#  A Reconfigurable Subarray Architecture and Hybrid Beamforming for
Millimeter-Wave Dual-Function-Radar-Communication Systems

Xin Jin, Tiejun Lv, , Wei Ni, , Zhipeng Lin, , Qiuming Zhu, , Ekram Hossain, ,
and H. Vincent Poor Manuscript received March 6, 2024; accepted April 22,
2024. This paper was supported in part by the National Natural Science
Foundation of China under No. 62271068, the Beijing Natural Science Foundation
under Grant No. L222046, the U.S National Science Foundation under Grants
CNS-2128448 and ECCS-2335876. (_corresponding author: Tiejun Lv_.)X. Jin and
T. Lv are with the School of Communication and Information Engineering,
Beijing University of Posts and Telecommunications, Beijing 100876, China
(e-mail: {jxzoe, lvtiejun}@bupt.edu.cn). W. Ni is with the Data61,
Commonwealth Scientific and Industrial Research Organization, Sydney,
Marsfield, NSW 2122, Australia. (e-mail: wei.ni@ieee.org). Z. Lin and Q. Zhu
are with the College of Electronic and Information Engineering, Nanjing
University of Aeronautics and Astronautics, Nanjing 211106, China (e-mail:
{linlzp, zhuqiuming}@nuaa.edu.cn). E. Hossain is with the Department of
Electrical and Computer Engineering, University of Manitoba, Canada (e-mail:
ekram.hossain@umanitoba.ca). H. V. Poor is with the Department of Electrical
and Computer Engineering, Princeton University, Princeton, NJ 08544 USA
(e-mail: poor@princeton.edu).

###### Abstract

Dual-function-radar-communication (DFRC) is a promising candidate technology
for next-generation networks. By integrating hybrid analog-digital (HAD)
beamforming into a multi-user millimeter-wave (mmWave) DFRC system, we design
a new reconfigurable subarray (RS) architecture and jointly optimize the HAD
beamforming to maximize the communication sum-rate and ensure a prescribed
signal-to-clutter-plus-noise ratio for radar sensing. Considering the non-
convexity of this problem arising from multiplicative coupling of the analog
and digital beamforming, we convert the sum-rate maximization into an
equivalent weighted mean-square error minimization and apply penalty dual
decomposition to decouple the analog and digital beamforming. Specifically, a
second-order cone program is first constructed to optimize the fully digital
counterpart of the HAD beamforming. Then, the sparsity of the RS architecture
is exploited to obtain a low-complexity solution for the HAD beamforming. The
convergence and complexity analyses of our algorithm are carried out under the
RS architecture. Simulations corroborate that, with the RS architecture, DFRC
offers effective communication and sensing and improves energy efficiency by
83.4% and 114.2% with a moderate number of radio frequency chains and phase
shifters, compared to the persistently- and fully-connected architectures,
respectively.

###### Index Terms:

Reconfigurable subarray (RS) architecture, hybrid beamforming (HBF), dual-
function-radar-communication (DFRC), penalty dual decomposition (PDD).

##  I Introduction

The sixth generation (6G) of wireless systems is anticipated to go beyond
traditional mobile networks in continuous and ubiquitous sensing capabilities
[1]. A promising concept for realizing simultaneous sensing and communication
capabilities is dual-functional-radar-communication (DFRC). By sharing
hardware, software, and information, DFRC enables these capabilities while
coping with resource constraints [2, 3]. Benefiting from advances in
millimeter-wave (mmWave) technologies, mmWave DFRC can support simultaneous
high-speed communications and high-resolution sensing in environment-aware
scenarios [4, 5], e.g., smart homes and Internet of Things (IoT). For
instance, the authors of [4] proposed a novel dual-function waveform design
scheme that can estimate velocity and range at ultra-high resolutions in noisy
environments.

To address severe path-loss of mmWave signals, massive multiple-input
multiple-output (mMIMO) antenna arrays with significant beamforming gains have
been studied for both communication and radar systems [6, 7, 8]. The authors
of [8] provided a parameter design approach for high mobility scenarios to
enhance the resilience to Doppler shift. Unfortunately, a fully digital (FD)
mMIMO architecture is of limited practical value due to its prohibitive
hardware and energy costs. To this end, hybrid analog-digital (HAD)
architectures, offering flexible connectivity between radio frequency (RF)
chains and antennas, become promising to balance hardware complexity and
system performance [9, 10, 11].

A similar concept in radar systems, i.e., a phased-MIMO radar, is used to join
the high resolution of a MIMO radar and the coherent processing gain of a
phased-array radar [12, 13]. For instance, the authors of [12] developed a
hybrid beamforming (HBF) phased-array architecture for active sensing
applications, using a hybrid beamformer system for transmitting and receiving
images. In [13], an approach to synthesize probing beampatterns for mMIMO
radars with fully-connected (FC) structures was presented based on learning-
based RF chains and antenna selection.

As communication and sensing converge in mmWave mMIMO systems, it is
imperative to design an effective transceiver architecture to avoid high
hardware costs and enhance the integration gain of DFRC systems. To this end,
we present a new DFRC system with a reconfigurable subarray (RS) architecture
for improved spectral efficiency (SE), energy efficiency (EE), and sensing
performance.

###  I-A Related Works

Many works have considered HAD DFRC systems, including FC [14, 15, 16, 17, 18]
and persistently-connected (PC) architectures [19, 20, 21, 22]. In [14], two-
stage alternating minimization was proposed to optimize hybrid transmit beams
and phase vectors for detecting several targets and communicating with several
users, subject to the quality-of-service (QoS) requirements, e.g., signal-to-
interference-plus-noise ratio (SINR), for communication. The authors of [16]
conducted weighted optimization of optimal FD transmission and radar
beampattern for an Internet of Vehicles (IoV) scenario to improve SE of
vehicular communication as well as sensing accuracy. However, these studies
focused on radar similarity constraints or beampattern approximation, and
overlooked the sensing performance of the radar receiver. Considering that the
radar detection probability directly depends on the signal-to-clutter-plus-
noise ratio (SCNR) of the radar receiver, the authors of [17] optimized hybrid
transmit beamforming to maximize the receive SCNR. Additionally, the authors
of [18] opted for a communication-centric transmitter design under the Cramer-
Rao bound (CRB). These studies were generally based on an FC architecture,
where every RF chain is hardwired to all antennas via a phase shift (PS)
network, which could be prohibitive in mMIMO.

Another hybrid architecture is the PC architecture, where RF chains connect
individual antenna subarrays. In [19], an effective trade-off between sub-
array MIMO radar and communication was achieved by minimizing a weighted sum
of beamforming errors for both. The EE of the integrated system was maximized
by activating a small number of RF chains [20]. In [21], the communication
rate was maximized under a sensing beampattern mismatch requirement in both
single- and multi-user multi-carrier systems. More recently, the authors of
[22] considered a more realistic clutter interference environment and used a
double-phase-shifter (DPS) structure, providing a well-tolerated compromise
between the number of PSs and the performance gain of the DPS HBF. Although
the PC architecture significantly reduces hardware costs compared to the FC
architecture, these HBF designs offer no flexibility.

RS architecture-based HBF offers a promising solution to balance system
performance and hardware costs in communication systems [23, 24, 25]. This is
achieved by flexibly connecting an RF chain to a non-overlapping subarray
through a switch network and PSs [23]. Hybrid subarrays were reconfigurable
based on channel statistics in [24], where a greedy algorithm was implemented
to approach the SE achieved by exhaustive search. In multi-user scenarios, the
authors of [25] designed HBF based on sub-channel differences. However, these
discussions were limited to communication scenarios. A summary of these
existing studies is provided in Table I.

Table I: Comparison between the proposed scheme and the existing HBF designs

Ref. | Architecture | Communication Metric | Sensing Metric
---|---|---|---
FC | PC | RS
[14] | ✓ |  |  | SINR | Desired beampattern
[15] | ✓ |  |  | Sum-rate | Beampattern matching error
[16] | ✓ |  |  |  FD matching error; Multiuser interference (MUI) |  Desired beampattern; Power distribution error
[17] | ✓ |  |  | SCNR | SINR
[18] | ✓ |  |  | Weighted sum-rate | CRB
[19] |  | ✓ |  | FD matching error | Desired beampattern
[21] |  | ✓ |  | Weighted sum-rate | Waveform similarity
[22] |  | ✓ |  | Sum-rate | SCNR
[23] |  |  | ✓ | Sum-rate | Not applicable (n.a.)
[24] |  |  | ✓ | SE | n.a.
[25] |  |  | ✓ | Sum-rate | n.a.
Ours |  |  | ✓ | Sum-rate | SCNR

###  I-B Challenges and Contributions

The incorporation of an RS architecture and HBF in a mmWave DFRC system is not
straightforward. On the one hand, the tight multiplicative coupling of the
analog and digital beamforming makes their joint optimization challenging. The
need to selectively connect the RF chains and antennas in the RS architecture
could result in an NP-hard mixed integer programming problem [23]. Also, a
direct use of techniques, such as alternating optimization and block
decomposition, is likely to get trapped in inferior solutions or even fail to
converge. On the other hand, many existing DFRC designs focus on the transmit
HBF, which can penalize the sensing performance in cluttered environments. To
cope with clutter interference, the authors of [26] considered the processing
of radar receive filters and their optimization on a DFRC system, which
effectively utilizes the available degrees of freedom (DoFs) in radar receive
arrays. But only small-scale FD beamforming was considered in [26].

This paper presents a new framework for HAD DFRC systems with hybrid transmit
beamforming and radar receive beamforming, where the PSs and antennas are
reconfigurably connected through adaptive switching networks. A dual-function
base station (BS) delivers communication services to multiple users in the
downlink, meanwhile detecting a potential target amidst signal-dependent
interference sources, i.e., clutter. Unlike conventional approaches focusing
on radar beampattern matching errors, we adopt the radar receive SCNR to
quantify and constrain the sensing performance. We propose an effective
algorithm that optimizes the digital and analog beamformers to maximize the
achievable sum-rate of the HAD DFRC system under a prescribed SCNR requirement
of its radar sensing performance.

Following is the list of contributions of this paper:

  * •

We propose a new RS architecture for mmWave HAD DFRC systems in which every RF
chain can be flexibly linked to a non-overlapping subset of antennas. This
architecture can significantly reduce the hardware costs while maintaining the
DFRC performance.

  * •

We present a new problem formulation to maximize the non-convex sum-rate under
the SCNR constraints of radar sensing in an HAD DFRC system with an RS
architecture.

  * •

Using penalty dual decomposition (PDD) and weighted minimum mean square error
(WMMSE) as an interpretation of the non-convex sum-rate, we put forth a new
WMMSE-PDD (WPDD) algorithm to solve the new problem through block coordination
descent (BCD). As per iteration, a second-order cone program (SOCP) optimizes
the FD transmit counterpart of the HBF. The sparsity of the RS architecture is
exploited to obtain a low-complexity solution for the HBF.

  * •

The proposed WPDD algorithm is extended to DFRC systems with PC architectures.
Its convergence and complexity are analyzed, indicating the benefit of HBF in
DFRC systems in terms of EE.

Extensive simulations are carried out to verify our analysis and assess the
effectiveness of a DFRC system with an RS architecture between performance and
cost by comparing it with the FC and PC architectures. RS-based HBF leads to a
83.4% improvement in EE compared to the PC-based HBF, and surpasses the EE
achieved under the FC architectures by more than twofold.

The subsequent sections of this paper are arranged as follows. The studied
problem is cast in Section II. The new WPDD algorithm is presented in Section
III, followed by its convergence, complexity, and EE analyses in Section IV. A
summary of simulation results is given in Section V. Lastly, the conclusions
are presented in Section VI.

Notation: Bold-face upper- and lower-cases indicate matrices and vectors,
respectively. $(\cdot)^{T}$, $(\cdot)^{H}$, $(\cdot)^{\dagger}$, and
$\text{tr}(\cdot)$ denote transpose, conjugate transpose, pseudo-inverse, and
trace, respectively, ${\mathbb{C}^{M\times N}}$ denotes all $M\times N$
complex matrices, ${\cal R}{\\{\cdot\\}}$ takes the real part of a complex
value, $\|\cdot\|$ provides Euclidean or Frobenius norm for a vector or
matrix, $\|\cdot\|_{0}$ is the 0-norm of a vector, $\mathbb{E(\cdot)}$ takes
the statistical expectation, ${\cal{CN}}(\cdot,\cdot)$ represents the complex
normal distribution, blkdiag($\cdot$) denotes block diagonal matrix,
$\left\lceil\cdot\right\rceil$ stands for ceiling. The notation used is
collated in Table II.

Table II: Notation and definitions

Notation | Definition
---|---
${M_{\mathrm{T}}}/{M_{\mathrm{R}}}$ | Number of antennas at the DFRC transmitter/receiver;
$N_{\mathrm{RF}}^{\mathrm{t}}/N_{\mathrm{RF}}^{\mathrm{r}}$ | Number of RF chains at the DFRC transmitter/receiver;
$K$ | Number of users;
$d_{s}$ | Number of data streams per user;
${M_{\mathrm{U}}}$ | Number of receive antennas at the $k$-th user;
${P_{\mathrm{T}}}$ | Total transmit power of the DFRC BS;
$\alpha_{k,l}$ | The complex amplitude of the $l$-th path for the $k$-th user;
${\beta_{0}}/{\beta_{j}}$ | The complex reflection coefficients of the target/clutter;
${{\bf{T}}_{\mathrm{D}}}/{{\bf{W}}_{\mathrm{D}}}$ | Digital transmit/receive beamformer at the DFRC BS;
${{\bf{T}}_{\mathrm{RF}}}/{{\bf{W}}_{\mathrm{RF}}}$ | Analog transmit/receive beamformer at the DFRC BS;
${\bf{U}}_{k}$ | Receive beamformer at the $k$-th user;
${{\bf{H}}_{k}}$ | CSI from the BS to the $k$-th user;
${{\bf{a}}_{\mathrm{t}}}\left({{\theta}}\right)/{{\bf{a}}_{\mathrm{r}}}\left({\theta}\right)$ | Transmit/receive steering vector at the angle $\theta$;
$R$ | The achievable sum-rate of $K$ users;
$\varGamma$ | The receive SCNR at the DFRC receiver;
${{\bf{T}}_{k}}/{\bf{W}}$ | Auxiliary variables introduced for the WPDD algorithm;
$\rho$ | The penalty parameter of the WPDD algorithm;
${\bf{D}}_{k}/\tilde{\bf{D}}$ | The dual variables of the WPDD algorithm.

