# A Primer on Rate-Splitting Multiple Access:
Tutorial, Myths, and Frequently Asked Questions

Bruno Clerckx  Fellow, IEEE Yijie Mao  Member, IEEE Eduard A. Jorswieck
Fellow, IEEE Jinhong Yuan  Fellow, IEEE David J. Love  Fellow, IEEE Elza Erkip
Fellow, IEEE and Dusit Niyato  Fellow, IEEE
(Invited Paper) Manuscript received August 25, 2022; accepted Dec 15, 2022.
This work has been supported in part by the National Nature Science Foundation
of China under Grant 62201347. (Corresponding author: Yijie Mao.) B. Clerckx
is with the Department of Electrical and Electronic Engineering at Imperial
College London, London SW7 2AZ, UK and with Silicon Austria Labs (SAL), Graz
A-8010, Austria (email: b.clerckx@imperial.ac.uk). Y. Mao is with the School
of Information Science and Technology, ShanghaiTech University, Shanghai
201210, China (e-mail: maoyj@shanghaitech.edu.cn). E. A. Jorswieck is with the
Institute for Communications Technology at TU Braunschweig, Brunswick, Germany
(email: e.jorswieck@tu-bs.de). J. Yuan is with the School of Electrical
Engineering and Telecommunications, University of New South Wales, Sydney, NSW
2052, Australia (e-mail: j.yuan@unsw.edu.au). D. J. Love is with the School of
Electrical and Computer Engineering, Purdue University, West Lafayette, IN,
USA (email: djlove@purdue.edu). E. Erkip is with the Electrical and Computer
Engineering Department, New York University Tandon School of Engineering,
Brooklyn, NY 11201, USA (email: elza@nyu.edu). D. Niyato is with the School of
Computer Engineering, Nanyang Technological University, Singapore (email:
dniyato@ntu.edu.sg).

###### Abstract

Rate-Splitting Multiple Access (RSMA) has emerged as a powerful multiple
access, interference management, and multi-user strategy for next generation
communication systems. In this tutorial, we depart from the orthogonal
multiple access (OMA) versus non-orthogonal multiple access (NOMA) discussion
held in 5G, and the conventional multi-user linear precoding approach used in
space-division multiple access (SDMA), multi-user and massive MIMO in 4G and
5G, and show how multi-user communications and multiple access design for 6G
and beyond should be intimately related to the fundamental problem of
interference management. We start from foundational principles of interference
management and rate-splitting, and progressively delineate RSMA frameworks for
downlink, uplink, and multi-cell networks. We show that, in contrast to past
generations of multiple access techniques (OMA, NOMA, SDMA), RSMA offers
numerous benefits: 1) enhanced spectral, energy and computation efficiency; 2)
universality by unifying and generalizing OMA, SDMA, NOMA, physical-layer
multicasting, multi-user MIMO under a single framework that holds for any
number of antennas at each node (SISO, SIMO, MISO, and MIMO settings); 3)
flexibility by coping with any interference levels (from very weak to very
strong), network loads (underloaded, overloaded), services (unicast,
multicast), traffic, user deployments (channel directions and strengths); 4)
robustness to inaccurate channel state information (CSI) and resilience to
mixed-critical quality of service; 5) reliability under short channel codes
and low latency. We then discuss how those benefits translate into numerous
opportunities for RSMA in over forty different applications and scenarios of
6G, e.g., multi-user MIMO with statistical/quantized CSI, FDD/TDD/cell-free
massive MIMO, millimeter wave and terahertz, cooperative relaying, physical
layer security, reconfigurable intelligent surfaces, cloud-radio access
network, internet-of-things, massive access, joint communication and jamming,
non-orthogonal unicast and multicast, multigroup multicast, multibeam
satellite, space-air-ground integrated networks, unmanned aerial vehicles,
integrated sensing and communications, grant-free access, network slicing,
cognitive radio, optical/visible light communications, mobile edge computing,
machine/federated learning, etc. We finally address common myths and answer
frequently asked questions, opening the discussions to interesting future
research avenues. Supported by the numerous benefits and applications, the
tutorial concludes on the underpinning role played by RSMA in next generation
networks, which should inspire future research, development, and
standardization of RSMA-aided communication for 6G.

###### Index Terms:

Rate-Splitting, Rate-Splitting Multiple Access, Next Generation Multiple
Access, Non Orthogonal Multiple Access, Space Division Multiple Access, Multi-
user MIMO, Interference Management, 6G

##  I Introduction

Communication systems are inherently multi-user systems. Multiple access (MA)
techniques play the crucial role of deciding how to make use of the resources
(e.g., time, frequency, power, antenna, and code) to serve those multiple
users. Next generation communications systems, e.g., 6G and beyond, will have
to cope with increasing demands for high throughput, reliability,
heterogeneity of quality of service (QoS), and massive connectivity to satisfy
the requirements of further-enhanced mobile broadband (FeMBB), extremely ultra
reliable and low-latency communication (eURLLC), ultra massive machine type
communication (umMTC), mixture thereof, and of new services such as integrated
sensing and communications (ISAC), integrated satellite-terrestrial, and
extended reality. To that end, it is critical to understand how next
generation MA can fulfill those demands and requirements by going beyond the
conventional orthogonal versus non-orthogonal discussion held in 5G.

Table I details the main abbreviations used throughout this work.

TABLE I: List of abbreviations. BC | Broadcast Channel | MU–LP | Multi-User Linear Precoding
---|---|---|---
CDMA | Code Division Multiple Access | MU-MIMO | Multi-User Multiple-Input Multiple-Output
CoMP | Coordinated Multi-Point | NOMA | Non-Orthogonal Multiple Access
CSI | Channel State Information | NOUM | Non-Orthogonal Unicast and Multicast
CSIT/R | Channel State Information at the Transmitter/Receiver | OFDMA | Orthogonal Frequency Division Multiple Access
C-RAN | Cloud-Radio Access Networks | OMA | Orthogonal Multiple Access
DoF | Degree-of-Freedom | QoS | Quality of Service
DPC | Dirty Paper Coding | RF | Radio Frequency
DPCRS | Dirty Paper Coded Rate-Splitting | RIS | Reconfigurable Intelligent Surfaces
D2D | Device-to-Device | RS | Rate-Splitting
EE | Energy Efficiency | RSMA | Rate-Splitting Multiple Access
(F)eMBB | (further-)enhanced Mobile Broadband Service | SC | Superposition Coding
ER | Ergodic Rate | SDMA | Space Division Multiple Access
ESR | Ergodic Sum Rate | SE | Spectral Efficiency
FDD | Frequency Division Duplex | SIC | Successive Interference Cancellation
FDMA | Frequency Division Multiple Access | SISO | Single-Input Single-Output
F-RAN | Fog-Radio Access Networks | SNR | Signal-to-Noise Ratio
HK | Han and Kobayashi | SWIPT | Simultaneous Wireless Information and Power Transfer
IC | Interference Channel | SIMO | Single-Input Multiple-Output
IRS | Intelligent Reconfigurable Surface | TDD | Time Division Duplex
LLS | Link-Level Simulation | TDMA | Time-Division Multiple Access
MA | Multiple Access | THz | TeraHertz
MAC | Multiple Access Channel | UAV | Unmanned Aerial Vehicles
MIMO | Multiple-Input Multiple-Output | (e)URLLC | (extremely) Ultra-Reliable Low-Latency Communication
MISO | Multiple-Input Single-Output | VLC | Visible Light Communication
MMF | Max-Min Fairness | V2X | Vehicle-to-Everything
(u)mMTC | (ultra) massive Machine-Type Communication | WSR | Weighted Sum Rate
mmWave | millimeter-Wave | ZFBF | Zero-Forcing Beamforming

###  I-A Beyond Orthogonal versus Non-Orthogonal

In the past decade, MA schemes have often been classified into two categories,
namely orthogonal (serving a single user per resource) versus non-orthogonal
(serving multiple users per resource). This classification has triggered the
question Is a non-orthogonal approach to MA better than an orthogonal
approach? and has led to the emergence of the rich literature on non-
orthogonal multiple access (NOMA) versus orthogonal multiple access (OMA) [1,
2]. This question was motivated by the claim that prior generations of
cellular communication networks are based on OMA serving multiple users on
orthogonal resources using time division multiple access (TDMA), frequency
division multiple access (FDMA), code division multiple access (CDMA), or
orthogonal frequency division multiple access (OFDMA) [3].

Classifying MA schemes into non-orthogonal vs orthogonal is however over-
simplistic as it does not fully reflect modern communication designs. Indeed,
those systems are equipped with multiple antennas, and spatial domain
processing in the form of multi-user linear precoding (MU–LP), space division
multiple access (SDMA), multi-user multiple-input multiple-output (MU-MIMO),
massive MIMO, is an integral part of 4G and 5G for twenty years. Importantly,
MU-LP/SDMA/MU-MIMO serves users in a non-orthogonal manner since multiple
users are allocated different precoders/beams in the same time-frequency grid
and interfere with each other in the same cell. Hence both 4G and 5G already
use a combination of orthogonal (in the time-frequency domains) and non-
orthogonal (in the spatial domain) approaches in the form of OFDMA combined
with SDMA/MU-MIMO/massive MIMO. It is well documented that non-orthogonality
can be beneficial in both single and multi-antenna settings [4, 5, 6].

A major drawback of such classification is that it tends to amalgamate many
different MA schemes under the non-orthogonal umbrella without contrasting
them or truly understanding the essence of those schemes. This has caused
unnecessary confusions and misunderstandings in the past few years [7]. For
instance, SDMA and power-domain NOMA111In the sequel, we simply use NOMA to
refer to power-domain NOMA. are two different non-orthogonal approaches to MA
but are fundamentally different. Indeed, SDMA and NOMA can be seen as two
extreme interference management strategies where the former treats
interference as noise and the latter fully decodes interference [8]. An
alternative interpretation of this difference is to note that SDMA (and other
forms of linearly precoded and non-linearly precoded MU-MIMO schemes) relies
on a transmit-side interference cancellation strategy while NOMA can be seen
as a receive-side interference cancellation strategy [9]. Such major
differences have however not been captured and not been addressed when
answering the aforementioned question, though they lead to drastic performance
and complexity gaps between the two schemes [7]. Hence, instead of contrasting
orthogonal vs non-orthogonal, a different classification should be considered
in next generation wireless networks. In this paper, we will show that the
fundamental question behind MA design should instead be how to manage multi-
user interference? Answering this question will shed the light on the
differences between non-orthogonal approaches to MA designs and on a new
classification of MA schemes based on how the interference is managed. Even
more importantly, this exercise will bring to light the powerful and newly
emerging Rate-Splitting Multiple Access (RSMA) for downlink and uplink
communications.

###  I-B Toward Rate-Splitting Multiple Access

RSMA refers to a broad class of multi-user schemes whose commonality is to
rely on the rate-splitting (RS) principle [10, 11, 12]. RS consists in
splitting a user message (e.g., information bits) into two or multiple parts
such that each of those parts can be decoded flexibly at one or multiple
receivers. A receiver would have to retrieve each part to reconstruct the
original message. A key benefit of RS and its message splitting capability is
to flexibly manage inter-user interference, as it will appear clear throughout
this paper.

Though the RS principle first appeared in the information theoretic literature
in the late 70’s and early 80’s, RS (and consequently the emerging RSMA) has
received a renewed interest in the past decade in the broader communication
community. What triggered this renewed interest is a different line of
research, seemingly unrelated to the MA literature at first, on understanding
the fundamental limits of robust interference management, i.e., how to manage
interference in a multi-user multi-antenna communications system in the
presence of imperfect channel state information at the transmitter (CSIT)
[13].

Conventional multi-user multi-antenna approaches such as SDMA/MU-MIMO heavily
rely on timely and highly-accurate CSIT. Unfortunately, in practice, CSIT is
always imperfect due to pilot reuse, channel estimation errors, pilot
contamination, limited and quantized feedback accuracy, delay/latency,
mobility (due to ever-increasing speeds of vehicle/trains/satellite/flying
objects and emerging applications as Vehicle-to-Everything), radio frequency
(RF) impairments (e.g., phase noise), inaccurate calibrations of RF chains,
subband level estimation [14]. Consequently SDMA/MU-MIMO are inherently non-
robust. The classical approach to dealing with this practical limitation takes
a “robustification” stance – techniques that have been developed under the
assumption of perfect CSIT are tweaked to account for imperfect CSIT [15, 16].

The challenge is that any CSIT inaccuracy results in a residual multi-user
interference that needs proper management instead of “robustification”.
Despite its significance in realistic wireless deployments, the fundamental
limits of multi-user multi-antenna communications with imperfect CSIT are
still an open problem, i.e., the capacity and capacity achieving schemes
remain to be found. Consequently, this lack of understanding of how to design
robust communication schemes has led to modern systems like 4G and 5G designs
to be fundamentally designed for perfect CSIT (using SDMA and treat
interference as noise) instead of being designed from scratch to be truly
robust to imperfect CSIT [14, 9].

Interestingly, we are now in a much better position to design truly robust
MIMO wireless networks accounting for imperfect CSIT and its resulting multi-
user interference [13, 14, 9]. It is indeed known that to benefit from
imperfect CSIT and tackle the multi-user interference, the transmitter should
take an RS approach that splits the messages into common and private parts,
encodes the common parts into a common stream, and private parts into private
streams and superposes in a non-orthogonal manner the common stream on top of
all private streams [14]. The common stream is decodable by all receivers,
while the private streams are to be decoded by their corresponding receivers
only. Such an approach is optimal from an information theoretic perspective
(Degrees-of-Freedom, DoF222The Degrees-of-Freedom (DoF), or multiplexing gain,
is a first-order approximation of the rate at high signal-to-noise ratio
(SNR). It can be viewed as the pre-log factor of the rate at high SNR and be
interpreted as the number or fraction of interference-free stream(s) that can
be simultaneously communicated to a user (or multiple users). The DoF achieved
depends on the communication strategy used. The larger the DoF, the faster the
rate increases with the SNR. Hence, ideally a communication strategy should
achieve the highest DoF possible. Readers are referred to [7] for more details
on various definitions used to assess the DoF performance of multiple access
schemes.) for downlink multi-user mutliple-input single-output (MISO) and MIMO
transmissions with imperfect CSIT [17, 18, 19, 20, 21, 22].

This literature opens various doors that have a major impact on MA designs.
First, since imperfect CSIT is more general than perfect CSIT333Perfect CSIT
is obtained by setting the channel estimation error in the imperfect CSIT
model to zero., finding efficient schemes for imperfect CSIT leads to
discovering a broader and general class of communication strategies that would
subsume perfect CSIT strategies as particular instances. Second, it gives
communication engineers clear and fundamentally grounded guidelines on how to
design robust schemes. Third, it provides refreshing and new thoughts about
low complexity non-orthogonal scheme that are applicable and beneficial even
in perfect CSIT settings. Fourth, it brings RS, originally developed for the
two-user single-antenna interference channel [10, 11], into MU-MIMO, which was
never investigated despite the rich literature on MU-MIMO schemes in the past
two decades [6]. Fifth, RS serves users in a non-orthogonal manner by
partially treating interference as noise and partially decoding interference.
This highlights the usefulness to depart from the extremes of (fully) treat
interference as noise (as in SDMA) and (fully) decode interference (as in
NOMA) in multi-antenna networks, but also the usefulness to bridge and unify
those two extremes [8]. Sixth, RS can be seen as a smart combination of
transmit-side and receive-side interference cancellation strategy where the
contribution of the common stream is adjusted to benefit from the best balance
between transmit and receive cancellation. This departs from the transmit-side
only and receive-side only interference cancellation strategies of SDMA (and
the rich literature on MU-MIMO) and NOMA, respectively [9].

