# Fundamental CRB-Rate Tradeoff in Multi-Antenna ISAC Systems with Information
Multicasting and Multi-Target Sensing

Zixiang Ren, Yunfei Peng, Xianxin Song, Yuan Fang, Ling Qiu, Liang Liu,
Derrick Wing Kwan Ng, and Jie Xu Part of this paper has been presented at the
2022 IEEE Global Communications Conference workshop, 4-8 December 2022, Rio de
Janeiro, Brazil [1].Z. Ren is with Key Laboratory of Wireless-Optical
Communications, Chinese Academy of Sciences, School of Information Science and
Technology, University of Science and Technology of China, and the Future
Network of Intelligence Institute (FNii), The Chinese University of Hong Kong
(Shenzhen), Shenzhen, China (e-mail: rzx66@mail.ustc.edu.cn).Y. Peng, X. Song,
Y. Fang, and J. Xu are with the School of Science and Engineering (SSE) and
the FNii, The Chinese University of Hong Kong (Shenzhen), Shenzhen, China
(e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>xujie@cuhk.edu.cn). L. Qiu is with Key Laboratory of
Wireless-Optical Communications, Chinese Academy of Sciences, School of
Information Science and Technology, University of Science and Technology of
China (e-mail: lqiu@ustc.edu.cn).L. Liu is with the Department of Electronic
and Information Engineering, The Hong Kong Polytechnic University, Hong Kong
SAR, China (e-mails: liang-eie.liu@polyu.edu.hk). D. W. K. Ng is with the
University of New South Wales, Sydney, NSW 2052, Australia (e-mail:
w.k.ng@unsw.edu.au).L. Qiu and J. Xu are the corresponding authors.

###### Abstract

This paper investigates the performance tradeoff for a multi-antenna
integrated sensing and communication (ISAC) system with simultaneous
information multicasting and multi-target sensing, in which a multi-antenna
base station (BS) sends the common information messages to a set of single-
antenna communication users (CUs) and estimates the parameters of multiple
sensing targets based on the echo signals concurrently. We consider two target
sensing scenarios without and with prior target knowledge at the BS, in which
the BS is interested in estimating the complete multi-target response matrix
and the target reflection coefficients/angles, respectively. First, we
consider the capacity-achieving transmission and characterize the fundamental
tradeoff between the achievable rate and the multi-target estimation Cramér-
Rao bound (CRB) accordingly. To this end, we design the optimal transmit
signal covariance matrix at the BS to minimize the estimation CRB for each of
the two scenarios, subject to the minimum multicast rate requirement and the
maximum transmit power constraint. It is shown that the optimal covariance
matrix consists of two parts for ISAC and dedicated sensing, respectively.
Next, we consider the transmit beamforming designs, in which the BS sends one
information beam together with multiple a-priori known dedicated sensing beams
for effective ISAC and each CU can cancel the interference caused by the
sensing signals. By exploiting the successive convex approximation (SCA)
technique, we develop efficient algorithms to obtain the joint information and
sensing beamforming solutions to the resultant rate-constrained CRB
minimization problems. Finally, we provide numerical results to validate the
CRB-rate (C-R) tradeoff achieved by our proposed designs, as compared to two
benchmark schemes, namely the isotropic transmission and the joint beamforming
without sensing interference cancellation. It is shown that the proposed
optimal transmit covariance solution achieves much better C-R performance than
the benchmark schemes and the proposed joint beamforming with sensing
interference cancellation performs close to the optimal transmit covariance
solution when the number of CUs is small. We also conduct simulations to show
the practical estimation performance achieved by our proposed designs, by
considering randomly generated information signals and practical estimators.

###### Index Terms:

Integrated sensing and communications (ISAC), multicast channel, multi-target
sensing, Cramér-Rao bound (CRB), transmit beamforming, optimization.

##  I Introduction

Future beyond fifth-generation (B5G) and sixth-generation (6G) wireless
networks are envisioned to support abundant intelligent applications such as
the metaverse, auto-driving, smart homes, and smart cities, which demand
ultra-high-capacity, ultra-low-latency, and ultra-high reliable
communications, as well as high-accuracy and high-resolution sensing. Towards
this end, integrated sensing and communication (ISAC) has emerged as a
promising technology for realizing B5G/6G, which enables the dual use of radio
signals and wireless network infrastructures for providing both sensing and
communication services simultaneously [2]. By seamlessly coordinating and
integrating sensing and communications [3], ISAC is expected to significantly
enhance both communication and sensing performances, while improving the
spectrum utilization efficiency and the cost efficiency. As such, ISAC has
recently attracted tremendous research interests in both academia and industry
[4], motivating extensive studies on ISAC from different research perspectives
such as unified ISAC waveforms [2], multi-antenna ISAC [3], networked ISAC [5,
6], wideband ISAC [2], and intelligent reflecting surface (IRS)-enabled ISAC
[7, 8].

Among various enabling techniques, multiple-input multiple-output (MIMO) has
been recognized as an important solution to enhance the performance of
wireless networks, by equipping multiple antennas at the wireless
transceivers, e.g., base stations (BSs). In particular, MIMO can provide
potential spatial multiplexing and diversity gains to increase the
communication rate and reliability [9], while offering spatial and waveform
diversity gains to enhance the sensing accuracy and resolution [10, 11].
Indeed, how to optimize the transmit waveform and beamforming design based on
proper sensing and communication performance metrics is a key technical
challenge to be tackled for multi-antenna ISAC. On the one hand, for
communication, the signal-to-interference-plus-noise ratio (SINR) and data
rate are widely adopted as performance measures. On the other hand, for
sensing, various performance measures may apply depending on the specific
sensing tasks. For instance, when the sensing estimation tasks for target
detection and tracking [12] are considered, the estimation mutual information
[13], radar sensing SINR [14], transmit beampattern [15], and Cramér-Rao bound
(CRB) [16] are widely applied, among which the beampattern and CRB-based
designs are most promising. The basic idea of beampattern-based design is to
match the transmitted beampattern with a pre-defined sensing beampattern such
that the transmit signal beams are focused towards the desired target
directions and those leaked to the other undesired directions are suppressed
as possible. There have been extensive existing works considering the
waveform/beamforming design based on the transmit beampattern. For example,
the authors in [17, 15, 18] studied the downlink ISAC over broadcast channels
by considering different setups, in which the transmit beamforming was
optimized to properly balance the tradeoff between the transmit beampattern
for sensing and the received SINR for communication. These designs were then
extended to the case of downlink non-orthogonal multiple access (NOMA) in
[19]. However, aiming to mimic the transmit beampattern may not be able to
capture the quality of sensing directly and thus may lead to sacrificed
sensing performances.

In contrast to the transmit beampattern, the CRB has been recognized as a
practically relevant and analytically tractable metric to directly measure the
sensing performance limits for estimating target parameters. In general, the
CRB provides the lower bound of variance for any unbiased estimators, which
can be derived by exploiting the Fisher information matrix to measure the
amount of information that the data provides about the parameter of interest
[12, 16, 20, 21]. As a result, directly optimizing the CRB is a viable new
approach that is suitable for designing ISAC systems, which not only
facilitates the characterization of the fundamental CRB-rate (C-R) performance
tradeoff, but also leads to potentially enhanced parameter estimation
performances. For instance, the authors in [16] studied the multiuser ISAC
over a broadcast channel with one single-target, in which the transmit
beamforming was optimized to minimize the estimation CRB, subject to the
individual SINR (or equivalently rate) constraints at multiple communication
users (CUs). Furthermore, [20] and [21] investigated the C-R tradeoff in a
point-to-point MIMO ISAC system with one CU and one target. In particular, the
authors in [21] optimized the transmit covariance to characterize the complete
Pareto boundary of the C-R region for the MIMO ISAC system in two specific
scenarios with point and extended target models, respectively, by considering
the radar coherent processing interval to be sufficiently long. As for [20],
the authors revealed the C-R tradeoff for a more general case with finite
radar coherent processing intervals. In addition, [22] investigated the C-R
tradeoff for ISAC in multi-antenna broadcast channels, in which the emerging
rate-splitting multiple access (RSMA) technique was exploited to further
improve the ISAC performance.

With recent advancements in webcast and content broadcasting applications, it
has become crucial for future B5G/6G wireless networks to effectively support
simultaneous data transmission to multiple users through multicast channels
[23, 24]. Additionally, the ability to conduct multi-target sensing and
tracking plays a vital role in various multicast applications, such as
surveillance, object detection, and autonomous vehicles [4]. Consequently,
supporting efficient ISAC over multi-antenna systems during such information
multicasting has become an increasingly important problem. This problem,
however, is particularly challenging. This is due to the fact that the
transmission principle for multicast channels significantly differs from that
for broadcast channels, thus rendering the prior ISAC designs [16, 22, 21, 20]
over broadcast channels not applicable. To illustrate this, we consider a
multiple-input single-output (MISO) multicast channel as an example. First, it
has been demonstrated in [23] that achieving the MISO multicast capacity
typically requires a high-rank transmit covariance matrix for transmitting
common messages. This stands in contrast to the MISO broadcast channel, where
the capacity is achievable by employing a rank-one covariance matrix for each
user’s individual message along with dirty paper coding [25]. Next, it has
been shown in [24] that in MISO multicast channels, the transmit beamforming
optimization problem for minimizing the transmit power while ensuring
individual users’ signal-to-noise ratio (SNR) requirements is an NP-hard
problem. This is distinct from the MISO broadcast channels, in which the
transmit beamforming design for SINR-constrained power minimization has been
proven to be a convex problem [26, 27]. Due to such significant differences,
new design approaches and principles are needed for ISAC over multicast
channels, thus motivating our investigation in this work.

This paper investigates the multi-antenna ISAC over a multicast channel with
multi-target sensing, in which a multi-antenna BS sends common messages to a
set of single-antenna CUs and simultaneously exploits the received echo
signals to estimate the parameters of multiple sensing targets. In particular,
we consider two scenarios for multi-target sensing without and with prior
target knowledge at the BS, namely Scenario I and Scenario II, respectively.
In practice, Scenarios I and II may correspond to the target detection and
target tracking stages for multi-target sensing, respectively.111In general,
the BS often lacks prior knowledge of the targets before transmission.
Therefore, an initial target detection stage is required, in which the ISAC
signal transmission is designed to estimate the complete multi-target response
matrix for extracting multi-target parameters. With the obtained target
parameters at hand, a subsequent target tracking stage is implemented, in
which the ISAC signal transmission is designed by leveraging such a piece of
prior information to facilitate the tracking of these targets or the
estimation of their reflection coefficients and angles. The main results of
our work are summarized as follows.

  * •

First, we consider the capacity-achieving transmission, based on which we
characterize the fundamental performance tradeoff between the estimation CRB
for multi-target sensing versus the channel capacity for information
multicasting in the two scenarios, which generalizes the target sensing study
compared with [1]. To this end, we minimize the corresponding estimation CRB
for each scenario, by optimizing the transmit covariance matrix at the BS,
subject to the minimum multicast rate constraint and the maximum transmit
power constraint. We obtain the globally optimal solutions to the two rate-
constrained CRB minimization problems by applying advanced convex optimization
techniques. For both scenarios, the optimal transmit covariance is shown to
consist of two signal parts for ISAC and dedicated sensing, respectively.

  * •

Next, to facilitate the implementation, we present new joint information and
sensing beamforming designs for efficient ISAC. In such designs, the BS sends
one common information beam together with multiple dedicated a-priori known
sensing beams such that each CU is capable of canceling the interference
caused by these sensing beams. For each of the two scenarios, we develop a
computationally efficient algorithm by adopting the successive convex
approximation (SCA) technique to obtain a high-quality joint beamforming
solution for minimizing the multi-target estimation CRB while ensuring the
minimum multicast rate requirement.

  * •

Finally, we provide numerical results to validate the C-R tradeoff performance
achieved by our proposed designs, as compared to two benchmark schemes, namely
the isotropic transmission and the joint beamforming without sensing
interference cancellation. It is shown that for each scenario, the proposed
designs significantly outperform the two benchmark schemes and the proposed
joint beamforming with sensing interference cancellation achieves performance
close to the upper bound achieved by the optimal transmit covariance when the
number of CUs is small. We also perform simulations to assess the estimation
performance using randomly generated information signals and practical
estimators. The results clearly demonstrate that our CRB minimization designs
significantly reduce the root mean squared error (RMSE) of estimation compared
to conventional beampattern-based designs. This highlights the superior
performance and feasibility of our proposed designs in the practical ISAC
implementation and provides more comprehensive performance comparison than
[1].

The remainder of this paper is organized as follows. Section II introduces the
multicast multi-target ISAC system model and formulates the multicast-rate-
constrained CRB minimization problems for Scenarios I and II when the BS does
not and does have the prior multi-target knowledge, respectively. Sections III
and IV develop the optimal solutions to the formulated multicast-rate-
constrained CRB minimization problems for Scenarios I and II, respectively.
Section V proposes joint information and sensing beamforming designs with
sensing interference pre-cancellation at the CUs. Section VI provides
numerical results. Finally, Section VII concludes this paper.

Notations: Vectors and matrices are denoted by bold lower- and upper-case
letters, respectively. $\mathbb{C}^{N\times M}$ denotes the space of $N\times
M$ complex matrices. $\boldsymbol{I}$ and $\boldsymbol{0}$ represent an
identity matrix and an all-zero matrix with appropriate dimensions,
respectively. For a square matrix $\boldsymbol{A}$,
$\textrm{tr}(\boldsymbol{A})$ denotes its trace and
$\boldsymbol{A}\succeq\boldsymbol{0}$ means that $\boldsymbol{A}$ is positive
semi-definite. For a complex arbitrary-size matrix $\boldsymbol{B}$,
$\boldsymbol{B}[i,j]$, $\textrm{rank}(\boldsymbol{B})$, $\boldsymbol{B}^{T}$,
$\boldsymbol{B}^{H}$, and $\boldsymbol{B}^{c}$ denote its $(i,j)$-th element,
rank, transpose, conjugate transpose, and complex conjugate, respectively, and
$\mathrm{vec}(\boldsymbol{B})$ denotes the vectorization of $\boldsymbol{B}$.
For a vector $\boldsymbol{a}$, $\boldsymbol{a}[i]$ denotes its $i$-th element.
$\mathbb{E}(\cdot)$ denotes the statistical expectation. $\|\cdot\|$ denotes
the Euclidean norm of a vector. $|\cdot|$, $\mathrm{Re}(\cdot)$, and
$\mathrm{Im}(\cdot)$ denote the absolute value, the real component, and the
imaginary component of a complex entry.
$\mathcal{CN}(\boldsymbol{x},\boldsymbol{Y})$ denotes a circularly symmetric
complex Gaussian (CSCG) random vector with mean vector $\boldsymbol{x}$ and
covariance matrix $\boldsymbol{Y}$. $\boldsymbol{A}\otimes\boldsymbol{B}$ and
$\boldsymbol{A}\odot\boldsymbol{B}$ represent the Kronecker product and
Hadamard product of two matrices $\boldsymbol{A}$ and $\boldsymbol{B}$,
respectively.

