# Rate-Splitting Multiple Access for Multi-Antenna Joint Radar and
Communications

Chengcheng Xu, Bruno Clerckx, , Shiwa Chen, Yijie Mao, , and Jianyun Zhang
This paper was presented in part at 2020 IEEE International Conference on
Communications Workshops (ICC Workshops), Dublin, Ireland, June 2020
[1].Chengcheng Xu, Shiwa Chen and Jianyun Zhang are with the College of
Electronic Engineering, National University of Defense Technology, Hefei
230037, China (email<EMAIL_ADDRESS><EMAIL_ADDRESS>zjy.eei@163.com).Bruno Clerckx and Yijie Mao are with the Communications and
Signal Processing Group, Department of Electrical and Electronic Engineering,
Imperial College London, London SW7 2AZ, U.K. (email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Dual-Functional Radar-Communication (DFRC) system is an essential and
promising technique to achieve Integrated Sensing and Communication (ISAC) for
beyond 5G. In this work, we propose a powerful and unified multi-antenna DFRC
transmission framework, where an additional radar sequence is transmitted
apart from communication streams to enhance radar beampattern matching
capability, and Rate-Splitting Multiple Access (RSMA) is adopted to better
manage the interference. RSMA relies on multi-antenna Rate-Splitting (RS) with
Successive Interference Cancellation (SIC) receivers, and the split and
encoding of messages into common and private streams. We design the message
split and the precoders of the radar sequence, common and private streams so
as to jointly maximize the Weighted Sum Rate (WSR) and minimize the radar
beampattern approximation Mean Square Error (MSE) subject to the per antenna
power constraint. An iterative algorithm based on Alternating Direction Method
of Multipliers (ADMM) is developed to solve the problem. Numerical results
first show that RSMA-assisted DFRC achieves a better tradeoff between WSR and
beampattern approximation than Space-Division Multiple Access (SDMA)-assisted
DFRC with or without radar sequence, and other simpler radar-communication
strategies using orthogonal resources. We also show that the RSMA-assisted
DFRC frameworks with and without radar sequence achieve the same tradeoff
performance. This is because that the common stream is better exploited in the
proposed framework. The common stream of RSMA fulfils the triple function of
managing interference among communication users, managing interference between
communication and radar, and beampattern approximation. Uniquely, the SIC
receiver of RSMA is exploited for the dual purpose of managing interference
among communication users as well as interference between communication and
radar. Therefore, by enabling RSMA in DFRC, the system performance is enhanced
while the system architecture is simplified since there is no need to use
additional radar sequence and SIC. We conclude that RSMA is a more powerful
multiple access for DFRC.

###### Index Terms:

Dual-functional Radar-communication (DFRC), Rate-Splitting Multiple Access
(RSMA), Alternating Direction Method of Multipliers (ADMM), beampattern
design, Integrated Sensing and Communication (ISAC)

##  I Introduction

###  I-A Background and Literature Review

In 5G and beyond, many promising applications, such as autonomous vehicles,
Wi‐Fi sensing and extended reality have encouraged the Integration of Sensing
and Communication (ISAC). As a typical and reliable means of sensing, radar
has been widely used in both civil and military scenarios. Due to the scarcity
of the spectrum resources, the 4th and 5th generation wireless communication
systems are competing with radar applications in the S-band (2-4GHz), C-band
(4-8GHz) and possibly millimeter-wave (mmWave) band, which may result in
severe spectrum congestion and hamper the higher data rate demand in future
wireless communications[2]. As a long-term solution, communication and radar
spectrum sharing raises wide interests and shows a significant potential for
ISAC. There are two main research topics in the field of communication and
radar spectrum sharing, namely, 1) coexistence of existing radar and
communication devices, 2) Dual-Functional Radar-Communication (DFRC) system
design [3].

For coexistence of multi-antenna radar and communication devices, the authors
of [4] design waveforms of Multiple-Input Multiple-Output (MIMO) radar to
suppress the spectral energy leaked to the bands of communication devices,
while the authors of [5] design the precoders for the Multi-User MIMO (MU-
MIMO) communication to maximize the probing probability in a MIMO radar
coexisting scenario and ensure the Signal-to-Interference-plus-Noise-Ratio
(SINR) of downlink users. Instead of solely designing the multi-antenna radar
device or communication devices, [6] jointly designs the precoders of a MIMO
radar and the codebook of a MIMO communication system to maximize radar
effective SINR and ensure the average communication rate. [7] also jointly
designs the transmit waveforms and receiver filters of a MIMO radar and the
codebook of a MIMO communication system for the same purpose to achieve
coexistence. However, in the literature of the coexistence of radar and
communication devices, some important phenomena such as the cluster in radar
detection are not considered in the simplified system models [8]. Moreover,
governmental and military agencies would be unwilling to make major changes in
their radar deployments, which may impose huge costs on their financial
budgets [9].

Accounting for the possible drawbacks mentioned above, designing a DFRC system
that makes the best use of the spectrum for both detecting and communicating
is a promising alternative. Based on the waveform diversity of MIMO radar and
the concept of Space-Division Multiple Access (SDMA) in MIMO communication,
[10, 11] embed the information stream into radar pulses via a multi-antenna
platform to detect targets at the mainlobe and transmit information streams at
the sidelobe. The sidelobe level is modulated via Amplitude Shift Keying
(ASK), where different power levels correspond to different communication
symbols in [10]. The authors in [11] further develop Phase Shift Keying (PSK)
in this system by representing the symbols as different phases of the signals
received at the angle of the sidelobe. One significant restriction of such
dual-functional system is that the communication rate is limited by the Pulse
Repetition Frequency (PRF), which is far from satisfactory for communication
requirements. To overcome this problem, [12] proposes a multi-antenna DFRC
system that utilizes the communication streams for target detection,
simultaneously transmits probing signals to radar targets and serves multiple
downlink users. Specifically, the precoders are designed using the Zero-
Forcing (ZF) beamforming technique to form a desired radar beampattern and
meet the SINR requirements for communication users. Based on the transmission
structure of [12], the authors in [13] further investigate the tradeoff
between Weighted Sum-Rate (WSR) and probing power at target for the multi-
antenna DFRC, and show that a shared deployment with an integrated DFRC
transmission outperforms the separated deployment (which is a special case of
radar-communication coexistence). To the best of our knowledge, all existing
works in the realm of multi-antenna DFRC consider the use of SDMA. Managing
interference in DFRC can however be greatly improved by adopting more advanced
multiple access techniques.

Rate-Splitting Multiple Access (RSMA) has been recognized as a generalized and
powerful physical-layer transmission framework for multi-antenna multi-user
transmission networks that encompasses (and outperforms) SDMA and (power-
domain) Non-Orthogonal Multiple Access (NOMA) as sub-schemes [14, 15, 16, 17].
As a building block of RSMA, Rate-Splitting (RS) splits the messages of users
into private and common parts, jointly encodes the common parts into common
streams to be decoded by multiple users and independently encodes the private
parts into the private streams to be decoded by the corresponding users only,
which enables to partially decode the interference and partially treat the
remaining interference as noise. This significantly contrasts with SDMA that
fully treats interference as noise and NOMA that fully decodes the
interference. RSMA has been shown to achieve higher spectral and energy
efficiencies than SDMA, NOMA, and Orthogonal Multiple Access (OMA) in multi-
group multicast [18], massive MIMO [19], millimeter-wave systems[20],
Simultaneous Wireless Information and Power Transfer (SWIPT)[21], etc. We are
thus motivated to enable RSMA in DFRC to better manage the interference
between radar and communication functions and also among communication users.

###  I-B Contributions

In this paper, we propose a RSMA-assisted multi-antenna DFRC system that
simultaneously communicates with downlink users as a Base Station (BS) and
probes targets with a desired beampattern as a collocated MIMO radar within
the same frequency band. The major contributions are summarized as follows:

####  I-B1 We propose a novel multi-antenna DFRC architecture built upon RSMA

We exploit the powerful RSMA framework to further manage the interference
between radar and communication functions as well as among communication users
in multi-antenna DFRC. Specifically, the messages are split into the common
and private parts and encoded respectively into the common stream, which is
decoded by all users, and private streams, which are decoded by the
corresponding users only. The common stream and private streams are precoded
and transmit as dual-functional signals. By utilizing the common stream to
partially decode the interference, RSMA achieves a significant advantage in
DFRC to manage the interference between the dual functions and among
communication users. To the best of our knowledge, this is the first work that
enables RSMA in DFRC.

####  I-B2 We propose to enable the radar sequence apart from the
communication streams in DFRC

To improve the beampattern matching capability, we enable an additional radar
sequence which is precoded and transmitted together with communication streams
in the proposed multi-antenna DFRC. To the best of our knowledge, this is the
first work to consider joint transmission of radar and communication signals
in multi-antenna DFRC. We show that by using radar sequence, the tradeoff
performance of conventional SDMA-assisted DFRC is improved. It is also shown
that in RSMA-assisted DFRC, enabling and disabling radar sequence share the
same dual-functional performance, which is explicitly much better than that of
SDMA-assisted DFRC with radar sequence. This reveals that the common stream of
RSMA not only has the same functions as the radar sequence in achieving
desired beampattern and managing interference between radar and communication,
but also has an advantage of managing the interference among communication
users. Therefore, there is no need to use an additional radar sequence in
RSMA-assisted DFRC.

####  I-B3 We design an Alternating Direction Method of Multipliers
(ADMM)-based optimization framework to solve the joint WSR maximization and
Mean Square Error (MSE) of beampattern approximation minimization problem in
DFRC

The proposed DFRC is designed via solving a highly non-convex and complicated
optimization problem whose objective function is a combination of maximizing
the logarithmic WSR and minimizing the quartic MSE. In order to solve this
problem, we propose an iterative framework based on ADMM that transforms the
intractable problem into tractable iterative sub-problems. To solve the non-
convex sub-problems in each iteration of ADMM, we devise an algorithm based on
Majorization-Minimization (MM). We show that the proposed framework solves the
non-convex optimization problem efficiently.

####  I-B4 We present SDMA-assisted DFRC, Time-Division Radar-Communication
(TDRC) and Frequency-Division Radar-Communication (FDRC) as baselines for a
thorough comparison

First, to illustrate the advantages of integrated DFRC transmission using
dual-functional signals, we introduce two practically simpler radar-
communication strategies that operate the radar and communication functions
via orthogonal resources in time and frequency domain, namely, TDRC and FDRC.
Second, we develop the SDMA-assisted DFRC by turning off the common stream as
baselines. The proposed DFRC has an advantage of being able to realizing dual
functions within the same time-frequency resources compared with TDRC and
FDRC. We show that the proposed RSMA-assisted DFRC outperforms SDMA-assisted
DFRC as well as FDRC, and in certain conditions surpasses TDRC.

###  I-C Organization

The rest of the paper is organized as follows. The system model of the
proposed DFRC is illustrated in Section II. In Section III, the metrics used
are described, and the optimization problems are formulated. The ADMM-based
iterative framework is specified in Section IV, and the algorithms for solving
the updates in each ADMM iteration are illustrated in Section V. Section VI
demonstrates the simulation results. Section VII concludes the paper.

###  I-D Notation

Boldface upper-case and lower-case letters denote matrices and column vectors,
respectively. The scalars are denoted by italic letters. Scalar $x_{i}$
denotes the $i$th entry of the vector $\bf x$. ${\bf I}_{N}$ is the
$N$-dimensional identity matrix, ${\bf 0}^{N\times K}$ is an all-zero matrix
with dimension $N\times K$, and ${\bf 1}^{N\times K}$ is an all-one matrix
with dimension $N\times K$. $(\cdot)^{T}$, $(\cdot)^{H}$, $(\cdot)^{-1}$ and
$(\cdot)^{{\dagger}}$ denote transpose, conjugate-transpose, inverse and
pseudo inverse, respectively. $\lvert\cdot\lvert$ and $\lVert\cdot\lVert_{2}$
refer to the absolute value of a scalar and the 2-norm of a vector. The trace
is $\text{tr}(\cdot)$, $\mathfrak{R}\\{\cdot\\}$ and $\mathfrak{J}\\{\cdot\\}$
are the real and conjugate part of a complex vector, respectively. Vector
$\text{vec}({\bf X})$ is constructed by stacking the columns of ${\bf X}$. The
vector $\text{diag}({\bf X})$ is the diagonal entries of ${\bf X}$.
$\lambda_{\text{max}}({\bf X})$ and $\lambda_{\text{min}}({\bf X})$
respectively represent the largest and smallest eigenvalue of ${\bf X}$.
Notation ${\bf A}\succeq{\bf B}$ stands for matrix ${\bf A}-{\bf B}$ is
positive semidefinite.

