$% [8]. The transmission duration
is set as $L=256$.

Fig. 3 shows the Pareto boundaries of the C-R-E regions for the point target
case with $N_{\text{ID}}=1$ and $N_{\text{EH}}=2$. We consider the line-of-
sight (LoS) ID and EH channels with
$\alpha_{\text{ID}}=\alpha_{\text{EH}}=10^{-3}$. For comparison, for the
separate setup, we set $\sin\theta=\frac{1}{6}$,
$\sin\theta_{\text{ID}}=\frac{5}{6}$, and
$\sin\theta_{\text{EH}}=\frac{3}{2}$. For the co-located C-R and C-E setups,
we set $\theta_{\text{ID}}=\theta$ and $\theta_{\text{EH}}=\theta$,
respectively, such that the Pareto boundary surfaces collapse into curves. To
be specific, with co-located C-R, the original C-R-E tradeoff boils down to
the C&R-E tradeoff,121212By decomposing
$\boldsymbol{S}=\boldsymbol{s}\boldsymbol{s}^{H}$, problem (25) can be
reexpressed as
$\max_{\boldsymbol{s}}|\boldsymbol{a}_{t}^{T}\boldsymbol{s}|^{2},\
\mathrm{s.t.}\
\sum_{i=1}^{N_{\text{EH}}}\lambda_{\text{EH},i}|\boldsymbol{v}_{\text{EH},i}^{H}\boldsymbol{s}|^{2}\geq\Gamma_{\text{EH}},\
\|\boldsymbol{s}\|^{2}\leq P$, where $\\{\lambda_{\text{EH},i}\\}$ and
$\\{\boldsymbol{v}_{\text{EH},i}\\}$ are the eigenvalues and corresponding
eigenvectors of $\boldsymbol{H}_{\text{EH}}^{H}\boldsymbol{H}_{\text{EH}}$,
respectively. In this case, the optimal solution satisfies
$\boldsymbol{s}\in\mathrm{span}\big{\\{}\boldsymbol{a}_{t}^{\dagger},\\{\boldsymbol{v}_{\text{EH},i}\\}\big{\\}}$.
which enables improved ISAC performance than the separate setup under the same
harvested energy; with co-located C-E, the C-R-E tradeoff is merged into C&E-R
tradeoff,131313Similar to (25), (26) can be reexpressed as
$\max_{\boldsymbol{s}}|\boldsymbol{h}_{\text{ID}}^{H}\boldsymbol{s}|^{2},\
\mathrm{s.t.}\
|\boldsymbol{a}_{t}^{T}\boldsymbol{s}|^{2}\geq\max(\frac{\Gamma_{\text{EH}}}{\alpha_{\text{EH}}^{2}N_{\text{EH}}},\frac{1}{\|\boldsymbol{\dot{a}}_{r}\|^{2}\Gamma_{\text{S,1}}}),\
\|\boldsymbol{s}\|^{2}\leq P$. The closed-form optimal solution has been given
in [26], which satisfies
$\boldsymbol{s}\in\mathrm{span}\\{\boldsymbol{a}_{t}^{\dagger},\boldsymbol{h}_{\text{ID}}\\}$.
which enhances sensing and WPT performance compared to the separate setup at a
given communication rate.

##  V Optimal Solution to Problem (P2) with Extended Target

In this section, we address problem (P2) for the extended target case. For
convenience, we rewrite the constraint in (11c) for $i=2$ as

$\mathrm{tr}(\boldsymbol{S}^{-1})\leq\Gamma_{\text{S},2},$ (27)

where
$\Gamma_{\text{S},2}\triangleq\frac{L}{\sigma_{\text{S}}^{2}N_{\text{S}}}\Gamma_{\text{S}}$.
Notice that (P2) is convex and thus can be solved by using standard convex
optimization techniques [28]. In the following, we develop an efficient
algorithm to obtain a partitionable solution of $\boldsymbol{S}$ to (P2).

###  V-A Optimal Solution to (P2) Based on Problem Reformulation

This subsection finds the optimal solution to problem (P2) based on its
reformulation. We define
$\boldsymbol{H}\triangleq\big{[}\boldsymbol{H}_{\text{ID}}^{T}\
\boldsymbol{H}_{\text{EH}}^{T}\big{]}^{T}\in\mathbb{C}^{(N_{\text{ID}}+N_{\text{EH}})\times
M}$, which corresponds to the combined ID and EH channel matrix with
$\mathrm{rank}(\boldsymbol{H})=r$. The SVD of $\boldsymbol{H}$ is expressed as

$\boldsymbol{H}=\begin{bmatrix}\boldsymbol{U}^{\overline{\mathrm{null}}}\
\boldsymbol{U}^{\mathrm{null}}\end{bmatrix}\begin{bmatrix}\boldsymbol{\Lambda}^{\overline{\mathrm{null}}}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{0}\end{bmatrix}\begin{bmatrix}{\boldsymbol{V}^{\overline{\mathrm{null}}}}^{H}\\\
{\boldsymbol{V}^{\mathrm{null}}}^{H}\end{bmatrix},$ (28)

where
$\boldsymbol{\Lambda}^{\overline{\mathrm{null}}}=\mathrm{diag}(\lambda_{1},\dots,\lambda_{r})$,
and ${\boldsymbol{V}^{\overline{\mathrm{null}}}}\in\mathbb{C}^{M\times r}$ and
${\boldsymbol{V}^{\mathrm{null}}}\in\mathbb{C}^{M\times(M-r)}$ denote the
right singular vectors corresponding to the $r$ non-zero singular values and
the $M-r$ zero singular values of $\boldsymbol{H}$, respectively. Without loss
of generality, we express $\boldsymbol{S}$ as

$\boldsymbol{S}=\begin{bmatrix}\boldsymbol{V}^{\overline{\mathrm{null}}}\
\boldsymbol{V}^{\mathrm{null}}\end{bmatrix}\begin{bmatrix}\boldsymbol{S}_{1}&\boldsymbol{B}\\\
\boldsymbol{B}^{H}&\boldsymbol{S}_{0}\end{bmatrix}\begin{bmatrix}{\boldsymbol{V}^{\overline{\mathrm{null}}}}^{H}\\\
{\boldsymbol{V}^{\mathrm{null}}}^{H}\end{bmatrix},$ (29)

where $\boldsymbol{S}_{1}\in\mathbb{S}^{r}$,
$\boldsymbol{S}_{0}\in\mathbb{S}^{M-r}$, and
$\boldsymbol{B}\in\mathbb{C}^{r\times(M-r)}$ are variables to be optimized.
Note that since $\boldsymbol{H}\boldsymbol{V}^{\mathrm{null}}=\boldsymbol{0}$,
problem (P2) is equivalent to

$\displaystyle(\text{P}2.1):\max_{\boldsymbol{S}_{1},\boldsymbol{S}_{0},\boldsymbol{B}}$
$\displaystyle\
\log_{2}\det\Big{(}\boldsymbol{I}_{N_{\text{ID}}}+\frac{\boldsymbol{H}_{\text{ID}}\boldsymbol{V}^{\overline{\mathrm{null}}}\boldsymbol{S}_{1}{\boldsymbol{V}^{\overline{\mathrm{null}}}}^{H}\boldsymbol{H}_{\text{ID}}^{H}}{\sigma_{\text{ID}}^{2}}\Big{)}$
(30a) $\displaystyle\mathrm{s.t.}$ $\displaystyle\
\mathrm{tr}(\boldsymbol{H}_{\text{EH}}\boldsymbol{V}^{\overline{\mathrm{null}}}\boldsymbol{S}_{1}{\boldsymbol{V}^{\overline{\mathrm{null}}}}^{H}\boldsymbol{H}_{\text{EH}}^{H})\geq\Gamma_{\text{EH}},$
(30b) $\displaystyle\
\mathrm{tr}\left(\begin{bmatrix}\boldsymbol{S}_{1}&\boldsymbol{B}\\\
\boldsymbol{B}^{H}&\boldsymbol{S}_{0}\end{bmatrix}^{-1}\right)\leq\Gamma_{\text{S},2},$
(30c) $\displaystyle\
\mathrm{tr}(\boldsymbol{S}_{1})+\mathrm{tr}(\boldsymbol{S}_{0})\leq P,$ (30d)
$\displaystyle\ \begin{bmatrix}\boldsymbol{S}_{1}&\boldsymbol{B}\\\
\boldsymbol{B}^{H}&\boldsymbol{S}_{0}\end{bmatrix}\succeq\boldsymbol{0}.$
(30e)

To obtain a more insightful solution, we eliminate $\boldsymbol{B}$ and
rewrite $\boldsymbol{S}_{0}$ in constraints (30c), (30d), and (30e) as
follows.

###### Proposition 3

Problem (P2.1) is equivalent to the following problem.