##  II System Model and Problem Formulation

Figure 1: Illustration of the multi-antenna ISAC system for simultaneous
information multicasting and multi-target sensing.

As shown in Fig. 1, we consider a multi-antenna ISAC system for information
multicasting and multi-target sensing, which consists of a multi-antenna BS
equipped with $N_{t}$ transmit antennas and $N_{r}$ receive antennas, $K$
single-antenna CUs, and $M$ sensing targets. Let
$\mathcal{K}\overset{\bigtriangleup}{=}\\{1,2,\ldots,K\\}$ denote the set of
CUs and $\mathcal{M}\overset{\bigtriangleup}{=}\\{1,2,\ldots,M\\}$ denote that
of sensing targets. The BS sends common information signals to all the CUs and
exploits the received echo signals to sense the $M$ targets simultaneously. In
particular, we focus on the ISAC period consisting of $L$ symbols. In this
paper, we focus on a quasi-static environment, where the communication and
sensing channels are assumed to remain invariant during the ISAC period,
similar to [16, 28]. Let $\boldsymbol{x}(n)\in\mathbb{C}^{N_{t}\times 1}$
denote the transmitted ISAC signal by the BS in symbol
$n\in\mathcal{L}\triangleq\\{1,2,\ldots,L\\}$. For achieving the multicast
capacity, we consider that the transmitted signal $\boldsymbol{x}(n)$ is a
CSCG random vector with zero mean and covariance
$\mathbb{E}[\boldsymbol{x}(n)\boldsymbol{x}^{H}(n)]=\boldsymbol{S}_{x}\succeq\boldsymbol{0}$,
i.e.,
$\boldsymbol{x}(n)\sim\mathcal{CN}(\boldsymbol{0},\boldsymbol{S}_{x}),\forall
n$. Supposing that the BS is subject to a maximum transmit power budget $P$,
we have

$\mathrm{tr}(\boldsymbol{S}_{x})\leq P.$ (1)

###  II-A Multicasting Communication Model

First, we consider the multicast channel for communication. Let
$\boldsymbol{h}_{k}\in\mathbb{C}^{N_{t}\times 1}$ denote the channel vector
from the BS to CU $k\in\mathcal{K}$. The received signal at the receiver of CU
$k\in\mathcal{K}$ is given by [23]

$y_{k}(n)=\boldsymbol{h}_{k}^{H}\boldsymbol{x}(n)+z_{k}(n),\forall
n\in\mathcal{L},$ (2)

where $z_{k}(n)$ denotes the noise at the receiver of CU $k$ that is a CSCG
random variable with zero mean and variance $\sigma^{2}$, i.e.,
$z_{k}(n)\sim\mathcal{CN}(0,\sigma^{2}),\forall k\in\mathcal{K}$. Based on the
received signal in (2), the received SNR at CU $k\in\mathcal{K}$ is222Since
all the receivers require the same common message from one single BS, there is
no multi-user interference. On the other hand, the co-channel interference
from other radio transmitters in the environment can be included in $z_{k}(n)$
such that $\sigma^{2}$ captures the impacts caused by background interference.

$\gamma_{k}(\boldsymbol{S}_{x})=\mathbb{E}\bigg{(}\frac{\big{|}\boldsymbol{h}_{k}^{H}\boldsymbol{x}(n)\big{|}^{2}}{\big{|}z_{k}(n)\big{|}^{2}}\bigg{)}=\frac{\boldsymbol{h}_{k}^{H}\boldsymbol{S}_{x}\boldsymbol{h}_{k}}{\sigma^{2}}.$
(3)

Accordingly, the achievable rate of the multicast channel with a given
transmit covariance $\boldsymbol{S}_{x}$ is given by [23, 24]

$R(\boldsymbol{S}_{x})=\underset{k\in\mathcal{K}}{\min}\Big{\\{}\log_{2}\Big{(}1+\frac{\boldsymbol{h}_{k}^{H}\boldsymbol{S}_{x}\boldsymbol{h}_{k}}{\sigma^{2}}\Big{)}\Big{\\}}.$
(4)

As a result, the capacity of the multicast channel is defined as the maximum
achievable rate, given by [23, 24]

$C_{0}=\underset{\boldsymbol{S}_{x}\succeq\boldsymbol{0},\mathrm{tr}(\boldsymbol{S}_{x})\leq
P}{\max}R(\boldsymbol{S}_{x}).$ (5)

###  II-B Multi-target Sensing Model

Next, we consider the multi-target radar sensing, in which the BS adopts its
$N_{r}$ receive antennas for receiving the echo signals to estimate the
parameters of the $M$ sensing targets. Suppose that these $M$ targets are
located in the same range bin of the far-field region of the BS. In this case,
the received target echo signals at the BS in symbol $n\in\mathcal{L}$ is
expressed as333Similar as in prior works on ISAC and radar sensing [16, 21],
we ignore the self-interference due to the signal transmission in (5), as the
sensing receiver can handle such an issue properly [29]. This is different
from the communication receivers, for which the self-interference might be a
severe issue. Furthermore, similar to prior works on MIMO radar sensing [11,
28] and MIMO ISAC [2], we consider the estimation of target angles and complex
coefficients based on the echo signals in (6), in which the Doppler effects
are omitted.

$\boldsymbol{y}(n)=\sum_{m=1}^{M}\beta_{m}\boldsymbol{a}_{r}^{c}(\theta_{m})\boldsymbol{a}_{t}^{H}(\theta_{m})\boldsymbol{x}(n)+\boldsymbol{\boldsymbol{z}}(n),$
(6)

where $\beta_{m}\in\mathbb{C}$ denotes the complex coefficient of target
$m\in\mathcal{M}$ whose amplitude captures the round-trip path loss and is
proportional to its radar cross section (RCS) [30, 28], $\theta_{m}$ denotes
the angle of departure (AoD)/angle of arrival (AoA) of target
$m\in\mathcal{M}$, $\boldsymbol{a}_{r}(\theta)\in\mathbb{C}^{N_{r}\times 1}$
and $\boldsymbol{a}_{t}(\theta)\in\mathbb{C}^{N_{t}\times 1}$ denote the
receive and transmit steering vectors with angle $\theta$, respectively, and
$\boldsymbol{z}(n)\sim\mathcal{CN}(\boldsymbol{0},\sigma_{r}^{2}\boldsymbol{I})$
is the noise at the BS receiver with $\sigma_{r}^{2}$ being the noise power at
each receive antenna. Let
$\boldsymbol{X}=[\boldsymbol{x}(1),\dots,\boldsymbol{x}(L)]$ denote the
transmitted signal over the $L$ symbols. By assuming that $L$ is sufficiently
large, the sample covariance matrix of $\boldsymbol{X}$ can be approximated as
the statistical covariance matrix444We will show that the approximation is
sufficiently accurate for practical values of $L$ (e.g., $L\geq 64$) in the
numerical results in Section VI.  $\boldsymbol{S}_{x}$, i.e.,

$\frac{1}{L}\boldsymbol{X}\boldsymbol{X}^{H}\approx\boldsymbol{S}_{x}.$ (7)

To facilitate the derivation of the CRB matrix for estimating the parameters
of targets, we rewrite (6) as

$\boldsymbol{Y}=\boldsymbol{A}_{r}^{c}\boldsymbol{B}\boldsymbol{A}_{t}^{H}\boldsymbol{X}+\boldsymbol{Z},$
(8)

where $\boldsymbol{Y}=[\boldsymbol{y}(1),...,\boldsymbol{y}(L)]$,
$\boldsymbol{Z}=[\boldsymbol{z}(1),...,\boldsymbol{z}(L)]$,
$\boldsymbol{A}_{r}=[\boldsymbol{a}_{r}(\theta_{1})$,
$\boldsymbol{a}_{r}(\theta_{2}),\dots,\boldsymbol{a}_{r}(\theta_{M})]$,
$\boldsymbol{A}_{t}=[\boldsymbol{a}_{t}(\theta_{1}),\boldsymbol{a}_{t}(\theta_{2}),\dots,\boldsymbol{a}_{t}(\theta_{M})]$,
$\boldsymbol{\theta}=[\theta_{1},\theta_{2},\dots,\theta_{M}]^{T},\boldsymbol{\beta}=[\beta_{1},\beta_{2},\dots,\beta_{M}]^{T}$,
and $\boldsymbol{B}=\mathrm{diag}(\boldsymbol{\beta})$. Let
$\boldsymbol{G}=\boldsymbol{A}_{r}^{c}\boldsymbol{B}\boldsymbol{A}_{t}^{H}$
denote the multi-target response matrix. Accordingly, we vectorize (8) as

$\tilde{\boldsymbol{y}}=(\boldsymbol{X}^{T}\otimes\boldsymbol{I}_{N_{r}})\tilde{\boldsymbol{g}}+\tilde{\boldsymbol{z}},$
(9)

where
$\tilde{\boldsymbol{y}}=\mathrm{vec}(\boldsymbol{Y})\in\mathbb{C}^{N_{r}L\times
1}$,
$\tilde{\boldsymbol{g}}=\mathrm{vec}(\boldsymbol{G})\in\mathbb{C}^{N_{r}N_{t}\times
1},$ and
$\tilde{\boldsymbol{z}}=\mathrm{vec}(\boldsymbol{Z})\in\mathbb{C}^{N_{r}L\times
1}$. It follows from (9) that the received signal vector
$\tilde{\boldsymbol{r}}$ is a CSCG random vector with mean
$(\boldsymbol{X}^{T}\otimes\boldsymbol{I})\tilde{\boldsymbol{g}}$ and
covariance $\sigma_{r}^{2}\boldsymbol{I}$, i.e.,
$\tilde{\boldsymbol{y}}\sim\mathcal{CN}\big{(}(\boldsymbol{X}^{T}\otimes\boldsymbol{I})\tilde{\boldsymbol{g}},\sigma_{r}^{2}\boldsymbol{I}\big{)}$.

In particular, we consider two different sensing scenarios, namely Scenario I
and Scenario II, in which the BS does not and does have a-prior knowledge
about the multiple targets such that the BS aims to estimate the complete
multi-target response matrix
$\boldsymbol{G}=\boldsymbol{A}_{r}^{c}\boldsymbol{B}\boldsymbol{A}_{t}^{H}$
and the targets’ coefficients/angles (i.e., $\\{\beta_{m}\\}_{m=1}^{M}$ and
$\\{\theta_{m}\\}_{m=1}^{M}$), respectively. In practice, Scenario I may
correspond to the target detection phase, in which the BS tries to identify
the number of targets and obtain their parameters based on matrix
$\boldsymbol{G}$ via spatial spectrum algorithms such as multiple signal
classification (MUSIC). On the other hand, Scenario II may correspond to the
target tracking phases, in which the BS needs to track the $M$ targets by
estimating their coefficients/angles, with their initial parameters known in
advance [31, 32], respectively.

####  II-B1 CRB for Estimating $\boldsymbol{G}$ in Scenario I

First, we consider Scenario I, in which the BS lacks prior knowledge of the
targets in the initial target detection stage [33, 2]. In this scenario, the
BS is interested in estimating the complete multi-target response matrix
$\boldsymbol{G}=\boldsymbol{A}_{r}^{c}\boldsymbol{B}\boldsymbol{A}_{t}^{H}$
based on the received signal $\boldsymbol{Y}$ in (8), or equivalently
estimating $\tilde{\boldsymbol{g}}$ based on $\tilde{\boldsymbol{r}}$ in (9).
After obtaining the estimate of $\boldsymbol{G}$, the BS can further implement
spatial spectrum algorithms such as MUSIC[34] to further extract the angle
information of these targets. In this scenario, the BS needs to estimate
$N_{r}N_{t}$ complex parameters in $\tilde{\boldsymbol{g}}$ or
$\boldsymbol{G}$. Based on the complex linear model in (9), it has been
established in [35] that the CRB matrix for estimating
$\tilde{\boldsymbol{g}}$ is

$\displaystyle\boldsymbol{C}_{1}\negthickspace=$
$\displaystyle\big{(}(\boldsymbol{X}^{T}\otimes\boldsymbol{I}_{N_{r}})^{H}(\sigma_{r}^{2}\boldsymbol{I})^{-1}(\boldsymbol{X}^{T}\otimes\boldsymbol{I}_{N_{r}})\big{)}^{-1}\negthickspace\negthickspace=\sigma_{r}^{2}\big{(}\boldsymbol{X}^{c}\boldsymbol{X}^{T}\otimes\boldsymbol{I}_{N_{r}}\big{)}^{-1}\negthickspace\overset{(\mathrm{a})}{=}\frac{\sigma_{r}^{2}}{L}\big{(}\boldsymbol{S}_{x}^{T}\otimes\boldsymbol{I}_{N_{r}}\big{)}^{-1},$
(10)

where (a) follows from the approximation in (7). For facilitating the sensing
performance optimization, scalar functions of the CRB matrix are normally
adopted as the performance metrics, some examples include trace, determinant,
and minimum eigenvalue (10). In this paper, we adopt its trace as the scalar
performance metric, i.e.