##  II System Model

In this work, we focus on a novel multi-antenna DFRC that supports ISAC.
Specifically, the transmitter is equipped with $N_{\text{t}}$ antennas, aiming
at serving $K$ single-antenna downlink communication users as a BS and
detecting multiple radar targets in the azimuth angles of interests as a
collocated MIMO radar. For simplicity of exposure, we assume the transmitter
adopts a Uniform Linear Array (ULA). The communication users are indexed by
$\mathcal{K}=\\{1,\dots,K\\}$. In the following subsections, we will specify
the proposed DFRC system model with RSMA and radar sequence enabled, followed
by the three baseline schemes, namely, SDMA-assisted DFRC, TDRC, FDRC.

###  II-A Proposed RSMA-assisted DFRC

Figure 1: System architecture of the proposed RSMA-assisted DFRC

As is shown in Fig. 1, the proposed DFRC enables the simultaneous transmission
of the precoded information streams and the radar sequence to fulfil both
radar detection and communication functions. RSMA proposed in [14] is enabled
at the transmitter to process the messages intended for the communication
users. Specifically, the message $W_{k}$ of the $k$th user is split into a
common part $W_{\text{c},k}$ and a private part $W_{\text{p},k}$, $\forall
k\in\mathcal{K}$. The common parts of all users
$\\{W_{c,1},\dots,W_{\text{c},k}\\}$ are jointly encoded into the common
stream $s_{\text{c}}$, while the private parts
$\\{W_{p,1},\dots,W_{\text{p},k}\\}$ are respectively encoded into private
streams $\\{s_{1},\dots,s_{K}\\}$. Then the data streams as well as the
transmit radar sequence, which are denoted together as ${\bf
s}=[s_{\text{c}},s_{1},\dots,s_{k},s_{\text{r}}]^{T}\in\mathbb{C}^{K+2}$, are
linearly precoded before transmission using the precoder ${\bf
P}=[{\bf{p}}_{\text{c}},{\bf p}_{1},\dots,{\bf{p}}_{K},{\bf{p}}_{\text{r}}]$,
where ${\bf{p}}_{\text{c}}\in\mathbb{C}^{N_{t}}$ is the precoder of the common
stream and ${\bf p}_{\text{r}}\in\mathbb{C}^{N_{\text{t}}}$ is the precoder of
the radar sequence. We denote the power of the common stream as
$P_{\text{c}}=\lVert{\bf p}_{\text{c}}\lVert_{2}$ and that of the radar
sequence as $P_{\text{r}}=\lVert{\bf p}_{\text{r}}\lVert_{2}$. The baseband
transmit signal can be fully expressed as

${\bf x}=\underbrace{\overbrace{{\bf p}_{\text{c}}s_{\text{c}}}^{\text{Common
stream}}+\overbrace{\sum_{k\in\mathcal{K}}{\bf p}_{k}{s}_{k}}^{\text{Private
streams}}}_{\text{Communication streams}}+\underbrace{{\bf
p}_{\text{r}}s_{\text{r}}}_{\text{Radar sequence}}.$ (1)

Note that the transmit radar sequence $s_{\text{r}}$ contains no useful
messages for users and thus is not random. To improve detection performance,
the waveform of the radar sequence is usually designed and determined
previously before transmission. We set aside the radar sequence waveform
design problem as future work and focus on the precoder design. We assume that
${\bf s}$ follows $\mathbb{E}\\{{\bf s}{\bf s}^{H}\\}={\bf I}_{(K+2)}$. The
total transmit power budget is $P_{\text{t}}$, i.e., $\text{tr}({\bf P}{\bf
P}^{H})\leq P_{\text{t}}$.

Based on (1), the received signal at the $k$th downlink communication user is

$\displaystyle y_{k}=$ $\displaystyle{\bf h}_{k}^{H}{\bf x}+n_{k}$ (2)
$\displaystyle=$ $\displaystyle{\bf h}_{k}^{H}{\bf
p}_{\text{r}}s_{\text{r}}+{\bf h}_{k}^{H}{\bf p}_{\text{c}}s_{\text{c}}+{\bf
h}_{k}^{H}\sum_{j\in\mathcal{K}}{\bf p}_{j}{s}_{j}+n_{k}$

where ${\bf h}_{k}\in\mathbb{C}^{{N_{\text{t}}}\times 1}$ is the channel
vector between the DFRC transmitter and the $k$th user, which is assumed to be
perfectly known at the transmitter and the users. Received noise $n_{k}$ at
the $k$th user is modelled as a complex Gaussian random variable with zero
mean and variance $\sigma_{n,k}^{2}$. Without loss of generality, we assume
the noise variances of all users equal to 1, i.e., $\sigma_{n,k}^{2}=1,\forall
k\in\mathcal{K}$.

Depending on whether to utilize SIC to cancel the radar sequence interference
at users, there are different radar sequence modes of the proposed DFRC, which
lead to different metric expressions in the subsequent problem formulation.

###  II-B Different Modes of Radar Sequence

By introducing $\delta_{\text{c}}$ as an index variable, all three possible
modes of the radar sequence are specified in this part.

####  II-B1 Radar sequence is disabled

In this mode, the radar sequence is completely turned off in the proposed DFRC
transmission framework. This is equivalent to allocating zero power to ${\bf
p}_{\text{r}}$ in all above expressions, i.e., $P_{\text{r}}=0$. In the
following metrics and problem formulation, we specify $\delta_{c}=0$ and
$P_{\text{r}}=0$ to represent this mode. We define this mode as DFRC without
radar sequence in the following parts.

####  II-B2 Radar sequence is enabled with SIC

The DFRC system transmits the radar sequence in this mode. However, since the
radar sequence does not contain messages, we assume that $s_{\text{r}}$ is
pre-stored and known to the users. Therefore, its interference is successfully
canceled via SIC before information streams are decoded. We let
$\delta_{\text{c}}=0$ to denote this mode and define it as DFRC with radar
sequence and SIC.

####  II-B3 Radar sequence is enabled without SIC

For easier implementation, SIC is not applied to $s_{\text{r}}$, which means
the radar sequence will be treated as interference at users when information
streams are decoded. We use $\delta_{\text{c}}=1$ to denote this mode in the
following parts and define it as DFRC with radar sequence.

###  II-C SDMA-assisted DFRC Baselines

In order to show the advantages of RSMA, we also study DFRC enabling SDMA
based on Multi-User Linear Precoding (MU-LP) [14] as a baseline. SDMA-assisted
DFRC can be realised by disabling the common stream in (1) and (2), i.e.,
letting $P_{\text{c}}=0$. However, in the proposed DFRC system, switching
between RSMA and SDMA does not influence the working mode of the radar
sequence.

To be clear, we list variable settings for all possible radar sequence modes
and multiple access methods in Table I.

TABLE I: Variable Settings For Different Combinations of Multiple Access Techniques and Radar Sequence Modes in the Proposed DFRC | RSMA-assisted DFRC | SDMA-assisted DFRC
---|---|---
| Radar sequence
---
disabled
$P_{\text{r}}=0$, $\delta_{\text{c}}=0$ |  |  $P_{\text{c}}=0$, $P_{\text{r}}=0$,
---
$\delta_{\text{c}}=0$
| Radar sequence
---
enabled with SIC
$\delta_{\text{c}}=0$ |  $P_{\text{c}}=0$, $\delta_{\text{c}}=0$
| Radar sequence
---
enabled without SIC
$\delta_{\text{c}}=1$ |  $P_{\text{c}}=0$, $\delta_{\text{c}}=1$

###  II-D TDRC and FDRC Baselines

To show the advantages of the proposed DFRC transmission architecture, we
compare it with practically simpler radar-communication strategies using
orthogonal resources in time and frequency domain, i.e., TDRC and FDRC. Both
strategies are specified as follows. We denote the power budget of radar and
communication functions as $P_{\text{R}},P_{\text{C}}$, respectively.

TABLE II: Resources Used in the Proposed DFRC, TDRC and FDRC  | Time Allocation |  Within the Same Frequency Band |  Power Budget for Each Function
---|---|---|---
Comms | Radar
Proposed DFRC | 1 | 1 | Yes | $P_{\text{R}}=P_{\text{C}}=P_{\text{t}}$
TDRC | $\alpha$ | $1-\alpha$ | Yes | $P_{\text{R}}=P_{\text{C}}=P_{\text{t}}$
FDRC | 1 | 1 | No | $P_{\text{R}}+P_{\text{C}}=P_{\text{t}}$

####  II-D1 Time-Division Radar-Communication

The transmitter of the TDRC baseline spends a fraction $\alpha$ of time purely
working as a $N_{\text{t}}$-antenna BS and $1-\alpha$ of time working as a
$N_{\text{t}}$-antenna MIMO radar within the same frequency band. The working
power budget is $P_{\text{t}}$ for either function duration. To ensure the
fairness between TDRC and the proposed RSMA-assisted DFRC, we assume that RSMA
is enabled when the transmitter acts as a BS. The value of $\alpha$ depends on
real task demands. Since different $\alpha$ leads to different tradeoffs, we
only adopt representative values of $\alpha$ as baselines in numerical
experiments for an explicit comparison.

####  II-D2 Frequency-Division Radar-Communication

The transmitter of the FDRC simultaneously transmits precoded communication
streams and radar probing signals within different frequency bands. Since FDRC
operates radar and communication functions both for the whole time duration,
the total power budget is split for both functions, i.e.,
$P_{\text{R}}+P_{\text{C}}=P_{\text{t}}$. There is thus no interference
between the radar and communication functions because of the frequency
orthogonality. RSMA is enabled for the transmission of the communication
streams.

The resource allocation for the proposed DFRC, TDRC and FDRC are compared in
Table II. It is obvious that DFRC has an advantage of simultaneously
fulfilling dual functions within the same frequency band.

##  III Metrics and problem formulation

In this section, we introduce two classic metrics, i.e., WSR for communication
and MSE of beampattern approximation for radar. These two metrics are used to
formulate the optimization problem of DFRC transmission design.

###  III-A Metric for Communication: WSR

To evaluate the communication performance of DFRC, we use WSR in our work,
since this is the most representative performance metric of modern multiuser
communication systems.

According to the RSMA-assisted DFRC strategy proposed in Subsection II-A and
different modes of radar sequence defined in Subsection II-B, the respective
SINRs of decoding $s_{\text{c}}$ and $s_{k}$ at the $k$th user are

$\displaystyle\gamma_{\text{c},k}({\bf P},\delta_{\text{c}})=$
$\displaystyle\frac{\left|{\bf h}_{k}^{H}{\bf
p}_{\text{c}}\right|^{2}}{\sum_{j\in\mathcal{K}}\left|{\bf h}_{k}^{H}{\bf
p}_{j}\right|^{2}+\delta_{\text{c}}\left|{\bf h}_{k}^{H}{\bf
p}_{\text{r}}\right|^{2}+1},\forall k\in\mathcal{K},$ (3)
$\displaystyle\gamma_{k}({\bf P},\delta_{\text{c}})=$
$\displaystyle\frac{\left|{\bf h}_{k}^{H}{\bf
p}_{k}\right|^{2}}{\sum_{j\in\mathcal{K},j\neq k}\left|{\bf h}_{k}^{H}{\bf
p}_{j}\right|^{2}+\delta_{\text{c}}\left|{\bf h}_{k}^{H}{\bf
p}_{\text{r}}\right|^{2}+1},\forall k\in\mathcal{K},$ (4)

where the index variable $\delta_{\text{c}}$ is used to unify the expressions
for different radar sequence processing modes. Specifically, when the radar
sequence is enabled and is not removed via SIC, i.e., $\delta_{\text{c}}=1$,
it is fully treated as interference when each user-$k$ decodes $s_{\text{c}}$
and $s_{k}$. Otherwise, when $\delta_{c}=0$, it is fully removed from the
received signal at each user. Based on (3) and (4), the achievable rates of
$s_{\text{c}}$ and $s_{k}$ at the $k$th user are

$\displaystyle R_{\text{c},k}({\bf P},\delta_{\text{c}})=$
$\displaystyle\log_{2}\left(1+\gamma_{\text{c},k}({\bf
P},\delta_{\text{c}})\right),$ (5) $\displaystyle R_{k}({\bf
P},\delta_{\text{c}})=$ $\displaystyle\log_{2}\left(1+\gamma_{k}({\bf
P},\delta_{\text{c}})\right).$ (6)

To ensure that the common stream is successfully decoded by all users, the
rate of decoding the common stream $s_{\text{c}}$ should not exceed
$R_{\text{c}}({\bf P},\delta_{\text{c}})=\min\left\\{R_{\text{c},1}({\bf
P},\delta_{\text{c}}),\dots,R_{\text{c},k}({\bf
P},\delta_{\text{c}})\right\\}$[14]. As $R_{\text{c}}({\bf
P},\delta_{\text{c}})$ is shared by $K$ users, we have
$\sum_{k\in\mathcal{K}}C_{k}=R_{\text{c}}({\bf P},\delta_{\text{c}})$, where
$C_{k}$ is the portion of common rate at user-$k$ transmitting
$W_{\text{c},k}$. The total achievable rate of user-$k$ contains the portion
of common rate transmitting $W_{\text{c},k}$ and the private rate transmitting
$W_{\text{p},k}$. Thus, given the rate weight allocated to user-$k$ as
$\mu_{k}$, the WSR for the proposed RSMA-assisted DFRC system is

${\text{WSR}({\bf
P},\delta_{\text{c}})}=\sum_{k\in\mathcal{K}}\mu_{k}\left(C_{k}+R_{k}({\bf
P},\delta_{\text{c}})\right).$ (7)

For SDMA-assisted DFRC, the corresponding WSR expression is obtained from (7)
by turning off $C_{k},\forall k\in\mathcal{K}$, which is the WSR of the
private streams only.

