# An Overview of Signal Processing Techniques for Joint Communication and
Radar Sensing

J. Andrew Zhang, , Fan Liu, ,
Christos Masouros, , Robert W. Heath Jr., ,
Zhiyong Feng, , Le Zheng, , Athina Petropulu,
J. A. Zhang is with the University of Technology Sydney, Australia. Email:
<EMAIL_ADDRESS>F. Liu, the corresponding author, is with Southern
University of Science and Technology, China. Email<EMAIL_ADDRESS>C.
Masouros is with University College London, UK. Email<EMAIL_ADDRESS>R. W. Heath Jr. is with the North Carolina State University, USA. Email:
<EMAIL_ADDRESS>Z. Feng is with Beijing University of Posts and
Telecommunications, China. Email<EMAIL_ADDRESS>L. Zheng is with the
Aptiv Company, USA. Email<EMAIL_ADDRESS>A. Petropulu is with The
State University of New Jersey, USA. Email<EMAIL_ADDRESS>
###### Abstract

Joint communication and radar sensing (JCR) represents an emerging research
field aiming to integrate the above two functionalities into a single system,
sharing a majority of hardware and signal processing modules and, in a typical
case, sharing a single transmitted signal. It is recognised as a key approach
in significantly improving spectrum efficiency, reducing device size, cost and
power consumption, and improving performance thanks to potential close
cooperation of the two functions. Advanced signal processing techniques are
critical for making the integration efficient, from transmission signal design
to receiver processing. This paper provides a comprehensive overview of JCR
systems from the signal processing perspective, with a focus on state-of-the-
art. A balanced coverage on both transmitter and receiver is provided for
three types of JCR systems, communication-centric, radar-centric, and joint
design and optimization.

###### Index Terms:

Dual-function Radar-Communications (DFRC), RadCom, Joint Radar-Communications
(JRC), Joint Communications-Radar (JCR), Joint Communication and Radio/Radar
sensing (JCAS).

##  I Introduction

###  I-A Background

Wireless communication and radar sensing (C&R) have been advancing in parallel
yet with limited intersections for decades. They share many commonalities in
terms of signal processing algorithms, devices and, to a certain extent,
system architecture. They can also potentially share the spectrum. These have
recently motivated significant research interests in the coexistence,
cooperation, and joint design (or co-design) of the two systems [1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13].

The coexistence of C&R systems has been extensively studied in the past
decade, with a focus on developing efficient interference management
techniques so that the two individually deployed systems can operate smoothly
without interfering with each other [3, 14, 15, 16, 6, 17, 18]. Although C&R
systems may be co-located and even physically integrated, they transmit two
different signals overlapped in time and/or frequency domains. They operate
cooperatively to minimize interference to each other. Great research efforts
have been devoted to mutual interference cancellation in this case, using, for
example, beamforming (BF) design in [18] and cooperative spectrum sharing in
[17]. However, effective interference cancellation typically has stringent
requirements on the mobility of nodes and information exchange between them.
The spectral efficiency improvement is hence limited in practice.

The joint design of colocated C&R systems considers the integration of these
two functions in one system. The initial concept may be traced back to 1960s
[19], and the research had been primarily on multimode or multi-function
military radars until 2010s. Recently, we are witnessing a booming interest
from both academia and industry on the joint system, thanks to its great
potentials in emerging defence applications and more recently in a range of
Smart Cities applications such as intelligent vehicular networks [20, 21] and
more general, the Internet of Things (IoT) [5]. Where there is a recognised
congestion of sensors and transceivers, integrating the two systems in one can
achieve reduced device size, power consumption, and cost, lead to more
efficient radio spectrum usage, and is expected to significantly expand the
capabilities and performance of both communication and radar sensing.

The integration may be classified into the following two classes: (1) The two
functions are physically integrated in one system, but they use two sets of
dedicated hardware components, and/or two different waveforms, superimposed or
separated in time, frequency or spatial domains [2]; (2) The two functions are
more firmly integrated by sharing the majority of hardware components and are
delivered by the same waveform, which is designed to optimize both
communication and radar performance [7, 8, 12]. The first class represents a
loose integration and may only achieve limited benefits such as reduction in
signalling overheads and removal of interference [12]. The second class is a
step change beyond collocation, and is the main focus of this paper.

Based on the design priorities and the underlying signal and systems, current
joint C&R systems may be classified into the following three categories:

  * •

Communication-centric design. In this class, radar sensing is an add on to a
communication system, where the design priority is on communications. The aim
of such design is to exploit communication waveform to extract radar
information through target echoes. Enhancements to hardware and algorithms are
required to support radar sensing. Possible enhancements to communication
standards may be introduced to enable better reuse of the communication
waveform for radar sensing purposes. In this design, the communication
performance can be largely unaffected, however, the sensing performance may be
scenario-dependent and difficult-to-tune;

  * •

Radar-centric design. Conversely, such approaches aim at modulating or
introducing information signalling in known radar waveform. Since the radar
signalling remains largely unaltered, the resulting approaches benefit from a
near optimal radar performance. The main drawback of such approaches is the
limited data rates achieved. Some performance loss may be tolerated by the
radar to enable better communication functionality; and

  * •

Joint design and Optimization. This class encompasses systems that are jointly
designed from the start, to offer a tunable trade-off between C&R performance.
Such systems may not be limited by any of the existing communication or radar
standards.

Owing to the significant differences between traditional C&R systems, the
design problems in these three categories are quite different. In the first
two categories, the design and research focus is typically on how to realize
the secondary radar (communication) function based on the signal formats of
the primary communication (radar) system, in a way that does not significantly
affect the primary system. The last category considers the joint design and
optimization of the signal waveform, system, and network architecture, with a
flexible tradeoff achievable between C&R.

These categories of joint systems have been receiving strong and growing
research interest, with results having been reported under various names, such
as Radar-communications (RadCom) [1], joint radar (and) communications (JRC)
[9, 7, 11], joint communication (and) radar (JCR) [20, 22], joint
communication and radar/radio sensing (JCAS) [23, 24], and dual-function(al)
radar communications (DFRC) [4, 25, 10, 26]. The first three typically refer
to a general joint system and can be used interchangeably. Sometimes JRC and
JCR are used to differentiate between radar-centric and communication-centric
designs. The term JCAS is introduced to stress the evolution of radar towards
more general radio sensing applications of communication-centric joint
systems. These sensing applications go beyond localization, tracking and
object recognition of traditional radar functions, such as human behaviour
recognition and atmosphere monitoring using radio signals [5]. DFRC is
specifically used for joint systems with a shared, single transmitted
waveform. It has been widely used for radar-centric joint systems [4, 8], and
has also been recently used for communication-centric systems [25, 10]. In the
rest of this paper, we use the term JCR to refer to a general joint system,
and use the term DFRC to refer to joint systems with a shared single transmit
waveform.

###  I-B Contributions and Structure of this Paper

There exist some excellent overview articles on JCR, particularly on system
and signal modelling. For example, [1] provides a foundational review of
signal models for basic single carrier and multicarrier JCR systems; [3, 6,
11] cover C&R systems from coexistence, cooperation, to co-design; [4, 8]
survey radar-centric DFRC systems, with a focus on signal embedding; [7]
delivers an excellent review on the evolution of JCR systems and their
potential applications; [12] provides an overview on mobile network JCR
systems; [9, 10, 13] overview millimeter-wave JCR systems for automotive
networks, providing detailed signal models. However, the topic of receiver
signal processing, which is critical for the viability and performance of the
JCR systems, has not been adequately covered by the aforementioned literature.

The objective of this paper is to provide a comprehensive overview on JCR from
the signal processing perspective, with balanced coverage on both transmitter
and receiver. For each of the first two categories of JCR designs, we review
main systems and typical signal models, and then discuss critical signal
processing techniques at the receiver. For the third category, we provide a
detailed overview of joint design and optimization techniques. To reduce the
overlap with existing articles, we also focus on more recent technologies. The
detailed structure of the rest of this paper is as follows.

  * •

Section II describes the signal and channel models of typical C&R systems,
separately. Via comparing these models, we disclose the connections and
differences between C&R, which are very important for the understanding and
development of JCR technologies;

  * •

Section III provides a review on communication-centric JCR technologies. We
first introduce two main JCR systems based on 802.11ad and mobile networks,
and then discuss signal processing technologies on sensing parameter
estimation, resolution of clock asynchrony, and sensing assisted
communications;

  * •

Section IV reviews radar-centric JCR technologies. Well-known information
embedding technologies are first summarized, and the recent promising
technology of index modulation (IM) is elaborated, with reference to several
emerging radar systems. We then describe signal reception and processing
techniques for communications, including information demodulation and channel
estimation for IM. Codebook design for IM is also discussed;

  * •

A detailed review of joint design and optimization technologies for JCR is
provided in Section V. These technologies include waveform optimization via
spatial precoding, multibeam optimization for analog array, and signal
optimization in other domains. With these technologies, balanced performance
between C&R can be achieved as desired; and

  * •

Concluding remarks are provided in Section VI.

Notations: $\mathbb{C}$ denotes the set of complex numbers. $(\cdot)^{H}$,
$(\cdot)^{*}$, $(\cdot)^{T}$ denote the Hermitian transpose, conjugate, and
transpose of a matrix or vector. $n!$ denotes $n$ factorial and
$C_{k}^{n}=n!/(k!(n-k)!)$ denotes the binomial coefficient. $\lfloor x\rfloor$
rounds towards negative infinity. $\\{x_{m}\\},m=1,\ldots,M$ denotes a column
vector with elements $x_{1},\ldots,x_{M}$.

##  II System and Signal Models

In this section, we describe some typical C&R systems and their signal models.
Since these systems and models are generally well known in respective areas,
we only provide brief descriptions as necessary to illustrate the JCR
technologies. The basic pulsed radar, continuous-wave radar, and communication
systems are illustrated in Fig. 1. We first present a beamspace channel model
that can be used for both C&R. We then describe signal models for C&R
separately, and highlight the differences between them. Signal models for JCR
will be described in later sections, based on the models presented here. For
the simplicity of notation, we assume the total transmitted power of all
systems is $1$, unless noted otherwise.

We consider a general system with $Q\geq 2$ nodes and each node has a uniform
linear antenna array (ULA). Let $B$ denote the total signal bandwidth. For
MIMO-OFDM systems, $B$ is divided into $N$ subcarriers and the subcarrier
interval is $f_{0}=B/N$. The OFDM symbol period is $T_{s}=T_{0}+T_{cp}$ where
$T_{0}=N/B$ and $T_{cp}$ is the period of cyclic prefix.

In this section, for the clarity of illustration, we describe the signal
models with reference to a general transmitter and receiver without indexing
them. Later, when these models need to be differentiated for different nodes,
we will add a subscript of the node index to the variables in the models.

Figure 1: Illustration of basic pulsed radar, continuous-wave radar, and
communication systems. Tx: transmitter; Rx: Receiver; BPF: bandpass filter;
PRI: Pulse repetition interval.

###  II-A Beam-space Channel Models

Let the angle-of-departure (AoD) and angle-of-arrival (AoA) of a multipath be
$\theta_{\ell}$ and $\phi_{\ell}$, $\ell\in[1,L]$, respectively. Assume a
planar wave-front in signal propagation. The array steering/response vector of
a ULA is given by

$\displaystyle\mathbf{a}(M,\alpha)=[1,e^{j2\pi
d/\lambda\sin(\alpha)},\cdots,e^{j(M-1)2\pi d/\lambda\sin(\alpha)}]^{T},$

where $M$ is the number of antennas, $\lambda$ is the wavelength, $d$ is the
interval of antennas, and $\alpha$ is either AoD or AoA.

For $M_{T}$ transmitting and $M_{R}$ receiving antennas, the $M_{R}\times
M_{T}$ time-domain baseband channel matrix at time $t$ can be represented as

$\displaystyle{\mathbf{H}}(t)=\sum_{\ell=1}^{L}b_{\ell}\delta(t-\tau_{\ell}-\tau_{o}(t))$
$\displaystyle e^{j2\pi(f_{D,\ell}+f_{o}(t))t}\cdot$
$\displaystyle\mathbf{a}(M_{R},\phi_{\ell})\mathbf{a}^{T}(M_{T},\theta_{\ell}),$
(1)

where for the $\ell$-th multipath, $b_{\ell}$ is its amplitude of complex
value, accounting for both signal attenuation and initial phase difference;
$\tau_{\ell}$ is the propagation delay; $f_{D,\ell}$ is the associated Doppler
frequency; and $\tau_{o}(t)$ and $f_{o}(t)$ denote the potential time-varying
timing offset and carrier frequency offset (CFO) due to possibly unlocked
clocks between transmitter and receiver, respectively. Here, we ignore the
“beam squint” effect in BF and assume $b_{\ell}$ is frequency independent.