$\mathrm{CRB}_{1}(\boldsymbol{S}_{x})=\frac{N_{r}\sigma_{r}^{2}}{L}\mathrm{tr}(\boldsymbol{S}_{x}^{-1}).$
(11)

Intuitively, $\mathrm{CRB}_{1}(\boldsymbol{S}_{x})$ corresponds to the sum of
the CRBs for estimating the elements in $\boldsymbol{G}$.

####  II-B2 CRB for Estimating $\boldsymbol{\theta}$ and $\boldsymbol{\beta}$
in Scenario II

Next, we consider Scenario II, in which the BS has a priori knowledge of the
multiple targets (e.g., from the target detection stage), and thus is
interested in estimating the target coefficients $\boldsymbol{\beta}$ and
angles $\boldsymbol{\theta}$ as unknown parameters. This may correspond to the
target tracking stage in practice [33, 2]. Let
$\boldsymbol{\beta}_{R}=[\mathrm{Re}(\beta_{1}),\mathrm{Re(}\beta_{2}),\dots,\mathrm{Re(}\beta_{M})]^{T}$
and
$\boldsymbol{\beta}_{I}=[\mathrm{Im}(\beta_{1}),\mathrm{Im(}\beta_{2}),\dots,\mathrm{Im(}\beta_{M})]^{T}$
denote the real and imaginary parts of $\boldsymbol{\beta}$. Accordingly, we
have a total of $3M$ real parameters to be estimated, given by
$\boldsymbol{\xi}=[\boldsymbol{\theta}^{T},\boldsymbol{\text{$\beta$}}_{R}^{T},\boldsymbol{\beta}_{I}^{T}]$T.
To facilitate the CRB derivation, we have the following lemma [35].

######  Lemma 1.

[35] Consider the estimation of real-valued parameters
$\boldsymbol{\varsigma}\in\mathbb{R}^{T\times 1}$ based on the CSCG data
vector $\tilde{\boldsymbol{v}}\in\mathbb{C}^{N\times 1}$, i.e.,
$\tilde{\boldsymbol{v}}\sim\mathcal{CN}(\tilde{\boldsymbol{\mu}}(\boldsymbol{\varsigma}),\boldsymbol{C}_{\tilde{\boldsymbol{v}}})$,
where the covariance $\boldsymbol{C}_{\tilde{\boldsymbol{v}}}$ is independent
from parameters $\boldsymbol{\varsigma}$. The Fisher information matrix (FIM)
for estimating $\boldsymbol{\varsigma}$ is given by
$\boldsymbol{F}_{\boldsymbol{\varsigma}}$, with

$\boldsymbol{F}_{\boldsymbol{\varsigma}}[i,j]=2\mathrm{Re}\Big{(}\frac{\partial\tilde{\boldsymbol{\mu}}^{H}(\boldsymbol{\varsigma})}{\partial\boldsymbol{\varsigma}[i]}\boldsymbol{C}_{\tilde{\boldsymbol{v}}}^{-1}\frac{\partial\tilde{\boldsymbol{\mu}}(\boldsymbol{\varsigma})}{\partial\boldsymbol{\varsigma}[j]}\Big{)},\forall
i,j\in\\{1,\ldots,T\\},$ (12)

where $\frac{\partial}{\partial}(\cdot)$ denotes the partial derivative
operator.

Notice that based on (9), we have
$\tilde{\boldsymbol{y}}\sim\mathcal{CN}\big{(}\mathrm{vec}(\boldsymbol{A}_{r}^{c}\boldsymbol{B}\boldsymbol{A}_{t}^{H}\boldsymbol{X}),\sigma_{r}^{2}\boldsymbol{I}\big{)}$.
Let $\boldsymbol{F}_{\boldsymbol{\xi}}\in\mathbb{R}^{3M\times 3M}$ denote the
FIM for estimating $\boldsymbol{\xi}$. Based on Lemma 1, the element in the
$i$-th row and $j$-th column of $\boldsymbol{F}_{\boldsymbol{\xi}}$, $\forall
i,j\in\\{1,\ldots,3M\\}$, is given by

$\displaystyle\boldsymbol{F}_{\boldsymbol{\xi}}[i,j]$ $\displaystyle=$
$\displaystyle\frac{2}{\sigma_{r}^{2}}\mathrm{Re}\Big{(}\frac{\partial\mathrm{vec}(\boldsymbol{A}_{r}^{c}\boldsymbol{B}\boldsymbol{A}_{t}^{H}\boldsymbol{X})^{H}}{\partial\boldsymbol{\boldsymbol{\xi}}[i]}\cdot\frac{\partial\mathrm{vec}(\boldsymbol{A}_{r}^{c}\boldsymbol{B}\boldsymbol{A}_{t}^{H}\boldsymbol{X})}{\partial\boldsymbol{\boldsymbol{\xi}}[j]}\Big{)},$
(13)

where
$\dot{\boldsymbol{A}_{r}}=[\frac{\partial\boldsymbol{a}_{r}(\theta_{1})}{\partial\theta_{1}},\frac{\partial\boldsymbol{a}_{r}(\theta_{2})}{\partial\theta_{2}},\cdots,\frac{\partial\boldsymbol{a}_{r}(\theta_{M})}{\partial\theta_{M}}],$
and
$\dot{\boldsymbol{A}_{t}}=[\frac{\partial\boldsymbol{a}_{t}(\theta_{1})}{\partial\theta_{1}},\frac{\partial\boldsymbol{a}_{t}(\theta_{2})}{\partial\theta_{2}},\cdots,\frac{\partial\boldsymbol{a}_{t}(\theta_{M})}{\partial\theta_{M}}].$
Following the similar derivation procedures in [28], we have the FIM in a
block matrix form:

$\boldsymbol{F}_{\boldsymbol{\xi}}=\frac{2}{\sigma_{r}^{2}}\left[\begin{array}[]{ccc}\mathrm{Re}(\boldsymbol{F}_{11})&\mathrm{Re}(\boldsymbol{F}_{12})&-\mathrm{Im}(\boldsymbol{F}_{12})\\\
\mathrm{Re}(\boldsymbol{F}_{12})^{T}&\mathrm{Re}(\boldsymbol{F}_{22})&-\mathrm{Im}(\boldsymbol{F}_{22})\\\
-\mathrm{Im}(\boldsymbol{F}_{12})^{T}&-\mathrm{Im}(\boldsymbol{F}_{22})^{T}&\mathrm{Re}(\boldsymbol{F}_{22})\end{array}\right],$
(14)

where

$\displaystyle\boldsymbol{F}_{11}$ $\displaystyle=$ $\displaystyle
L(\dot{\boldsymbol{A}_{r}}^{T}\dot{\boldsymbol{A}_{r}}^{c})\odot(\boldsymbol{B}^{c}\boldsymbol{A}_{t}^{T}\boldsymbol{S}_{x}^{T}\boldsymbol{A}_{t}^{c}\boldsymbol{B}^{T})+L(\dot{\boldsymbol{A}_{r}}^{T}\boldsymbol{A}_{r}^{c})\odot(\boldsymbol{B}^{c}\boldsymbol{A}_{t}^{T}\boldsymbol{S}_{x}^{T}\dot{\boldsymbol{A}_{t}}^{c}\boldsymbol{B}^{T})$
(15)
$\displaystyle+L(\boldsymbol{A}_{r}^{T}\dot{\boldsymbol{A}_{r}}^{c})\odot(\boldsymbol{B}^{c}\dot{\boldsymbol{A}_{t}}^{T}\boldsymbol{S}_{x}^{T}\boldsymbol{A}_{t}^{c}\boldsymbol{B}^{T})+L(\boldsymbol{A}_{r}^{T}\boldsymbol{A}_{r}^{c})\odot(\boldsymbol{B}^{c}\dot{\boldsymbol{A}_{t}}^{T}\boldsymbol{S}_{x}^{T}\dot{\boldsymbol{A}_{t}}^{c}\boldsymbol{B}^{T}),$
$\displaystyle\boldsymbol{F}_{12}$ $\displaystyle=$ $\displaystyle
L(\dot{\boldsymbol{A}_{r}}^{T}\boldsymbol{A}_{r}^{c})\odot(\boldsymbol{B}^{c}\boldsymbol{A}_{t}^{T}\boldsymbol{S}_{x}^{T}\boldsymbol{A}_{t}^{c})+L(\boldsymbol{A}_{r}^{T}\boldsymbol{A}_{r}^{c})\odot(\boldsymbol{B}^{c}\dot{\boldsymbol{A}_{t}}^{T}\boldsymbol{S}_{x}^{T}\boldsymbol{A}_{t}^{c}),$
(16) $\displaystyle\boldsymbol{F}_{22}$ $\displaystyle=$ $\displaystyle
L(\boldsymbol{A}_{r}^{T}\boldsymbol{A}_{r}^{c})\odot(\boldsymbol{A}_{t}^{T}\boldsymbol{S}_{x}^{T}\boldsymbol{A}_{t}^{c}).$
(17)

Then, the corresponding CRB matrix for estimating $\boldsymbol{\xi}$ is

$\boldsymbol{C}_{2}=\boldsymbol{F}_{\boldsymbol{\xi}}^{-1}.$ (18)

Similar as in (10), we adopt the trace of the CRB matrix $\boldsymbol{C}_{2}$
as the scalar performance metric for estimating $\boldsymbol{\xi}$, i.e.,

$\mathrm{CRB}_{2}(\boldsymbol{S}_{x})=\mathrm{tr}(\boldsymbol{C}_{2})=\mathrm{tr}(\boldsymbol{F}_{\boldsymbol{\xi}}^{-1}).$
(19)

###  II-C Problem Formulation

We are interested in revealing the fundamental C-R performance tradeoff limits
of the multicast multi-target ISAC system for both Scenarios I and II. Notice
that in our considered setup, the transmit multi-beam signals $\boldsymbol{X}$
are reused for both sensing and communications. This ensures the optimal
performance, as adopting separate communication and sensing signal beams would
lead to the degradation in both communication and sensing performances, due to
the potential mutual interference between communication and sensing signals.
Let $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ denote the achievable C-R regions
of the ISAC system in Scenario I and II, respectively, which are defined as

$\displaystyle\mathcal{C}_{i}\overset{\triangle}{=}\underset{\boldsymbol{S}_{x}\succeq\boldsymbol{0}}{\bigcup}\big{\\{}(\hat{R},\hat{\Psi})|\hat{R}\leq
R(\boldsymbol{S}_{x}),$
$\displaystyle\hat{\Psi}\geq\mathrm{CRB}_{i}(\boldsymbol{S}_{x}),\mathrm{tr}(\boldsymbol{S}_{x})\leq
P\big{\\}},i\in\\{1,2\\}.$ (20)

Our objective is to characterize the Pareto boundary of each region
$\mathcal{C}_{i},i\in\\{1,2\\}$, at each point of which we cannot improve one
criteria without sacrificing the other [21, 20]. Note that the multicast rate
$R(\boldsymbol{S}_{x})$ in (4), $\mathrm{CRB}_{1}(\boldsymbol{S}_{x})$ in
(11), and $\mathrm{CRB}_{2}(\boldsymbol{S}_{x})$ in (18) are both convex with
respect to (w.r.t.) $\boldsymbol{S}_{x}$. Therefore, the C-R regions
$\mathcal{C}_{i},i\in\\{1,2\\}$, are both convex sets. As a result, we can
characterize the whole Pareto boundary by finding each boundary point via
minimizing the CRB subject to varying multicast rate thresholds [21, 20, 36].

In particular, we optimize the transmit covariance matrix $\boldsymbol{S}_{x}$
to minimize the estimation CRB for multi-target estimation while ensuring a
minimum multicast communication rate $\bar{R}$, subject to a maximum transmit
power constraint in (1). First, consider Scenario I, for which the multicast-
rate-constrained CRB minimization problem is formulated as

$\displaystyle(\mathrm{P1}):$
$\displaystyle\underset{\boldsymbol{S}_{x}\succeq\boldsymbol{0}}{\min}$
$\displaystyle\frac{N_{r}\sigma_{r}^{2}}{L}\mathrm{tr}(\boldsymbol{S}_{x}^{-1})$
(21) $\displaystyle\mathrm{s.t.}$
$\displaystyle\underset{k\in\mathcal{K}}{\min}\Big{\\{}\log_{2}\Big{(}1+\frac{\boldsymbol{h}_{k}^{H}\boldsymbol{S}_{x}\boldsymbol{h}_{k}}{\sigma^{2}}\Big{)}\Big{\\}}\geq\bar{R},$
$\displaystyle\mathrm{tr}(\boldsymbol{S}_{x})\leq P.$

By introducing $\Gamma=\sigma^{2}(2^{\bar{R}}-1)$, problem (P1) is
equivalently reformulated as

$\displaystyle(\mathrm{P1.1}):$
$\displaystyle\underset{\boldsymbol{S}_{x}\succeq\boldsymbol{0}}{\min}$
$\displaystyle\mathrm{tr}(\boldsymbol{S}_{x}^{-1})$ (22b)
$\displaystyle\mathrm{s.t.}$
$\displaystyle\boldsymbol{h}_{k}^{H}\boldsymbol{S}_{x}\boldsymbol{h}_{k}\geq\Gamma,\forall
k\in\mathcal{K},$ $\displaystyle\mathrm{tr}(\boldsymbol{S}_{x})\leq P.$

In problem (P1.1), the objective is a trace inverse function that is strictly
convex w.r.t. $\boldsymbol{S}_{x}$ [37], and the constraints in (22b) and
(22b) are both affine. As a result, problem (P1.1) is a convex optimization
problem. In Section III, we will obtain its globally optimal solution in a
semi-closed form by applying the Lagrange duality method [36].