###  III-B Metric for Radar: MSE of Beampattern Approximation

It is well-known that to get higher Signal-to-Noise-Ratio (SNR) for target
tracking or narrow-beam scanning, a specially desired rather than a nominally
omnidirectional MIMO radar beampattern is expected to be achieved via
correlated waveform design [22, 23]. Hence, we focus on the transmit
beampattern design to represent radar performance in this work. According to
existing literature [22, 23], beampattern design is generally formulated into
a desired beampattern approximation problem via least squares. Based on the
metric of these works, we introduce MSE for beampattern approximation as
$\sum_{m=1}^{M}\left|P_{\text{d}}(\theta_{m})-{\bf a}^{H}(\theta_{m}){\bf
R}_{\bf x}{\bf a}(\theta_{m})\right|^{2},$ where $\theta_{m}$ is the $m$th
azimuth angle grid among all $M$ grids, $P_{\text{d}}(\theta_{m})$ is the
desired beampattern level at $\theta_{m}$, $\bf R_{x}$ is the covariance
matrix of transmit waveforms, ${\bf
a}(\theta_{m})=[1,e^{\text{j}2\pi\Delta\sin(\theta_{m})},\dots,e^{\text{j}2\pi(N_{\text{t}}-1)\Delta\sin(\theta_{m})}]^{T}\in\mathbb{C}^{N_{\text{t}}\times
1}$ is the transmit steering vector for ULA and $\Delta$ is the normalized
distance (relative to wavelength) between adjacent array elements. To improve
robustness, $P_{\text{d}}(\theta_{m})$ is assumed to be generated previously
via classical beampattern synthesis approaches such as in [23] according to
the radar prior knowledge including azimuth angles of interest and the array
setting.

For the transmission of our proposed DFRC system, we thus have the beampattern
approximation MSE as

$\text{MSE}({\bf P})=\sum_{m=1}^{M}\left|P_{\text{d}}(\theta_{m})-{\bf
a}^{H}(\theta_{m}){\bf P}{\bf P}^{H}{\bf a}(\theta_{m})\right|^{2}.$ (8)

When $P_{\text{c}}=0$, (8) is the MSE for SDMA-assisted DFRC. Likewise, by
forcing $P_{\text{r}}=0$, (8) is the MSE when radar sequence is disabled.

###  III-C Problem Formulation

In this work, we focus on jointly designing the message split, precoders of
communication streams and radar sequence with the aim of jointly maximizing
WSR for communication users and minimizing MSE for radar beampattern
approximation. To keep the tradeoff between WSR and MSE, a regularization
parameter $\lambda\in[0,1]$ is introduced to combine WSR and MSE in the
objective function. The optimization problem of our proposed RSMA-assisted
DFRC strategy is formulated as

$\displaystyle\max\limits_{{\bf c},{\bf P}}\quad$
$\displaystyle(1-\lambda)\sum_{k\in\mathcal{K}}\mu_{k}\left(C_{k}+R_{k}({\bf
P},\delta_{\text{c}})\right)$
$\displaystyle-\lambda\sum_{m=1}^{M}\left|P_{\text{d}}(\theta_{m})-{\bf
a}^{H}(\theta_{m}){\bf P}{\bf P}^{H}{\bf a}(\theta_{m})\right|^{2}$ (9a)
$\displaystyle s.t.\quad$
$\displaystyle\sum_{k^{\prime}\in\mathcal{K}}C_{k^{\prime}}\leq
R_{\text{c},k}({\bf P},\delta_{\text{c}}),\forall k\in\mathcal{K}$ (9b)
$\displaystyle{\bf c}\geq 0$ (9c) $\displaystyle\text{diag}({\bf P}{\bf
P}^{H})=\frac{P_{\text{t}}{\bf 1}^{N_{\text{t}}\times 1}}{N_{\text{t}}},$ (9d)

where ${\bf c}=[C_{1},C_{2},\dots,C_{K}]^{T}$. (9b) ensures the common stream
is successfully decoded at each user. (9d) ensures the transmit average power
of each antenna to be the same, which is commonly adopted for MIMO radar
beampattern design problem [22, 23].

Note that by applying the variable settings in Table I to (9), we obtain the
optimization problems of SDMA-assisted baseline schemes with different radar
sequence modes.

##  IV ADMM-based Framework for Solving the Problem

The combination of logarithmic WSR and quartic MSE of beampattern
approximation in the objective function is non-convex. The equality power
constraint is also non-convex and intractable. Since (9) combines both the
objective of maximizing the WSR for communication users and minimizing the MSE
for radar beampattern approximation, our idea is to decompose the problem into
two communication and radar sub-problems, which are then solved alternatively
until a local optimal solution is obtained.

To give an explicit expression of the framework, we first denote a new vector
that contains all optimization variables in (9), i.e., ${\bf v}=[{\bf
c}^{T},{\bf
p}_{\text{vec}}^{T}]^{T}\in\mathbb{R}_{+}^{K}\times\mathbb{C}^{N_{\text{t}}(K+2)}$,
with ${\bf p}_{\text{vec}}=\text{vec}({\bf P})$. We then define selection
matrices as ${\bf D}_{\text{p}}=\left[{\bf 0}^{(K+2)N_{\text{t}}\times K},{\bf
I}_{(K+2)N_{\text{t}}}\right]$, ${\bf D}_{\text{c}}=\left[{\bf
0}^{N_{\text{t}}\times K},{\bf I}_{N_{\text{t}}},{\bf
0}^{N_{\text{t}}\times(K+1)N_{\text{t}}}\right]$, ${\bf
D}_{\text{r}}=\left[{\bf 0}^{N_{\text{t}}\times[K+(K+1)N_{\text{t}}]},{\bf
I}_{N_{\text{t}}}\right]$, ${\bf D}_{k}=\left[{\bf
0}^{N_{\text{t}}\times(K+kN_{\text{t}})},{\bf I}_{N_{\text{t}}}\quad{\bf
0}^{N_{\text{t}}\times(K+1-k)N_{\text{t}}}\right]$. We also define selection
vectors ${\bf e}_{k}=[{\bf 0}^{1\times(k-1)},1,{\bf
0}^{1\times[(K+2)N_{\text{t}}+K-k]}]^{T}$, such that we have $C_{k}={\bf
e}_{k}^{T}{\bf v}$.

To unify the expressions, when the radar sequence mode is determined, we
denote the rates at user-$k$ as $\eta^{\delta_{\text{c}}}_{\text{c},k}({\bf
p}_{\text{vec}})=\eta^{\delta_{\text{c}}}_{\text{c},k}({\bf D}_{\text{p}}{\bf
v})=R_{\text{c},k}({\bf P},\delta_{\text{c}})$,
$\eta^{\delta_{\text{c}}}_{k}({\bf
p}_{\text{vec}})=\eta^{\delta_{\text{c}}}_{k}({\bf D}_{\text{p}}{\bf
v})=R_{k}({\bf P},\delta_{\text{c}}),$ and rewrite (9) into a tractable ADMM
formulation as

$\begin{split}\min\limits_{{\bf v},{\bf u}}\quad&f_{\text{c}}({\bf
v})+g_{\text{c}}({\bf v})+f_{\text{r}}({\bf u})+g_{\text{r}}({\bf u})\\\
s.t.\quad&{\bf D}_{\text{p}}({\bf v}-{\bf u})=0,\end{split}$ (10)

where $f_{\text{c}}({\bf
v})=-(1-\lambda)\sum_{k\in\mathcal{K}}\mu_{k}\left({\bf e}_{k}^{T}{\bf
v}+\eta^{\delta_{\text{c}}}_{k}({\bf D}_{\text{p}}{\bf v})\right)$.
$g_{\text{c}}({\bf v})$ is the indicator function of the communication
feasible set $\mathcal{C}=\left\\{{\bf v}\bigg{|}\sum_{k\in\mathcal{K}}{\bf
e}_{k}^{T}{\bf v}\leq\eta^{\delta_{\text{c}}}_{\text{c},k}({\bf
D}_{\text{p}}{\bf v}),{\bf v}^{H}{\bf D}_{\text{p}}^{H}{\bf D}_{\text{p}}{\bf
v}\leq P_{\text{t}}\right\\}$, where the first condition corresponds to (9b).
The second condition is added as a power constraint only to improve robustness
of convergence, which does not change the original problem as it is always met
if (9d) holds. ${\bf
u}\in\mathbb{R}_{+}^{K}\times\mathbb{C}^{N_{\text{t}}(K+2)}$ is introduced as
a new vector variable. $f_{\text{r}}({\bf
u})=\lambda\sum_{m=1}^{M}|P_{\text{d}}(\theta_{m})-{\bf
a}^{H}(\theta_{m})\big{(}{\bf D}_{\text{c}}{\bf u}{{\bf u}}^{H}{\bf
D}_{\text{c}}^{H}+\sum_{k\in\mathcal{K}}{\bf D}_{k}{\bf u}{{\bf u}}^{H}{\bf
D}_{k}^{H}+{\bf D}_{\text{r}}{\bf u}{{\bf u}}^{H}{\bf
D}_{\text{r}}^{H}\big{)}{\bf a}(\theta_{m})|^{2}$, and $g_{\text{r}}({\bf u})$
is the indicator function of radar feasible set $\mathcal{R}=\left\\{{\bf
u}\bigg{|}\text{diag}({\bf D}_{\text{c}}{\bf u}{{\bf u}}^{H}{\bf
D}_{\text{c}}^{H}+\sum_{k\in\mathcal{K}}{\bf D}_{k}{\bf u}{{\bf u}}^{H}{\bf
D}_{k}^{H}\right.$ $\left.+{\bf D}_{\text{r}}{\bf u}{{\bf u}}^{H}{\bf
D}_{\text{r}}^{H})=\frac{P_{\text{t}}{\bf 1}^{N_{\text{t}}\times
1}}{N_{\text{t}}}\right\\}$ corresponding to (9d).

According to [24], (10) can be solved by iteratively updating ${\bf v},{\bf
u},{\bf d}$, which is given as

$\displaystyle{\bf v}^{t+1}:=$ $\displaystyle\arg\min\limits_{{\bf
v}}\big{(}f_{\text{c}}({\bf v})+g_{\text{c}}({\bf v})$
$\displaystyle+(\rho/2)\lVert{\bf D}_{\text{p}}({\bf v}-{\bf u}^{t})+{\bf
d}^{t}\rVert_{2}^{2}\big{)},$ (11) $\displaystyle{\bf u}^{t+1}:=$
$\displaystyle\arg\min\limits_{{\bf u}}\big{(}f_{\text{r}}({\bf
u})+g_{\text{r}}({\bf u})$ $\displaystyle+(\rho/2)\lVert{\bf
D}_{\text{p}}({\bf v}^{t+1}-{\bf u})+{\bf d}^{t}\rVert_{2}^{2}\big{)},$ (12)
$\displaystyle{\bf d}^{t+1}:=$ $\displaystyle{\bf d}^{t}+{\bf
D}_{\text{p}}({\bf v}^{t+1}-{\bf u}^{t+1}),$ (13)

where ${\bf d}\in\mathbb{C}^{N_{\text{t}}(K+2)}$ is the scaled dual variable.
It is worth clarifying that (11) – (13) can be rewritten into real-value
formations following the rules in [25] in order to rigorously meet the real-
value definition of ADMM [24].

