††thanks: M. Shell was with the Department of Electrical and Computer
Engineering, Georgia Institute of Technology, Atlanta, GA, 30332 USA e-mail:
(see http://www.michaelshell.org/contact.html).††thanks: J. Doe and J. Doe are
with Anonymous University.††thanks: Manuscript received April 19, 2005;
revised August 26, 2015.

# Hybrid Precoding for Multi-Group Multicasting
in mmWave Systems

Luis F. Abanto-Leon, Matthias Hollick, and Gek Hong (Allyson) Sim {labanto,
mhollick<EMAIL_ADDRESS>Secure Mobile Networking (SEEMOO) Lab,
Technische Universität Darmstadt, Germany

###### Abstract

Multicast beamforming is known to improve spectral efficiency. However, its
benefits and challenges for hybrid precoders design in millimeter-wave
(mmWave) systems remain understudied. To this end, this paper investigates the
first joint design of hybrid transmit precoders (with an arbitrary number of
finite-resolution phase shifts) and receive combiners for mmWave multi-group
multicasting. Our proposed design leverages semidefinite relaxation (SDR),
alternating optimization and Cholesky matrix factorization to sequentially
optimize the digital/analog precoders at the transmitter and the combiners at
each receiver. By considering receivers with multiple-antenna architecture,
our design remarkably improves the overall system performance. Specifically,
with only two receive antennas the average transmit power per received message
improves by $16.8\%$ while the successful information reception is boosted by
$60\%$. We demonstrate by means of extensive simulations that our hybrid
precoder design performs very close to its fully-digital counterpart even
under challenging scenarios (i.e., when co-located users belong to distinct
multicast groups).

###### Index Terms:

hybrid precoding, millimeter-wave, multicast, semidefinite relaxation,
alternating optimization.

##  I Introduction

In recent years, millimeter-wave (mmWave) has emerged as a promising
technology to fuel the ever-increasing consumer demands for extremely fast
(i.e., up to multi-Gbps) connectivity. In delivering such requirements for
dense networks scenarios (due to extreme densification in next-generation
networks), a system can leverage the benefits of multicast communications [1].
Indeed, recent studies in [2, 3] demonstrate the potential of multicast to
significantly improve the network throughput and spectral efficiency of mmWave
systems. To guarantee these performances, an appropriate design for multicast
precoders is crucial _(i)_ to compensate for severe channel attenuation, and
_(ii)_ to minimize the interference between simultaneous transmissions.

An early effort on multicast precoder design is presented in [4], where the
authors investigated single-group multicast precoding with a multi-antenna
base station and several single-antenna receivers. Aiming at minimizing the
transmit power subject to predefined quality of service (QoS) requirements
(i.e., QoS problem), the problem is formulated as a relaxed semidefinite
program (SDP) where befitting solutions are obtained via randomization [5].
Since their work only considers single-group multicast, the problem
formulation thus excludes the interference aspect that is relevant in
designing multi-group multicast precoders. Expanding on [4], the authors of
[6, 7] investigate a scenario with multiple co-channel multicast groups, which
allows transmissions of simultaneous multicast signals by exploiting spatial
multiplexing. Furthermore, to mitigate the interference between the distinct
signals (and thus increase the number of served users), [6, 7] an additional
stage of power control is incorporated. The QoS problem is also considered in
[8, 9, 10] with diverse extensions to the formulation. A related formulation
known as the max-min fair (MMF) problem is studied in [11, 12, 13].

The works mentioned above are developed within the framework of fully-digital
multicast precoders. Given the affordable hardware and moderate computational
complexity of hybrid precoders, a shift of interest has been observed in
departing from fully-digital to adopting hybrid antenna arrays architectures.
Hybrid precoders are composed of a low-dimensional digital beamformer in
cascade with a high-dimensional network of cost-efficient constant-modulus
phase shifters that admit a limited number of phase rotations. In general,
hybrid precoders are less flexible than their fully-digital counterparts, thus
rendering the design of an optimal hybrid precoder a challenging task.
Besides, they pose a compromise in terms of beamforming capabilities and
interference management. On the other hand, the versatility of digital
precoders comes at the expense of highly complicated and expensive hardware,
wherein a dedicated radio frequency (RF) chain is required for each antenna
element.

To date, the body of works that has studied hybrid precoding for physical
layer multicasting includes single-group multicasting (in [14]) and multi-
group multicasting (in [15, 16, 17]). The authors of [14] consider the MMF
problem in single-group multicast settings, wherein a codebook-based design is
presented. On the other hand, the multi-group multicast QoS and MMF problems
are revisited in [15], where the authors propose a customized hybrid
architecture with improved performance. In [16], the authors investigate the
QoS problem by considering a high-resolution lens array with adjustable power.
However, such design circumvents the constant-modulus discrete phase shifts
characteristics of analog circuitry components of hybrid precoders. On the
contrary, the authors in [17] design a scheme to support joint power and
information transfer with hybrid precoders. Their formulation considers
discrete phase shifts, but restrains the set of phase shifts to only four
choices. In addition, the existing studies on multicast precoders for mmWave
systems only consider receivers with single-antenna architecture. This
conditioning prevents the mitigation of undesired signals (e.g.,
interference), especially when users from different multicast groups have
correlated channel vectors. In particular, endowing receivers with multiple
antennas: _(i)_ mitigates interference from other sources, _(ii)_ reduces the
power expenditure from the transmitter, and _(iii)_ improves the service
ubiquitousness.

To the best of our knowledge, we are the first to investigate the _joint
design of hybrid multicast precoders with an arbitrary number of finite-
resolution constant-modulus phase shifts at the transmitter while considering
multiple antennas at the receivers_. Related art on hybrid precoding for
multi-user scenarios (e.g., in [18]) are fundamentally different as each RF
chain at the transmitter is matched to the channel of one dedicated user. In
the multi-group multicast scenario we consider in this paper—due to the
limitation of RF chains—several users with distinct channel conditions need to
be served by a single RF unit, thus complicating the design of the hybrid
precoder. Our proposed formulation focuses on the QoS problem, for which we
present an SDR-based approach to optimize the digital precoder, analog phase
shifts and receive combiners. Due to the existence of several design
parameters, our proposed formulation is divided into a set of sub-problems
that we approach adopting alternating optimization (as in e.g., [19]).
Moreover, we incorporate a set of slack parameters to promote coherent
parameter binding among the decoupled sub-problems. Since alternating
optimization requires each sub-problem to be solvable to guarantee the
continuity of the optimization process, such a set of slack parameters ensures
that each sub-problem always yields a feasible solution for the succeeding
stages. Finally, due to the selection of finite-resolution constant-modulus
phase shifts, the problem is inherently of combinatorial nature. To circumvent
this matter, we propose a scheme where the phase shifts selection is recast as
an SDR program followed by a stage consisting of Cholesky matrix
factorization, least squares, and randomization.

The paper is structured as follows. In Section II, we model and elaborate on
the problem of multi-group multicast hybrid transmit precoders with finite-
resolution phase shifts and multi-antenna receivers in mmWave systems. In
Section III, we formulate the problem and present the proposed solution in
Section IV. We analyze and compare the performance of our design in Section V.
Also, we include an insightful discussion in Section VI. Finally, we conclude
with the contributions of this paper in Section VII.