Next, we consider Scenario II, for which the multicast-rate-constrained CRB
minimization problem is formulated as

$\displaystyle(\mathrm{P2}):$
$\displaystyle\underset{\boldsymbol{S}_{x}\succeq\boldsymbol{0}}{\min}$
$\displaystyle\mathrm{CRB}_{2}(\boldsymbol{S}_{x})$ (23)
$\displaystyle\mathrm{s.t.}$
$\displaystyle\underset{k\in\mathcal{K}}{\min}\Big{\\{}\log_{2}\Big{(}1+\frac{\boldsymbol{h}_{k}^{H}\boldsymbol{S}_{x}\boldsymbol{h}_{k}}{\sigma^{2}}\Big{)}\Big{\\}}\geq\bar{R},$
$\displaystyle\mathrm{tr}(\boldsymbol{S}_{x})\leq P.$

Then, problem (P2) is equivalently reformulated as

$\displaystyle\mathrm{(P2.1):}$
$\displaystyle\underset{\boldsymbol{S}_{x}\succeq\boldsymbol{0}}{\min}$
$\displaystyle\mathrm{CRB}_{2}(\boldsymbol{S}_{x})$ (24b)
$\displaystyle\mathrm{s.t.}$
$\displaystyle\boldsymbol{h}_{k}^{H}\boldsymbol{S}_{x}\boldsymbol{h}_{k}\geq\Gamma,\forall
k\in\mathcal{K},$ $\displaystyle\mathrm{tr}(\boldsymbol{S}_{x})\leq P.$

In Section IV, we will find the optimal solution to problem (P2.1) or
equivalently (P2) by using the semi-definite programing (SDP) technique.

It is worth noting that by solving the multicast-rate-constrained CRB
minimization problem (P1) or (P2) under one given rate threshold $\bar{R}$, we
can generally find one Pareto boundary point at C-R region $\mathcal{C}_{1}$
for Scenario I or $\mathcal{C}_{2}$ for Scenario II. By adjusting the value of
$\bar{R}$, we can obtain the complete Pareto boundary points on C-R region
$\mathcal{C}_{i}$, $i\in\\{1,2\\}$, for achieving different C-R tradeoffs [21,
20].

######  Remark 1.

Before proceeding to solve problems (P1) and (P2) for ISAC, we discuss the two
special cases with sole information multicasting and sole multi-target
sensing, respectively. First, considering the information multicasting only,
the maximum multicast capacity can be found by solving the following problem:

$\displaystyle\underset{\boldsymbol{S}_{x}\succeq\boldsymbol{0}}{\max}$
$\displaystyle\underset{k\in\mathcal{K}}{\min}\Big{\\{}\log_{2}\Big{(}1+\frac{\boldsymbol{h}_{k}^{H}\boldsymbol{S}_{x}\boldsymbol{h}_{k}}{\sigma^{2}}\Big{)}\Big{\\}}$
(25) $\displaystyle\mathrm{s.t.}$
$\displaystyle\mathrm{tr}(\boldsymbol{S}_{x})\leq P.$

It has been shown in [23] that problem (25) is optimally solvable via SDP. Let
$\boldsymbol{S}_{x}^{\mathrm{com}}$ denote the optimal solution to problem
(25). Accordingly, we have the maximum multicast capacity as
$R_{\mathrm{max}}=R(\boldsymbol{S}_{x}^{\mathrm{com}})$, and the corresponding
CRB in Scenario $i\in\\{1,2\\}$ as
$\mathrm{CRB}_{i}(\boldsymbol{S}_{x}^{\mathrm{com}})$. As such, we obtain a
corner Pareto boundary point of the C-R region $\mathcal{C}_{i}$ as
$(R_{\mathrm{max}},\mathrm{CRB}_{i}(\boldsymbol{S}_{x}^{\mathrm{com}}))$,
which corresponds to the maximum multicast rate. Note that in Scenario I, if
$\boldsymbol{S}_{x}^{\mathrm{com}}$ is rank deficient, i.e., not a full-rank
matrix, we have
$\mathrm{CRB}_{1}(\boldsymbol{S}_{x}^{\mathrm{com}})\rightarrow\infty$, which
means the transmit degrees of freedom (DoFs) are not sufficient to estimate
the complete multi-target response matrix $\boldsymbol{G}$. Next, we consider
the multi-target sensing only, in which the CRB minimization problem is
expressed as follows, where $i=1$ and $i=2$ correspond to Scenarios I and II,
respectively.

$\displaystyle\underset{\boldsymbol{S}_{x}\succeq\boldsymbol{0}}{\min}$
$\displaystyle\mathrm{CRB}_{i}(\boldsymbol{S}_{x})$ (26)
$\displaystyle\mathrm{s.t.}$ $\displaystyle\mathrm{tr}(\boldsymbol{S}_{x})\leq
P.$

For Scenario I, it has been shown in [16] that the optimal solution to problem
(26) is
$\boldsymbol{S}_{x}^{\mathrm{sen},\mathrm{1}}=\frac{P}{N_{t}}\boldsymbol{I}$,
i.e., the isotropic transmission is optimal. For Scenario II, it has been
proved in [28] that the optimal solution
$\boldsymbol{S}_{x}^{\mathrm{sen},\mathrm{2}}$ to problem (26) can be obtained
via semidefinite program (SDP). With $\boldsymbol{S}_{x}^{\mathrm{sen},i}$ at
hand, the minimum CRB is obtained as
$\mathrm{CRB}_{\min}^{i}=\mathrm{CRB}_{i}(\boldsymbol{S}_{x}^{\mathrm{sen},i})$,
and the corresponding multicast rate becomes
$R(\boldsymbol{S}_{x}^{\mathrm{sen},i})$. We thus obtain another corner Pareto
boundary point
$\big{(}R(\boldsymbol{S}_{x}^{\mathrm{sen},i}),\mathrm{CRB}_{\min}^{i}\big{)}$
at the C-R region $\mathcal{C}_{i}$, which corresponds to multi-target CRB
minimization.

##  III Optimal Solution to Problem (P1.1) or (P1) for Scenario I

In this section, we present the optimal solution to the multicast-rate-
constrained CRB minimization problem (P1.1) for Scenario I. Notice that
problem (P1.1) is convex and satisfies the Slater’s condition [36]. As a
result, the strong duality holds between problem (P1.1) and its dual problem.
Therefore, we adopt the Lagrange duality method to find the optimal solution
to (P1.1) and analyze its structure to gain insights. The optimal solution to
(P1.1) or equivalently (P1) is obtained in the following proposition.

######  Proposition 1.

Suppose that $\lambda^{\mathrm{opt}}$ and $\\{\mu_{k}^{\mathrm{opt}}\\}$
denote the optimal dual solutions to the dual problem of (P1.1), which are
associated to constraints (22b) and (22b), respectivaly. Define
$\boldsymbol{A}(\lambda^{\mathrm{opt}},\\{\mu_{k}^{\mathrm{opt}}\\})\overset{\triangle}{=}\lambda^{\mathrm{opt}}\boldsymbol{I}-\sum_{k=1}^{K}\mu_{k}^{\mathrm{opt}}\boldsymbol{h}_{k}\boldsymbol{h}_{k}^{H}\succeq\boldsymbol{0}$,
for which the eigenvalue decomposition (EVD) is
$\boldsymbol{A}(\lambda^{\mathrm{opt}},\\{\mu_{k}^{\mathrm{opt}}\\})=\boldsymbol{U}^{\mathrm{opt}}\boldsymbol{\Lambda}^{\mathrm{opt}}\boldsymbol{U}^{\mathrm{opt}H}$,
where
$\boldsymbol{\Lambda}^{\mathrm{opt}}=\mathrm{diag}(\alpha_{1}^{\mathrm{opt}},\dots,\alpha_{N_{t}}^{\mathrm{opt}})$
. Then it must follow that
$\mathrm{rank}(\boldsymbol{A}(\lambda^{\mathrm{opt}},\\{\mu_{k}^{\mathrm{opt}}\\}))=N_{t}$
and accordingly
$\alpha_{1}^{\mathrm{opt}}\geq\dots\geq\alpha_{N_{t}}^{\mathrm{opt}}>0$. The
optimal solution $\boldsymbol{S}_{x}^{\mathrm{opt}}$ to problem (P1.1) is
given by

$\displaystyle\boldsymbol{S}_{x}^{\mathrm{opt}}=$
$\displaystyle\boldsymbol{U}^{\mathrm{opt}}{\boldsymbol{\Sigma}^{\mathrm{opt}}}\boldsymbol{U}^{\mathrm{opt}H},$
(27)

where
${\boldsymbol{\Sigma}^{\mathrm{opt}}}=\mathrm{diag}(\tau_{1}^{\mathrm{opt}},\dots,\tau_{N_{t}}^{\mathrm{opt}}$)
and $\tau_{i}^{\mathrm{opt}}=(\alpha_{i}^{\mathrm{opt}})^{-1/2}$, $\forall
i\in\\{1,\dots,N_{t}\\}$.

###### Proof.

Please refer to Appendix A. ∎

Based on Proposition 1, we have the following remark to reveal more insights
on the optimal solution to (P1.1) or (P1).

######  Remark 2.

With the optimal dual solution $\lambda^{\mathrm{opt}}$ and
$\\{\mu_{k}^{\mathrm{opt}}\\}$, we define the weighted communication channel
of the $K$ CUs as
$\boldsymbol{H}=\sum_{k=1}^{K}\mu_{k}^{\mathrm{opt}}\boldsymbol{h}_{k}\boldsymbol{h}_{k}^{H}\succeq\boldsymbol{0}$,
with rank $N_{\text{com}}=\text{rank}(\boldsymbol{H})\leq N_{t}$. The EVD of
$\boldsymbol{H}$ is then expressed as
$\boldsymbol{H}=[{\boldsymbol{U}}_{\text{sen}}~{}{\boldsymbol{U}}_{\text{com}}]{\boldsymbol{\Delta}}[{\boldsymbol{U}}_{\text{sen}}~{}{\boldsymbol{U}}_{\text{com}}]^{H}$,
where
${\boldsymbol{\Delta}}=\mathrm{diag}(0,\ldots,0,\delta_{1},\ldots,\delta_{N_{\text{com}}})$
with $\delta_{1}\geq\ldots\geq\delta_{N_{\text{com}}}\geq 0$ denoting the
$N_{\text{com}}$ eigenvalues, and
$\boldsymbol{U}_{\text{com}}\in\mathbb{C}^{N_{t}\times N_{\text{com}}}$ and
${\boldsymbol{U}}_{\text{sen}}\in\mathbb{C}^{N_{t}\times(N_{t}-N_{\text{com}})}$
consist of the eigenvectors corresponding to the non-zero and zero
eigenvalues, thus spanning the communication subspaces and sensing subspaces,
respectively. In this case, recall that
$\boldsymbol{A}(\lambda^{\mathrm{opt}},\\{\mu_{k}^{\mathrm{opt}}\\})=\lambda^{\mathrm{opt}}\boldsymbol{I}-\sum_{k=1}^{K}\mu_{k}^{\mathrm{opt}}\boldsymbol{h}_{k}\boldsymbol{h}_{k}^{H}=\lambda^{\mathrm{opt}}\boldsymbol{I}-\boldsymbol{H}\succ\boldsymbol{0}$.
It is evident that the EVD of
$\boldsymbol{A}(\lambda^{\mathrm{opt}},\\{\mu_{k}^{\mathrm{opt}}\\})$ can be
expressed as

$\displaystyle\boldsymbol{A}(\lambda^{\mathrm{opt}},\\{\mu_{k}^{\mathrm{opt}}\\})=$
$\displaystyle\boldsymbol{U}^{\mathrm{opt}}\boldsymbol{\Lambda}^{\mathrm{opt}}\boldsymbol{U}^{\mathrm{opt}H}=[{\boldsymbol{U}}_{\text{sen}}~{}{\boldsymbol{U}}_{\text{com}}]\big{(}\lambda^{\mathrm{opt}}\boldsymbol{I}-{\boldsymbol{\Delta}}\big{)}[{\boldsymbol{U}}_{\text{sen}}~{}{\boldsymbol{U}}_{\text{com}}]^{H},$
(28)

where
$\boldsymbol{U}^{\mathrm{opt}}=[{\boldsymbol{U}}_{\text{sen}}~{}\boldsymbol{U}_{\text{com}}]$
and
$\boldsymbol{\Lambda}^{\mathrm{opt}}=\mathrm{diag}(\alpha_{1}^{\mathrm{opt}},\dots,\alpha_{N_{t}}^{\mathrm{opt}})$
with

$\alpha_{i}^{\mathrm{opt}}=\left\\{\begin{array}[]{cc}\lambda^{\mathrm{opt}},&{\text{if~{}}}i=1,\ldots,N_{t}-N_{\text{com}}\\\
\lambda^{\mathrm{opt}}-\delta_{i-(N_{t}-N_{\text{com}})},&{\text{if~{}}}i=N_{t}-N_{\text{com}}+1,\ldots,N_{t}\end{array}\right..$
(29)

In this case, it follows from Proposition 1 that the optimal transmit
covariance is given by

$\displaystyle\boldsymbol{S}_{x}^{\mathrm{opt}}$
$\displaystyle=\boldsymbol{U}{\boldsymbol{\Sigma}^{\mathrm{opt}}}\boldsymbol{U}^{\mathrm{opt}H}={\boldsymbol{U}}_{\text{sen}}{\boldsymbol{\Sigma}}_{\text{sen}}^{\mathrm{opt}}{\boldsymbol{U}}_{\text{sen}}^{H}+\boldsymbol{U}_{\text{com}}{\boldsymbol{\Sigma}}_{\text{com}}^{\mathrm{opt}}\boldsymbol{U}_{\text{com}}^{H},$
(30)

where
${\boldsymbol{\Sigma}}_{\text{sen}}^{\mathrm{opt}}=\mathrm{diag}(\sqrt{1/\lambda^{\mathrm{opt}}},\ldots,\sqrt{1/\lambda^{\mathrm{opt}}})$
and
${\boldsymbol{\Sigma}}_{\text{com}}^{\mathrm{opt}}=\mathrm{diag}(\sqrt{1/(\lambda^{\mathrm{opt}}-\delta_{1})},\ldots,\sqrt{1/(\lambda^{\mathrm{opt}}-\delta_{N_{\text{com}}})})$.
It is clear that the transmit covariance can be decomposed into two parts,
including
$\boldsymbol{U}_{\text{com}}{\boldsymbol{\Sigma}}_{\text{com}}^{\mathrm{opt}}\boldsymbol{U}_{\text{com}}^{H}$
towards the CUs for ISAC, and
${\boldsymbol{U}}_{\text{sen}}{\boldsymbol{\Sigma}}_{\text{sen}}^{\mathrm{opt}}{\boldsymbol{U}}_{\text{sen}}^{H}$
for dedicated sensing. In the first signal part for dedicated sensing, equal
power allocation is adopted. In contrast, in the second signal part for ISAC,
more transmit power is allocated over the sub-channels with stronger combined
channel gains ($\alpha_{i}^{\mathrm{opt}}$).