The proposed ADMM-based framework is summarized in Algorithm 1, where ${\bf
r}^{t+1}$ and ${\bf q}^{t+1}$ are the primal and dual residuals, and
$\epsilon_{0}$ is the feasibility tolerance. Based on the proposed framework,
we are able to solve the intractable problem (9) through updating ${\bf
v},{\bf u},{\bf d}$, respectively.

The update of $\bf{v}$ ($\bf v$-update) in (11) is achieved by using the
Weighted Minimum Mean Square Error (WMMSE) algorithm while the update of
$\bf{u}$ ($\bf u$-update) in (12) is based on MM algorithm. Both algorithms
will be specified in Section V followed by the computational complexity
analysis of the entire ADMM framework.

Input: $t\leftarrow 0,{\bf v}^{t},{\bf u}^{t},{\bf d}^{t}$

1repeat

2 Update ${\bf v}^{t+1}\leftarrow\arg\min\limits_{{\bf
v}}\big{(}f_{\text{c}}({\bf v})+g_{\text{c}}({\bf v})+(\rho/2)\lVert{\bf
D}_{\text{p}}({\bf v}-{\bf u}^{t})+{\bf d}^{t}\rVert_{2}^{2}\big{)}$ via WMMSE
algorithm;

3 Update ${\bf u}^{t+1}\leftarrow\arg\min\limits_{{\bf
u}}\big{(}f_{\text{r}}({\bf u})+g_{\text{r}}({\bf u})+(\rho/2)\lVert{\bf
D}_{\text{p}}({\bf v}^{t+1}-{\bf u})+{\bf d}^{t}\rVert_{2}^{2}\big{)}$ via MM-
based algorithm;

4 Update ${\bf d}^{t+1}\leftarrow{\bf d}^{t}+{\bf D}_{\text{p}}({\bf
v}^{t+1}-{\bf u}^{t+1})$;

5 ${\bf r}^{t+1}={\bf D}_{\text{p}}({\bf v}^{t+1}-{\bf u}^{t+1})$;

6 ${\bf q}^{t+1}={\bf D}_{\text{p}}({\bf u}^{t+1}-{\bf u}^{t})$;

7 t++;

8until  _$\lVert{\bf r}^{t+1}\lVert_{2}\leq\epsilon_{0}$ and $\lVert{\bf
q}^{t+1}\lVert_{2}\leq\epsilon_{0}$_;

Algorithm 1 ADMM-based Framework

##  V Algorithms for solving ADMM updates

The ${\bf v}$-update and ${\bf u}$-update in the ADMM framework are still
challenging. Thus, we adopt the WMMSE algorithm for ${\bf v}$-update and MM-
based algorithm for ${\bf u}$-update in this section.

###  V-A WMMSE Algorithm for $\bf v$-update

In this subsection, we illustrate the procedure of solving the ${\bf
v}$-update in each ADMM iteration. Since ${\bf v}=[{\bf c}^{T},{\bf
p}_{\text{vec}}^{T}]^{T}$, (11) is equivalently rewritten in a tractable
manner as

$\begin{split}\min\limits_{{\bf c},{\bf
p}_{\text{vec}}}\quad&-(1-\lambda)\sum_{k\in\mathcal{K}}\mu_{k}\left(C_{k}+\eta^{\delta_{\text{c}}}_{k}({\bf
p}_{\text{vec}})\right)\\\ &+\frac{\rho}{2}\lVert{\bf p}_{\text{vec}}-{\bf
D}_{\text{p}}{\bf u}^{t}+{\bf d}^{t}\rVert_{2}^{2}\\\
s.t.\quad&\sum_{k^{\prime}\in\mathcal{K}}C_{k^{\prime}}\leq\eta^{\delta_{\text{c}}}_{\text{c},k}({\bf
p}_{\text{vec}}),\forall k\in\mathcal{K}\\\ &{\bf p}_{\text{vec}}^{H}{\bf
p}_{\text{vec}}\leq P_{\text{t}}\\\ &{\bf c}\geq 0.\\\ \end{split}$ (14)

This problem can be reformulated with WMMSE approach and solved through the
WMMSE algorithm following [21].

The mode index $\delta_{\text{c}}$ is used to unify the derivation for
different radar sequence modes. The common stream $s_{\text{c}}$ is first
decoded at user-$k$ via an equalizer $g^{\delta_{\text{c}}}_{\text{c},k}$.
Equalizer $g^{\delta_{\text{c}}}_{k}$ is employed to decode the private stream
$s_{k}$ after removing the decoded common stream. For both modes, we get the
estimations $\hat{s}^{\delta_{\text{c}}}_{\text{c}}$,
$\hat{s}^{\delta_{\text{c}}}_{k}$ of $s_{\text{c}}$, $s_{k}$ , which are
respectively given
as$\hat{s}^{\delta_{\text{c}}}_{\text{c}}=g^{\delta_{\text{c}}}_{k}\tilde{y}^{\delta_{\text{c}}}_{k}$,
$\hat{s}^{\delta_{\text{c}}}_{k}=g^{\delta_{\text{c}}}_{k}(\tilde{y}^{\delta_{\text{c}}}_{k}-{\bf
h}_{k}^{H}{\bf p}_{\text{c}}s_{\text{c}})$, where

$\tilde{y}^{\delta_{\text{c}}}_{k}=\left\\{\begin{array}[]{ll}y_{k}-{\bf
h}_{k}^{H}{\bf p}_{\text{r}}s_{\text{r}},&\delta_{\text{c}}=0\\\
y_{k},&\delta_{\text{c}}=1\end{array}\right..$ (15)

Subsequently, the MSEs of both common and private stream estimation, defined
respectively as
$\varepsilon^{\delta_{\text{c}}}_{\text{c},k}\triangleq\mathbb{E}\left\\{\left|\hat{s}^{\delta_{\text{c}}}_{\text{c}}-s_{\text{c}}\right|^{2}\right\\}$
and
$\varepsilon^{\delta_{\text{c}}}_{k}\triangleq\mathbb{E}\left\\{\left|\hat{s}^{\delta_{\text{c}}}_{k}-s_{k}\right|^{2}\right\\}$,
are expressed as

$\displaystyle\varepsilon^{\delta_{\text{c}}}_{\text{c},k}$
$\displaystyle=\left|g_{\text{c},k}\right|^{2}T^{\delta_{\text{c}}}_{\text{c},k}-2\mathfrak{R}\left\\{g^{\delta_{\text{c}}}_{\text{c},k}{\bf
h}_{k}^{H}{\bf p}_{\text{c}}\right\\}+1,$ (16)
$\displaystyle\varepsilon^{\delta_{\text{c}}}_{k}$
$\displaystyle=\left|g_{k}\right|^{2}T^{\delta_{\text{c}}}_{k}-2\mathfrak{R}\left\\{g^{\delta_{\text{c}}}_{k}{\bf
h}_{k}^{H}{\bf p}_{k}\right\\}+1,$ (17)

where

$\displaystyle T^{\delta_{\text{c}}}_{\text{c},k}\triangleq$
$\displaystyle\left\\{\begin{array}[]{ll}\left|{\bf h}_{k}^{H}{\bf
p}_{\text{c}}\right|^{2}+\sum_{j\in\mathcal{K}}\left|{\bf h}_{k}^{H}{\bf
p}_{j}\right|^{2}+1,&\delta_{\text{c}}=0\\\ \left|{\bf h}_{k}^{H}{\bf
p}_{\text{r}}\right|^{2}+\left|{\bf h}_{k}^{H}{\bf
p}_{\text{c}}\right|^{2}+\sum_{j\in\mathcal{K}}\left|{\bf h}_{k}^{H}{\bf
p}_{j}\right|^{2}+1,&\delta_{\text{c}}=1\end{array}\right.,$ (20)
$\displaystyle
T^{\delta_{\text{c}}}_{k}\triangleq\left\\{\begin{array}[]{ll}\sum_{j\in\mathcal{K}}\left|{\bf
h}_{k}^{H}{\bf p}_{j}\right|^{2}+1,&\delta_{\text{c}}=0\\\ \left|{\bf
h}_{k}^{H}{\bf p}_{\text{r}}\right|^{2}+\sum_{j\in\mathcal{K}}\left|{\bf
h}_{k}^{H}{\bf p}_{j}\right|^{2}+1,&\delta_{\text{c}}=1\end{array}\right..$
(23)

Optimal equalizers are obtained by letting
$\frac{\partial\varepsilon^{\delta_{\text{c}}}_{\text{c},k}}{\partial
g^{\delta_{\text{c}}}_{\text{c},k}}=0$ and
$\frac{\partial\varepsilon^{\delta_{\text{c}}}_{k}}{\partial
g^{\delta_{\text{c}}}_{k}}=0$, which are the Minimized MSE (MMSE) equalizers
given by $g^{\text{MMSE},{\delta_{\text{c}}}}_{\text{c},k}={\bf
p}_{\text{c}}^{H}{\bf h}_{k}(T^{\delta_{\text{c}}}_{\text{c},k})^{-1}$,
$g^{\text{MMSE},{\delta_{\text{c}}}}_{k}={\bf p}_{k}^{H}{\bf
h}_{k}(T^{\delta_{\text{c}}}_{k})^{-1}.$ The MMSEs based on
$g^{\text{MMSE},{\delta_{\text{c}}}}_{\text{c},k}$ and
$g^{\text{MMSE},{\delta_{\text{c}}}}_{k}$ are given by

$\begin{split}\varepsilon^{\text{MMSE},{\delta_{\text{c}}}}_{\text{c},k}&\triangleq\min\limits_{g^{\delta_{\text{c}}}_{\text{c},k}}\varepsilon^{\delta_{\text{c}}}_{\text{c},k}=(T^{\delta_{\text{c}}}_{\text{c},k})^{-1}\left(T^{\delta_{\text{c}}}_{\text{c},k}-\left|{\bf
h}_{k}^{H}{\bf p}_{\text{c}}\right|^{2}\right),\\\
\varepsilon^{\text{MMSE},{\delta_{\text{c}}}}_{k}&\triangleq\min\limits_{g^{\delta_{\text{c}}}_{k}}\varepsilon^{\delta_{\text{c}}}_{k}=(T^{\delta_{\text{c}}}_{k})^{-1}\left(T^{\delta_{\text{c}}}_{k}-\left|{\bf
h}_{k}^{H}{\bf p}_{k}\right|^{2}\right).\end{split}$ (24)

Hence, by comparing (24) with (3) and (4), we rewrite SINRs of decoding the
intended streams as
$\gamma^{\delta_{\text{c}}}_{\text{c},k}=(1/\varepsilon_{\text{c},k}^{\text{MMSE},{\delta_{\text{c}}}})-1$
and
$\gamma^{\delta_{\text{c}}}_{k}=(1/\varepsilon_{k}^{\text{MMSE},{\delta_{\text{c}}}})-1$.
The common and private rates at user-$k$ are thus
$R^{\delta_{\text{c}}}_{\text{c},k}=\log_{2}(1+\gamma^{\delta_{\text{c}}}_{\text{c},k})=-\log_{2}(\varepsilon_{\text{c},k}^{\text{MMSE},{\delta_{\text{c}}}})$
and
$R^{\delta_{\text{c}}}_{k}=\log_{2}(1+\gamma^{\delta_{\text{c}}}_{k})=-\log_{2}(\varepsilon_{k}^{\text{MMSE},{\delta_{\text{c}}}})$.