$\displaystyle(\text{P}2.2):\max_{\boldsymbol{S}_{1}\succeq\boldsymbol{0},p_{0}\geq
0}$ $\displaystyle\
\log_{2}\det\Big{(}\boldsymbol{I}_{N_{\text{ID}}}+\frac{1}{\sigma_{\text{ID}}^{2}}\widetilde{\boldsymbol{H}}_{\text{ID}}\boldsymbol{S}_{1}\widetilde{\boldsymbol{H}}_{\text{ID}}^{H}\Big{)}$
$\displaystyle\mathrm{s.t.}$ $\displaystyle\
\mathrm{tr}(\widetilde{\boldsymbol{H}}_{\text{EH}}\boldsymbol{S}_{1}\widetilde{\boldsymbol{H}}_{\text{EH}}^{H})\geq\Gamma_{\text{EH}},$
$\displaystyle\
\mathrm{tr}(\boldsymbol{S}_{1}^{-1})+\frac{M-r}{p_{0}}\leq\Gamma_{\text{S},2},$
$\displaystyle\ \mathrm{tr}(\boldsymbol{S}_{1})+(M-r)p_{0}\leq P,$

where
$\widetilde{\boldsymbol{H}}_{\text{ID}}\triangleq\boldsymbol{H}_{\text{ID}}\boldsymbol{V}^{\overline{\mathrm{null}}}$
and
$\widetilde{\boldsymbol{H}}_{\text{EH}}\triangleq\boldsymbol{H}_{\text{EH}}\boldsymbol{V}^{\overline{\mathrm{null}}}$
denotes the projection of the ID and EH channels on the range space of the
combined channel $\boldsymbol{H}$, respectively.

Proof: See Appendix B. $\square$

Let $\boldsymbol{S}_{1}^{\mathrm{opt}}$ and $p_{0}^{\mathrm{opt}}$ denote the
optimal solution to (P2.2). Then the optimal solution to (P2) is

$\boldsymbol{S}^{\mathrm{opt},2}=\boldsymbol{V}^{\overline{\mathrm{null}}}\boldsymbol{S}_{1}^{\mathrm{opt}}{\boldsymbol{V}^{\overline{\mathrm{null}}}}^{H}+p_{0}^{\mathrm{opt}}\boldsymbol{V}^{\mathrm{null}}{\boldsymbol{V}^{\mathrm{null}}}^{H}.$
(31)

###### Remark 2

Based on (31), the optimal transmit covariance matrix
$\boldsymbol{S}^{\mathrm{opt},2}$ for the extended target case differs from
that of $\boldsymbol{S}^{\mathrm{opt},1}$ to problem (P1) for the point target
case in several aspects. Firstly, while $\boldsymbol{S}^{\mathrm{opt},1}$ is
derived based on SVD of the ID channel $\boldsymbol{H}_{\text{ID}}$ and the
composite sensing and EH channel $\boldsymbol{D}$,
$\boldsymbol{S}^{\mathrm{opt},2}$ is obtained based on SVD of the combined ID
and EH channel $\boldsymbol{H}=\big{[}\boldsymbol{H}_{\text{ID}}^{T}\
\boldsymbol{H}_{\text{EH}}^{T}\big{]}^{T}$. Consequently,
$\boldsymbol{S}^{\mathrm{opt},2}$ is divided into two parts. The first part,
$\boldsymbol{V}^{\overline{\mathrm{null}}}\boldsymbol{S}_{1}^{\mathrm{opt}}{\boldsymbol{V}^{\overline{\mathrm{null}}}}^{H}$,
balances the tradeoff among the WF power allocation for communication, the
strongest EMT for WPT, and the isotropic transmission for sensing. This part
requires $\boldsymbol{S}_{1}^{\mathrm{opt}}$ to be a general positive
semidefinite matrix, which differs from the diagonal structure of
$\boldsymbol{S}_{11}^{\mathrm{opt}}$ to (P1). The second part,
$p_{0}^{\mathrm{opt}}\boldsymbol{V}^{\mathrm{null}}{\boldsymbol{V}^{\mathrm{null}}}^{H}$,
is dedicated to sensing only, which differs from
$\boldsymbol{C}^{\mathrm{opt}}$ and $\boldsymbol{S}_{00}^{\mathrm{opt}}$ to
(P1) for both sensing and WPT. Furthermore, the CRB constraint in (27)
requires $\boldsymbol{S}^{\mathrm{opt},2}$ to be full-rank and thus the
average transmit power allocated for dedicated sensing $p_{0}^{\mathrm{opt}}$
is always necessary for cases with $M>r$, whereas
$\boldsymbol{C}^{\mathrm{opt}}$ and $\boldsymbol{S}_{00}^{\mathrm{opt}}$ to
(P1) are generally not required for randomly generated channels.

###  V-B Semi-Closed-Form Solution to Problem (P2) in Special Case with Low
SNR

To obtain more insights, this subsection further considers the special case
with low communication SNR (i.e.,
$\frac{P}{\sigma_{\text{ID}}^{2}}\mathrm{tr}(\boldsymbol{H}_{\text{ID}}^{H}\boldsymbol{H}_{\text{ID}})\to
0$) and obtains a structured solution to problem (P2) in this case. Towards
this end, we transform problem (P2) as problem (P2.2) similarly as in Section
V-A, and then focus on solving (P2.2). In (P2.2), maximizing
$\log_{2}\det\Big{(}\boldsymbol{I}_{N_{\text{ID}}}+\frac{1}{\sigma_{\text{ID}}^{2}}\widetilde{\boldsymbol{H}}_{\text{ID}}\boldsymbol{S}_{1}\widetilde{\boldsymbol{H}}_{\text{ID}}^{H}\Big{)}$
is equivalent to maximizing
$\mathrm{tr}(\widetilde{\boldsymbol{H}}_{\text{ID}}\boldsymbol{S}_{1}\widetilde{\boldsymbol{H}}_{\text{ID}}^{H})$
due to the consideration of low SNR. Accordingly, (P2.2) is reformulated as

$\displaystyle(\text{P}3):\max_{\boldsymbol{S}_{1}\succeq\boldsymbol{0},p_{0}\geq
0}$ $\displaystyle\
\mathrm{tr}(\widetilde{\boldsymbol{H}}_{\text{ID}}\boldsymbol{S}_{1}\widetilde{\boldsymbol{H}}_{\text{ID}}^{H})$
(32a) $\displaystyle\mathrm{s.t.}$ $\displaystyle\
\mathrm{tr}(\widetilde{\boldsymbol{H}}_{\text{EH}}\boldsymbol{S}_{1}\widetilde{\boldsymbol{H}}_{\text{EH}}^{H})\geq\Gamma_{\text{EH}},$
(32b) $\displaystyle\
\mathrm{tr}(\boldsymbol{S}_{1}^{-1})+(M-r)\frac{1}{p_{0}}\leq\Gamma_{\text{S},2},$
(32c) $\displaystyle\ \mathrm{tr}(\boldsymbol{S}_{1})+(M-r)p_{0}\leq P.$ (32d)

###### Proposition 4

The solution to problem (P3) is

$\boldsymbol{S}_{1}^{\mathrm{low}}=\boldsymbol{Q}_{\text{e}}\mathrm{diag}\Big{(}\sqrt{\frac{\mu^{*}}{\sigma_{\text{e},1}}},\dots,\sqrt{\frac{\mu^{*}}{\sigma_{\text{e},r}}}\Big{)}{\boldsymbol{Q}_{\text{e}}}^{H},\
p_{0}^{\mathrm{low}}=\sqrt{\frac{\mu^{*}}{\nu^{*}}},$ (33)

where $\lambda^{*}\geq 0$, $\mu^{*}>0$, and $\nu^{*}>0$ denote the optimal
dual variables associated with the constraints in (32b), (32c), and (32d) in
(P3), respectively, and ${\boldsymbol{Q}_{\text{e}}}$ and
$\\{\sigma_{\text{e},k}\\},\forall k\in\\{1,\dots,r\\}$, are obtained based on
the EVD
$\nu^{*}\boldsymbol{I}_{r}-\widetilde{\boldsymbol{H}}_{\text{ID}}^{H}\widetilde{\boldsymbol{H}}_{\text{ID}}-\lambda^{*}\widetilde{\boldsymbol{H}}_{\text{EH}}^{H}\widetilde{\boldsymbol{H}}_{\text{EH}}=\boldsymbol{Q}_{\text{e}}\mathrm{diag}(\sigma_{\text{e},1},\dots,\sigma_{\text{e},r}){\boldsymbol{Q}_{\text{e}}}^{H}$.

Proof: See Appendix C. $\square$

Based on Proposition 4 and (31), we obtain the optimal transmit covariance
matrix in the low SNR case as
$\boldsymbol{S}^{\mathrm{low}}=\boldsymbol{V}^{\overline{\mathrm{null}}}\boldsymbol{S}_{1}^{\mathrm{low}}{\boldsymbol{V}^{\overline{\mathrm{null}}}}^{H}+p_{0}^{\mathrm{low}}\boldsymbol{V}^{\mathrm{null}}{\boldsymbol{V}^{\mathrm{null}}}^{H}$,
which is also of full rank to satisfy the CRB requirement. It is observed in
(33) that $\boldsymbol{S}_{1}^{\mathrm{low}}$ follows the EMT structure based
on the composite channel matrix
$\nu^{*}\boldsymbol{I}_{r}-\widetilde{\boldsymbol{H}}_{\text{ID}}^{H}\widetilde{\boldsymbol{H}}_{\text{ID}}-\lambda^{*}\widetilde{\boldsymbol{H}}_{\text{EH}}^{H}\widetilde{\boldsymbol{H}}_{\text{EH}}$,
together with a channel-inversion-like power allocation, to balance the
competing between the isotropic transmission for sensing and the strongest EMT
for combined communication and WPT (or SWIPT), with
$\\{\sigma_{\text{e},k}\\},\forall k\in\\{1,\dots,r\\}$, denoting the
equivalent channel power gains.