Equation (1) represents a general channel model that can be used for both C&R,
although the physical meaning and names of these parameters are slightly
different. Our description above is mainly based on the terminologies in
communications. For radar sensing,
$\\{\tau_{\ell},f_{D,\ell},\phi_{\ell},\theta_{\ell},b_{\ell}\\}$ are the
sensing parameters to be estimated. These parameters can be used to determine
a target/reflector’s spatial and moving information. In particular,
$\phi_{\ell}$ and $\theta_{\ell}$ are the AoA and AoD of the target in
relation to the receiving and transmitting ULA, resepctively;
$\tau_{\ell}=R_{\ell}/c$ and $f_{D,\ell}=v_{\ell}f_{c}/c$ where $R_{\ell}$ is
the signal propagation distance, $c$ is the speed of light, $v_{\ell}$ is the
radial velocity of the reflector, and $f_{c}$ is the carrier frequency; and
$b_{\ell}$ is related to the radar cross-section (RCS) and material property
of the target. We define a coherent processing interval (CPI) when all these
parameters remain almost unchanged. The length of CPI depends on the mobility
of objects in the channels and is typically a few milliseconds when objects
move at speeds of tens of meters per second.

For a broadband OFDM system, the frequency-domain channel matrix at the $n$-th
subcarrier corresponding to (1) is

$\displaystyle\tilde{\mathbf{H}}_{n}(t)=\sum_{\ell=1}^{L}b_{\ell}$
$\displaystyle e^{-j2\pi
n(\tau_{\ell}+\tau_{o}(t))f_{0}}e^{j2\pi(f_{D,\ell}+f_{o}(t))t}\cdot$
$\displaystyle\mathbf{a}(M_{R},\phi_{\ell})\mathbf{a}^{T}(M_{T},\theta_{\ell}),$
(2)

where we have approximated the slightly varying phases due to Doppler
frequency and CFO over one OFDM block as a constant value. For the $k$-th OFDM
symbol with $t=kT_{s}$, we denote
$\tilde{\mathbf{H}}_{n,k}=\tilde{\mathbf{H}}_{n}(t)_{t=kT_{s}}$.

Note that for communications, we generally only need to know the composite
values of the matrix ${\mathbf{H}}(t)$ or $\tilde{\mathbf{H}}_{n}(t)$. They
can typically be obtained by directly estimating channel coefficients, or for
OFDM, directly estimating at some subcarriers and obtaining the rest via
interpolation. For radar sensing, however, the system needs to resolve the
detailed channel structure and estimate the sensing parameters. Note that this
beam-space channel model is also widely used in millimeter wave (mmWave)
communication systems.

When the oscillator clocks of the transmitter and receiver are not
locked/synchronized, both the timing offset $\tau_{o}(t)$ and CFO $f_{o}(t)$
are nonzero. The values of $\tau_{o}(t)$ can also be fast time-varying in
terms of ranging due to crystal oscillator’s instability, while CFO changes
relatively slower. For example, for a typical clock stability of $20$ parts-
per-million (PPM), the accumulated maximal variation of $\tau_{o}(t)$ over $1$
millisecond can be $20$ nanoseconds, which translates to a ranging error of
$6$ meters. For communications, these offsets are generally not a big problem,
as $\tau_{o}(t)$ can be absorbed into channel estimates after synchronization,
and $f_{o}(t)$ can be estimated and compensated. Their residual values become
relatively small compared to the baseband signal parameters. However, for
radar sensing, when these offsets are unknown to the receiver, they can cause
ambiguity in range and speed estimation, and become obstacles for processing
signals across packets coherently.

###  II-B Basic Communication Systems and Signals

We consider a node transmitting $M_{S}$ spatial streams with $M_{T}\geq M_{S}$
antennas. The description here is with reference to a single user MIMO system,
and it will be extended to multiuser MIMO later.

####  II-B1 Single Carrier MIMO

For a general single carrier (SC) MIMO system, we can represent the baseband
signal vector at time $t$ from the transmitter as

$\displaystyle\mathbf{x}(t)=\mathbf{P}\mathbf{s}(t),$ (3)

where $\mathbf{P}$ is the spatial precoding matrix of size $M_{T}\times M_{S}$
and is typically fixed over a packet, and $\mathbf{s}(t)$ are the data vector
of $M_{S}\times 1$. The symbols $\mathbf{s}(t)$ can be either the directly
modulated constellation points that are unknown to the receiver, or known
pilots. In the case of spread spectrum signals, each element in
$\mathbf{s}(t)$ can be the product of a pseudo-random code and the
constellation point (or pilot).

For a narrowband system, after propagating over the channel $\mathbf{H}(t)$,
the received signals are given by

$\displaystyle\mathbf{y}(t)=\mathbf{H}(t)\circledast\mathbf{x}(t)+\mathbf{z}(t),$
(4)

where $\circledast$ denotes convolution, and $\mathbf{z}(t)$ is the AWGN.

####  II-B2 MIMO-OFDM

For a MIMO-OFDM system, the baseband transmitting signals at subcarrier $n$ in
the $k$-th OFDM symbol over all antennas can be represented as

$\displaystyle\tilde{\mathbf{x}}_{n,k}=\mathbf{P}_{n,k}\mathbf{s}_{n,k},$ (5)

where these variables are similarly defined as those in (3), but could have
different values for different subcarriers. In the case of MIMO-OFDMA, each
node may be allocated to a resource block occupying groups of the antennas,
subcarriers, and OFDM symbols, that are typically non-overlapping. These
blocks can be discontinuous and irregular in these domains, which can cause
significant challenges in sensing parameter estimation, as will be discussed
in Section III-C.

After propagating over the channel, the received frequency-domain baseband
signals at subcarrier $n$ over all antennas are given by

$\displaystyle\tilde{\mathbf{y}}_{n,k}=\tilde{\mathbf{H}}_{n,k}\tilde{\mathbf{x}}_{n,k}+\tilde{\mathbf{z}}_{n,k},$
(6)

where $\tilde{\mathbf{z}}_{n,k}$ is the AWGN.

###  II-C MIMO Radar Signals and Systems

In a MIMO radar, the waveforms transmitted from different antennas are
typically orthogonal, and the waveform can be either pulsed or continuous
waveforms. The MIMO radar baseband waveform at a transmitter with $M_{T}$
antennas can be expressed as

$\displaystyle\mathbf{x}_{R}(t)=\sum_{m=1}^{M_{T}}\mathbf{w}_{m}\psi_{m}(t)=\mathbf{W}\bm{\psi}(t),$
(7)

where $\mathbf{x}_{R}(t)=\\{x_{R,m}(t)\\},m=1,\ldots,M_{T}$ with $x_{R,m}(t)$
being the signal at the $m$-th antenna, $\psi_{m}(t)$ is the basic radar
waveform, $\mathbf{w}_{m}$ is the $M_{T}\times 1$ precoding/BF vector, and
$\mathbf{W}=[\mathbf{w}_{1},\cdots,\mathbf{w}_{M_{T}}]$ and
$\bm{\psi}(t)=\\{\psi_{m}(t)\\}$, $m=1,\cdots,M_{T}$. Note that in radar, the
peak to average power ratio (PAPR) of the transmitted signal is typically
required to be very low, so that high power efficiency can be achieved. Thus
$\mathbf{W}$ is typically an identity matrix.

The basic waveforms $\psi_{m}(t)$ can be any set of signals that are
orthogonal in either one or multiple domains of time, frequency, space, and
code. It can also take any waveform of pulsed and continuous-wave radars [27,
28]. In pulsed radar systems, short pulses of large bandwidth are transmitted
either individually or in a group, followed by a silent period for receiving
the echoes of the pulses, as can be seen from Fig. 1. Continuous-wave radars
transmit waveforms continuously, typically scanning over a large range of
frequencies. In both systems, the waveforms are typically non-modulated.

Referring to the channel model in (1), $\tau_{o}(t)$ and $f_{o}(t)$ are zeros
for radar since the clocks for transmitter and receiver are typically locked.
With the transmitting signal in (7), the received noise-free radar signal is
given by

$\displaystyle\mathbf{y}_{R}(t)=\sum_{m=1}^{M_{T}}\sum_{\ell=1}^{L}$
$\displaystyle b_{\ell}\psi_{m}(t-\tau_{\ell})e^{j2\pi f_{D,\ell}t}\cdot$
$\displaystyle\mathbf{a}(M_{R},\phi_{\ell})\mathbf{a}^{T}(M_{T},\theta_{\ell})\mathbf{w}_{m},$
(8)

Applying matched filtering with $\psi_{m^{\prime}}(t)$ to $\mathbf{y}_{R}(t)$,
we obtain

$\displaystyle\mathbf{r}_{m}(t)=\sum_{\ell=1}^{L}$ $\displaystyle
b_{\ell}\rho_{m}(t-\tau_{\ell})e^{j2\pi
f_{D,\ell}t}\mathbf{a}(M_{R},\phi_{\ell})\mathbf{a}^{T}(M_{T},\theta_{\ell})\mathbf{w}_{m},$

where $\rho_{m}(t)$ is the non-zero output of the matched filtering of
$\psi_{m}(t)$ when $m=m^{\prime}$, as all other outputs for $m\neq m^{\prime}$
are zeros.

Assume that $\rho_{m}(t)=\rho(t)$ is the same for all $m\in[1,M_{T}]$. Staking
all $\mathbf{r}_{m}(t),m=1,\cdots,M_{T}$ to a matrix, we get

$\displaystyle\mathbf{R}(t)=\sum_{\ell=1}^{L}$ $\displaystyle
b_{\ell}\rho(t-\tau_{\ell})e^{j2\pi
f_{D,\ell}t}\mathbf{a}(M_{R},\phi_{\ell})\mathbf{a}^{T}(M_{T},\theta_{\ell})\mathbf{W}.$

When $\mathbf{W}$ is an identity matrix (if not, multiplying both sides with
$\mathbf{W}^{-1}$), we can vectorize $\mathbf{R}(t)$ and get

$\displaystyle\text{vec}(\mathbf{R}(t))=\sum_{\ell=1}^{L}$ $\displaystyle
b_{\ell}\rho(t-\tau_{\ell})e^{j2\pi
f_{D,\ell}t}\mathbf{a}(M_{R},\phi_{\ell})\otimes\mathbf{a}^{T}(M_{T},\theta_{\ell}),$

where $\otimes$ denotes the Kronecker product.

In MIMO radars, particularly mono-static radars, the antenna intervals of
transmitter and receiver, $d_{T}$ and $d_{R}$, are typically set as
$d_{T}=M_{T}d_{R}$ or $d_{R}=M_{T}d_{T}$. Then when
$\phi_{\ell}=\theta_{\ell}$, which is generally true for a monostatic radar,
we have
$\mathbf{a}(M_{R},\phi_{\ell})\otimes\mathbf{a}^{T}(M_{T},\theta_{\ell})=\mathbf{a}(M_{R}M_{T},\phi_{\ell})$.
This enables a MIMO radar to achieve the spatial resolution corresponding to a
virtual ULA with $M_{T}M_{R}$ antennas [28]. Note that the increased aperture
of the virtual array is only meaningful when $\phi_{\ell}$ is related to
$\theta_{\ell}$ in some way, although $\phi_{\ell}=\theta_{\ell}$ is not a
necessary condition.

In this paper, we mainly consider the following emerging MIMO radars, which
have not been widely implemented in practice, but have great potentials for
realizing JCR systems with balanced C&R performance.

####  II-C1 MIMO-OFDM Radar

In a MIMO-OFDM radar, the waveform $\psi_{m}(t)$ is in the form of time-domain
OFDM signals. Without considering the cyclic prefix, the baseband signal of
$\psi_{m}(t)$ can be represented as

$\displaystyle\psi_{m}(t)=\sum_{n\in\mathcal{S}_{m}}\tilde{w}_{m,n}e^{j2\pi
nf_{0}t}g(t-kT_{0}),$ (9)

where $\mathcal{S}_{m}$ is the set of used subcarriers, $g(t)$ is a windowing
function, and $\tilde{w}_{m,n}$ can be a complex variable combining orthogonal
coding, subcarrier-dependent spatial precoding/BF, and/or other processing
such as PAPR reduction.

To achieve orthogonality, $\mathcal{S}_{m}$s are typically selected to be
orthogonal for different $m$. The set of subcarriers can be allocated in
various forms, such as the interleaved pattern [29], and nonequidistant
subcarrier interleaving [30]. Different allocations may lead to different
signal properties and ambiguity functions. Alternatively, the orthogonality
may also be achieved over time domain via the use of orthogonal codes, while
different antennas share the subcarriers.

MIMO-OFDM signals for radar are very similar to those for communications,
except that they are typically non-modulated and/or orthogonal across
antennas. Hence MIMO-OFDM signals are excellent options for JCR systems. In
particular, the training sequence in the preamble of MIMO-OFDM communication
systems holds the desired characteristic of orthogonality, and can be directly
used for radar sensing.

####  II-C2 Frequency-Hopping MIMO Radar and Frequency Agile Radar

The frequency-hopping (FH) MIMO radar [31] and the frequency agile radar (FAR)
[32], as well as its extensions to multicarrier [33], all use the FH
technologies - the total bandwidth is divided into many subbands, only a
subset of subbands are used at a time, and the subset at each antenna randomly
varies over time. FH can be implemented in either fast time or slow time, and
in the form of either pulse or continuous-wave. We consider pulsed fast FH
here, i.e., the signals are continuously transmitted with frequencies being
changed rapidly and in multiple times over a PRI, followed by a silent period.
Using FH leads to major advantages such as better security, and lower
implementation cost by avoiding the use of costly instantaneous wideband
components, while with negligible degradation in sensing performance, compared
to using full bandwidth signals. FH-MIMO radar and FAR and its variants differ
in how the frequencies are used at each antenna. Next, we briefly present
their signal models, using notations similar to those for OFDM: $B$ for the
radar bandwidth, $N$ for the number of sub-bands, and $T$ for the hop
duration.