##  II System Model

We adopt a mmWave system where a gNodeB serves $K$ users distributed into $G$
different co-channel multicast groups. The sets of users and groups are
denoted by $\mathcal{K}=\left\\{1,2,\dots,K\right\\}$ and
$\mathcal{I}=\left\\{1,2,\dots,G\right\\}$, respectively. Each multicast group
$\mathcal{G}_{i}\left(i\in\mathcal{I}\right)$ contains the indices of users
that constitute it. The amount of users in each multicast group is represented
by $\left|\mathcal{G}_{i}\right|$, such that
$\sum^{G}_{i=1}\left|\mathcal{G}_{i}\right|=K$. As in [6], we assume that
$\mathcal{G}_{i}\cap\mathcal{G}_{j}=\left\\{\emptyset\right\\},\forall i\neq
j$. The gNodeB is equipped with $N_{\mathrm{tx}}$ transmit antennas and
$N^{\mathrm{RF}}_{\mathrm{tx}}$ RF chains, with $G\leq
N^{\mathrm{RF}}_{\mathrm{tx}}\leq N_{\mathrm{tx}}$. The downlink signal is
represented by $\mathbf{x}=\mathbf{F}\mathbf{M}\mathbf{s}$, where
$\mathbf{F}\in\mathbb{C}^{N_{\mathrm{tx}}\times
N^{\mathrm{RF}}_{\mathrm{tx}}}$ is the analog precoder whereas
$\mathbf{M}=\left[\mathbf{m}_{1},\mathbf{m}_{2},\dots,\mathbf{m}_{G}\right]\in\mathbb{C}^{N^{\mathrm{RF}}_{\mathrm{tx}}\times
G}$ assembles the digital precoders for each of the multicast group. The
collection of data symbols for the intended groups is denoted by
$\mathbf{s}=\left[s_{1},s_{2},\dots,s_{G}\right]^{T}\in\mathbb{C}^{G\times
1}$, where each entry has unit power on average, i.e.,
$\mathbb{E}\left\\{\mathbf{s}\mathbf{s}^{H}\right\\}=\mathbf{I}$. Also, every
element $\left(q,r\right)$ of the analog precoder is a phase rotation with
constant modulus. Therefore, $\left[\mathbf{F}\right]_{q,r}\in\mathcal{F}$,
where $q\in\mathcal{Q}=\left\\{1,2,\dots,N_{\mathrm{tx}}\right\\}$,
$r\in\mathcal{R}=\left\\{1,2,\dots,N^{\mathrm{RF}}_{\mathrm{tx}}\right\\}$,
$\mathcal{F}=\left\\{\sqrt{\delta},\sqrt{\delta}e^{\frac{2\pi}{L}},\dots,\sqrt{\delta}e^{\frac{2\pi\left(L-1\right)}{L}}\right\\}$,
$L$ denotes the number of different phase shifts that are allowed, and
$\delta$ is a scaling factor. Each multicast receiver has a finite number of
receive antennas $N_{\mathrm{rx}}\ll N_{\mathrm{tx}}$, and an equal number of
RF chains. Under the assumption of narrowband flat-fading, the signal received
by the $k$-th user ($k\in\mathcal{G}_{i}$) is given by

$\displaystyle\begin{split}y_{k}=\underbrace{\mathbf{w}^{H}_{k}\mathbf{H}_{k}\mathbf{F}\mathbf{m}_{i}s_{i}}_{\text{desired
multicast
signal}}+\underbrace{\mathbf{w}^{H}_{k}\mathbf{H}_{k}\sum^{G}_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}\mathbf{F}\mathbf{m}_{j}s_{j}}_{\text{interference}}+\underbrace{\mathbf{w}^{H}_{k}\mathbf{n}_{k}}_{\text{noise}},\end{split}$
(1)

where $i$ is the index of group $\mathcal{G}_{i}$, and
$\mathbf{w}_{k}\in{\mathbb{C}}^{N_{\mathrm{rx}}\times 1}$ represents the
digital receive beamformer of the $k$-th user. Also,
$\mathbf{H}_{k}\in{\mathbb{C}}^{N_{\mathrm{rx}}\times N_{\mathrm{tx}}}$
denotes the channel between the gNodeB and the $k$-th user, whereas
$\mathbf{n}_{k}\sim\mathcal{CN}\left(\mathbf{0},{\sigma}^{2}\mathbf{I}\right)$
denotes additive white Gaussian noise. The signal–to–interference–plus-noise
ratio (SINR) at user $k$ is defined as

$\mathrm{SINR}_{k}=\frac{\left|\mathbf{w}^{H}_{k}\mathbf{H}_{k}\mathbf{F}\mathbf{m}_{i}\right|^{2}}{\sum_{j\neq
i}\left|\mathbf{w}^{H}_{k}\mathbf{H}_{k}\mathbf{F}{\mathbf{m}_{j}}\right|^{2}+{\sigma}^{2}\left\|\mathbf{w}_{k}\right\|^{2}_{2}}.$
(2)

##  III Problem Formulation

Aiming to optimize the transmit power, we formulate

$\displaystyle\mathcal{P}^{\mathrm{hyb}}:$
$\displaystyle\min_{\begin{subarray}{c}\mathbf{F},\left\\{\mathbf{m}_{i}\right\\}^{G}_{i=1},\\\
\left\\{\mathbf{w}_{k}\right\\}^{K}_{k=1}\end{subarray}}$
$\displaystyle{\sum^{G}_{i=1}\left\|\mathbf{F}\mathbf{m}_{i}\right\|^{2}_{2}}$
(3a) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{s.t.}$
$\displaystyle\frac{\left|\mathbf{w}^{H}_{k}\mathbf{H}_{k}\mathbf{F}\mathbf{m}_{i}\right|^{2}}{\sum_{j\neq
i}\left|\mathbf{w}^{H}_{k}\mathbf{H}_{k}\mathbf{F}{\mathbf{m}_{j}}\right|^{2}+{\sigma}^{2}\left\|\mathbf{w}_{k}\right\|^{2}_{2}}\geq\gamma_{i},$
(3b)
$\displaystyle\left\|\mathbf{w}_{k}\right\|^{2}_{2}=P^{\mathrm{max}}_{\mathrm{rx}},$
(3c)
$\displaystyle\left\|\mathbf{F}\right\|^{2}_{\mathrm{F}}=N^{\mathrm{RF}}_{\mathrm{tx}},$
(3d)
$\displaystyle\left[\mathbf{F}\right]_{q,r}\in\mathcal{F},q\in\mathcal{Q},r\in\mathcal{R},\forall
k\in\mathcal{G}_{i},i\in\mathcal{I},$ (3e)

where (3a) targets the minimization of the transmit power. Constraint (3b)
imposes specific QoS requirements for each multicast group, whereas (3c)
restricts the power expenditure for receive beamforming at each user.
Constraint (3d) limits the power associated with the analog precoder. In
addition, (3e) enforces every phase shift to belong to $\mathcal{F}$. The
target SINR of every group $\mathcal{G}_{i}$ is denoted by $\gamma_{i}$. Note
that (3a) is non-convex due to multiplicative coupling between $\mathbf{F}$
and $\mathbf{m}_{i}$. Constraint (3b) is non-convex since it is defined as the
ratio of two non-convex expressions. On the other hand, (3c) is quadratic and
non-convex on $\mathbf{w}_{k}$. Constraint (3e) is inherently of combinatorial
nature, therefore non-convex. Thus, (3d) is also non-convex due to its
dependence on (3e). As a result, $\mathcal{P}^{\mathrm{hyb}}$ is classified as
a non-convex quadratically constrained quadratic program (QCQP), which is
known to be NP-hard.

##  IV Proposed Solution

In this section, we propose an approach based on alternating optimization,
where the unknown parameters $\mathbf{F}$,
$\left\\{\mathbf{m}_{i}\right\\}^{G}_{i=1}$ and
$\left\\{\mathbf{w}_{k}\right\\}^{K}_{k=1}$ are optimized sequentially and
iteratively. Due to sequential (and independent) parameter optimization, the
suitability of the solution can be compromised. Therefore, we include an
additional set of slack parameters $\left\\{x_{k}\right\\}^{K}_{k=1}$ to
reinforce the linkage between $\mathbf{F}$,
$\left\\{\mathbf{m}_{i}\right\\}^{G}_{i=1}$ and
$\left\\{\mathbf{w}_{k}\right\\}^{K}_{k=1}$. Thus, the resulting problem
formulation is defined as follows,

$\displaystyle\mathcal{P}^{\mathrm{hyb}}_{0}:$
$\displaystyle\min_{\begin{subarray}{c}\mathbf{F},\left\\{\mathbf{m}_{i}\right\\}^{G}_{i=1},\\\
\left\\{\mathbf{w}_{k}\right\\}^{K}_{k=1},\\\
\left\\{x_{k}\right\\}^{K}_{k=1}\end{subarray}}$
$\displaystyle{\sum^{G}_{i=1}\left\|\mathbf{F}\mathbf{m}_{i}\right\|^{2}_{2}+\beta\sum^{K}_{k=1}x_{k}}$
(4a) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{s.t.}$
$\displaystyle\frac{\left|\mathbf{w}^{H}_{k}\mathbf{H}_{k}\mathbf{F}\mathbf{m}_{i}\right|^{2}+x_{k}}{\sum_{j\neq
i}\left|\mathbf{w}^{H}_{k}\mathbf{H}_{k}\mathbf{F}{\mathbf{m}_{j}}\right|^{2}+{\sigma}^{2}\left\|\mathbf{w}_{k}\right\|^{2}_{2}}\geq\gamma_{i},$
(4b)
$\displaystyle\left\|\mathbf{w}_{k}\right\|^{2}_{2}=P^{\mathrm{max}}_{\mathrm{rx}},$
(4c)
$\displaystyle\left\|\mathbf{F}\right\|^{2}_{\mathrm{F}}=N^{\mathrm{RF}}_{\mathrm{tx}},$
(4d) $\displaystyle\left[\mathbf{F}\right]_{q,r}\in\mathcal{F},$ (4e)
$\displaystyle x_{k}\geq 0,q\in\mathcal{Q},r\in\mathcal{R},\forall
k\in\mathcal{G}_{i},i\in\mathcal{I}.$ (4f)