###  V-C Special Case with Co-located R-E

In the co-located R-E setup, we perform SWIPT for a co-located ID&EH receiver
and locate a separate target concurrently. Accordingly, the communication and
WPT channels are identical, i.e.,
$\boldsymbol{H}_{\text{ID}}=\boldsymbol{H}_{\text{EH}}=\boldsymbol{G}$ and
$N_{\text{ID}}=N_{\text{EH}}=N$. In the following, we investigate two co-
located receiver designs with time-switching and power-splitting.

####  V-C1 Time-switching design

For this type of receiver, we define the ratio of time allocated for ID and EH
as $1-\tau$ and $\tau$, respectively, where $0<\tau<1$. We also define the
transmit covariance matrices for ID and EH as $\boldsymbol{\bar{S}}_{1}$ and
$\boldsymbol{\bar{S}}_{2}$, respectively. With the flexible power constraint,
the SVD of
$\boldsymbol{G}=\boldsymbol{U}\boldsymbol{\Lambda}\boldsymbol{V}^{H}$, and
defining
$\boldsymbol{\widetilde{S}}_{1}=\boldsymbol{V}\boldsymbol{\bar{S}}_{1}\boldsymbol{V}^{H}$
and
$\boldsymbol{\widetilde{S}}_{2}=\boldsymbol{V}\boldsymbol{\bar{S}}_{2}\boldsymbol{V}^{H}$,
(P2) becomes
$\max_{\boldsymbol{\widetilde{S}}_{1}\succeq\boldsymbol{0},\boldsymbol{\widetilde{S}}_{2}\succeq\boldsymbol{0}}\
(1-\tau)\log_{2}\det\Big{(}\boldsymbol{I}_{N}+\frac{1}{\sigma_{\text{ID}}^{2}}\boldsymbol{\Lambda}^{H}\boldsymbol{\Lambda}\boldsymbol{\widetilde{S}}_{1}\Big{)}$,
$\mathrm{s.t.}\
\tau\mathrm{tr}(\boldsymbol{\Lambda}^{H}\boldsymbol{\Lambda}\boldsymbol{\widetilde{S}}_{2})\geq\Gamma_{\text{EH}},\
\mathrm{tr}\Big{(}\big{(}(1-\tau)\boldsymbol{\widetilde{S}}_{1}+\tau\boldsymbol{\widetilde{S}}_{2}\big{)}^{-1}\Big{)}\leq\Gamma_{\text{S},2},\
\mathrm{tr}((1-\tau)\boldsymbol{\widetilde{S}}_{1}+\tau\boldsymbol{\widetilde{S}}_{2})\leq
P$. According to Proposition 3, the optimal solutions to
$\boldsymbol{\widetilde{S}}_{1}$ and $\boldsymbol{\widetilde{S}}_{2}$ admit
the form of
$\boldsymbol{\widetilde{S}}_{i}=\mathrm{diag}(p_{i,1},\dots,p_{i,\bar{r}},p_{i,0},\dots,p_{i,0})$,
$\forall i=1,2$, where $\bar{r}=\mathrm{rank}(\boldsymbol{G})$. In particular,
if the receiver is equipped with a single antenna, i.e., $\boldsymbol{G}$
becomes $\boldsymbol{g}^{H}\in\mathbb{C}^{1\times M}$, then the optimal
solution admits the form of
$\boldsymbol{S}_{i}=p_{i,1}\frac{\boldsymbol{g}\boldsymbol{g}^{H}}{\|\boldsymbol{g}\|^{2}}+p_{i,0}(\boldsymbol{I}_{M}-\frac{\boldsymbol{g}\boldsymbol{g}^{H}}{\|\boldsymbol{g}\|^{2}}),\forall
i=1,2$, and (P2) is equivalent to $\max_{p_{1,1},p_{1,0},p_{2,1},p_{2,0}}\
p_{1,1},\ \mathrm{s.t.}\
\tau\|\boldsymbol{g}\|^{2}p_{2,1}\geq\Gamma_{\text{EH}},\
\frac{1}{(1-\tau)p_{1,1}+\tau p_{2,1}}+\frac{M-1}{(1-\tau)p_{1,0}+\tau
p_{2,0}}\leq\Gamma_{\text{S},2},\ (1-\tau)p_{1,1}+\tau
p_{2,1}+(M-1)\big{(}(1-\tau)p_{1,0}+\tau p_{2,0}\big{)}\leq P$. Applying the
Lagrangian dual method, we find that all the three constraints are tight at
the optimality, thus the Pareto boundary with given $\tau$ is
$\big{\\{}(\mathrm{CRB}_{\text{TS}},R_{\text{TS}},E_{\text{TS}})\big{|}\mathrm{CRB}_{\text{TS}}=\frac{\sigma_{\text{S}}^{2}N_{\text{S}}\Gamma_{\text{S},2}}{L},R_{\text{TS}}=(1-\tau)\log_{2}(1+\frac{\Gamma_{1}}{(1-\tau)\sigma_{\text{ID}}^{2}}),E_{\text{TS}}=\Gamma_{2},P=\frac{\Gamma_{1}+\Gamma_{2}}{\|\boldsymbol{g}\|^{2}}+\frac{(M-1)^{2}}{\Gamma_{\text{S},2}-\frac{\|\boldsymbol{g}\|^{2}}{\Gamma_{1}+\Gamma_{2}}}\big{\\}}$.

####  V-C2 Power-splitting design

For this type of receiver, we define the ratio of power allocated for ID and
EH at each receive antenna $n\in\\{1,\dots,N\\}$ as $1-\rho_{n}$ and
$\rho_{n}$, respectively, where $0<\rho_{n}<1$. Then, (P2) becomes
$\max_{\boldsymbol{S}\succeq\boldsymbol{0}}\
\log_{2}\det\Big{(}\boldsymbol{I}_{N}+\frac{1}{\sigma_{\text{ID}}^{2}}\boldsymbol{\bar{\Delta}}^{1/2}\boldsymbol{G}\boldsymbol{S}\boldsymbol{G}^{H}\boldsymbol{\bar{\Delta}}^{1/2}\Big{)}$,
$\mathrm{s.t.}\
\mathrm{tr}(\boldsymbol{\Delta}^{1/2}\boldsymbol{G}\boldsymbol{S}\boldsymbol{G}^{H}\boldsymbol{\Delta}^{1/2})\geq\Gamma_{\text{EH}},\
\mathrm{tr}(\boldsymbol{S}^{-1})\leq\Gamma_{\text{S},2},\
\mathrm{tr}(\boldsymbol{S})\leq P$, where
$\boldsymbol{\Delta}=\mathrm{diag}(\rho_{1},\dots,\rho_{N})$ and
$\boldsymbol{\bar{\Delta}}=\boldsymbol{I}_{N}-\boldsymbol{\Delta}$. By
treating $\boldsymbol{\bar{\Delta}}^{1/2}\boldsymbol{G}$ and
$\boldsymbol{\Delta}^{1/2}\boldsymbol{G}$ as equivalent ID and EH channels,
this problem can be solved similarly as (P2) with given
$\rho_{1},\dots,\rho_{N}$. If the receiver is equipped with a single antenna,
then similar to the time-switching design, the optimal solution admits the
form of
$\boldsymbol{S}=p_{1}\frac{\boldsymbol{g}\boldsymbol{g}^{H}}{\|\boldsymbol{g}\|^{2}}+p_{0}(\boldsymbol{I}_{M}-\frac{\boldsymbol{g}\boldsymbol{g}^{H}}{\|\boldsymbol{g}\|^{2}})$,
and (P2) becomes $\max_{p_{1},p_{0}}\ p_{1},\ \mathrm{s.t.}\
\frac{1}{p_{1}}+\frac{M-1}{p_{0}}\leq\Gamma_{\text{S},2},\
p_{1}+(M-1)p_{0}\leq P$, whose feasibility requires
$\rho_{1}\|\boldsymbol{g}\|^{2}p_{1}\geq\Gamma_{\text{EH}}$. According to
Proposition 4, both the CRB and transmit power constraints are tight at the
optimality, thus the Pareto boundary with given $\rho_{1}$ is expressed as
$\big{\\{}(\mathrm{CRB}_{\text{PS}},R_{\text{PS}},E_{\text{PS}})\big{|}\mathrm{CRB}_{\text{PS}}=\frac{\sigma_{\text{S}}^{2}N_{\text{S}}\Gamma_{\text{S},2}}{L},R_{\text{PS}}=\log_{2}(1+\frac{(1-\rho_{1})\Gamma}{\sigma_{\text{ID}}^{2}}),E_{\text{PS}}=\rho_{1}\Gamma,P=\frac{\Gamma}{\|\boldsymbol{g}\|^{2}}+\frac{(M-1)^{2}}{\Gamma_{\text{S},2}-\frac{\|\boldsymbol{g}\|^{2}}{\Gamma}}\big{\\}}$.

(a) Cross-section view on
$\Gamma_{\text{S}}=20\mathrm{CRB}_{\mathrm{min},2}$. (b) Cross-section view on
$\Gamma_{\text{EH}}=0.2E_{\mathrm{max}}$.

Figure 4: The Pareto boundary of the C-R-E region for the co-located R-E setup
with the time-switching design.

(a) Cross-section view on
$\Gamma_{\text{S}}=20\mathrm{CRB}_{\mathrm{min},2}$. (b) Cross-section view on
$\Gamma_{\text{EH}}=0.2E_{\mathrm{max}}$.

Figure 5: The Pareto boundary of the C-R-E region for the co-located R-E setup
with the power-splitting design.