The above design makes RS a fundamental building block of a powerful MA
framework for downlink and uplink communications, namely, RSMA [8, 23].

In the downlink, RSMA uses linearly or non-linearly precoded RS at the
transmitter to split each user message into one or multiple common messages
and a private message. The common messages are combined and encoded into
common streams for the intended users. Successive interference cancellation
(SIC) - or any other forms of joint decoding - is used at each user to
sequentially decode the intended common streams (and therefore decode part of
the interference). Such linearly or non-linearly precoded generalized RSMA has
been demonstrated to be a powerful framework to bridge, reconcile, and
generalize SDMA, NOMA, OMA, and physical-layer multicasting and further boost
system spectral and energy efficiencies for downlink transmissions with both
perfect and partial CSIT [8, 24, 25].

In the uplink, users split their message into multiple streams and allocate
proper transmit power to each stream. The receiver at the base station
performs SIC to retrieve each stream and reconstruct the original messages. By
splitting messages, inter-user interference can be dynamically managed without
utilizing time sharing among users to achieve the capacity. Here again, NOMA
is a subset of RSMA and is obtained when user messages are not split. This
message splitting property can enable core services where users have
intermittent access behaviour such as URLLC and mMTC, as well as enable
different combinations of core services to serve users with heterogeneous
profiles [23, 26].

###  I-C Challenges and Opportunities for RSMA

In our view, MA schemes will have to address new challenges and requirements
in next generation networks:

  * •

Efficient: Time, frequency, power, spatial, and code-domain resources should
be used to serve users in the most efficient way, from both spectral
efficiency and energy efficiency perspectives, while enhancing QoS and
fairness across devices and services (e.g., unicast, multicast) and making the
best use of the available computational resources (transmitter and receiver
complexity). This calls for performance evaluations of MA schemes beyond OMA
vs NOMA [7].

  * •

Universal: Simpler is better, i.e., a single unified and general MA scheme
would be easier to implement and optimize than a combination of multiple MA
schemes, each optimized for specific conditions. This calls for a deeper
understanding of non-orthogonality in MA designs and of how non-orthogonal
approaches to MA schemes relate to each other [7, 24].

  * •

Flexible: Wireless networks are dynamic and MA schemes should be flexible or
versatile enough to cope with various interference levels, network density and
load (underloaded, overloaded), topology, services, user distribution, channel
directions and strengths in general deployments. This calls for MA schemes
that depart from those two extreme interference management strategies, namely
fully treat interference as noise and fully decode interference [8].

  * •

Robust and Resilient: MA designs need to depart from the conventional
aforementioned “robustification” stance and adopt a true robustness to
practical scenarios subject to imperfect CSIT [14]. Aside being robust, MA
schemes need to also become increasingly resilient to emerging services
involving mixed-criticality, i.e., services with different priorities. This is
particularly relevant for applications in safety-critical contexts such as
driverless traffic [27].

  * •

Low Latency and Reliable: Providing reliable and low-latency communications
for intelligent transportation and industrial automation is key in next
generation communications. Critical sources of latency relate to link
establishment, packet re-transmissions, and data blocklength. Hence, an MA
scheme needs to be able to perform reliably under short channel codes and
decrease the number of re-transmissions [28, 29].

RSMA uniquely appeared in recent years to fulfill all these requirements [25],
thanks to its inherent message splitting capability, which is not featured in
any other MA schemes. This split capability provides several benefits at once
as it allows to: 1) partially decode interference and partially treat
interference as noise (hence its efficiency, flexibility, reliability, and
resilience), 2) reconcile the two extreme strategies of interference
management and multiple MA schemes into a single framework (hence its
generality/universality), 3) achieve the optimal DoF in practical scenarios
subject to imperfect CSIT (hence its robustness). This contrasts with
conventional approaches like OMA, SDMA, and NOMA, that achieve only some of
the aforementioned features [7]. Indeed, SDMA has low complexity and works
well in perfect CSIT conditions especially in underloaded regimes, while
downlink multi-antenna NOMA is inefficient in general as it incurs a severe
DoF loss despite an increased receiver complexity due to an inefficient use of
SIC receivers [7]. OMA, SDMA, and NOMA are not general as they are suited for
particular propagation conditions [24]. Similarly they are not as flexible as
RSMA since they rely on eliminating interference by orthogonalization (OMA),
treat interference as noise (SDMA), and decode interference (NOMA), which are
suboptimal strategies in general settings [8, 30]. Though NOMA can be made
more robust than SDMA in the presence of imperfect CSIT, the DoF of NOMA and
SDMA are both suboptimal, therefore incurring a rate loss compared to RSMA
[7]. Similarly SDMA has also been shown to be less resilient than RSMA [31].
Finally, SDMA and NOMA are less reliable with finite block code lengths [32,
33, 34].

###  I-D Objectives and Contributions

It is worth to point out (again) that SDMA, NOMA, and RSMA are all non-
orthogonal approaches to MA design. However, classifying them all as non-
orthogonal does not give us a clue as to where the difference in efficiency,
universality, flexibility, robustness and resilience, reliability and low
latency comes from. Answering the fundamental question how to manage multi-
user interference? is crucial because it helps us to design efficient MA
schemes, to reveal how MA schemes manage interference, and identify and
predict under what conditions a given MA scheme may be efficient, universal,
flexible, robust, resilient, reliable and have low latency.

In this paper, we provide a tutorial on RSMA and address common myths and
frequently asked questions. To that end, we make the following contributions:

  * •

We start this tutorial by departing from the orthogonal vs non-orthogonal
classification and rather classify schemes as a function of how they manage
interference. We go back to the basics of RS proposed for the two-user
interference channel, and show step by step, how to build downlink, uplink,
and multi-cell RSMA and its various schemes. We also show how interference
management and MA designs are closely related and how OMA, SDMA, NOMA, and
RSMA schemes differ in terms of how to manage interference. We then explain
that by understanding the interference management capability of a scheme, we
can understand the drawbacks and benefits of the MA schemes.

  * •

We generalize the downlink, uplink, and multi-cell RSMA frameworks to MIMO
settings with multiple antennas at all nodes. This builds upon recent efforts
to design MIMO RSMA [35], contrast with recent survey and works [25, 7] that
were primarily limited to MISO settings, and confirm that the generality and
universality of RSMA are not limited to single-input single-output (SISO),
MISO, or single-input multiple-output (SIMO) settings, but hold for general
MIMO settings. Consequently, recent MIMO NOMA schemes [36, 37] are shown
particular instances of MIMO RSMA.

  * •

We demonstrate that, in contrast to previous generation MA schemes (as OMA,
SDMA, NOMA), RSMA is spectrally and energy efficient, universal, flexible,
robust and resilient, reliable and low latency. Those unique features of RSMA
are discussed in over 40 applications of RSMA in different systems, scenarios,
and services. We briefly describe how RSMA is beneficial for each of those
applications to demonstrate how powerful RSMA is for next generation networks.

  * •

We discuss numerous myths/misunderstandings and frequently asked questions
about RSMA that we have personally witnessed. Myths and questions cover a wide
range of topics, ranging from RSMA relationship with NOMA, gain over MU-MIMO
and massive MIMO, role and content of common streams, standardization status
of RSMA, implementation complexity of RSMA, interplay with other techniques,
role of machine learning in RSMA, etc. Those myths and frequently asked
questions about RSMA nicely complement the tutorial part.

This paper also differs from the recent survey paper on RSMA [25] by providing
a tutorial flavour instead of survey, by diving further into the key role of
interference management, by addressing myths and frequently asked questions,
and by expanding the framework to general MIMO settings. It also differs from
[26, 38, 39] that give an overview of RSMA and focus on the specific interplay
between RSMA and ISAC and reconfigurable intelligent surfaces (RIS).

###  I-E Organization and Notations

The remainder of this paper is organized as follows. Section II introduces
fundamentals about how to manage interference, delineates the RS strategy for
the 2-user interference channel, relates the problem of MA design with
interference management, and draws important observations. Section III builds
upon important lessons drawn from previous section and introduces various
architectures of 2-user RSMA for both downlink and uplink for general MIMO
settings at all nodes. Important lessons about the relationships between
existing MA schemes and RSMA are summarized. Section IV extends the RSMA
design to general $K$-user settings for downlink, uplink and multi-cell
deployments. Section V shows how the RSMA architectures find applications in
40 different scenarios relevant in 6G. The breadth and depth of those
applications and scenarios showcase how powerful and underpinning RSMA is for
6G. Section VI debunks some myths that have appeared in the RSMA literature.
Section VII presents answers to several common questions that are often asked
about RSMA. Section VIII concludes this tutorial overview.

Notations: Bold upper and lower case letters denote matrices and column
vectors, respectively. $(\cdot)^{T}$, $(\cdot)^{H}$, $|\cdot|$, $\|\cdot\|$,
$\mathbb{E}\\{\cdot\\}$, and $\mathrm{tr}(\cdot)$ represent the transpose,
Hermitian, absolute value, Euclidean norm, expectation, and trace operators,
respectively. $\mathcal{CN}(0,\sigma^{2})$ denotes the circularly symmetric
complex Gaussian (CSCG) distribution with zero mean and variance $\sigma^{2}$.

##  II Key Question behind Multiple Access Design: How to Manage Interference?

In this section, we present fundamentals about how to manage interference. To
that end, we focus on an example of a symmetric two-user interference channel,
which is the simplest scenario that enables to capture the essence of
interference management. We then use that simple example to draw major lessons
for MA design.

###  II-A Interference Channel

Figure 1: Two-user symmetric Gaussian interference channel.

Let us consider a two-user symmetric Gaussian interference channel (IC) of
Fig. 1. We have four nodes, two transmitters (Tx) and two receivers (Rx), each
equipped with a single antenna. Tx-1 (resp. Tx-2) wants to transmit a message
$W_{1}$ (resp. $W_{2}$) to Rx-1 (resp. Rx-2). To simplify the number of
parameters and obtain some insight into the role of interference, we assume
some symmetry in that the direct channel from a Tx to its intended Rx (Tx-1
$\rightarrow$ Rx-1, Tx-2 $\rightarrow$ Rx-2) is $h_{\mathrm{d}}$ and the cross
channel (Tx-1 $\rightarrow$ Rx-2, Tx-2 $\rightarrow$ Rx-1) is
$h_{\mathrm{c}}$. Hence, the stronger $h_{\mathrm{c}}$ compared to
$h_{\mathrm{d}}$, the stronger the interference from Tx-$k$ to Rx-$j$, $j\neq
k$.

A few strategies come to mind to manage this interference:

  * •

Orthogonalize: Tx-1 and Tx-2 transmit on orthogonal resources (e.g., time or
frequency) so that that they do not interfere with each other at all. This is
suboptimal since this strategy does not care about how weak the interference
is.

  * •

Treat Interference as (additive white Gaussian) Noise: This is a natural
approach whenever $h_{\mathrm{c}}$ is very weak compared to $h_{\mathrm{d}}$
since the interference created by Tx-$k$ would be buried in the noise of
Rx-$j$, $j\neq k$. The drawback of this strategy is that the interference
actually carries information and has a structure that could potentially be
exploited in mitigating its effect [30].

  * •

Decode Interference: Fully decode the interference at the receivers is
especially relevant and actually optimal when the interference is strong
enough that $h_{\mathrm{c}}$ is large compared to $h_{\mathrm{d}}$ [40]. In
that case, Rx-$k$ has a better reception of Tx-$j$’s signal than the intended
receiver Rx-$j$, $k\neq j$.

However, we can do more than those seemingly three different strategies by
adopting a rate-splitting (RS) approach [10, 11, 30]. RS splits the
transmitted information into two parts, namely a common part to be decoded by
both receivers and a private part to be decoded only by the intended receiver.
The key benefit of RS is that by enabling a receiver to decode the common
part, part of the interference is cancelled off, while the remaining private
part from the other transmitter is treated as noise. This enables RS (by
properly adjusting the power to the common and private parts) to partially
decode interference and partially treat the remaining interference as noise,
therefore bridging, unifying, and actually outperforming the two extremes of
treat interference as noise and decode interference, as it will appear clearer
in the sequel.

###  II-B Rate-Splitting for Two-User Interference Channel

Figure 2: RS for two-user SISO IC (HK scheme) [11].

We adopt an RS strategy444In the two-user IC, RS is also known as Han-
Kobayashi strategy [11]. to split each transmitter message $W_{k}$ into two
parts denoted as a common part $W_{\mathrm{c},k}$ and a private part
$W_{\mathrm{p},k}$ as per Fig. 2 [10, 11, 30]. On the one hand, a codebook
shared between both transmitters555This is not an issue in modern systems
since, for example, in an 4G Long Term Evolution (LTE)/5G New Radio (NR)
system, all codebooks are shared since all users use the same family of
modulation and coding schemes (MCS) specified in the standard. is used to
encode the common parts $W_{\mathrm{c},k}$, $k=1,2$, into the common stream
$s_{\mathrm{c},k}$, $k=1,2$, such that they are decodable by both receivers.
The private parts $W_{\mathrm{p},k}$, $k=1,2$ are on the other hand encoded
into private streams $s_{k}$, $k=1,2$, constructed from independent codebooks.
Tx-$k$ then transmits the superposition of its common stream
$s_{\mathrm{c},k}$ and private stream $s_{k}$ with proper power allocation to
both streams as

$x_{k}=\sqrt{P_{\mathrm{c}}}s_{\mathrm{c},k}+\sqrt{P_{k}}s_{k}.$ (1)

By defining $\mathbf{s}_{k}=[s_{\mathrm{c},k},s_{k}]^{T}$ and assuming that
$\mathbb{E}[\mathbf{s}_{k}\mathbf{s}_{k}^{H}]=\mathbf{I}$, the average
transmit (sum) power constraint at each transmitter is $P$ with
$P_{\mathrm{c}}$ and $P_{k}=P-P_{\mathrm{c}}$ being the power allocated to the
common and private stream respectively. Let us also denote by
$t=\frac{P_{\mathrm{c}}}{P}$ the fraction of the transmit power allocated to
the common stream.

The received signals at both receivers can be written as

$\displaystyle y_{1}=h_{\mathrm{d}}x_{1}+h_{\mathrm{c}}x_{2}+n_{1},$ (2)
$\displaystyle y_{2}=h_{\mathrm{c}}x_{1}+h_{\mathrm{d}}x_{2}+n_{2},$ (3)

where $n_{k}\sim\mathcal{CN}(0,\sigma_{n,k}^{2})$ is the additive white
Gaussian noise (AWGN). Without loss of generality, we assume the variance
$\sigma_{n,k}^{2}=1$. We also assume perfect CSIT and perfect CSI at the
receivers (CSIR).

Each Rx decodes the common streams and its intended private stream as
illustrated in Fig. 2. To that end, each Rx jointly decodes the common streams
into messages $\widehat{W}_{\mathrm{c},1}$ and $\widehat{W}_{\mathrm{c},2}$ by
treating the private streams as noise. Then Rx-$k$ cancels the common streams
from the received signal and decodes the intended private stream $s_{k}$ into
message $\widehat{W}_{\mathrm{p},k}$ by treating the other private stream
$s_{j}$, $j\neq k$, as noise. Rx-$k$ finally reconstructs the original message
by recombining $\widehat{W}_{\mathrm{p},k}$ and $\widehat{W}_{\mathrm{c},k}$
into $\widehat{W}_{k}$, which is the same as $W_{k}$ in the absence of any
decoding error. By doing so, Rx-$k$ has partially decoded the message of
Rx-$j$, namely $\widehat{W}_{\mathrm{c},j}$, and therefore cancelled off part
of the interference.