In a FAR, all antennas use one common frequency at a hop, and a BF weight is
applied to each antenna so that the array forms steerable beam [32]. The basic
concept of FAR is extended to multi-subband signaling in [33], that is, a
subset of more than one frequencies are used in each hop. In particular, in
the multi-Carrier AgilE phaSed ArrayRadar (CAESAR) scheme proposed in [33],
the whole array is divided into multiple non-overlapped subarrays, and each
antenna in one subarray only uses one common frequency from the frequency
subset. CAESAR randomizes both the frequencies and their allocation among the
antenna elements, and induces both frequency and spatial agility. It also
maintains narrowband transmission from each antenna and introduces the BF
capability. These capabilities make CAESAR more attractive than the original
FAR. Let $\mathcal{A}_{s}$ denote the index set of the antennas in the $s$-th
subarray, and $S$ be the number of total subarrays. Let $\mathcal{F}_{k}$ be
the set of frequencies selected for hop $k$, and $f_{k,m}\in\mathcal{F}_{k}$
denote the centroid frequency of the sub-band selected by antenna $m$ at hop
$k$. The transmitted signal of CAESAR at the $m$-th antenna can be represented
as

$\displaystyle\psi_{m}(t)=\tilde{w}_{k,m}e^{j2\pi f_{k,m}t}g(t-kT),$ (10)
$\displaystyle\text{s.t.}\ \left\\{\
\begin{array}[]{l}f_{k,m}=f_{k,m^{\prime}},\text{when}\
\\{m,m^{\prime}\\}\in\mathcal{A}_{s},\forall s;\\\ f_{k,m}\neq
f_{k,m^{\prime}},\ \text{otherwise}.\end{array}\right.$ (13)

where $\tilde{w}_{k,m}$ is the BF weight.

Although FH-MIMO was developed before CAESAR, it can be treated as a special
case of CAESAR where each subarray has only one antenna and each frequency is
only used by one antenna, that is $S=M_{T}$ and $|\mathcal{A}_{s}|=1$. CAESAR
can also be regarded as a generalization of FH-MIMO radar by introducing the
BF capability.

Both FH-MIMO radar and CAESAR are based on frequency division. They can also
be realized on the framework of MIMO-OFDM, with the frequency hopping concept
being introduced to subcarriers.

###  II-D Major Differences between C&R Systems and Signals

TABLE I: Brief comparison between C&R signals and systems. Properties |  Radar |  Communications
---|---|---
Typical Signal Waveforms |  Signals are unmodulated and have large bandwidth and Low PAPR; Orthogonal if MIMO-radar. |  Mix of unmodulated (pilots) and modulated symbols; High PAPR; complicated and diverse signal waveforms.
Signal Structure |  • A silent period follows each pulse transmission in pulsed radar to allow the reception of echo signals; • In continuous-wave radar, signals can be transmitted continuously, enabled by special hardware designs; • Signal repeats every PRI within CPI to increase received signal power and enable Doppler frequency estimation. |  • Typically packet-based. Typically no repetition; • Packet length and interval can be time-varying; • Signal may occupy discontinuous resources in time, frequency and space domains.
Transmission Capability (Duplex) |  • In continuous-wave radar, transmitted signal is used as local oscillator input at Rx to realize full duplex. This outputs “beat” signals only, characterizing the variation of the signals; • Pulsed radar operates in half duplex mode with a silent period following each pulse transmission. |  Time division duplex or frequency division duplex mode. Full duplex is immature for communications. Short-term solutions can be used to enable communication-centric JCR [12].
Clock Synchronization |  Transmitter and receiver are clock-locked in most radar setups, including monostatic, bistatic and multi-static systems. |  Co-located transmitter and receiver share the same timing clock, but non-colocated nodes typically do not.
Receiver Signal Sampling |  A conventional continuous-wave radar samples received signals at a rate much smaller than the scanning bandwidth, proportional to the desired detection capability of the maximal ranging and moving speed. This makes information conveying difficult. |  Sampling rate corresponds to the signal bandwidth. Full-bandwidth information available.
Performance Metrics |  Detection probability, Cramer-Rao lower bound (CRLB), Mutual information (MI), Ambiguity function |  Capacity, Rate, Spectral efficiency, Signal-to-interference-and-noise ratio (SINR), and Bit error rate (BER)

As can be partially observed from the signal models in preceding subsections,
there are some significant differences between C&R systems and signals,
despite of their potentials for integration. A brief comparison is summarized
in Table I, where radar waveforms are mainly referred to traditional pulsed
and continuous waveforms such as chirp. Next, we elaborate two aspects that
have considerable impacts on the joint system design.

Firstly, it is a fundamental challenge to address the potential requirements
for full duplex operation of JCR systems. On one hand, (mono-static) radar
addresses the requirement for full duplex operation in mainly two approaches,
as illustrated in Fig. 1, which may not be replicable in communication
systems. One approach is typically applied in a pulsed radar, by applying a
long silent period to receive echo signals, which essentially bypasses full-
duplex operation and makes the radar work in a time-division duplex mode; the
other is typically used in a continuous-wave radar, via using the transmitter
signal as the local template signal to the oscillator at the receiver, and
detecting only the “beat” signal, the difference between the transmitted and
received signals. Such designs enable low-complexity and efficient radar
sensing. However, they constrain the options of integrating communications and
limit the achievable communication rates. For example, there is large
uncertainty with the availability and bandwidth of the beat signal, hence
information conveying will be unreliable. On the other hand, full-duplex
operation is still immature for communications, and there is typically clock
asynchronism between spatially separated transmitting and receiving nodes.
These impose significant limits on integrating radar sensing into
communications.

Secondly, C&R signals are originally designed and optimized for different
applications, and are generally not directly applicable to each other. Radar
signals are typically designed to optimize localization and tracking accuracy,
and enable simple sensing parameter estimation. The following properties of
radar signals are desired: low PAPR to enable high-efficiency power amplifier
and long-range operation; and an waveform ambiguity function with steep and
narrow mainlobes for high resolution; Comparatively, communication signals are
designed to maximize the information-carrying capabilities, and are typically
modulated and packet-based. To support diverse devices and meet various
quality-of-services requirements, communication signals can have complicated
structures, with advanced modulations applied across time, frequency, and
spatial domains, and being discontinuous and fragmented over these domains.

These differences make the integration of C&R an interesting and challenging
task. As we will see from the following three sections, a good JCR design
always seek and exploit the commonalities between C&R, while taking these
differences into consideration.

##  III JCR: Communication-Centric Design

In communication-centric JCR systems, radar sensing is integrated into
existing communication systems as a secondary function. Revision and
enhancement to communication infrastructure and systems may be required, but
the primary communication signals and protocols largely remain unchanged.

Considering the topology of communication networks, communication-centric JCR
systems can be classified into two types, namely, those realizing sensing in
point-to-point communication systems and in large networks such as mobile
networks. Two good examples are the IEEE 802.11ad JCR systems for vehicular
networks [22, 13, 34] and the perceptive mobile networks [24, 12],
respectively. They use the single carrier and multiuser-MIMO OFDM signals as
described in Section II-B, respectively. Both are DFRC systems where a single
transmitted signal is used for both C&R.

In this section, we first describe the two types of DFRC systems, and then
review some signal processing techniques for general communication-centric JCR
systems.

###  III-A 802.11ad DFRC Systems

The 802.11ad standard defines a millimeter-wave packet communication system
operating in the 60 GHz unlicensed band. There are three types of physical-
layer (PHY) packets: single-carrier, OFDM, and control, with OFDM being
optional. The preamble in each packet is the main signal that has been
exploited for radar sensing [20, 35, 22, 13, 34] in an 802.11ad DFRC system.
Although the DFRC system can be applied in many scenarios, it has been mainly
investigated for vehicular networks. In a typical setup, the sensing receiver
is co-located with the DFRC transmitter, using two separated analog arrays.
The DFRC device can be located either on the road side unit (RSU) or on a
vehicle.

Referring to (1), the noise-free time-domain echo signal at the sensing
receiver can be represented as

$\displaystyle y(t)=\sum_{\ell=1}^{L}h_{\ell}(t)s(t-\tau_{\ell})e^{j2\pi
f_{D,\ell}t},$ (14)

where $h_{\ell}(t)\triangleq
b_{\ell}\mathbf{w}_{R}(t)^{T}\mathbf{a}(M_{R},\theta_{\ell})\mathbf{a}^{T}(M_{T},\theta_{\ell})\mathbf{w}_{T}(t)$,
$\mathbf{w}_{T}(t)$ and $\mathbf{w}_{R}(t)$ are the beamforming vector in the
transmitter and receiver respectively, and the AoA and AoD are assumed to be
the same. Note that the clock between the transmitter and sensing receiver is
locked, therefore $\tau_{o}(t)=0$ and $f_{o}(t)=0$ in (1). The primary goal of
sensing here is estimating the location and velocity of objects via estimating
$\tau_{\ell}$, $\theta_{\ell}$, and $\mathbf{f}_{D,\ell}$.

The three PHYs in 802.11ad have a similar preamble structure, consisting of
short training field (STF) and channel estimation field (CEF). The STF
consists of tens of repeated 128-sample Golay sequences, followed by its
binary complement. The CEF consists of two 512-sample Golay complementary
pair, which has the property of perfect aperiodic autocorrelation, i.e.,
$s(t-\tau)\circledast s(t)\neq 0$ if and only if $\tau=0$. Both the STF and
CEF can be used for sensing, in either a hierarchical or joint manner [20].
The hierarchical strategy processes the STF and CEF separately, exploiting
their respective properties. For example, the repetition pattern of STF is
typically used for packet detection in communications, and hence it is ideal
for target detection in sensing; while the perfect aperiodic autocorrelation
of CEF can lead to excellent channel estimation and sensing performance, based
on, e.g., the generalized likelihood ratio test [35]. The joint strategy uses
both STF and CEF for common tasks of sensing, based on, e.g., matched
filtering [20]. The sensing performance bounds are also derived in [20, 35].
More advanced sensing algorithms will be discussed in Section III-C.

Both the single-carrier PHY, which has an identical preamble with the OFDM
PHY, and the control PHY have been explored for sensing [35, 20]. There are
some differences between their sensing efficiency. On one hand, in the
standard, a beamforming training protocol is defined to align the transmit and
receive beams, using beam scanning and the control-PHY signals. Single-carrier
or OFDM PHY signals are typically used after beamforming training. On the
other hand, the control PHY has a longer STF. So in terms of sensing, the
control PHY enables a wider field-of-view (FoV) and potentially better
accuracy, while the other two are generally limited to the fixed direction of
communications. A multibeam approach, as will be discussed in Section V-B, can
be applied to relax this limitation.

###  III-B Mobile network DFRC Systems

In [24], the framework of perceptive mobile networks (PMNs) is introduced by
applying the JCR, more specifically DFRC, techniques, to cellular networks.
Downlink sensing and uplink sensing are defined, corresponding to sensing
using the received downlink and uplink communication signals, respectively, as
illustrated in Fig. 2. In the scenario of cloud radio access networks (CRANs)
where distributed remote radio units (RRUs) cooperatively communicate with
user equipment (UE), the received downlink communication signals from one RRU
itself and other cooperative RRUs can be used for downlink active sensing and
downlink passive sensing, respectively.

Extend the single user MIMO-OFDM model in Section II-B to multiuser MIMO-
OFDMA. Suppose that one node receives signals transmitted from a set of nodes
$q,q\in\mathcal{Q}_{T}$, and uses the signals for sensing. Let $Q_{T}$ be the
cardinality of $\mathcal{Q}_{T}$. Referring to the transmitting signal model
in (5) and the channel model in (2), we can represent the received noise-free
$k$-th OFDM symbol at the $n$-th subcarrier as

$\displaystyle\tilde{\mathbf{y}}_{n,k}=\sum_{q\in\mathcal{Q}_{T}}\sum_{\ell=1}^{L_{q}}$
$\displaystyle b_{q,\ell}e^{-j2\pi n(\tau_{q,\ell}+\tau_{o,q,k})f_{0}}e^{j2\pi
k(f_{D,q,\ell}+f_{o,q,k})T_{s}}\cdot$
$\displaystyle\quad\mathbf{a}(M_{R},\phi_{q,\ell})\mathbf{a}^{T}(M_{q},\theta_{q,\ell})\tilde{\mathbf{x}}_{q,n,k}$
(15)

Scenarios represented by this model are exemplified below:

  1. 1.

Downlink sensing in a standalone base station (BS): This is the case where the
BS uses its own reflected transmitted signals for sensing, similar to a mono-
static radar. In this case, $Q_{T}=1$ and $\tau_{o,q,k}=f_{o,q,k}=0$;

  2. 2.

Uplink sensing in a standalone BS: $\mathcal{Q}_{T}$ denotes the set of
$Q_{T}$ UEs sharing the same subcarriers via SDMA, and $\tau_{o,q,k}\neq 0$
and $f_{o,q,k}\neq 0$.. Each UE only occupies partial of the total
subcarriers;

  3. 3.