##  II System Architecture And Problem Statement

Figure 1: Illustration of the considered DFRC system with an RS architecture.

As depicted in Fig. 1, an mmWave DFRC BS and users both have uniform linear
arrays (ULAs) installed. The DFRC BS has $M_{\mathrm{T}}$ transmit antennas
and $N_{\mathrm{RF}}^{\mathrm{t}}$ RF chains to serve $K$ users in the
downlink. It also has $M_{\mathrm{R}}$ receive antennas and
$N_{\mathrm{RF}}^{\mathrm{r}}$ RF chains to detect a target of interest in
cluttered environments. Each user has $M_{\mathrm{U}}$ receive antennas. The
angle-of-departure (AoD) of the radar target is the same as the angle-of-
arrival (AoA), as considered in the literature [27]. An RS-based HAD
architecture is considered at both the receive and transmit sides of the DFRC
BS, where every RF chain can be linked to a non-overlapping subset of transmit
antennas via an adaptive switching network.

Let
${\bf{s}}(t)={\left[{{\bf{s}}_{1}^{T}(t),\cdots,{\bf{s}}_{K}^{T}(t)}\right]^{T}}\in{\mathbb{C}^{N_{s}}}$
($N_{s}\leq N_{\mathrm{RF}}^{\mathrm{t}}\ll{M_{\mathrm{T}}}$) denote the
symbol sequence sent in the $t$-th slot for $K$ users and the radar target.
$d_{s}$ is the number of data streams for user $k$
($\mathbb{E}\\{{{\bf{s}}_{k}(t){\bf{s}}_{k}{{(t)}^{H}}}\\}={{\bf{I}}_{d_{s}}}$).
The symbols are independent among the users. The transmit signal is obtained
as

${\bf{x}}(t)={{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{\mathrm{D}}}{\bf{s}}(t)=\sum\limits_{k=1}^{K}{{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{{\mathrm{D}},k}}{{\bf{s}}_{k}}(t)},$
(1)

where
${\bf{T}}_{{\mathrm{D}},k}\in{\mathbb{C}^{{N_{\mathrm{RF}}^{\mathrm{t}}}\times
d_{s}}}$ stands for the digital beamfomer for the $k$-th user,
${{\bf{T}}_{\mathrm{D}}}=\left[{{{\bf{T}}_{{\mathrm{D}},1}},\cdots,{{\bf{T}}_{{\mathrm{D}},K}}}\right]$,
and ${\mathbf{T}}_{\mathrm{RF}}\in{\mathbb{C}^{M_{\mathrm{T}}\times
N_{\mathrm{RF}}^{\mathrm{t}}}}$ stands for the analog beamformer.

###  II-A Communication Model

Following the extended Saleh-Valenzuela model [19, 28], mmWave channel between
the DFRC BS and user $k$ is modeled as

${{\bf{H}}_{k}}=\sqrt{\frac{{{M_{\mathrm{T}}}{M_{\mathrm{U}}}}}{{{L_{k}}}}}\sum\limits_{l=1}^{{L_{k}}}{{\alpha_{k,l}}{\bf{a}}_{\mathrm{r}}({\psi_{k,l}}){\bf{a}}_{\mathrm{t}}^{H}\left({{\phi_{k,l}}}\right)},\forall
k,$ (2)

where $L_{k}$ specifies the number of propagation paths to the user $k$,
$\alpha_{k,l}\sim{\cal{CN}}({0,10^{-0.1PL(d_{k})}})$ is the complex gain of
the $l$-th path, $l=1,\cdots,L_{k}$. $PL(d_{k})$ is the path loss over the
distance, $d_{k}$, between the BS and user $k$.
${\bf{a}}_{\mathrm{t}}({{\phi_{k,l}}})$ and
${\bf{a}}_{\mathrm{r}}({{\psi_{k,l}}})$ give the transmit and receive array
response vectors, respectively:

${\bf{a}}_{\mathrm{t}}(\\!{{\phi_{k,l}}})\\!\\!=\\!\\!\frac{1}{{\sqrt{M_{\mathrm{T}}}}}{{\\!\left[\\!{1,{e^{-\\!j\frac{{2\pi}}{\lambda}\\!d\sin\\!{{\phi_{k,l}}}}}\\!,\\!\cdots\\!,\\!{e^{-\\!j\frac{{2\pi}}{\lambda}\\!d\left(\\!{M_{\mathrm{T}}\\!-\\!1}\\!\right)\sin\\!{{\phi_{k,l}}}}}}\\!\right]\\!}^{T}},$
(3)
${\bf{a}}_{\mathrm{r}}(\\!{{\psi_{k,l}}})\\!\\!=\\!\\!\frac{1}{{\sqrt{M_{\mathrm{U}}}}}{{\\!\left[\\!{1,{e^{-\\!j\frac{{2\pi}}{\lambda}\\!d\sin\\!{{\psi_{k,l}}}}}\\!,\\!\cdots\\!,\\!{e^{-\\!j\frac{{2\pi}}{\lambda}\\!d\left(\\!{M_{\mathrm{U}}\\!-\\!1}\\!\right)\\!\sin\\!{{\psi_{k,l}}}}}}\\!\right]\\!}^{T}}\\!.$
(4)

Here, $\phi_{k,l}$ and $\psi_{k,l}$ are the AoD/AoA of the $l$-th path to the
$k$-th user, respectively; $\lambda$ indicates the signal wavelength;
$d=\lambda/2$ specifies antenna spacing. Suppose ${\bf{H}}_{k}$, $\forall k$,
is known at the DFRC BS. In practice, the channels can be estimated, e.g.,
using the algorithms developed in [29, 30, 31].

The received signal of user $k$ is

${{\bf{y}}_{k}}(t)={\bf{H}}_{k}\sum\limits_{k=1}^{K}{{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{{\mathrm{D}},k}}{{\bf{s}}_{k}}(t)}+{{\bf{n}}_{k}}(t),$
(5)

where ${{\bf{n}}_{k}}(t)\sim{\cal
CN}({{\bf{0}},{\sigma^{2}}{\bf{I}}_{M_{\mathrm{U}}}})$ stands for the additive
white Gaussian noise (AWGN) of the $k$-th user. ${{\bf{n}}_{k}}(t)$ and
${{\bf{s}}_{k}}(t)$ are independent of each other. For the brevity of
notation, we suppress the time index $t$ in the remainder of this paper.

After FD receive beamforming,
${\bf{U}}_{k}\in{\mathbb{C}^{M_{\mathrm{U}}\times d_{s}}}$, the detected
signal of user $k$ is

$\displaystyle{\hat{\bf{s}}_{k}}$
$\displaystyle\\!=\\!{\bf{U}}_{k}^{H}{{\bf{y}}_{k}}$
$\displaystyle\\!=\\!\underbrace{{\bf{U}}_{k}^{H}{\bf{H}}_{k}{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{{\mathrm{D}},k}}{{\bf{s}}_{k}}}_{\text{desired
signal}}\\!+\\!\underbrace{\sum\limits_{i\neq
k}^{K}\\!{{\bf{U}}_{k}^{H}{\bf{H}}_{k}{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{{\mathrm{D}},i}}{{\bf{s}}_{i}}}}_{\text{multi-
user
interference}}\\!+\\!\underbrace{{\bf{U}}_{k}^{H}{{\bf{n}}_{k}}}_{\text{noise}}.$
(6)

Consequently, the overall achievable rate of the $K$ users is

$\displaystyle
R({{{\bf{T}}_{\mathrm{RF}}},\\!{{\bf{T}}_{{\mathrm{D}},k}}},{\bf{U}}_{k})\\!=\\!$
$\displaystyle\sum\limits_{k=1}^{K}{\log\det}({\bf{I}}_{d_{s}}+{\bf{U}}_{k}^{H}{\bf{H}}_{k}{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{{\mathrm{D}},k}}$
$\displaystyle\times{{\bf{T}}_{{\mathrm{D}},k}^{H}}{{\bf{T}}_{\mathrm{RF}}^{H}}{{\bf{H}}_{k}^{H}}{{\bf{U}}_{k}}{\bf{R}}_{k}^{-1}),$
(7)

where
${\bf{R}}_{k}\\!=\\!{\bf{U}}_{k}^{H}({\sigma^{2}}{\bf{I}}_{M_{\mathrm{U}}}\\!+\\!\sum\limits_{i\neq
k}^{K}\\!{{\bf{H}}_{k}{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{{\mathrm{D}},i}}{{\bf{T}}_{{\mathrm{D}},i}^{H}}{{\bf{T}}_{\mathrm{RF}}^{H}}{{\bf{H}}_{k}^{H}}}){{\bf{U}}_{k}}$.

###  II-B Radar Model

Apart from sending communication symbols to the $K$ users, the transmitted
waveform is also exploited to accomplish a radar detection task. Considering
an obstacle detection scenario, a target (e.g., a slowly moving pedestrian) is
located at angle $\theta_{0}$. There are also $J$ stationary clutterers (e.g.,
trees and buildings) at angles $\theta_{j},j=1,\cdots,J$. In this case, the
time delay and Doppler shift are relatively negligible, as discussed in [32].
The prior knowledge of $\theta_{0}$ and $\theta_{j}$ is obtainable from an
environmental dynamic database using a cognitive paradigm [33, 32].

Suppose that the potential self-interference from the transmit array to the
receive array of the MIMO radar is adequately addressed by implementing
appropriate techniques, for instance, the one developed in [34]. According to
(1), the echo signal received by the BS can be written as

${{\bf{y}}_{\mathrm{R}}}={\beta_{0}}{{\bf{a}}_{\mathrm{r}}}({{\theta_{0}}}){\bf{a}}_{\mathrm{t}}^{T}({{\theta_{0}}}){\bf{x}}+\sum\limits_{j=1}^{J}{{\beta_{j}}{{\bf{a}}_{\mathrm{r}}}({{\theta_{j}}}){\bf{a}}_{\mathrm{t}}^{T}({{\theta_{j}}}){\bf{x}}}+{\bf{n}}_{\mathrm{R}},$
(8)

where $\beta_{0}$ and $\beta_{j}$ denote the round-trip attenuation and
complex reflection coefficients concerning the target and the $j$-th clutter,
respectively; $\mathbb{E}({{\left|{\beta_{0}}\right|}^{2}})=\sigma_{0}^{2}$;
$\mathbb{E}({{\left|{{\beta_{j}}}\right|}^{2}})=\sigma_{\mathrm{C}}^{2}$;
${\bf{a}}_{\mathrm{r}}({\theta})$ gives the $M_{\mathrm{R}}\times 1$ receive
steering vector; ${\bf{a}}_{\mathrm{t}}({\theta})$ specifies the
$M_{\mathrm{T}}\times 1$ transmit steering vector;
${\bf{a}}_{\mathrm{r}}(\theta)$ and ${\bf{a}}_{\mathrm{t}}(\theta)$ are
obtained in the same way as (3); ${\bf{n}}_{\mathrm{R}}\sim{\cal
CN}({{\bf{0}},\sigma_{\mathrm{R}}^{2}{{\bf{I}}_{M_{\mathrm{R}}}}})$ is the
AWGN.

An RS-based HAD architecture is also considered at the DFRC BS receiver. The
received echo signal after analog receive beamforming
${\mathbf{W}}_{\mathrm{RF}}\in{\mathbb{C}^{M_{\mathrm{R}}\times
N_{\mathrm{RF}}^{\mathrm{r}}}}$ and digital receive beamforming
${\bf{W}}_{{\mathrm{D}}}\in{\mathbb{C}^{N_{\mathrm{RF}}^{\mathrm{r}}\times
N_{\mathrm{s}}}}$, is given by

$\displaystyle{\bf{y}}^{\prime}_{\mathrm{R}}$
$\displaystyle={{\bf{W}}_{{\mathrm{D}}}^{H}}{{\mathbf{W}}_{\mathrm{RF}}^{H}}{{\bf{y}}_{\mathrm{R}}}$
$\displaystyle=\underbrace{{\beta_{0}}{{\bf{W}}_{{\mathrm{D}}}^{H}}{{\mathbf{W}}_{\mathrm{RF}}^{H}}{\bf{A}}({{\theta_{0}}}){\bf{x}}}_{\text{target}}+\underbrace{{{\bf{W}}_{{\mathrm{D}}}^{H}}{{\mathbf{W}}_{\mathrm{RF}}^{H}}\sum\limits_{j=1}^{J}\\!{{\beta_{j}}{\bf{A}}({{\theta_{j}}}){\bf{x}}}}_{\text{clutters}}$
$\displaystyle+\underbrace{{{\bf{W}}_{{\mathrm{D}}}^{H}}{{\mathbf{W}}_{\mathrm{RF}}^{H}}{\bf{n}}_{\mathrm{R}}}_{\text{noise}},$
(9)

where
${\bf{A}}(\theta)\\!=\\!{{\bf{a}}_{\mathrm{r}}}(\theta){\bf{a}}_{\mathrm{t}}^{T}(\theta)$.