Because of symmetry, the rates of the common streams $s_{\mathrm{c},1}$ and
$s_{\mathrm{c},2}$ are the same and are simply denoted as $R_{\mathrm{c}}$.
Similarly, the rates of the private streams $s_{1}$ and $s_{2}$ are also the
same and are denoted as $R_{\mathrm{p}}$. Assuming Gaussian signaling and
infinite blocklength, $R_{\mathrm{c}}$ must satisfy the following inequalities
at Rx-1 (and equivalently at Rx-2 due to symmetry)

$\displaystyle R_{\mathrm{c}}$
$\displaystyle\leq\log_{2}\left(1+\frac{tP\left|h_{\mathrm{d}}\right|^{2}}{1+I}\right),$
(4) $\displaystyle R_{\mathrm{c}}$
$\displaystyle\leq\log_{2}\left(1+\frac{tP\left|h_{\mathrm{c}}\right|^{2}}{1+I}\right),$
(5) $\displaystyle 2R_{\mathrm{c}}$
$\displaystyle\leq\log_{2}\left(1+\frac{tP\left[\left|h_{\mathrm{d}}\right|^{2}+\left|h_{\mathrm{c}}\right|^{2}\right]}{1+I}\right),$
(6)

where
$I=\left(1-t\right)P\left[\left|h_{\mathrm{d}}\right|^{2}+\left|h_{\mathrm{c}}\right|^{2}\right]$.
Inequalities (4), (5), and (6) originate from the fact that the common streams
are jointly decoded first at each receiver by treating the private streams as
noise. Those three inequalities are obtained by writing the rate constraints
of a two-user multiple access channel (MAC) formed by two virtual transmitters
sending respectively $s_{\mathrm{c},1}$ and $s_{\mathrm{c},2}$ to Rx-1 (or 2
due to symmetry) subject to the additional noise power $I$ created by the
private streams $s_{1}$ and $s_{2}$. Hence, (4) refers to the individual rate
constraint of decoding $s_{\mathrm{c},1}$ at Rx-1 by treating $s_{1}$ and
$s_{2}$ as noise. Similarly, (5) refers to the individual rate constraint of
decoding $s_{\mathrm{c},2}$ at Rx-2 by treating $s_{1}$ and $s_{2}$ as noise.
The sum-rate constraint writes as (6).

The rate $R_{\mathrm{p}}$ of the private streams in (7) is obtained by
noticing that a private stream is decoded after cancelling the common streams
while treating the other interfering private stream as noise. We can therefore
write

$R_{\mathrm{p}}\leq\log_{2}\left(1+\frac{\left(1-t\right)P\left|h_{\mathrm{d}}\right|^{2}}{1+\left(1-t\right)P\left|h_{\mathrm{c}}\right|^{2}}\right),$
(7)

where the presence of $\left(1-t\right)P\left|h_{\mathrm{c}}\right|^{2}$ at
the denominator expresses the interference from the private stream of Tx-$k$
owed to Rx-$j$ private stream.

Combining (4)-(6) and (7), the achievable (symmetric) rate in this two-user IC
is written as the sum of the private and common rates
$R_{\mathrm{sym}}=R_{\mathrm{p}}+R_{\mathrm{c}}$ in (8). The terminology rate-
splitting appears clearly here where we note that $R_{\mathrm{sym}}$ is split
into two parts, namely, the private rate $R_{\mathrm{p}}$ and the common rate
$R_{\mathrm{c}}$. The benefit of this RS architecture is the ability to adjust
the power allocation to the common and private streams through parameter $t$
(and therefore the amount of information to be carried by the common and
private streams) to maximize $R_{\mathrm{sym}}$ depending on the channel
strengths of $h_{\mathrm{d}}$ and $h_{\mathrm{c}}$.

$R_{\mathrm{sym}}=\log_{2}\left(1+\frac{\left(1-t\right)P\left|h_{\mathrm{d}}\right|^{2}}{1+\left(1-t\right)P\left|h_{\mathrm{c}}\right|^{2}}\right)+\min\left\\{\log_{2}\left(1+\frac{tP\left|h_{\mathrm{d}}\right|^{2}}{1+I}\right),\log_{2}\left(1+\frac{tP\left|h_{\mathrm{c}}\right|^{2}}{1+I}\right),\frac{1}{2}\log_{2}\left(1+\frac{tP\left[\left|h_{\mathrm{d}}\right|^{2}+\left|h_{\mathrm{c}}\right|^{2}\right]}{1+I}\right)\right\\}$
(8)

###### Remark 1

The terminology common and private is taken from the information theory
literature. In simple words, common (also sometimes called public) simply
refers to a message or stream that is to be decoded by multiple users though
the content of the message is not necessarily intended to all those users.
Indeed, $s_{\mathrm{c},k}$, $k=1,2$, are decoded by both receivers though the
content of $s_{\mathrm{c},k}$ is only intended to Rx-$k$. Common also
contrasts with a multicast message that is a message decoded by multiple users
but that is genuinely intended to all those users. Private on the other hand
refers to a message or stream that is only decoded by its intended receiver
and is treated as noise at other receivers. Indeed, $s_{k}$ is decoded only by
Rx-$k$ and is treated as noise by Rx-$j$ with $j\neq k$. Messages
$W_{k},k=1,2$, are unicast messages (each intended to a single user), but they
are split into a common and private parts for interference management
benefits.

###  II-C Interference Regimes

We can identify four interference regimes depending on the relative
strengths666In the asymmetric IC, the interference regimes take a more
complicated form [30]. of $h_{\mathrm{d}}$ and $h_{\mathrm{c}}$:

Very weak ($\left|h_{\mathrm{c}}\right|\ll\left|h_{\mathrm{d}}\right|$):
Whenever $\left|h_{\mathrm{c}}\right|$ is much smaller than
$\left|h_{\mathrm{d}}\right|$, the maximization of $R_{\mathrm{sym}}$ ends up
allocating all the transmit power to the private stream such that $t=0$ and
$R_{\mathrm{sym}}=\log_{2}\left(1+\frac{P\left|h_{d}\right|^{2}}{1+P\left|h_{c}\right|^{2}}\right)$.
In such a regime, message $W_{k}$ is entirely encoded in the private stream
$s_{k}$ (i.e., no splitting of the messages occurs) such that any interference
from Tx-$k$ to Rx-$j$ is treated as noise. This corresponds to the regime
where the Treat Interference as Noise strategy is optimal.

Weak ($\left|h_{\mathrm{c}}\right|\leq\left|h_{\mathrm{d}}\right|$): As the
strength of $h_{\mathrm{c}}$ increases relatively to $h_{\mathrm{d}}$ but as
long as $\left|h_{\mathrm{c}}\right|$ remains smaller than
$\left|h_{\mathrm{d}}\right|$, one enters the weak interference regime where
splitting the messages becomes necessary to maximize $R_{\mathrm{sym}}$. In
that regime, inequality (4) is inactive (since r.h.s. of (5) < r.h.s. of (4))
and $t>0$ is to be chosen, i.e., a non-zero power is allocated to the common
stream. There exists a tradeoff between achieving a large private rate and
minimizing the interference caused to the other receiver. A qualitative and
insightful way of allocating power to the common stream is to choose $t$ so
that the interference level caused by the private stream has roughly the same
level as the other receiver’s noise level, i.e., from (7) choose $t$ such that
$\left(1-t\right)P\left|h_{\mathrm{c}}\right|^{2}\approx 1$ or equivalently
$t\approx\frac{P\left|h_{\mathrm{c}}\right|^{2}-1}{P\left|h_{\mathrm{c}}\right|^{2}}$
[30]. By doing so, the interference caused by a private stream has little
impact on the other receiver’s performance (compared to the impairments
already caused by the noise). At the same time, it does not prevent each
transmitter from experiencing a relatively large private rate as long as
$\left|h_{\mathrm{c}}\right|\leq\left|h_{\mathrm{d}}\right|$.

Strong
($\left|h_{\mathrm{d}}\right|^{2}\leq\left|h_{\mathrm{c}}\right|^{2}\leq\left|h_{\mathrm{d}}\right|^{2}\big{(}1+P\left|h_{\mathrm{d}}\right|^{2}\big{)}$):
As the strength of $h_{c}$ further increases relatively to $h_{\mathrm{d}}$
and enters the regime where $\left|h_{\mathrm{c}}\right|$ is larger than
$\left|h_{\mathrm{d}}\right|$, each receiver is able to decode both the
interfering signal and the desired signal by performing for instance SIC or
joint decoding. In other words, $R_{\mathrm{sym}}$ is maximized by choosing
$t=1$ and messages $W_{k}$, $k=1,2$, are entirely encoded in the common
streams $s_{\mathrm{c},k}$, respectively, so that they are both decoded by
both receivers (i.e., there are no private streams in this regime and $I=0$).
With $\left|h_{\mathrm{c}}\right|\geq\left|h_{\mathrm{d}}\right|$, inequality
(5) is inactive and
$R_{\mathrm{sym}}=\frac{1}{2}\log_{2}\big{(}1+P\big{[}\left|h_{d}\right|^{2}+\left|h_{c}\right|^{2}\big{]}\big{)}$
as long as
$\frac{1}{2}\log_{2}\big{(}1+P\big{[}\left|h_{\mathrm{d}}\right|^{2}+\left|h_{\mathrm{c}}\right|^{2}\big{]}\big{)}\leq\log_{2}\big{(}1+P\left|h_{d}\right|^{2}\big{)}$,
i.e., $P\left|h_{\mathrm{d}}\right|^{2}\leq
P\left|h_{\mathrm{c}}\right|^{2}\leq
P\left|h_{\mathrm{d}}\right|^{2}\big{(}1+P\left|h_{\mathrm{d}}\right|^{2}\big{)}$.
In other words, the sum-rate constraint (6) is active and forces each
transmitter to transmit at a rate smaller than
$\log_{2}\big{(}1+P\left|h_{d}\right|^{2}\big{)}$, i.e., smaller than the rate
achievable without any interference. It is worth noting that
$\frac{1}{2}\log_{2}\big{(}1+P\big{[}\left|h_{\mathrm{d}}\right|^{2}+\left|h_{\mathrm{c}}\right|^{2}\big{]}\big{)}\leq\log_{2}\big{(}1+P\left|h_{\mathrm{d}}\right|^{2}\big{)}$
can equivalently be written as
$\log_{2}\big{(}1+\frac{P\left|h_{\mathrm{c}}\right|^{2}}{1+P\left|h_{\mathrm{d}}\right|^{2}}\big{)}\leq\log_{2}\big{(}1+P\left|h_{\mathrm{d}}\right|^{2}\big{)}$,
which expresses that when Rx-$k$ performs SIC to decode the interfering signal
$s_{\mathrm{c},j}$ ($j\neq k$) before decoding $s_{\mathrm{c},k}$, the
decodability of $s_{\mathrm{c},j}$ at Rx-$k$ puts a constraint on the
transmission rate of $s_{\mathrm{c},j}$ at Tx-$j$.

Very strong
($\left|h_{\mathrm{c}}\right|^{2}\geq\left|h_{\mathrm{d}}\right|^{2}\big{(}1+P\left|h_{\mathrm{d}}\right|^{2}\big{)}$):
In this regime, the interference link is even stronger such that
$\frac{1}{2}\log_{2}\big{(}1+P\big{[}\left|h_{d}\right|^{2}+\left|h_{c}\right|^{2}\big{]}\big{)}\geq\log_{2}\big{(}1+P\left|h_{d}\right|^{2}\big{)}$
and the sum-rate constraint (6) becomes inactive. In other words, when
performing SIC, the decodability of $s_{\mathrm{c},j}$ at Rx-$k$ does not
restrict the transmission rate of $s_{\mathrm{c},j}$ at transmitter $j$.
Instead the rates are only limited by the direct links and each transmitter
can transmit at a rate
$R_{\mathrm{sym}}=\log_{2}\big{(}1+P\left|h_{d}\right|^{2}\big{)}$ equal to
the one achievable without any interference. The strong and very strong
interference regimes correspond to the regimes where the Decode Interference
strategy is optimal.

What distinguishes the weak interference regime from the strong (and very
strong) regime is that the interference in the former is not strong enough so
that only a part of the interference (through the common message) can be
decoded by a receiver while the interference in the latter is so strong that
the receiver can fully decode the interference. Consequently, moving from the
very weak interference regime to the strong interference regime, the messages
are respectively encoded into private streams only, a mixture of common and
private streams, and into common streams only as further summarized in Table
II.

TABLE II: Messages-to-streams mapping in the two-user IC. | $s_{k}$ | $s_{\mathrm{c},k}$
---|---|---
Very weak | $\,\,\quad\quad W_{1}\quad\quad\,\,$ | –
Weak | $W_{\mathrm{p},k}$ | $W_{\mathrm{c},k}$
Strong/Very strong | – | $W_{2}$
| decoded by its intended Rx-$k$ and | decoded by
| treated as noise by Rx-$j$ |  both Rx

Though the capacity of the two-user IC remains unknown, RS can achieve a rate
that is within a single bit per second per hertz (bit/s/Hz) of the capacity
for all values of the channel parameters (even in the non-symmetric case) and
is information-theoretically optimal at asymptotic high SNR regimes [30].

Figure 3: Symmetric rate versus INR/SNR.
INR=$P\left|h_{\mathrm{c}}\right|^{2}$,
SNR=$P\left|h_{\mathrm{d}}\right|^{2}$=1000.

Fig. 3 illustrates the symmetric rate of the various interference management
strategies using a concrete numerical example. The x-axis is the ratio between
the interference-to-noise ratio (INR) and SNR; hence the higher INR/SNR, the
stronger the interference. The different interference regimes are clearly
visible. We note how RS can softly bridge all strategies and outperform them
in the weak interference regime777Note that if the two-user IC is used to
represent a two-cell network, the inter-cell interference level is unlikely to
be in the strong regime since a user is associated with its closest base
station.. In contrast, the treat interference as noise strategy (resp. decode
interference) quickly becomes suboptimal as INR/SNR increases (resp.
decreases).

###### Remark 2

If each transmitter is equipped with a directive antenna, each transmitter
could increase the direct channel $h_{\mathrm{d}}$ and decrease the cross
channel $h_{\mathrm{c}}$ by suitably pointing the directive antenna. If this
operation can be done accurately, the interference level is more likely to
shift toward weaker regimes. This suggests that it would make more sense to
combine multi-antenna processing (enabling directive beams) with an
interference management strategy designed for weaker interference regimes
(treat interference as noise and RS) rather than stronger interference regimes
(decode interference).

###  II-D Lessons Learned

• Conventional interference management strategies rely on orthogonalization,
treat interference as noise, or decode interference. • Orthogonalize the
resources so as to completely eliminate multi-user interference is clearly
suboptimal. • Treat interference as noise is an efficient strategy whenever
the interference level is very weak but is inefficient whenever the
interference level grows to weak, strong or very strong. • Decode interference
is an efficient strategy whenever the interference level is strong and very
strong but is inefficient whenever the interference level is very weak or
weak. • RS splits messages into common and private parts so as to partially
decode interference and partially treat interference as noise. This allows RS
to bridge, unify, and generalize the two extremes of treat interference as
noise and decode interference and being efficient in all four interference
regimes. • RS is a superset of treat interference as noise and decode
interference strategies, and can specialize to each of them depending on how
messages are mapped to streams.