Each $x_{k}\in\mathbb{R}_{+}$ penalizes the objective function (with a
sufficiently large $\beta$) whenever $x_{k}>0$ needs to be added to the left-
hand side numerator of (4b) for the QoS inequality to hold. Thus, an increment
of $\sum^{G}_{i=1}\left\|\mathbf{F}\mathbf{m}_{i}\right\|^{2}_{2}$ will be
prioritized instead of letting $\sum^{K}_{k=1}x_{k}$ augment. This
regularization promotes more QoS inequalities to be satisfied by action of
$\mathbf{F}$, $\left\\{\mathbf{m}_{i}\right\\}^{G}_{i=1}$ and
$\left\\{\mathbf{w}_{k}\right\\}^{K}_{k=1}$. The slack parameters
$\left\\{x_{k}\right\\}^{K}_{k=1}$ ensure that a feasible solution always
exists as $x_{k}$ will absorb any surplus that is required for (4b) to hold.
In the following we optimize the three sets of parameters by separating
$\mathcal{P}^{\mathrm{hyb}}_{0}$ into 3 sub-problems
$\mathcal{P}^{\mathrm{hyb}}_{1}$, $\mathcal{P}^{\mathrm{hyb}}_{2}$ and
$\mathcal{P}^{\mathrm{hyb}}_{3}$, which are sequentially and alternately
solved.

Observation: Even with fully-digital precoders and single-antenna receivers
(i.e., $\mathbf{F}=\mathbf{I}$, $\mathbf{w}_{k}=1$), a feasible solution to
(4) cannot always be guaranteed. This usually occurs when $N_{\mathrm{tx}}<K$
(as in our case). As a consequence, $\mathcal{P}^{\mathrm{hyb}}_{1}$,
$\mathcal{P}^{\mathrm{hyb}}_{2}$ or $\mathcal{P}^{\mathrm{hyb}}_{3}$ may
render infeasible, thus interrupting the optimization procedure. To prevent
this, we include $x_{k}$ to ensure the existence of a feasible solution
(without raising infeasibility certificates), thereby guaranteeing the
continuity of the sequential optimization process.

Observation: In contrast to adaptive hybrid precoding, where the architectures
changes dynamically (i.e., some phase shifters activate/deactivate), in our
case the fixed fully-connected architecture allows us to determine
$\delta=1/N^{\mathrm{RF}}_{\mathrm{tx}}$ from (4d). Thus, (4d) is removed in
the sequel.

###  IV-A Optimization of $\mathbf{F}$

Assuming that $\left\\{\mathbf{m}_{i}\right\\}^{G}_{i=1}$ and
$\left\\{\mathbf{w}_{k}\right\\}^{K}_{k=1}$ are known, we optimize over
$\mathbf{F}$. Thus,

$\displaystyle{\mathcal{P}}^{\mathrm{hyb}}_{1}:$
$\displaystyle\min_{\begin{subarray}{c}\mathbf{F},\left\\{x_{k}\right\\}^{K}_{k=1}\end{subarray}}$
$\displaystyle{\sum^{G}_{i=1}\left\|\mathbf{F}\mathbf{m}_{i}\right\|^{2}_{2}+\beta\sum^{K}_{k=1}x_{k}}$
(5a) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{s.t.}$
$\displaystyle\gamma_{i}\left(\sum_{j\neq
i}\left|\mathbf{w}^{H}_{k}\mathbf{H}_{k}\mathbf{F}{\mathbf{m}_{j}}\right|^{2}+{\sigma}^{2}\left\|\mathbf{w}_{k}\right\|^{2}_{2}\right)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-\left|\mathbf{w}^{H}_{k}\mathbf{H}_{k}\mathbf{F}{\mathbf{m}_{i}}\right|^{2}\leq
x_{k},$ (5b) $\displaystyle\left[\mathbf{F}\right]_{q,r}\in\mathcal{F},$ (5c)
$\displaystyle x_{k}\geq 0,q\in\mathcal{Q},r\in\mathcal{R},\forall
k\in\mathcal{G}_{i},i\in\mathcal{I}.$ (5d)

Notice that we can express
$\mathbf{F}\mathbf{m}_{i}=\mathbf{J}_{i}\mathbf{f}$, where
$\mathbf{J}_{i}=\mathbf{m}^{T}_{i}\otimes\mathbf{I}$ and
$\mathbf{f}=\mathrm{vec}\left(\mathbf{F}\right)$. With this redefinition, (5)
can be equivalently expressed as,

$\displaystyle\mathcal{P}^{\mathrm{hyb}}_{1}:$
$\displaystyle\min_{\begin{subarray}{c}\mathbf{f},\left\\{x_{k}\right\\}^{K}_{k=1}\end{subarray}}$
$\displaystyle{\sum^{G}_{i=1}\left\|\mathbf{J}_{i}\mathbf{f}\right\|^{2}_{2}+\beta\sum^{K}_{k=1}x_{k}}$
(6a) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{s.t.}$
$\displaystyle\gamma_{i}\left(\sum_{j\neq
i}\left|\mathbf{w}^{H}_{k}\mathbf{H}_{k}\mathbf{J}_{j}\mathbf{f}\right|^{2}+{\sigma}^{2}\left\|\mathbf{w}_{k}\right\|^{2}_{2}\right)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-\left|\mathbf{w}^{H}_{k}\mathbf{H}_{k}\mathbf{J}_{i}\mathbf{f}\right|^{2}\leq
x_{k},$ (6b) $\displaystyle\left[\mathbf{f}\right]_{n}\in\mathcal{F},$ (6c)
$\displaystyle x\geq 0,n\in\mathcal{N},\forall
k\in\mathcal{G}_{i},i\in\mathcal{I},$ (6d)

where
$\mathcal{N}=\left\\{1,2,\dots,N^{\mathrm{RF}}_{\mathrm{tx}}N_{\mathrm{tx}}\right\\}$.
Note that
$\left\|\mathbf{J}_{i}\mathbf{f}\right\|^{2}_{2}=\mathrm{Tr}\left(\mathbf{R}_{i}\mathbf{D}\right)$,
with $\mathbf{R}_{i}=\mathbf{J}^{H}_{i}\mathbf{J}_{i}$ and
$\mathbf{D}=\mathbf{f}\mathbf{f}^{H}$. Also,
$\left[\mathbf{D}\right]_{n,n}=\delta$ since
$\left[\mathbf{f}\right]_{n}\in\mathcal{F}$. Furthermore, since
$\left|\mathbf{w}^{H}_{k}\mathbf{H}_{k}\mathbf{J}_{i}\mathbf{f}\right|^{2}=\mathrm{Tr}\left(\mathbf{V}_{i,k}\mathbf{D}\right)$,
with
$\mathbf{V}_{i,k}=\mathbf{J}^{H}_{i}\mathbf{H}^{H}_{k}\mathbf{w}_{k}\mathbf{w}^{H}_{k}\mathbf{H}_{k}\mathbf{J}_{i}$,
we can recast (6) in its SDP form as,

$\displaystyle{\mathcal{P}}^{\mathrm{hyb}}_{\mathrm{SDP},1}:$
$\displaystyle\min_{\begin{subarray}{c}\mathbf{D},\left\\{x_{k}\right\\}^{K}_{k=1}\end{subarray}}$
$\displaystyle{\sum^{G}_{i=1}\mathrm{Tr}\left(\mathbf{D}\mathbf{R}_{i}\right)+\beta\sum^{K}_{k=1}x_{k}}$
(7a) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{s.t.}$
$\displaystyle\mathrm{Tr}\left(\mathbf{D}\left(\gamma_{i}\sum_{j\neq
i}\mathbf{V}_{j,k}-\mathbf{V}_{i,k}\right)\right)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
+{\sigma}^{2}\gamma_{i}\left\|\mathbf{w}_{k}\right\|^{2}_{2}\leq x_{k},$ (7b)
$\displaystyle\left[\mathbf{D}\right]_{n,n}=\delta,$ (7c)
$\displaystyle\mathrm{rank}\left(\mathbf{D}\right)=1,$ (7d)
$\displaystyle\mathbf{D}\succcurlyeq\mathbf{0},$ (7e) $\displaystyle x_{k}\geq
0,n\in\mathcal{N},\forall k\in\mathcal{G}_{i},i\in\mathcal{I}.$ (7f)

The SDP program in (7) has a linear objective subject to affine constraints
except for the non-convex constraint (7d), which imposes a rank-one condition
on $\mathbf{D}$ (as it is originally obtained from
$\mathbf{D}=\mathbf{f}\mathbf{f}^{H}$). Constraint (7e) restricts $\mathbf{D}$
to be Hermitian positive semidefinite. It is worth noticing that (6c) is the
only constraint not strongly enforced in (7). Thus, while the constant-modulus
requirement of (6c) is satisfied by (7c), its phase has been ignored.
Nevertheless, the phase will be optimized through the following procedure
[20].