We adopt the SCNR as the metric to measure the radar’s target detection and
localization capabilities, as given by

$\displaystyle\varGamma({{{\bf{T}}_{\mathrm{RF}}},{{\bf{T}}_{\mathrm{D}}},{\bf{W}}_{{\mathrm{RF}}},{\bf{W}}_{{\mathrm{D}}}})$
$\displaystyle\\!=\\!\frac{\mathbb{E}{\left({\left\|{{\beta_{0}}{{\bf{W}}_{{\mathrm{D}}}^{H}}{{\mathbf{W}}_{\mathrm{RF}}^{H}}{\bf{A}}({{\theta_{0}}}){\bf{x}}}\right\|}^{2}\right)}}{\mathbb{E}\left({{{{\left\|{{{\bf{W}}_{{\mathrm{D}}}^{H}}{{\mathbf{W}}_{\mathrm{RF}}^{H}}\\!\sum\limits_{j=1}^{J}\\!{{\beta_{j}}{\bf{A}}\left({{\theta_{j}}}\right){\bf{x}}}}\right\|}^{2}}}}\\!\right)+\\!\mathbb{E}\left({\left\|{{\bf{W}}_{{\mathrm{D}}}^{H}}{{\mathbf{W}}_{\mathrm{RF}}^{H}}{\bf{n}}_{\mathrm{R}}\right\|}^{2}\right)}$
$\displaystyle\\!=\\!\frac{\mathrm{tr}({{{\bf{W}}_{{\mathrm{D}}}^{H}}{{\mathbf{W}}_{\mathrm{RF}}^{H}}{{\bf{\Sigma}}_{\mathrm{t}}}{{\bf{W}}_{{\mathrm{RF}}}}{{\mathbf{W}}_{\mathrm{D}}}})}{\mathrm{tr}({{{\bf{W}}_{{\mathrm{D}}}^{H}}{{\mathbf{W}}_{\mathrm{RF}}^{H}}{{\bf{\Sigma}}_{\mathrm{cn}}}{{\bf{W}}_{{\mathrm{RF}}}}{{\mathbf{W}}_{\mathrm{D}}}})},$
(10)

where
${\bf{\Sigma}}_{\mathrm{t}}\\!=\\!{\sigma_{0}^{2}{{\bf{A}}\\!({{\theta_{0}}}){{\bf{T}}_{\mathrm{RF}}}{\sum\limits_{k=1}^{K}({{{\bf{T}}_{{\mathrm{D}},k}}{{\bf{T}}_{{\mathrm{D}},k}^{H}}})}{{\bf{T}}_{\mathrm{RF}}^{H}}{{\bf{A}}^{H}}\\!({{\theta_{0}}})}}$,
and
${\bf{\Sigma}}_{\mathrm{cn}}\\!\\!=\\!\\!\sum\limits_{j=1}^{J}\\!{\sigma_{\mathrm{C}}^{2}{\bf{A}}\\!({{\theta_{j}}}){{\bf{T}}_{\mathrm{RF}}}{\sum\limits_{k=1}^{K}({{{\bf{T}}_{{\mathrm{D}},k}}{{\bf{T}}_{{\mathrm{D}},k}^{H}}})}{{\bf{T}}_{\mathrm{RF}}^{H}}{{\bf{A}}^{H}}\\!({{\theta_{j}}})}+\sigma_{\mathrm{R}}^{2}{\bf{I}}$.

We note that SCNR is a measurable sensing performance indicator at the
receiver, and can be reasonably adapted to different signal characteristics
and application scenarios. Moreover, the radar detection probability is known
to be a non-decreasing function of the SCNR111 The target detection
probability (PD) increases monotonically with the received SCNR, i.e.,
$P_{\mathrm{D}}(\gamma)=Q(\sqrt{2{\gamma}},\sqrt{-2\ln P_{\mathrm{FA}}})$ [26,
35], where $Q(\cdot,\cdot)$ is the Marcum Q function and $P_{\mathrm{FA}}$
gives the false-alarm probability., indicating the relevance of the SCNR in
DFRC systems. The measurability and relevance of SCNR offer excellent
practicality, leading it to be broadly considered in radar and DFRC systems
[35].

###  II-C Problem Statement

We holistically design the hybrid transmit beamformers
${\bf{T}}_{{\mathrm{D}},k}$ and ${\bf{T}}_{\mathrm{RF}}$, and the hybrid
receive beamformers ${\bf{W}}_{\mathrm{D}}$ and ${\bf{W}}_{\mathrm{RF}}$ under
the RS-based HAD architecture, to elevate the downlink sum-rate while ensuring
the radar detection performance. This problem is cast as follows:

$\displaystyle\mathrm{P}_{1}:\mathop{\max}\limits_{\mathcal{X}}\ $
$\displaystyle
R({{{\bf{T}}_{\mathrm{RF}}},\\!{{\bf{T}}_{{\mathrm{D}},k}}},{\bf{U}}_{k})$
(11a) s.t.
$\displaystyle\varGamma({{{\bf{T}}_{\mathrm{RF}}},{{\bf{T}}_{\mathrm{D}}},{{\bf{W}}_{\mathrm{RF}}},{{\bf{W}}_{\mathrm{D}}}})\geq\gamma,$
(11b)
$\displaystyle\left\|{{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{\mathrm{D}}}}\right\|_{F}^{2}\leq
P_{\mathrm{T}},$ (11c)
$\displaystyle{{\bf{T}}_{\mathrm{RF}}}({m,n})\\!\in\\!\left\\{{0,{\cal
F}}\right\\},\forall m,n,$ (11d)
$\displaystyle{\left\|{{{\bf{T}}_{\mathrm{RF}}}({m,:})}\right\|_{0}}=1,\forall
m,$ (11e)
$\displaystyle{{\bf{W}}_{\mathrm{RF}}}({\tilde{m},\tilde{n}})\\!\in\\!\left\\{{0,{\cal
F}}\right\\},\forall\tilde{m},\tilde{n},$ (11f)
$\displaystyle{\left\|{{{\bf{W}}_{\mathrm{RF}}}({\tilde{m},:})}\right\|_{0}}=1,\forall\tilde{m},$
(11g)

where
$\mathcal{X}=\\{{{{\bf{T}}_{\mathrm{RF}}},{{\bf{T}}_{\mathrm{D},k}},{\bf{U}}_{k},{{\bf{W}}_{\mathrm{RF}}},{{\bf{W}}_{\mathrm{D}}}}\\}$
collects all of the optimization variables. Constraint (11b) ensures the
minimum SCNR $\gamma$. Constraint (11c) specifies the total transmit power
budget. Constraints (11d)-(11g) specify the HBF mapping criteria. More
specifically, if the $n$-th RF chain is associated with the $m$-th antenna via
a PS, ${{\bf{T}}_{\mathrm{RF}}}\left({m,n}\right)\in{\cal F}$,
$m=1,\cdots,{M_{\mathrm{T}}},n=1,\cdots,{N_{\mathrm{RF}}^{\mathrm{t}}}$,
contains a nonzero phase, where ${\cal F}$ collects all possible values of a
PS; otherwise, ${{\bf{T}}_{\mathrm{RF}}}\left({m,n}\right)=0$. Similarly, if
the $\tilde{n}$-th RF chain is connected to the $\tilde{m}$-th antenna via a
PS, ${{\bf{W}}_{\mathrm{RF}}}\left({\tilde{m},\tilde{n}}\right)\in{\cal F}$,
$\tilde{m}=1,\cdots,{M_{\mathrm{R}}},\tilde{n}=1,\cdots,{N_{\mathrm{RF}}^{\mathrm{r}}}$;
otherwise, ${{\bf{W}}_{\mathrm{RF}}}\left({\tilde{m},\tilde{n}}\right)=0$.
Since no subarrays overlap, the analog beamformer has only one nonzero element
per row, i.e.,
${\left\|{{{\bf{T}}_{\mathrm{RF}}}\left({m,:}\right)}\right\|_{0}}=1$ for
$m=1,\cdots,{M_{\mathrm{T}}}$;
${\left\|{{{\bf{W}}_{\mathrm{RF}}}\left({\tilde{m},:}\right)}\right\|_{0}}=1$
for $\tilde{m}=1,\cdots,{M_{\mathrm{R}}}$.

Problem $\mathrm{P}_{1}$ poses significant challenges ascribed to non-convex
objectives and constraints, and the strong coupling of digital and analog
beamformers. The non-convexity makes it difficult to attain a globally optimal
solution. A direct attempt to tackle the coupling effect with block
decomposition methods, e.g., block successive upper-bound minimization and
inexact flexible parallel algorithm, can also lead to difficulty in ensuring
convergence [36]. In what follows, we develop a tractable solution to this
problem.

##  III WPDD Algorithm for DFRC with RS Architecture

###  III-A Reformulation of Problem $\mathrm{P}_{1}$

To overcome the difficulty in solving the nonconvex sum-rate problem
$\mathrm{P}_{1}$ with coupled variables, we develop a tractable objective
according to the WMMSE method to decrease the complexity of solving problem
$\mathrm{P}_{1}$. As revealed in Proposition 1, problem $\mathrm{P}_{1}$ can
be rewritten equivalently as a WMMSE minimization problem [37].

######  Proposition 1.

Problem $\mathrm{P}_{1}$ and the following problem of minimizing weighted sum
mean square error (MSE) have the same global optimum:

$\displaystyle\mathrm{P}_{2}:\mathop{\max}\limits_{\mathcal{X},\\{{\bf{G}}_{k}\\},\\{{\bf{E}}_{k}\\}}$
$\displaystyle\sum\limits_{k=1}^{K}{(\log\det({\bf{G}}_{k})-\mathrm{tr}({\bf{G}}_{k}{\bf{E}}_{k}))}$
(12a) $\displaystyle\mathrm{s.t.}\ $
$\displaystyle(\mathrm{\ref{eq:problem1B}})-(\mathrm{\ref{eq:problem1G}}),$

where ${\bf{G}}_{k}$ denotes the weighting factor for user $k$, ${\bf{U}}_{k}$
is the receive beamforming for user $k$, and ${\bf{E}}_{k}$ is the MSE of user
$k$, as given by

$\displaystyle{{\bf{E}}_{k}}$
$\displaystyle=\mathbb{E}{\\{({\hat{\bf{s}}_{k}}-{\bf{s}}_{k})({\hat{\bf{s}}_{k}}-{\bf{s}}_{k})^{H}\\}}$
$\displaystyle={\bf{I}}_{d_{s}}-2{\cal
R}({{\bf{U}}_{k}^{H}{\bf{H}}_{k}{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{{\mathrm{D}},k}}})+{\sigma^{2}}{{\bf{U}}_{k}^{H}}{{\bf{U}}_{k}}$
$\displaystyle\ \
+\sum\limits_{i=1}^{K}\\!{\bf{U}}_{k}^{H}{{\bf{H}}_{k}{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{{\mathrm{D}},i}}{{\bf{T}}_{{\mathrm{D}},i}^{H}}{{\bf{T}}_{\mathrm{RF}}^{H}}{{\bf{H}}_{k}^{H}}}{{\bf{U}}_{k}}.$
(13)

###### Proof.

Since ${\bf{G}}_{k}$ and ${\bf{U}}_{k}$ appear only in (12a), the equivalence
between problems $\mathrm{P}_{2}$ and $\mathrm{P}_{1}$ can be established by
substituting their optimal solutions (by first-order optimality condition)
into (12a). For more details, please refer to [37, Thm 1]. ∎

Although the WMMSE-based transformation makes the objective function easy to
handle, the analog and digital beamforming are coupled in constraints (11b)
and (11c). Moreover, the non-convex SCNR constraint in (11b), and the $L_{0}$
constraints in (11e) and (11g) make the problem challenging. Jointly
optimizing analog and digital beamforming is difficult. We introduce auxiliary
variables,
${{\bf{T}}_{k}}={{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{{\mathrm{D}},k}},\,\forall
k$, and ${\bf{W}}={{\bf{W}}_{\mathrm{RF}}}{{\bf{W}}_{\mathrm{D}}}$. Problem
$\mathrm{P}_{2}$ is recast as

$\displaystyle\mathrm{P}_{3}:\mathop{\max}\limits_{\mathcal{\tilde{X}}}$
$\displaystyle\sum\limits_{k=1}^{K}{(\log\det({\bf{G}}_{k})-\mathrm{tr}({\bf{G}}_{k}{\bf{E}}_{k}))}$
(14a) $\displaystyle\mathrm{s.t.}\ $
$\displaystyle\frac{\mathrm{tr}({{{\bf{W}}^{H}}{{\bf{\Sigma}}_{\mathrm{t}}}{\bf{W}}})}{\mathrm{tr}({{\bf{W}}^{H}}{\bf{\Sigma}}_{\mathrm{cn}}{\bf{W}})}\geq\gamma,$
(14b) $\displaystyle\sum\limits_{k=1}^{K}\left\|{{\bf{T}}_{k}}\right\|^{2}\leq
P_{\mathrm{T}},$ (14c)
$\displaystyle{{\bf{T}}_{k}}={{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{{\mathrm{D}},k}},\forall
k,$ (14d)
$\displaystyle{\bf{W}}={{\bf{W}}_{\mathrm{RF}}}{{\bf{W}}_{\mathrm{D}}},$ (14e)
$\displaystyle\mathrm{\eqref{eq:problem1D}}-\mathrm{\eqref{eq:problem1G}},$

where
$\mathcal{\tilde{X}}\\!=\\!\\{\\!{{{\bf{T}}_{\mathrm{RF}}},{{\bf{T}}_{\mathrm{D},k}},{\bf{U}}_{k},{{\bf{W}}_{\mathrm{RF}}},{{\bf{W}}_{\mathrm{D}}},{\bf{G}}_{k},{{\bf{T}}_{k}},{\bf{W}}}\\!\\}$.