Those lessons are summarized in Table III and Fig. 4.

TABLE III: Comparison of preferable interference levels of different
interference management strategies.

Interference levels | Very weak | Weak | Strong | Very strong
---|---|---|---|---
Orthogonalize | $\times$ | $\times$ | $\times$ | $\times$
Treat interf. as noise | $\surd$ | $\times$ | $\times$ | $\times$
Decode interference | $\times$ | $\times$ | $\surd$ | $\surd$
Rate-Splitting | $\surd$ | $\surd$ | $\surd$ | $\surd$

Notations: $\surd$: Suited. $\times$: Not well suited.

Figure 4: The relationship between RS, treat interference as noise and decode
interference strategies. Each set illustrates the optimization space of the
corresponding interference management strategy. The optimization space of RS
is larger such that other strategies are just subsets.

##  III Two-User Rate-Splitting Multiple Access

Building upon the RS scheme for two-user IC, we can obtain some insight into
how to design RSMA for two-user downlink (broadcast channel) and uplink
(multiple access channel) in the next two sub-sections. We can also relate the
above discussion to MA design and show a direct relationship/analogy between
interference management and conventional MA designs such as OMA, SDMA, and
NOMA.

###  III-A Two-User Downlink Rate-Splitting Multiple Access

Figure 5: Two-user MISO downlink (Gaussian MISO broadcast channel).

Let us consider a single transmitter potentially equipped with $M$ antennas
serving two users. To that end, suppose that the two transmit antennas in Fig.
1 cooperate by exchanging CSI and messages such that they effectively belong
to the same transmitter as per Fig. 5. The two-user IC then becomes a two-user
MISO broadcast channel (BC) with $M=2$ transmit antennas where the base
station transmits to two active receiving users whose channels are given by
$\mathbf{h}_{1}$ and $\mathbf{h}_{2}$, respectively. In the example of Fig. 5,
$\mathbf{h}_{1}=\left[\begin{array}[]{cc}h_{\mathrm{d}}&h_{\mathrm{c}}\end{array}\right]$
and
$\mathbf{h}_{2}=\left[\begin{array}[]{cc}h_{\mathrm{c}}&h_{\mathrm{d}}\end{array}\right]$.
Considering a transmit signal vector $\mathbf{x}$ spanning across the transmit
antennas, the received signal at user-$k$ can be written as

$y_{k}=\mathbf{h}_{k}\mathbf{x}+n_{k},$ (9)

where $n_{k}\sim\mathcal{CN}(0,\sigma_{n,k}^{2})$. Such a multi-antenna BC is
a basic building block of modern downlink communication systems. Assuming
perfect CSIT and CSIR, the capacity of this channel is known and is achieved
by Dirty Paper Coding (DPC) [41, 42, 43]. DPC is nevertheless complex to
implement due to its inherent nonlinear encoding/precoding mechanism. RSMA has
appeared in the past few years as an appealing strategy to achieve close
performance to DPC while maintaining the low complexity of linear precoding
[8]. In the presence of imperfect CSIT, the capacity and the capacity-
achieving strategy are unknown. It is nevertheless known that RS plays a
central role to achieve the optimal DoF (and generalized DoF) [17, 18, 19, 20,
21, 22], and RSMA can outperform DPC [9]. In the sequel, we delineate
progressively the key design principles and schemes of RSMA in the two-user
downlink scenario.

####  III-A1 MISO RSMA

Inspired by the IC, the first MISO RSMA architecture is obtained by splitting
the two messages $W_{1}$ and $W_{2}$ into common and private parts and
encoding each part into a corresponding stream such that
$W_{\mathrm{c},k}\rightarrow s_{\mathrm{c},k}$ and
$W_{\mathrm{p},k}\rightarrow s_{\mathrm{p},k}$, $k=1,2$. Given the presence of
multiple antennas, precoding/beamforming can be performed across the two
antennas and the transmit signal model can be written as

$\mathbf{x}=\sum_{k=1,2}\mathbf{p}_{\mathrm{c},k}s_{\mathrm{c},k}+\mathbf{p}_{k}s_{k},$
(10)

where $\mathbf{p}_{\mathrm{c},k}$ is the precoder of common stream $k$ and
$\mathbf{p}_{k}$ is the precoder of private stream $k$. Defining
$\mathbf{s}=[s_{\mathrm{c},1},s_{1},s_{\mathrm{c},2},s_{2}]^{T}$ and assuming
that $\mathbb{E}[\mathbf{s}\mathbf{s}^{H}]=\mathbf{I}$, the average transmit
sum power constraint888We assume a sum power constraint but a per-antenna
power constraint could also be considered. at the transmitter is
$\sum_{k=1,2}P_{\mathrm{c},k}+P_{k}\leq P$ with
$P_{\mathrm{c},k}=\left\|\mathbf{p}_{\mathrm{c},k}\right\|^{2}$ and
$P_{k}=\left\|\mathbf{p}_{k}\right\|^{2}$ the power allocated to common stream
$k$ and private stream $k$, respectively. Uniquely, since $s_{\mathrm{c},1}$
and $s_{\mathrm{c},2}$ are transmitted from the same transmitter and are
decoded by both users in this downlink setting, one can choose
$\mathbf{p}_{\mathrm{c},1}=\mathbf{p}_{\mathrm{c},2}=\mathbf{p}_{\mathrm{c}}$
so that we only have a single common precoder and
$\mathbf{x}=\mathbf{p}_{\mathrm{c}}\left(s_{\mathrm{c},1}+s_{\mathrm{c},2}\right)+\sum_{k=1,2}\mathbf{p}_{k}s_{k}$.
User-$k$ can now decode using SIC or joint decoding common streams
$s_{\mathrm{c},1}$ and $s_{\mathrm{c},2}$ and then private stream $s_{k}$.
Using SIC, this architecture would require each receiver $k$ to be equipped
with two SIC layers so as to decode three streams $s_{\mathrm{c},j}$,
$s_{\mathrm{c},k}$, and $s_{k}$, $j\neq k$. In summary, from two messages,
this architecture creates four streams, and two SIC are required to recover
the original messages. This RSMA architecture is illustrated999Only the
receiver of user-1 is detailed in Fig. 6. The receiver of user-2 follows the
same principle as that of user-1 where two SIC layers are needed to decode and
cancel the common streams before retrieving user-2’s private message. in Fig.
6.

Figure 6: Two-user downlink MISO RSMA without message combiner.

A second MISO RSMA architecture is obtained by noting that an additional
benefit of the downlink is that instead of encoding each common part into a
common stream $W_{\mathrm{c},k}\rightarrow s_{\mathrm{c},k}$, we can first
combine the common parts into a common message
$W_{\mathrm{c}}=\left\\{W_{\mathrm{c},1},W_{\mathrm{c},2}\right\\}$ that is
then encoded into a single common stream $W_{\mathrm{c}}\rightarrow
s_{\mathrm{c}}$ such that

$\mathbf{x}=\mathbf{p}_{\mathrm{c}}s_{\mathrm{c}}+\sum_{k=1,2}\mathbf{p}_{k}s_{k}.$
(11)

By defining $\mathbf{s}=[s_{\mathrm{c}},s_{1},s_{2}]^{T}$ and assuming that
$\mathbb{E}[\mathbf{s}\mathbf{s}^{H}]=\mathbf{I}$, the average transmit (sum)
power constraint at the transmitter is $P_{\mathrm{c}}+\sum_{k=1,2}P_{k}\leq
P$ with $P_{\mathrm{c}}=\left\|\mathbf{p}_{\mathrm{c},k}\right\|^{2}$ the
power allocated to the unique common stream. The received signal at user-$k$,
$k=1,2$, $j\neq k$, is written as

$y_{k}=\mathbf{h}_{k}\mathbf{p}_{\mathrm{c}}s_{\mathrm{c}}+\mathbf{h}_{k}\mathbf{p}_{k}s_{k}+\mathbf{h}_{k}\mathbf{p}_{j}s_{j}+n_{k}.$
(12)

This is the so-called 1-layer RS architecture of RSMA because it relies on a
single common stream and therefore a single SIC layer at each receiver, hence
simplifying the encoding complexity (three streams used instead of four) and
the decoding complexity (one SIC layer instead of two) compared to (10) and
(1). Indeed both users decode the single common stream $s_{\mathrm{c}}$ into
$\widehat{W}_{\mathrm{c}}$ by treating the interference from all private
streams as noise. Each user-$k$ then retrieves the estimate
$\widehat{W}_{\mathrm{c},k}$ from $\widehat{W}_{\mathrm{c}}$. Using SIC,
$\widehat{W}_{\mathrm{c}}$ is then re-encoded, precoded, and subtracted from
the received signal such that user-$k$ decodes its private stream $s_{k}$ into
$\widehat{W}_{\mathrm{p},k}$ by treating the remaining interference from the
other private streams as noise. Finally user-$k$ recombines
$\widehat{W}_{\mathrm{c},k}$ and $\widehat{W}_{\mathrm{p},k}$ into the message
$\widehat{W}_{k}$ which is the same as the original message $W_{k}$ if no
decoding error occurs. This architecture is illustrated in Fig. 7.

Figure 7: Two-user downlink MISO RSMA with message combiner (1-layer RS).

####  III-A2 Revisiting the Interference Regimes

It is worth relating (12) to the lessons learned in Section II. In (12), after
precoding, the two-user MISO BC can effectively be seen as a two-user IC where
user-$k$ decodes a common stream $s_{\mathrm{c}}$ and a private stream $s_{k}$
subject to interference from private stream $s_{j}$, $j\neq k$. The
interference regime experienced by user-$k$ in this effective two-user IC is
therefore determined by the strength of the precoded channels
$\mathbf{h}_{k}\mathbf{p}_{k}$ and $\mathbf{h}_{k}\mathbf{p}_{j}$ (with
$\mathbf{h}_{k}\mathbf{p}_{k}$ and $\mathbf{h}_{k}\mathbf{p}_{j}$ taking the
role of $h_{\mathrm{d}}$ and $h_{\mathrm{c}}$, respectively):

  * •

$|\mathbf{h}_{k}\mathbf{p}_{j}|\ll|\mathbf{h}_{k}\mathbf{p}_{k}|$: very weak
interference regime and interference should be treated as noise, i.e.,
$\mathbf{p}_{\mathrm{c}}=0$ and only private streams are used. Decoding
interference would perform badly. This scenario can typically occur when the
channels are close to being orthogonal.

  * •

$|\mathbf{h}_{k}\mathbf{p}_{j}|\leq|\mathbf{h}_{k}\mathbf{p}_{k}|$: weak
interference regime and non-zero power should be allocated to all common and
private streams. This would typically occur whenever the channels are neither
orthogonal nor aligned.

  * •

$|\mathbf{h}_{k}\mathbf{p}_{j}|\geq|\mathbf{h}_{k}\mathbf{p}_{k}|$: strong
interference regime and interference should be decoded. Treat interference as
noise would perform badly. This scenario can typically occur when the channels
are aligned.

This shows that depending on the propagation conditions in multi-antenna
settings (e.g., angle between user channel directions), the interference
regime can change from very weak to strong. Consequently, downlink MA schemes
therefore need the ability to softly evolve from the extreme of decode
interference to treat interference as noise. RSMA has the flexibility to cope
with all those interference regimes and propagation conditions.

It will now appear clear in the sequel how existing MA schemes such as SDMA
and NOMA are specifically tailored for one specific interference management
strategy (e.g., treat interference as noise, decode interference), one
specific interference regime (e.g., very weak, strong) and one type of channel
conditions (e.g., orthogonal, aligned), and how RSMA can unify them all.

####  III-A3 Unifying OMA, SDMA, NOMA, and Multicasting

Figure 8: The relationship between OMA, NOMA, SDMA, physical-layer
multicasting, and RSMA in the downlink two-user case. Each set illustrates the
optimization space of the corresponding communication strategy. The
optimization space of RSMA is larger such that SDMA, NOMA, and physical-layer
multicasting are just subsets. Figure 9: A beam representation of RSMA and its
sub-schemes SDMA, NOMA, OMA, and physical-layer multicasting.

In the simple two-user case, OMA, SDMA, NOMA, and physical-layer multicasting
are particular instances of RSMA, as illustrated by Fig. 8, the message to
stream mapping in Table IV [24], and Fig. 9.

SDMA is a special case of RSMA by forcing $\mathbf{p}_{\mathrm{c}}=0$. In this
way, $W_{k}$ is directly encoded into $s_{k}$ and the system model writes as
$\mathbf{x}=\sum_{k=1,2}\mathbf{p}_{k}s_{k}$. By doing so, each stream is
decoded by its intended user by treating any residual interference from the
other stream as noise. Recalling Tables II and III, from an interference
management strategy, SDMA is reminiscent of and builds upon the Treat
Interference as Noise strategy which would be efficient only if the residual
multi-user interference $\left|\mathbf{{h}}_{k}\mathbf{{p}}_{j}\right|$ is
sufficiently weak as in orthogonal channels [24].

NOMA is a special case of RSMA by forcing the encoding of message $W_{2}$
entirely into $s_{\mathrm{c}}$ (i.e., $W_{\mathrm{c}}=W_{2}$) and $W_{1}$ into
$s_{1}$ while turning off $s_{2}$ ($P_{2}=0$). By doing so, user-1 fully
decodes the message of user-2 (and therefore interference created by user-2
stream) and 1-layer RS system model becomes the NOMA system model
$\mathbf{x}=\mathbf{p}_{\mathrm{c}}s_{\mathrm{c}}+\mathbf{p}_{1}s_{1}$. Note
that NOMA also utilizes a common stream since the message of one of the two
users, namely $W_{2}$ in this example, is decoded by both users. Connecting
back to Tables II and III, from an interference management strategy, NOMA is
reminiscent of and builds upon the Decode Interference strategy which would be
efficient only if the multi-user interference level is sufficiently strong as
in aligned channels [24].

OMA is a special case of RSMA by forcing only one user to be scheduled, e.g.,
user-1 (i.e.,
$\left\|\mathbf{p}_{\mathrm{c}}\right\|^{2}=\left\|\mathbf{p}_{2}\right\|^{2}=0$).

Physical-layer multicasting is a special case of RSMA obtained when messages
$W_{1},W_{2}$ are both encoded into $s_{\mathrm{c}}$ (i.e.,
$W_{\mathrm{c}}=\\{W_{1},W_{2}\\}$) and the private streams are turned off
($P_{1}=P_{2}=0$).

TABLE IV: Messages-to-streams mapping in the two-user MISO BC [24]. | $s_{1}$ | $s_{2}$ | $s_{\mathrm{c}}$
---|---|---|---
SDMA | $\,\,\quad\quad W_{1}\quad\quad\,\,$ | $W_{2}$ | –
NOMA | $W_{1}$ | – | $W_{2}$
OMA | $W_{1}$ | – | –
Multicasting | – | – | $W_{1},W_{2}$
RSMA | $W_{\mathrm{p},1}$ | $W_{\mathrm{p},2}$ | $W_{\mathrm{c},1},W_{\mathrm{c},2}$
| decoded by its intended user and | decoded by
| treated as noise by the other user |  both users
TABLE V: Comparison of preferable residual multi-user interference levels of
different MA schemes.

Interference levels | Very weak | Weak | (Very) Strong
---|---|---|---
SDMA | $\surd$ | $\times$ | $\times$
NOMA | $\times$ | $\times$ | $\surd$
RSMA | $\surd$ | $\surd$ | $\surd$

Notations: $\surd$: Suited. $\times$: Not well suited.