Stage A1: Notice that any element $\left(n_{1},n_{2}\right)$ of matrix
$\mathbf{D}$ can be represented as
$\left[\mathbf{D}\right]_{n_{1},n_{2}}=\left[\mathbf{f}\right]_{n_{1}}\left[\mathbf{f}\right]^{*}_{n_{2}}$.
Now, let us define a vector
$\mathbf{u}\in\mathbb{C}^{N^{\mathrm{RF}}_{\mathrm{tx}}N_{\mathrm{tx}}\times
1}$ such that $\left\|\mathbf{u}\right\|^{2}_{2}=\mathbf{u}^{H}\mathbf{u}=1$.
As a consequence, we can express $\left[\mathbf{D}\right]_{n_{1},n_{2}}$ in
terms of $\mathbf{u}$, i.e.,
$\left[\mathbf{D}\right]_{n_{1},n_{2}}=\left(\left[\mathbf{f}\right]_{n_{1}}\mathbf{u}^{T}\right)\left(\left[\mathbf{f}\right]^{*}_{n_{2}}\mathbf{u}^{*}\right)$.
Assuming that $\mathbf{q}_{n}=\left[\mathbf{f}\right]_{n}\mathbf{u}$,
$\mathbf{D}$ can be recast as $\mathbf{D}=\mathbf{Q}^{T}\mathbf{Q}^{*}$ with
$\mathbf{Q}=\left[\mathbf{q}_{1},\mathbf{q}_{2},\dots,\mathbf{q}_{N^{\mathrm{RF}}_{\mathrm{tx}}N_{\mathrm{tx}}}\right]$.

Stage A2: In (7), the only non-convex constraint is (7d). Thus, we define
${\mathcal{P}}^{\mathrm{hyb}}_{\mathrm{SDR},1}$ as the resultant SDR surrogate
of (7) obtained upon dropping (7d). The solution returned by
$\mathcal{P}^{\mathrm{hyb}}_{\mathrm{SDR},1}$ is denoted by
$\widehat{\mathbf{D}}$. Then, via Cholesky matrix factorization we obtain
$\widehat{\mathbf{D}}=\widehat{\mathbf{Q}}^{T}\widehat{\mathbf{Q}}^{*}$, where
$\widehat{\mathbf{Q}}=\left[\widehat{\mathbf{q}}_{1},\widehat{\mathbf{q}}_{2},\dots,\widehat{\mathbf{q}}_{N^{\mathrm{RF}}_{\mathrm{tx}}N_{\mathrm{tx}}}\right]$.
Although we have derived a relation that associates the unknown phase shifts
$\widehat{\mathbf{f}}$ with the known vectors
$\left\\{\widehat{\mathbf{q}}_{n}\right\\}^{N^{\mathrm{RF}}_{\mathrm{tx}}N_{\mathrm{tx}}}_{n=1}$
(via
$\widehat{\mathbf{q}}_{n}=\left[\widehat{\mathbf{f}}\right]_{n}\widehat{\mathbf{u}}$),
the vector $\widehat{\mathbf{u}}$ also remains unknown. Moreover, the initial
premise was that every $\widehat{\mathbf{q}}_{n}$ could be obtained from the
same $\widehat{\mathbf{u}}$. However, this cannot be guaranteed as a solution
$\left(\widehat{\mathbf{f}},\widehat{\mathbf{u}}\right)$ for
$\widehat{\mathbf{q}}_{n}=\left[\widehat{\mathbf{f}}\right]_{n}\widehat{\mathbf{u}},\forall
n\in\mathcal{N}$ may not exist. Thus, we aim at finding approximate
$\widehat{\mathbf{f}}$ and $\widehat{\mathbf{u}}$, such that
$\widehat{\mathbf{q}}_{n}\approx\left[\widehat{\mathbf{f}}\right]_{n}\widehat{\mathbf{u}}$,
and whose error is minimum in the 2-norm sense. Mathematically,

$\displaystyle\mathcal{P}^{\mathrm{hyb}}_{\mathrm{LS}}:\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\min_{\begin{subarray}{c}\widehat{\mathbf{u}},\left[\widehat{\mathbf{f}}\right]_{n}\end{subarray}}$
$\displaystyle{\sum^{N^{\mathrm{RF}}_{\mathrm{tx}}N_{\mathrm{tx}}}_{n=1}\left\|\widehat{\mathbf{q}}_{n}-\left[\widehat{\mathbf{f}}\right]_{n}\widehat{\mathbf{u}}\right\|^{2}_{2}}$
(8a) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{s.t.}$
$\displaystyle\left\|\widehat{\mathbf{u}}\right\|^{2}_{2}=1,\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (8b)
$\displaystyle\left[\widehat{\mathbf{f}}\right]_{n}\in\mathcal{F},n\in\mathcal{N}.$
(8c)

Stage A3: Minimizing simultaneously over both $\widehat{\mathbf{q}}_{n}$ and
$\widehat{\mathbf{u}}$ is challenging. If we assume that
$\widehat{\mathbf{u}}$ is known such that (8b) is satisfied, then we are
required to solve

$\displaystyle\widetilde{\mathcal{P}}^{\mathrm{hyb}}_{\mathrm{LS}}:\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\min_{\begin{subarray}{c}\left[\widehat{\mathbf{f}}\right]_{n}\end{subarray}}$
$\displaystyle{\sum^{N^{\mathrm{RF}}_{\mathrm{tx}}N_{\mathrm{tx}}}_{n=1}\left\|\widehat{\mathbf{q}}_{n}-\left[\widehat{\mathbf{f}}\right]_{n}\widehat{\mathbf{u}}\right\|^{2}_{2}}$
(9a) $\displaystyle\leavevmode\nobreak\ \mathrm{s.t.}$
$\displaystyle\left[\widehat{\mathbf{f}}\right]_{n}\in\mathcal{F},n\in\mathcal{N}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (9b)

By expanding (9a), we realize that
$\left\|\widehat{\mathbf{q}}_{n}-\left[\widehat{\mathbf{f}}\right]_{n}\widehat{\mathbf{u}}\right\|^{2}_{2}=\widehat{\mathbf{q}}^{H}_{n}\widehat{\mathbf{q}}_{n}-2\mathfrak{Re}\left(\left[\widehat{\mathbf{f}}\right]_{n}\widehat{\mathbf{q}}^{H}_{n}\widehat{\mathbf{u}}\right)+\left|\left[\widehat{\mathbf{f}}\right]_{n}\right|^{2}\widehat{\mathbf{u}}^{H}\widehat{\mathbf{u}}$.
Thus, (9) is equivalent to

$\displaystyle\widetilde{\mathcal{P}}^{\mathrm{hyb}}_{\mathrm{LS}}:\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ $
$\displaystyle\max_{\begin{subarray}{c}\left[\widehat{\mathbf{f}}\right]_{n}\end{subarray}}$
$\displaystyle{\sum^{N^{\mathrm{RF}}_{\mathrm{tx}}N_{\mathrm{tx}}}_{n=1}\mathfrak{Re}\left(\left[\widehat{\mathbf{f}}\right]_{n}\widehat{\mathbf{q}}^{H}_{n}\widehat{\mathbf{u}}\right)}$
(10a) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{s.t.}$
$\displaystyle\left[\widehat{\mathbf{f}}\right]_{n}\in\mathcal{F},n\in\mathcal{N}.\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ $ (10b)

Note that (10) can be decomposed into
$N^{\mathrm{RF}}_{\mathrm{tx}}N_{\mathrm{tx}}$ independent sub-problems. Thus,
since $z_{n}=\widehat{\mathbf{q}}^{H}_{n}\widehat{\mathbf{u}}$ is known, we
need to select $\left[\mathbf{f}\right]_{n}$ such that the real part of (10a)
is maximized. This is equivalent to choosing
$\left[\widehat{\mathbf{f}}\right]_{n}\in\mathcal{F}$ with the closest phase
to $z^{*}_{n}$. After finding $\widehat{\mathbf{f}}$, it can be reshaped in
order to obtain $\widehat{\mathbf{F}}$. As shown in Algorithm 1,
$N_{\mathrm{rand}}$ candidate vectors $\widehat{\mathbf{u}}$ are generated and
the best-performing option is maintained.