##  IV Optimal Solution to Problem (P2.1) or (P2) for Scenario II

In this section, we present the optimal solution to problem (P2.1) or (P2) by
exploiting the SDP technique. Towards this end, we re-express
$\boldsymbol{F}_{\boldsymbol{\xi}}^{-1}[i,i]$ as
$\boldsymbol{e}_{i}^{T}\boldsymbol{F}_{\boldsymbol{\xi}}^{-1}\boldsymbol{e}_{i}$,
$\forall i\in\\{1,...,3M\\}$, where $\boldsymbol{e}_{i}$ denotes the $i$-th
column vector of an identity matrix with proper dimension. By introducing a
set of auxiliary variables $\\{t_{i}\\}_{i=1}^{3M}$, problem (P2.1) is
equivalently reformulated as

$\displaystyle\mathrm{(P2.2):}$
$\displaystyle\underset{\boldsymbol{S}_{x}\succeq\boldsymbol{0},\\{t_{i}\\}_{i=1}^{3M}}{\min}$
$\displaystyle\sum_{i=1}^{3M}t_{i}$ $\displaystyle\mathrm{s.t.}$
$\displaystyle
t_{i}\geq\boldsymbol{e}_{i}^{T}\boldsymbol{F}_{\boldsymbol{\xi}}^{-1}\boldsymbol{e}_{i},\forall
i\in\\{1,\dots,3M\\},$ $\displaystyle\textrm{\eqref{eq:rate constraint 2} and
\eqref{eq:power constraint2}}.$

Next, by exploiting the Schur component, the constraints in (31) are
equivalently re-expressed as

$\left[\begin{array}[]{cc}\boldsymbol{F}_{\boldsymbol{\xi}}&\boldsymbol{e}_{i}\\\
\boldsymbol{e}_{i}^{T}&t_{i}\end{array}\right]\succeq\boldsymbol{0},\forall
i\in\\{1,\dots,3M\\}.$ (32)

By replacing (31) in (P2.2) as (32), problem (P2.2) is further equivalently
reformulated into

$\displaystyle(\mathrm{P2.3}):$
$\displaystyle\underset{\boldsymbol{S}_{x}\succeq\boldsymbol{0},\\{t_{i}\\}_{i=1}^{3M}}{\min}$
$\displaystyle\sum_{i=1}^{3M}t_{i}$ (33a) $\displaystyle\mathrm{s.t.}$
$\displaystyle\textrm{{\eqref{eq:rate constraint 2}, \eqref{eq:power
constraint2}}, and \eqref{eq:Auxiliary variables 2}}.$

Notice that the constraints in (32) are linear matrix inequality (LMI)
constraints w.r.t. $\boldsymbol{S}_{x}$ and the constraints in (24b) and (24b)
are affine. As a result, (P2.3) is a standard convex SDP, which can be
efficiently solved via convex optimization tools, such as CVX [38]. Therefore,
we obtain the optimal solution of $\boldsymbol{S}_{x}$ to problem (P2.3), and
thus problem (P2), denoted by $\boldsymbol{S}_{x}^{\mathrm{opt,2}}$.

######  Remark 3.

It is worth discussing the structure of the optimal solution
$\boldsymbol{S}_{x}^{\mathrm{opt,2}}$ for gaining insights. First, it is
observed from (18) that only the transmit signal components lying in the
subspaces spanned by
$\left[\boldsymbol{A}_{t},\dot{\boldsymbol{A}_{t}}\right]$ contribute to
$\mathrm{CRB}_{2}(\boldsymbol{S}_{x})$. Similarly, it is observed from (4)
that only the signal components lying in the subspaces spanned by
$\left[\boldsymbol{h}_{1},\dots,\boldsymbol{h}_{K}\right]$ contribute to
multicast communication rate $R(\boldsymbol{S}_{x})$. In this case, suppose
that the rank of the accumulated matrix
$\left[\boldsymbol{A}_{t},\dot{\boldsymbol{A}_{t}},\boldsymbol{h}_{1},\dots,\boldsymbol{h}_{K}\right]\in\mathbb{C}^{N_{t}\times(2M+K)}$
is $J$, i.e.,
$\mathrm{rank}\big{(}\left[\boldsymbol{A}_{t},\dot{\boldsymbol{A}_{t}},\boldsymbol{h}_{1},\dots,\boldsymbol{h}_{K}\right]\big{)}=J\leq
N_{t}$, and its truncated singular value decomposition (SVD) is

$\left[\boldsymbol{A}_{t},\dot{\boldsymbol{A}_{t}},\boldsymbol{h}_{1},\dots,\boldsymbol{h}_{K}\right]=\boldsymbol{U}\boldsymbol{\Lambda}\boldsymbol{V}^{H},$
(34)

where $\boldsymbol{U}\in\mathbb{C}^{N_{t}\times J}$ and
$\boldsymbol{V}\in\mathbb{C}^{(2M+K)\times J}$ collect the left and right
singular vectors corresponding to the non-zero singular values, respectively.
In this case, we can express the transmit covariance matrix
$\boldsymbol{S}_{x}$ as
$\boldsymbol{S}_{x}={\boldsymbol{U}}^{H}\bar{\boldsymbol{S}_{x}}{\boldsymbol{U}}$,
where $\bar{\boldsymbol{S}_{x}}\in\mathbb{C}^{J\times J}$ corresponds to the
equivalent transmit covariance matrix to be decided. This intuitively shows
that the transmit covariance matrix $\boldsymbol{S}_{x}$ should lie in the
subspaces spanned by the sensing and communication channels for efficient
ISAC. In the case when $M$ and $K$ (and thus $J$) are small but $N_{t}$ is
large, we can optimize $\bar{\boldsymbol{S}_{x}}$ instead of
$\boldsymbol{S}_{x}$, in order to significantly reduce the dimension of
optimization variables and thus reduce the solution complexity.

##  V Joint Information and Sensing Beamforming Design

Sections III and IV presented the optimal transmit covariance solutions to the
multicast-rate-constrained CRB minimization problems (P1) and (P2) for
Scenarios I and II, respectively, which are of high rank in general,
especially when the number of CUs $K$ and the number of targets $M$ become
large. Note that this high-rank covariance is typically achieved through
multi-stream transmission. While the transmitted streams may interfere with
each other, it is remarkable that separate codeword decoding can still achieve
capacity by utilizing a linear minimum mean-square error (LMMSE) successive
interference cancellation (SIC) receiver. The basic idea behind the LMMSE-SIC
receiver is as follows: for each target stream, we convey the observed
sequence to a corresponding LMMSE filter and then decode that particular
stream. Subsequently, we reconstruct the interference caused by that stream
and cancel it before decoding the next stream. Note that although this LMMSE-
SIC approach is capacity achieving, it requires high decoding complexity[9].
To resolve this issue, this section presents joint transmit information and
sensing beamforming designs, in which the BS adopts one single information
beam for delivering the common message, together with multiple dedicated
sensing beams555Notice that for problems (P1) and (P2) introducing the
dedicated sensing beams is not necessary. This is due to the fact that the
transmit covariance $\boldsymbol{S}_{x}$ in (P1) and (P2) is of general rank,
and thus adding dedicated sensing beams does not improve the ISAC performance.
.

###  V-A Joint Transmit Beamforming

In the joint transmit beamforming, the BS uses one information beam to deliver
the common message. Let $\boldsymbol{w}\in\mathbb{C}^{N_{t}\times 1}$ denote
the information beamforming vector and $s_{\text{com}}(n)$ denote the common
message carried by symbol $n$ with $s_{\text{com}}(n)\sim\mathcal{CN}(0,1)$.
In addition, the BS also employs dedicated sensing beams for providing
additional DoFs to facilitate the multi-target sensing. Let
$\boldsymbol{x}_{\mathrm{sen}}(n)$ denote the dedicated sensing signals to
provide additional DoFs for estimating the targets parameters, which is a
pseudo random vector with mean zero and covariance matrix
$\mathbb{E}(\boldsymbol{x}_{\mathrm{sen}}(n)\boldsymbol{x}_{\mathrm{sen}}^{H}(n))=\boldsymbol{S}_{\mathrm{sen}}\succeq\boldsymbol{0}$.
Without loss of generality, we assume that
$\mathrm{rank}(\boldsymbol{S}_{\mathrm{sen}})=r_{\mathrm{sen}}\geq 0$, which
corresponds to a number of $r_{\mathrm{sen}}$ sensing beams that can be
obtained by performing the EVD on $\boldsymbol{S}_{\mathrm{sen}}$ [18]. In
this case, the transmit signal is given by

$\boldsymbol{x}(n)=\boldsymbol{w}s_{\text{com}}(n)+\boldsymbol{x}_{\mathrm{sen}}(n),$
(35)

for which the corresponding transmit covariance matrix is given by

$\boldsymbol{S}_{x}=\boldsymbol{S}_{\mathrm{sen}}+\boldsymbol{w}\boldsymbol{w}^{H}.$
(36)

The transmit power constraint in (1) thus becomes

$\mathrm{tr}(\boldsymbol{S}_{x})=\mathrm{tr}(\boldsymbol{S}_{\mathrm{sen}})+\|\boldsymbol{w}\|^{2}\leq
P.$ (37)

First, consider the multicast communication. The received signal at CU
$k\in\mathcal{K}$ is given as

$\tilde{y_{k}}(n)=\boldsymbol{h}_{k}^{H}\boldsymbol{w}s_{\text{com}}(n)+\boldsymbol{h}_{k}^{H}\boldsymbol{x}_{\mathrm{sen}}(n)+z_{k}(n).$
(38)

In general, the dedicated sensing signals
$\boldsymbol{h}_{k}^{H}\boldsymbol{x}_{\mathrm{sen}}(n)$ may introduce harmful
interference at the receiver of the CUs. We impose an assumption that the CU
possesses a priori knowledge of the radar sensing sequence and is aware of the
CSI such that the interference caused by the radar signal can be efficiently
eliminated from $\tilde{y_{k}}(n)$ in (38). As a result, we consider that the
CU is capable of such interference cancellation before information decoding
[18].666The case without sensing interference cancellation will be considered
as a benchmark for performance evaluation in Section VI. In this case, the SNR
at the receiver of CU $k\in\mathcal{K}$ is denoted as

$\hat{\gamma_{k}}=\frac{\boldsymbol{h}_{k}^{H}\boldsymbol{w}\boldsymbol{w}^{H}\boldsymbol{h}_{k}}{\sigma^{2}},$
(39)

and the corresponding achievable multicast rate is

$\hat{R}(\boldsymbol{w})\overset{\triangle}{=}\underset{k\in\mathcal{K}}{\min}~{}\Big{\\{}\log_{2}\Big{(}1+\frac{\boldsymbol{h}_{k}^{H}\boldsymbol{w}\boldsymbol{w}^{H}\boldsymbol{h}_{k}}{\sigma^{2}}\Big{)}\Big{\\}}.$
(40)

Next, we consider the multi-target sensing, in which both information and
sensing beams are jointly employed for target estimation. In this case, the
estimation CRB for Scenarios I and II are expressed as
$\mathrm{CRB}_{1}(\boldsymbol{S}_{\mathrm{sen}}+\boldsymbol{w}\boldsymbol{w}^{H})$
and
$\mathrm{CRB}_{2}(\boldsymbol{S}_{\mathrm{sen}}+\boldsymbol{w}\boldsymbol{w}^{H})$,
respectively.

Based on the derived multicast rate and estimation CRB above, the multicast-
rate-constrained CRB minimization problems via joint information and sensing
beamforming are formulated as problems (P3) and (P4) for Scenarios I and II,
respectively:

$\displaystyle(\mathrm{P3}):$
$\displaystyle\underset{\boldsymbol{w},\boldsymbol{S}_{\mathrm{sen}}\succeq\boldsymbol{0}}{\min}$
$\displaystyle\mathrm{CRB}_{1}(\boldsymbol{S}_{\mathrm{sen}}+\boldsymbol{w}\boldsymbol{w}^{H})$
(42) $\displaystyle\mathrm{s.t.}$
$\displaystyle\underset{k\in\mathcal{K}}{\min}~{}\Big{\\{}\log_{2}\Big{(}1+\frac{\boldsymbol{h}_{k}^{H}\boldsymbol{w}\boldsymbol{w}^{H}\boldsymbol{h}_{k}}{\sigma^{2}}\Big{)}\Big{\\}}\geq\bar{R},$
$\displaystyle\mathrm{tr}(\boldsymbol{S}_{\mathrm{sen}}+\boldsymbol{w}\boldsymbol{w}^{H})\leq
P.$ $\displaystyle(\mathrm{P4}):$
$\displaystyle\underset{\boldsymbol{w},\boldsymbol{S}_{\mathrm{sen}}\succeq\boldsymbol{0}}{\min}$
$\displaystyle\mathrm{CRB}_{2}(\boldsymbol{S}_{\mathrm{sen}}+\boldsymbol{w}\boldsymbol{w}^{H})$
(44) $\displaystyle\mathrm{s.t.}$
$\displaystyle\underset{k\in\mathcal{K}}{\min}~{}\Big{\\{}\log_{2}\Big{(}1+\frac{\boldsymbol{h}_{k}^{H}\boldsymbol{w}\boldsymbol{w}^{H}\boldsymbol{h}_{k}}{\sigma^{2}}\Big{)}\Big{\\}}\geq\bar{R},$
$\displaystyle\mathrm{tr}(\boldsymbol{S}_{\mathrm{sen}}+\boldsymbol{w}\boldsymbol{w}^{H})\leq
P.$

###  V-B SCA-Based Solution to Joint Beamforming Problems (P3) and (P4)

This subsection addresses problems (P3) and (P4). Notice that the two problems
are both non-convex and thus are generally difficult to be optimally solved.
To tackle this issue, we propose to apply the SCA technique to find high-
quality solutions. As the SCA algorithms for solving problems (P3) and (P4)
are similar, we only focus on solving problem (P3) in the following. The
similar algorithm can be adopted to solve problem (P4), for which the details
are omitted for brevity.