###  III-B PDD-Based Framework for Problem $\mathrm{P}_{3}$

We resort to the PDD to solve $\mathrm{P}_{3}$, as problem $\mathrm{P}_{3}$
has the form of the problem ($\mathcal{P}$) in [36]. The PDD method integrates
the penalty method, the augmented Lagrangian method, and the alternating
direction method of multipliers (ADMM) and delivers a general algorithmic
framework for minimizing a non-convex and non-smooth function while dealing
with non-convex coupling constraints. The PDD comprises double-loop
iterations, with the outer loops updating the dual variables and penalty
parameters, and the inner loops executing the augmented Lagrangian method. The
PDD suits problem $\mathrm{P}_{3}$ with a differentiable objective function.

Through the incorporation of two penalty terms for constraint (14d) and (14e)
into (14a), the inner loop can be formulated as

$\displaystyle\mathrm{P}_{4}:\\!\mathop{\min}\limits_{\mathcal{\tilde{X}}}$
$\displaystyle\sum\limits_{k=1}^{K}\\{{\mathrm{tr}({\bf{G}}_{k}{\bf{E}}_{k})\\!-\\!\log\det({\bf{G}}_{k})}$
$\displaystyle\ \
+{\frac{1}{2\rho}{\left\|{{{\bf{T}}_{k}}\\!-\\!{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{{\mathrm{D}},k}}\\!+\\!\rho{{\bf{D}}_{k}}}\right\|}^{2}}\\}$
$\displaystyle\ \
+{\frac{1}{2\rho}\\!{\left\|{{\bf{W}}\\!-\\!{{\bf{W}}_{\mathrm{RF}}}\\!{{\bf{W}}_{\mathrm{D}}}\\!+\\!\rho{\bf{\tilde{D}}}}\right\|}^{2}}$
(15a) $\displaystyle\mathrm{s.t.}\ $
$\displaystyle\mathrm{\eqref{eq:problem1D}}-\mathrm{\eqref{eq:problem1G},\,\eqref{eq:problem3B}}-\mathrm{\eqref{eq:problem3C}},$

where $\rho$ denotes the penalty parameter; ${{\bf{D}}_{k}}$ and
${\bf{\tilde{D}}}$ are the dual variables of constraints (14d) and (14e),
respectively. $\rho$, ${{\bf{D}}_{k}},\forall k$, and ${\bf{\tilde{D}}}$ are
fixed throughout the inner-loop iterations.

Next, we solve problem $\mathrm{P}_{4}$ utilizing the PDD framework, where BCD
is used to solve problem $\mathrm{P}_{4}$.

_1) Equivalent FD Sensing Receive Beamformer ${\bf{W}}$_:

By letting ${{\bf{t}}_{k}}={\mathrm{vec}}({{\bf{T}}_{k}})$,
${\bf{t}}=[{\bf{t}}_{1}^{T},\cdots,{\bf{t}}_{K}^{T}]^{T}$, and
${\bf{w}}={\mathrm{vec}}({\bf{W}})$, the SCNR in (II-B) can be rewritten as

$\displaystyle\varGamma({{{\bf{T}}_{\mathrm{RF}}},{{\bf{T}}_{\mathrm{D}}},{\bf{W}}_{{\mathrm{RF}}},{\bf{W}}_{{\mathrm{D}}}})\\!=\\!\frac{{\sigma_{0}^{2}}{\left|{\bf{w}}^{H}\tilde{\bf{A}}\left({{\theta_{0}}}\right){\bf{t}}\right|^{2}}}{{{{\bf{w}}^{H}}{\tilde{\bf{\Sigma}}_{\mathrm{cn}}}{{\bf{w}}}}},$
(16)

where $\tilde{\bf{A}}(\theta)\\!=\\!\bf{I}\otimes{\bf{A}}(\theta)$,
$\tilde{\bf{\Sigma}}_{\mathrm{cn}}\\!\\!=\\!\\!\sum\limits_{j=1}^{J}\\!{\sigma_{\mathrm{C}}^{2}\tilde{\bf{A}}\\!({{\theta_{j}}}){\bf{t}}{\bf{t}}^{H}{\tilde{\bf{A}}}^{H}\\!({{\theta_{j}}})}\\!+\\!\sigma_{\mathrm{R}}^{2}{\bf{I}}$.

When the transmit beamforming is given and fixed, we employ the well-known
minimum variance distortionless response (MVDR) beamformer [38] to obtain the
maximum SCNR at the MIMO radar, as given by

${\bf{w}}=\frac{{{\tilde{\bf{\Sigma}}_{\mathrm{cn}}^{-1}}\tilde{\bf{A}}({{\theta_{0}}}){\bf{t}}}}{{{\bf{t}}^{H}}{\tilde{\bf{A}}}^{H}({{\theta_{0}}}){\tilde{\bf{\Sigma}}_{\mathrm{cn}}^{-1}}\tilde{\bf{A}}({{\theta_{0}}}){\bf{t}}},$
(17)

then $\bf{W}$ can be obtained by reshaping $\bf{w}$. According to this optimal
radar receive beamformer $\bf{W}$, the SCNR in (16) is further expressed as

$\tilde{\varGamma}({{{\bf{T}}_{\mathrm{RF}}},{{\bf{T}}_{\mathrm{D},k}}})=\sum\limits_{k=1}^{K}\mathrm{tr}\left({\bf{T}}_{{\mathrm{D}},k}^{H}{\bf{T}}_{\mathrm{RF}}^{H}{\bf{\Phi}}{\bf{T}}_{\mathrm{RF}}{\bf{T}}_{{\mathrm{D}},k}\right),$
(18)

where
${\bf{\Phi}}=\sigma_{0}^{2}{{\bf{A}}^{H}}({{\theta_{0}}}){{\bf{\Sigma}}_{\mathrm{cn}}^{-1}}{\bf{A}}({{\theta_{0}}})$.

_2) Equivalent FD Transmit Beamformer ${{\bf{T}}_{k}}$_:

With the rest of variables fixed, the subproblem of problem $\mathrm{P}_{4}$
for solving the auxiliary variable ${{\bf{T}}_{k}}$ is written as

$\displaystyle\mathop{\min}\limits_{\left\\{{{\bf{T}}_{k}}\right\\}}$
$\displaystyle\sum\limits_{k=1}^{K}(\mathrm{tr}({\bf{G}}_{k}{\bf{E}}_{k})\\!+\\!{\frac{1}{2\rho}{\left\|{{{\bf{T}}_{k}}\\!-\\!{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{{\mathrm{D}},k}}\\!+\\!\rho{{\bf{D}}_{k}}}\right\|}^{2}})$
(19a) $\displaystyle\mathrm{s.t.}\ $
$\displaystyle\sum\limits_{k=1}^{K}\mathrm{tr}\left({{\bf{T}}_{k}^{H}{\bf{\Phi}}{{\bf{T}}_{k}}}\right)\geq\gamma,$
(19b)
$\displaystyle\sum\limits_{k=1}^{K}{\left\|{{\bf{T}}_{k}}\right\|^{2}}\leq{P_{\mathrm{T}}},$
(19c)

where ${{\bf{E}}_{k}}$ is obtained by plugging (14d) into (1), as given by

$\displaystyle{{\bf{E}}_{k}}$ $\displaystyle={\bf{I}}_{d_{s}}-2{\cal
R}({{\bf{U}}_{k}^{H}{\bf{H}}_{k}{{\bf{T}}_{k}}})+{\sigma^{2}}{{\bf{U}}_{k}^{H}}{{\bf{U}}_{k}}$
$\displaystyle\ \
+\sum\limits_{i=1}^{K}\\!{\bf{U}}_{k}^{H}{{\bf{H}}_{k}{{\bf{T}}_{i}}{{\bf{T}}_{i}^{H}}{{\bf{H}}_{k}^{H}}}{{\bf{U}}_{k}}.$
(20)

Notice that (19) provides a non-convex quadratic constraint quadratic
programming (QCQP). Semi-definite relaxation (SDR) is often applied to resolve
such a problem. However, SDR conducts a relaxation of rank-one constraints,
resulting in an additional process like Gaussian randomization to restore the
rank-one constraints on obtained solutions [39]. Extra computational
complexity occurs.

We propose to transform (19) into a series of SOCPs [40], using viable
substitution and successive convex approximation (SCA) techniques and solve
the SOCPs recursively to obtain a sub-optimal solution to (19). Let
${\bf{Z}}_{k}={\bf{T}}_{k}+\rho{\bf{D}}_{k}$ and
${\bf{P}}_{k}={\bf{H}}_{k}^{H}{\bf{U}}_{k}{{\bf{G}}_{k}}{{\bf{U}}_{k}^{H}}{{\bf{H}}_{k}}$.
We introduce slack variables ${a_{k,i}}>0$ and ${b_{k}}>0$,
$i,k\in\\{1,\cdots,K\\}$ such that
${{{\bf{t}}_{i}^{H}}({\bf{I}}\otimes{\bf{P}}_{k}^{H}){{\bf{t}}_{i}}}\leq{a_{k,i}}$
and
${\left\|{{{\bf{Z}}_{k}}\\!-\\!{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{{\mathrm{D}},k}}}\right\|}^{2}\leq{b_{k}}$,
which can be further rewritten in the following second-order cone (SOC)
constraints:

$\left\|{\left[{\begin{array}[]{*{20}{c}}{{\bf{t}}_{i}^{H}}({\bf{I}}\otimes{\bf{P}}_{k}^{H})^{1/2}\\\
{\frac{{{a_{k,i}}-1}}{2}}\end{array}}\right]}\right\|\leq\frac{{{a_{k,i}}+1}}{2},\forall
i,k;$ (21)
$\left\|{\left[{\begin{array}[]{*{20}{c}}{{{\bf{Z}}_{k}}\\!-\\!{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{{\mathrm{D}},k}}}\\\
{\frac{{{b_{k}}-1}}{2}}\end{array}}\right]}\right\|\leq\frac{{{b_{k}}+1}}{2},\forall
k.$ (22)

In the non-convex SCNR constraint (19b), we can rewrite
$\mathrm{tr}\left({{\bf{T}}_{k}^{H}{\bf{\Phi}}{{\bf{T}}_{k}}}\right)={{\bf{t}}_{k}^{H}}({\bf{I}}\otimes{\bf{\Phi}}^{H}){{\bf{t}}_{k}}$.
Then, we resort to the SCA technique to replace this non-convex term with its
linear lower bound (e.g., the first-order expansion), i.e.,

${\bar{\varGamma}}_{k}={2{\cal
R}({{\bf{\bar{t}}}_{k}^{H}\bar{{\bf{\Phi}}}{{\bf{t}}_{k}}})-{\bf{\bar{t}}}_{k}^{H}\bar{\bf{\Phi}}{{{\bf{\bar{t}}}}_{k}}},$
(23)

where $\bar{\bf{\Phi}}={\bf{I}}\otimes{\bf{\Phi}}^{H}$, and
${{{\bf{\bar{t}}}}_{k}}$ is the value of ${\bf{t}}_{k}$ in the previous SCA
iteration.

Thus, the approximate convex problem of (19) is

$\displaystyle\mathop{\min}\limits_{\left\\{{{\bf{t}}_{k}},{a_{k,i}},{b_{k}}\right\\}}$
$\displaystyle\sum\limits_{k=1}^{K}\\!{\left(\\!{{\sum\limits_{i=1}^{K}{a_{k,i}}}\\!-\\!{2{\cal
R}(\mathrm{vec}^{H}({\bf{G}}_{k}{\bf{U}}_{k}^{H}{\bf{H}}_{k})^{H}{{\bf{t}}_{k}})}\\!+\\!\frac{1}{2\rho}{b_{k}}}\\!\right)}$
(24a) $\displaystyle\mathrm{s.t.}\ $
$\displaystyle\sum\limits_{k=1}^{K}{2{\cal
R}({{\bf{\bar{t}}}_{k}^{H}\bar{{\bf{\Phi}}}{{\bf{t}}_{k}}})-{\bf{\bar{t}}}_{k}^{H}\bar{\bf{\Phi}}{{{\bf{\bar{t}}}}_{k}}}\geq\gamma,$
(24b) $\displaystyle{\left\|{\bf{t}}\right\|}\leq\sqrt{P_{\mathrm{T}}},$ (24c)
$\displaystyle\eqref{eq:aki}-\eqref{eq:bk}.$

We can use off-the-shelf toolboxes, e.g., CVX [41], to efficiently solve (24).
By recursively solving (24), we can obtain at least a local optimum of (19).

Through the use of the PDD framework, we can decouple hybrid transmit and
receive beamforming for DFRC into the subproblems of optimizing the equivalent
sensing receive beamformer ${\bf{W}}$ and the equivalent DFRC transmit
beamformers ${{\bf{T}}_{k}},\,\forall k$, which can be solved sequentially.
The inter-user interference in the considered system leads to non-convexity in
the objective function in (19a). The clutter interference leads to non-
convexity in the SCNR constraint in (19b). To deal with this, variable
substitution and SCA are conducted to convexify both the objective function
and the SCNR constrain in the subproblem of ${{\bf{T}}_{k}},\,\forall k$; see
(21), (22) and (23). As a result, the subproblem can be solved using off-the-
peg CVX toolkits.

Remark: PDD is a general algorithmic framework for solving problems with
multiplicatively coupled variables. It has been applied to optimize hybrid
transmit and receive beamforming for a single user in a MIMO channel, e.g., in
[23]. By contrast, the studied problem of multi-user mMIMO can undergo non-
negligible inter-user interference; see (II-A). Moreover, our studied system
suffers from interference from clutters to the radar sensing signals received
by the BS; see (II-B). This differs substantially from the work presented in
[23], where no such interference exists. In this sense, our studied problem is
distinct from [23] in objectives and constraints.