Figs. 4 and 5 show the Pareto boundaries of the C-R-E regions for the extended
target case with co-located R-E setup, where $N_{\text{ID}}=N_{\text{EH}}=1$.
We consider the Rayleigh fading channel vector $\boldsymbol{g}$, which is
generated as a CSCG random vector with each element being zero mean and
variance of $10^{-6}$. As shown in Figs. 4 and 5, the upper bound of the
Pareto boundary for the time-switching design is achieved when $\tau=0$
(without the peak power constraint), which is the same as the union of Pareto
boundaries for the power-splitting design with $\rho_{1}$ ranging from 0 to 1.
We also show the Pareto boundary of the separate setup with $\boldsymbol{g}$
modeling both ID and EH channels, which collapses to a curve, similar to the
co-located C-R/C-E setups in Section IV-D. By contrast, for the co-located R-E
setup, there is an R-E tradeoff at the receiver, which impairs the SWIPT
performance. This tradeoff, along with the tradeoff between SWIPT and sensing,
contributes to the gap between the Pareto boundary of the co-located R-E setup
and that of the separate setup.

##  VI Numerical Results

This section evaluates the performance of our proposed optimal designs. In the
following simulation, in addition to the system setups mentioned in Section
IV-D, we consider $\alpha_{\text{ID}}=10^{-4}$ and
$\alpha_{\text{EH}}=10^{-2}$ for the ID and EH channels, corresponding to the
path loss of $80$ dB and $40$ dB [8], respectively.

Figure 6: The Pareto boundaries of the C-R-E regions for the point target case
with $N_{\text{ID}}=N_{\text{EH}}=1$.

Fig. 6 shows the Pareto boundaries of the C-R-E regions for the point target
case with $N_{\text{ID}}=N_{\text{EH}}=1$ and target angle $\theta=0$. In this
figure, we consider the LoS channels for the ID and EH channels, i.e.,
$\boldsymbol{H}_{\text{ID}}=\alpha_{\text{ID}}\boldsymbol{a}_{t}^{T}(\theta_{\text{ID}})$
and
$\boldsymbol{H}_{\text{EH}}=\alpha_{\text{EH}}\boldsymbol{a}_{t}^{T}(\theta_{\text{EH}})$,
where $\sin\theta_{\text{ID}}=\frac{2\gamma}{M}$ and
$\sin\theta_{\text{EH}}=\frac{4\gamma}{M}$ correspond to the angles of the ID
and EH receivers, with $\gamma=0$, $0.6$, and $1$ corresponding to the cases
when the sensing, ID, and EH channels are identical, correlated, and
orthogonal, respectively. It is observed that for the identical channels with
$\gamma=0$, the Pareto boundary collapses to a point, which means that the
R-max, E-max, and C-min strategies are identical and the three performance
metrics are optimized simultaneously. It is also observed that for the
orthogonal channels with $\gamma=1$, optimizing one metric (e.g., E-max)
always leads to poor performances for the other two (e.g., a close-to-zero
rate and the highest CRB), thus showing that the three objectives are
conflicting with each other. Furthermore, for the correlated channels with
$\gamma=0.6$, the C-R-E region boundary is observed to lie between those with
$\gamma=1$ and $\gamma=0$, thus showing the non-trivial C-R-E tradeoff in this
case.

(a) $M=6$. (b) $M=2$.

Figure 7: The Pareto boundaries of the C-R-E regions for the extended target
case with $N_{\text{ID}}=N_{\text{EH}}=2$ and $P=40$ dBm.

Figs. 7(a) and 7(b) show the Pareto boundaries of the C-R-E regions for the
extended target case with $M=6$ and $M=2$, respectively, where $P=40$ dBm and
$N_{\text{ID}}=N_{\text{EH}}=2$. We consider Rician fading for the ID and EH
channels, i.e.,
$\boldsymbol{H}_{\text{ID}}=\sqrt{\frac{\kappa_{\text{ID}}}{\kappa_{\text{ID}}+1}}\boldsymbol{H}_{\text{ID}}^{\text{los}}+\sqrt{\frac{1}{\kappa_{\text{ID}}+1}}\boldsymbol{H}_{\text{ID}}^{w}$
and
$\boldsymbol{H}_{\text{EH}}=\sqrt{\frac{\kappa_{\text{EH}}}{\kappa_{\text{EH}}+1}}\boldsymbol{H}_{\text{EH}}^{\text{los}}+\sqrt{\frac{1}{\kappa_{\text{EH}}+1}}\boldsymbol{H}_{\text{EH}}^{w}$,
where $\kappa_{\text{ID}}=\kappa_{\text{EH}}=20$ are the Rician factors,
$\boldsymbol{H}_{\text{ID}}^{w}$ and $\boldsymbol{H}_{\text{EH}}^{w}$ are
generated as CSCG random matrices with each element being zero mean and
variance of $\alpha_{\text{ID}}^{2}$ and $\alpha_{\text{EH}}^{2}$,
respectively, and
$\boldsymbol{H}_{\text{ID}}^{\text{los}}=\alpha_{\text{ID}}\boldsymbol{a}_{\text{ID}}(\theta_{\text{ID}})\boldsymbol{a}_{t}^{T}(\theta_{\text{ID}})$
and
$\boldsymbol{H}_{\text{EH}}^{\text{los}}=\alpha_{\text{EH}}\boldsymbol{a}_{\text{EH}}(\theta_{\text{EH}})\boldsymbol{a}_{t}^{T}(\theta_{\text{EH}})$
with $\theta_{\text{ID}}=\frac{\pi}{6}$ and
$\theta_{\text{EH}}=\frac{2\pi}{3}$. It is observed in Fig. 7(a) that when
$M=6$, the R-max vertex, the E-max vertex, and the R-E tradeoff edge are not
achievable. This is due to the fact that $\mathrm{CRB}_{\text{ID},2}\to\infty$
and $\mathrm{CRB}_{\text{EH},2}\to\infty$ when $M>N_{\text{ID}}>1$ (cf.
Section III-A). When the required CRB $\Gamma_{\text{S}}$ becomes sufficiently
large, the Pareto boundary contracts and approaches a boundary that is
parallel to the CRB-axis. This is because the CRB constraint in (P2) can be
easily satisfied, as if is not considered. Thus, the required CRB will hardly
affect the C-R tradeoff. Next, it is observed in Fig. 7(b) that when
$M=N_{\text{ID}}=2$, only the R-max vertex is achieved and the E-max vertex is
not achievable. This is due to the fact that $\mathrm{CRB}_{\text{ID},2}$ is
finite as $\boldsymbol{S}_{\text{ID}}$ is of full rank, but
$\mathrm{CRB}_{\text{EH},2}\to\infty$ as $M>1$.

Next, we evaluate the performance of our proposed designs as compared to the
following benchmark schemes based on the time division and the EMT.

  * •

Time division protocol: The transmission duration is divided into three
orthogonal portions, $\tau_{\text{ID}}\geq 0$, $\tau_{\text{EH}}\geq 0$, and
$\tau_{\text{S}}\geq 0$, such that
$\tau_{\text{ID}}+\tau_{\text{EH}}+\tau_{\text{S}}=1$, during which the H-AP
employs the transmit covariance matrices $\boldsymbol{S}_{\text{ID}}$,
$\boldsymbol{S}_{\text{EH}}$, and $\boldsymbol{S}_{\text{S},i}$, respectively.
For case $i\in\\{1,2\\}$, the corresponding communication rate, harvested
energy, and estimation CRB are expressed as
${R}_{\text{TD}}(\tau_{\text{ID}},\tau_{\text{EH}},\tau_{\text{S}})=\tau_{\text{ID}}{R}(\boldsymbol{S}_{\text{ID}})$,
${E}_{\text{TD},i}(\tau_{\text{ID}},\tau_{\text{EH}},\tau_{\text{S}})=\tau_{\text{ID}}{E}(\boldsymbol{S}_{\text{ID}})+\tau_{\text{EH}}{E}(\boldsymbol{S}_{\text{EH}})+\tau_{\text{S}}{E}(\boldsymbol{S}_{\text{S},i})$,
and
$\mathrm{CRB}_{\text{TD},i}(\tau_{\text{ID}},\tau_{\text{EH}},\tau_{\text{S}})=\mathrm{CRB}_{i}(\tau_{\text{ID}}\boldsymbol{S}_{\text{ID}}+\tau_{\text{EH}}\boldsymbol{S}_{\text{EH}}+\tau_{\text{S}}\boldsymbol{S}_{\text{S},i})$,
respectively. The time periods $\tau_{\text{ID}}$, $\tau_{\text{EH}}$, and
$\tau_{\text{S}}$ are optimized to achieve different C-R-E tradeoffs.