Drawing an analogy with Table III, we can conclude that MA schemes operate in
some preferred residual multi-user interference regimes as shown in Table V.
The above clearly shows how three different non-orthogonal approaches to MA
designs, namely SDMA, NOMA, and RSMA, fundamentally differ based on how multi-
user interference is managed. NOMA is such that at least one user is forced to
fully decode the message(s) of other co-scheduled user. SDMA and RSMA do not
follow this approach since they both do not force a user to fully decode the
messages of another co-scheduled user. SDMA actually treats any residual
interference as noise, and RSMA is built upon the principle of splitting the
messages so as to partially treat interference as noise and partially decode
the remaining interference. Consequently, this difference in managing
interference has deep consequences on the universality of MA schemes, with
RSMA being a superset of SDMA and NOMA as per Fig. 8, but also on the
performance of those MA schemes as a function of the propagation conditions
[24].

RSMA can also be seen as a smart combination of transmit-side and receive-side
interference cancellation strategy where the contribution of the common stream
is adjusted according to the level of interference that needs to be canceled
by the receiver, therefore departing from the transmit-side only and receive-
side only interference cancellation strategies of SDMA (and DPC) and NOMA,
respectively. This is further summarized in Table VI.

TABLE VI: Comparison of different MA schemes in terms of transmit-side vs
receive-side interference cancellation.

Interf. cancel. | transmit-side | receive-side | both sides
---|---|---|---
SDMA/DPC | $\surd$ | $\times$ | $\times$
NOMA | $\times$ | $\surd$ | $\times$
RSMA | $\times$ | $\times$ | $\surd$

Notations: $\surd$: Relevant. $\times$: Not relevant.

###### Example 1

To further illustrate the split of the messages and the flexibility of RSMA,
let us imagine that the message of user-1
$W_{1}=\left(a_{1}\>a_{2}\>a_{3}\>a_{4}\right)\in\mathcal{W}_{1}=\left\\{0000,0001,0010,\ldots,1111\right\\}$,
where $\left|\mathcal{W}_{1}\right|=16$. Similarly, the message of user-2 is
$W_{2}=\left(b_{1}\>b_{2}\>b_{3}\right)\in\mathcal{W}_{2}=\left\\{000,001,010,\ldots,111\right\\}$,
where $\left|\mathcal{W}_{2}\right|=8$. In SDMA, $W_{1}$ would be encoded into
$s_{1}$ and $W_{2}$ into $s_{2}$. Assuming uncoded transmission for
simplicity, $s_{1}$ and $s_{2}$ would then be a 16-QAM symbol and a 8-PSK
symbol, respectively. In NOMA, $W_{1}$ would be encoded into $s_{1}$ and
$W_{2}$ into $s_{\mathrm{c}}$, also using a 16-QAM symbol and a 8-PSK symbol,
respectively. In RS, we split user-1’s message in, e.g.,
$W_{\mathrm{c},1}=\left(a_{1}\>a_{2}\right)$,
$W_{\mathrm{p},1}=\left(a_{3}\>a_{4}\right)$, and user-2’s message in, e.g.,
$W_{\mathrm{c},2}=\left(b_{1}\right)$,
$W_{\mathrm{p},2}=\left(b_{2}\>b_{3}\right)$. The common message is then
constructed as
$W_{\mathrm{c}}=\left(W_{\mathrm{c},1}\>W_{\mathrm{c},2}\right)=\left(a_{1}\>a_{2}\>b_{1}\right)$,
which is then encoded into $s_{\mathrm{c}}$ using a 8-PSK symbol.
$W_{\mathrm{p},1}$ and $W_{\mathrm{p},2}$ are encoded into $s_{1}$ and $s_{2}$
using QPSK symbols, respectively. The interested reader is referred to [25]
for more examples.

####  III-A4 Rate Analysis

Under the assumption of Gaussian signaling and infinite blocklength, and
perfect CSIT and CSIR, the instantaneous rates for decoding the common and
private streams at user-$k$ are given as

$\displaystyle R_{\mathrm{c},k}$
$\displaystyle=\log_{2}\left(1+\frac{\left|\mathbf{{h}}_{k}\mathbf{{p}}_{\mathrm{c}}\right|^{2}}{\left|\mathbf{{h}}_{k}\mathbf{{p}}_{1}\right|^{2}+\left|\mathbf{{h}}_{k}\mathbf{{p}}_{2}\right|^{2}+1}\right),$
(13) $\displaystyle R_{k}$
$\displaystyle=\log_{2}\left(1+\frac{\left|\mathbf{{h}}_{k}\mathbf{{p}}_{k}\right|^{2}}{\left|\mathbf{{h}}_{k}\mathbf{{p}}_{j}\right|^{2}+1}\right),$

where the noise variance was normalized $\sigma_{n,k}^{2}=1$ without loss of
generality. To ensure that $s_{\mathrm{c}}$ is successfully decoded by both
users, its rate cannot exceed

$R_{\mathrm{c}}=\min\left\\{R_{\mathrm{c},1},R_{\mathrm{c},2}\right\\}.$ (14)

As $s_{\mathrm{c}}$ contains sub-messages $W_{\mathrm{c},1},W_{\mathrm{c},2}$
of the two users, the rate distribution of $R_{\mathrm{c}}$ among the users is
adapted to the amount of sub-messages that each user contributed. Let $C_{k}$
denote the portion of rate $R_{\mathrm{c}}$ allocated to user-$k$ for
$W_{\mathrm{c},k}$. Then, we have

$C_{1}+C_{2}=R_{\mathrm{c}}.$ (15)

The overall achievable rate of user-$k$ is

$R_{k,\mathrm{tot}}=C_{k}+R_{k}.$ (16)

Here again, the terminology rate-splitting appears clearly in (16) where we
note that the rate of each user is split into two parts, namely, the rate of
$s_{k}$ (i.e., the private rate) and part of the rate of $s_{\mathrm{c}}$
(i.e., the common rate).

Common metrics to design the systems include 1) weighted sum rate (WSR)
$u_{1}R_{1,\mathrm{tot}}+u_{2}R_{2,\mathrm{tot}}$ where $u_{1}$ and $u_{2}$
are weights set to account for fairness among users (for instance when
conducting proportional fair scheduling); 2) max-min fair (MMF) that aims at
maximizing $\min_{k=1,2}R_{k,\mathrm{tot}}$; 3) energy efficiency (EE)
$\frac{R_{1,\mathrm{tot}}+R_{2,\mathrm{tot}}}{\frac{1}{\eta}(P_{1}+P_{2}+P_{\mathrm{c}})+P_{\mathrm{cir}}}$
where $\eta\in(0,1]$ and $P_{\mathrm{cir}}$ are respectively the power
amplifier efficiency and the circuit power consumption. All three metrics
could also be subject to a QoS constraint $R_{k,\mathrm{tot}}\geq R_{k}^{th}$
with $R_{k}^{th}$ a minimum rate threshold to be achieved by user-$k$.

###### Remark 3

We can wonder what happens if $M=1$, namely downlink SISO. RSMA strategy (11)
yields

$x=\sqrt{P_{c}}s_{\mathrm{c}}+\sqrt{P_{1}}s_{1}+\sqrt{P_{2}}s_{2}$ (17)

and the common and private rates write as

$\displaystyle R_{\mathrm{c},k}$
$\displaystyle=\log_{2}\left(1+\frac{\left|h_{k}\right|^{2}P_{\mathrm{c}}}{\left|h_{k}\right|^{2}P_{1}+\left|h_{k}\right|^{2}P_{2}+1}\right),$
(18) $\displaystyle R_{k}$
$\displaystyle=\log_{2}\left(1+\frac{\left|h_{k}\right|^{2}P_{k}}{\left|h_{k}\right|^{2}P_{j}+1}\right),$

for $k=1,2$. Without loss of generality, we consider
$\left|h_{2}\right|\leq\left|h_{1}\right|$ so that

$R_{\mathrm{c}}=\log_{2}\left(1+\frac{\left|h_{2}\right|^{2}P_{\mathrm{c}}}{\left|h_{2}\right|^{2}P_{1}+\left|h_{2}\right|^{2}P_{2}+1}\right).$
(19)

Since the capacity of the Gaussian SISO BC with perfect CSIT and CSIR is
obtained by performing superposition coding (SC) with SIC (SISO NOMA),
optimizing $P_{1},P_{2},P_{c}$ to maximize the WSR would lead to choosing
$P_{2}=0$ so that the message of the weaker user (user-2) $W_{2}$ is entirely
encoded in $s_{\mathrm{c}}$ and SISO RSMA becomes SISO NOMA
$x=\sqrt{P_{c}}s_{\mathrm{c}}+\sqrt{P_{1}}s_{1}$ and its achievable rates are

$\displaystyle R_{1}$
$\displaystyle=\log_{2}\left(1+\left|h_{1}\right|^{2}P_{1}\right),$ (20)
$\displaystyle R_{\mathrm{c}}$
$\displaystyle=\log_{2}\left(1+\frac{\left|h_{2}\right|^{2}P_{\mathrm{c}}}{\left|h_{2}\right|^{2}P_{1}+1}\right).$
(21)

This shows how RSMA can act as NOMA in Gaussian SISO downlink.

###### Remark 4

The capacity region of a Gaussian SISO BC is known for both cases with fixed
and fading channels when there are perfect CSIT and CSIR. However, the problem
is open in general when there is imperfect CSIT and only limited cases are
known [44]. The fading SISO BC with only perfect CSIR but imperfect CSIT lacks
the degraded structure in general for arbitrary fading distributions. In these
cases, SC and SIC (and therefore NOMA) can not achieve the capacity region. In
[45, Theorem 1], the sufficient condition that the second channel is
stochastically larger than or equal to the first channel, i.e.,
$H_{2}\geq_{st}H_{1}$ 101010Stochastic order $H_{2}\geq_{st}H_{1}$ means that
$\mathbb{P}(H_{2}\geq s)\geq\mathbb{P}(H_{1}\geq s)$ for all $s$ from the
support of $H_{1},H_{2}$., is derived for which the capacity region is given
by

$\displaystyle R_{1}$ $\displaystyle\leq$ $\displaystyle I(V;Y_{1}|H_{1}),$
$\displaystyle R_{2}$ $\displaystyle\leq$ $\displaystyle I(X;Y_{2}|V,H_{2}),$
(22)

with Markov chain $V-X-Y_{2}-Y_{1}$. Even though the capacity expressions in
(22) look familiar to the rates achievable with SC and SIC (and NOMA), the
optimal coding and decoding strategy to achieve (22) remains unknown because
the same marginal property of the BC is applied to transform the joint
distribution of the underlying channel to fully coupled.

####  III-A5 Precoder Design and Power Allocation

Recall from Section II-C that a qualitative and insightful way of allocating
power to the common stream is so that the interference level caused by the
private stream has roughly the same level as the other receiver’s noise level,
i.e., $\left(1-t\right)P\left|h_{\mathrm{c}}\right|^{2}\approx 1$. In the
downlink scenario, looking at (13), we can think in a similar way and choose
the power allocation $P_{1}$ and $P_{2}$ to the two private streams such that
$\left|\mathbf{{h}}_{2}\mathbf{{p}}_{1}\right|^{2}\approx 1$ and
$\left|\mathbf{{h}}_{1}\mathbf{{p}}_{2}\right|^{2}\approx 1$, i.e., instead of
allocating all the transmit power $P$ to the private streams (as done in
SDMA), we allocate a fraction of the total available power to them such that
the signal-to-interference-plus-noise ratios (SINRs) of the private streams
are not interference limited. If we were allocating a higher power to the
private streams with $\left|\mathbf{{h}}_{k}\mathbf{{p}}_{j}\right|^{2}\gg 1$,
the SINR
$\frac{\left|\mathbf{{h}}_{k}\mathbf{{p}}_{k}\right|^{2}}{\left|\mathbf{{h}}_{k}\mathbf{{p}}_{j}\right|^{2}+1}\approx\frac{\left|\mathbf{{h}}_{k}\mathbf{{p}}_{k}\right|^{2}}{\left|\mathbf{{h}}_{k}\mathbf{{p}}_{j}\right|^{2}}$
would saturate and the private rates would not increase further. Hence, the
key is for the private streams not to enter the interference limited regime,
and allocate any remaining power to the common stream whose SINR will keep
increasing linearly as $P_{\mathrm{c}}$ increases. Doing so, $R_{1}+R_{2}$
would be roughly equivalent to the sum-rate achieved by SDMA, but the common
rate $R_{\mathrm{c}}$ would provide an additional rate increase over what SDMA
can offer.

The above is particularly insightful when the CSIT is imperfect and the
transmitter only has an estimate $\hat{\mathbf{h}}_{k}$ of user-$k$ channel
$\mathbf{h}=\hat{\mathbf{h}}_{k}+\tilde{\mathbf{h}}_{k}$ with
$\tilde{\mathbf{h}}_{k}$ the channel acquisition error. In such setting,
residual multi-user interference scaling as
$\big{|}\tilde{\mathbf{{h}}}_{k}\mathbf{{p}}_{j}\big{|}^{2}$ would be
unavoidable regardless of how the precoder $\mathbf{{p}}_{j}$ is designed.
Following such an approach, we can demonstrate that RSMA is information
theoretically optimal from the DoF perspective in the presence of imperfect
CSIT [18, 19, 46] while SDMA and NOMA are not [7]. Because of its inherent
robustness to imperfect CSIT, RSMA can also afford smaller feedback overhead
than conventional SDMA [47].

More systematic methods to design the precoder and power allocation can be
obtained using either closed form low complexity suboptimal techniques or
using optimization techniques detailed in [25]. Extensive results have
demonstrated that the power allocation to the common stream vs private stream
depends on a number of factors, including the angle between user channels [8,
24], the disparity of channel strengths [8, 24], the objective function to
maximize [18, 48], the quality of CSIT [18, 48], the QoS, the network load,
etc.

####  III-A6 Cooperative RSMA

An interesting extension is achieved by considering that the transmitter can
opportunistically ask one of the users to act as a relay. This is known as a
cooperative relay BC in information theory [49, 50, 51]. Specifically, in our
setting, the common stream is decoded by both users and the transmitter can
ask the relay user to forward the decoded common message/stream to the other
user to efficiently cope with a wide range of propagation conditions
(disparity of user channel strengths and directions) and compensate for the
performance degradation due to deep fading [52, 53], as illustrated in Fig.
10. The parameter $0<\theta\leq 1$ refers to the time split between the direct
transmission phase and the relaying phase, i.e., $\theta=1$ indicates that no
time is allocated to relaying, hence boiling down to the conventional downlink
transmission of Fig. 5.

Figure 10: Cooperative downlink RSMA with user relaying [52].

The user relaying feature of cooperative RSMA enables to enlarge the pool of
possible schemes within the RSMA framework as shown by the messages-to-streams
mapping in the two-user cooperative MISO downlink of Table VII. We note how
the mapping of Table IV has been extended to include cooperation where
conventional decode and forward (DF) and cooperative NOMA (C-NOMA) are
particular instances of cooperative RSMA (C-RSMA).