Downlink sensing in an RRU: $\mathcal{Q}_{T}$ denotes the set of RRUs whose
downlink communication signals are seen by the sensing RRU, including its own
echo signals.

The sensing can be based on (15) with the signals $\tilde{\mathbf{x}}_{q,n,k}$
corresponding to both the pilots and data symbols, as will be detailed next.
For more details on the signals usable for sensing, as well as solutions to
the full-duplex problems in PMNs, the readers are referred to [24, 12].

Figure 2: Network setup of PMNs (top subfigure) and the brief system block
diagram of a DFRC transceiver based on MIMO-OFDM (bottom subfigure).

###  III-C Sensing Parameter Estimation

Sensing parameter estimation in communication-centric JCR is generally
different to that in traditional radar systems, due to the significant
differences between the two types of signals as described in Section II. Next,
referring to the MIMO-OFDM signal models in Section III-B, we review key
techniques in sensing parameter estimation. Most of them are also applicable
to single carrier systems such as the IEEE802.11ad DFRC. We first discuss two
important problems to be resolved, before we review optional sensing
algorithms.

####  III-C1 Direct and Indirect Sensing

The first problem is how to deal with $\tilde{\mathbf{x}}_{q,n,k}$ in the
received signals, which can represent either known pilots or (unknown) data
payload in a packet. Both can be used for sensing. For unknown data payload,
it can be demodulated after channel estimation, as in conventional
communications. Using data payload can significantly extend the sensing
capability such as the range, as it is much longer than the pilot. Here, the
receiver is assumed to know $\tilde{\mathbf{x}}_{q,n,k}$ or its estimate via
demodulating $\tilde{\mathbf{s}}_{q,n,k}$. For multiuser-MIMO signals, for
example, signals received at an RRU from multiple RRUs in downlink sensing, or
signals received at a standalone BS from multiple UEs, we can use two methods
to formulate the estimation problem.

One method, which may be called as direct sensing, directly feeds the received
signals to sensing algorithms. In some cases, e.g., sensing using the data
payload in a MIMO system, this is the only option as
$\tilde{\mathbf{x}}_{q,n,k}$ cannot be readily removed even when they are
known. The presence of $\tilde{\mathbf{x}}_{q,n,k}$ often limits the optional
algorithms for sensing parameter estimation.

Let us have a look at one example in [24], where direct sensing is conducted
via the block compressive sensing (CS) techniques [36], and the symbols
$\tilde{\mathbf{x}}_{q,n,k}$ are used as part of the sensing matrix. For the
clarify of presentation, we consider the case of $Q_{k}=1$ in (15), and ignore
the timing offset $\tau_{o,q,k}$. Ignore the noise and rewrite (15) in a more
compact matrix form as

$\displaystyle\tilde{\mathbf{y}}_{n,k}=\mathbf{A}(M_{R},\bm{\phi}_{q})\mathbf{C}_{n}\mathbf{D}_{k}\mathbf{A}^{T}(M_{q},\bm{\theta}_{q})\tilde{\mathbf{x}}_{q,n,k},$
(16)

where $\mathbf{A}(M_{R},\bm{\phi}_{q})$ and
$\mathbf{A}(M_{q},\bm{\theta}_{q})$ are matrices with the $\ell$-th column
being $\mathbf{a}(M_{R},\phi_{q,\ell})$ and
$\mathbf{a}(M_{q},\theta_{q,\ell})$), respectively, and $\mathbf{D}_{k}$ and
$\mathbf{C}_{n}$ are diagonal matrices with the diagonal element being
$b_{q,\ell}e^{j2\pi kf_{D,q,\ell}T_{s}}$ and $e^{-j2\pi n\tau_{q,\ell}f_{0}}$,
respectively.

In order to apply sensing algorithms, we need to re-organize signals so that
we can stack more measurements over the same domain. Consider the case of
collecting samples from all the subcarriers for the estimation of delay and
AoA. Take the transpose of $\tilde{\mathbf{y}}_{n,k}$ in (16), and rewrite it
as

$\displaystyle\tilde{\mathbf{y}}_{n,k}^{T}$
$\displaystyle=\tilde{\mathbf{x}}^{T}_{q,n,k}(\mathbf{c}^{T}_{n}\otimes\mathbf{I}_{M_{q}})\mathbf{V}_{q}\mathbf{A}^{T}(M_{R},\bm{\phi}_{q}).$
(17)

where $\mathbf{c}_{n}$ is a column vector containing the diagonal elements of
$\mathbf{C}_{n}$, $\mathbf{I}_{M_{q}}$ is an $M_{q}\times M_{q}$ identity
matrix, and $\mathbf{V}_{q}$ is a block diagonal matrix

$\displaystyle\mathbf{V}_{q}=\text{diag}\\{b_{\ell}e^{-j2\pi
kf_{D,q,\ell}T_{s}}\mathbf{a}(M_{q},\theta_{q,\ell})\\},\ell=1,\ldots,L_{q}.$

We have now separated signals
$\tilde{\mathbf{x}}^{T}_{q,n,k}(\mathbf{c}^{T}_{n}\otimes\mathbf{I}_{M_{q}})$
that depend on $n$ from those on other variables. We can then stack the row
vectors $\tilde{\mathbf{y}}_{n,k}^{T}$ from all available subcarriers to a
matrix, and obtain

$\displaystyle\tilde{\mathbf{Y}}_{k}$
$\displaystyle\triangleq[\tilde{\mathbf{y}}_{1,k},\cdots,\tilde{\mathbf{y}}_{n,k},\cdots]^{T}=\mathbf{W}\mathbf{V}_{q}\mathbf{A}^{T}(M_{R},\bm{\phi}_{q}),$
(18)

where the $n$-th row of $\mathbf{W}$ is
$\tilde{\mathbf{x}}^{T}_{q,n,k}(\mathbf{c}^{T}_{n}\otimes\mathbf{I}_{M_{q}})$.

The signal model in (18) enables the applications of both 1-D multi-
measurement vector (MMV) CS and 2-D CS techniques. For 1-D MMV CS,
$\mathbf{c}_{n}$ is expanded to a quantized on-grid vector, and then
$\mathbf{W}$ is used as the sensing matrix. The delay and
$\mathbf{V}_{q}\mathbf{A}^{T}(M_{R},\bm{\phi}_{q})$ will then be the outputs
of the algorithm, and the Doppler frequency and AoA can be further estimated
from the estimates of $\mathbf{V}_{q}\mathbf{A}^{T}(M_{R},\bm{\phi}_{q})$. For
2D CS, both delay and AoD can be estimated together by expanding both
$\mathbf{c}_{n}$ and $\mathbf{A}(M_{R},\bm{\phi}_{q})$ to on-grid models. It
is easy to see that if we swap the terms of Doppler frequency and delay in
(17), samples across OFDM symbols can be stacked and the Doppler frequencies
can be estimated first.

The direct sensing method has a high computational complexity. Due to the
presence of $\tilde{\mathbf{x}}_{q,n,k}$, the applicable sensing solutions are
also limited. However, it is the only option, when
$\tilde{\mathbf{x}}_{q,n,k}$ cannot be removed due to, e.g, insufficient
measurements.

The other method, indirect sensing, first estimate the elements of channel
matrix for each node. It decorrelates signals from multiple nodes, removes
$\tilde{\mathbf{x}}_{q,n,k}$ or $\tilde{\mathbf{s}}_{q,n,k}$ from the received
signals, and applies sensing parameter estimation to the estimated channel
matrix. In multi-user MIMO systems, referring to (15), this can be achieved by
decorrelating signals collected from $K$ $\tilde{\mathbf{y}}_{n,k}$s,
$k=k^{\prime},k^{\prime}+1,\cdots,k^{\prime}+K-1,K\geq M_{T}Q_{T}$ at
subcarrier $n$. Mathematically, this can be represented as

$\displaystyle\left[\tilde{\mathbf{H}}_{1,n,k},\cdots,\tilde{\mathbf{H}}_{Q_{T},n,k}\right]$
$\displaystyle=$
$\displaystyle\left[\breve{\mathbf{x}}_{n,1},\cdots,\breve{\mathbf{x}}_{n,K}\right]^{-1}\left[\tilde{\mathbf{y}}_{n,1},\cdots,\tilde{\mathbf{y}}_{n,K}\right],$
(19)

where
$\breve{\mathbf{x}}_{n,k}=[\tilde{\mathbf{x}}_{1,n,k}^{T},\cdots,\tilde{\mathbf{x}}_{Q_{T},n,k}^{T}]^{T}$
is a $M_{T}Q_{T}\times 1$ vector. Note that the channel matrix is assumed to
be constant over this interval. Equation (19) indicates that the decorrelation
is only possible when $K\geq M_{T}Q_{T}$ $\breve{\mathbf{x}}_{n,k}$s are
available during a CPI and the inversion of
$\left[\breve{\mathbf{x}}_{n,1},\cdots,\breve{\mathbf{x}}_{n,K}\right]$
exists.

The decorrelation involves high computational complexity due to matrix
inversion and may cause significant noise enhancement, unless
$\tilde{\mathbf{x}}_{q,n,k}$ or $\tilde{\mathbf{s}}_{q,n,k}$ are orthogonal.
Hence indirect sensing is particularly suitable for training and pilot symbols
which are typically orthogonal. After $\tilde{\mathbf{x}}_{q,n,k}$ is removed,
we can equivalently work on the single user channel matrix
$\tilde{\mathbf{H}}_{n}(t)$ in (2). This can largely simplify sensing
parameter estimation, and offer great flexibility in problem formulation. Note
that if the precoding matrix $\mathbf{P}_{n,k}$ is unknown to the receiver,
$\mathbf{a}^{T}(M_{T},\theta_{\ell})$ in (2) will be replaced by
$\mathbf{a}^{T}(M_{T},\theta_{\ell})\mathbf{P}_{n,k}$. This will make the
estimation of AoD challenging.

####  III-C2 Clutter Removal

The second problem is how to deal with clutter signals that are useless for
sensing. Communication systems are typically deployed in an environment with
dense multipath, where many signal propagation paths involve static objects
only and are not of the interest of sensing. Removing clutter before sensing
may distort the signal, however, it can largely reduce the number of
parameters to be estimated. In this sense, it is generally a better strategy
to remove clutter before the application of sensing algorithms, using, e.g.,
background subtraction or filtering techniques. For detailed discussions on
potential clutter removal techniques, the readers are referred to [24].

####  III-C3 Sensing Algorithms

We now discuss options for sensing algorithms based on the indirect method.
Traditional radar typically applies matched filtering for sensing parameter
estimation, which has also been adopted in some DFRC systems, e.g., 802.11ad
DFRC [20, 22, 34]. However, the accuracy and resolution capability of these
methods largely depend on the signal correlation properties (i.e., ambiguity
functions). More options that are less affected by the correlation property
can be explored for communication-centric DFRC signals.

From the decorrelated estimates of $\tilde{\mathbf{H}}_{n,k}$s in (2), we can
represent the $(m_{R},m_{T})$-th element in $\tilde{\mathbf{H}}_{n,k}$ as

$\displaystyle\tilde{h}_{n,k,m_{R},m_{T}}$
$\displaystyle=\sum_{\ell=1}^{L}b_{\ell}e^{-j2\pi n\tau_{\ell}f_{0}}e^{j2\pi
kf_{D,\ell}T_{s}}$ $\displaystyle\quad\cdot e^{j2\pi
d_{R}m_{R}\sin(\phi_{\ell})/\lambda}e^{j2\pi
d_{T}m_{T}\sin(\theta_{\ell})/\lambda},$ (20)

where $m_{R}$ and $m_{T}$ represent the indexes of the receiving and
transmitting antennas, respectively. This is also known as a 4-D Harmonic
retrieval problem [37], where the observation signals in each domain can be
represented as a Vandermonde matrix when the samples are equally spaced. The
4-D harmonic retrieval problem can be reduced to multiple-snapshot lower-
dimensional problems by combining one or more of the exponential functions
with the unknown variable $b_{\ell}$. Thus, we can rewrite them to different
matrix and Tensor forms so that sensing parameters can be estimated in
different ways and orders.

Since solutions to the classical harmonic retrieval problems are generally
well studied, we ignore the details and only provide a comparison of typical
techniques in Table II with reference to our sensing problems. More details of
such techniques for JCR can be referred to [37, 38, 24, 39, 40].

When selecting sensing algorithms, the following issues need to be further
considered:

  1. 1.

Modern communication signals are typically very complicated in terms of
resource usage, and they may be discontinuous in one and more domains of
space, frequency, and time. See [12] for the available sensing signals and
their properties in 5G-based PMNs. This requires sensing algorithms with the
capability of processing discontinuous and varying-interval samples;

  2. 2.

Higher-dimension algorithms can generally identify more parameters and achieve
better estimation performance at the cost of higher complexity. However, when
there are insufficient samples in one domain, it could be better to use lower-
dimension algorithms.