  * •

EMT over the ID channel: This scheme is motivated by the communication-optimal
EMT design for MIMO communication [25]. For the point target case, supposing
that the SVD of the ID channel $\boldsymbol{H}_{\text{ID}}$ is
$\boldsymbol{H}_{\text{ID}}=\boldsymbol{U}_{\text{ID}}\boldsymbol{\Lambda}_{\text{ID}}\boldsymbol{V}_{\text{ID}}^{H}$,
we design the transmit covariance matrix as
$\boldsymbol{S}=\boldsymbol{V}_{\text{ID}}\boldsymbol{\widehat{S}}^{\text{ID}}\boldsymbol{V}_{\text{ID}}^{H}$,
where $\boldsymbol{\widehat{S}}^{\text{ID}}$ is a diagonal matrix that is
optimized based on problem (P1). For the extended target case, supposing the
SVD of the projected ID channel $\widetilde{\boldsymbol{H}}_{\text{ID}}$ in
problem (P2.2) is
$\widetilde{\boldsymbol{H}}_{\text{ID}}=\widetilde{\boldsymbol{U}}_{\text{ID}}\widetilde{\boldsymbol{\Lambda}}_{\text{ID}}\widetilde{\boldsymbol{V}}_{\text{ID}}^{H}$,
we design the transmit covariance as
$\boldsymbol{S}=\boldsymbol{V}^{\overline{\mathrm{null}}}\boldsymbol{S}_{1}{\boldsymbol{V}^{\overline{\mathrm{null}}}}^{H}+p_{0}\boldsymbol{V}^{\mathrm{null}}{\boldsymbol{V}^{\mathrm{null}}}^{H}=\boldsymbol{V}^{\overline{\mathrm{null}}}\widetilde{\boldsymbol{V}}_{\text{ID}}\boldsymbol{\widehat{S}}_{1}^{\text{ID}}\widetilde{\boldsymbol{V}}_{\text{ID}}^{H}{\boldsymbol{V}^{\overline{\mathrm{null}}}}^{H}+p_{0}\boldsymbol{V}^{\mathrm{null}}{\boldsymbol{V}^{\mathrm{null}}}^{H}$
based on (31), where $\boldsymbol{\widehat{S}}_{1}^{\text{ID}}$ is a diagonal
matrix that is optimized together with $p_{0}$ based on problem (P2.2).

  * •

EMT over combined ID and EH channels: This scheme is motivated by the optimal
solution to (P2), in which the SVD of the combined ID and EH channels
$\boldsymbol{H}=\big{[}\boldsymbol{H}_{\text{ID}}^{T}\
\boldsymbol{H}_{\text{EH}}^{T}\big{]}^{T}$ is employed to facilitate the
optimization. As a result, for the point target case, by using the SVD of
$\boldsymbol{H}$ in (28), we design the transmit covariance matrix as
$\boldsymbol{S}=\boldsymbol{V}^{\overline{\mathrm{null}}}\boldsymbol{\widehat{S}}{\boldsymbol{V}^{\overline{\mathrm{null}}}}^{H}$,
where $\boldsymbol{\widehat{S}}$ is a diagonal matrix that is optimized based
on problem (P1). For the extended target case, supposing the SVD of the
combined projected ID and EH channels
$\widetilde{\boldsymbol{H}}=\big{[}\widetilde{\boldsymbol{H}}_{\text{ID}}^{T}\
\widetilde{\boldsymbol{H}}_{\text{EH}}^{T}\big{]}^{T}$ is
$\widetilde{\boldsymbol{H}}=\widetilde{\boldsymbol{U}}\widetilde{\boldsymbol{\Lambda}}\widetilde{\boldsymbol{V}}^{H}$,
we design the transmit covariance as
$\boldsymbol{S}=\boldsymbol{V}^{\overline{\mathrm{null}}}\boldsymbol{S}_{1}{\boldsymbol{V}^{\overline{\mathrm{null}}}}^{H}+p_{0}\boldsymbol{V}^{\mathrm{null}}{\boldsymbol{V}^{\mathrm{null}}}^{H}=\boldsymbol{V}^{\overline{\mathrm{null}}}\widetilde{\boldsymbol{V}}\boldsymbol{\widehat{S}}_{1}\widetilde{\boldsymbol{V}}^{H}{\boldsymbol{V}^{\overline{\mathrm{null}}}}^{H}+p_{0}\boldsymbol{V}^{\mathrm{null}}{\boldsymbol{V}^{\mathrm{null}}}^{H}$
based on (31), where $\boldsymbol{\widehat{S}}_{1}$ is a diagonal matrix that
is optimized together with $p_{0}$ based on problem (P2.2).

(a) Point target case. (b) Extended target case.

Figure 8: The communication rate ${R}$ versus the energy constraint
$\Gamma_{\text{EH}}$ with $N_{\text{ID}}=N_{\text{EH}}=2$.

Figs. 8(a) and 8(b) show the obtained communication rate ${R}$ by solving
problems (P1) and (P2) versus the EH constraint $\Gamma_{\text{EH}}$ for the
point and extended target cases, respectively, where the sensing constraints
are set as $\Gamma_{\text{S}}=50\mathrm{CRB}_{\mathrm{min},1}$ and
$\Gamma_{\text{S}}=50\mathrm{CRB}_{\mathrm{min},2}$, respectively. Here, we
set $N_{\text{ID}}=N_{\text{EH}}=2$ and consider Rician fading for
$\boldsymbol{H}_{\text{ID}}$ and $\boldsymbol{H}_{\text{EH}}$. It is observed
that for each target case, our proposed optimal design outperforms the other
three benchmark schemes. The EMT over combined ID and EH channels is observed
to outperform the other two benchmarks in the medium regime of
$\Gamma_{\text{EH}}$. This shows the benefit of considering both ID and EH
channels for EMT in this case. In addition, the time division protocol and EMT
over the ID channel are observed to perform close to the proposed design when
$\Gamma_{\text{EH}}$ is low. This is because the three schemes behave as the
communication-optimal design in this case, as both the CRB and EH constraints
become less stringent.

(a) Point target case. (b) Extended target case.

Figure 9: The communication rate ${R}$ versus the CRB constraint
$\Gamma_{\text{S}}$ with $N_{\text{ID}}=N_{\text{EH}}=2$.

Figs. 9(a) and 9(b) show the obtained communication rate ${R}$ by problems
(P1) and (P2) versus the estimation CRB constraint $\Gamma_{\text{S}}$ for the
point and extended target cases, respectively, in which two EH constraints
$\Gamma_{\text{EH}}=0.5{E}_{\mathrm{max}}$ (high $\Gamma_{\text{EH}}$ case)
and $\Gamma_{\text{EH}}=0.05{E}_{\mathrm{max}}$ (low $\Gamma_{\text{EH}}$
case) are considered, with $N_{\text{ID}}=N_{\text{EH}}=2$. It is observed
that for each case with different EH constraints, our proposed optimal design
outperforms the other three benchmark schemes. The EMT over combined ID and EH
channels exhibits superior performance compared with EMT over the ID channel
when $\Gamma_{\text{EH}}$ is high, while the opposite results hold with low
$\Gamma_{\text{EH}}$, due to the similar reasons for the results in Figs. 8(a)
and 8(b). Furthermore, with a low $\Gamma_{\text{EH}}$, the performance of
time division protocol and EMT over the ID channel are observed to approach
that of the optimal design when $\Gamma_{\text{S}}$ becomes large. This is due
to the fact that both schemes can achieve the maximum communication rate with
less restrictive CRB and EH constraints in this case.

##  VII Conclusion

This paper studied the fundamental C-R-E performance limits for a new multi-
functional MIMO system integrating triple functions of sensing, communication,
and powering. For both the point and extended target cases, we characterized
the Pareto boundaries of the C-R-E performance regions, by optimally solving
novel MIMO communication rate maximization problems subject to both EH and
estimation CRB constraints. It was shown that for the point target case, the
resultant C-R-E tradeoff highly depends on the correlations among the ID, EH,
and sensing channels; while for the extended target case, the resultant C-R-E
tradeoff relies on the system configuration such as the number of antennas
equipped at the transceivers. Furthermore, the proposed optimal design was
shown to significantly outperform the benchmark schemes based on time division
and EMT. It is our hope that this paper can open new avenues for leveraging
radio signals for multiple purposes towards 6G.

###  -A Proof of Proposition 1

Substituting (18) and (19), problem (17) is re-expressed as

$\displaystyle\max_{\boldsymbol{S}_{11},\boldsymbol{C},\boldsymbol{S}_{00}}$
$\displaystyle\
\log_{2}\det\Big{(}\boldsymbol{I}_{N_{\text{ID}}}+\frac{1}{\sigma_{\text{ID}}^{2}}\boldsymbol{H}_{\text{ID}}(\boldsymbol{Q}^{\overline{\mathrm{null}}}\boldsymbol{S}_{11}{\boldsymbol{Q}^{\overline{\mathrm{null}}}}^{H}$
(34) $\displaystyle\
+\boldsymbol{Q}^{\overline{\mathrm{null}}}\boldsymbol{C}{\boldsymbol{Q}^{\mathrm{null}}}^{H}+\boldsymbol{Q}^{\mathrm{null}}\boldsymbol{C}^{H}{\boldsymbol{Q}^{\overline{\mathrm{null}}}}^{H}$
$\displaystyle\
+\boldsymbol{Q}^{\mathrm{null}}\boldsymbol{S}_{00}{\boldsymbol{Q}^{\mathrm{null}}}^{H})\boldsymbol{H}_{\text{ID}}^{H}\Big{)}-\mathrm{tr}(\boldsymbol{\Sigma}^{\overline{\mathrm{null}}}\boldsymbol{S}_{11})$
$\displaystyle\mathrm{s.t.}$ $\displaystyle\
\begin{bmatrix}\boldsymbol{S}_{11}&\boldsymbol{C}\\\
\boldsymbol{C}^{H}&\boldsymbol{S}_{00}\end{bmatrix}\succeq\boldsymbol{0}.$