###  IV-B Optimization of $\mathbf{m}_{i}$

We assume herein that $\mathbf{F}$ and
$\left\\{\mathbf{w}_{k}\right\\}^{K}_{k=1}$ are known. Thus, the original
problem in (4) collapses to

$\displaystyle\mathcal{P}^{\mathrm{hyb}}_{2}:$
$\displaystyle\min_{\begin{subarray}{c}\left\\{\mathbf{m}_{i}\right\\}^{G}_{i=1},\\\
\left\\{x_{k}\right\\}^{K}_{k=1}\end{subarray}}$
$\displaystyle{\sum^{G}_{i=1}\left\|\mathbf{F}\mathbf{m}_{i}\right\|^{2}_{2}+\beta\sum^{K}_{k=1}x_{k}}$
(11a) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{s.t.}$
$\displaystyle\gamma_{i}\left(\sum_{j\neq
i}\left|\mathbf{w}^{H}_{k}\mathbf{H}_{k}\mathbf{F}{\mathbf{m}_{j}}\right|^{2}+{\sigma}^{2}\left\|\mathbf{w}_{k}\right\|^{2}_{2}\right)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-\left|\mathbf{w}^{H}_{k}\mathbf{H}_{k}\mathbf{F}{\mathbf{m}_{i}}\right|^{2}\leq
x_{k},$ (11b) $\displaystyle x_{k}\geq 0,\forall
k\in\mathcal{G}_{i},i\in\mathcal{I}.$ (11c)

The SDP equivalent formulation of (11) is expressed as

$\displaystyle\mathcal{P}^{\mathrm{hyb}}_{\mathrm{SDP},2}:$
$\displaystyle\min_{\begin{subarray}{c}\left\\{\mathbf{M}_{i}\right\\}^{G}_{i=1},\\\
\left\\{x_{k}\right\\}^{K}_{k=1}\end{subarray}}$
$\displaystyle{\sum^{G}_{i=1}\mathrm{Tr}\left(\mathbf{Y}\mathbf{M}_{i}\right)+\beta\sum^{K}_{k=1}x_{k}}$
(12a) $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{s.t.}$
$\displaystyle\mathrm{Tr}\left(\mathbf{X}_{k}\left(\gamma_{i}\sum_{j\neq
i}\mathbf{M}_{j}-\mathbf{M}_{i}\right)\right)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\
+{\sigma}^{2}\gamma_{i}\left\|\mathbf{w}_{k}\right\|^{2}_{2}\leq x_{k},$ (12b)
$\displaystyle\mathbf{M}_{i}\succcurlyeq\mathbf{0},$ (12c)
$\displaystyle\mathrm{rank}\left(\mathbf{M}_{i}\right)=1,$ (12d)
$\displaystyle x_{k}\geq 0,\forall k\in\mathcal{G}_{i},i\in\mathcal{I},$ (12e)

where $\mathbf{Y}=\mathbf{F}^{H}\mathbf{F}$,
$\mathbf{X}_{k}=\mathbf{F}^{H}\mathbf{H}^{H}_{k}\mathbf{w}_{k}\mathbf{w}^{H}_{k}\mathbf{H}_{k}\mathbf{F}$
and $\mathbf{M}_{i}=\mathbf{m}_{i}\mathbf{m}^{H}_{i}$. Similarly as before,
(12) has a linear objective with affine constraints except for (12d). Thus, we
define $\mathcal{P}^{\mathrm{hyb}}_{\mathrm{SDR},2}$ as the SDR surrogate of
(12), where (12d) is neglected.

###  IV-C Optimization of $\mathbf{w}_{k}$

Now, we assume that $\mathbf{F}$ and
$\left\\{\mathbf{m}_{i}\right\\}^{G}_{i=1}$ are given. Therefore, we optimize
over $\left\\{\mathbf{w}_{k}\right\\}^{K}_{k=1}$ as shown in (13)

$\displaystyle\mathcal{P}^{\mathrm{hyb}}_{3}:$
$\displaystyle\min_{\begin{subarray}{c}\left\\{\mathbf{w}_{k}\right\\}^{K}_{k=1},\\\
\left\\{x_{k}\right\\}^{K}_{k=1}\end{subarray}}$
$\displaystyle{\sum^{K}_{k=1}x_{k}}$ (13a) $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{s.t.}$
$\displaystyle\gamma_{i}\left(\sum_{j\neq
i}\left|\mathbf{w}^{H}_{k}\mathbf{H}_{k}\mathbf{F}{\mathbf{m}_{j}}\right|^{2}+{\sigma}^{2}\left\|\mathbf{w}_{k}\right\|^{2}_{2}\right)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
-\left|\mathbf{w}^{H}_{k}\mathbf{H}_{k}\mathbf{F}\mathbf{m}_{i}\right|^{2}\leq
x_{k},$ (13b)
$\displaystyle\left\|\mathbf{w}_{k}\right\|^{2}_{2}=P^{\mathrm{max}}_{\mathrm{rx}},$
(13c) $\displaystyle x_{k}\geq 0,\forall k\in\mathcal{G}_{i},i\in\mathcal{I},$
(13d)

In SDP form, (13) can be recast as

$\displaystyle\mathcal{P}^{\mathrm{hyb}}_{\mathrm{SDP},3}:$
$\displaystyle\min_{\begin{subarray}{c}\left\\{\mathbf{W}_{k}\right\\}^{K}_{k=1},\\\
\left\\{x_{k}\right\\}^{K}_{k=1}\end{subarray}}$
$\displaystyle{\sum^{K}_{k=1}x_{k}}$ (14a) $\displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \mathrm{s.t.}$
$\displaystyle\mathrm{Tr}\left(\mathbf{W}_{k}\left(\gamma_{i}\sum_{j\neq
i}\mathbf{Z}_{k,j}-\mathbf{Z}_{k,i}\right)\right)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
+{\sigma}^{2}\gamma_{i}\mathrm{Tr}\left(\mathbf{W}_{k}\right)\leq x_{k},$
(14b)
$\displaystyle\mathrm{Tr}\left(\mathbf{W}_{k}\right)=P^{\mathrm{max}}_{\mathrm{rx}},$
(14c) $\displaystyle\mathbf{W}_{k}\succcurlyeq\mathbf{0},$ (14d)
$\displaystyle\mathrm{rank}\left(\mathbf{W}_{k}\right)=1,$ (14e)
$\displaystyle x_{k}\geq 0,\forall k\in\mathcal{G}_{i},i\in\mathcal{I},$ (14f)

where $\mathbf{W}_{k}=\mathbf{w}_{k}\mathbf{w}^{H}_{k}$ and
$\mathbf{Z}_{k,i}=\mathbf{H}_{k}\mathbf{F}\mathbf{m}_{i}\mathbf{m}^{H}_{i}\mathbf{F}^{H}\mathbf{H}^{H}_{k}$.
Now, we define $\mathcal{P}^{\mathrm{hyb}}_{\mathrm{SDR},3}$ as (14) without
the non-convex constraint (14e). Furthermore, since the optimization of
$\mathbf{W}_{k}$ only affects $\mathrm{SINR}_{k}$, then
$\mathcal{P}^{\mathrm{hyb}}_{\mathrm{SDR},3}$ can be split into $K$ parallel
sub-problems $\mathcal{P}^{\mathrm{hyb}}_{\mathrm{SDR},3,k}$. For
completeness, we summarize our proposed scheme in Algorithm 1 with more
implementation details. Note that
${\mathcal{P}}^{\mathrm{hyb}}_{\mathrm{SDR},1}$,
$\mathcal{P}^{\mathrm{hyb}}_{\mathrm{SDR},2}$ and
$\mathcal{P}^{\mathrm{hyb}}_{\mathrm{SDR},3}$ can be recast as linear programs
and can therefore be efficiently solved in polynomial time by numerical
solvers. In our case, we employed CVX and SDPT3. In Algorithm 1, $g^{(t)}$
computes the total transmit power, whereas $\mathcal{K}^{(t)}$ counts the
number of users whose QoS requirement has been satisfied at iteration $t$. In
the initialization stage, every $\mathbf{w}_{k}$ is set in omnidirectional
reception mode, i.e., only one antenna is active. Similarly, every multicast
precoder $\mathbf{m}_{i}$ is in omnidirectional mode. Then, $\mathbf{F}$,
$\left\\{\mathbf{m}_{i}\right\\}^{G}_{i=1}$ and
$\left\\{\mathbf{w}_{k}\right\\}^{K}_{k=1}$ are alternately optimized for a
number of iterations $N_{\mathrm{iter}}$. At each iteration, the SDR-based
solutions are used to generate $N_{\mathrm{rand}}$ potentially befitting
solutions.