TABLE II: Comparison of Sensing Parameter Estimation Algorithms Algorithms |  Advantages |  Main limitations
---|---|---
Periodogram such as 2D DFT (typically based on the outputs of matched filtering) |  Traditional technique. Simple and easy to implement. May be used as the starting point for other algorithms. |  • Low resolution; • Generally require a full set of continuous samples in each domain, which may not always be satisfied.
Maximal Likelihood Estimation [35] |  Statistically optimal formulation; Particularly suitable for low-dimension signals. |  Typically require searching to find the solutions and hence complexity is high; Complexity also increases with signal dimensions exponentially.
Subspace methods such as ESPRIT and MUSIC [40, 41] |  • Separate signal and noise subspaces and hence is resilient to noise; • ESPRIT can achieve very high resolution and can do off-grid estimation; • MUSIC can flexibly work with non-continuous samples. |  • ESPRIT requires a large segment of consecutive samples, which may not always be satisfied; • Resolution of MUSIC depends on searching granularity; • High complexity associated with singular value decomposition.
Compressive sensing (On-grid) [38, 24] |  • Flexible and does not require consecutive samples; • Various recovery algorithms available, allowing good tradeoff between complexity and performance; • Different dimensions of formulation can be used, adapting to sensing requirements and conditions; • Dense dictionaries can be used to improve resolution. |  Although it may even work well for estimating a small amount of off-grid parameters, performance can degrade significantly when the number of parameters to be estimated is large.
Compressive Sensing (Off-grid) such as atomic norm minimization [38] |  Have all the advantages of on-grid CS algorithms. Capable of estimating off-grid values. |  Limitation in real time operation due to very high complexity. Still require sufficient separation between parameter values.
Tensor based algorithms [37] |  High-dimension formulation and estimation are made easy. Reduce computational complexity and provide capability in resolving multipath with repeated parameter values. |  Need to be combined with other algorithms such as ESPRIT and CS, thus facing their inherent problems.

###  III-D Resolution of Clock Asynchrony

As described in Section II-A, when the oscillator clocks between the
transmitter and the sensing receiver are not locked, the timing offset
$\tau_{o}(t)$ and CFO $f_{o}(t)$ in (1) and (2), are typically non-zero and
time-varying. They can directly cause ambiguity in range and speed estimation.
They also prevent from aggregating measurements over a relatively long
interval, e.g., preamble signals from two packets, for joint processing, which
is otherwise important for parameter estimation, particularly Doppler
frequencies. This is a critical and challenging problem in communication-
centric JCR systems.

There have been a limited number of works that address this problem in passive
WiFi sensing [42, 43, 44], based on the cross-antenna cross-correlation (CACC)
method. The basic assumption is that timing offsets and CFO across multiple
antennas in the receiver are the same, as the same clock is used. Therefore
these offsets can be removed by computing the cross-correlation between
signals from multiple receiving antennas. Considering a single transmitter
with a single antenna and referring to (15), the received noise-free signal at
the $m$-th antenna can be rewritten as

$\displaystyle\tilde{y}_{n,k,m}=\sum_{\ell=1}^{L}b_{\ell}e^{-j2\pi
n(\tau_{\ell}+\tau_{o,k})f_{0}}$ $\displaystyle e^{j2\pi
k(f_{D,\ell}+f_{o,k})T_{s}}\dot{}$ $\displaystyle
e^{jm\pi\sin(\phi_{\ell})}\tilde{x}_{n,k}.$

Let the $m_{0}$-th antenna be the reference. Computing the cross-correlation
between $\tilde{y}_{n,k,m}$ and $\tilde{y}_{n,k,m_{0}}$ yields

$\displaystyle R(n,k,m)=\tilde{y}_{n,k,m}\tilde{y}_{n,k,m_{0}}^{*}$ (21)
$\displaystyle=$
$\displaystyle\sum_{\ell_{m}=1}^{L}\sum_{\ell_{m_{0}}=1}^{L}b_{\ell_{m}}b_{\ell_{m_{0}}}^{*}e^{-j2\pi
n(\tau_{\ell_{m}}-\tau_{\ell_{m_{0}}})f_{0}}$ $\displaystyle\cdot e^{j2\pi
k(f_{D,\ell_{m}}-f_{D,\ell_{m_{0}}})T_{s}}e^{jm\pi(\sin(\phi_{\ell_{m}})-\sin(\phi_{\ell_{m_{0}}}))}|\tilde{x}_{n,k}|^{2}.$

Note that in (21), $\tau_{o,k}$ and $f_{o,k}$ are removed. However, cross-
correlation causes doubled terms and sensing parameters become relative.

Two assumptions, which limit the applications, are necessary for subsequent
processing: (1) The transmitter and sensing receiver are relatively static and
the relative location of the transmitter is known to the receiver; and (2)
there exists a line-of-sight (LOS) path between them and it has much larger
magnitude than non-LOS paths. The cross-product between LOS paths is invariant
over the CPI and can be removed by passing $R(n,k,m)$ through a high pass
filter. The cross-terms between NLOS and LOS paths thus dominate in the output
of the filter. The sensing parameters can then be estimated, with respect to
the known parameters of the LOS path.

However, the outputs after CACC contain cross-product terms that include
signals and their images, and hence the number of unknown parameters to be
estimated is actually doubled. This will not only cause degraded estimation
accuracy, but also ambiguity between the actual value and its image. The
authors in [43] proposed an add-minus method to suppress the image signals by
adding a constant to $\tilde{y}_{n,k,m}$ and subtracting another one to
$\tilde{y}_{n,k,m_{0}}$. However, this method is found to be susceptible to
the number and power distribution of static and dynamic signal propagation
paths. To resolve these problems, a method is proposed in [41] by introducing
a mirrored MUSIC algorithm.

Observing from (21), we can see that the relative delays and Doppler
frequencies have values symmetric to zero. Exploiting this symmetric, the
mirrored MUSIC algorithm first constructs new signals from the CACC outputs,
or the outputs of a further high pass filter to remove the dominating static
components, and then define new basis vectors for MUSIC algorithms. Both of
the new signals and basis vectors are constructed by adding the original ones
with their sample-reversed versions. The mirrored MUSIC algorithm equivalently
reduces the number of unknown parameters by half, and is shown to
significantly improve the estimation accuracy, and simplify the ambiguity
resolution problem associated with image signals.

###  III-E Sensing Assisted Communications

In many location-aware services and applications, e.g., V2X network, sensing
and communication are recognized as a pair of intertwined functions, where
communication-centric JCR can be applied so as to reduce the costs and improve
spectral-, energy-, and hardware-efficiency. In addition to those general
advantages, one may also leverage the sensing results to facilitate
communication, where significant performance gain can be obtained over that of
the conventional communication-only schemes. Examples are sensing assisted
secure communications [45] and BF [46, 47, 48, 49]. Below we briefly review
the recent state-of-the-art on sensing-assisted BF.

BF for mmWave communications relies on the AoA and AoD parameters of signal
propagation paths. With such angular information at hand, the Tx and the Rx
can accurately align their beams, such that a high-quality communication link
can be established. Conventionally, the AoA and AoD are acquired by _beam
training_ or _beam tracking_ techniques, i.e., communication-only approaches.
The basic idea of these schemes is to send pilots from the Tx to the Rx before
data transmission, with each pilot being beamformed to a different direction.
The Rx then estimates the angles from the received pilot signal and feeds them
back to the Tx.

It can be observed from the above procedure that there exists an inherent
tradeoff between the estimation accuracy and the signalling overhead. Indeed,
the beam information can be more precisely attained by transmitting more
pilots, which, however, is at the price of reducing the effective transmission
time of the useful data, as well as of increasing the latency. This problem is
particularly pronounced in high-mobility V2X scenarios, where the trained
beams can be easily outdated. To cope with this issue, radar sensing is
exploited to enhance the BF performance via providing accurate positioning
information for vehicles, such that the beam search interval can be narrowed
down. Recent results show that with the aid of an extra radar sensor mounted
on the road side unit (RSU), the beam training overhead incurred in the
vehicle-to-infrastructure (V2I) links can be significantly reduced to 6.5% of
that of the Global Navigation Satellite System assisted approach [46, 47].

As a step beyond using stand-alone radar sensors, JCR signalling is envisioned
to play a unique role in V2I communications, offering not only considerable
reduction in the overhead, but also the capability to beamform towards
predicted directions of the vehicles, in order to adapt the fast-changing
vehicular channels. To show this, let us consider a mmWave RSU equipped with
$M_{T}$ transmit and $M_{R}$ receive antennas, which acts as a mono-static
radar, and is serving a single-antenna vehicle driving at a nearly constant
speed on a straight road. We assume that the RSU communicates with the vehicle
over a single LoS path, and that all the antenna arrays are adjusted to be
parallel to the road. As a consequence, the AoA equals to the AoD in the V2I
LoS channel.

At the $k$th epoch, the RSU transmits a DFRC signal
$\mathbf{x}_{k}\left(t\right)={{\mathbf{f}}_{k}}s_{k}\left(t\right)$ from the
RSU to the vehicle, with ${{\mathbf{f}}_{k}}$ being the JCR beamformer, and
$s_{k}\left(t\right)$ being the data stream. The signal is partially received
by the vehicle’s antenna array, and is partially reflected back to the RSU.
The received echo signal can be expressed in the form of

$\begin{gathered}{{\mathbf{y}}_{R}}\left(t\right)=b_{k}{e^{j2\pi{f_{D,k}}t}}{\mathbf{a}}\left({{M_{R}},{\theta_{k}}}\right){{\mathbf{a}}^{T}}\left({{M_{T}},{\theta_{k}}}\right)\cdot\hfill\\\
\quad\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{{\mathbf{f}}_{k}}s_{k}\left({t-{\tau_{k}}}\right)+{\mathbf{z}}_{R}\left(t\right),\hfill\\\
\end{gathered}$ (22)

where the beamformer $\mathbf{f}_{k}$ is designed based on a predicted angle,
which is
${{\mathbf{f}}_{k}}={{\mathbf{a}}^{*}}\left({{M_{T}},{{\hat{\theta}}_{k\left|{k-1}\right.}}}\right)$.
Moreover, ${{\hat{\theta}}_{k\left|{k-1}\right.}}$ is the $k$th predicted
angle based on the $\left(k-1\right)$th estimate, and $b_{k}$, $f_{D,k}$,
$\theta_{k}$, and $\tau_{k}$ denote the reflection coefficient, the Doppler
frequency, the AoA, and the round-trip delay for the vehicle at the $k$th
epoch, respectively. In particular, $f_{D,k}$ and $\tau_{k}$ can be further
written as functions of the distance $d_{k}$, the velocity $v_{k}$, and the
AoA $\theta_{k}$, which are
${f_{D,k}}=\frac{{2{v_{k}}\cos{\theta_{k}}{f_{c}}}}{c},{\tau_{k}}=\frac{{2{d_{k}}}}{c}$.
By matched-filtering (22) with a delayed and Doppler shifted counterpart of
$s_{k}\left(t\right)$, one obtains the estimates of the Doppler and the time-
delay, denoted as $\hat{f}_{D,k}$ and $\hat{\tau}_{k}$. Let us denote the
vehicle’s state as
${\mathbf{q}_{k}}={\left[{{\theta_{k}},{d_{k}},{v_{k}},{b_{k}}}\right]^{T}}$.
The sensing measurement and the vehicle state can be connected by a function
${{{\mathbf{\tilde{r}}}}_{k}}={\mathbf{h}}\left({{{\mathbf{q}}_{k}}}\right)+{{\mathbf{z}}_{k}}$,
which can be expanded as [48]

$\begin{gathered}\left\\{\begin{gathered}{{{\mathbf{\tilde{y}}}}_{R}}={b_{k}}{E_{k}}{\mathbf{a}}\left({{M_{R}},{\theta_{k}}}\right){{\mathbf{a}}^{T}}\left({{M_{T}},{\theta_{k}}}\right){{\mathbf{a}}^{*}}\left({{M_{T}},{{\hat{\theta}}_{k\left|{k-1}\right.}}}\right)+{{\mathbf{z}}_{\theta}},\hfill\\\
{{\hat{f}}_{D,k}}=\frac{{2{v_{k}}\cos{\theta_{k}}{f_{c}}}}{c}+{z_{f}},\hfill\\\
{{\hat{\tau}}_{k}}=\frac{{2{d_{k}}}}{c}+{z_{\tau}},\hfill\\\
\end{gathered}\right.\hfill\\\ \end{gathered}$ (23)

where $E_{k}$ is the matched filtering gain, and
$\mathbf{z}_{k}=\left[z_{\theta}^{T},z_{f},z_{\tau}\right]^{T}$ denotes the
measurement noise.

Recalling the assumption that the vehicle is driving on a straight road with a
nearly constant speed, the state transition can be modeled as
${{\mathbf{q}}_{k}}={\mathbf{g}}\left({{{\mathbf{q}}_{k-1}}}\right)+{{\bm{\omega}}_{k}}$,
which can be expressed in the form of [48]

$\begin{gathered}\left\\{\begin{gathered}{\theta_{k}}={\theta_{k-1}}+d_{k-1}^{-1}{v_{k-1}}\Delta
T\sin{\theta_{k-1}}+{\omega_{\theta,k}},\hfill\\\
{d_{k}}={d_{k-1}}-{v_{k-1}}\Delta T\cos{\theta_{k-1}}+{\omega_{d,k}},\hfill\\\
{v_{k}}={v_{k-1}}+{\omega_{v,k}},\hfill\\\
{b_{k}}={b_{k-1}}\left({1+d_{k-1}^{-1}{v_{k-1}}\Delta
T\cos{\theta_{k-1}}}\right)+{\omega_{b,k}},\hfill\\\
\end{gathered}\right.\hfill\\\ \end{gathered}$ (24)

where
${{\bm{\omega}}_{k}}=\left[\omega_{\theta,k},\omega_{d,k},\omega_{v,k},\omega_{b,k}\right]^{T}$
represents the state noise, and $\Delta_{T}$ is the duration of one epoch.