By allocating positive weights
$(w^{\delta_{\text{c}}}_{\text{c},k},w^{\delta_{\text{c}}}_{k})$ to user-$k$’s
common and private rates, we define the augmented WMSEs as
$\xi^{\delta_{\text{c}}}_{\text{c},k}\triangleq
w^{\delta_{\text{c}}}_{\text{c},k}\varepsilon^{\delta_{\text{c}}}_{\text{c},k}-\log_{2}(w^{\delta_{\text{c}}}_{\text{c},k}),\xi^{\delta_{\text{c}}}_{k}\triangleq
w^{\delta_{\text{c}}}_{k}\varepsilon^{\delta_{\text{c}}}_{k}-\log_{2}(w^{\delta_{\text{c}}}_{k})$.
After optimizing over the equalizers and weights, the rate-WMMSE relationships
are established as

$\begin{split}\xi_{\text{c},k}^{\text{MMSE},{\delta_{\text{c}}}}&\triangleq\min\limits_{w^{\delta_{\text{c}}}_{\text{c},k},g^{\delta_{\text{c}}}_{\text{c},k}}\xi^{\delta_{\text{c}}}_{\text{c},k}=1-R^{\delta_{\text{c}}}_{\text{c},k}\\\
\xi_{k}^{\text{MMSE},{\delta_{\text{c}}}}&\triangleq\min\limits_{w^{\delta_{\text{c}}}_{k},g^{\delta_{\text{c}}}_{k}}\xi^{\delta_{\text{c}}}_{k}=1-R^{\delta_{\text{c}}}_{k},\end{split}$
(25)

where the optimal equalizers are
$g_{\text{c},k}^{{\delta_{\text{c}}}*}=g_{\text{c},k}^{\text{MMSE},{\delta_{\text{c}}}}$
and $g_{k}^{{\delta_{\text{c}}}*}=g_{k}^{\text{MMSE},{\delta_{\text{c}}}}$,
and the optimal weights are
$w_{\text{c},k}^{{\delta_{\text{c}}}*}=w_{\text{c},k}^{\text{MMSE},{\delta_{\text{c}}}}=\left(\varepsilon_{\text{c},k}^{\text{MMSE},{\delta_{\text{c}}}}\right)^{-1}$
and
$w_{k}^{{\delta_{\text{c}}}*}=w_{k}^{\text{MMSE},{\delta_{\text{c}}}}=\left(\varepsilon_{k}^{\text{MMSE},{\delta_{\text{c}}}}\right)^{-1}$.
They are obtained by meeting the first-order optimality conditions.

Using the rate-WMMSE relationships (25), we then reformulate (14) as

$\begin{split}\min\limits_{\begin{array}[]{l}{\bf c},{\bf p}_{\text{vec}}\\\
{\bf w}^{\delta_{\text{c}}},{\bf
g}^{\delta_{\text{c}}}\end{array}}&-(1-\lambda)\sum_{k\in\mathcal{K}}\mu_{k}\left(-C_{k}+\xi^{\delta_{\text{c}}}_{k}({\bf
p}_{\text{vec}})\right)\\\ &+\frac{\rho}{2}\lVert{\bf p}_{\text{vec}}-{\bf
D}_{\text{p}}{\bf u}^{t}+{\bf d}^{t}\lVert_{2}^{2}\\\
s.t.\quad&\sum_{k^{\prime}\in\mathcal{K}}C_{k^{\prime}}+\xi^{\delta_{\text{c}}}_{\text{c},k}({\bf
p}_{\text{vec}})\leq 1,\forall k\in\mathcal{K}\\\ &{\bf
p}_{\text{vec}}^{H}{\bf p}_{\text{vec}}\leq P_{\text{t}},\\\ &{\bf c}\geq
0,\end{split}$ (26)

where ${\bf
w}^{\delta_{\text{c}}}=\left[w^{\delta_{\text{c}}}_{1},\dots,w^{\delta_{\text{c}}}_{K},w^{\delta_{\text{c}}}_{\text{c},1},\dots,w^{\delta_{\text{c}}}_{\text{c},K}\right]$
is the vector of all MSE weights. ${\bf
g}^{\delta_{\text{c}}}=\left[g^{\delta_{\text{c}}}_{1},\dots,g^{\delta_{\text{c}}}_{K},g^{\delta_{\text{c}}}_{\text{c},1},\dots,g^{\delta_{\text{c}}}_{\text{c},K}\right]$
is the vector of all equalizers.

Note that when ${\bf w}^{\delta_{\text{c}}}$ and ${\bf g}^{\delta_{\text{c}}}$
are fixed, (26) is a convex Quadratic Constrained Quadratic Programming (QCQP)
according to $\\{{\bf p}_{\text{vec}},{\bf c}\\}$, which can be efficiently
solved by CVX toolbox[26]. The WMMSE algorithm to solve (11) is summarized in
Algorithm 2. $\epsilon_{0}$ is the tolerance of this algorithm. The weights,
equalizers, and precoders are iteratively updated till convergence. Readers
are referred to [21] for more details on the WMMSE algorithm.

Input: ${\bf u}^{t},{\bf d}^{t}$.

Output: ${\bf v}^{t+1}$.

1 $k\leftarrow 0$;

2 ${\bf p}_{\text{vec}}^{[k]}={\bf D}_{\text{p}}{\bf u}^{t},{\bf
c}^{[k]}=({\bf I}_{(K+2)N_{\text{t}}}-{\bf D}_{\text{p}}){\bf u}^{t}$;

3 $\text{WSR}^{[k]}$ is calculated from ${\bf p}_{\text{vec}}^{[k]}$ and ${\bf
c}^{[k]}$;

4 repeat

5 ${\bf w}^{\delta_{\text{c}}*}\leftarrow{\bf
w}^{\text{MMSE},\delta_{\text{c}}}({\bf p}_{\text{vec}}^{[k]})$;

6 ${\bf g}^{\delta_{\text{c}}*}\leftarrow{\bf
g}^{\text{MMSE},\delta_{\text{c}}}({\bf p}_{\text{vec}}^{[k]})$;

7 update $({\bf p}_{\text{vec}}^{[k+1]},{\bf c}^{[k+1]})$ by solving QCQP (26)
with the updated ${\bf w}^{\delta_{\text{c}}*},{\bf g}^{\delta_{\text{c}}*}$;

8 update $\text{WSR}^{[k+1]}$ by using ${\bf p}_{\text{vec}}^{[k+1]}$ and
${\bf c}^{[k+1]}$

9 $k++$;

10until  _$|\text{WSR}^{[k]}-\text{WSR}^{[k-1]}|\leq\epsilon_{1}$_ ;

${\bf v}^{t+1}=[{\bf c}^{[k]};{\bf p}_{\text{vec}}^{[k]}]$;

Algorithm 2 WMMSE algorithm

###  V-B MM-based Algorithm for ${\bf u}$-update

In this subsection, we propose an approach based on MM algorithm to solve
(12). By defining ${\bf p}_{\text{u}}={\bf D}_{\text{p}}{\bf u}$ and

$\begin{split}{\bf D}_{\text{p},k}=\left[{\bf
0}^{N_{\text{t}}\times(k-1)N_{\text{t}}}\quad{\bf I}_{N_{\text{t}}}\quad{\bf
0}^{N_{\text{t}}\times(K+2-k)N_{\text{t}}}\right],\\\
k=1,2,\dots,K+2,\end{split}$ (27)

(12) is equivalently formulated as

$\begin{split}\min_{{\bf p}_{\text{u}}}\quad&\lambda\sum_{m=1}^{M}\lvert
P_{\text{d}}(\theta_{m})-{\bf a}^{H}(\theta_{m})\big{(}\sum_{k=1}^{K+2}{\bf
D}_{\text{p},k}{\bf p}_{\text{u}}{{\bf p}_{\text{u}}}^{H}{\bf
D}_{\text{p},k}^{H}\big{)}\\\ &{\bf
a}(\theta_{m})\lvert^{2}+\frac{\rho}{2}\lVert{\bf D}_{\text{p}}{\bf
v}^{t+1}-{\bf p}_{\text{u}}+{\bf d}^{t}\rVert_{2}^{2}\\\
s.t.\quad&\text{diag}(\sum_{k=1}^{K+2}{\bf D}_{\text{p},k}{\bf
p}_{\text{u}}{{\bf p}_{\text{u}}}^{H}{\bf
D}_{\text{p},k}^{H})=\frac{P_{\text{t}}{\bf 1}^{N_{\text{t}}\times
1}}{N_{\text{t}}}.\end{split}$ (28)

Since the complicated quartic formulation makes it difficult to find a tight
relaxation of (12), we propose an MM-based iterative approach to solve this
problem referring to [27].

Recall Lemma 1 in [28] that will be used in our MM-based algorithm.

###### Lemma 1

Let ${\bf L}$, ${\bf M}$ be the $n\times n$ Hermitian matrices and ${\bf
M}\succeq{\bf L}$. For any point ${\bf x}_{0}\in\mathbb{C}^{n}$, there is
${\bf x}^{H}{\bf L}{\bf x}\leq{\bf x}^{H}{\bf M}{\bf x}+2\mathfrak{R}\\{{\bf
x}^{H}({\bf L-M}){\bf x}_{0}\\}+{\bf x}_{0}^{H}({\bf M-L}){\bf
x}_{0}$.$\hfill\blacksquare$

Minimizing the objective function of (28) is equivalent to minimizing

$f_{\text{u}}({\bf p}_{\text{u}})=\lambda\sum_{m=1}^{M}\lvert a_{m}-{\bf
p}_{\text{u}}^{H}{\bf Z}_{m}{\bf
p}_{\text{u}}\lvert^{2}-\rho\mathfrak{R}\\{{\bf p}_{\text{u}}^{H}\hat{\bf
d}^{t}\\},$ (29)

where $a_{m}=P_{\text{d}}(\theta_{m})$, $\hat{\bf d}^{t}={\bf
D}_{\text{p}}{\bf v}^{t+1}+{\bf d}^{t}$, ${\bf Z}_{m}=\sum_{k=1}^{K+2}{\bf
D}_{\text{p},k}^{H}{\bf a}(\theta_{m}){\bf a}^{H}(\theta_{m}){\bf
D}_{\text{p},k}$. Let $q_{m}({\bf p}_{\text{u}})={\bf p}_{\text{u}}^{H}{\bf
Z}_{m}{\bf p}_{\text{u}}$ and ${\bf Q}_{m}=\text{vec}({\bf
Z}_{m})\text{vec}({\bf Z}_{m})^{H}$, we use Lemma 1 and get

$\begin{split}&q_{m}({\bf p}_{\text{u}})^{2}\leq 2\lambda_{\text{max}}({\bf
Q}_{m})P_{\text{t}}^{2}-q_{m}({\bf p}_{\text{u}}^{k})^{2}\\\
&+2\mathfrak{R}\\{q_{m}({\bf p}_{\text{u}}^{k})q_{m}({\bf
p}_{\text{u}})-\lambda_{\text{max}}({\bf Q}_{m})({\bf p}_{\text{u}}^{H}{\bf
p}_{\text{u}}^{k}({\bf p}_{\text{u}}^{k})^{H}{\bf
p}_{\text{u}})\\}.\end{split}$ (30)

Substituting (30) into (29), we then have the majorization function of (29) as

$\begin{split}f_{\text{u}}({\bf p}_{\text{u}})\leq f_{\text{u}}^{1}({\bf
p}_{\text{u}},{\bf p}_{\text{u}}^{k})=C_{\text{u}}+{\bf p}_{\text{u}}^{H}{\bf
Q^{\prime}}{\bf p}_{\text{u}}-\rho\mathfrak{R}\\{{\bf
p}_{\text{u}}^{H}\hat{\bf d}^{t}\\}\end{split}$ (31)

where $C_{\text{u}}=\lambda\sum_{m=1}^{M}\big{[}2\lambda_{\text{max}}({\bf
Q}_{m})P_{\text{t}}^{2}-q_{m}({\bf p}_{\text{u}}^{k})^{2}+a_{m}^{2}\big{]}$,
${\bf Q^{\prime}}=\lambda\sum_{m=1}^{M}\big{[}2\big{(}q_{m}({\bf
p}_{\text{u}}^{k})-a_{m}\big{)}{\bf Z}_{m}-2\lambda_{\text{max}}({\bf
Q}_{m}){\bf p}_{\text{u}}^{k}({\bf p}_{\text{u}}^{k})^{H}\big{]}$.