Algorithm 1 Proposed Iterative Approach

Define
Let
$g^{(t)}=\sum^{G}_{i=1}\left\|\mathbf{F}^{(t)}\mathbf{m}^{(t)}_{i}\right\|^{2}_{2}$
be the total transmit power. Let $\mathcal{K}^{(t)}$ be the number of users
that satisfy (5b) at iteration $t$. Initialize
Set $\mathbf{w}^{(0)}_{k}\leftarrow\left[1\leavevmode\nobreak\
\mathbf{0}\right]^{T},\forall k\in\mathcal{K}$,
$\mathbf{m}^{(0)}_{i}\leftarrow\left[1\leavevmode\nobreak\
\mathbf{0}\right]^{T},\forall i\in\mathcal{I}$. Set
$\widetilde{\mathcal{K}}\leftarrow 0$, $\tilde{g}\leftarrow 10^{5}$,
$t\leftarrow 1$.

Iterate
Set $C_{1}\leftarrow 0$, $C_{2}\leftarrow 0$ and
$\left\\{C_{3,k}\right\\}^{K}_{k=1}\leftarrow 0$. Optimize $\mathbf{F}$: Solve
${\mathcal{P}}^{\mathrm{hyb}}_{\mathrm{SDR},1}$ to obtain $\mathbf{D}^{(t)}$.
repeat Generate $\mathbf{u}$ with uniform distribution in the sphere
$\left\|\mathbf{u}\right\|^{2}_{2}=1$. Solve
$\widetilde{\mathcal{P}}^{\mathrm{hyb}}_{\mathrm{LS}}$ and compute
$\mathbf{F}^{(t)}$. if $\mathcal{K}^{(t)}>\widetilde{\mathcal{K}}$ or
$\left(\mathcal{K}^{(t)}=\widetilde{\mathcal{K}}\text{ and
}g^{(t)}\leq\tilde{g}\right)$ Assign $\mathbf{F}\leftarrow\mathbf{F}^{(t)}$,
$\tilde{g}\leftarrow g^{(t)}$,
$\widetilde{\mathcal{K}}\leftarrow\mathcal{K}^{(t)}$. end Increase the counter
$C_{1}$, $C_{1}\leftarrow C_{1}+1$. while $C_{1}\leq N_{\mathrm{rand}}$
Optimize $\mathbf{m}_{i}$: Solve $\mathcal{P}^{\mathrm{hyb}}_{\mathrm{SDR},2}$
and obtain $\left\\{\mathbf{M}^{(t)}_{i}\right\\}_{i=1}^{G}$. repeat Generate
$\widetilde{\mathbf{m}}^{(t)}_{i}\sim\mathcal{CN}\left(\mathbf{0},\mathbf{M}^{(t)}_{i}\right),\forall
i\in\mathcal{I}$. if $\mathcal{K}^{(t)}>\widetilde{\mathcal{K}}$ or
$\left(\mathcal{K}^{(t)}=\widetilde{\mathcal{K}}\text{ and
}g^{(t)}\leq\tilde{g}\right)$ Assign
$\left\\{\mathbf{m}_{i}\right\\}_{i=1}^{G}\leftarrow\left\\{\mathbf{m}^{(t)}_{i}\right\\}_{i=1}^{G}$,
$\tilde{g}\leftarrow g^{(t)}$,
$\widetilde{\mathcal{K}}\leftarrow\mathcal{K}^{(t)}$. end Increase the counter
$C_{2}$, $C_{2}\leftarrow C_{2}+1$. while $C_{2}\leq N_{\mathrm{rand}}$
Optimize $\mathbf{w}_{k}$: Solve $\mathcal{P}^{\mathrm{hyb}}_{\mathrm{SDR},3}$
and obtain $\left\\{\mathbf{W}^{(t)}_{k}\right\\}_{k=1}^{K}$. repeat for each
$k$ Generate
$\mathbf{w}^{(t)}_{k}\leftarrow\mathbf{W}^{(t)}_{k}\mathbf{v}_{k},\forall
k\in\mathcal{K}$ with $\mathbf{v}_{k}$ uniformly distributed in the sphere
$\left\|\mathbf{v}_{k}\right\|^{2}_{2}=1$. if
$\mathcal{K}^{(t)}>\widetilde{\mathcal{K}}$ or
$\left(\mathcal{K}^{(t)}=\widetilde{\mathcal{K}}\text{ and
}g^{(t)}\leq\tilde{g}\right)$ Assign
$\mathbf{w}_{k}\leftarrow\mathbf{w}^{(t)}_{k}$, $\tilde{g}\leftarrow g^{(t)}$,
$\widetilde{\mathcal{K}}\leftarrow\mathcal{K}^{(t)}$ end Increase the counter
$C_{3,k}$, $C_{3,k}\leftarrow C_{3,k}+1$ while $C_{3,k}\leq\lfloor
N_{\mathrm{rand}}/K\rfloor$
Until $t>N_{\mathrm{iter}}$

##  V Simulation Results

To evaluate our proposed design, we consider the geometric channel model with
$M_{p}=8$ propagation paths between the transmitter and each user. The maximum
receive power for all the users is $P^{\mathrm{max}}_{\mathrm{rx}}=10$ dBm,
and $\mathcal{F}$ consists of $L=8$ different phase shifts equally spaced in a
circle with radius $\sqrt{\delta}$. The regularization hyper-parameter in (4)
is defined as
$\beta=G^{3}N^{\mathrm{RF}}_{\mathrm{tx}}N_{\mathrm{tx}}N_{\mathrm{rx}}$. In
the following scenarios, we compare the performance of fully-digital and
hybrid precoders in terms of the number of decoded packets
$N_{\mathrm{packets}}$ and the required transmit power $P_{\mathrm{tx}}$ for
several configurations of $\gamma_{i}$, $N^{\mathrm{RF}}_{\mathrm{tx}}$,
$N_{\mathrm{tx}}$, $N_{\mathrm{rx}}$, $N_{\mathrm{rand}}$ and
$N_{\mathrm{iter}}$. In the sequel, we consider $K=60$ users evenly
distributed among $G=4$ multicast groups. All groups have the same SINR
requirements, i.e., $\gamma=\gamma_{i}\left(\forall i\in\mathcal{I}\right)$
and $\sigma^{2}=10$ dBm. All the numerical results show the average over 100
channel realizations.

####  V-1 Impact of $N^{\mathrm{RF}}_{\mathrm{tx}}$

The objective of this experiment is to evaluate the performance of the hybrid
precoder with respect to its fully-digital counterpart, when
$N^{\mathrm{RF}}_{\mathrm{tx}}=\left\\{5,6,7,8,9,10,11\right\\}$ is varied for
different $\gamma=\left\\{4,6,8\right\\}$. We assume that
$N_{\mathrm{tx}}=12$, $N_{\mathrm{rx}}=2$, $N_{\mathrm{iter}}=3$ and
$N_{\mathrm{rand}}=500$. The results for this setting are shown in Fig. 1,
where the hybrid precoder is denoted by HY and the fully-digital by FD. We
observe that for any specific $\gamma$, the number of decoded packets
$N_{\mathrm{packets}}$ augments when $N_{\mathrm{RF}}$ increases. By observing
$N_{\mathrm{packets}}$, it is evident that it suffices to only have
$N^{\mathrm{RF}}_{\mathrm{tx}}=8$ to yield a similar performance as that of
the fully-digital precoder (which requires
$N^{\mathrm{RF}}_{\mathrm{tx}}=12$). However, due to the reduced number of RF
chains in the hybrid precoder, interference management becomes more
challenging at the transmitter. Thus, we observe that in general, the hybrid
precoder requires more transmit power $P_{\mathrm{tx}}$ to attain a similar
performance. As more RF chains are added, the required transmit power
decreases as interference can be more effectively mitigated. It is also worth
noting an apparently abnormal behavior that, for instance, occurs when
$\gamma=6$ dB for $N^{\mathrm{RF}}_{\mathrm{tx}}=7$ and
$N^{\mathrm{RF}}_{\mathrm{tx}}=8$. Observe that $P_{\mathrm{tx}}$ is higher
when $N^{\mathrm{RF}}_{\mathrm{tx}}=8$ than when
$N^{\mathrm{RF}}_{\mathrm{tx}}=7$. However, $N_{\mathrm{packets}}$ is larger
by (approximately) one unit when $N^{\mathrm{RF}}_{\mathrm{tx}}=8$. The reason
is that some users experiencing high interference cannot be served when
$N^{\mathrm{RF}}_{\mathrm{tx}}=7$. Nevertheless, when an additional RF chain
is incorporated, the number of degrees of freedom increases and oftentimes a
subset of the uncatered users can be served at the expense of boosting the
transmit power. The maximum value of $N_{\mathrm{packets}}$ is $60$ as we
consider one transmitted message per user. Due to the highly interfering
scenario we have considered, not even the fully-digital precoder with
$N_{\mathrm{rand}}=500$ can guarantee $100$% successful reception.