TABLE VII: Messages-to-streams mapping in the two-user cooperative MISO
downlink [52].

| $s_{1}$ | $s_{2}$ | $s_{\mathrm{c}}$ |  $\theta$
---|---|---|---|---
SDMA | $\,\,\quad\quad W_{1}\quad\quad\,\,$ | $W_{2}$ | – |  $\theta=1$
NOMA | $W_{1}$ | – | $W_{2}$ |  $\theta=1$
OMA | $W_{1}$ | – | – |  $\theta=1$
Multic. | – | – | $W_{1},W_{2}$ |  $\theta=1$
RSMA | $W_{\mathrm{p},1}$ | $W_{\mathrm{p},2}$ | $W_{\mathrm{c},1},W_{\mathrm{c},2}$ |  $\theta=1$
C-NOMA | $W_{1}$ | – | $W_{2}$ |  $\theta<1$
DF | – | – | $W_{2}$ |  $\theta<1$
C-RSMA | $W_{\mathrm{p},1}$ | $W_{\mathrm{p},2}$ | $W_{\mathrm{c},1},W_{\mathrm{c},2}$ |  $\theta<1$
| decoded by its intended user and | decoded by |
| treated as noise by the other user |  both users |

Notations: $\theta=1$ refers to non-cooperative schemes as in Table IV.
$\theta<1$ refers to cooperative RSMA schemes.

####  III-A7 Space-Time / Space-Frequency RSMA

Most RSMA literature deal with a single channel use (be it in time or
frequency). This means the signal model as expressed in (11) is applied in a
given time slot or given frequency subband. For instance, in the non-
alternating CSIT pattern of Table VIII, (11) would be applied separately on
subbands A and B, i.e., each subband determining the split of the message,
power allocation and precoders based on the available quality of CSIT in that
subband (good on subband A and bad on subband B). One could also apply the
same strategy to the alternating CSIT pattern of Table VIII. However, because
of the alternating feature, simply doing the spatial domain RSMA transmission
(11) would not be as efficient as doing an RSMA transmission across the two
subbands. In other words, RSMA can benefit from a multi-channel transmission
in space-time (ST) or in space-frequency (SF) depending on the CSIT pattern
[47, 54, 55]. Such a multi-channel RSMA transmission is particularly helpful
in the presence of alternating CSIT, i.e., whenever the CSIT changes across
time or frequency in an alternating user-specific manner. The alternating CSIT
pattern of Table VIII is a typical (and practical) scenario where the
transmitter wants to serve two users but the CSIT of user-1 (resp. user-2) is
better on time/frequency B (resp. A) and worse on time/frequency A (resp. B).
In such scenario, an ST/SF RSMA scheme can further increase the DoF over
conventional RSMA [54, 55].

TABLE VIII: Two different CSIT patterns.

| User-1 | User-2
---|---|---
subband A | $\surd$ | $\surd$
subband B | $\times$ | $\times$

Non-alternating CSIT

| User-1 | User-2
---|---|---
subband A | $\times$ | $\surd$
subband B | $\surd$ | $\times$

Alternating CSIT

Notations:$\surd$: Good CSIT quality. $\times$: Poor CSIT quality.

Compared to the RSMA scheme of (11), ST/SF-RSMA scheme transmits an additional
common stream (obtained from a further split of the messages), i.e., $s_{0}$,
across the two channel uses. Specifically, considering the alternating CSIT
pattern of Table VIII, the transmitted signals in subbands A and B write can
be expressed as follows:

$\displaystyle\mathbf{x}^{(A)}$
$\displaystyle=\mathbf{p}_{\mathrm{0}}^{(A)}s_{\mathrm{0}}+\mathbf{p}_{\mathrm{c}}^{(A)}s_{\mathrm{c}}^{(A)}+\sum_{k=1,2}\mathbf{p}_{k}^{(A)}s_{k}^{(A)},$
(23) $\displaystyle\mathbf{x}^{(B)}$
$\displaystyle=\mathbf{p}_{\mathrm{0}}^{(B)}s_{\mathrm{0}}+\mathbf{p}_{\mathrm{c}}^{(B)}s_{\mathrm{c}}^{(B)}+\sum_{k=1,2}\mathbf{p}_{k}^{(B)}s_{k}^{(B)},$
(24)

where the superscript (i) refers to the subband. We note the addition of the
new common stream $s_{0}$ that has been repeated across the two channel uses
(precoded by $\mathbf{p}_{\mathrm{0}}^{(i)}$ in channel use $i=1,2$). If the
CSIT quality becomes non-alternating, the common stream $s_{\mathrm{0}}$
becomes useless (zero power is allocated to $s_{\mathrm{0}}$) and SF-RSMA
boils down to (11) in each subband. The receiver at both users is more
complicated since two common streams have to be decoded at each user before
decoding the respective private stream.

The decoding works as follows. Let us focus on user-1 for simplicity. First,
user-1 decodes $s_{\mathrm{c}}^{(A)}$ and $s_{\mathrm{0}}$ sequentially using
subband A observation by treating the private streams as noise. Secondly,
after removing $s_{\mathrm{0}}$ from subband B observation, user-1 recovers
$s_{\mathrm{c}}^{(B)}$ by treating the private streams as noise. Thirdly, by
removing all common streams, user-1 decodes $s_{1}^{(A)}$ and $s_{1}^{(B)}$ by
treating the other private stream as noise.

More details on the CSIT pattern conditions needed for the optimality of
separate RSMA in each subband can be found from [56]. Another interesting use
of ST-RSMA, where RS is combined with space-time block coding to address the
lack of knowledge of the channel phase information at the transmitter, has
been developed in [57].

####  III-A8 MIMO RSMA

Let us now consider a two-user MIMO downlink with $M$ transmit antennas and
$N$ receive antennas at each user. The transmitter wants to transmit two
$N$-dimensional vectors of messages $\mathbf{w}_{1}$ and $\mathbf{w}_{2}$ to
user-1 and user-2, respectively. To that end, for each user, each of those $N$
messages is split into a common part and a private part. Common parts of both
users are combined and encoded into a $N$-dimensional vector of common streams
$\mathbf{s}_{\mathrm{c}}$ and private parts are encoded into two
$N$-dimensional vectors of private streams $\mathbf{s}_{k}$. $M\times N$
precoding are performed on the common and private stream vectors such that the
transmit signal is written as

$\mathbf{x}=\mathbf{P}_{\mathrm{c}}\mathbf{s}_{\mathrm{c}}+\sum_{k=1,2}\mathbf{P}_{k}\mathbf{s}_{k}.$
(25)

Let us consider an example with $N=2$ to illustrate the universality of the
framework. We have four messages to transmit (two for each user)
$\mathbf{w}_{1}=[W_{1}^{(1)},W_{1}^{(2)}]$ and
$\mathbf{w}_{2}=[W_{2}^{(1)},W_{2}^{(2)}]$, where $W_{i}^{(j)}$ refers to the
$j$th message of user-$i$, $i=1,2$, $j=1,2$. We split the four messages into
common and private parts such that

$\displaystyle W_{1}^{(1)}$
$\displaystyle=\left\\{W_{\mathrm{c},1}^{(1)},W_{\mathrm{p},1}^{(1)}\right\\},$
(26) $\displaystyle W_{1}^{(2)}$
$\displaystyle=\left\\{W_{\mathrm{c},1}^{(2)},W_{\mathrm{p},1}^{(2)}\right\\},$
(27) $\displaystyle W_{2}^{(1)}$
$\displaystyle=\left\\{W_{\mathrm{c},2}^{(1)},W_{\mathrm{p},2}^{(1)}\right\\},$
(28) $\displaystyle W_{2}^{(2)}$
$\displaystyle=\left\\{W_{\mathrm{c},2}^{(2)},W_{\mathrm{p},2}^{(2)}\right\\}.$
(29)

Common parts are then combined into $W_{\mathrm{c}}^{(1)}$ and
$W_{\mathrm{c}}^{(2)}$ and then respectively encoded into common stream
$s_{\mathrm{c}}^{(1)}$ and $s_{\mathrm{c}}^{(2)}$

$\displaystyle W_{\mathrm{c}}^{(1)}$
$\displaystyle=\left\\{W_{\mathrm{c},1}^{(1)},W_{\mathrm{c},2}^{(1)}\right\\}\rightarrow
s_{\mathrm{c}}^{(1)},$ (30) $\displaystyle W_{\mathrm{c}}^{(2)}$
$\displaystyle=\left\\{W_{\mathrm{c},1}^{(2)},W_{\mathrm{c},2}^{(2)}\right\\}\rightarrow
s_{\mathrm{c}}^{(2)},$ (31)

so as to form the 2-dimensional common stream vector
$\mathbf{s}_{\mathrm{c}}=\big{[}\begin{array}[]{cc}s_{\mathrm{c}}^{(1)}&s_{\mathrm{c}}^{(2)}\end{array}\big{]}^{T}$.
Private parts are encoded into private streams

$\displaystyle W_{\mathrm{p},1}^{(1)}$ $\displaystyle\rightarrow
s_{1}^{(1)},\hskip 5.69046ptW_{\mathrm{p},1}^{(2)}\rightarrow s_{1}^{(2)},$
(32) $\displaystyle W_{\mathrm{p},2}^{(1)}$ $\displaystyle\rightarrow
s_{2}^{(1)},\hskip 5.69046ptW_{\mathrm{p},2}^{(2)}\rightarrow s_{2}^{(2)},$
(33)

so as to create two 2-dimensional private stream vectors
$\mathbf{s}_{1}=\big{[}\begin{array}[]{cc}s_{1}^{(1)}&s_{1}^{(2)}\end{array}\big{]}^{T}$
and
$\mathbf{s}_{2}=\big{[}\begin{array}[]{cc}s_{2}^{(1)}&s_{2}^{(2)}\end{array}\big{]}^{T}$.
At the receivers, both users decode the vector of common streams first using
the two receive antennas, perform SIC, and then decode their respective
private stream vector while treating the co-scheduled user’s private stream
vector as noise. Fig. 11 illustrates the MIMO RSMA architecture.

Figure 11: Downlink MIMO RSMA [35]. TABLE IX: Messages-to-streams mapping in the two-user MIMO BC. | $\mathbf{s}_{1}$ | $\mathbf{s}_{2}$ | $\mathbf{s}_{\mathrm{c}}$
---|---|---|---
| $s_{1}^{(1)}$ | $s_{1}^{(2)}$ | $s_{2}^{(1)}$ | $s_{2}^{(2)}$ | $s_{\mathrm{c}}^{(1)}$ | $s_{\mathrm{c}}^{(2)}$
MU-MIMO | $\,\,\quad\quad W_{1}^{(1)}\quad\quad\,\,$ | $\,\,\quad\quad W_{1}^{(2)}\quad\quad\,\,$ | $\,\,\quad\quad W_{2}^{(1)}\quad\quad\,\,$ | $\,\,\quad\quad W_{2}^{(2)}\quad\quad\,\,$ | – | –
NOMA | $W_{1}^{(1)}$ | $W_{1}^{(2)}$ | – | – | $\,\,\quad\quad W_{2}^{(1)}\,\,\quad\quad$ | $\,\,\quad\quad W_{2}^{(2)}\,\,\quad\quad$
OMA | $W_{1}^{(1)}$ | $W_{1}^{(2)}$ | – | – | – | –
Multicasting | – | – | – | – | $W_{1}^{(1)},W_{2}^{(1)}$ | $W_{1}^{(2)},W_{2}^{(2)}$
RSMA | $W_{\mathrm{p},1}^{(1)}$ | $W_{\mathrm{p},1}^{(2)}$ | $W_{\mathrm{p},2}^{(1)}$ | $W_{\mathrm{p},2}^{(2)}$ | $W_{\mathrm{c},1}^{(1)},W_{\mathrm{c},2}^{(1)}$ | $W_{\mathrm{c},1}^{(2)},W_{\mathrm{c},2}^{(2)}$
subscheme 1 | $W_{1}^{(1)}$ | – | $W_{2}^{(1)}$ | – | $W_{1}^{(2)}$ | $W_{2}^{(2)}$
subscheme 2 | – | $W_{1}^{(2)}$ | – | $W_{2}^{(2)}$ | $W_{1}^{(1)}$ | $W_{2}^{(1)}$
subscheme 3 | $W_{1}^{(1)}$ | $W_{1}^{(2)}$ | $W_{2}^{(1)}$ | – | – | $W_{2}^{(2)}$
|  decoded by its intended user and treated as noise by the other user | decoded by both users

The mapping of messages to streams of Table IV is now further extended to
account for MIMO RSMA in Table IX. RSMA again is a superset of all schemes. We
note that SDMA is replaced by MU-MIMO where a vector of private streams is
transmitted to each user. NOMA in the form of having one user (user-1)
decoding all the messages of the other user (user-2) is illustrated, along
with OMA. Additionally, other subschemes of RSMA are listed for the sake of
illustration. For instance, in subscheme 1, both users decode the 2nd messages
and, in subscheme 2, both users decode the 1st messages, i.e., each user
decodes one message of the other user. Subscheme 3 is when both users decode
the 2nd message of user-2. Note that the MIMO NOMA scheme proposed in [36, 37]
is also a subscheme of MIMO RSMA, and it switches among MU-MIMO, NOMA, or the
three additional subschemes of RSMA when different numbers of transmit
antennas and receive antennas are considered.

Optimization of MIMO RSMA with both perfect and imperfect CSIT can be
performed as in [35]. Similar to the MISO case, RSMA outperforms MU-MIMO and
NOMA in the MIMO case. Note that RSMA is information theoretically optimal
from a DoF perspective in MIMO BC with imperfect CSIT [22, 21]. In the
asymmetric case where the number of receive antennas is not the same at each
user, RSMA with a multi-channel transmission (similar to ST/SF RSMA) needs to
be used to achieve optimality. The reader is referred to [22] for more details
on such a ST RSMA scheme for asymmetric MIMO BC.

###  III-B Two-User Uplink Rate-Splitting Multiple Access

In the uplink, we consider two single-antenna users simultaneously
transmitting their messages to a receiver equipped with $M$ antennas. When
$M=1$, such two-user MAC can be considered as a special case of the two-user
IC in Fig. 1 where the two receivers are colocated and cooperatively decode
the messages of the two users.

####  III-B1 Two-User Architectures

Inspired by the RSMA design in IC, the uplink RSMA architecture has been
proposed in [23] by splitting the message of one user into two parts. Without
loss of generality, we assume that user-1’s message $W_{1}$ is split into
$W_{1,1}$ and $W_{1,2}$. By independently encoding the two parts into
$s_{1,1},s_{1,2}$, respectively allocating transmit power $P_{1,1},P_{1,2}$,
and superposing the two streams, the transmit signal at user-1 is given by

$x_{1}=\sqrt{P_{1,1}}s_{1,1}+\sqrt{P_{1,2}}s_{1,2}.$ (34)

At user-2, the message $W_{2}$ is directly encoded into $s_{2}$. By allocating
certain power $P_{2}$, the transmit signal at user-2 is
$x_{2}=\sqrt{P_{2}}s_{2}$.

The signal received at the receiver is

$\displaystyle\mathbf{y}$
$\displaystyle=\mathbf{h}_{1}x_{1}+\mathbf{h}_{2}x_{2}+\mathbf{n}$ (35)
$\displaystyle=\sqrt{P_{1,1}}\mathbf{h}_{1}s_{1,1}+\sqrt{P_{1,2}}\mathbf{h}_{1}s_{1,2}+\sqrt{P_{2}}\mathbf{h}_{2}s_{2}+\mathbf{n},$

where $\mathbf{h}_{1},\mathbf{h}_{2}\in\mathbb{C}^{M\times 1}$ are the channel
vectors and $\mathbf{n}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{M})$ is the
AWGN vector. The receiver can employ SIC or joint decoding to decode the three
streams $s_{1,1},s_{1,2},s_{2}$. If using SIC, two layers of SIC are required
at the receiver. In summary, two-user uplink RSMA creates two virtual users
from the user that splits its message into two parts and in total three
streams are sent to the receiver. Two SIC layers are required to recover the
original messages of the two users. This uplink RSMA architecture is
illustrated in Fig. 12.