According to Lemma 1, to solve (17), we only need to deal with problem (34)
for the case with
$\boldsymbol{H}_{\text{ID}}(\boldsymbol{Q}^{\overline{\mathrm{null}}}\boldsymbol{C}{\boldsymbol{Q}^{\mathrm{null}}}^{H}+\boldsymbol{Q}^{\mathrm{null}}\boldsymbol{C}^{H}{\boldsymbol{Q}^{\overline{\mathrm{null}}}}^{H}+\boldsymbol{Q}^{\mathrm{null}}\boldsymbol{S}_{00}{\boldsymbol{Q}^{\mathrm{null}}}^{H})\boldsymbol{H}_{\text{ID}}^{H}=\boldsymbol{0}$.
Therefore, we can simply choose $\boldsymbol{C}^{*}=\boldsymbol{0}$ and
$\boldsymbol{S}_{00}^{*}=\boldsymbol{0}$ as the optimal solution of
$\boldsymbol{C}$ and $\boldsymbol{S}_{00}$ for obtaining the dual function
$g(\lambda,\nu,\boldsymbol{Z})$ only, and thus (34) is simplified as the
optimization of $\boldsymbol{S}_{11}$ in the following:

$\displaystyle\max_{\boldsymbol{S}_{11}\succeq\boldsymbol{0}}$ $\displaystyle\
\log_{2}\det\Big{(}\boldsymbol{I}_{N_{\text{ID}}}+\frac{1}{\sigma_{\text{ID}}^{2}}\boldsymbol{H}_{\text{ID}}\boldsymbol{Q}^{\overline{\mathrm{null}}}\boldsymbol{S}_{11}{\boldsymbol{Q}^{\overline{\mathrm{null}}}}^{H}\boldsymbol{H}_{\text{ID}}^{H}\Big{)}$
(35) $\displaystyle\
-\mathrm{tr}(\boldsymbol{\Sigma}^{\overline{\mathrm{null}}}\boldsymbol{S}_{11}).$

Define
$\boldsymbol{S}_{11}=(\boldsymbol{\Sigma}^{\overline{\mathrm{null}}})^{-\frac{1}{2}}\boldsymbol{V}_{\text{p}}\boldsymbol{\widetilde{S}}_{11}\boldsymbol{V}_{\text{p}}^{H}(\boldsymbol{\Sigma}^{\overline{\mathrm{null}}})^{-\frac{1}{2}}$.
Problem (35) is re-expressed as

$\max_{\boldsymbol{\widetilde{S}}_{11}\succeq\boldsymbol{0}}\
\log_{2}\det\Big{(}\boldsymbol{I}_{\widetilde{r}_{\text{p}}}+\frac{1}{\sigma_{\text{ID}}^{2}}\boldsymbol{\Lambda}_{\text{p}}^{H}\boldsymbol{\Lambda}_{\text{p}}\boldsymbol{\widetilde{S}}_{11}\Big{)}-\mathrm{tr}(\boldsymbol{\widetilde{S}}_{11}).$
(36)

As shown in [8], the optimal solution to problem (36) is given by
$\boldsymbol{\widetilde{S}}_{11}^{*}=\mathrm{diag}(\widetilde{p}_{1},\dots,\widetilde{p}_{\widetilde{r}_{\text{p}}})$,
where $\widetilde{p}_{k}=(\frac{1}{\ln
2}-\frac{\sigma_{\text{ID}}^{2}}{\lambda_{\text{p},k}^{2}})^{+}$, $\forall
k\in\\{1,\dots,\widetilde{r}_{\text{p}}\\}$. Thus, the optimal solution of
$\boldsymbol{S}_{11}$ to (34) is

$\boldsymbol{S}_{11}^{*}=(\boldsymbol{\Sigma}^{\overline{\mathrm{null}}})^{-\frac{1}{2}}\boldsymbol{V}_{\text{p}}\boldsymbol{\widetilde{S}}_{11}^{*}\boldsymbol{V}_{\text{p}}^{H}(\boldsymbol{\Sigma}^{\overline{\mathrm{null}}})^{-\frac{1}{2}}.$
(37)

Thus, this completes the proof.

###  -B Proof of Proposition 3

To facilitate the proof of this proposition, we first present the following
lemmas.

###### Lemma 2

If $\begin{bmatrix}\boldsymbol{S}_{1}&\boldsymbol{B}\\\
\boldsymbol{B}^{H}&\boldsymbol{S}_{0}\end{bmatrix}\succeq\boldsymbol{0}$, then
we have $\mathrm{tr}\left(\begin{bmatrix}\boldsymbol{S}_{1}&\boldsymbol{B}\\\
\boldsymbol{B}^{H}&\boldsymbol{S}_{0}\end{bmatrix}^{-1}\right)\geq\mathrm{tr}\left(\begin{bmatrix}\boldsymbol{S}_{1}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{S}_{0}\end{bmatrix}^{-1}\right)$.

Proof: Based on the Schur complement,
$\begin{bmatrix}\boldsymbol{S}_{1}&\boldsymbol{B}\\\
\boldsymbol{B}^{H}&\boldsymbol{S}_{0}\end{bmatrix}\succeq\boldsymbol{0}$ is
equivalent to $\boldsymbol{S}_{1}\succeq\boldsymbol{0}$ and
$\boldsymbol{F}_{1}=\boldsymbol{S}_{0}-\boldsymbol{B}^{H}\boldsymbol{S}_{1}^{-1}\boldsymbol{B}\succeq\boldsymbol{0}$,
while is also equivalent to $\boldsymbol{S}_{0}\succeq\boldsymbol{0}$ and
$\boldsymbol{F}_{0}=\boldsymbol{S}_{1}-\boldsymbol{B}\boldsymbol{S}_{0}^{-1}\boldsymbol{B}^{H}\succeq\boldsymbol{0}$.
According to the partitioned matrix formula, we have
$\begin{bmatrix}\boldsymbol{S}_{1}&\boldsymbol{B}\\\
\boldsymbol{B}^{H}&\boldsymbol{S}_{0}\end{bmatrix}^{-1}=\begin{bmatrix}\boldsymbol{F}_{0}^{-1}&-\boldsymbol{S}_{1}^{-1}\boldsymbol{B}\boldsymbol{F}_{1}^{-1}\\\
-\boldsymbol{F}_{1}^{-1}\boldsymbol{B}^{H}\boldsymbol{S}_{1}^{-1}&\boldsymbol{F}_{1}^{-1}\end{bmatrix}$
and $\begin{bmatrix}\boldsymbol{S}_{1}&\boldsymbol{0}\\\
\boldsymbol{0}^{H}&\boldsymbol{S}_{0}\end{bmatrix}^{-1}=\begin{bmatrix}\boldsymbol{S}_{1}^{-1}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{S}_{0}^{-1}\end{bmatrix}$. Since
$\boldsymbol{F}_{1}\succeq\boldsymbol{0}$ and
$\boldsymbol{F}_{0}\succeq\boldsymbol{0}$, we have
$\mathrm{tr}(\boldsymbol{F}_{0}^{-1})=\mathrm{tr}(\boldsymbol{S}_{1}^{-1}+\boldsymbol{S}_{1}^{-1}\boldsymbol{B}\boldsymbol{F}_{1}^{-1}\boldsymbol{B}^{H}\boldsymbol{S}_{1}^{-1})\geq\mathrm{tr}(\boldsymbol{S}_{1}^{-1})$
and
$\mathrm{tr}(\boldsymbol{F}_{1}^{-1})=\mathrm{tr}(\boldsymbol{S}_{0}^{-1}+\boldsymbol{S}_{0}^{-1}\boldsymbol{B}^{H}\boldsymbol{F}_{0}^{-1}\boldsymbol{B}\boldsymbol{S}_{0}^{-1})\geq\mathrm{tr}(\boldsymbol{S}_{0}^{-1})$,
respectively. Therefore, it follows that
$\mathrm{tr}\left(\begin{bmatrix}\boldsymbol{S}_{1}&\boldsymbol{B}\\\
\boldsymbol{B}^{H}&\boldsymbol{S}_{0}\end{bmatrix}^{-1}\right)=\mathrm{tr}(\boldsymbol{F}_{0}^{-1})+\mathrm{tr}(\boldsymbol{F}_{1}^{-1})\geq\mathrm{tr}(\boldsymbol{S}_{1}^{-1})+\mathrm{tr}(\boldsymbol{S}_{0}^{-1})=\mathrm{tr}\left(\begin{bmatrix}\boldsymbol{S}_{1}&\boldsymbol{0}\\\
\boldsymbol{0}&\boldsymbol{S}_{0}\end{bmatrix}^{-1}\right)$, where the
equality holds when $\boldsymbol{B}=\boldsymbol{0}$. $\square$

###### Lemma 3

Suppose that $\boldsymbol{S}_{0}\in\mathbb{S}_{+}^{M-r}$ with diagonal
elements $p_{0,1},\dots,p_{0,M-r}$, where $p_{0,k}\geq 0,\forall
k\in\\{1,\dots,M-r\\}$. Then it follows that
$\mathrm{tr}(\boldsymbol{S}_{0}^{-1})\geq\mathrm{tr}\big{(}\mathrm{diag}(p_{0,1},\dots,p_{0,M-r})^{-1}\big{)}$.