$\gamma=4\leavevmode\nobreak\ dB$ $\gamma=6\leavevmode\nobreak\ dB$
$\gamma=8\leavevmode\nobreak\ dB$
$30$$40$$50$$60$$58.49$$58.34$$58.38$$49.35$$48.66$$41.9$$53.7$$51.3$$45.76$$56.72$$55.56$$51.94$$57.14$$56.49$$53.29$$57.67$$57.04$$56.67$$57.89$$57.69$$56.94$$58.21$$58.15$$57.96$$N_{\mathrm{packets}}$

$\gamma=4\leavevmode\nobreak\ dB$ $\gamma=6\leavevmode\nobreak\ dB$
$\gamma=8\leavevmode\nobreak\ dB$
$10$$20$$30$$40$$24.11$$27.67$$32.23$$31.1$$32.79$$31.44$$34.52$$37.07$$35.69$$32.35$$33.81$$35.94$$30.7$$37.43$$38.55$$29.8$$31.58$$36.17$$28.05$$32.16$$34.22$$25.7$$30.59$$34.71$$P_{\mathrm{tx}}$
[dBm]  FD $|N_{\mathrm{RF}}=12$ HY $|N_{\mathrm{RF}}=5$ HY
$|N_{\mathrm{RF}}=6$ HY $|N_{\mathrm{RF}}=7$ HY $|N_{\mathrm{RF}}=8$ HY
$|N_{\mathrm{RF}}=9$ HY $|N_{\mathrm{RF}}=10$ HY $|N_{\mathrm{RF}}=11$

Figure 1: Evaluation of the number of decoded packets and transmit power for
$N_{\mathrm{tx}}=12$ when $\gamma$ and $N^{\mathrm{RF}}_{\mathrm{tx}}$ are
varied. $1$ $2$ $3$ $4$ $5$
$20$$30$$40$$50$$60$$54.24$$59.16$$59.7$$59.92$$59.96$$36.15$$57.81$$59.19$$59.6$$59.75$$N_{\mathrm{rx}}$
$N_{\mathrm{packets}}$ Fully-digital (FD) Hybrid (HY) $1$ $2$ $3$ $4$ $5$
$15$$20$$25$$30$$35$$26.8$$25.45$$25.04$$24.08$$23.7$$24.86$$33.06$$33.49$$31.66$$29.37$$N_{\mathrm{rx}}$
$P_{\mathrm{tx}}$ [dBm]  Figure 2: Evaluation of the number of decoded packets
and transmit power when $N_{\mathrm{rx}}$ is varied.

####  V-2 Impact of $N_{\mathrm{rx}}$

The objective of this configuration is to observe the performance improvement
of $N_{\mathrm{packets}}$ and the importance of the multi-antenna architecture
at the receiver. We consider that $\gamma=5$ dB and $N_{\mathrm{iter}}=4$.
Since we vary $N_{\mathrm{rx}}$, the number of randomization
$N_{\mathrm{rand}}$ should scale with the dimensionality of
$\mathbf{H}_{k},\mathbf{F},\left\\{\mathbf{m}_{i}\right\\}^{G}_{i=1},\left\\{\mathbf{w}_{k}\right\\}^{K}_{k=1}$.
Thus, for this scenario, we select
$N_{\rm{rand}}=400+300\left(N_{\mathrm{tx}}+N_{\mathrm{rx}}-11\right)$ with
$N_{\mathrm{tx}}=12$ and $N_{\mathrm{rx}}=\left\\{1,2,3,4,5\right\\}$. For the
hybrid precoder, we assume that $N_{\mathrm{RF}}=8$. On the other hand, for
the fully-digital version $N_{\mathrm{tx}}=N^{\mathrm{RF}}_{\mathrm{tx}}$. The
results in Fig. 2 demonstrate that, with only $N_{\mathrm{rx}}=2$ antennas at
the receiver, it is possible to mitigate the interference and improve
$N_{\mathrm{packets}}$ considerably. The gain is more noticeable for the
hybrid precoder as $N_{\mathrm{packets}}$ improves by $60\%$. For the fully-
digital precoder, there is also a moderate gain of $9\%$. Moreover, the
average transmit power per successfully received message improves by $12.9\%$
and $16.8\%$ for the digital and hybrid precoders, respectively. It is evident
from this scenario that, at the transmitter side, $N_{\mathrm{packets}}$
cannot be further improved when the receivers operate with a single
omnidirectional antenna (i.e., $N_{\mathrm{rx}}=1$), as interference and
desired signals are equally amplified. However, when $N_{\mathrm{rx}}=2$, the
receivers can enforce limited selectivity by rejecting the unwanted
interference to a certain extent, thereby improving $N_{\mathrm{packets}}$.
Finally, we observe that $N_{\mathrm{packets}}$ for both types of precoders
are very similar when $N_{\mathrm{rx}}\geq 2$ although the consumed power
differs in $6-8$ dBm.

####  V-3 Impact of $N_{\mathrm{rand}}$ and $N_{\mathrm{iter}}$

The objective of this setting is to analyze the performance sensitivity of the
fully-digital and hybrid precoders to the selection of $N_{\mathrm{rand}}$ and
$N_{\mathrm{iter}}$. To this purpose, we consider $N_{\mathrm{rx}}=2$ and
$\gamma=5$ dB. For the fully-digital precoder, we assume that
$N_{\mathrm{tx}}=N^{\mathrm{RF}}_{\mathrm{tx}}=12$. On the other hand, the
hybrid precoder is endowed with $N^{\mathrm{RF}}_{\mathrm{tx}}=8$ and
$N_{\mathrm{tx}}=12$. We evaluate the performance variation when
$N_{\mathrm{iter}}=\left\\{1,2,3,4,5\right\\}$ and
$N_{\mathrm{rand}}=\left\\{1,10,25,50,75,100,500,1000\right\\}$. The results
in terms of $N_{\mathrm{packets}}$ and $P_{\mathrm{tx}}$ for both types of
precoder are shown in Fig. 3. We observe that in the fully-digital precoder
case, $N_{\mathrm{rand}}$ is more influential than $N_{\mathrm{iter}}$ since
$N_{\mathrm{packets}}$ improves noticeably when $N_{\mathrm{rand}}$ is
augmented, whereas a small improvement can be observed between the cases
$N_{\mathrm{iter}}=1$ and $N_{\mathrm{iter}}=2$. Conversely, for the hybrid
precoder, $N_{\mathrm{iter}}$ promotes performance gap reduction with respect
to the fully-digital implementation. The fully-digital precoder converges
faster since only $\left\\{\mathbf{m}_{i}\right\\}^{G}_{i=1}$ and
$\left\\{\mathbf{w}_{k}\right\\}^{K}_{k=1}$ are optimized. In the hybrid
precoder case, we need to design $\mathbf{F}$,
$\left\\{\mathbf{m}_{i}\right\\}^{G}_{i=1}$ and
$\left\\{\mathbf{w}_{k}\right\\}^{K}_{k=1}$ with even more limiting
constraints (finite-resolution constant-modulus phase shifts), thus requiring
more iterations to obtain an stable solution.

$N_{\rm{iter}}=1$ (FD) $N_{\rm{iter}}=2$ (FD) $N_{\rm{iter}}=3$ (FD)
$N_{\rm{iter}}=4$ (FD) $N_{\rm{iter}}=1$ (HY) $N_{\rm{iter}}=2$ (HY)
$N_{\rm{iter}}=3$ (HY) $N_{\rm{iter}}=4$ (HY)
$20$$40$$60$$42.85$$45.31$$46.43$$47.23$$32.4$$41.31$$44.73$$45.54$$51.68$$52.5$$53.19$$53.7$$39.78$$50.19$$51.92$$52.71$$53.59$$55$$55.8$$56.15$$40.12$$52.94$$53.49$$53.9$$55.18$$55.95$$56.5$$56.69$$42.03$$53.75$$54.52$$54.97$$56.08$$56.4$$56.74$$56.98$$42.76$$54.87$$55.32$$55.64$$56.3$$56.89$$57.07$$57.28$$42.84$$55.01$$55.18$$55.75$$57.49$$57.94$$58.47$$58.6$$43.55$$56.15$$56.73$$56.99$$58.25$$58.63$$58.76$$58.89$$43.44$$56.45$$57.39$$57.58$$N_{\mathrm{packets}}$
$N_{\rm{rand}}$ = $1$ $N_{\rm{rand}}$ = $10$ $N_{\rm{rand}}$ = $25$
$N_{\rm{rand}}$ = $50$ $N_{\rm{rand}}$ = $75$ $N_{\rm{rand}}$ = $100$
$N_{\rm{rand}}$ = $500$ $N_{\rm{rand}}$ = $1000$ $N=1$ $N=3$ $N=5$ $N=7$ $N=2$
$N=4$ $N=6$ $N=8$
$10$$20$$30$$23.46$$24.04$$24.22$$24.38$$22.87$$28.46$$30.69$$30.85$$24.84$$25.03$$25.07$$25.18$$23.81$$30.97$$31.93$$32.28$$25.47$$25.59$$25.62$$25.64$$23.69$$31.64$$32.14$$32.16$$25.88$$25.86$$25.8$$25.77$$24.44$$31.75$$32.18$$32.31$$26.01$$26.06$$25.98$$25.96$$24.57$$32.29$$32.94$$32.87$$26.04$$25.99$$25.91$$25.73$$24.39$$32.45$$33.04$$33.04$$26.59$$26.36$$25.86$$25.65$$24.13$$32.2$$33.07$$33.18$$26.64$$26.36$$26.14$$26.03$$24.22$$33.43$$33.18$$33.26$$P_{\mathrm{tx}}$
[dBm]  Figure 3: Evaluation of the number of decoded packets and transmit
power for $N_{\mathrm{tx}}=12$ when $N_{\mathrm{iter}}$ and
$N_{\mathrm{rand}}$ are varied.