It is obvious that $C_{\text{u}}$ and ${\bf Q^{\prime}}$ are constant when
${\bf p}_{\text{u}}^{k}$ is determined in each iteration. Since that there is
still a quadratic term in (31), we apply Lemma 1 again and have

$\begin{split}{\bf p}_{\text{u}}^{H}{\bf Q^{\prime}}{\bf
p}_{\text{u}}\leq&\lambda_{\text{max}}({\bf Q}^{\prime})P_{\text{t}}+({\bf
p}_{\text{u}}^{k})^{H}(\lambda_{\text{max}}({\bf Q}^{\prime}){\bf I}-{\bf
Q^{\prime}}){\bf p}_{\text{u}}^{k}\\\ &+2\mathfrak{R}\\{{\bf
p}_{\text{u}}^{H}({\bf Q^{\prime}}-\lambda_{\text{max}}({\bf Q}^{\prime}){\bf
I}){\bf p}_{\text{u}}^{k}\\}.\end{split}$ (32)

Therefore, we obtain

$\begin{split}&f_{\text{u}}({\bf p}_{\text{u}})\leq f_{\text{u}}^{1}({\bf
p}_{\text{u}},{\bf p}_{\text{u}}^{k})\leq f_{\text{u}}^{2}({\bf
p}_{\text{u}},{\bf p}_{\text{u}}^{k})\\\
&=C_{\text{u}}^{\prime}+\mathfrak{R}\big{\\{}{\bf
p}_{\text{u}}^{H}\big{[}2({\bf Q^{\prime}}-\lambda_{\text{max}}({\bf
Q}^{\prime}){\bf I}){\bf p}_{\text{u}}^{k}-\rho\hat{\bf
d}^{t}\big{]}\big{\\}}\end{split}$ (33)

where

$C_{\text{u}}^{\prime}=C_{\text{u}}+\lambda_{\text{max}}({\bf
Q}^{\prime})P_{\text{t}}+({\bf
p}_{\text{u}}^{k})^{H}(\lambda_{\text{max}}({\bf Q}^{\prime}){\bf I}-{\bf
Q^{\prime}}){\bf p}_{\text{u}}^{k}.$ (34)

We now obtain the linear majorization function of the quartic objective
function $f_{\text{u}}({\bf p}_{\text{u}})$ at the point ${\bf
p}_{\text{u}}^{k}$ as $f_{\text{u}}^{2}({\bf p}_{\text{u}},{\bf
p}_{\text{u}}^{k})$. After omitting the constant term, the majorized problem
of (28) is denoted as

$\begin{split}\min_{{\bf p}_{\text{u}}}\quad&\mathfrak{R}\big{\\{}{\bf
p}_{\text{u}}^{H}\big{[}2({\bf Q^{\prime}}-\lambda_{\text{max}}({\bf
Q}^{\prime}){\bf I}){\bf p}_{\text{u}}^{k}-\rho\hat{\bf
d}^{t}\big{]}\big{\\}}\\\ s.t.\quad&\text{diag}(\sum_{k=1}^{K+1}{\bf
D}_{\text{p},k}{\bf p}_{\text{u}}{{\bf p}_{\text{u}}}^{H}{\bf
D}_{\text{p},k}^{H})=\frac{P_{\text{t}}{\bf 1}^{N_{\text{t}}\times
1}}{N_{\text{t}}}.\end{split}$ (35)

To further simplify (35), we denote $\hat{\bf k}^{\prime}={\bf
f}_{\hat{\text{k}}}({\bf p}_{\text{u}}^{k})=-2({\bf
Q^{\prime}}-\lambda_{\text{max}}({\bf Q}^{\prime}){\bf I}){\bf
p}_{\text{u}}^{k}-\rho\hat{\bf d}^{t},$ and split ${\bf p}_{\text{u}}$ into
$N_{\text{t}}$ sub-vectors as

$\begin{split}\hat{\bf
p}_{j}=\left[[p_{\text{u}}]_{j},[p_{\text{u}}]_{N_{\text{t}}+j},\dots,[p_{\text{u}}]_{((K+1)N_{\text{t}}+j}\right]^{T},\\\
j=1,2,\dots,N_{\text{t}},\end{split}$ (36)

where $[p_{\text{u}}]_{j}$ is the $j$th entry in ${\bf p}_{\text{u}}$. We then
split $\hat{\bf k}^{\prime}$ by the same means into $\hat{\bf
k}_{j}=\left[\hat{k}^{\prime}_{j},\hat{k}^{\prime}_{N_{\text{t}}+j},\dots,\hat{k}^{\prime}_{(K+1)N_{\text{t}}+j}\right]^{T},j=1,2,\dots,N_{\text{t}}.$
where $\hat{k}^{\prime}_{j}$ is the $j$th entry in $\hat{\bf k}^{\prime}$.
According to [13], (35) has a closed-form solution as

$\hat{\bf
p}_{j}^{*}=\frac{\sqrt{\frac{P_{\text{t}}}{N_{\text{t}}}}}{\lVert\hat{\bf
k}_{j}\lVert_{2}}\hat{\bf k}_{j},j=1,2,\dots,N_{\text{t}},$ (37)

and ${\bf p}_{\text{u}}^{*}$ can be recovered from $\hat{\bf p}_{j}^{*}$
according to (36).

Therefore, (28) is solved by iteratively computing $\hat{\bf p}_{j}^{*}$ based
on (37) until convergence. The initial point for the proposed MM-based
algorithm can be chosen via classic Semi-definite Relaxation (SDR) [25]. The
MM-based algorithm is summarized in Algorithm 3, where $\epsilon_{2}$ is its
tolerance. The first $K$ elements of the output ${\bf u}^{t+1}$ are directly
copied from ${\bf v}^{t+1}$ as they do not participate in the optimization
(28) and have no relations with the whole ${\bf u}$-update.

Input: ${\bf v}^{t+1},{\bf d}^{t}$, $k\leftarrow 0$, ${\bf
p}_{\text{u}}^{[k]}$.

Output: ${\bf u}^{t+1}$.

1 repeat

2 $\hat{\bf k}^{\prime}={\bf f}_{\hat{\text{k}}}({\bf p}_{\text{u}}^{[k]})$;

3 for  _$j=1$ to $N_{\text{t}}$_ do

4 $\hat{\bf
k}_{j}=\left[\hat{k}^{\prime}_{j},\hat{k}^{\prime}_{N_{\text{t}}+j},\dots,\hat{k}^{\prime}_{(K+1)N_{\text{t}}+j}\right]^{T}$;

5 $\hat{\bf
p}_{j}^{*}=\frac{\sqrt{\frac{P_{\text{t}}}{N_{\text{t}}}}}{\lVert\hat{\bf
k}_{j}\lVert_{2}}\hat{\bf k}_{j}$;

6 end for

7 ${\bf p}_{\text{u}}^{[k+1]}$ is obtained by doing the inverse operation of
(36) using $\hat{\bf p}_{j}^{*}$;

8 $k++$;

9until  _$\lVert{\bf p}_{\text{u}}^{[k]}-{\bf
p}_{\text{u}}^{[k-1]}\lVert_{2}\leq\epsilon_{2}$_ ;

${\bf u}^{t+1}=[({\bf I}_{(K+2)N_{\text{t}}}-{\bf D}_{\text{p}}){\bf
v}^{t+1};{\bf p}_{\text{u}}^{[k]}]$;

Algorithm 3 MM-based algorithm

###  V-C Computational Complexity Analysis

In this part, we analyse the complexity of the proposed ADMM-based framework
to solving (9). From Algorithm 1, it is clear that the complexity in one
iteration consists of running the WMMSE algorithm and the MM-based algorithm.

According to [29], the polynomial complexity of implementing WMMSE-AO to solve
the QCQP problem (26) via interior-point method is
$\mathcal{O}([3+(K+2)(N_{\text{t}})]\times[(K+2)(N_{\text{t}})]^{2})=\mathcal{O}([(K+2)(N_{\text{t}})]^{3})$.

For MM-based algorithm, the complexity of calculating the largest eigenvalue
of ${\bf Q}^{\prime}$ in each MM iteration is
$\mathcal{O}([(K+2)(N_{\text{t}})]^{3})$, while that for other steps of linear
operation is $\mathcal{O}([(K+2)(N_{\text{t}})]^{2})$. Hence, the complexity
of the MM-based iteration is $\mathcal{O}([(K+2)(N_{\text{t}})]^{3})$.

Therefore, for the proposed DFRC, the complexity of each iteration of our
ADMM-based framework is polynomial as
$\mathcal{O}([(K+2)(N_{\text{t}})]^{3})$. Following the same analysis, we can
obtain the complexity of RSMA-assisted DFRC without radar sequence as
$\mathcal{O}([(K+1)(N_{\text{t}})]^{3})$.

##  VI Numerical Results

In this section, we provide numerical results to validate the performance of
the proposed RSMA-assisted DFRC. We mainly reveal the superior advantages of
using RSMA over SDMA in DFRC and further demonstrate that the common stream of
RSMA outperforms radar sequence. TDRC and FDRC are also compared as baselines.

###  VI-A Settings of Numerical Experiments

We assume the proposed DFRC employs a ULA where $N_{\text{t}}=8$ with half-
wavelength spacing, i.e., $\Delta=0.5$. The total transmit power budget is
$P_{\text{t}}=20\text{dBm}$ and the noise power at each user is $0\text{dBm}$.
We consider four users and three targets of interest. These users are numbered
as User-1 to User-4. We generate four channel vectors randomly obeying the
i.i.d. standard complex Gaussian distribution to represent the four user
channels. The three targets of interest are Target-1 with the azimuth angle
range $[-6,6]$°, Target-2 at $[-56,-44]$°and Target-3 at $[44,56]$°. All four
users and their channel vectors, together with the three targets will be
chosen and combined by different means in the following experiments for
thorough analysis. TDRC and FDRC are included as baselines and we assume the
radar and communication function share the same power budget $P_{\text{t}}/2$
in FDRC.

For the optimization variables, precoders $[{\bf{p}}_{\text{c}},{\bf
p}_{1},\dots,{\bf{p}}_{K}]$ are initialized through Maximum Ratio Combining
(MRC) method following [30], and ${\bf p}_{\text{r}}$ and ${\bf d}$ are
initialized randomly. ${\bf c}$ is initialized as ${\bf 1}^{K\times 1}$. The
desired beampattern is generated following the matching method in [23].

In the following results, in order to achieve a clearer comparison, we use
Root Mean Squared Error (RMSE) instead of MSE for beampattern approximation.
In the figures, “RSMA, no Rad-Seq, DFRC” represents the scheme “RSMA-assisted
DFRC without radar sequence”, “RSMA, noSIC Rad-Seq, DFRC” represents “RSMA-
assisted DFRC with radar sequence”, and “RSMA, SIC Rad-Seq, DFRC” represents
“RSMA-assisted DFRC with radar sequence and SIC”. The three SDMA-assisted DFRC
modes are denoted in the same manner.

###  VI-B Metric for Radar-to-Communication Interference: LB-IBR

Since we adopt WSR and beampattern approximation MSE as optimization
objectives, they are unable to quantitatively depict the dual-functional
interference.

From this perspective, we define Interference-to-Beampattern-Ratio (IBR) and
derive its lower bound as LB-IBR. The definitions are based on SDMA-assisted
DFRC without radar sequence, such that the advantage of enabling the common
stream and radar sequence can be analysed accordingly. IBR is defined as
follows.

$\text{IBR}(\theta_{m})=\frac{\sum_{k\in\mathcal{K}}\sum_{j\neq k}{\bf
h}_{j}^{H}{\bf p}_{k}{\bf p}_{k}^{H}{\bf h}_{j}}{\sum_{k\in\mathcal{K}}{\bf
a}(\theta_{m})^{H}{\bf p}_{k}{\bf p}_{k}^{H}{\bf a}(\theta_{m})}.$ (38)

The denominator is the transmit beampattern value of SDMA-assisted DFRC
without radar sequence at $\theta_{m}$, while the numerator is the total
interference leakage towards the communication users. A higher
$\text{IBR}(\theta_{m})$ means more interference will be leaked to users when
a unit-power beampattern at $\theta_{m}$ is achieved, which reflects stronger
radar-to-communication interference in DFRC.

Interestingly, by exploiting the precoder power constraint (9d), we can obtain
the lower bound of $\text{IBR}(\theta_{m})$ that does not contain the
optimization variable ${\bf p}_{k}$. This lower bound is denoted as LB-IBR and
can be expressed as

$\text{LB-
IBR}(\theta_{m})=(1/N_{\text{t}}^{2})\min_{k\in\mathcal{K}}(\sum_{j\neq
k}\lVert{\bf a}(\theta_{m})^{H}{\bf
h}_{j}\lVert_{2}^{2})\leq\text{IBR}(\theta_{m}).$ (39)

The derivation is shown in Appendix. LB-IBR can be used to evaluate the basic
radar-to-communication interference level for SDMA-assisted DFRC without radar
sequence in the operating environment.

###  VI-C Beampattern Approximation Performance

We first show the beampattern approximation performance of the proposed DFRC.
We take the DFRC enabling RSMA and radar sequence with SIC as an example,
where the DFRC serves all four users and detects all three targets.

(a) $\lambda=1\times 10^{-5}$, $\text{WSR}=3.85$bps/Hz, RMSE=5.5.

(b) $\lambda=1\times 10^{-6}$, $\text{WSR}=5.65$bps/Hz, RMSE=$4.7\times
10^{2}$.

Figure 2: Beampattern approximation comparisons among proposed DFRC with
different WSR and RMSE, MIMO radar, TDRC and FDRC.