_3) Digital Receive Beamformer ${\bf{U}}_{k}$ for User $k$_:

We optimize $\\{{\bf{U}}_{k}\\}$ by fixing all other variables, leading to an
unconstrained problem:

$\mathop{\min}\limits_{{\bf{U}}_{k}}\sum\limits_{k=1}^{K}{\mathrm{tr}({\bf{G}}_{k}{\bf{E}}_{k})}.$
(25)

By substituting (III-B) into (25) and then assessing the first-order
derivative of (25), the optimum receive combiner ${\bf{U}}_{k}$ for user $k$
is obtained in closed form as follows:

${\bf{U}}_{k}=\left({\sum\limits_{i=1}^{K}{{\bf{H}}_{k}}{{\bf{T}}_{i}}{{\bf{T}}_{i}^{H}}{{\bf{H}}_{k}^{H}}+\sigma^{2}{\bf{I}}_{M_{\mathrm{U}}}}\right)^{-1}\\!{{\bf{H}}_{k}{{\bf{T}}_{k}}},\forall
k.$ (26)

_4) WMMSE Weighting Matrix ${\bf{G}}_{k}$_:

By substituting (26) into (III-B), the mean square error ${{\bf{E}}_{k}}$ is
given by $({{\bf{I}}-{\bf{U}}_{k}^{H}{\bf{H}}_{k}{{\bf{T}}_{k}}})$. Then,
assessing the first-order optimality condition of problem $\mathrm{P}_{4}$
concerning ${\bf{G}}_{k}$, the closed-form solution of ${\bf{G}}_{k}$ is

${\bf{G}}_{k}={({{\bf{E}}_{k}})^{-1}}=({{\bf{I}}-{\bf{U}}_{k}^{H}{\bf{H}}_{k}{{\bf{T}}_{k}}})^{-1},\forall
k.$ (27)

_5) Analog Transmit Beamformer ${{\bf{T}}_{\mathrm{RF}}}$_:

Recall that ${\bf{Z}}_{k}={\bf{T}}_{k}+\rho{\bf{D}}_{k}$, and the analog
transmit beamformer ${{\bf{T}}_{\mathrm{RF}}}$ is only in the second term of
(15a). Therefore, the subproblem concerning ${{\bf{T}}_{\mathrm{RF}}}$ can be
given by

$\displaystyle\mathop{\min}\limits_{{\bf{T}}_{\mathrm{RF}}}$
$\displaystyle\sum\limits_{k=1}^{K}{{\left\|{{\bf{Z}}_{k}}-{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{{\mathrm{D}},k}}\right\|^{2}}}$
(28) $\displaystyle\mathrm{s.t.}$
$\displaystyle{{\bf{T}}_{\mathrm{RF}}}\left({m,n}\right)\in\left\\{{0,{\cal
F}}\right\\},\forall m,n,$
$\displaystyle{\left\|{{{\bf{T}}_{\mathrm{RF}}}\left({m,:}\right)}\right\|_{0}}=1,\forall
m.$

Let ${\bf{Z}}=\left[{{{\bf{Z}}_{1}},\cdots,{{\bf{Z}}_{K}}}\right]$. Then, the
objective
$\sum\nolimits_{k=1}^{K}{{\left\|{{\bf{Z}}_{k}}-{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{{\mathrm{D}},k}}\right\|^{2}}}$
can be rewritten as
$\left\|{{\bf{Z}}-{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{\mathrm{D}}}}\right\|^{2}$.
The constraints in (28) indicate there is only one non-zero element with a
constant magnitude in every row of ${{\bf{T}}_{\mathrm{RF}}}$, allowing (28)
to be solved row-by-row.

Let ${{\cal S}_{n}},n=1,\cdots,{N_{\mathrm{RF}}^{\mathrm{t}}}$ denote the
transmit antennas linked to the $n$-th RF chain. Because each of the antennas
is linked to only an RF chain and the subarrays do not intersect each other,
we can have

$\cup_{n=1}^{N_{\mathrm{RF}}^{\mathrm{t}}}{{\cal
S}_{n}}=\left\\{{1,\cdots,{M_{\mathrm{T}}}}\right\\},\forall n;$ (29) ${{\cal
S}_{p}}\cap{{\cal S}_{q}}=\emptyset,\forall p\neq q.$ (30)

Based on (29) and (30), we establish the following proposition to solve
problem (28).

######  Proposition 2.

Define
${\bf{Z}}({m,:}){{\bf{T}}_{\mathrm{D}}}{({n,:})^{H}}=\left|{{\zeta_{m,n}}}\right|{e^{j{\tilde{\phi}_{m,n}}}}$
for $m\in{{\cal S}_{n}}$ and $n=1,\cdots,{N_{\mathrm{RF}}^{\mathrm{t}}}$.
Problem (28) is rewritten as

$\displaystyle\mathop{\max}\limits_{\left\\{{\varphi_{m,n}}\right\\}\left\\{{{\cal
S}_{n}}\right\\}}$
$\displaystyle\sum\limits_{n=1}^{N_{\mathrm{RF}}^{\mathrm{t}}}{\sum\limits_{{\forall
m}\in{\cal
S}_{n}}{\left|{\zeta_{m,n}}\right|\cos({{\tilde{\phi}_{m,n}}-{\varphi_{m,n}}})}}$
(31) $\displaystyle\mathrm{s.t.}$ $\displaystyle\text{\eqref{eq:DCcons1},
\eqref{eq:DCcons2}},$

where ${\varphi_{m,n}}$ stands for the phase of the $({m,n})$-th element of
${{\bf{T}}_{\mathrm{RF}}}$, i.e.,
${{\bf{T}}_{\mathrm{RF}}}({m,n})={e^{j{\varphi_{m,n}}}}$.

###### Proof.

See Appendix A. ∎

Clearly, the objective of problem (31) is maximized when
${\varphi_{m,n}}={\tilde{\phi}_{m,n}}$. Therefore, for each antenna $m$, we
map it to the $n$-th RF chain maximizing
$\left|{\zeta_{m,n}}\right|\cos({{\tilde{\phi}_{m,n}}-{\varphi_{m,n}}})$:

$n_{m}^{\star}=\arg\mathop{\max}\limits_{n}\left|{\zeta_{m,n}}\right|\cos({{\tilde{\phi}_{m,n}}-{\varphi_{m,n}}}),$
(32)

and refresh the subset of transmit antennas for the $n$-th RF chain: ${{\cal
S}_{n^{\star}}}={{\cal S}_{n^{\star}}}\cup\left\\{m\right\\}$.

The optimal subarray mapping scheme is obtained by performing (32) for each
antenna. Then the corresponding optimal analog beamformer is obtained as

${\bf{T}}_{\mathrm{RF}}{({m,n})}\\!=\\!\begin{cases}{e^{j{\varphi_{m,n}}}},&\mathrm{if}\
m\\!\in\\!{{\cal
S}_{n^{\star}}},\,n\\!=\\!1,\\!\cdots,\\!{N_{\mathrm{RF}}^{\mathrm{t}}};\\\
{0,}&{\mathrm{otherwise.}}\end{cases}$ (33)

Based on the sparsity in the reconfigurable connection constraint (11d), we
guarantee the optimality of the analog beamformer ${\bf{T}}_{\mathrm{RF}}$
since rows are enumerated one after another.

_6) Analog Sensing Receive Beamformer ${\bf{W}}_{\mathrm{RF}}$_:

Define ${\bf{Q}}={{\bf{W}}\\!+\\!\rho{\bf{\tilde{D}}}}$. The subproblem
concerning ${{\bf{W}}_{\mathrm{RF}}}$ is

$\displaystyle\mathop{\min}\limits_{{\bf{W}}_{\mathrm{RF}}}$
$\displaystyle{\left\|{\bf{Q}}-{{\bf{W}}_{\mathrm{RF}}}{{\bf{W}}_{\mathrm{D}}}\right\|^{2}}$
(34) $\displaystyle\mathrm{s.t.}$
$\displaystyle{{\bf{W}}_{\mathrm{RF}}}({\tilde{m},\tilde{n}})\\!\in\\!\left\\{{0,{\cal
F}}\right\\},\forall\tilde{m},\tilde{n},$
$\displaystyle{\left\|{{{\bf{W}}_{\mathrm{RF}}}({\tilde{m},:})}\right\|_{0}}=1,\forall\tilde{m}.$

Since problem (34) has the same form as problem (28), we can solve the radar
receive analog beamforming ${\bf{W}}_{\mathrm{RF}}$ in the same way as
described in Proposition 2, as given by

$\displaystyle\mathop{\max}\limits_{\left\\{{\bar{\varphi}_{\tilde{m},\tilde{n}}}\right\\},\left\\{{\bar{\cal
S}}_{\tilde{n}}\right\\}}$
$\displaystyle\sum\limits_{{\tilde{n}}=1}^{N_{\mathrm{RF}}^{\mathrm{r}}}{\sum\limits_{{\forall{\tilde{m}}}\in{\bar{\cal
S}}_{\tilde{n}}}{\left|{\bar{\zeta}_{{\tilde{m}},{\tilde{n}}}}\right|\cos({{\bar{\phi}_{{\tilde{m}},{\tilde{n}}}}-{\bar{\varphi}_{{\tilde{m}},{\tilde{n}}}}})}}$
(35) $\displaystyle\mathrm{s.t.}$
$\displaystyle\cup_{\tilde{n}=1}^{N_{\mathrm{RF}}^{\mathrm{r}}}\bar{\cal
S}_{\tilde{n}}=\left\\{{1,\cdots,{M_{\mathrm{R}}}}\right\\},\forall\tilde{n},$
$\displaystyle\bar{{\cal S}_{p}}\cap{\bar{\cal S}_{q}}=\emptyset,\forall p\neq
q,$

where $\bar{\cal S}_{\tilde{n}}$,
${\tilde{n}}=1,\cdots,{N_{\mathrm{RF}}^{\mathrm{r}}}$ stands for the set of
receive antennas linked to the ${\tilde{n}}$-th RF chain;
${\bar{\zeta}_{{\tilde{m}},{\tilde{n}}}}$ and
${\bar{\phi}_{{\tilde{m}},{\tilde{n}}}}$ stand for the amplitude and phase of
${\bf{Q}}({{\tilde{m}},:}){{\bf{W}}_{\mathrm{D}}}{({{\tilde{n}},:})^{H}}$,
respectively; ${\bar{\varphi}_{{\tilde{m}},{\tilde{n}}}}$ is the phase of the
$({{\tilde{m}},{\tilde{n}}})$-th element of ${{\bf{W}}_{\mathrm{RF}}}$. The
corresponding optimal receive analog beamformer is given by

${\bf{W}}_{\mathrm{RF}}{({{\tilde{m}},{\tilde{n}}})}\\!=\\!\begin{cases}{e^{j{\bar{\varphi}_{{\tilde{m}},{\tilde{n}}}}}},&\\!\mathrm{if}\
{\tilde{m}}\\!\in\\!{\bar{\cal
S}_{{\tilde{n}}^{\star}}},\,{\tilde{n}}\\!=\\!1,\\!\cdots,\\!{N_{\mathrm{RF}}^{\mathrm{r}}};\\\
{0,}&\\!{\mathrm{otherwise.}}\end{cases}$ (36)

Here, ${\bar{\cal S}_{{\tilde{n}}^{\star}}}$ is the set of optimal antennas
connected to the ${\tilde{n}}$-th RF chain obtained from (35).

_7) Digital Transmit Beamformer ${\bf{T}}_{{\mathrm{D}},k}$ and Sensing
Receive Beamformer ${\bf{W}}_{\mathrm{D}}$_:

The subproblem concerning the digital transmit beamforming
${\bf{T}}_{{\mathrm{D}},k}$ is an unconstrained least squares (LS) problem:

$\mathop{\min}\limits_{{\bf{T}}_{{\mathrm{D}},k}}\sum\limits_{k=1}^{K}{{\left\|{{\bf{Z}}_{k}}-{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{{\mathrm{D}},k}}\right\|^{2}}}.$
(37)

Subsequently, the optimal solution to (37) is obtained as

${{\bf{T}}_{{\mathrm{D}},k}}={\bf{T}}_{\mathrm{RF}}^{\dagger}{{\bf{Z}}_{k}},\forall
k.$ (38)

The subproblem concerning the radar digital receive beamforming
${\bf{W}}_{\mathrm{D}}$ can be formulated in the same way as (37). With
reference to (38), the optimal ${\bf{W}}_{\mathrm{D}}$ can be given by:

${{\bf{W}}_{\mathrm{D}}}={\bf{W}}_{\mathrm{RF}}^{\dagger}{\bf{Q}}.$ (39)

###  III-C Summary of the WPDD Algorithm

Algorithm 1 summarizes the new WPDD algorithm for designing the HBF in the
considered DFRC system by solving problem $\mathrm{P}_{1}$. Let
${\bf{D}}=\left[{\bf{D}}_{1},\cdots,{\bf{D}}_{K}\right]$. In the outer-loop
iteration, we refresh the dual variable sets $\\{{\bf{D}},\tilde{\bf{D}}\\}$
and the penalty parameter $\rho$, based on the following constraint violation
condition:

$h({\mathcal{\tilde{X}}})\\!\\!=\\!\\!\max({\left\|{{{\bf{T}}_{k}}\\!\\!-\\!\\!{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{{\mathrm{D}},k}}}\right\|,\left\|{{\bf{W}}\\!\\!-\\!\\!{{\bf{W}}_{\mathrm{RF}}}{{\bf{W}}_{\mathrm{D}}}}\right\|}).$
(40)

If $h({\mathcal{\tilde{X}}})\leq\eta$, we update the dual variables by

$\displaystyle{\bf{D}}_{k}$
$\displaystyle\leftarrow{\bf{D}}_{k}+\frac{1}{\rho}{({{\bf{T}}_{k}-{\bf{T}}_{\mathrm{RF}}{\bf{T}}_{\mathrm{D},k}})},\forall
k,$ (41) $\displaystyle\tilde{\bf{D}}$
$\displaystyle\leftarrow\tilde{\bf{D}}+\frac{1}{\rho}{({{\bf{W}}-{\bf{W}}_{\mathrm{RF}}{\bf{W}}_{\mathrm{D}}})}.$
(42)

If $h({\mathcal{\tilde{X}}})>\eta$, we reduce the penalty factor $\rho$ by
$\rho\leftarrow c\rho$, with $0<c<1$. Algorithm 1 terminates if
$h({\mathcal{\tilde{X}}})<\varepsilon_{2}$, where $\varepsilon_{2}$ is a
prespecified threshold. The WPDD stabilizes at a stationary point of problem
$\mathrm{P}_{1}$ [42], as discussed later in Section IV-A.