Consider problem (P3). First, by substituting
$\boldsymbol{S}_{\mathrm{sen}}=\boldsymbol{S}_{x}-\boldsymbol{w}\boldsymbol{w}^{H}$
and introducing $\Gamma=\sigma^{2}(2^{\bar{R}}-1)$, we equivalently
reformulate problem (P3) as

$\displaystyle(\mathrm{P3.1}):\underset{\boldsymbol{w},\boldsymbol{S}_{x}}{\min}\
$ $\displaystyle\mathrm{tr}(\boldsymbol{S}_{x}^{-1})$
$\displaystyle\mathrm{s.t.}$ $\displaystyle\
\boldsymbol{h}_{k}^{H}\boldsymbol{w}\boldsymbol{w}^{H}\boldsymbol{h}_{k}\geq\Gamma,\forall
k\in\mathcal{K},$ (45a) $\displaystyle\mathrm{tr}(\boldsymbol{S}_{x})\leq P,$
(45b)
$\displaystyle\boldsymbol{S}_{x}-\boldsymbol{w}\boldsymbol{w}^{H}\succeq\boldsymbol{0}.$
(45c)

Based on the Schur component, problem (P3.1) is equivalently reformulated as

$\displaystyle(\mathrm{P3.2}):\underset{\boldsymbol{w},\boldsymbol{S}_{x}}{\min}\
$ $\displaystyle\mathrm{tr}(\boldsymbol{S}_{x}^{-1})$
$\displaystyle\mathrm{s.t.}~{}$ $\displaystyle\textrm{\eqref{eq:SINR
constraintCancel} and \eqref{eq:22-2}},$
$\displaystyle\left[\begin{array}[]{cc}\boldsymbol{S}_{x}&\boldsymbol{w}\\\
\boldsymbol{w}^{H}&1\end{array}\right]\succeq\boldsymbol{0},$ (46c)

where constraint (45c) in (P3.1) is replaced by the LMI constraint in (46c).
However, the reformulated problem (P3.2) is still non-convex due to the non-
convex quadratic constraints.

Next, we propose a SCA-based algorithm to find a high-quality solution to
(P3.2) in an iterative manner, in which problem (P3.2) is approximated as a
series of convex problems. Specifically, consider a particular iteration $i$,
with $\boldsymbol{w}^{(i)}$ and $\boldsymbol{S}_{x}^{(i)}$ denoting the local
solution point. Based on the first-order approximation, it follows that

$\boldsymbol{h}_{k}^{H}\boldsymbol{w}\boldsymbol{w}^{H}\boldsymbol{h}_{k}\geq
2\mathrm{Re}(\boldsymbol{w}^{H}\boldsymbol{h}_{k}\boldsymbol{h}_{k}^{H}\boldsymbol{w}^{(i)})-\boldsymbol{w}^{(i)H}\boldsymbol{h}_{k}\boldsymbol{h}_{k}^{H}\boldsymbol{w}^{(i)}.$
(47)

By replacing
$\boldsymbol{h}_{k}^{H}\boldsymbol{w}\boldsymbol{w}^{H}\boldsymbol{h}_{k}$ as
$\mathrm{Re}(\boldsymbol{w}^{H}\boldsymbol{h}_{k}\boldsymbol{h}_{k}^{H}\boldsymbol{w}^{(i)})-\boldsymbol{w}^{(i)H}\boldsymbol{h}_{k}\boldsymbol{h}_{k}^{H}\boldsymbol{w}^{(i)}$,
we approximate the non-convex quadratic constraints in (45a) as

$\displaystyle
2\mathrm{Re}(\boldsymbol{w}^{H}\boldsymbol{h}_{k}\boldsymbol{h}_{k}^{H}\boldsymbol{w}^{(i)})-\boldsymbol{w}^{(i)H}\boldsymbol{h}_{k}\boldsymbol{h}_{k}^{H}\boldsymbol{w}^{(i)}\geq\Gamma,\forall
k\in\mathcal{K}.$ (48)

Accordingly, we obtain a restricted convex version of problem (P3.2) in
iteration $i$ as

$\displaystyle(\mathrm{P3.3\textrm{-}}i):$
$\displaystyle\underset{\boldsymbol{w},\boldsymbol{S}_{x}}{\max}$
$\displaystyle\mathrm{tr}(\boldsymbol{S}_{x}^{-1})$
$\displaystyle\mathrm{s.t.}$ (48), (45b), and (46c).

Notice that problem ($\textrm{P3.3-}i$) is convex and thus can be optimally
solved via numerical tools such as CVX. Let
$\hat{\boldsymbol{w}}_{\mathrm{opt}}^{(i)}$ denote the optimal solution to
problem ($\textrm{P3.3-}i$) in iteration $i$, which is updated as the local
point for the next iteration, i.e.,
$\boldsymbol{w}^{(i+1)}=\hat{\boldsymbol{w}}_{\mathrm{opt}}^{(i)}$. Note that
in order to implement the SCA based algorithm, we need to find a feasible
initial point. In particular, we first find the transmit information
beamforming to maximize the multicast rate by using the algorithm in [24]
based on the semi-definite relaxation (SDR) and Gaussian randomization.
Suppose that $\hat{\boldsymbol{w}}$ denote the obtained transmit information
beamformer. Accordingly, we use $\boldsymbol{w}^{(0)}=\hat{\boldsymbol{w}}$ as
the initial point for the SCA-based algorithms.

###  V-C Convergence and Computational Complexity

In this subsection, we provide the convergence analysis for the proposed SCA
based algorithm. Due to the lower-bound approximation in (48), it can be shown
that the optimal solution $\hat{\boldsymbol{w}}_{\mathrm{opt}}^{(i)}$ in each
iteration $i$ corresponds to a feasible point for next iteration the next
iteration $i+1$. As such, the SCA iterations are ensured to result in
monotonically decreasing CRB objective values. Therefore, the convergence of
the SCA-based algorithm for handling problem (P3.2) and thus (P3) is
guaranteed [39]. Also, following [39], the iteration can converge to a Karush-
Kuhn-Tucker (KKT) point.

Next, we analyze the computational complexity. First, with a maximum iteration
number $N_{\max}$, the iteration complexity is given as
$\mathcal{O}(N_{\max})$, where $\mathcal{O}(\cdot)$ is the big-O operator.
Next, we analyze the worst-case computational complexity of solving problem
(P3.2). Problem (P3.2) is a convex problem that can be effectively solved by
CVX via the interior-point method. Problem (P3.2) involves $(N+1)$ affine
constraints and one LMI constraints with dimension $(N+1)$. Based on [40], the
order of computational complexity is
$\mathcal{O}\big{(}\sqrt{N+1}N^{3}\ln(\frac{1}{\varepsilon})\big{)}$ where
$\varepsilon>0$ is the duality gap accuracy. By combining them, we finally
have the total computational complexity as
$\mathcal{O}\Big{(}N_{\max}N^{3.5}\ln(\frac{1}{\varepsilon})\Big{)}$.

##  VI Numerical Results

This section provides numerical results to validate the performance of our
proposed designs. In the following, we first show the C-R tradeoff
performances achieved by our proposed designs as compared to other benchmark
schemes and then show the practical target estimation performances by
considering practical estimators. For convenience, we normalize the noise
powers as $\sigma^{2}=\sigma_{r}^{2}=0\mathrm{\textrm{ dBm}}$.

###  VI-A C-R Tradeoff Performance

This subsection presents the C-R tradeoff performance achieved by our proposed
optimal transmit covariance solution and the joint beamforming design, as
compared to the following two benchmark schemes.

  * •

Isotropic transmission: The BS adopts the isotropic transmission by setting
the transmit covariance as
$\boldsymbol{S}_{x}=\frac{P}{N_{t}}\boldsymbol{I}=\boldsymbol{S}_{x}^{\mathrm{sen},\mathrm{1}}$,
which is optimal in minimizing the estimation CRB in Scenario I (see Remark
1). This scheme is only feasible when the achieved multicast rate
$R(\boldsymbol{S}_{x}^{\mathrm{sen},\mathrm{1}})$ is no smaller than the rate
threshold $\bar{R}$.

  * •

Beamforming without sensing interference cancellation: The BS sends one
information beam $\boldsymbol{w}s_{\text{com}}(n)$ together with dedicated
sensing beams $\boldsymbol{x}_{\mathrm{sen}}(n)$. Different from the design in
Section IV, we consider that the CUs are not capable of canceling the
interference caused by sensing signals
$\boldsymbol{h}_{k}^{H}\boldsymbol{x}_{\mathrm{sen}}(n)$. As a result, the
received SINR becomes
$\frac{\boldsymbol{h}_{k}^{H}\boldsymbol{w}\boldsymbol{w}^{H}\boldsymbol{h}_{k}}{\boldsymbol{h}_{k}^{H}\boldsymbol{S}_{s}\boldsymbol{h}_{k}+\sigma^{2}}$.
The joint beamforming designs in this case correspond to problems (P3) and
(P4) by replacing (42) as

$\underset{k\in\mathcal{K}}{\min}\Big{\\{}\log_{2}\Big{(}1+\frac{\boldsymbol{h}_{k}^{H}\boldsymbol{w}\boldsymbol{w}^{H}\boldsymbol{h}_{k}}{\boldsymbol{h}_{k}^{H}\boldsymbol{S}_{s}\boldsymbol{h}_{k}+\sigma^{2}}\Big{)}\Big{\\}}\geq\bar{R}.$
(49)

Although the resultant beamforming optimization problems are non-convex, they
can be solved similarly as for (P3) and (P4) by using the SCA technique. We
skip the details for brevity and the interested readers can refer to [1] for
the detailed algorithm for solving the problem for Scenario I.

First, we consider Scenario I, in which Rayleigh fading is adopted [16] for
modeling the communication channels from the BS to the CUs.
$\boldsymbol{h}_{k}$ is a CSCG random vector with zero mean and identity
covariance matrix. The total transmit power budget at the BS is set as
$P=10\textrm{ dBm}$ unless otherwise stated. Fig. 2 shows the achieved C-R
tradeoff performance, where we set $N_{t}=N_{r}=10$ and $K=3$. It is observed
that as the CRB increases, the achievable rate by the optimal covariance
solution approaches the asymptotical multicast capacity but can never reach
it. This is due to the fact that in this setup, the rate-maximization transmit
covariance $\boldsymbol{S}_{x}^{\mathrm{com}}$ (see Remark 1) is rank-
deficient such that $\mathrm{CRB_{com}}\rightarrow\infty$. It is also observed
that the optimal covariance solution significantly outperforms the other three
schemes and the proposed transmit beamforming with sensing interference
cancellation outperforms that without sensing interference cancellation. This
is because in this scenario, full-rank transmission with multiple dedicated
sensing beams is necessary for effective target estimation, which may result
in harmful sensing interference making the sensing interference cancellation
crucial.

Figure 2: C-R tradeoff in Scenario I with $N_{t}=N_{r}=10$ and $K=3$.

Figs. 4 and 4 show the achieved C-R tradeoff performance with $N_{t}=N_{r}=4$,
where $K=50$ and $K=500$, respectively. In both figures, it is observed that
when the CRB becomes sufficiently large, the optimal covariance solution
reaches the multicast capacity. This is because the rate-maximization transmit
covariance $\boldsymbol{S}_{x}^{\mathrm{com}}$ is of full-rank such that
$\mathrm{CRB_{com}}$ is finite. Furthermore, it is observed that the two
transmit beamforming designs perform much worse than the optimal covariance
solution and the isotropic transmission (w.r.t. the rate and CRB), while the
performance gaps become more substantial when $K$ becomes larger (i.e.,
$K=500$ in Fig. 4 compared with $K=50$ in Fig. 4). This is due to the fact it
is more likely that some CUs may have wireless channels orthogonal to the
information beam with the increasing number of CUs, thus making the transmit
beamforming design less efficient.

Figure 3: C-R tradeoff in Scenario I with $N_{t}=N_{r}=4$ and $K=50$.

Figure 4: C-R tradeoff in Scenario I with $N_{t}=N_{r}=4$ and $K=500$.

Fig. 5 shows the estimation CRB versus the transmit power $P$, where
$\bar{R}=1.2\textrm{ bps/Hz}$, $N_{t}=N_{r}=10$, and $K=10$. It is observed
that in the low transmit power regime, the transmit beamforming with sensing
interference cancellation performs close to that without sensing interference
cancellation, and inferior to the optimal covariance solution. This is becasue
in this case, most power needs to be allocated for communication, thus making
the sensing interference cancellation less efficient. By contrast, in the high
transmit power regime, the transmit beamforming with sensing interference
cancellation is observed to perform close to the optimal covariance solution
and significantly outperform that without sensing interference cancellation.
This is due to the fact that in high SNR regime, a single communication beam
with equal power allocation is able to satisfy the communication requirement.
As a result, the sensing-optimal isotropic transmission is adopted for both BF
with sensing interference cancellation and optimal design. This highlights the
importance of sensing interference cancellation in this case.

Figure 5: The CRB versus the transmit power $P$ in Scenario I, where
$\bar{R}=1.2\textrm{ bps/Hz}$, $N_{t}=N_{r}=10,$ and $K=10$.

Next, we consider Scenario II. In this scenario, there are $M=2$ target
located at $30{{}^{\circ}}$ and $-30{{}^{\circ}}$, with the reflection
coefficients being normalized to be $\beta_{1}=\beta_{2}=1$ [41]. We consider
Rician fading for the wireless channel from the BS to each CU, i.e.,
$\boldsymbol{h}_{k}=\sqrt{\frac{K_{r}}{K_{r}+1}}\boldsymbol{a}(\boldsymbol{\theta}_{K}[k])+\sqrt{\frac{1}{K_{r}+1}}\boldsymbol{h}_{k,\mathrm{NLoS}}$,
where $K_{r}=4\textrm{ dB}$ denotes the Rician factor,
$\boldsymbol{\theta}_{K}[k]$ denotes the AoD for CU $k\in\mathcal{K}$, and
$\boldsymbol{h}_{k,\mathrm{NLoS}}$ is a CSCG random vector with zero mean and
identity covariance matrix.