Fig. 2 shows the transmit beampattern approximation performance of the
proposed DFRC system compared with pure MIMO radar with $N_{\text{t}}$
antennas, TDRC and FDRC. The beampattern of DFRC is obtained by solving (9).
The desired beampattern is obtained via beampattern matching methods for a
pure MIMO radar, which can be achieved in TDRC. In Fig. 2(a), it is clear that
DFRC is able to fully approximate MIMO radar’s beampattern, and has near 3dB
gain over FDRC. This is because the latter has only half the power budget when
doing the radar function. Fig. 2(b) shows that by decreasing the
regularization parameter to $\lambda=1\times 10^{-6}$, DFRC experiences near
1.8bps/Hz increase in WSR but also near 19.3dB increase in beampattern
approximation RMSE. This reveals that there is a tradeoff between beampattern
approximation MSE and WSR.

###  VI-D RSMA vs. SDMA in DFRC

In this subsection, we analyse the advantages of enabling RSMA in DFRC
compared with using SDMA in terms of beampattern approximation RMSE and WSR
tradeoff performance. We also demonstrate how the common stream contributes to
DFRC and explain why RSMA outperforms SDMA when user’s channels, target
location, user number and target number change.

Too eliminate influence of radar sequence, we only investigate RSMA-assisted
and SDMA-assisted DFRC without radar sequence in this part. Performance of
both TDRC and FDRC is included as baselines in the simulation results.

####  VI-D1 RSMA Gain vs. Users’ Channels

To show how RSMA benefits DFRC when users’ channels change, we first fix the
user number, target number and the target location. We then set up the first
scenario, denoted as Scenario-UC, where different user pairs are served, i.e.,
User-1 and User-4, User-2 and User-4, User-3 and User-4. In these three
scenarios, only one target, i.e., Target-1, is supposed to be detected.

(a) Tradeoffs when serving User-1 and User-4.

(b) Tradeoffs when serving User-2 and User-4.

(c) Tradeoffs when serving User-3 and User-4.

(d) LB-IBR when serving different user pairs.

Figure 3: Tradeoff and LB-IBR for Scenario-UC.

In Fig. 3(a)–3(c), the tradeoffs between RMSE and WSE for RSMA-assisted and
SDMA-assisted DFRC without radar sequence, FDRC and TDRC strategies are
compared in this scenario. It is clear that RSMA-assisted DFRC without radar
sequence outperforms FDRC with 0.8bps/Hz WSR gain on average given the same
beampattern approximation RMSE in all scenarios, and surpasses TDRC when
$\alpha$ is smaller than around 0.8. Additionally, RSMA-assisted DFRC without
radar sequence outperforms SDMA-assisted DFRC without radar sequence as well,
but the tradeoff gain varies according to different user pairs. Specifically,
when serving User-2&4, the former achieves the largest WSR gain of 1.2bps/Hz
when RMSE$\leq 1$, while the WSR gain almost disappears when serving
User-1&4\. Then we use LB-IBR to analyze the radar-to-communication
interference level in this scenario, so as to explain why RSMA-assisted DFRC
outperforms SDMA-assisted DFRC when radar sequence is disabled.

Fig. 3(d) demonstrates the LB-IBR in Scenario-UC, where the azimuth angle
range of Target-1 is highlighted. Obviously, at the location of Target-1, LB-
IBR is the highest when User-2&4 are served but lowest when User-1&4 are
served, which means detecting Target-1 leads to the highest and lowest
interference on the former and latter user pair, respectively. Fig. 4 further
displays the beampattern contribution of RSMA-assisted DFRC without radar
sequence in Scenario-UC with the same RMSE=$10$. We observe that the common
stream of RSMA contributes more to target detection when the user pair with
higher LB-IBR is served. This means that the common stream can further
mitigate radar-to-communication interference compared to SDMA, and when such
interference is larger, the common stream contributes even more and thus
achieves a larger tradeoff gain (see for instance Fig. 4(b) where the
contribution of the common stream to the beampattern is significant, which
leads to a larger tradeoff gain in Fig. 3(b)).

(a) Beampattern contribution when serving User-1 and User-4. (b) Beampattern
contribution when serving User-2 and User-4. (c) Beampattern contribution when
serving User-3 and User-4.

Figure 4: Beampattern contribution of RSMA-assisted DFRC without radar
sequence for Scenario-UC.

####  VI-D2 RSMA Gain vs. Target’s Location

The second scenario denoted as Scenario-TL is set to serve the same two users,
i.e., User-3 and User-4, and detect one target varying among all three
prepared targets.

(a) Tradeoffs when detecting Target-1.

(b) Tradeoffs when detecting Target-2.

(c) Tradeoffs when detecting Target-3.

(d) LB-IBR for different target’s locations.

Figure 5: Tradeoff and LB-IBR for Scenario-TL.

Fig 5(a)–5(c) first illustrate the tradeoffs in this scenario. Again, we see
that RSMA-assisted DFRC without radar sequence outperforms FDRC as well as
TDRC when $\alpha$ is smaller than around 0.8, and also has different tradeoff
gains over SDMA-assisted DFRC without radar sequence. Likewise, we show in
Fig. 5(d) the LB-IBR at the different target locations. It shows again that
RSMA helps to further mitigate the radar-to-communication interference.

####  VI-D3 RSMA Gain vs. User Number

The third scenario denoted as Scenario-UN aims at detecting Target-1, while
serving a different number of users, i.e., two users (User-1,4), three users
(User-1,2,4) and four users (User-1,2,3,4).

Fig. 6 shows the tradeoffs in all scenarios. Similarly, RSMA-assisted DFRC
without radar sequence still outperforms FDRC, but when the number of user
increases, it outperforms TDRC with smaller $\alpha$. It is also clear that
RSMA-assisted DFRC without radar sequence has an advantage over SDMA-assisted
DFRC without radar sequence. Fig. 6(d) demonstrates the LB-IBR for this
scenario, and again proves that RSMA brings more gain in the scenario with
higher LB-IBR.

(a) Tradeoffs when serving two users. (b) Tradeoffs when serving three users.

(c) Tradeoffs when serving four users.

(d) LB-IBR for different user numbers.

Figure 6: Tradeoff and LB-IBR comparison for Scenario-UN.

####  VI-D4 RSMA Performance in Multi-target Detection

Next, we show the performance of RSMA to detect multiple targets. The scenario
is denoted as Scenario-MUT where User-2 and User-4 are served and all three
targets are detected.

Figure 7: LB-IBR for Scenario-MUT.

Figure 8: Beampattern contribution of RSMA-assisted DFRC without radar
sequence for Scenario-MUT.

Fig. 8 displays the LB-IBR in this scenario, where we can see LB-IBR is the
highest at Target-1 but lowest at Target-3. We then show in Fig. 8 the
beampattern contribution of RSMA-assisted DFRC without radar sequence when
$\text{RMSE}=10^{1}$, where the common stream contributes more to the target
with higher LB-IBR.

All numerical experiments in this subsection show that RSMA-assisted DFRC
without radar sequence outperforms FDRC, and in certain conditions surpasses
TDRC. The common stream of RSMA also shows the capability of fulfilling the
beampattern requirements of radar and mitigating radar-to-communication
interference compared to SDMA in DFRC, which supports the observation that
RSMA outperforms SDMA in DFRC.

Figure 9: Beampattern contribution of SDMA-assisted DFRC with radar sequence
and SIC for Scenario-MUT.

Figure 10: Tradeoffs of all combinations of multiple access techniques and
radar sequence modes for Scenario-MUT.

Figure 11: IPR comparison between modes of SDMA-assisted DFRC with radar
sequence.

###  VI-E RSMA vs. Radar Sequence in DFRC

After investigating the advantage of enabling RSMA over SDMA in DFRC, we
analyse DFRC’s performance when radar sequence is enabled.

Here, we adopt Scenario-MUT and obtain the tradeoffs of different combinations
of multiple access techniques and radar sequence modes in Fig. 11. It is clear
that SDMA-assisted DFRC with radar sequence brings an around 0.4bps/Hz WSR
gain over that without radar sequence when RMSE$\leq 1$, while enabling SIC to
radar sequence leads to an additional 1.1bps/Hz WSR gain. However, when RSMA
is used, it is interesting that whatever radar sequence mode is enabled, the
tradeoff is the same with RSMA-assisted DFRC without radar sequence, which
outperforms all SDMA-assisted DFRC systems, FDRC and TDRC when $\alpha<0.78$.

We first explain in SDMA-assisted DFRC with radar sequence, why SIC brings
benefits. We represent the total interference power of radar sequence imposed
on communication users as $\text{IPR}=\sum_{k\in\mathcal{K}}\lVert{\bf
h}^{H}_{k}{\bf p}_{\text{r}}\lVert_{2}$. In Scenario-MUT, IPRs of both modes
of SDMA-assisted DFRC with radar sequence are demonstrated corresponding to
RMSE in Fig. 11. We can see that when SIC is disabled, IPR is significantly
low after optimization, which indicates that ${\bf p}_{\text{r}}$ is designed
obeying ZF to mitigate interference as much as possible. In contrast, when SIC
is used, IPR is comparatively high, because it can always be canceled via SIC
and thus ${\bf p}_{\text{r}}$ can be designed freely without obeying ZF. Since
the restriction on ${\bf p}_{\text{r}}$ is much weaker in SDMA-assisted DFRC
with radar sequence and SIC, the feasible region is enlarged.

We then illustrate why RSMA outperforms SDMA-assisted DFRC with radar
sequence. Fig. 11 illustrates the beampattern contribution of SDMA-assisted
DFRC with radar sequence and SIC in Scenario-MUT with RMSE=$10^{1}$, where
radar sequence acts quite similarly compared to the common stream in RSMA-
assisted DFRC without radar sequence in Fig. 8. Despite the same beampattern
behaviour, RSMA-assisted DFRC without radar sequence still has a higher WSR.
This is because SIC to the radar sequence is not used for managing the
multiuser communications, but only the interference between radar and
communication. However, apart from fulfilling the radar beampattern and
managing the radar-to-communication interference, the common stream in RSMA
can further manage the interference among communication users as it can be
decoded partly at users. Therefore, the SIC receiver in RSMA is used for the
dual purpose of managing interference between communication users but also the
interference between communication and radar.

To summarize, the common stream of RSMA has the triple function of 1) managing
interference between communication users, 2) managing interference between
communication and radar, and 3) fulfilling the beampattern requirements of the
radar. As a consequence, there is no need for an additional radar sequence and
SIC. This coincides with the observation that RSMA-assisted DFRC with and
without radar sequence achieve the same tradeoff performance.

###  VI-F Convergence

Next, we show the convergence performance of the proposed ADMM framework
solving the DFRC design problem. We consider the scenario where User-2 and
User-4 are served while Target-1 is to be detected. In this scenario, Fig. 12
shows the convergence of the ADMM framework that solves the optimization
problem of designing SDMA-assisted DFRC without radar sequence, RSMA-assisted
DFRC without radar sequence, SDMA-assisted DFRC with radar sequence and SIC,
and RSMA-assisted DFRC with radar sequence and SIC. We set $\lambda=10^{-5}$
in (9) in Fig. 12. It is clear that the convergence of the objective values of
the proposed ADMM framework is guaranteed, and the framework converges to a
good solution in around ten times of iteration.

Figure 12: Convergence performance of the ADMM framework for solving the DFRC
design problem, $\lambda=10^{-5}$.

##  VII Conclusion

To conclude, we propose a multi-antenna DFRC system that enables RSMA and
different modes of radar sequence. The message split, precoders of information
streams and radar sequence are optimized together with the aim of jointly
maximizing WSR and minimizing MSE of beampattern approximation under the
average transmit power constraint at each antenna. We propose the ADMM-based
framework to effectively solve the intractable non-convex optimization problem
of the proposed RSMA-assisted DFRC. SDMA-assisted DFRC, practically simpler
TDRC and FDRC are compared as baselines. The DFRC tradeoff between WSR and
beampattern approximation MSE is investigated. Numerical results show that
without radar sequence, RSMA-assisted DFRC has a better tradeoff performance
than SDMA-assisted DFRC, DFRC and TDRC in certain conditions. Most
importantly, although enabling radar sequence improves the performance of
SDMA-assisted DFRC and its interference on communication can be removed by
SIC, RSMA-assisted DFRC without radar sequence even outperforms SDMA-assisted
DFRC with radar sequence and SIC, and shares the same performance with RSMA-
assisted DFRC with radar sequence and SIC. This is because the common stream
of RSMA not only has the same function with radar sequence to match the
transmit beampattern and manage the interference between radar and
communication, but also has the unique capability to manage the interference
among communication users. The SIC receiver of RSMA is thus better exploited
to manage the interference between radar and communication as well as among
communication users. As a result, by using RSMA, the DFRC performance is
enhanced but the architecture is simplified because there is no need to use an
additional radar sequence and SIC. We conclude that RSMA is a more powerful
and promising technique for DFRC.