Algorithm 1 The proposed WPDD algorithm

0: ${I_{\rm{max}}}$, $\varepsilon_{1}$, $\varepsilon_{2}$,
${\bf{T}}_{\mathrm{RF}}^{(0)}$, ${\bf{T}}_{\mathrm{D},k}^{(0)}$,
${\bf{U}}_{k}^{(0)}$, ${\bf{W}}_{\mathrm{RF}}^{(0)}$,
${\bf{W}}_{\mathrm{D}}^{(0)}$, ${\rho^{(0)}}>0$, $0<c<1$

1: repeat

2: repeat

3: Update $\bf{W}$ by (17), ${{\bf{T}}_{k}}$ by (24), ${\bf{U}}_{k}$ by (26),
${\bf{G}}_{k}$ by (27), ${\bf{T}}_{\mathrm{RF}}$ by (33),
${\bf{W}}_{\mathrm{RF}}$ by (36), ${\bf{T}}_{{\mathrm{D}},k}$ by (38),
${\bf{W}}_{\mathrm{D}}$ by (39)

4: until the convergence with $\varepsilon_{1}$ or ${I_{\rm{max}}}$ inner
iterations

5: if $h({\mathcal{\tilde{X}}})\leq\eta$ then

6: Update ${\bf{D}}_{k}$ by (41) and $\tilde{\bf{D}}$ by (42)

7: else

8: Update $\rho$ by $\rho\leftarrow c\rho$

9: end if${\eta\leftarrow 0.8{h({\mathcal{\tilde{X}}})}}$

10: until ${h({\mathcal{\tilde{X}}})}<\varepsilon_{2}$

10: ${\bf{T}}_{\mathrm{RF}},{\bf{T}}_{\mathrm{D},k}$, ${\bf{U}}_{k}$,
${\bf{W}}_{\mathrm{RF}}$, ${\bf{W}}_{\mathrm{D}}$

###  III-D Adaptation Under PC Architectures

A PC configuration represents a specific instance of RS connection, where
every RF chain is persistently hardwired to a set of antennas. Specifically,
the analog beamformer ${\bf{T}}_{\mathrm{RF}}$ and ${\bf{W}}_{\mathrm{RF}}$
are block diagonal matrices. For example, every block of transmit analog
beamforming ${\bf{T}}_{\mathrm{RF}}$ has an $M$-dimensional vector with a
constant-modulus (CM) value, i.e.,

${{\bf{T}}_{\mathrm{RF}}}={\rm{blkdiag}}\\{{{{\bf{p}}_{1}},\cdots,{{\bf{p}}_{N_{\mathrm{RF}}^{\mathrm{t}}}}}\\}\in{\mathbb{C}}^{{{M_{\mathrm{T}}}\times{N_{\mathrm{RF}}^{\mathrm{t}}}}},$
(43)

where ${{\bf{p}}_{i}}\in{\mathbb{C}}^{M\times 1}$ for
$i=1,\cdots,{N_{\mathrm{RF}}^{\mathrm{t}}}$,
$M={{M_{\mathrm{T}}}\mathord{\left/{\vphantom{{{M_{\mathrm{T}}}}{N_{\mathrm{RF}}^{\mathrm{t}}}}}\right.\kern-1.2pt}{N_{\mathrm{RF}}^{\mathrm{t}}}}$,
and $\left|{{{\bf{p}}_{i}}(\nu)}\right|=1$ for $\nu=1,\cdots,M.$

By replacing the reconfigurable connection constraints (11d) and (11e) in
problem $\mathrm{P}_{1}$ with (43), we can obtain the DFRC HBF design under
the PC architecture. Fortunately, Algorithm 1 is still applicable to this
problem, except that the subproblems for updating the digital and analog
beamformers, ${{\bf{T}}_{\mathrm{RF}}}$ and ${{\bf{T}}_{\mathrm{D},k}}$
differ, that is, Step 3 in Algorithm 1, becomes different, as described below.

The subproblem of ${{\bf{T}}_{\mathrm{RF}}}$, i.e., (28), can be rewritten as

$\displaystyle\mathop{\min}\limits_{{\bf{T}}_{\mathrm{RF}}}\left\|{{\bf{Z}}-{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{\mathrm{D}}}}\right\|^{2}$
$\displaystyle=$
$\displaystyle\mathop{\min}\limits_{{\varphi_{m,n}}}\sum\limits_{\forall
n}{\sum\limits_{\forall
m}{\left\|{{\bf{Z}}({m,:})-{e^{j{\varphi_{m,n}}}}{{\bf{T}}_{\mathrm{D}}}({n,:})}\right\|^{2}}}.$
(44)

Here, $m=1,\cdots,{M_{\mathrm{T}}}$, and
$n=1,\cdots,\left\lceil{m\frac{N_{\mathrm{RF}}^{\mathrm{t}}}{M_{\mathrm{T}}}}\right\rceil$.
Then, the phase of the $(m,n)$-th element of ${\bf{T}}_{\mathrm{RF}}$ is given
by

${\varphi_{m,n}}=\arg\left({{\bf{Z}}\left({m,:}\right){{\bf{T}}_{\mathrm{D}}}{{({n,:})}^{H}}}\right).$
(45)

On the other hand, the subproblem concerning ${{\bf{T}}_{\mathrm{D}}}$, i.e.,
(37), can be rewritten as

$\displaystyle\mathop{\min}\limits_{{{\bf{T}}_{\mathrm{D}}}}$
$\displaystyle\left\|{{\bf{Z}}-{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{\mathrm{D}}}}\right\|^{2},\;\;\mathrm{s.t.}\
\left\|{{\bf{T}}_{\mathrm{D}}}\right\|^{2}=\frac{{N_{\mathrm{RF}}^{\mathrm{t}}}{P_{\mathrm{T}}}}{{M_{\mathrm{T}}}},$
(46)

which is a non-convex QCQP. Here, we employ a computationally efficient
algorithm to acquire the optimal ${{{\bf{T}}_{\mathrm{D}}}}$.

By defining $\bar{\bf{Q}}={\bf{T}}_{\mathrm{RF}}^{H}{{\bf{T}}_{\mathrm{RF}}}$,
and $\bar{\bf{G}}={\bf{T}}_{\mathrm{RF}}^{H}{\bf{Z}}$, the objective of (46)
is reformulated as

${\mathrm{tr}}({\bf{T}}_{\mathrm{D}}^{H}\bar{\bf{Q}}{{\bf{T}}_{\mathrm{D}}})-2{\cal
R}({\mathrm{tr}}({\bf{T}}_{\mathrm{D}}^{H}\bar{\bf{G}})).$ (47)

By introducing the Lagrange multiplier $\lambda_{L}$, the Lagrangian function
is formulated as

$\displaystyle\mathcal{L}({\bf{T}}_{\mathrm{D}},\lambda_{L})=$
$\displaystyle{\mathrm{tr}}({\bf{T}}_{\mathrm{D}}^{H}\bar{\bf{Q}}{{\bf{T}}_{\mathrm{D}}})-2{\cal
R}({\mathrm{tr}}({\bf{T}}_{\mathrm{D}}^{H}{\bar{\bf{G}}}))$ (48)
$\displaystyle+\lambda_{L}(\left\|{{\bf{T}}_{\mathrm{D}}}\right\|^{2}-\frac{{N_{\mathrm{RF}}^{\mathrm{t}}}{P_{\mathrm{T}}}}{{M_{\mathrm{T}}}}).$

According to the optimality condition of (48), we obtain

${\bf{T}}_{\mathrm{D}}^{\mathrm{opt}}={{({\bar{\bf{Q}}+{\lambda_{L}^{\mathrm{opt}}}{\bf{I}}})}^{-1}}\bar{\bf{G}},$
(49)

where $\lambda_{L}^{\mathrm{opt}}$ can be obtained by bisection search [43].

In general, the HBF for DFRC systems with PC architectures can be obtained by
replacing the solution process of ${{\bf{T}}_{\mathrm{RF}}}$ and
${{\bf{T}}_{\mathrm{D}}}$ in Algorithm 1, i.e., replacing $\eqref{TRF}$ and
$\eqref{eq:TDk}$ using $\eqref{eq:TRF2}$ and $\eqref{eq:TD}$ in Step 3,
respectively. The description is omitted because the PC-based receive
beamforming can be obtained similarly.

##  IV System Performance Analysis

###  IV-A Convergence Analysis

Algorithm 1 converges to an effective solution to problem $\mathrm{P}_{1}$ by
iteratively running the proposed double-loop process. Recall that
$\mathcal{\tilde{X}}\\!=\\!\\{{{{\bf{T}}_{\mathrm{RF}}},{{\bf{T}}_{\mathrm{D},k}},{\bf{U}}_{k},{{\bf{W}}_{\mathrm{RF}}},{{\bf{W}}_{\mathrm{D}}},{\bf{G}}_{k},{{\bf{T}}_{k}},{\bf{W}}}\\}$
collects all variables to be updated in the inner-loop, where ${\bf{W}}$,
${\bf{U}}_{k}$, and ${\bf{G}}_{k}$ depend uniquely and deterministically on
${\bf{T}}_{k}$, followed by finding
${{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{\mathrm{D},k}}$ and
${{\bf{W}}_{\mathrm{RF}}}{{\bf{W}}_{\mathrm{D}}}$ to approximate
${\bf{T}}_{k}$ and $\bf{W}$, respectively. In this sense, we can partition
$\mathcal{\tilde{X}}$ into three subsets:
${\mathcal{\tilde{X}}}_{1}=\\{{\bf{W}},{\bf{T}}_{k},{\bf{U}}_{k},{\bf{G}}_{k}\\}$,
${\mathcal{\tilde{X}}}_{2}=\\{{{\bf{T}}_{\mathrm{RF}}},{{\bf{W}}_{\mathrm{RF}}}\\}$,
and
${\mathcal{\tilde{X}}}_{3}=\\{{{\bf{T}}_{\mathrm{D},k}},{{\bf{W}}_{\mathrm{D}}}\\}$,
the three of which are updated alternately in each inner-loop iteration.

Given $\rho$, $\bf{D}$, and $\tilde{\bf{D}}$ specified at the beginning of the
current outer-loop iteration, ${\mathcal{\tilde{X}}}_{1}$ depends solely on
the SCA process of ${\bf{T}}_{k},\forall k$. The objective function (15a) is
certainly non-increasing during the process. Given
${\mathcal{\tilde{X}}}_{1}$, ${\mathcal{\tilde{X}}}_{2}$ and
${\mathcal{\tilde{X}}}_{3}$ are updated, which would further decrease (15a)
due to the fact that (33) and (38) provide the optimal solutions to (28) and
(37), respectively, and hence ensure the objective function of (28) or (37) is
strictly non-increasing. Likewise, (36) and (39) can ensure the objective
function of (34) is strictly non-increasing. As a result, (15a) is strictly
non-increasing during an inner-loop iteration and hence throughout the inner-
loop iterations during an outer-loop iteration. On the other hand, (15a) is
apparently lower bounded due to the finite and non-negative transmit power of
the BS.

In the outer loop, $\rho$, $\bf{D}$, and $\tilde{\bf{D}}$ are updated under
the constraint violation condition, ${h(\mathcal{\tilde{X}})}$, until the PDD
convergence. While (15a) may not be strictly non-increasing in the outer loop,
the outer-loop iterations can increasingly penalize the violation of (14d) and
(14e). When ${\bf{T}}_{k}$ stops changing in the inner loop, so do the rest of
the variables in the inner loop, and the outer-loop iterations also terminate
according to (41) and (42). With the optimal $\rho^{*}$, $\bf{D}^{*}$, and
$\tilde{\bf{D}}^{*}$ in the outer loop, Algorithm 1 eventually converges.

###  IV-B Discussion on Optimality

Due to the NP-hardness of problem $\mathrm{P}_{1}$, existing techniques fall
short of delivering global optima. Nevertheless, the FD counterpart of the
considered system, i.e., problem P1 with (11d)–(11g) dropped, can be solved
using Algorithm 1 without (33), (36), (38) and (39) in Step 3, reach a Karush-
Kuhn-Tucker (KKT) point, and serve as an upper bound for the proposed HBF
algorithm. As evidenced numerically in Section V, the HBF design in Algorithm
1 significantly outperforms the existing alternative approaches, much closer
to the FD counterpart. In the FD counterpart of the considered problem, only
${\mathcal{\tilde{X}}}_{1}$ remains to be optimized. As discussed earlier,
${\mathcal{\tilde{X}}}_{1}$ depends uniquely and deterministically on
${\bf{T}}_{k},\forall k$, with the rest of the variables in
${\mathcal{\tilde{X}}}_{1}$ all given in closed-form functions of
${\bf{T}}_{k},\forall k$. To this end, the optimality of
${\mathcal{\tilde{X}}}_{1}$ depends on that of ${\bf{T}}_{k},\forall k$.