Proof: We decompose $\boldsymbol{S}_{0}$ as
$\boldsymbol{S}_{0}=\begin{bmatrix}\boldsymbol{S}_{0,1}&\boldsymbol{s}_{0,1}\\\
\boldsymbol{s}_{0,1}^{H}&p_{0,M-r}\end{bmatrix}$, where
$\boldsymbol{S}_{0,1}\in\mathbb{S}^{M-r-1}$ and
$\boldsymbol{s}_{0,1}\in\mathbb{C}^{(M-r-1)\times 1}$. From Lemma 2, we have
$\mathrm{tr}\left(\begin{bmatrix}\boldsymbol{S}_{0,1}&\boldsymbol{s}_{0,1}\\\
\boldsymbol{s}_{0,1}^{H}&p_{0,M-r}\end{bmatrix}^{-1}\right)\geq\mathrm{tr}\left(\begin{bmatrix}\boldsymbol{S}_{0,1}&\boldsymbol{0}\\\
\boldsymbol{0}&p_{0,M-r}\end{bmatrix}^{-1}\right)$, where
$\boldsymbol{S}_{0,1}\succeq\boldsymbol{0}$ and $p_{0,M-r}\geq 0$. We further
decompose $\boldsymbol{S}_{0,1}$ to
$\boldsymbol{S}_{0,2}\in\mathbb{S}^{M-r-2},\dots,$
$\boldsymbol{S}_{0,M-r-1}=p_{0,1}$. After repeating the process above, this
lemma can be proved. $\square$

According to Lemma 2, subject to constraint in (30d), the left-hand side of
(30c) is minimized to
$\mathrm{tr}(\boldsymbol{S}_{1}^{-1})+\mathrm{tr}(\boldsymbol{S}_{0}^{-1})$
when $\boldsymbol{B}=\boldsymbol{0}$. We assume that $\boldsymbol{S}_{0}$ is
not diagonal and define $p_{0}\triangleq\frac{\sum_{k=1}^{M-r}p_{0,k}}{M-r}$.
According to Lemma 3, we have
$\mathrm{tr}(\boldsymbol{S}_{0}^{-1})\geq\mathrm{tr}\big{(}\mathrm{diag}(p_{0,1},\dots,p_{0,M-r})^{-1}\big{)}=\sum_{k=1}^{M-r}\frac{1}{p_{0,k}}\geq\frac{M-r}{p_{0}}=\mathrm{tr}\big{(}(p_{0}\boldsymbol{I}_{M-r})^{-1}\big{)}$,
where the second inequality holds based on the harmonic inequality. Thus, with
$\mathrm{tr}(\boldsymbol{S}_{0})$ unchanged,
$\mathrm{tr}(\boldsymbol{S}_{0}^{-1})$ is minimized by removing the non-
diagonal elements of $\boldsymbol{S}_{0}$ and averaging its diagonal elements.
Therefore, the optimal solution to problem (P2.1) satisfies that
$\boldsymbol{B}=\boldsymbol{0}$ and
$\boldsymbol{S}_{0}=p_{0}\boldsymbol{I}_{M-r}$. Thus, this completes the
proof.

###  -C Proof of Proposition 4

Let $\lambda\geq 0$, $\mu\geq 0$, and $\nu\geq 0$ denote the dual variables
associated with the constraints in (32b), (32c), and (32d), respectively. The
Lagrangian of problem (P3) is given by

$\displaystyle\mathcal{L}_{3}$
$\displaystyle(\boldsymbol{S}_{1},p_{0},\lambda,\mu,\nu)=-\mathrm{tr}(\boldsymbol{E}\boldsymbol{S}_{1})-\mu\mathrm{tr}(\boldsymbol{S}_{1}^{-1})$
(38) $\displaystyle-(M-r)(\frac{\mu}{p_{0}}+\nu
p_{0})-\lambda\Gamma_{\text{EH}}+\mu\Gamma_{\text{S},2}+\nu P,$

where
$\boldsymbol{E}\triangleq\nu\boldsymbol{I}_{r}-\widetilde{\boldsymbol{H}}_{\text{ID}}^{H}\widetilde{\boldsymbol{H}}_{\text{ID}}-\lambda\widetilde{\boldsymbol{H}}_{\text{EH}}^{H}\widetilde{\boldsymbol{H}}_{\text{EH}}$.
Accordingly, the dual function of (P3) is defined as

$g_{3}(\lambda,\mu,\nu)=\max_{\boldsymbol{S}_{1}\succeq\boldsymbol{0},p_{0}\geq
0}\ \mathcal{L}_{3}(\boldsymbol{S}_{1},p_{0},\lambda,\mu,\nu).$ (39)

We have the following lemma, which can be verified similarly as Lemma 1.

###### Lemma 4

In order for the dual function $g_{3}(\lambda,\mu,\nu)$ to be bounded from
above, it must hold that $\boldsymbol{E}\succeq\boldsymbol{0}$.

Based on Lemma 4, the dual problem of (P3) is defined as

$(\text{D}3):\min_{\lambda\geq 0,\mu\geq 0,\nu\geq 0}\
g_{3}(\lambda,\mu,\nu),\ \mathrm{s.t.}\ \boldsymbol{E}\succeq\boldsymbol{0}.$
(40)

Since (P3) is convex and satisfies the Slater’s condition, strong duality
holds between (P3) and its dual problem (D3) [30]. Therefore, we can solve
(P3) by equivalently solving (D3).

First, we find the dual function $g_{3}(\lambda,\mu,\nu)$ in (39) under given
$\lambda\geq 0$, $\mu\geq 0$, and $\nu\geq 0$. By dropping the constant terms
$-\lambda\Gamma_{\text{EH}}+\mu\Gamma_{\text{S},2}+\nu P$, the problem in (39)
is equivalent to

$\max_{\boldsymbol{S}_{1}\succeq\boldsymbol{0},p_{0}\geq 0}\
-\mathrm{tr}(\boldsymbol{E}\boldsymbol{S}_{1})-\mu\mathrm{tr}(\boldsymbol{S}_{1}^{-1})-(M-r)(\frac{\mu}{p_{0}}+\nu
p_{0}).$ (41)

The EVD of $\boldsymbol{E}$ is expressed as
$\boldsymbol{E}=\boldsymbol{Q}_{\text{e}}\boldsymbol{\Sigma}_{\text{e}}\boldsymbol{Q}_{\text{e}}^{H}$,
where
$\boldsymbol{\Sigma}_{\text{e}}=\mathrm{diag}(\sigma_{\text{e},1},\dots,\sigma_{\text{e},r})$.
Without loss of generality, we assume
$\sigma_{\text{e},1}\geq\dots\geq\sigma_{\text{e},r}\geq 0$. Define
$\boldsymbol{S}_{1}=\boldsymbol{Q}_{\text{e}}\boldsymbol{\widetilde{S}}_{1}\boldsymbol{Q}_{\text{e}}^{H}$.
Problem (41) is re-expressed as

$\max_{\boldsymbol{\widetilde{S}}_{1}\succeq\boldsymbol{0},p_{0}\geq 0}\
-\mathrm{tr}(\boldsymbol{\Sigma}_{\text{e}}\boldsymbol{\widetilde{S}}_{1})-\mu\mathrm{tr}(\boldsymbol{\widetilde{S}}_{1}^{-1})-(M-r)(\frac{\mu}{p_{0}}+\nu
p_{0}).$ (42)

According to Lemma 3 in Appendix B, the optimal solution of
$\boldsymbol{\widetilde{S}}_{1}$ to problem (42) is diagonal, i.e., we have
$\boldsymbol{\widetilde{S}}_{1}=\mathrm{diag}(\widetilde{p}_{1,1},\dots,\widetilde{p}_{1,r})$.
Therefore, (42) is re-expressed as

$\max_{\\{\widetilde{p}_{1,k}\geq 0\\},p_{0}\geq 0}\
-\sum_{k=1}^{r}(\sigma_{\text{e},k}\widetilde{p}_{1,k}+\frac{\mu}{\widetilde{p}_{1,k}})-(M-r)(\frac{\mu}{p_{0}}+\nu
p_{0}).$ (43)

Notice that for the case with $\sigma_{\text{e},r}=0$, i.e., $\boldsymbol{E}$
is rank-deficient, the optimality of problem (43) cannot be achieved unless
$\mu=0$. However, it is easy to verify that at the optimality of problem (P3),
the constraints (32c) and (32d) should be tight. Therefore, it follows that
$\mu>0$ and $\nu>0$, which means that we should have $\sigma_{\text{e},r}>0$,
i.e., the dual problem (D3) should also satisfy the constraint
$\boldsymbol{E}\succ\boldsymbol{0}$ to achieve the optimality.

By checking the first derivative, the optimal solution to problem (43) is

$\displaystyle\widetilde{p}_{1,k}^{*}$
$\displaystyle=\sqrt{\frac{\mu}{\sigma_{\text{e},k}}},\forall
k\in\\{1,\dots,r\\},$ (44) $\displaystyle p_{0}^{*}$
$\displaystyle=\sqrt{\frac{\mu}{\nu}}.$ (45)

Thus, the optimal solution of $\boldsymbol{S}_{1}$ to problem (41) is

$\boldsymbol{S}_{1}^{*}=\boldsymbol{Q}_{\text{e}}\mathrm{diag}(\widetilde{p}_{1,1}^{*},\dots,\widetilde{p}_{1,r}^{*})\boldsymbol{Q}_{\text{e}}^{H}.$
(46)

Next, we solve the dual problem (D3), which is convex but not necessarily
differentiable, and thus can be solved by applying the subgradient-based
method such as the ellipsoid method. For the objective function
$g_{3}(\lambda,\mu,\nu)$, the subgradient at $(\lambda,\mu,\nu)$ is
$\Big{[}\mathrm{tr}(\widetilde{\boldsymbol{H}}_{\text{EH}}^{H}\widetilde{\boldsymbol{H}}_{\text{EH}}\boldsymbol{S}_{1}^{*})-\Gamma_{\text{EH}},\Gamma_{\text{S},2}-\mathrm{tr}\big{(}(\boldsymbol{S}_{1}^{*})^{-1}\big{)}-(M-r)\frac{1}{p_{0}^{*}},P-\mathrm{tr}(\boldsymbol{S}_{1}^{*})-(M-r)p_{0}^{*}\Big{]}^{T}$.
Furthermore, let $\boldsymbol{q}_{3}$ denote the eigenvector of
$\boldsymbol{E}$ corresponding to its minimum eigenvalue
$\sigma_{\text{e},r}$. With the similar analysis as in Section IV, the
subgradient of constraint $\boldsymbol{E}\succ\boldsymbol{0}$ at
$(\lambda,\mu,\nu)$ is
$\big{[}\boldsymbol{q}_{3}^{H}\widetilde{\boldsymbol{H}}_{\text{EH}}^{H}\widetilde{\boldsymbol{H}}_{\text{EH}}\boldsymbol{q}_{3},0,-1\big{]}^{T}$.