Figs. 7 and 7 show the achieved C-R tradeoff with $N_{t}=N_{r}=10,$
$P=10\textrm{ dBm}$, and $K=10$ CUs, where the CU channels are correlated and
uncorrelated with the sensing channels (i.e.,
$\\{\boldsymbol{\theta}_{K}[k]\\}$ are randomly sampled in $\pm
30{{}^{\circ}}\pm 2{{}^{\circ}}$ and $\pm 60{{}^{\circ}}\pm 2{{}^{\circ}}$),
respectively. It is observed in Fig. 7 that the achieved CRB almost remains
unchanged when the rate threshold $\bar{R}$ is less than $3.5\textrm{
bps/Hz}$. This is due to the fact that the rate constraint is inactive in this
regime, and thus we can minimize the CRB and achieve a satisfactory
communication rate concurrently by exploiting the correlated communication and
sensing channels. On the other hand, it is observed in Fig. 7 that the
achieved CRB monotonically increases as the required achievable rate becomes
larger. This is because when the communication and sensing channels are less
correlated, the communication and sensing signals can hardly be reused for
optimizing both performances, thus leading to a more evident C-R tradeoff.

Figure 6: C-R tradeoff in Scenario II with correlated communication and
sensing channels, where $K=10$, $N_{t}=N_{r}=10,$ and $P=10\textrm{ dBm}$.

Figure 7: C-R tradeoff in Scenario II with uncorrelated communication and
sensing channels, where $K=10$, $N_{t}=N_{r}=10,$ and $P=10\textrm{ dBm}$.

Fig. 8 shows the CRB versus the transmit power ${P}$ for Scenario II, with
$\bar{R}=1.2\textrm{ bps/Hz}$, $N_{t}=N_{r}=10,$ and $K=10$, where the CUs’
angles $\\{\boldsymbol{\theta}_{K}[k]\\}$ are randomly sampled between
$-90{{}^{\circ}}$ and $90{{}^{\circ}}$. It is observed that the proposed
transmit beamforming with sensing interference cancellation performs close to
that without sensing interference cancellation when $P$ is small and
approaches that of the optimal covariance solution when $P$ is large. This
phenomenon is similar to Fig. 5 for Scenario I, which can be similarly
explained.

Figure 8: The CRB versus the transmit power $P$ in Scenario II, where
$\bar{R}=1.2\textrm{ bps/Hz}$, $N_{t}=N_{r}=10$, and $K=10$.

###  VI-B Target Estimation Performance

This subsection shows the exact estimation performance by considering the
practical estimators, i.e., least square (LS) estimator for Scenario I and the
Capon approximate maximum likelihood (CAML) estimator for Scenario II [16,
41]. In the simulation, we generate the transmitted signal $\boldsymbol{x}(n)$
based on the given transmit covariance $\boldsymbol{S}_{x}$ as

$\boldsymbol{x}(n)=\boldsymbol{V}\boldsymbol{u},$ (50)

where $\boldsymbol{S}_{x}=\boldsymbol{V}\boldsymbol{V}^{H}$ is the Cholesky
decomposition for $\boldsymbol{S}_{x}$ and $\boldsymbol{u}$ is a CSCG random
vector with zero mean and identity covariance. In this case, we obtain the
received signal in (6). Accordingly, for Scenario I, the LS estimation of
$\tilde{\boldsymbol{g}}$ based on (6) is obtained as

$\tilde{\boldsymbol{g}}_{\mathrm{est}}=\big{(}\boldsymbol{X}^{c}\boldsymbol{X}^{T}\otimes\boldsymbol{I}_{N_{r}}\big{)}^{-1}(\boldsymbol{X}^{T}\otimes\boldsymbol{I}_{N_{r}})^{H}\tilde{\boldsymbol{y}}.$
(51)

In particular, we can find that
$\tilde{\boldsymbol{g}}_{\mathrm{est}}\sim\mathcal{CN}\big{(}\tilde{\boldsymbol{g}},\sigma_{r}^{2}(\boldsymbol{X}^{c}\boldsymbol{X}^{T}\otimes\boldsymbol{I}_{N_{r}})^{-1}\big{)}$.
As a result, $\tilde{\boldsymbol{g}}_{\mathrm{est}}$ is a minimum variance
unbiased (MVU) estimator thus is CRB-achieving. For Scenario II, we estimate
$\boldsymbol{\beta}$ and $\boldsymbol{\theta}$ simultaneously by adopting the
CAML estimation [41], for which 20,000 angle grids in
$\left[-90{{}^{\circ}},90{{}^{\circ}}\right]$ are considered to ensure the
resolution.

For comparison, we consider the beampattern-based design [33] as the benchmark
scheme. The basic idea of beampattern-based design is to allocate more power
in desired target directions. Let
$\boldsymbol{\theta}_{N}\in\mathbb{R}^{N\times 1}$ denote the angle grid
vector of interest, which is given as
$[-\frac{\pi}{2}:\frac{\pi}{200}:\frac{\pi}{2}]$, and
$\boldsymbol{q}\in\mathbb{R}^{N\times 1}$ denote the desired beampattern in
this grid. For Scenario I, we set

$\boldsymbol{q}[n]=1,\forall n.$ (52)

For Scenario II, we set

$\boldsymbol{q}[n]=\begin{cases}1,&\exists
m\in\mathcal{M},|\theta_{m}-\boldsymbol{\theta}_{N}[n]|\leq\Delta_{\theta},\\\
0,&\textrm{others},\end{cases}$ (53)

where $\Delta_{\theta}$ is a pre-defined angle width. Accordingly, the
beampattern-based design aims to minimize the beampattern matching error while
ensuring the multicast rate constraint at CUs, which is formulated as

$\displaystyle\underset{\boldsymbol{S}_{x}\succeq\boldsymbol{0},\eta}{\min}$
$\displaystyle\sum_{i=1}^{N}|\eta\boldsymbol{q}[i]-\boldsymbol{a}^{H}(\boldsymbol{\theta}_{N}[i])\boldsymbol{S}_{x}\boldsymbol{a}^{H}(\boldsymbol{\theta}_{N}[i])|^{2}$
(54) $\displaystyle\mathrm{s.t.}$
$\displaystyle\underset{k\in\mathcal{K}}{\min}\Big{\\{}\log_{2}\Big{(}1+\frac{\boldsymbol{h}_{k}^{H}\boldsymbol{S}_{x}\boldsymbol{h}_{k}}{\sigma^{2}}\Big{)}\Big{\\}}\geq\bar{R},$
$\displaystyle\mathrm{tr}(\boldsymbol{S}_{x})\leq P.$

Problem (54) can be transformed as a convex problem that can be optimally
solved via CVX.

In the simulation, we obtain the results by averaging over 1000 Monte Carlo
realizations. We suppose that there are $M=2$ targets located at
$-30{{}^{\circ}},30{{}^{\circ}}$. We set $N_{t}=N_{r}=10,K=10$, and
$\bar{R}=0.5\textrm{ bps/Hz}$.

First, we evaluate the estimation performance in Scenario I. Fig. 10 shows the
estimation RMSE and the CRB versus the transmit power $P$ with $L=64$, in
which the actual RMSE and the theoretical CRB using $\boldsymbol{S}_{x}$, and
actually average CRB using $\frac{1}{L}\boldsymbol{XX}^{H}$ are shown. It is
observed that our considered CRB-based design achieves much better sensing
performance than the beampattern-based benchmark. Furthermore, the actual RMSE
and actual CRB are observed to be almost identical. This is due to the fact
that the LS estimator is adopted to estimate $\tilde{\boldsymbol{g}}$, which
is generally a CRB-achieving estimator.

Fig. 10 shows the RMSE and the CRB versus the transmit sequence length $L$ for
Scenario I, where $P=15\textrm{ dBm}$. It is observed that the gap between the
RMSE/CRB achieved by the sample covariance $\frac{1}{L}\boldsymbol{XX}^{H}$
versus that by covariance $\boldsymbol{S}_{x}$ becomes negligible when the
sequence length $L$ is greater than 64. This means that the approximation in
(7) is quite accurate when $L\geq 64$ in Scenario I.

Figure 9: Estimation RMSE or root CRB versus transmit power $P$ for Scenario
I, where $L=64$.

Figure 10: Estimation RMSE or root CRB versus transmit sequence length $L$ for
Scenario I, where $P=15\textrm{ dBm}$.

Next, we evaluate the estimation performance for Scenario II. Fig. 12 shows
the RMSE or the root CRB of angle estimation versus the transmit power $P$
with $L=64$. It is observed that the CRB-based design improves the practical
estimation performance as compared to the beampattern-based design. It is also
observed that the actual CRB using $\frac{1}{L}\boldsymbol{XX}^{H}$ is well
approached by the estimation RMSE. In this case, the gap between the RMSE and
the theoretical CRB using $\boldsymbol{S}_{x}$ mostly originates from the
approximation error.

Fig. 12 shows the estimation RMSE or root CRB of angle estimation versus the
transmit sequence length $L$ with $P=15\textrm{ dBm}$. It is observed that for
the CRB-based design, the achieved estimation RMSE nearly achieves the CRB
using $\boldsymbol{S}_{x}$ with $L\geq 64$. However, for the beampattern-based
design, the achieved RMSE is unable to achieve the corresponding CRB. This
shows the benefits of CRB-based design again.

Fig. 14 shows the RMSE or root CRB of amplitude estimation with $L=64$. It is
observed that there always exists a gap between RMSE and CRB both in CRB- and
beampattern-based designs. This gap comes from the CAML estimator. It is also
observed that the CRB-based design actually improves the estimation RMSE
around $4\textrm{ dB }$ compared to beampattern-based design.

Fig. 14 shows the estimation RMSE or root CRB of amplitude estimation versus
the transmit sequence length $L$ with $P=15\textrm{ dBm}$. It is observed that
there is almost no gap of CRB using $\frac{1}{L}\boldsymbol{XX}^{H}$ and
$\boldsymbol{S}_{x}$ with $L\geq 16$ while there is always around
$2~{}\textrm{dB}$ gap between the RMSE of amplitude estimation and CRB, which
is due to the CAML estimator.

Figure 11: Estimation RMSE or root CRB of angle estimation versus transmit
power $P$ with $L=64$ in Senario II.

Figure 12: Estimation RMSE or root CRB of angle estimation versus transmit
sequence length $L$ with $P=15\textrm{ dBm}$ in Senario II.

Figure 13: Estimation RMSE or root CRB of amplitude estimation versus transmit
power $P$ with $L=64$ in Senario II.

Figure 14: Estimation RMSE or root CRB of amplitude estimation versus transmit
sequence length $L$ with $P=15\textrm{ dBm}$ in Senario II.

##  VII Conclusion

We studied the fundamental C-R tradeoff in multi-antenna multi-target ISAC
over multicast channels. In particular, we considered two scenarios, namely
Scenario I and Scenario II, in which the BS does not and does have prior
knowledge about targets, respectively. For both scenarios, we characterized
the fundamental C-R tradeoff by solving multicast-rate-constrained CRB
minimization problems via optimizing the transmit covariance matrix. Next, we
considered joint information and sensing beamforming designs, by exploiting
dedicated sensing signal beams together with sensing interference
cancellation. Finally, we provided numerical results to reveal the C-R
tradeoff performance and practical estimation performance in our considered
scenarios.

There are several interesting research directions for future multicast ISAC
networks. One particularly promising direction is expanding the design to
multi-group multicast channels, where the presence of inter-group interference
cannot be overlooked. Next, extending our work to a multi-cell scenario is
another interesting direction, in which different BSs need to coordinate their
signals effectively to mitigate multi-cell interference. Moreover,
investigating multicast ISAC with other sensing tasks represents another
interesting aspect, as it involves employing diverse sensing metrics.
Additionally, regarding to secure multicast ISAC networks, taking into account
secrecy capacity, holds potential for future exploration. This specific area
requires further investigation to ensure robust and secure communications.

##  Appendix A Proof of Proposition 1

Let $\\{\mu_{k}\geq 0\\}$ and $\lambda\geq 0$ denote the dual variables
associated with the constraints in (22b) and (22b), respectively. Then, the
Lagrangian of problem (P1.1) is

$\displaystyle\mathcal{L}(\boldsymbol{S}_{x},\lambda,\\{\mu_{k}\\})=$
$\displaystyle\mathrm{tr}(\boldsymbol{S}_{x}^{-1})+\Gamma\sum_{k=1}^{K}\mu_{k}+\mathrm{tr}\big{(}(\lambda\boldsymbol{I}-\sum_{k=1}^{K}\mu_{k}\boldsymbol{h}_{k}\boldsymbol{h}_{k}^{H})\boldsymbol{S}_{x}\big{)}-\lambda
P.$ (55)

Accordingly, the dual function is

$\displaystyle g\big{(}\lambda,\\{\mu_{k}\\}\big{)}=$
$\displaystyle\underset{\boldsymbol{S}_{x}\succeq\boldsymbol{0}}{\min}$
$\displaystyle\mathcal{L}(\boldsymbol{S}_{x},\lambda,\\{\mu_{k}\\}),$ (56)

for which we have the following lemma.

######  Lemma 2.

For $g\big{(}\lambda,\\{\mu_{k}\\}\big{)}$ to be bounded from below, it must
hold that
$\boldsymbol{A}(\lambda,\\{\mu_{k}\\})\overset{\triangle}{=}\lambda\boldsymbol{I}-\sum_{k=1}^{K}\mu_{k}\boldsymbol{h}_{k}\boldsymbol{h}_{k}^{H}\succeq\boldsymbol{0}$.