Under the FD architecture, the objective function of the ${\bf{T}}_{k}$
subproblem contains only the first term of (19a). The solution yields
${a_{k,i}}={{{\bf{t}}_{i}^{H}}({\bf{I}}\otimes{\bf{P}}_{k}^{H}){{\bf{t}}_{i}}},\,\forall
i,k$. Or, one could reduce the auxiliary variable ${a_{k,i}}$ to reduce (19a).
After variable substitution and SCA, the ${\bf{T}}_{k}$ subproblem is
convexified. The non-convex constraint
$\mathrm{tr}({{\bf{T}}_{k}^{H}{\bf{\Phi}}{{\bf{T}}_{k}}}),\forall k$ is
replaced by the first-order Taylor approximation ${2{\cal
R}({{\bf{\bar{t}}}_{k}^{H}\bar{{\bf{\Phi}}}{{\bf{t}}_{k}}})-{\bf{\bar{t}}}_{k}^{H}\bar{\bf{\Phi}}{{{\bf{\bar{t}}}}_{k}}},\forall
k$. Here, $\mathrm{tr}({{\bf{T}}_{k}^{H}{\bf{\Phi}}{{\bf{T}}_{k}}})$ and
${2{\cal
R}({{\bf{\bar{t}}}_{k}^{H}\bar{{\bf{\Phi}}}{{\bf{t}}_{k}}})-{\bf{\bar{t}}}_{k}^{H}\bar{\bf{\Phi}}{{{\bf{\bar{t}}}}_{k}}}$
yield the same value and the same gradient at the local point
${\bf{\bar{t}}}_{k}$. Moreover, the Slater’s condition is satisfied in every
iteration of SCA; i.e., there is a feasible solution to the ${\bf{T}}_{k}$
subproblem such that all inequality constraints are strictly satisfied.
According to [44], this solution is a KKT stationary point, which is a local
minimum if it is in the interior of the feasible set of the ${\bf{T}}_{k}$
subproblem. As a result, the WPDD can achieve a KKT point for the FD
counterpart of the considered problem, which is a local minimum if it is
inside the feasible region.

###  IV-C Computational Complexity

Since there are typically many more transmit antennas than receive antennas in
DFRC systems, the complexity of the WPDD algorithm is predominantly
contributed by the solution of ${\bf{T}}_{k}$ in Step 3. Solving for
${\bf{T}}_{k}$ in (24) involves $(K{M_{\mathrm{T}}}+{K^{2}}+K)$ variables and
$({K^{2}}+K+1)$ SOC constraints. The complexity of running the interior point
method to solve for an $\epsilon$-optimal solution for ${\bf{T}}_{k}$ is
$\mathcal{O}({{K^{4}}{M_{\mathrm{T}}^{3}}}\log(1/\epsilon)),\,\forall k$ [45].
The worst-case complexity of WPDD under the RS architecture is
$\mathcal{O}\left({{I_{\mathrm{O}}}{I_{\mathrm{I}}}{\left({{K^{4}}{M_{\mathrm{T}}^{3}}\log(1/\epsilon)}\right)}}\right)$,
where ${I_{\mathrm{O}}}$ and ${I_{\mathrm{I}}}$ indicate the iteration numbers
in the outer and inner loops, respectively.

###  IV-D Energy Efficiency (EE)

Table III: The total power consumption under different beamforming
architectures

Architecture | Total Power Consumption
---|---
RS | $P_{\mathrm{tot}}^{\mathrm{RS}}={P_{\mathrm{T}}}\\!+\\!{P_{\mathrm{BB}}}\\!+\\!({N_{\mathrm{RF}}^{\mathrm{t}}}\\!+\\!{N_{\mathrm{RF}}^{\mathrm{r}}}){P_{\mathrm{RF}}}\\!+\\!({M_{\mathrm{T}}}\\!+\\!{M_{\mathrm{R}}}){P_{\mathrm{PS}}}\\!+\\!({M_{\mathrm{T}}}\\!+\\!{M_{\mathrm{R}}}){P_{\mathrm{SW}}}$
PC | $P_{\mathrm{tot}}^{\mathrm{PC}}={P_{\mathrm{T}}}+{P_{\mathrm{BB}}}+({N_{\mathrm{RF}}^{\mathrm{t}}}\\!+\\!{N_{\mathrm{RF}}^{\mathrm{r}}}){P_{\mathrm{RF}}}+({M_{\mathrm{T}}}\\!+\\!{M_{\mathrm{R}}}){P_{\mathrm{PS}}}$
DPC | $P_{\mathrm{tot}}^{\mathrm{DPC}}={P_{\mathrm{T}}}+{P_{\mathrm{BB}}}+({N_{\mathrm{RF}}^{\mathrm{t}}}\\!+\\!{N_{\mathrm{RF}}^{\mathrm{r}}}){P_{\mathrm{RF}}}+2({M_{\mathrm{T}}}\\!+\\!{M_{\mathrm{R}}}){P_{\mathrm{PS}}}$
FC | $P_{\mathrm{tot}}^{\mathrm{FC}}={P_{\mathrm{T}}}+{P_{\mathrm{BB}}}+({N_{\mathrm{RF}}^{\mathrm{t}}}\\!+\\!{N_{\mathrm{RF}}^{\mathrm{r}}}){P_{\mathrm{RF}}}+({M_{\mathrm{T}}}{N_{\mathrm{RF}}^{\mathrm{t}}}+{M_{\mathrm{R}}}{N_{\mathrm{RF}}^{\mathrm{r}}}){P_{\mathrm{PS}}}$
FD | $P_{\mathrm{tot}}^{\mathrm{FD}}={P_{\mathrm{T}}}+{P_{\mathrm{BB}}}+({M_{\mathrm{T}}}\\!+\\!{M_{\mathrm{R}}}){P_{\mathrm{RF}}}$

To evaluate the effect of the RS HAD DFRC system from the perspectives of
hardware costs and power consumption, the average EE of each user is defined
as

$\eta\buildrel\Delta\over{=}\frac{1}{K}\frac{{R}}{P_{\mathrm{tot}}},$ (50)

where $P_{\mathrm{tot}}$ is the total energy consumption of the DFRC BS.

The overall power consumption under various beamforming architectures with
transmit power $P_{\mathrm{T}}$ is summarized in Table III, where
$P_{\mathrm{BB}}$ specifies the circuit power consumed by the baseband
circuit; $P_{\mathrm{RF}}$ and $P_{\mathrm{SW}}$ are the powers consumed by an
RF chain and a switch, respectively; $P_{\mathrm{PS}}$ gives the power
consumed by a PS; $N_{\mathrm{PS}}$ is the number of PSs. Clearly, the FD has
the maximum power consumption because there are the same number of RF chains
and transmit/receive antennas. Moreover, the number of PSs is proportional to
that of the transmit/receive antennas and depends on the specific connection
mode in the HAD architecture.

##  V Numerical Results

In this section, the developed algorithm for DFRC systems with HBF
architectures is numerically assessed by MATLAB simulations. The path loss
$PL(d_{k})$ is modeled as

$PL(d_{k})\ [\text{dB}]=\alpha+10\beta\log_{10}(d_{k})+\xi,$ (51)

where $\xi\sim{\cal{N}}(0,\sigma^{2})$, $\alpha=72.0$, $\beta=2.92$, and
$\sigma=8.7$ dB for a non-line-of-sight (NLoS) path [28]. The noise power of
user $k$ is $\sigma_{k}^{2}=-90$ dBm. The AoDs $\phi_{k,l},\,\forall k,l$,
follow a uniform distribution within $[-\frac{\pi}{2},\frac{\pi}{2}]$. The
other simulation parameters are collated in Table IV. For the WPDD algorithm,
the initial penalty factor is $\rho=1$, the control parameter is $c=0.6$, and
the termination tolerances are $\varepsilon_{1}=\varepsilon_{2}=10^{-4}$,
where $\varepsilon_{1}$ controls the accuracy of inner iterations. Moreover,
we set the maximum number of iterations $I_{\mathrm{max}}$ = 30 to prevent
slow convergence in the inner loop. All of the simulations are conducted using
the Matlab 2018b version using a standard PC with an i7-8700 3.20 GHz CPU, 16
GB RAM, and a 64-bit operating system.

Table IV: Simulation parameters

Parameter | Value | Parameter | Value
---|---|---|---
$M_{\mathrm{T}}$ | 64 | $M_{\mathrm{R}}$ | 16
${N_{\mathrm{RF}}^{\mathrm{t}}}$ | 8 | ${N_{\mathrm{RF}}^{\mathrm{r}}}$ | 8
$M_{\mathrm{U}}$ | 2 | $P_{\mathrm{T}}$ | 40 dBm
$K$ | 4 | $d_{s}$ | 2
$L_{k}$ | 3 | $d_{k}$ | 80 m
$\theta_{0}$ | $0^{\circ}$ | $\theta_{j}$ | $\\{-30^{\circ},30^{\circ}\\}$
$\sigma_{\mathrm{0}}^{2}$ | 10 dB | $\sigma_{\mathrm{C}}^{2}$ | 20 dB

The schemes used for evaluating and benchmarking the system performance are
outlined as follows:

  * •

RS-WPDD: This is the proposed WPDD algorithm for HBF design in multi-user DFRC
systems with the RS architectures (see Section III-C).

  * •

PC-WPDD: This is the adaptation of the proposed WPDD algorithm for multi-user
DFRC systems with the PC architectures (see Section III-D).

  * •

DPC-THEREON: This is the joint hybrid waveform and radar receiver optimization
algorithm in [22] based on the DPS architectures.

  * •

FC-TwoStage: This is the FC-based HBF algorithm developed in [14], which first
constructs an FD beamformer $\mathbf{T}$ and then iteratively optimizes the
analog and digital beamformers to approximate the achieved $\mathbf{T}$.

  * •

PC-TwoStage: This is the HBF algorithm developed under the PC architectures in
[19], which can be implemented similarly to FC-TwoStage algorithm.

  * •

FD/FA: This is the fully-digital/analog-based beamforming algorithm that
provides an upper/lower bound for all HBF algorithms.

Figure 2: Convergence behaviors of the WPDD algorithm with $\gamma$ = 10 dB,
and 15 dB.

Fig. 2 plots the convergence of the proposed WPDD algorithm for the HBF design
with the increase of outer iterations, where the radar SCNR threshold $\gamma$
is set to 10 dB and 15 dB. It is noticed that the communication sum-rate
converges within about ten iterations under both the RS-WPDD and PC-WPDD
algorithms. Moreover, RS-WPDD always obtains better communication performance
than PC-WPDD.

Figure 3: Trade-off between the communication sum-rate and the radar receive
SCNR $\gamma$, under different beamforming design schemes. $M_{\mathrm{T}}$ =
64, $M_{\mathrm{R}}$ = 16,
${N_{\mathrm{RF}}^{\mathrm{t}}}={N_{\mathrm{RF}}^{\mathrm{r}}}=8$, $K=4$ and
$M_{\mathrm{U}}=d_{s}=2$. Figure 4:  Trade-off between the communication sum-
rate and the radar receive SCNR $\gamma$, under hybrid/FD DFRC receivers.
$M_{\mathrm{T}}$ = 64, $M_{\mathrm{R}}$ = 16,
${N_{\mathrm{RF}}^{\mathrm{t}}}={N_{\mathrm{RF}}^{\mathrm{r}}}$ = 8, $K=4$ and
$M_{\mathrm{U}}=d_{s}=2$. Figure 5: Achievable sum-rate of communication
versus different transmit powers $P_{\mathrm{T}}$, with $\gamma$ = 15 dB,
$M_{\mathrm{T}}$ = 32, 64, and 128.

Fig. 3 plots the trade-off between the achievable sum-rate and the radar SCNR
constraint under different beamforming design schemes, where $\gamma$ ranges
from 10 dB to 20 dB. The sum-rate declines as $\gamma$ grows. The reason is
that when the sensing demand is higher, fewer DoFs are used to improve
communication performance. When $\gamma\leq 12$ dB, the multi-user
communication sum-rate remains approximately constant since the system is
saturated when the sensing performance is easily satisfied with small
$\gamma$. When $\gamma\geq 16$ dB, the communication performance degrades
rapidly. In addition, Fig. 3 shows that the proposed RS-WPDD algorithm
outperforms the PC-based algorithms. Even the proposed PC-WPDD has comparable
performance with DPC-THEREON. Here, FD offers an upper bound for the
performance of our proposed system, while FA gives a lower bound.

Fig. 4 shows the trade-off between the communication and sensing of the WPDD
framework under different DFRC radar receive architectures, including the
hybrid receiver (HR) architecture and the fully digital receiver (FDR)
architecture. The RS-based DFRC hybrid receive beamforming is closer to the FD
receive beamforming than the PC architecture. Fig. 5 plots the achievable sum-
rate of the DFRC system with the increasing total transmit power
$P_{\mathrm{T}}$, where the number of transmit antennas is
$M_{\mathrm{T}}=32,\,64$, and 128. It is shown that the proposed RS-WPDD is
better than PD-WPDD, as observed in Fig. 3. The system performance improves as
$M_{\mathrm{T}}$ increases, enhancing antenna diversity and better beamforming
gain. Furthermore, the difference between RS-WPDD and PC-WPDD decreases as
$M_{\mathrm{T}}$ increases.

Figure 6: Beampattern for different architectures with $M_{\mathrm{T}}$ = 32,
$M_{\mathrm{R}}$ = 16, $\gamma$ = 18 dB, ${\theta}_{0}=0^{\circ}$, and
${\theta}_{j}=\\{-30^{\circ}$, $30^{\circ}\\}$.  Figure 7:  Detection
probability with $M_{\mathrm{T}}$ = 64, $M_{\mathrm{R}}$ = 16,
${\theta}_{0}=0^{\circ}$, and ${\theta}_{j}=\\{-30^{\circ}$, $30^{\circ}\\}$.

We adopt the spatial beampattern,
$P(\theta)={\left|{{{\bf{w}}^{H}}{\bf{A}}(\theta){\bf{x}}}\right|^{2}}$ [26],
as a sensing performance metric. The optimized beampatterns of the proposed
WPDD algorithm and the benchmarks with $M_{\mathrm{T}}=32$,
$M_{\mathrm{R}}=16$, and $\gamma=18$ dB are depicted in Fig. 6. Clearly, all
schemes can be pointed at a $0^{\circ}$ target and achieve good suppression of
clutters at $-30^{\circ}$ and $30^{\circ}$. Clutters can be effectively
suppressed through receive beamforming at the sensing receiver. Moreover, the
RS-WPDD scheme has a peak-to-side-lobe ratio of about 30 dB, which is much
better than the rest of the schemes.