Finally, we present the optimal solution to the primal problem (P3). With the
optimal dual variables $\lambda^{*}$, $\mu^{*}$, and $\nu^{*}$ at hand, the
corresponding optimal solutions $\boldsymbol{S}_{1}^{*}$ and $p_{0}^{*}$ in
(46) and (45) to problem (41) can be directly used for constructing the
optimal primal solutions to (P3), denoted by
$\boldsymbol{S}_{1}^{\mathrm{low}}$ and $p_{0}^{\mathrm{low}}$, respectively.
This thus completes the proof of this proposition.

## References

  * [1] Y. Chen, H. Hua, and J. Xu, “Transmit optimization for multi-functional MIMO systems integrating sensing, communication, and powering,” _presented at IEEE ICC 2023, arXiv preprint arXiv:2210.16716_, 2022.
  * [2] Y. Cui, F. Liu, X. Jing, and J. Mu, “Integrating sensing and communications for ubiquitous IoT: Applications, trends, and challenges,” _IEEE Netw._ , vol. 35, no. 5, pp. 158–167, Sep. 2021.
  * [3] W. Saad, M. Bennis, and M. Chen, “A vision of 6G wireless systems: Applications, trends, technologies, and open research problems,” _IEEE Netw._ , vol. 34, no. 3, pp. 134–142, May 2020.
  * [4] W. Tong and P. Zhu, _6G: The Next Horizon: From Connected People and Things to Connected Intelligence_. Cambridge University Press, 2021, pp. 24–26.
  * [5] Y. Zeng, B. Clerckx, and R. Zhang, “Communications and signals design for wireless power transmission,” _IEEE Trans. Commun._ , vol. 65, no. 5, pp. 2264–2290, May 2017.
  * [6] B. Clerckx, R. Zhang, R. Schober, D. W. K. Ng, D. I. Kim, and H. V. Poor, “Fundamentals of wireless information and power transfer: From RF energy harvester models to signal and system designs,” _IEEE J. Sel. Areas Commun._ , vol. 37, no. 1, pp. 4–33, Jan. 2019.
  * [7] F. Liu, Y. Cui, C. Masouros, J. Xu, T. X. Han, Y. C. Eldar, and S. Buzzi, “Integrated sensing and communications: Toward dual-functional wireless networks for 6G and beyond,” _IEEE J. Sel. Areas Commun._ , vol. 40, no. 6, pp. 1728–1767, Jun. 2022.
  * [8] R. Zhang and C. K. Ho, “MIMO broadcasting for simultaneous wireless information and power transfer,” _IEEE Trans. Wireless Commun._ , vol. 12, no. 5, pp. 1989–2001, May 2013.
  * [9] J. Xu, L. Liu, and R. Zhang, “Multiuser MISO beamforming for simultaneous wireless information and power transfer,” _IEEE Trans. Signal Process._ , vol. 62, no. 18, pp. 4798–4810, Sep. 2014.
  * [10] S. Luo, J. Xu, T. J. Lim, and R. Zhang, “Capacity region of MISO broadcast channel for simultaneous wireless information and power transfer,” _IEEE Trans. Commun._ , vol. 63, no. 10, pp. 3856–3868, Oct. 2015.
  * [11] J. Park and B. Clerckx, “Joint wireless information and energy transfer in a two-user MIMO interference channel,” _IEEE Trans. Wireless Commun._ , vol. 12, no. 8, pp. 4210–4221, Aug. 2013.
  * [12] D. W. K. Ng, E. S. Lo, and R. Schober, “Robust beamforming for secure communication in systems with wireless information and power transfer,” _IEEE Trans. Wireless Commun._ , vol. 13, no. 8, pp. 4599–4615, Aug. 2014\.
  * [13] E. Boshkovska, D. W. K. Ng, N. Zlatanov, and R. Schober, “Practical non-linear energy harvesting model and resource allocation for SWIPT systems,” _IEEE Commun. Lett._ , vol. 19, no. 12, pp. 2082–2085, Dec. 2015.
  * [14] Y. Xiong, F. Liu, Y. Cui, W. Yuan, and T. X. Han, “Flowing the information from Shannon to Fisher: Towards the fundamental tradeoff in ISAC,” in _2022 IEEE Global Commun. Conf. (GLOBECOM)_ , Dec. 2022, pp. 5601–5606.
  * [15] H. Hua, T. X. Han, and J. Xu, “MIMO integrated sensing and communication: CRB-rate tradeoff,” _arXiv preprint arXiv:2209.12721_ , 2022.
  * [16] F. Liu, C. Masouros, A. Li, H. Sun, and L. Hanzo, “MU-MIMO communications with MIMO radar: From co-existence to joint transmission,” _IEEE Trans. Wireless Commun._ , vol. 17, no. 4, pp. 2755–2770, Apr. 2018.
  * [17] X. Liu, T. Huang, N. Shlezinger, Y. Liu, J. Zhou, and Y. C. Eldar, “Joint transmit beamforming for multiuser MIMO communications and MIMO radar,” _IEEE Trans. Signal Process._ , vol. 68, pp. 3929–3944, Jun. 2020.
  * [18] H. Hua, J. Xu, and T. X. Han, “Optimal transmit beamforming for integrated sensing and communication,” _IEEE Trans. Veh. Technol._ , pp. 1–16, Mar. 2023.
  * [19] J. A. Zhang, X. Huang, Y. J. Guo, J. Yuan, and R. W. Heath, “Multibeam for joint communication and radar sensing using steerable analog antenna arrays,” _IEEE Trans. Veh. Technol._ , vol. 68, no. 1, pp. 671–685, Jan. 2019.
  * [20] C. Xu, B. Clerckx, S. Chen, Y. Mao, and J. Zhang, “Rate-splitting multiple access for multi-antenna joint radar and communications,” _IEEE J. Sel. Topics Signal Process._ , vol. 15, no. 6, pp. 1332–1347, Nov. 2021.
  * [21] L. Yin, Y. Mao, O. Dizdar, and B. Clerckx, “Rate-splitting multiple access for 6G–part II: Interplay with integrated sensing and communications,” _IEEE Commun. Lett._ , vol. 26, no. 10, pp. 2237–2241, Oct. 2022.
  * [22] Z. Wang, Y. Liu, X. Mu, Z. Ding, and O. A. Dobre, “NOMA empowered integrated sensing and communication,” _IEEE Commun. Lett._ , vol. 26, no. 3, pp. 677–681, Mar. 2022.
  * [23] J. Li, L. Xu, P. Stoica, K. W. Forsythe, and D. W. Bliss, “Range compression and waveform optimization for MIMO radar: A Cramér-Rao bound based study,” _IEEE Trans. Signal Process._ , vol. 56, no. 1, pp. 218–232, Jan. 2008.
  * [24] I. Bekkerman and J. Tabrikian, “Target detection and localization using MIMO radars and sonars,” _IEEE Trans. Signal Process._ , vol. 54, no. 10, pp. 3873–3883, Oct. 2006.
  * [25] E. Telatar, “Capacity of multi-antenna Gaussian channels,” _Eur. Trans. Telecommun._ , vol. 10, no. 6, pp. 585–595, Nov. 1999.
  * [26] F. Liu, Y.-F. Liu, A. Li, C. Masouros, and Y. C. Eldar, “Cramér-Rao bound optimization for joint radar-communication beamforming,” _IEEE Trans. Signal Process._ , vol. 70, pp. 240–253, Dec. 2021.
  * [27] X. Song, J. Xu, F. Liu, T. X. Han, and Y. C. Eldar, “Intelligent reflecting surface enabled sensing: Cramér-Rao bound optimization,” in _2022 IEEE Globecom Workshops (GC Wkshps)_ , Dec. 2022, pp. 413–418.
  * [28] M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex programming, version 2.1,” 2014.
  * [29] Z. Ren, X. Song, Y. Fang, L. Qiu, and J. Xu, “Fundamental CRB-rate tradeoff in multi-antenna multicast channel with ISAC,” in _2022 IEEE Globecom Workshops (GC Wkshps)_ , Dec. 2022, pp. 1261–1266.
  * [30] S. Boyd and L. Vandenberghe, _Convex Optimization_. Cambridge University Press, 2004.
  * [31] Y. Chen, H. Hua, and J. Xu, “ISAC meets SWIPT: Multi-functional wireless systems integrating sensing, communication, and powering,” _arXiv preprint arXiv:2211.10605_ , 2022.