With both models (23) and (24) above, the RSU can predict and estimate the
vehicle’s state at each epoch via various approaches, e.g., Kalman filtering
and factor graph based message passing algorithms [48, 49, 50]. The predicted
angle is then employed to design the JCR beamformer for the next epoch. At the
$k$th epoch, the received signal at the vehicle can be written as

${{\mathbf{y}}_{C}}\left(t\right)={\beta_{k}}{{\mathbf{a}}^{T}}\left({{M_{T}},{\theta_{k}}}\right){{\mathbf{a}}^{*}}\left({{M_{T}},{{\hat{\theta}}_{k\left|{k-1}\right.}}}\right)s_{k}\left(t\right)+{z_{C}}\left(t\right),$

where $\beta_{k}$ is the path-loss of the LoS path, and
${z_{C}}\left(t\right)$ is the noise with variance $\sigma_{C}^{2}$.
Accordingly, the achievable rate is obtained as

${R_{k}}=\log\left({1+{{{{\left|{{\beta_{k}}{{\mathbf{a}}^{T}}\left({{M_{T}},{\theta_{k}}}\right){{\mathbf{a}}^{*}}\left({{M_{T}},{{\hat{\theta}}_{k\left|{k-1}\right.}}}\right)}\right|}^{2}}{p_{k}}}}/{{\sigma_{C}^{2}}}}\right),$

where $p_{k}$ is the power of the data stream $s_{k}\left(t\right)$. It can be
seen that the achievable rate relies critically on the sensing and prediction
accuracy of the AoA. Once the next AoA is accurately predicted, the RSU can
keep tracking the vehicle while offering high-quality communication service
when it is within the RSU’s coverage.

_Remark:_ We conclude here the superiorities of the JCR based predictive BF
over conventional communication-only approaches, i.e., beam training and
tracking:

  * •

First of all, JCR signalling removes the necessity of dedicated downlink
pilots, as the whole JCR downlink block is exploited both for beam sensing and
communication. This reduces the downlink overhead.

  * •

Secondly, the uplink feedback is not required, since the RSU estimates the
angle from the returned target echo signal instead of from the feedback, which
reduces the uplink overhead.

  * •

Thirdly, the quantization error generated in the uplink feedback is avoided.
As such, the estimation of the vehicle’s state can be performed in a
continuous manner.

  * •

Finally, JCR signalling achieves higher matched-filtering gain than that of
beam training and tracking approaches, as the whole downlink frame, rather
than a part of it, is tailored for both downlink sensing and data
transmission.

##  IV JCR: Radar-Centric Design

Radar systems, particularly military radar, have the extraordinary capability
of long-range operation, up to hundreds of kilometers. Therefore, a major
advantage of implementing communication in radar systems is the possibility of
achieving very long range communications, with much lower latency compared to
satellite communications. However, the achievable data rates for such systems
are typically limited, due to the inherent limitation in the radar waveform
[51, 3, 4, 8].

Research on radar-centric JCR has been mainly focused on the information
embedding technologies, and there are only limited works on other aspects such
as communication protocol and receiver design based on the radar-centric JCR
signals. In this section, we concentrate on more recent DFRC systems based on
MIMO-OFDM, CAESAR and FH-MIMO radar, because of the remarkable benefits they
can offer as described in Section II-C.

###  IV-A Embedding Information in Radar Waveform

Realization of communication in radar systems needs to be based on either
pulsed or continuous-wave radar signals. Hence information embedding with
little interference on radar operation is one of the major challenges. This
topic has been widely investigated, as reviewed in, e.g., [4, 6, 8]. Here, we
summarize these techniques in Table III.

TABLE III: Summary of information embedding methods in radar-centric DFRC systems. Modulations | Methods | Advantages | Disadvantages
---|---|---|---
Modified Waveforms |  Time-frequency Embedding |  Apply various combinations of amplitude, phase and/or frequency shift keying to radar chirp signals [52, 4, 6], or map data to multiple chirp sub-carriers via the use of fractional Fourier Transform [53]. |  • The chirp signal form remains when the inter pulse modulation is used, which is prefered in many radar applications. • The waveform can be implemented in many existing radar systems with only modifications to the software. |  • The slow time coding is restricted by the PRF of the radar, thereby limiting the maximum rate of communication.
Code-domain Embedding |  Modulate binary/poly-phased codes in radar signals using direct spread spectrum sequences [54]. |  • Naturally coexist with the CDMA / DSSS communication signal form. • Enables covert communication by spreading the signal over the bandwidth of radar. |  • Phase modulation will inevitably lead to spectrum alteration of the radar waveform, which may result in energy leakage outside the assigned bandwidth
Spatial embedding |  Modulate information bits to the sidelobes of the radar beampattern [4, 6]. |  • Has little impact on the radar sensing performance in the mainlobe. |  • The performance is sentive to the accuracy of array calibration and BF • The multi-path of radar signal may incur interference to the communication.
Index Modulation (No waveform modification) |  Represent information by the indexes of antennas, frequencies, and/or codes of the signals [8, 55, 10, 56]. |  • Naturally coexist with the radar functionality, with negligible impact on radar performance; • Generally achieve higher data rates compared to modulation with modified waveform. |  • Demodulation may be complicated; • Demodulation performance is sensitive to channel if IM is applied to spatial domain; • Codebook design could be a challenge.

One of the particular techniques of interest is index modulation (IM). IM
embeds information to different combinations of radar signals’ parameters,
over one or more domains of space, time, frequency and code [26, 8, 10]. Thus
IM does not change the basic radar waveform and signal structure, and has
negligible influence on radar operation. For MIMO-OFDM, CAESAR and FH-MIMO
radar, IM can be realized via frequency selection/combination and/or antenna
selection/permutation [57, 58, 59, 56]. Frequency combination selects
different sets of frequencies, and antenna permutation allocates the selected
frequencies to different antennas. Information is represented by the
combinations and permutations. Let the number of combinations and permutations
be $N_{c}$ and $N_{p}$, respectively. Then the number of bits can be
represented is $\log_{2}N_{c}$ and $\log_{2}N_{p}$, respectively.
Mathematically, frequency combination and antenna permutation can generally be
combined. However, decoupling them is consistent with the way that the
information is demodulated, as will be discussed later.

For MIMO-OFDM radar with orthogonal frequency allocation, frequency
combination allocates the total $N$ subcarriers to $M_{T}$ groups without
repetition, with each group having at least one subcarrier [57, 60]. If
without additional constraint on subcarrier allocation, there are a total of
$N_{c}=C^{N}_{M_{T}}M_{T}^{N-M_{T}}$ combinations (i.e., Selecting $M_{T}$ out
of $N$ subcarriers first to ensure each group to have at least one, and then
the remained $N-M_{T}$ subcarriers can go to any of the $M_{T}$ groups); if
each antenna needs to have the same number of $L_{s}=N/M_{T}$ subcarriers, the
total number of combinations is $N_{c}=C^{N}_{L_{s}}C^{N-L_{s}}_{L_{s}}\ldots
C^{L_{s}}_{L_{s}}=N!/(L_{s}!)^{M_{T}}$. The number of permutations of
allocating $M_{T}$ groups of subcarriers to $M_{T}$ antennas is
$N_{p}=M_{T}!$.

The DFRC system extended from CAESAR is proposed in [58], where each virtual
subarray is assumed to have the same number of antennas and use one frequency.
Hence frequency combination selects $S$ out of $N$ frequencies, and
$N_{c}=C^{N}_{S}$. The number of total antenna permutations is shown to be
$N_{p}=M_{T}!/((M_{T}/S)!)^{S}$.

For FH-MIMO DFRC systems [59, 61], the total number of frequency combinations
and antenna permutations are $N_{c}=C^{N}_{M_{T}}$ and $N_{p}=M_{T}!$,
respectively. This corresponds to $S=M_{T}$ in [58]. Let $\bm{B}_{k}$ denote
the $M_{T}\times M_{T}$ antenna permutation matrix at hop $k$, which has only
a single non-zero element, $1$, in each row and column. The transmitted signal
of the FH-MIMO DFRC can be represented as

$\displaystyle\mathbf{x}_{R}(t)$ $\displaystyle=\bm{B}_{k}\bm{\psi}(t).$ (25)

Note that both $\bm{B}_{k}$ and the frequency set $\mathcal{F}_{k}$ vary with
$k$ and are determined by the information bits. One simple example of
information embedded FH-MIMO DFRC for packet communication is shown in Fig. 3.

Figure 3: A simple example showing the packet structure of FH-MIMO DFRC with
$N=8$ and $M_{T}=2$, consisting of a preamble with two identical hops and 3
hops with embedded information. A total of 4 bits can be conveyed in each hop
with IM.

It is noted that the values of $N_{c}$ and $N_{p}$ above define the maximum
achievable bit rates only, without considering the communication reception
performance. In practice, the number of actually used combinations and
permutations may be reduced, due to the overall system design and the
consideration of the demodulation complexity and performance. In particular,
demodulating the bits embedded in antenna permutation is much more difficult
and subject to higher demodulation error, compared to demodulating those
embedded in frequency combination. These issues will be discussed in detail in
Section IV-B and IV-C. In addition, the impacts of information embedding on
radar performance should also be evaluated, such as the ambiguity functions in
the time domain [56] and in the angular domain [60],.

###  IV-B Signal Reception and Processing for Communications

Since IM in all the three systems involve FH, and their receiver processing
methods are similar in many aspects, we use FH-MIMO DFRC as an example to
illustrate the signal reception and processing for its relative simplicity.
Overall, the research on the receiver design for these systems are still
limited. Our overview here is mainly based on [62, 56, 61], and also
incorporates works on MIMO-OFDM and CAESAR DFRCs [60, 58] into the framework
of FH-MIMO DFRC.

Consider a receiver with $M_{R}$ antennas. The signal received from each
antenna is passed to a mixer with local oscillator frequency $f_{c}$, which is
generally the central frequency of the $N$ subbands. Assume narrowband
communications and the difference between multipath delays
$|\tau_{\ell}-\tau_{\ell^{\prime}}|\ll T$. Referring to (1) and (4) and
ignoring the variation of Doppler frequencies, the noise-free baseband
received signal can be approximated as

$\displaystyle\mathbf{y}(t)$
$\displaystyle=\sum_{m=1}^{M_{T}}\sum_{\ell=1}^{L}c_{\ell}e^{j2\pi(f_{k,m}-f_{c})(t-\tau-\tau_{o}(t))}$
(26)
$\displaystyle\quad\mathbf{a}(M_{R},\phi_{\ell})\mathbf{a}^{T}(M_{T},\theta_{\ell})\bm{\beta}_{k,m}g(t-\tau-\tau_{o}(t)-kT),$

where $\bm{\beta}_{k,m}$ is the $m$-th column of $\bm{B}_{k}$,
$c_{\ell}\in\mathbb{C}$ is the equivalent path coefficient, subsuming multiple
terms, and $\tau_{\ell}\approx\tau$ is used. The baseband signal is then
sampled at $T_{s}=1/B$, generating $L_{p}=\lfloor T/T_{s}\rfloor$ samples per
hop. Let

$\displaystyle\mathbf{H}\triangleq\sum_{\ell=1}^{L}c_{\ell}\mathbf{a}(M_{R},\phi_{\ell})\mathbf{a}^{T}(M_{T},\theta_{\ell}).$
(27)

For the simplicity of presentation, assume that $g(t)$ is a rectangular
windowing function. Assume that synchronization is done perfectly. We can
stack $L_{p}$ measurements from all $M_{R}$ antennas to a matrix
${\mathbf{Y}}_{k}$, which is given by

$\displaystyle{\mathbf{Y}}_{k}=\mathbf{H}\bm{B}_{k}\bm{\Phi}^{T}=\sum_{m=1}^{M_{T}}\mathbf{H}\bm{\beta}_{k,m}\bm{\psi}_{k,m}^{T},$
(28)

where $\bm{\Phi}=[\bm{\psi}_{k,1},\ldots,\bm{\psi}_{k,M_{T}}]$,
$\bm{\beta}_{k,m}$ is the $m$-th column of $\bm{B}_{k}$, and
$\bm{\psi}_{k,m}=\\{e^{j2\pi(f_{k,m}-f_{c})\ell_{p}}\\},\ell_{p}=0,\ldots,L_{p}-1$.

####  IV-B1 Demodulation

The task of demodulation is to retrieve information bits from
${\mathbf{Y}}_{k}$. We assume perfect synchronization and channel estimation
here. It will become clear that channel estimation may not be necessary if
only frequency combination needs to be identified.

An optimal formulation of the demodulator can be based on the maximum
likelihood principle [58]. But it has very high computational complexity and
is infeasible for practical implementation. Alternatively, we can apply sub-
optimal methods such as CS techniques.