We also plot the detection probability of the proposed RS-WPDD with the SCNR
threshold $\gamma$, as shown in Fig. 7. When $\gamma$ = 15 dB, the detection
probability is 99.72% under a false alarm probability of $10^{-6}$, and 99.99%
under a false alarm probability of $10^{-4}$. As shown in Fig. 3, RS-WPDD can
obtain a satisfactory sensing performance with marginal or even negligible
communication rate losses when $\gamma$ = 15 dB.

Figure 8: Average EE vs. the threshold of radar receive SCNR $\gamma$, with
$K$ = 4. Figure 9: Average EE vs. the number of users $K$, with $\gamma$ = 10
dB and $d_{s}$ = 1.

We evaluate EE to show the effect of our proposed HBF on the system cost and
performance, where $P_{\mathrm{BB}}=200$ mW, $P_{\mathrm{RF}}=300$ mW,
$P_{\mathrm{PS}}=50$ mW, and $P_{\mathrm{SW}}=5$ mW [23]. Fig. 8 plots the
average EE of the DRFC systems with the increase of $\gamma$. We see that the
EE of the proposed RS-WPDD is better than that of PC-WPDD and significantly
surpasses the other benchmarks. Specifically, when $\gamma=15$ dB, RS-WPDD
improves the EE by 23.9% compared to PC-WPDD, 83.4% compared to PC-TwoStage,
and 114.2% compared to FC-TwoStage. Fig. 9 shows the change in EE with the
number of users $K\,(K\leq{N_{\mathrm{RF}}^{\mathrm{t}}})$, when each user has
only one data stream, i.e., $d_{s}=1$, and ${N_{\mathrm{RF}}^{\mathrm{t}}}=8$.
The RS-WPDD still yields the best EE across the spectrum of the user number,
as observed in Fig. 8. The system parameters can be selected to balance the
hardware cost and system performance of DFRC with the RS architectures.

##  VI Conclusion

We have designed HBF for a multi-user mmWave DFRC system with a new RS HAD
architecture. We have jointly optimized digital and analog beamformers to
maximize the communication sum-rate under a prescribed SCNR of target
detection. An efficient WPDD algorithm has been designed based on WMMSE and
PDD methods, which can guarantee convergence to a stationary point. The
effectiveness of WPDD has been confirmed in terms of convergence,
communication SE, target beampattern, and system EE. Simulations have shown
that with the RS architecture and HBF, DFRC systems excel in both SE and EE
compared to the conventional PC architecture. The EE of the systems is
dramatically improved compared to the FC architecture by 114.2%.

##  Appendix A Proof Of Proposition 2

Because the reconfigurable subarrays do not overlap and every transmit antenna
is linked to only an RF chain, we can solve problem (28) row by row, as
follows:

$\displaystyle\mathop{\min}\limits_{{\bf{T}}_{\mathrm{RF}}}\left\|{{\bf{Z}}-{{\bf{T}}_{\mathrm{RF}}}{{\bf{T}}_{\mathrm{D}}}}\right\|^{2}$
(52) $\displaystyle\mathop{=}\limits^{(a)}$
$\displaystyle\mathop{\min}\limits_{{\bf{T}}_{\mathrm{RF}}}\sum\limits_{n=1}^{N_{\mathrm{RF}}^{\mathrm{t}}}{\sum\limits_{{\forall
m}\in{{\cal
S}_{n}}}{\left\|{{\bf{Z}}\left({m,:}\right)-{{\bf{T}}_{\mathrm{RF}}}\left({m,n}\right){{\bf{T}}_{\mathrm{D}}}\left({n,:}\right)}\right\|^{2}}}$
$\displaystyle=$
$\displaystyle\mathop{\min}\limits_{{{\bf{T}}_{\mathrm{RF}}}}\sum\limits_{n=1}^{N_{\mathrm{RF}}^{\mathrm{t}}}\sum\limits_{{\forall
m}\in{{\cal
S}_{n}}}\\{\left\|{{\bf{Z}}\left({m,:}\right)}\right\|^{2}+{{{\left|{{{\bf{T}}_{\mathrm{RF}}}\left({m,n}\right)}\right|}^{2}}\left\|{{{\bf{T}}_{\mathrm{D}}}\left({n,:}\right)}\right\|^{2}\\}}$
$\displaystyle-2{\cal
R}\left({{\bf{Z}}\left({m,:}\right){{\bf{T}}_{\mathrm{D}}}{{\left({n,:}\right)}^{H}}{{\bf{T}}_{\mathrm{RF}}}{{\left({m,n}\right)}^{H}}}\right),$

where the equality $(a)$ holds when the objective function is expanded in
rows.

Define ${\varphi_{m,n}}$ as the phase of the ($m,n$)-th entry in
${{\bf{T}}_{\mathrm{RF}}}$, i.e.,
${{\bf{T}}_{\mathrm{RF}}}\left({m,n}\right)=\frac{1}{{\sqrt{M_{\mathrm{T}}}}}{e^{j{\varphi_{m,n}}}}$,
and
${\bf{Z}}\left({m,:}\right){{\bf{T}}_{\mathrm{D}}}{\left({n,:}\right)^{H}}=\left|{{\zeta_{m,n}}}\right|{e^{j{\phi_{m,n}}}},n=1,\cdots,{N_{\mathrm{RF}}^{\mathrm{t}}},m\in{{\cal
S}_{n}}$. By suppressing the constant term, (52) is further equivalent to

$\displaystyle\mathop{\max}\limits_{{\bf{T}}_{\mathrm{RF}}}\sum\limits_{n=1}^{N_{\mathrm{RF}}^{\mathrm{t}}}{\sum\limits_{{\forall
m}\in{{\cal S}_{n}}}{{\cal
R}\left({{\bf{Z}}\left({m,:}\right){{\bf{T}}_{\mathrm{D}}}{{\left({n,:}\right)}^{H}}{{\bf{T}}_{\mathrm{RF}}}{{\left({m,n}\right)}^{H}}}\right)}}$
(53) $\displaystyle\mathop{=}\limits^{(b)}$
$\displaystyle\mathop{\max}\limits_{\left\\{{\varphi_{m,n}}\right\\}}\sum\limits_{n=1}^{N_{\mathrm{RF}}^{\mathrm{t}}}{\sum\limits_{{\forall
m}\in{{\cal S}_{n}}}{{\cal
R}\left({{e^{-j{\varphi_{m,n}}}}{\bf{Z}}\left({m,:}\right){{\bf{T}}_{\mathrm{D}}}{{\left({n,:}\right)}^{H}}}\right)}}$
$\displaystyle=$
$\displaystyle\mathop{\max}\limits_{\left\\{{\varphi_{m,n}}\right\\}}\sum\limits_{n=1}^{N_{\mathrm{RF}}^{\mathrm{t}}}{\sum\limits_{{\forall
m}\in{{\cal
S}_{n}}}{\left|{\zeta_{m,n}}\right|\cos\left({{\phi_{m,n}}-{\varphi_{m,n}}}\right)}}.$

Since the ($m,n$)-th entry of ${{\bf{T}}_{\mathrm{RF}}}$ has a constant
modulus value, the equality in (b) holds. The final establishment of the
equivalence between problems (28) and (31) is achieved by integrating the
objective (53) with the reconfigurable connection constraints (29) and (30).

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|  Xin Jin received the B.S. degree in communication engineering from Sichuan
Normal University (SICNU), Chengdu, China, in 2020. She is currently pursuing
the Ph.D. degree with the School of Information and Communication Engineering,
Beijing University of Posts and Telecommunications (BUPT), Beijing, China. Her
research interests include integrated sensing and communications, massive
MIMO, hybrid beamforming, and array signal processing.
---|---
|  Tiejun Lv (Senior Member, IEEE) received the M.S. and Ph.D. degrees in
electronic engineering from the University of Electronic Science and
Technology of China (UESTC), Chengdu, China, in 1997 and 2000, respectively.
From January 2001 to January 2003, he was a Postdoctoral Fellow at Tsinghua
University, Beijing, China. In 2005, he was promoted to Full Professor at the
School of Information and Communication Engineering, Beijing University of
Posts and Telecommunications (BUPT). From September 2008 to March 2009, he was
a Visiting Professor with the Department of Electrical Engineering at Stanford
University, Stanford, CA, USA. He is the author of four books, more than 150
published journal papers and 200 conference papers on the physical layer of
wireless mobile communications. His current research interests include signal
processing, communications theory and networking. He was the recipient of the
Program for New Century Excellent Talents in University Award from the
Ministry of Education, China, in 2006. He received the Nature Science Award
from the Ministry of Education of China for the hierarchical cooperative
communication theory and technologies in 2015. He has won best paper award at
CSPS 2022.
---|---
|  Wei Ni (Fellow, IEEE) received the B.E. and Ph.D. degrees in Electronic
Engineering from Fudan University, Shanghai, China, in 2000 and 2005,
respectively. He is a Principal Research Scientist at CSIRO, Sydney,
Australia, and a Conjoint Professor at the University of New South Wales. He
is also an Adjunct Professor at the University of Technology Sydney, and an
Honorary Professor at Macquarie University. He serves as a Technical Expert at
Standards Australia in support of the ISO standardization of AI and Big Data.
He was a Postdoctoral Research Fellow at Shanghai Jiaotong University from
2005 to 2008; Deputy Project Manager at Bell Labs, Alcatel/Alcatel-Lucent from
2005 to 2008; and Senior Researcher at Devices R&D, Nokia from 2008 to 2009.
He has co-authored one book, ten book chapters, more than 300 journal papers,
more than 100 conference papers, 26 patents, ten standard proposals accepted
by IEEE, and three technical contributions accepted by ISO. His research
interests include 6G security and privacy, machine learning, stochastic
optimization, and their applications to system efficiency and integrity. Dr.
Ni has been an Editor for IEEE Transactions on Wireless Communications since
2018, an Editor for IEEE Transactions on Vehicular Technology since 2022, and
an Editor for IEEE Transactions on Information Forensics and Security and IEEE
Communications Surveys and Tutorials since 2024. He served first as the
Secretary, then the Vice-Chair and Chair of the IEEE VTS NSW Chapter from 2015
to 2022, Track Chair for VTC-Spring 2017, Track Co-chair for IEEE VTC-Spring
2016, Publication Chair for BodyNet 2015, and Student Travel Grant Chair for
WPMC 2014.
---|---
|  Zhipeng Lin (Member, IEEE) received the Ph.D. degrees from the School of
Information and Communication Engineering, Beijing University of Posts and
Telecommunications, Beijing, China, and the School of Electrical and Data
Engineering, University of Technology of Sydney, NSW, Australia, in 2021.
Currently, He is an Associate Researcher in the College of Electronic and
Information Engineering, Nanjing University of Aeronautics and Astronautics,
Nanjing, China. His current research interests include signal processing,
massive MIMO, spectrum sensing, and UAV communications.
---|---
|  Qiuming Zhu (Senior Member, IEEE) received his B.S. in electronic
engineering from Nanjing University of Aeronautics and Astronautics (NUAA),
Nanjing, China, in 2002 and his M.S. and Ph.D. in communication and
information system from NUAA in 2005 and 2012, respectively. Since 2021, he
has been a professor in the Department of Electronic Information Engineering,
NUAA. From Oct. 2016 to Oct. 2017, June 2018 to Aug. 2018 and June 2018 to
Aug. 2018, he was also an academic visitor at Heriot Watt University,
Edinburgh, U. K. He has authored or coauthored more than 120 articles in
refereed journals and conference proceedings and holds over 40 patents. His
current research interests include channel sounding, modeling, and emulation
for the fifth/sixth generation (5G/6G) mobile communication, vehicle-to-
vehicle (V2V) communication and unmanned aerial vehicles (UAV) communication
systems.
---|---
|  Ekram Hossain (Fellow, IEEE) is a Professor and the Associate Head
(Graduate Studies) of the Department of Electrical and Computer Engineering,
University of Manitoba, Canada. He is a Member (Class of 2016) of the College
of the Royal Society of Canada. He is also a Fellow of the Canadian Academy of
Engineering and the Engineering Institute of Canada. He has won several
research awards, including the 2017 IEEE Communications Society Best Survey
Paper Award and the 2011 IEEE Communications Society Fred Ellersick Prize
Paper Award. He was listed as a Clarivate Analytics Highly Cited Researcher in
Computer Science in 2017-2023. Previously, he served as the Editor-in-Chief
(EiC) for the IEEE Press (2018–2021) and the IEEE Communications Surveys and
Tutorials (2012–2016). He was a Distinguished Lecturer of the IEEE
Communications Society and the IEEE Vehicular Technology Society. He served as
the Director of Magazines (2020-2021) and the Director of Online Content
(2022-2023) for the IEEE Communications Society.
---|---
|  H. Vincent Poor (Life Fellow, IEEE) received the Ph.D. degree in EECS from
Princeton University in 1977. From 1977 until 1990, he was on the faculty of
the University of Illinois at Urbana-Champaign. Since 1990 he has been on the
faculty at Princeton, where he is currently the Michael Henry Strater
University Professor. During 2006 to 2016, he served as the dean of
Princeton’s School of Engineering and Applied Science. He has also held
visiting appointments at several other universities, including most recently
at Berkeley and Cambridge. His research interests are in the areas of
information theory, machine learning and network science, and their
applications in wireless networks, energy systems and related fields. Among
his publications in these areas is the recent book Machine Learning and
Wireless Communications. (Cambridge University Press, 2022). Dr. Poor is a
member of the National Academy of Engineering and the National Academy of
Sciences and is a foreign member of the Chinese Academy of Sciences, the Royal
Society, and other national and international academies. He received the IEEE
Alexander Graham Bell Medal in 2017.
---|---