Figure 12: Two-user uplink SIMO RSMA. Figure 13: The relationship between OMA, NOMA, and RSMA in the uplink two-user case. Each set illustrates the optimization space of the corresponding communication strategy. The optimization space of RSMA is larger than OMA and NOMA. TABLE X: Messages-to-streams mapping in the two-user multiple access channel. | User-1 | User-2
---|---|---
| $s_{1,1}$ | $s_{1,2}$ | $s_{2}$
NOMA | $W_{1}$ | – | $W_{2}$
OMA | $W_{1}$ | – | –
RSMA | $W_{1,1}$ | $W_{1,2}$ | $W_{2}$

####  III-B2 Unifying OMA and NOMA

In the two-user MAC, both OMA and NOMA are subschemes of RSMA, as per Fig. 13.
The corresponding messages to streams mapping is illustrated in Table X.

NOMA is a subscheme of RSMA by forcing the encoding of message $W_{1}$
entirely into $s_{\mathrm{1,1}}$ and $W_{2}$ into $s_{2}$ while turning off
$s_{1,2}$ ($P_{1,2}=0$). There is no message splitting at user-1 and one layer
of SIC is required at the receiver to decode and remove the entire message of
one user (i.e., user-1) before decoding the message of the other user (i.e.,
user-2). Note that, though we here refer to this strategy as (uplink) NOMA,
this strategy is nothing else than the traditional SIC at the receiver to
achieve the corner points of the MAC capacity region.

OMA is also a subscheme of RSMA by forcing one user to be scheduled, i.e.,
$W_{1}$ is directly encoded into $s_{1,1}$ while $W_{2}$ is turned off.

####  III-B3 Uplink MIMO RSMA

An extension to the two-user uplink SIMO RSMA in Fig. 12 is the two-user
uplink MIMO RSMA with $N$ transmit antennas at each user and $M$ receive
antennas at the receiver. In this case, each user-$k$ transmits a
$N$-dimensional vector of messages
$\mathbf{w}_{k}=[W_{k}^{(1)},\ldots,W_{k}^{(N)}]$ to the receiver. At user-1,
each of its $N$ messages $W_{1}^{(j)},j\in\\{1,\ldots,N\\}$ is split into two
parts, $W_{1,1}^{(j)}$ and $W_{1,2}^{(j)}$, and encoded independently into
$s_{1,1}^{(j)}$ and $s_{1,2}^{(j)}$. The two $N$-dimensional vectors
$\mathbf{s}_{1,1}=[s_{1,1}^{(1)},\ldots,s_{1,1}^{(N)}]^{T}$ and
$\mathbf{s}_{1,2}=[s_{1,2}^{(1)},\ldots,s_{1,2}^{(N)}]^{T}$ are linearly
precoded by the precoders
$\mathbf{P}_{1,1},\mathbf{P}_{1,2}\in\mathbb{C}^{M\times N}$ and superposed
such that the transmit signal at user-$1$ is

$\mathbf{x}_{1}=\mathbf{P}_{1,1}\mathbf{s}_{1,1}+\mathbf{P}_{1,2}\mathbf{s}_{1,2}.$
(36)

At user-2, the message vector $\mathbf{w}_{2}$ is encoded into a
$N$-dimensional stream vector $\mathbf{s}_{2}$ and linearly precoded by
$\mathbf{P}_{2}$. The transmit signal at user-2 is
$\mathbf{x}=\mathbf{P}_{2}\mathbf{s}_{2}$.

An example of the two-user uplink MIMO RSMA when $N=2$ is illustrated in Fig.
14. Each message in the message vector $\mathbf{w}_{1}$ of user-1 is split
into two parts as

$\displaystyle W_{1}^{(1)}$
$\displaystyle=\left\\{W_{1,1}^{(1)},W_{1,2}^{(1)}\right\\},$ (37)
$\displaystyle W_{1}^{(2)}$
$\displaystyle=\left\\{W_{1,1}^{(2)},W_{1,2}^{(2)}\right\\}.$

The four submessages are respectively encoded into four streams and create two
2-dimensional stream vectors
$\mathbf{s}_{1,1}=\big{[}\begin{array}[]{cc}s_{1,1}^{(1)}&s_{1,1}^{(2)}\end{array}\big{]}^{T}$
and
$\mathbf{s}_{1,2}=\big{[}\begin{array}[]{cc}s_{1,2}^{(1)}&s_{1,2}^{(2)}\end{array}\big{]}^{T}$.
The two stream vectors are linearly precoded and superposed to form the
transmit signal $\mathbf{x}_{1}$. At user-2, the two messages in
$\mathbf{w}_{2}$ are directly encoded into two streams in
$\mathbf{s}_{2}=\big{[}\begin{array}[]{cc}s_{2}^{(1)}&s_{2}^{(2)}\end{array}\big{]}^{T}$
and linearly precoded to form the transmit signal $\mathbf{x}_{2}$. The
mapping of messages to streams for this example is illustrated in Table XI.
Again, MIMO NOMA and OMA are subschemes of MIMO RSMA in the uplink. Though we
here did not split user-2 messages, other variants of uplink RSMA schemes can
also be derived by also splitting $\mathbf{w}_{2}$.

Figure 14: Two-user uplink MIMO RSMA. TABLE XI: Messages-to-streams mapping in the two-user MIMO multiple access channel. | User-1 | User-2
---|---|---
| $\mathbf{s}_{1,1}$ | $\mathbf{s}_{1,2}$ | $\mathbf{s}_{2}$
| $s_{1,1}^{(1)}$ | $s_{1,1}^{(2)}$ | $s_{1,2}^{(1)}$ | $s_{1,2}^{(2)}$ | $s_{2}^{(1)}$ | $s_{2}^{(2)}$
NOMA | $W_{1}^{(1)}$ | $W_{1}^{(2)}$ | – | – | $W_{2}^{(1)}$ | $W_{2}^{(2)}$
OMA | $W_{1}^{(1)}$ | $W_{1}^{(2)}$ | – | – | – | –
RSMA | $W_{1,1}^{(1)}$ | $W_{1,1}^{(2)}$ | $W_{1,2}^{(1)}$ | $W_{1,2}^{(2)}$ | $W_{2}^{(1)}$ | $W_{2}^{(2)}$

###  III-C Lessons Learned

• OMA, SDMA, and NOMA are respectively based on the orthogonalization, treat
interference as noise, and decode interference strategy. This limits the
application of each of those MA schemes to a specific interference regime. •
RSMA relies on the RS interference strategy and is efficient in all
interference regimes. • RSMA bridges, unifies, and generalizes OMA, SDMA and
NOMA. • RSMA is a superset of OMA, SDMA and NOMA, and can specialize to each
of them depending on how messages are mapped to streams. • RSMA is applicable
to both downlink and uplink, to SISO, MISO and MIMO settings. • RSMA can be
extended to the cooperative relay setting and to space-time/frequency
transmission.

##  IV $K$-User RSMA

The schemes developed for two-user in the previous sections can all be
extended to $K$-user and conclusions derived for two-user hold for $K$-user as
well. Nevertheless, the $K$-user scenario also opens the door to other RSMA
schemes with a variable number of SIC layers. In this section, the state-of-
the-art $K$-user RSMA schemes are delineated in the downlink, uplink, and
multi-cell MIMO settings where each node is equipped with multiple antennas.

###  IV-A Downlink

We first consider a downlink symmetric MIMO BC where a transmitter equipped
with $M$ antennas serves $K$ users, each equipped with $N$ receive antennas.
The users are indexed by $\mathcal{K}=\\{1,\ldots,K\\}$. Without loss of
generality, we assume a $Q_{k}$-dimensional vector of messages is transmitted
to user-$k$, i.e., $\mathbf{w}_{k}=[W_{k}^{(1)},\ldots,W_{k}^{(Q_{k})}]^{T}$,
where $Q_{k}\leq\min(M,N)$. Depending on the different RSMA schemes adopted,
user messages $\\{\mathbf{w}_{k},k\in\mathcal{K}\\}$ are split, combined, and
encoded into different stream vectors, which are then mapped to the transmit
antennas using linear or non-linear precoders and forms the transmit signal
$\mathbf{x}$. The signal is transmitted through a MIMO BC and the receive
signal at user-$k$ is given by
$\mathbf{y}_{k}=\mathbf{H}_{k}^{H}\mathbf{x}+\mathbf{n}_{k}$, where
$\mathbf{H}_{k}\in\mathbb{C}^{M\times N}$ is the channel matrix between the
base station and user-$k$,
$\mathbf{n}_{k}\sim\mathcal{CN}(\mathbf{0},\sigma_{n,k}^{2}\mathbf{I}_{N})$ is
the AWGN vector at user-$k$. Without loss of generality, the noise variances
of all users are assumed to be equal to 1, i.e., $\sigma_{n,k}^{2}=1,\forall
k\in\mathcal{K}$. Next, we detail the transceiver architectures for different
RSMA schemes in such symmetric MIMO BC.

####  IV-A1 1-layer RS

The system model of 1-layer RS for MIMO BC is first proposed in [35]. As
illustrated in Fig. 15, each message $W_{k}^{(i)}$ in $\mathbf{w}_{k}$ for
user-$k$ is split into one common sub-message and one private sub-message as
$W_{k}^{(i)}=\\{W_{\mathrm{c},k}^{(i)},W_{\mathrm{p},k}^{(i)}\\},\forall
i\in\\{1,\ldots,Q_{k}\\}$, resulting in one common message vector
$\mathbf{w}_{\mathrm{c},k}=[W_{\mathrm{c},k}^{(1)},\ldots,W_{\mathrm{c},k}^{(Q_{k})}]^{T}$
and one private message vector
$\mathbf{w}_{\mathrm{p},k}=[W_{\mathrm{p},k}^{(1)},\ldots,W_{\mathrm{p},k}^{(Q_{k})}]^{T}$
for user-$k$. The common messages vectors of all users
$\\{\mathbf{w}_{\mathrm{c},1},\ldots,\mathbf{w}_{\mathrm{c},K}\\}$ are
combined into $Q_{\mathrm{c}}$ ($Q_{\mathrm{c}}\leq\min(M,N)$) common messages
$\mathbf{w}_{\mathrm{c}}\in\mathbb{C}^{Q_{\mathrm{c}}\times 1}$ and encoded
into one common stream vector
$\mathbf{s}_{\mathrm{c}}=[s_{\mathrm{c}}^{(1)},\ldots,s_{\mathrm{c}}^{(Q_{\mathrm{c}})}]^{T}$,
which is decoded by all users. The private message vector
$\mathbf{w}_{\mathrm{p},k}$ of user-$k$ is independently encoded into the
private stream vector
$\mathbf{s}_{k}=[s_{k}^{(1)},\ldots,s_{k}^{(Q_{k})}]^{T}$ to be decoded by
user-$k$ only. The $K+1$ data stream vectors
$\mathbf{s}_{\mathrm{c}},\mathbf{s}_{1},\ldots,\mathbf{s}_{K}$ are linearly
precoded by the precoders
$\mathbf{P}_{\mathrm{c}},\mathbf{P}_{1},\ldots,\mathbf{P}_{K}$ and superposed
at the transmitter. The resulting transmit signal is

$\mathbf{x}=\mathbf{P}_{\mathrm{c}}\mathbf{s}_{\mathrm{c}}+\sum_{k=1}^{K}\mathbf{P}_{k}\mathbf{s}_{k}.$
(38)

The signal received at user-$k$ is

$\mathbf{y}=\mathbf{H}_{k}^{H}\mathbf{P}_{\mathrm{c}}\mathbf{s}_{\mathrm{c}}+\sum_{j=1}^{K}\mathbf{H}_{k}^{H}\mathbf{P}_{j}\mathbf{s}_{j}+\mathbf{n}_{k}.$
(39)

User-$k$ employs SIC or joint decoding to decode two stream vectors
$\mathbf{s}_{\mathrm{c}}$ and $\mathbf{s}_{k}$. Only one single layer of SIC
is required at each user if employing SIC to decode the data streams. Assuming
Gaussian signalling, the instantaneous rates of decoding the common and
private stream vectors based on SIC are given by

$\displaystyle
R_{\mathrm{c},k}=\log_{2}\det\left(\mathbf{I}+\mathbf{P}_{\mathrm{c}}^{H}\mathbf{H}_{k}\left(\mathbf{R}_{\mathrm{c},k}\right)^{-1}\mathbf{H}_{k}^{H}\mathbf{P}_{\mathrm{c}}\right),$
(40) $\displaystyle
R_{k}=\log_{2}\det\left(\mathbf{I}+\mathbf{P}_{k}^{H}\mathbf{H}_{k}\left(\mathbf{R}_{k}\right)^{-1}\mathbf{H}_{k}^{H}\mathbf{P}_{k}\right),$

where $\mathbf{R}_{\mathrm{c},k}$ and $\mathbf{R}_{k}$ are noise plus
interference covariance matrices defined as

$\displaystyle\mathbf{R}_{\mathrm{c},k}=\mathbf{I}+\sum_{j=1}^{K}\mathbf{H}_{k}^{H}\mathbf{P}_{j}\mathbf{P}_{j}^{H}\mathbf{H}_{k},$
(41) $\displaystyle\mathbf{R}_{k}=\mathbf{I}+\sum_{j=1,j\neq
k}^{K}\mathbf{H}_{k}^{H}\mathbf{P}_{j}\mathbf{P}_{j}^{H}\mathbf{H}_{k}.$

To guarantee the common stream vector is successfully decoded by all users, it
is achievable rate is
$R_{\mathrm{c}}=\min_{k}\\{R_{\mathrm{c},1},\ldots,R_{\mathrm{c},K}\\}$. As
$\mathbf{s}_{\mathrm{c}}$ contains sub-messages of $K$ users, we have
$\sum_{k\in\mathcal{K}}C_{k}=R_{\mathrm{c}}$, where $C_{k}$ denotes the
portion of $R_{\mathrm{c}}$ allocated to user-$k$ for the transmission of
$\mathbf{w}_{{\mathrm{c}},k}$. The overall achievable rate of user-$k$ is
$R_{k,\mathrm{tot}}=C_{k}+R_{k}$.

Figure 15: $K$-user downlink 1-layer MIMO RSMA.