###### Proof.

Suppose that $\boldsymbol{A}(\lambda,\\{\mu_{k}\\})\succeq\boldsymbol{0}$ does
not hold. Let $\hat{\alpha}<0$ denote any one negative eigenvalue of
$\boldsymbol{A}(\lambda,\\{\mu_{k}\\})$ and $\hat{\boldsymbol{u}}$ denote its
associated eigenvector. In this case, we can set
$\boldsymbol{S}_{x}=\rho\hat{\boldsymbol{u}}\hat{\boldsymbol{u}}^{H}$ with
$\rho\rightarrow\infty$ such that
$\mathcal{L}(\boldsymbol{S}_{x},\lambda,\\{\mu_{k}\\})\rightarrow-\infty$,
i.e., we have $g\big{(}\lambda,\\{\mu_{k}\\}\big{)}\rightarrow-\infty$. This
thus contradicts the fact that $g\big{(}\lambda,\\{\mu_{k}\\}\big{)}$ is
bounded from below. Therefore,
$\boldsymbol{A}(\lambda,\\{\mu_{k}\\})\succeq\boldsymbol{0}$ must hold true. ∎

Based on Lemma 2, the dual problem of (P1.1) is given by

$\displaystyle\textrm{(D1.1)}:\underset{\lambda,\\{\mu_{k}\\}}{\max}$
$\displaystyle~{}g\big{(}\lambda,\\{\mu_{k}\\}\big{)}$
$\displaystyle\mathrm{s.t.}$
$\displaystyle~{}\boldsymbol{A}(\lambda,\\{\mu_{k}\\})\succeq\boldsymbol{0},$
(57a) $\displaystyle\lambda\geq 0,\mu_{k}\geq 0,\forall k\in\mathcal{K}.$
(57b)

For notational convenience, let $\mathcal{D}$ denote the feasible region of
$\lambda$ and $\\{\mu_{k}\\}$ characterized by (57a) and (57b). Let
$\boldsymbol{S}_{x}^{*}$ denote the optimal solution to problem (56) with
given $(\lambda,\\{\mu_{k}\\})\in\mathcal{D}$. Furthermore, let
$\boldsymbol{S}_{x}^{\mathrm{opt}}$ denote the optimal primal solution to
problem (P1), and $\lambda^{\mathrm{opt}}$ and $\\{\mu_{k}^{\mathrm{opt}}\\}$
denote the optimal dual solution to problem (D1.1).

As the strong duality holds between problem (P1.1) and its dual problem
(D1.1), problem (P1.1) can be solved by equivalently solving dual problem
(D1.1). In the following, we first solve problem (56) with a given set
$(\lambda,\\{\mu_{k}\\})\in\mathcal{D}$, then find $\lambda^{\mathrm{opt}}$
and $\\{\mu_{k}^{\mathrm{opt}}\\}$ for problem (D1.1), and finally obtain the
optimal primary solution $\boldsymbol{S}_{x}^{\mathrm{opt}}$ to problem
(P1.1).

First, we evaluate the dual function $g\big{(}\lambda,\\{\mu_{k}\\}\big{)}$
with any given $(\lambda,\\{\mu_{k}\\})\in\mathcal{D}$. To this end, we
suppose that $\mathrm{rank}(\boldsymbol{A}(\lambda,\\{\mu_{k}\\}))=N\leq
N_{t}$ without loss of generality. Accordingly, we express the EVD of
$\boldsymbol{A}(\lambda,\\{\mu_{k}\\})$ as
$\boldsymbol{A}(\lambda,\\{\mu_{k}\\})=\boldsymbol{U}\boldsymbol{\Lambda}\boldsymbol{U}^{H}$,
where
$\boldsymbol{U}\boldsymbol{U}^{H}=\boldsymbol{U}^{H}\boldsymbol{U}=\boldsymbol{I}$,
and $\boldsymbol{\Lambda}=\mathrm{diag}(\alpha_{1},\dots,\alpha_{N_{t}})$ with
$\alpha_{1}\geq\dots\geq\alpha_{N_{t}}$ being the eigenvalues of
$\boldsymbol{A}(\lambda,\\{\mu_{k}\\})$. We then have the following
Lemma777Note that Lemma 3 provides the optimal solution to problem (56) for
any given $(\lambda,\\{\mu_{k}\\})\in\mathcal{D}$, for which
$\mathrm{rank}(\boldsymbol{A}(\lambda,\\{\mu_{k}\\})<N_{t}$ may hold such that
the eigenvalues of $\boldsymbol{S}_{x}^{*}$ can be unbounded. This is
different from Proposition 1, which considers the optimal dual variables
$(\lambda^{\mathrm{opt}},\\{\mu_{k}^{\mathrm{opt}}\\})$, for which
$\mathrm{rank}(\boldsymbol{A}(\lambda^{\mathrm{opt}},\\{\mu_{k}^{\mathrm{opt}}\\}))=N_{t}$
follows..

######  Lemma 3.

For any given $(\lambda,\\{\mu_{k}\\})\in\mathcal{D}$, we have the optimal
solution $\boldsymbol{S}_{x}^{*}$ to problem (56) as

$\boldsymbol{S}_{x}^{*}=\boldsymbol{U}{\boldsymbol{\Sigma}}\boldsymbol{U}^{H},$
(58)

where ${\boldsymbol{\Sigma}}=\mathrm{diag}(\tau_{1},\dots,\tau_{N_{t}})$ with
$\tau_{i}=\alpha_{i}^{-1/2}$ for $i\leq N$ and $\tau_{i}\rightarrow+\infty$
for $N<i\leq N_{t}$.

###### Proof.

Please refer to Appendix B. ∎

Based on Lemma 3, the dual function $g\big{(}\lambda,\\{\mu_{k}\\}\big{)}$ is
obtained.

Next, we find the optimal dual solution $\lambda^{\mathrm{opt}}$ and
$\\{\mu_{k}^{\mathrm{opt}}\\}$ to solve dual problem (D1.1). As problem (D1.1)
is always convex but non-differentiable in general, we adopt sub-gradient
based methods such as the ellipsoid method to find its optimal solution. The
basic idea of the ellipsoid method is to first generate an ellipsoid
containing $\lambda^{\mathrm{opt}}$ and $\\{\mu_{k}^{\mathrm{opt}}\\}$, and
then iteratively construct new ellipsoids containing these variables with
reduced volumes, until convergence [42]. To successfully implement the
ellipsoid method, we only need to find the sub-gradients of the objective and
constraint functions in problem (D1.1).

First, we consider the objective function of problem (D1.1), one sub-gradient
of which at any given
$[\mu_{1},\dots,\mu_{K},\lambda]^{T}\in\mathbb{C}^{(K+1)\times 1}$ is
$[\mathrm{tr}\big{(}\boldsymbol{H}_{1}\boldsymbol{S}_{x}^{*}\big{)}-\Gamma,\dots,\mathrm{tr}\big{(}\boldsymbol{H}_{K}\boldsymbol{S}_{x}^{*}\big{)}-\Gamma,P-\mathrm{tr}(\boldsymbol{S}_{x}^{*})]^{T}$.
Then, we consider the constraints in (57b). It is clear that the subgradient
for constraint $\mu_{k}\geq 0$ is $-\boldsymbol{e}_{k},\forall
k\in\mathcal{K}$, and the subgradient for $\lambda\geq 0$ is
$-\boldsymbol{e}_{K+1}$. Finally, we consider constraint
$\boldsymbol{A}(\lambda,\\{\mu_{k}\\})\succeq\boldsymbol{0}$ in (57a), whose
subgradient is given in the following lemma. For notational convenience, we
define $\boldsymbol{H}_{k}=\boldsymbol{h}_{k}\boldsymbol{h}_{k}^{H},\forall
k\in\mathcal{K}$.

######  Lemma 4.

Let $\boldsymbol{v}\in\mathbb{C}^{N_{t}\times 1}$ denote the eigenvector of
$\boldsymbol{A}(\lambda,\\{\mu_{k}\\})$ corresponding to its smallest
eigenvalue. The sub-gradient of
$\boldsymbol{A}(\lambda,\\{\mu_{k}\\})\succeq\boldsymbol{0}$ in (57a) at given
$\big{(}\lambda,\\{\mu_{k}\\}\big{)}\in\mathcal{D}$ is

$\begin{array}[]{c}[\boldsymbol{v}^{H}\boldsymbol{H}_{1}\boldsymbol{v},\dots,\boldsymbol{v}^{H}\boldsymbol{H}_{K}\boldsymbol{v},-1]^{T}.\end{array}$
(59)

###### Proof.

This lemma follows by noting that
$\boldsymbol{A}(\lambda,\\{\mu_{k}\\})\succeq\boldsymbol{0}\Longleftrightarrow\boldsymbol{v}^{H}\boldsymbol{A}(\lambda,\\{\mu_{k}\\})\boldsymbol{v}\geq
0.$ ∎

With the sub-gradients of the objective function and all the constraint
functions obtained, we can efficiently implement the ellipsoid method to
obtain the optimal dual solution $\lambda^{\mathrm{opt}}$ and
$\\{\mu_{k}^{\mathrm{opt}}\\}$.

Notice that at the optimal dual solution $\lambda^{\mathrm{opt}}$ and
$\\{\mu_{k}^{\mathrm{opt}}\\}$, it must follow that $\lambda^{\mathrm{opt}}>0$
and $\boldsymbol{A}(\lambda^{\mathrm{opt}},\\{\mu_{k}^{\mathrm{opt}}\\})$ is
of full rank (i.e.,
$\mathrm{rank}(\boldsymbol{A}(\lambda^{\mathrm{opt}},\\{\mu_{k}^{\mathrm{opt}}\\}))=N_{t}$),
since otherwise, the maximum transmit power constraint in (22b) cannot be
satisfied. Then, Proposition 1 follows directly from Lemma 3\. This completes
the proof.

##  Appendix B Proof of Lemma 3

First, we have
$\mathrm{tr}(\boldsymbol{A}(\lambda,\\{\mu_{k}\\})\boldsymbol{S}_{x})=\mathrm{tr}(\boldsymbol{\Lambda}\boldsymbol{U}^{H}\boldsymbol{S}_{x}\boldsymbol{U})$.
Let
$\tilde{\boldsymbol{S}_{x}}=\boldsymbol{U}^{H}\boldsymbol{S}_{x}\boldsymbol{U}$.
It is easy to figure out that
$\mathrm{tr}(\boldsymbol{S}_{x}^{-1})=\mathrm{tr}(\tilde{\boldsymbol{S}_{x}}^{-1})$.
Recall that $\boldsymbol{S}_{x}$ is positive semi-definite. We denote
$(\tau_{1},\dots,\tau_{N_{t}})$ as the diagonal entries of
$\tilde{\boldsymbol{S}_{x}}$ to be determined. Note that
$\mathrm{tr}(\boldsymbol{A}(\lambda,\\{\mu_{k}\\})\boldsymbol{S}_{x})=\sum_{i=1}^{N}\alpha_{i}\tau_{i}$.
Here, we introduce the following lemma to find the minimum of
$\mathrm{tr}(\tilde{\boldsymbol{S}_{x}}^{-1})$ w.r.t.
$(\tau_{1},\dots,\tau_{N_{t}})$, for which the proof can be found in [43,
Appendix A] and thus is omitted.

######  Lemma 5.

[43] For a positive semi-definite matrix
$\boldsymbol{B}_{0}\in\mathbb{C}^{M\times M}$, with $(m,n)$-th entry $a(m,n),$
it holds that

$\mathrm{tr}(\boldsymbol{B}_{0}^{-1})\geq\sum_{i=1}^{M}\frac{1}{a(i,i)},$ (60)

where the equality holds if and only if $\boldsymbol{B}_{0}$ is diagonal.

Hence, $\tilde{\boldsymbol{S}_{x}}$ must be diagonal and we obtain

$\displaystyle\mathrm{tr}(\boldsymbol{A}(\lambda,\\{\mu_{k}\\})\boldsymbol{S}_{x})+\mathrm{tr}(\boldsymbol{S}_{x}^{-1})$
$\displaystyle=$
$\displaystyle\mathrm{tr}(\boldsymbol{\Lambda}\tilde{\boldsymbol{S}_{x}})+\mathrm{tr}(\tilde{\boldsymbol{S}_{x}}^{-1})=\sum_{i=1}^{N}\alpha_{i}\tau_{i}+\sum_{i=1}^{N_{t}}\frac{1}{\tau_{i}}.$
(61)

In this case, when $N<N_{t},$ for $i>N$, $\tau_{i}$ must approach infinity to
achieve the minimum value 0, i.e., for $i>N$, $\tau_{i}\rightarrow+\infty$. As
for $i\leq N$, by checking the first order differentiation, we have
$\tau_{i}=\alpha_{i}^{-1/2}$. Then, we obtain $\boldsymbol{S}_{x}^{*}$ as

$\boldsymbol{S}_{x}^{*}=\boldsymbol{U}{\boldsymbol{\Sigma}}\boldsymbol{U}^{H},$
(62)

where ${\boldsymbol{\Sigma}}=\mathrm{diag}(\tau_{1},\dots,\tau_{N_{t}})$ with
$\tau_{i}\rightarrow+\infty$, for $i>N$, and $\tau_{i}=\alpha_{i}^{-1/2},$ for
$i\leq N$.

When $\boldsymbol{A}(\lambda,\\{\mu_{k}\\})$ is full-rank, we have

$\displaystyle\mathrm{tr}(\boldsymbol{A}(\lambda,\\{\mu_{k}\\})\boldsymbol{S}_{x})+\mathrm{tr}(\boldsymbol{S}_{x}^{-1})$
$\displaystyle=$
$\displaystyle\mathrm{tr}(\boldsymbol{\Lambda}\tilde{\boldsymbol{S}_{x}})+\mathrm{tr}(\tilde{\boldsymbol{S}_{x}}^{-1})=\sum_{i=1}^{N_{t}}\alpha_{i}\tau_{i}+\sum_{i=1}^{N_{t}}\frac{1}{\tau_{i}}.$
(63)

For any $i\leq N_{t}$, we need to set $\tau_{i}=\alpha_{i}^{-1/2}$ to achieve
the minimum value. Then, the $\boldsymbol{S}_{x}^{*}$ is given by
$\boldsymbol{S}_{x}^{*}=\boldsymbol{U}^{H}{\boldsymbol{\Sigma}}\boldsymbol{U},$
where
$\tilde{\boldsymbol{\Lambda}}=\mathrm{diag}(\tau_{1},\dots,\tau_{N_{t}})$ with
$\tau_{i}=\alpha_{i}^{-1/2},$ for $i\leq N_{t}$.

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