The patterns of both frequency combination and antenna permutation can be
identified by formulating a sparse recovery problem. The basic idea is to
construct an $L_{p}\times N$ dictionary matrix $\bm{\Phi}_{d}$ by expanding
$\bm{\Phi}$ to cover all $N$ subband frequencies. Each column of
$\bm{\Phi}_{d}$ has the similar expression with $\bm{\psi}_{k,m}$ for each
subband frequency. Then we can get a 1-D MMV-CS formulation as

$\displaystyle{\mathbf{Y}}_{k}^{T}=\bm{\Phi}_{d}\mathbf{A}_{\text{MMV}},$ (29)

where only $M_{T}$ out of $N$ rows in $\mathbf{A}_{\text{MMV}}$ are non-zero,
corresponding to $(\mathbf{H}\bm{B}_{k})^{T}$. The $M_{T}$ frequency
estimates, which correspond to the frequency combination pattern, can then be
found by using one of the well-known MMV-CS recovery algorithms. The estimate
of the $M_{T}$ non-zero rows of $\mathbf{A}_{\text{MMV}}$, obtained in the
recovery process, can be used to find the antenna permutation pattern, by
matching them with $(\mathbf{H}\bm{B}_{k})^{T}$. This matching process can be
realized in either a simpler row-wise way or a more complicated process
jointly across all rows.

Another simpler sub-optimal method is to exploit the orthogonality of
frequencies across antennas and apply a discrete Fourier transform (DFT)
matrix $\mathbf{F}$:

$\displaystyle{\mathbf{Y}}_{k}\mathbf{F}=\mathbf{H}\bm{B}_{k}\bm{\Phi}_{k}^{T}\mathbf{F},$
(30)

where each row of $\bm{\Phi}_{k}^{T}\mathbf{F}$ is the windowed DFT output of
a single tone signal and its waveform has the shape of an impulse with a
single peak. Thus each row of ${\mathbf{Y}}_{k}\mathbf{F}$ represents the
weighted sum of these impulses. Therefore, the frequency combination pattern
may be identified via locating the peaks. This is particularly effective when
either the inverse of $\mathbf{H}$ exists or when the LOS path is dominating
in $\mathbf{H}$. In the former, we can compute
$\mathbf{H}^{-1}{\mathbf{Y}}_{k}$ and obtain
$\bm{B}_{k}\bm{\Phi}_{k}^{T}\mathbf{F}$. This leads to simple identification
of both frequency combination and antenna permutation patterns, as
$\bm{B}_{k}$ is a permutation matrix. In the latter, the frequency combination
pattern can be found via the peaks and the antenna permutation pattern is
determined via exhaustive searching, even in a single antenna receiver [56,
61].

####  IV-B2 Channel Estimation

An accurate estimate of $\mathbf{H}$ is critical for demodulation. However, it
is challenging to design and incorporate long training sequences, which is
essential for estimating $\mathbf{H}$, in FH-MIMO DFRC systems. This is
because training sequence requires certainty, which will affect the randomness
of FH radar operation.

There are very limited results on channel estimation for FH-DFRC systems with
IM. In [56], both synchronization and channel estimation are investigated for
a single antenna receiver, with the consideration of packet communications.
For channels with a dominating LOS-path, which could be a typical operating
condition for radar-centric DFRC, a frame structure is proposed with two
identical hops serving as preamble followed by hops with embedded information.
The two identical hops are designed to enable effective estimation of timing
offset, carrier frequency offset and channel. To simplify synchronization and
channel estimation, re-ordered hopping frequencies are used, which slightly
reduces the information embedding capability in terms of antenna permutation.
Timing offset and channel estimators are proposed by exploiting the signal
differences between two neighbouring antennas. The work is also extended to
NLOS channels by using incomplete sampled hops and judiciously designed
hopping frequencies to combat inter-hop and inter-antenna interference.

###  IV-C Codebook Design

The codebook determines how the patterns of frequency combination and antenna
permutation are selected and mapped to information bits. As was disclosed in
[58], the achievable communication rates are largely constrained by antenna
permutation, as its demodulation performance is sensitive to the differences
between different columns of $\mathbf{H}$.

The design criterion can hence be formulated based on the distance between two
codewords of antenna permutation:

$\displaystyle\lambda(m,m^{\prime})=\parallel\mathbf{H}\bm{\beta}_{k,m}-\mathbf{H}\bm{\beta}_{k,m^{\prime}}\parallel_{2}^{2}.$
(31)

Maximizing the minimal distance among all $\lambda(m,m^{\prime})$ is a typical
design criterion. Since directly searching via (31) is computational
complicated, a method of projection into a lower dimensional plane is proposed
in [58]. However, given that the design needs to be updated once $\mathbf{H}$
is changed, the complexity is still very high.

Such a complicated design may be avoided by using pre-compensation. For
example, for LOS-path dominating channels, the channel differences between
antenna permutations will be small when the AoD is small. In [61], an element-
wise phase compensation method is proposed to remove the AoD dependence of
demodulating antenna permutation. Thus the distances between different
codewords become identical.

In addition to its impact on communication performance, codebook may also
affect the radar performance, for example, the ambiguity function as evaluated
in [62, 56]. More specifically, it is demonstrated in [62] that the
probability of radar waveform degeneration can be reduced by spreading the
available frequency hops between waveforms as evenly as possible; and in [56],
it is shown that by constraining the codewords, the receiver processing can be
largely simplified, with negligible impact on the radar ambiguity function.

##  V JCR: Joint Design and Optimization

Although no clear boundary exists between the third category of JCR and the
other two, there is more freedom here in terms of signal and system design.
That is, JCR technologies can be developed without being constrained to
existing C&R systems, and they can be designed and optimized to balance the
requirements for C&R, potentially providing a better trade-off between the two
functions.

Joint waveform optimization is a key research problem here. It can be
conducted in multiple domains, using various performance metrics jointly for
C&R. In this section, we review several typical DFRC waveform optimization
schemes. For simplicity, we will mainly consider underlying signals based on
narrowband single carrier communications, and the extension to OFDM signals is
generally straightforward.

To avoid confusion, we use $\mathbf{H}_{C}$ and $\mathbf{H}_{R}$ for
communication and radar channels, respectively; and for simplicity, we assume
$M_{T}=M_{R}=M$. Note that $\mathbf{H}_{C}\neq\mathbf{H}_{R}$ in downlink
sensing and $\mathbf{H}_{C}=\mathbf{H}_{R}$ in uplink sensing, as detailed for
PMNs in [24, 12]. Referring to (4), collect $K$ received signal vectors over a
CPI and stake them to a matrix, generating
${\mathbf{Y}}_{C}=\mathbf{H}_{C}\mathbf{X}+\mathbf{Z}_{C}$ for communications,
and ${\mathbf{Y}}_{R}=\mathbf{H}_{R}\mathbf{X}+\mathbf{Z}_{R}$ for sensing. We
assume the AWGN matrices $\mathbf{Z}_{R}$ and $\mathbf{Z}_{C}$ both have zero
mean and element-wise variance $\sigma_{z}^{2}$.

###  V-A Waveform Optimization via Spatial Precoding

When different beams are needed for C&R, the precoding matrix $\mathbf{P}$ can
be optimized. The basic optimization formulation is as follows:

$\displaystyle\arg\max_{\mathbf{P}}\lambda(\mathbf{P}),\quad\text{s.t.
Constraints 1, 2}\cdots,$ (32)

where $\lambda(\mathbf{P})$ is the objective function. There could be various
methods and combinations in defining the objective functions and the
constraints. Each can be either for communication or sensing individually, or
a weighted joint function. Next, we review three types of optimizations that
consider mutual information (MI), waveform mismatch, and estimation accuracy
for the sensing performance, respectively.

####  V-A1 Mutual Information (MI) Based

MI is well known for communication systems, and the usage of MI for radar
waveform design can also be traced back to 1990s [63]. MI for radar sensing
measures how much information about the channel, the propagation environment,
is conveyed to the receiver. The conditional MI is defined as the entropy
between the sensing channels and the received signals, conditional on the
transmitted signal. Mathematically, this can be represented as [64]

$\displaystyle
I_{R}(\mathbf{H}_{R};{\mathbf{Y}}_{R}|\mathbf{X})=M\log_{2}\left(\text{det}\left(\frac{1}{\sigma_{z}^{2}}\mathbf{X}^{H}\bm{\Sigma}_{H_{R}}\mathbf{X}+\mathbf{I}_{K}\right)\right),$

where $\mathbf{I}_{K}$ is an identity matrix of size $K\times K$, and
$\bm{\Sigma}_{H_{R}}=\text{E}[\mathbf{H}_{R}\mathbf{H}_{R}^{H}]/M$. The
conditional MI for communication is given by

$\displaystyle
I_{C}(\mathbf{X};{\mathbf{Y}}_{C}|\mathbf{H}_{C})=K\log_{2}\left(\text{det}\left(\frac{1}{\sigma_{z}^{2}}\mathbf{H}_{C}^{H}\bm{\Sigma}_{X}\mathbf{H}_{C}+\mathbf{I}_{M}\right)\right),$

where $\bm{\Sigma}_{X}=\text{E}[\mathbf{X}\mathbf{X}^{H}]/K$.

Based on the two MI expressions, we can formulate various optimization
problems. In [3], the estimation rate, defined as the MI within a unit time,
is used for analyzing the radar performance, together with the capacity metric
for communications. In [65], a weighted sum of MI for both C&R is formulated
as the objective function of optimization for a single-antenna OFDM DFRC
system. In [64], a more complicated MI-based joint optimization is conducted
for MIMO DFRC systems, by taking into consideration practical packet structure
with orthogonal training sequences and random data symbols, and the channel
estimation error.

A general and flexible formulation can be based on a weighted sum of the two
MIs [65, 64]

$\displaystyle
F=\frac{w_{R}}{F_{R}}I_{R}(\mathbf{H}_{R};{\mathbf{Y}}_{R}|\mathbf{X})+\frac{1-w_{R}}{F_{C}}I_{C}(\mathbf{X};{\mathbf{Y}}_{C}|\mathbf{H}_{C}),$
(33)

where $F_{C}$ and $F_{R}$ are the maximal MI for C&R individually, and are
treated as two known constants in the optimization, and $w_{R}\in[0,1]$ is a
weighting factor. The function in (33) is concave, and can be maximized by
using, e.g., the Karush-Kuhn-Tucker (KKT) conditions. The optimal $\mathbf{P}$
turns out to be a water-filling type of solution that jointly considers the
distributions of the eigen-values of $\bm{\Sigma}_{X}$ and
$\bm{\Sigma}_{H_{R}}$.

####  V-A2 Waveform/Beampattern Similarity Based

The DFRC waveform is typically expected to possess some useful properties that
are beneficial for radar sensing, e.g., good auto- and cross-correlations,
high peak-to-sidelobe level ratio (PSLR), low peak-to-average power ratio
(PAPR), and resilience to clutter and interference. Nevertheless, it could be
quite challenging to implement all these features simultaneously in a single
waveform, especially in the case of JCR, where the randomness in the
communication data and channels may break down the structure of the waveform
tailored for sensing.

To cope with this issue, one may consider to optimize the DFRC
waveform/beampattern while approximating a well-designed benchmark radar
signal, which is known to have the desired characteristics above. This could
be achieved by imposing a figure-of-merit for the waveform/beampattern
similarity, either in the objective function or in the constraints. As an
example, the JCR BF design in [16] aims to approximate a baseline radar
beampattern while guaranteeing the individual signal-to-interference-and-noise
ratio (SINR) $\gamma_{k}$ for $K$ single-antenna downlink users. Accordingly,
the optimization problem can be formulated as

$\begin{gathered}\mathop{\min}\limits_{{\mathbf{\mathbf{P}}},\beta}\;\;\left\|{{\mathbf{P}}{{\mathbf{P}}^{H}}-\beta{\mathbf{R}}}\right\|_{F}^{2}\hfill\\\
s.t.\;\;\;\;\beta\geq 0,\quad{\gamma_{k}}\geq{\Gamma_{k}},\forall k,\hfill\\\
\;\;\;\;\;\;\;\;\;\operatorname{diag}\left({{\mathbf{P}}{{\mathbf{P}}^{H}}}\right)=\frac{{{P_{T}}}}{{{M_{T}}}}{{\mathbf{1}}_{{M_{T}}}},\hfill\\\
\end{gathered}$ (34)

where $\mathbf{P}\in\mathbb{C}^{M_{T}\times K}$ is the DFRC BF/precoding
matrix to be designed, $\mathbf{R}\in\mathbb{C}^{M_{T}\times M_{T}}$
represents the spatial covariance matrix for a benchmark radar waveform, which
generates a favorable MIMO radar beampattern $P_{d}\left(\theta\right)$, given
by

${P_{d}}\left(\theta\right)={{\mathbf{a}}^{H}}\left({{M_{T}},\theta}\right){\mathbf{Ra}}\left({{M_{T}},\theta}\right).$
(35)

It can be readily observed that the cost function in (34) is the Euclidean
distance between covaraince matrices for the DFRC waveform and its pure radar
benchmark. Moreover, $\beta\geq 0$ is a scaling factor. The remaining
constraints aim to ensure the SINR for each user, as well as to restrict the
per-antenna transmit power, with $P_{T}$ being the overall power budget
available. While problem (34) is non-convex, it can be sub-optimally solved
via the semidefinite relaxation (SDR) approach or manifold based algorithms
[16].