Generation of co-channel users: In order to gain more understanding on the
kind of scenario we are dealing with, we show in Fig. 4 the histogram of
channel correlations for _(i)_ users that belong to the same group (intra-
cluster) and _(ii)_ users that belong to different groups (inter-cluster). In
the first case, the average channel correlation is 0.24 whereas in the second
case is 0.10. The mean angles of departure
$\left\\{\bar{\theta}^{\mathrm{AoD}}_{i}\right\\}^{G}_{i=1}$ for the multicast
groups are distributed in the range $\left[-80\degree,80\degree\right]$ with
angular spread $\sigma_{\mathrm{AoD}}=30\degree$. The mean angles of arrival
$\left\\{\bar{\theta}^{\mathrm{AoA}}_{k}\right\\}^{K}_{k=1}$ for each receiver
are uniformly distributed in the range $\left[-360\degree,360\degree\right]$
with angular spread of $\sigma_{\mathrm{AoA}}=60\degree$. Thus, for a given
user $k^{\prime}$ that belongs to group $G_{i^{\prime}}$, the paths will have
angles of departure and arrival in the ranges
$\bar{\theta}^{\mathrm{AoD}}_{i^{\prime}}\pm\sigma_{\mathrm{AoD}}$ and
$\bar{\theta}^{\mathrm{AoA}}_{i^{\prime}}\pm\sigma_{\mathrm{AoD}}$,
respectively. To shed more light on this aspect, Fig. 5 shows a particular
channel realization when $N_{\mathrm{tx}}=8$,
$N^{\mathrm{RF}}_{\mathrm{tx}}=4$, $N_{\mathrm{rx}}=2$, $K=4$, $G=4$,
$\bar{\theta}^{\mathrm{AoD}}_{1}=-60\degree$,
$\bar{\theta}^{\mathrm{AoD}}_{2}=-20\degree$,
$\bar{\theta}^{\mathrm{AoD}}_{3}=20\degree$,
$\bar{\theta}^{\mathrm{AoD}}_{4}=60\degree$ and
$\sigma_{\mathrm{AoD}}=5\degree$. Due to existence of multiple paths, the
transmit and receive beams are not fully aligned as expected in line-of-sight
scenarios. Thus, each of the users orients their receive power in specific
directions that are coherent with the most meaningful beams of the transmitter
beam-pattern. Also, note that secondary lobes at the receiver have been shaped
to minimize amplification of interfering signals.

$0$ $0.1$ $0.2$ $0.3$ $0.4$ $0.5$ $0.6$ $0.7$ 0.00 0.05 0.10 0.15 0.20
Probability  Intra-cluster Inter-cluster Figure 4: Channel correlation
histogram.

|  |
---
|
---
|  |

$-80^{\circ}$$-60^{\circ}$$-40^{\circ}$$-20^{\circ}$$0^{\circ}$$20^{\circ}$$40^{\circ}$$60^{\circ}$$80^{\circ}$$0$$0.1$$0.2$$0.3$Transmitter${\scriptsize\text{U}_{1}}$${\scriptsize\text{U}_{2}}$${\scriptsize\text{U}_{3}}$${\scriptsize\text{U}_{4}}$

$135^{\circ}$$180^{\circ}$$225^{\circ}$$\cdot 10^{-2}$User 1
(${\scriptsize\text{U}_{1}}$)

$135^{\circ}$$180^{\circ}$$225^{\circ}$$\cdot 10^{-2}$User 3
(${\scriptsize\text{U}_{3}}$)

$135^{\circ}$$180^{\circ}$$225^{\circ}$$\cdot 10^{-2}$User 2
(${\scriptsize\text{U}_{2}}$)

$135^{\circ}$$180^{\circ}$$225^{\circ}$$\cdot 10^{-2}$User 4
(${\scriptsize\text{U}_{4}}$)

Figure 5: Radiation patterns

##  VI Discussion

Co-channel users: We have considered a very challenging scenario throughout
our simulations in order to examine the operational limits of our design. We
can observe from Fig. 4 that intra-cluster and inter-cluster users are not
easily separable as a subset of them have similar channel correlations. In our
scenarios, this is determined by the selection of
$\left\\{\bar{\theta}^{\mathrm{AoD}}_{i}\right\\}^{G}_{i=1}$,
$\left\\{\bar{\theta}^{\mathrm{AoA}}_{k}\right\\}^{K}_{k=1}$,
$\sigma_{\mathrm{AoD}}$ and $\sigma_{\mathrm{AoA}}$. We notice that highly
correlated users (co-located users) belonging to different groups were the
most challenging to cater, specifically for the hybrid precoder whose
beamforming flexibility is limited.

Hybrid precoder design: Different from the majority of works in hybrid
precoding (either multi-user or multicast), the proposed design has no
knowledge of the optimal fully-digital precoder (as in e.g., [19]). Thus, our
proposed design is not obtained as an approximation of the optimal fully-
digital implementation. Without an optimal reference, the design becomes more
challenging.

Initial points: We have considered naive initializations for the optimization
parameters. We leave for future work the exploitation of AoA/AoD to infer more
befitting initializations and thus improve the performance of the scheme.

Fully-digital precoder design: The fully-digital implementation is obtained by
assigning $\mathbf{F}=\mathbf{I}$ and then optimizing alternately over
$\left\\{\mathbf{m}_{i}\right\\}^{G}_{i=1}$ and
$\left\\{\mathbf{w}_{k}\right\\}^{K}_{k=1}$.

Algorithm convergence: There is no theoretical evidence supporting the
convergence of Algorithm 1, essentially due to the non-convexity of the
problem. However, the proposed scheme exhibits an stable behavior for both
digital and hybrid precoders since the solutions do not vary significantly as
$N_{\mathrm{iter}}$ and $N_{\mathrm{rand}}$ increase beyond a certain limit.

Computational complexity: Neglecting the complexity owing to randomization and
obviating the insignificant complexity increase due to the inclusion of slack
parameters, the computational complexity of the proposed scheme when
$N_{\mathrm{iter}}=1$ is
$\mathcal{O}\left(\left(N^{\mathrm{RF}}_{\mathrm{tx}}N_{\mathrm{tx}}\right)^{6}+K\left(N^{\mathrm{RF}}_{\mathrm{tx}}N_{\mathrm{tx}}\right)^{2}\right)+\mathcal{O}\left(G^{3}\left(N^{\mathrm{RF}}_{\mathrm{tx}}\right)^{6}+KG\left(N^{\mathrm{RF}}_{\mathrm{tx}}\right)^{2}\right)+\mathcal{O}\left(K\left(N_{\mathrm{rx}}\right)^{6}+K\left(N_{\mathrm{rx}}\right)^{4}\right)$.

##  VII Conclusion

In this paper, we investigated the optimization of multi-group multicast
hybrid precoders in mmWave systems. Our proposed solution is based on the
alternating optimization, semidefinite relaxation and Cholesky matrix
factorization, where the digital precoder, analog phase shifts, and receive
combiners are optimized sequentially in an iterative manner. Furthermore, our
formulation allows the employment of an arbitrary number of phase shifts. It
was corroborated through extensive simulations that the hybrid precoder can
indeed attain similar performance as its fully-digital counterpart, even in
very challenging scenarios with high inter-cluster user correlation. In
addition, we demonstrate that having receivers with two antennas suffices to
improve the number of decoded packets. Thus, our proposed design achieves up
to $60\%$ gain.

##  VIII Acknowledgment

This work has been funded by the LOEWE initiative (Hessen, Germany) within the
German Research Foundation (DFG) in the Collaborative Research Center (CRC)
1053 - MAKI.

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