[Proof of (39)] We denote ${\bf p}^{\prime}=\left[{\bf p}_{1}^{T},{\bf
p}_{2}^{T},\dots,{\bf p}_{K}^{T}\right]^{T}$, and the selection matrix in (27)
as ${\bf D}_{\text{p},k}=\left[{\bf
0}^{N_{\text{t}}\times(k-1)N_{\text{t}}}\quad{\bf I}_{N_{\text{t}}}\quad{\bf
0}^{N_{\text{t}}\times(K-k)N_{\text{t}}}\right],k=1,2,\dots,K.$ Then,
IBR$(\theta_{m})$ is rewritten as

$\begin{split}\text{IBR}(\theta_{m})&=\frac{{\bf p}^{\prime
H}\left(\sum_{k\in\mathcal{K}}\sum_{j\neq k}{\bf D}_{\text{p},k}^{H}{\bf
h}_{j}{\bf h}_{j}^{H}{\bf D}_{\text{p},k}\right){\bf p}^{\prime}}{{\bf
p}^{\prime H}\left(\sum_{k\in\mathcal{K}}{\bf D}_{\text{p},k}^{H}{\bf
a}(\theta_{m}){\bf a}(\theta_{m})^{H}{\bf D}_{\text{p},k}\right){\bf
p}^{\prime}}.\end{split}$ (40)

Denoting ${\bf M}=\sum_{k\in\mathcal{K}}{\bf D}_{\text{p},k}^{H}{\bf
a}(\theta_{m}){\bf a}(\theta_{m})^{H}{\bf D}_{\text{p},k}$ and ${\bf
L}=\sum_{k\in\mathcal{K}}\sum_{j\neq k}{\bf D}_{\text{p},k}^{H}{\bf h}_{j}{\bf
h}_{j}^{H}{\bf D}_{\text{p},k}$, we have $\text{IBR}(\theta_{m})=\frac{{\bf
p}^{\prime H}{\bf L}{\bf p}^{\prime}}{{\bf p}^{\prime H}{\bf M}{\bf
p}^{\prime}}$.

According to the property of Rayleigh-Ritz quotient, we have the following
inequality $\frac{{\bf p}^{\prime H}{\bf L}{\bf p}^{\prime}}{{\bf p}^{\prime
H}{\bf M}{\bf p}^{\prime}}\geq\lambda_{\text{min}}\left({\bf M}^{-1}{\bf
L}\right)$ where $\lambda_{\text{min}}$ is the minimum eigenvalue of the
matrix respectively. Since ${\bf M}$ does not have full rank, we substitute
the inverse with the pseudo inverse ${\bf M}^{{\dagger}}$. With the fact that
$\frac{1}{N_{\text{t}}}{\bf M}{\bf M}={\bf M},$ we have ${\bf
M}^{{\dagger}}=(1/N_{\text{t}}^{2}){\bf M}$. Then we get ${\bf
M}^{{\dagger}}{\bf L}=(1/N_{\text{t}}^{2}){\bf M}{\bf
L}=(1/N_{\text{t}}^{3}){\bf M}{\bf M}{\bf L}$. Since the eigenvalues of ${\bf
M}{\bf M}{\bf L}$ are the same with those of ${\bf M}{\bf L}{\bf M}$, we
further derive

$\begin{split}{\bf M}&{\bf L}{\bf M}\\\
=&\frac{1}{N_{\text{t}}^{3}}\sum_{i\in\mathcal{K}}\sum_{k\in\mathcal{K}}\sum_{j\neq
k}\sum_{i^{\prime}\in\mathcal{K}}\left[{\bf D}_{p,i}^{H}{\bf
a}(\theta_{m}){\bf a}(\theta_{m})^{H}{\bf D}_{p,i}{\bf D}_{\text{p},k}^{H}{\bf
h}_{j}{\bf h}_{j}^{H}\right.\\\ &\left.{\bf D}_{\text{p},k}{\bf
D}_{p,i^{\prime}}^{H}{\bf a}(\theta_{m}){\bf a}(\theta_{m})^{H}{\bf
D}_{p,i^{\prime}}\right]\\\
=&\frac{1}{N_{\text{t}}^{3}}\sum_{k\in\mathcal{K}}\left[\sum_{j\neq
k}\lVert{\bf a}(\theta_{m})^{H}{\bf h}_{j}\lVert_{2}^{2}\right]{\bf
D}_{\text{p},k}^{H}{\bf a}(\theta_{m}){\bf a}(\theta_{m})^{H}{\bf
D}_{\text{p},k}.\end{split}$ (41)

It is easy to get the $K$ eigenvalues of ${\bf M}^{{\dagger}}{\bf L}$ as
$\lambda_{k}=(1/N_{\text{t}}^{2})\sum_{j\neq k}\lVert{\bf
a}(\theta_{m})^{H}{\bf h}_{j}\lVert_{2}^{2}$. Then, we have
$\lambda_{\text{min}}\left({\bf M}^{{\dagger}}{\bf
L}\right)=(1/N_{\text{t}}^{2})\min_{k\in\mathcal{K}}(\sum_{j\neq k}\lVert{\bf
a}(\theta_{m})^{H}{\bf h}_{j}\lVert_{2}^{2})$. Thus, (39) is proved.

## References

  * [1] C. Xu, B. Clerckx, S. Chen, Y. Mao, and J. Zhang, “Rate-splitting multiple access for multi-antenna joint communication and radar transmissions,” in _2020 IEEE International Conference on Communications Workshops (ICC Workshops)_ , Dublin, Ireland, 2020.
  * [2] H. Griffiths, L. Cohen, S. Watts, E. Mokole, C. Baker, M. Wicks, and S. Blunt, “Radar spectrum engineering and management: Technical and regulatory issues,” _Proceedings of the IEEE_ , vol. 103, no. 1, pp. 85–102, Jan 2015.
  * [3] F. Liu, C. Masouros, A. P. Petropulu, H. Griffiths, and L. Hanzo, “Joint radar and communication design: Applications, state-of-the-art, and the road ahead,” _IEEE Transactions on Communications_ , vol. 68, no. 6, pp. 3834–3862, 2020.
  * [4] B. Tang and J. Li, “Spectrally constrained MIMO radar waveform design based on mutual information,” _IEEE Transactions on Signal Processing_ , vol. 67, no. 3, pp. 821–834, 2019.
  * [5] F. Liu, C. Masouros, A. Li, and T. Ratnarajah, “Robust MIMO beamforming for cellular and radar coexistence,” _IEEE Wireless Communications Letters_ , vol. 6, no. 3, pp. 374–377, 2017.
  * [6] B. Li and A. P. Petropulu, “Joint transmit designs for coexistence of MIMO wireless communications and sparse sensing radars in clutter,” _IEEE Transactions on Aerospace and Electronic Systems_ , vol. 53, no. 6, pp. 2846–2864, 2017.
  * [7] J. Qian, M. Lops, L. Zheng, X. Wang, and Z. He, “Joint system design for coexistence of MIMO radar and MIMO communication,” _IEEE Transactions on Signal Processing_ , vol. 66, no. 13, pp. 3504–3519, 2018.
  * [8] L. Zheng, M. Lops, Y. C. Eldar, and X. Wang, “Radar and communication coexistence: An overview: A review of recent methods,” _IEEE Signal Processing Magazine_ , vol. 36, no. 5, pp. 85–99, 2019.
  * [9] F. Liu, A. Garcia-Rodriguez, C. Masouros, and G. Geraci, “Interfering channel estimation in radar-cellular coexistence: How much information do we need?” _IEEE Transactions on Wireless Communications_ , vol. 18, no. 9, pp. 4238–4253, 2019.
  * [10] A. Hassanien, M. G. Amin, Y. D. Zhang, and F. Ahmad, “Dual-function radar-communications: Information embedding using sidelobe control and waveform diversity,” _IEEE Transactions on Signal Processing_ , vol. 64, no. 8, pp. 2168–2181, 2015.
  * [11] ——, “Phase-modulation based dual-function radar-communications,” _IET Radar, Sonar & Navigation_, vol. 10, no. 8, pp. 1411–1421, 2016.
  * [12] F. Liu, C. Masouros, A. Li, H. Sun, and L. Hanzo, “MU-MIMO communications with MIMO radar: From co-existence to joint transmission,” _IEEE Transactions on Wireless Communications_ , vol. 17, no. 4, pp. 2755–2770, 2018\.
  * [13] C. Xu, B. Clerckx, and J. Zhang, “Multi-antenna joint radar and communications: Precoder optimization and weighted sum-rate vs probing power tradeoff,” _IEEE Access_ , vol. 8, pp. 173 974–173 982, 2020.
  * [14] Y. Mao, B. Clerckx, and V. O. Li, “Rate-splitting multiple access for downlink communication systems: bridging, generalizing, and outperforming SDMA and NOMA,” _EURASIP journal on wireless communications and networking_ , vol. 2018, no. 1, p. 133, 2018.
  * [15] B. Clerckx, Y. Mao, R. Schober, and H. V. Poor, “Rate-splitting unifying SDMA, OMA, NOMA, and multicasting in MISO broadcast channel: A simple two-user rate analysis,” _IEEE Wireless Communications Letters_ , vol. 9, no. 3, pp. 349–353, 2020.
  * [16] Y. Mao, B. Clerckx, and V. O. K. Li, “Rate-splitting for multi-antenna non-orthogonal unicast and multicast transmission: Spectral and energy efficiency analysis,” _IEEE Transactions on Communications_ , vol. 67, no. 12, pp. 8754–8770, 2019.
  * [17] Y. Mao and B. Clerckx, “Beyond dirty paper coding for multi-antenna broadcast channel with partial CSIT: A rate-splitting approach,” _IEEE Transactions on Communications_ , pp. 1–1, 2020.
  * [18] H. Joudeh and B. Clerckx, “Rate-splitting for max-min fair multigroup multicast beamforming in overloaded systems,” _IEEE Transactions on Wireless Communications_ , vol. 16, no. 11, pp. 7276–7289, 2017.
  * [19] M. Dai, B. Clerckx, D. Gesbert, and G. Caire, “A rate splitting strategy for massive MIMO with imperfect CSIT,” _IEEE Transactions on Wireless Communications_ , vol. 15, no. 7, pp. 4611–4624, 2016.
  * [20] M. Dai and B. Clerckx, “Multiuser millimeter wave beamforming strategies with quantized and statistical CSIT,” _IEEE Transactions on Wireless Communications_ , vol. 16, no. 11, pp. 7025–7038, 2017.
  * [21] Y. Mao, B. Clerckx, and V. O. K. Li, “Rate-splitting for multi-user multi-antenna wireless information and power transfer,” in _2019 IEEE 20th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC)_ , July 2019, pp. 1–5.
  * [22] D. R. Fuhrmann and G. San Antonio, “Transmit beamforming for MIMO radar systems using signal cross-correlation,” _IEEE Transactions on Aerospace and Electronic Systems_ , vol. 44, no. 1, pp. 171–186, 2008.
  * [23] P. Stoica, J. Li, and Y. Xie, “On probing signal design for MIMO radar,” _IEEE Transactions on Signal Processing_ , vol. 55, no. 8, pp. 4151–4161, 2007.
  * [24] S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” _Foundations and Trends® in Machine Learning_ , vol. 3, no. 1, pp. 1–122, 2011. [Online]. Available: http://dx.doi.org/10.1561/2200000016
  * [25] Z.-Q. Luo, W.-k. Ma, A. So, Y. Ye, and S. Zhang, “Semidefinite relaxation of quadratic optimization problems,” _Signal Processing Magazine, IEEE_ , vol. 27, pp. 20 – 34, 06 2010.
  * [26] M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex programming, version 2.1,” http://cvxr.com/cvx, Mar. 2014.
  * [27] Y. Sun, P. Babu, and D. P. Palomar, “Majorization-minimization algorithms in signal processing, communications, and machine learning,” _IEEE Transactions on Signal Processing_ , vol. 65, no. 3, pp. 794–816, 2017\.
  * [28] J. Song, P. Babu, and D. P. Palomar, “Optimization methods for designing sequences with low autocorrelation sidelobes,” _IEEE Transactions on Signal Processing_ , vol. 63, no. 15, pp. 3998–4009, 2015.
  * [29] Y. Nesterov and A. Nemirovskii, _Interior-point polynomial algorithms in convex programming_. Siam, 1994, vol. 13.
  * [30] H. Joudeh and B. Clerckx, “Sum-rate maximization for linearly precoded downlink multiuser MISO systems with partial CSIT: A rate-splitting approach,” _IEEE Transactions on Communications_ , vol. 64, no. 11, pp. 4847–4861, 2016.