In addition to the above design that approximates the MIMO radar beampattern,
a more straightforward method is to directly approximate the MIMO radar
waveform itself. Under the same $K$-user MU-MIMO downlink scenario, let us
first rewrite the MIMO communication signal model (4) in a discrete matrix
form as

${\mathbf{Y}}_{C}={{\mathbf{H}}_{C}\mathbf{X}}+{\mathbf{Z}}_{C}={\mathbf{S}}+\underbrace{\left({{{\mathbf{H}}_{C}\mathbf{X}}-{\mathbf{S}}}\right)}_{{\text{MUI}}}+{\mathbf{Z}}_{C},$
(36)

where $\mathbf{S}\in\mathbb{C}^{K\times L}$ contains the information symbols
intended for $K$ communication users. The second term at the right-hand side
of the second equality is known to be the multi-user interference (MUI). If
the MUI is minimized to zero, then (36) boils down to an AWGN transmission
model, where the fading effect incurred by the channel $\mathbf{H}_{C}$
vanishes. By noting this fact, [25] formulates the following optimization
problem to design a DFRC waveform matrix $\mathbf{X}$

$\begin{gathered}\mathop{\min}\limits_{\mathbf{X}}\;\;\left\|{{{\mathbf{H}}_{C}\mathbf{X}}-{\mathbf{S}}}\right\|_{F}^{2}\hfill\\\
s.t.\;\;\;\;{\left\|{\operatorname{vec}\left({\mathbf{X}}\right)-\operatorname{vec}\left({{{\mathbf{X}}_{0}}}\right)}\right\|_{\infty}}\leq\varepsilon,\hfill\\\
\;\;\;\;\;\;\;\;\;{\left|{{x_{i,j}}}\right|^{2}}=\frac{{{P_{T}}}}{{{M_{T}}}},\forall
i,j,\hfill\\\ \end{gathered}$ (37)

where the first constraint is to explicitly control the distance between
$\mathbf{X}$ and the benchmark $\mathbf{X}_{0}$ in an $L_{\infty}$-norm sense,
with a given similarity coefficient $\varepsilon$. The second constraint, on
the other hand, requires $\mathbf{X}$ to be constant-modulus (CM), i.e., with
a 0dB PAPR. $\mathbf{X}_{0}$ could be any CM radar signal matrix, e.g.,
orthogonal chirp waveform. While problem (37) is again non-convex and NP-hard
in general, a branch-and-bound (BnB) algorithm is developed in [25], which
finds its global optimum within only tens of iterations.

####  V-A3 Estimation Accuracy Based

The accuracy of sensing parameter estimation is important for radar sensing.
Since the received signals are not a linear function of the sensing
parameters, it is generally difficult to get closed-form expressions for,
e.g., the mean square error (MSE) of the estimates, and to apply them in the
optimization directly. Alternatively, we can derive and use the CRLBs of the
estimates, which are low bounds of the MSE. The CRLBs of signal estimates can
be derived via the inverse of the Fisher information matrix (FIM). For
estimates based on radar signals, they are well known, e.g., as reported in
[66]. For communication signals, the CRLBs for some sensing parameters based
on the beamspace channel models have also been derived, e.g., in [67].

DFRC waveform optimization based on the CRLBs for sensing performance has also
been studied in the literature. However, due to the complicated expressions of
the CRLBs, it is generally challenging to obtain closed-form solutions in such
optimizations. In [68], considering a single-antenna OFDM DFRC system, Pareto-
optimal waveform design approaches are proposed to simultaneously improve the
estimation accuracy of range and velocity and the channel capacity for
communications. The Pareto-optimal solutions are obtained for a multiobjective
optimization problem, and cannot be improved with respect to any objective
function without deteriorating other objective functions, and hence the
solutions are suboptimal. In [69], waveform optimization is studied with the
application and comparison of multiple sensing performance metrics including
MI, MMSE and CRLB, together with an approximated SINR metric for
communications. It is shown that there are close connections between MI-based
and CRLB-based optimizations, and the MI-based method is more efficient and
less complicated compared to the CRLB-based method. Specific to the CRLB-base
method, a closed-form solution is obtained for some special cases, while an
iterative algorithm is proposed as a general solution.

###  V-B Multibeam Optimization for Analog Array

Steerable analog array, which is also the basic component of a hybrid array,
could be widely applied in mmWave JCR systems. A JCR system may need to
support communication and sensing at different directions, which is
challenging to address given the limits on the BF capability of analog array.
A good solution is the multibeam technology [23, 70, 71]. The multibeam
consists of a fixed subbeam dedicated to communication and a scanning subbeam
with directions varying over different communication packets. By optimizing
the beam with multiple subbeams, communication and radar sensing at different
directions can be supported simultaneously with a single signal. This can
largely extend the field of view of sensing, for example, in the 802.11ad DFRC
with the single carrier PHY. The multibeam can be applied at both transmitter
and receiver, while transmitter is considered here.

Two classes of methods have been proposed for the multibeam optimization: the
subbeam-combination method [23, 70, 71] via constructively combining two pre-
generated subbeams according to given criteria, and the global optimization
[71] by jointly considering the C&R BF requirements and optimizing a single BF
vector directly. These methods can also be extended to full digital arrays.

####  V-B1 Subbeam Combination

In the subbeam-combination method, two basic beams for C&R are separately
generated according to the desired BF waveform, using, e.g., the iterative
least squares method [23]. The two beams are further shifted to the desired
directions by multiplying a sequence, and then combined by optimizing a power
distribution factor $\rho$ and a phase shifting coefficient $\varphi$.

The BF vector $\mathbf{w}_{t}$ in (38) can be represented as

$\displaystyle\mathbf{w}_{T}=\sqrt{\rho}\mathbf{w}_{T,F}+\sqrt{1-\rho}e^{j\varphi}\mathbf{w}_{T,S},$
(38)

where $\mathbf{w}_{T,F}$ and $\mathbf{w}_{T,S}$ are the BF vectors that are
predetermined, corresponding to the fixed subbeam and scanning subbeam,
respectively. The value of $\rho$ can typically be determined via balancing
C&R distances [70], and the optimization is conducted mainly with respect to
$\varphi$, which can ensure the pre-generated subbeams are coherently combined
when generating the multibeam.

The multibeam optimization problem can then be formulated with desired
objective functions and constraints. Consider one example of maximizing the
received signal power for communications with constrained BF gain in scanning
directions. Let the threshold of the BF gain in the $i$-th sensing direction
$\phi_{i}$ be $C^{2}_{i}(1-\rho)M_{T}$, where $C_{s_{i}}\in[0,1]$ is a scaling
coefficient, representing the ratio between the gain of the scanning subbeam
in the interested direction and the maximum gain that the array can achieve,
i.e., $(1-\rho)M_{T}$. We can formulate the constrained optimization problem
as

$\displaystyle{\varphi}^{(1)}_{\text{opt}}=\arg\max_{\varphi}\dfrac{\mathbf{w}_{T}^{H}{\mathbf{H}_{C}}^{H}{\mathbf{H}_{C}}\mathbf{w}_{T}}{\|\mathbf{w}_{T}\|^{2}},$
(39) $\displaystyle\text{s.t. }\
\dfrac{|\mathbf{a}^{T}(M_{T},\phi_{i})\mathbf{w}_{T}|^{2}}{||\mathbf{w}_{T}||^{2}}\geq
C^{2}_{i}(1-\rho)M_{T},\ i=1,2,\cdots,N_{s},$

where $N_{s}$ is the number of the total constraints, $\phi_{i}$s are the
angles of interest where the BF gain is constrained, and a maximal ratio
combiner is assumed to be applied at the communication receiver.

With $\mathbf{w}_{T,F}$, $\mathbf{w}_{T,S}$ and $\rho$ being predetermined,
the optimization problem can be solved by first finding the unconstrained
optimal solution for the objective function and then looking for its
intersection with the intervals determined by the constraints. Closed-form
solutions can then be obtained as detailed in [71].

The subbeam-combination method enables simple and flexible multibeam
generation and optimization, and hence is promising for practical applications
that require fast adaptation to changing BF requirements. It is particularly
useful for mmwave systems where directional BF is used.

####  V-B2 Global Optimization

The subbeam-combination method is an efficient low-complexity solution, but it
is suboptimal. A globally optimal solution can be obtained by solving a
problem formulated directly with respect to $\mathbf{w}_{T}$. Considering a
similar example of maximizing the received signal power for communication with
constraints on BF waveform, the formulation can be represented as

$\displaystyle\mathbf{w}_{t,\text{opt}}=\operatorname*{argmax}_{\mathbf{w}_{T},\mathbf{w}_{T}^{H}\mathbf{w}_{T}=1}\
\mathbf{w}_{T}^{H}{\mathbf{H}_{C}}^{H}{\mathbf{H}_{C}}\mathbf{w}_{T},$ (40a)
$\displaystyle\text{s.t.}\quad\|\mathbf{A}(M_{T},\phi_{i})\mathbf{w}_{T}-\mathbf{d}_{v})\|^{2}\leq\varepsilon_{w},\
\text{and/or}$ (40b) $\displaystyle\qquad\
|\mathbf{a}^{T}(M_{T},\phi_{i})^{T}\mathbf{w}_{T}|^{2}\geq\varepsilon_{i},\
i=1,2,\cdots,N_{s},,$ (40c)

where (40b) bounds the mismatches between the generated BF waveform and the
desired one $\mathbf{d}_{v}$, (40c) constrains the gain of the scanning
subbeam in $N_{s}$ directions, and $\varepsilon_{w}$ and $\varepsilon_{i}$ are
the thresholds for these constraints. These constraints can be applied
individually or jointly.

The optimization in (40) is generally a nonconvex NP-hard problem. In [71],
this problem is converted to a homogeneous quadratically constrained quadratic
program (QCQP), which is then solved by the SDR technique.

The global optimization method provides a benchmark for suboptimal multibeam
optimization schemes. Simulation results in [71] demonstrate that there exist
a loss up to 10% in the received signal power for communications and the BF
waveform, when the subbeam combination method is applied.

###  V-C Signal Optimization in Other Domains

In addition to spatial optimization, communication signals can also be
optimized across the time and frequency domains, to improve the estimation
accuracy of sensing parameters. For example, non-uniform preamble is proposed
to improve the Doppler estimation performance in [22]; a modified Golay
complementary sequence is proposed for 802.11ad-based JCR systems in [34] to
reduce the sidelobe and achieve improved ranging and Doppler estimation; and
the idea of using and optimizing nonequidistant subcarriers in MIMO-OFDM radar
in [30] can also be extended to a JCR MIMO-OFDM system. Here, we briefly
illustrate the idea by referring to the work in [22].

In [22], for a single data-stream single-carrier JCR system based on 802.11ad,
non-uniformly placed preambles are proposed to enhance velocity estimation
accuracy. The signal has a packet structure, consisting preamble and data
payload. Radar sensing uses the preamble only. A novel metric of distortion
MMSE (DMMSE) is developed for communications in [22]. A log-scale is then
applied to the DMMSE and the CRLB for radar, to achieve proportional fairness
between C&R, such that the two log-scaled metrics can directly be added up in
the optimization objective function. The objective function is given by

$\displaystyle
w_{C}\frac{1}{K}\text{Tr}[\log_{2}\text{DMMSE}]+w_{R}\text{Conv}(\frac{1}{L_{v}}\text{Tr}[\log_{2}\text{CRLB}]),$

where $K$ is the number of symbols, $L_{v}$ is the number of velocities to be
estimated, $\text{Tr}[\cdot]$ denotes the trace of a matrix,
$\text{Conv}(\cdot)$ denotes the convex hull operation, and $w_{C}$ and
$w_{R}$ are weighting factors for C&R, respectively.

Based on the objective function and some constraints, the number and position
of preambles are optimized in [22]. It is found that when preambles are
equally spaced, the performance of radar or communications cannot be
effectively improved without affecting the other. Comparatively, non-uniform
preambles are found to achieve a better performance trade-off between C&R,
particularly at large radar distances.

##  VI Conclusion

JCR is a promising technology that can be used to revolutionize both
traditional communication and radar systems. Although the potentials and
prospects for integrating C&R are great, there are many challenges and open
research problems to be addressed due to the inherent differences of the
signal formats, and system and network structures between them. Signal
processing techniques are key enablers to make the integration happen. For
communication-centric JCR, accurate and practical sensing parameter estimation
algorithms are the key, and a viable solution to the clock asynchrony problem
can relax the full-duplex requirement. Sensing assisted communications is a
great way of exploiting the benefit of integration. For radar-centric JCR, how
to improve the communication rates without notable impact on radar operation
is a big challenge. While it is shown that applying index modulation to
frequency-hopping DFRC is an attractive solution, more signal processing
techniques are needed for improving the communication receiver. For joint
design and optimization, the journey is just started, with most work being
focused on waveform optimization in existing systems. A fresh look at the
joint requirements for C&R may lead to more efficient solutions, particularly
those underlying frequency hopping and millimeter wave signals that have
excellent potentials for both high data rates and radar resolution.

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