####  IV-A2 Hierarchical RS

Hierarchical RS (HRS) is proposed in [58] for MISO BC where the transmitter is
equipped with multiple antennas and each receiver is equipped with a single
antenna. In this subsection, we extend the system model to MIMO BC where each
node has multiple antennas. In HRS, users are clustered into $G$ separate
groups indexed by $\mathcal{G}=\\{1,\ldots,G\\}$ according to the similarity
of their channel covariance matrices. Each group-$g$ contains a subset of
users $\mathcal{K}_{g}$ such that
$\bigcup_{g\in\mathcal{G}}\mathcal{K}_{g}=\mathcal{K}$. In contrast to 1-layer
RS where each user message is split into two parts, in HRS, each user message
$W_{k}^{(i)},i\in\\{1,\ldots,Q_{k}\\}$ in $\mathbf{w}_{k}$ for user-$k$
(assuming user-$k$ is in group $g$) is split into three sub-messages as
$W_{k}^{(i)}=\\{W_{\mathcal{K},k}^{(i)},W_{\mathcal{K}_{g},k}^{(i)},W_{k,k}^{(i)}\\}$,
resulting in three sub-message vectors, namely, an inter-group common message
vector
$\mathbf{w}_{k}^{\mathcal{K}}=\\{W_{\mathcal{K},k}^{(1)},\ldots,W_{\mathcal{K},k}^{(Q_{k})}\\}$,
an inner-group common message vector
$\mathbf{w}_{k}^{\mathcal{K}_{g}}=\\{W_{\mathcal{K}_{g},k}^{(1)},\ldots,W_{\mathcal{K}_{g},k}^{(Q_{k})}\\}$,
and a private message vector
$\mathbf{w}_{k}^{k}=\\{W_{k,k}^{(1)},\ldots,W_{k,k}^{(Q_{k})}\\}$. The inter-
group common messages of all users
$\\{\mathbf{w}_{1}^{\mathcal{K}},\ldots,\mathbf{w}_{K}^{\mathcal{K}}\\}$ are
combined into $Q_{\mathrm{c}}$ ($Q_{\mathrm{c}}\leq\min(M,N)$) common messages
$\mathbf{w}_{\mathcal{K}}\in\mathbb{C}^{Q_{\mathrm{c}}\times 1}$ and encoded
into one common stream vector
$\mathbf{s}_{\mathcal{K}}=[s_{\mathcal{K}}^{(1)},\ldots,s_{\mathcal{K}}^{(Q_{\mathrm{c}})}]^{T}$,
which is decoded by all users. For group-$g,g\in\mathcal{G}$, the inner-group
common message vectors of users in group-$g$
$\\{\mathbf{w}_{k}^{\mathcal{K}_{g}},\forall k\in\mathcal{K}_{g}\\}$ are
combined into $Q_{\mathrm{c}g}$ ($Q_{\mathrm{c}g}\leq\min(M,N)$) common
messages $\mathbf{w}_{\mathcal{K}_{g}}\in\mathbb{C}^{Q_{\mathrm{c}g}\times 1}$
and encoded into one common stream vector
$\mathbf{s}_{\mathcal{K}_{g}}=[s_{\mathcal{K}_{g}}^{(1)},\ldots,s_{\mathcal{K}_{g}}^{(Q_{\mathrm{c}g})}]^{T}$
to be decoded by all users in group-$g$. The private message vector
$\mathbf{w}_{k}^{k}$ of user-$k$ is independently encoded into the private
stream vector $\mathbf{s}_{k}=[s_{k}^{(1)},\ldots,s_{k}^{(Q_{k})}]^{T}$ to be
decoded by user-$k$ only. The overall $K+G+1$ data stream vectors
$\mathbf{s}_{\mathcal{K}},\mathbf{s}_{\mathcal{K}_{1}},\ldots,\mathbf{s}_{\mathcal{K}_{G}},\mathbf{s}_{1},\ldots,\mathbf{s}_{K}$
are linearly precoded by the corresponding precoders
$\mathbf{P}_{\mathcal{K}},\mathbf{P}_{\mathcal{K}_{1}},\ldots,\mathbf{P}_{\mathcal{K}_{G}},\mathbf{P}_{1},\ldots,\mathbf{P}_{K}$.
The resulting transmit signal is

$\mathbf{x}=\mathbf{P}_{\mathcal{K}}\mathbf{s}_{\mathcal{K}}+\sum_{g=1}^{G}\mathbf{P}_{\mathcal{K}_{g}}\mathbf{s}_{\mathcal{K}_{g}}+\sum_{k=1}^{K}\mathbf{P}_{k}\mathbf{s}_{k}.$
(42)

The signal received at user-$k$ is

$\mathbf{y}=\mathbf{H}_{k}^{H}\mathbf{P}_{\mathcal{K}}\mathbf{s}_{\mathcal{K}}+\sum_{g=1}^{G}\mathbf{H}_{k}^{H}\mathbf{P}_{\mathcal{K}_{g}}\mathbf{s}_{\mathcal{K}_{g}}+\sum_{j=1}^{K}\mathbf{H}_{k}^{H}\mathbf{P}_{j}\mathbf{s}_{j}+\mathbf{n}_{k}.$
(43)

In contrast to 1-layer RS where each user only decodes two streams, in HRS,
each user-$k$ ($k\in\mathcal{K}_{g}$) employs SICs or joint decoding to decode
three stream vectors $\mathbf{s}_{\mathcal{K}}$,
$\mathbf{s}_{\mathcal{K}_{g}}$, and $\mathbf{s}_{k}$. Two layers of SIC are
required at each user if employing SIC. Assuming Gaussian signalling, the
instantaneous rates of decoding $\mathbf{s}_{\mathcal{K}}$,
$\mathbf{s}_{\mathcal{K}_{g}}$, and $\mathbf{s}_{k}$ based on SIC are given by

$\displaystyle
R_{\mathcal{K},k}=\log_{2}\det\left(\mathbf{I}+\mathbf{P}_{\mathcal{K}}^{H}\mathbf{H}_{k}\left(\mathbf{R}_{\mathcal{K},k}\right)^{-1}\mathbf{H}_{k}^{H}\mathbf{P}_{\mathcal{K}}\right),$
(44) $\displaystyle
R_{{\mathcal{K}_{g}},k}=\log_{2}\det\left(\mathbf{I}+\mathbf{P}_{{\mathcal{K}_{g}}}^{H}\mathbf{H}_{k}\left(\mathbf{R}_{{\mathcal{K}_{g}},k}\right)^{-1}\mathbf{H}_{k}^{H}\mathbf{P}_{{\mathcal{K}_{g}}}\right),$
$\displaystyle
R_{k}=\log_{2}\det\left(\mathbf{I}+\mathbf{P}_{k}^{H}\mathbf{H}_{k}\left(\mathbf{R}_{k}\right)^{-1}\mathbf{H}_{k}^{H}\mathbf{P}_{k}\right),$

where the noise plus interference covariance matrices
$\mathbf{R}_{\mathcal{K},k}$, $\mathbf{R}_{{\mathcal{K}_{g}},k}$, and
$\mathbf{R}_{k}$ are defined as

$\displaystyle\mathbf{R}_{\mathcal{K},k}=\mathbf{I}+\sum_{g=1}^{G}\mathbf{H}_{k}^{H}\mathbf{P}_{\mathcal{K}_{g}}\mathbf{P}_{\mathcal{K}_{g}}^{H}\mathbf{H}_{k}+\sum_{j=1}^{K}\mathbf{H}_{k}^{H}\mathbf{P}_{j}\mathbf{P}_{j}^{H}\mathbf{H}_{k},$
(45) $\displaystyle\mathbf{R}_{{\mathcal{K}_{g}},k}=\mathbf{I}+\sum_{i=1,i\neq
g}^{G}\mathbf{H}_{k}^{H}\mathbf{P}_{\mathcal{K}_{i}}\mathbf{P}_{\mathcal{K}_{i}}^{H}\mathbf{H}_{k}+\sum_{j=1}^{K}\mathbf{H}_{k}^{H}\mathbf{P}_{j}\mathbf{P}_{j}^{H}\mathbf{H}_{k},$
$\displaystyle\mathbf{R}_{k}=\mathbf{I}+\sum_{i=1,i\neq
g}^{G}\mathbf{H}_{k}^{H}\mathbf{P}_{\mathcal{K}_{i}}\mathbf{P}_{\mathcal{K}_{i}}^{H}\mathbf{H}_{k}+\sum_{j=1,j\neq
k}^{K}\mathbf{H}_{k}^{H}\mathbf{P}_{j}\mathbf{P}_{j}^{H}\mathbf{H}_{k}.$

To guarantee the successful decoding of the inter-group and inner-group common
stream vectors, the achievable rates follow
$R_{\mathcal{K}}=\min_{k}\\{R_{{\mathcal{K}},k},k\in\mathcal{K}\\}$ and
$R_{\mathcal{K}_{g}}=\min_{k}\\{R_{{\mathcal{K}_{g}},k},k\in{\mathcal{K}_{g}}\\}$.
As $\mathbf{s}_{\mathcal{K}}$ and $\mathbf{s}_{\mathcal{K}_{g}}$ contain sub-
messages of multiple users, we have
$\sum_{k\in\mathcal{K}}C_{k}^{\mathcal{K}}=R_{\mathcal{K}}$ and
$\sum_{k\in\mathcal{K}_{g}}C_{k}^{\mathcal{K}_{g}}=R_{\mathcal{K}_{g}}$, where
$C_{k}^{\mathcal{K}}$ and $C_{k}^{\mathcal{K}_{g}}$ respectively denote the
portion of $R_{\mathcal{K}}$ and $R_{\mathcal{K}_{g}}$ allocated to user-$k$
for the transmission of $\mathbf{w}_{k}^{{\mathcal{K}}}$ and
$\mathbf{w}_{k}^{{\mathcal{K}_{g}}}$. The overall achievable rate of user-$k$
is $R_{k,\mathrm{tot}}=C_{k}^{\mathcal{K}}+C_{k}^{\mathcal{K}_{g}}+R_{k}$. It
is easy to observe that HRS is a more general framework than 1-layer RS for
MIMO BC. HRS boils down to 1-layer RS when the inner-group common stream
vectors $\\{\mathbf{s}_{{\mathcal{K}_{g}}},\forall g\in{\mathcal{G}}\\}$ are
turned off, i.e., $\mathbf{P}_{{\mathcal{K}_{g}}}=\mathbf{0},\forall
g\in{\mathcal{G}}$. In this case, $\mathbf{s}_{{\mathcal{K}}}$ for HRS is
equivalent to $\mathbf{s}_{c}$ for 1-layer RS.

In Fig. 16, a four-user two-group HRS example is illustrated with user-1 and
user-2 in group-1, user-3 and user-4 in group-2. The user sets
$\mathcal{K}=\\{1,2,3,4\\}$, $\mathcal{K}_{1}=\\{1,2\\}$,
$\mathcal{K}_{2}=\\{3,4\\}$ are simply denoted as $1234$, $12$, $34$,
respectively. User-1 and user-2 are required to decode the inter-group common
stream vector $\mathbf{s}_{1234}$ and the inner-group common stream vector
$\mathbf{s}_{12}$ while user-3 and user-4 are required to decode the
$\mathbf{s}_{1234}$ and $\mathbf{s}_{34}$.

Figure 16: $4$-user downlink MIMO HRS.

####  IV-A3 Generalized RS

Generalized RS (GRS) is proposed in [8] for MISO BC to further enhance the
spectral efficiency by splitting and encoding multiple common streams, which
are required to be decoded by different subsets of users. It is a generalized
transmission framework that embraces 1-layer RS, HRS, linearly precoded MU-
MIMO, and MIMO NOMA as special cases. We here extend it to MIMO BC. At the
transmitter, each user message is split into $2^{K-1}$ parts, i.e.,
$W_{k}^{(i)}=\\{W_{\mathcal{A}^{\prime},k}^{(i)}\mid\mathcal{A}^{\prime}\in\mathcal{K},k\in\mathcal{A}^{\prime}\\}$.
The message vector $\mathbf{w}_{k}$ is therefore split into $2^{K-1}$ sub-
message vectors
$\mathbf{w}_{k}^{\mathcal{A}^{\prime}},\forall\mathcal{A}^{\prime}\in\mathcal{K},k\in\mathcal{A}^{\prime}\\}$
with
$\mathbf{w}_{k}^{\mathcal{A}^{\prime}}=\\{W_{\mathcal{A}^{\prime},k}^{(1)},\ldots,W_{\mathcal{A}^{\prime},k}^{(Q_{k})}\\}$.
The sub-message vectors $\\{\mathbf{w}_{k}^{\mathcal{A}}\mid
k\in\mathcal{A}\\}$ for all users in a given subset
$\mathcal{A}\in\mathcal{K}$ are combined and encoded into one stream vector
$\mathbf{s}_{\mathcal{A}}$ of dimension ${Q_{\mathrm{c}\mathcal{A}}\times 1}$
which is only decoded by users in ${\mathcal{A}}$ and treated as noise by the
remaining users. Following the concept of stream order introduced in GRS for
MISO BC [8], we denote the stream vectors to be decoded by $l$ number of users
as $l$-order stream vectors. All $l$-order stream vectors
$\\{\mathbf{s}_{\mathcal{A}^{\prime}}\mid\mathcal{A}^{\prime}\in\mathcal{K},|\mathcal{A}^{\prime}|=l\\}$
are wrapped up into a larger dimensional vector
$\mathbf{s}_{l}\in\mathbb{C}^{\sum_{\mathcal{A}^{\prime}\in\mathcal{K},\mathcal{|}\mathcal{A}^{\prime}|=l}Q_{\mathrm{c}\mathcal{A^{\prime}}}\times
1}$ and linearly precoded by the precoding matrix $\mathbf{P}_{l}$ composed by
$\\{\mathbf{P}_{\mathcal{A}^{\prime}}\mid\mathcal{A}^{\prime}\in\mathcal{K},|\mathcal{A}^{\prime}|=l\\}$.
The resulting transmit signal is

$\mathbf{{x}}=\sum_{l=1}^{K}\mathbf{{P}}_{{l}}\mathbf{{s}}_{{l}}=\sum_{l=1}^{K}\sum_{\mathcal{A}^{\prime}\subseteq\mathcal{K},|\mathcal{A}^{\prime}|=l}\mathbf{{P}}_{\mathcal{A}^{\prime}}{\mathbf{s}}_{\mathcal{A}^{\prime}}.$
(46)

At user-$k$, $2^{K-1}-1$ layers of SIC or joint decoding is employed to decode
the required streams vectors. The set of $l$-order stream vectors required to
be decoded at user-$k$ is denoted by
$\mathcal{S}_{l,k}=\\{\mathbf{s}_{\mathcal{A^{\prime}}}\mid\mathcal{A}^{\prime}\subseteq\mathcal{K},|\mathcal{A}^{\prime}|=l,k\in\mathcal{A}^{\prime}\\}$.
Following the decoding order from the $K$-order stream vector down to the
$1$-order stream vector as well as a certain decoding order $\pi_{l,k}$ to
decode the $l$-order stream vectors in $\mathcal{S}_{l,k}$ (i.e.,
$\mathbf{s}_{\pi_{l,k}(i)}$ is decoded before $\mathbf{s}_{\pi_{l,k}(j)}$ if
$i<j$), we obtain the rate of decoding $l$-order stream vector
$\mathbf{s}_{\pi_{l,k}(i)}$ at user-$k$ under Gaussian signalling and SIC
decoding as

$\displaystyle R_{\pi_{l,k}{(i)},k}=$ (47)
$\displaystyle\,\,\log_{2}\det\left(\mathbf{I}+\mathbf{P}_{\pi_{l,k}{(i)}}^{H}\mathbf{H}_{k}\left(\mathbf{R}_{\pi_{l,k}{(i)},k}\right)^{-1}\mathbf{H}_{k}^{H}\mathbf{P}_{\pi_{l,k}{(i)}}\right),$

where $\mathbf{R}_{\pi_{l,k}{(i)},k}$ is given as

$\displaystyle\mathbf{R}_{\pi_{l,k}{(i)},k}$
$\displaystyle=\mathbf{I}+\sum_{j>i}\mathbf{{H}}_{k}^{H}\mathbf{{P}}_{\pi_{l,k}(j)}\mathbf{{P}}_{\pi_{l,k}(j)}^{H}\mathbf{{H}}_{k}$
(48)
$\displaystyle+\sum_{l^{\prime}=1}^{l-1}\sum_{j=1}^{|\mathcal{S}_{l^{\prime},k}|}\mathbf{{H}}_{k}^{H}\mathbf{{P}}_{\pi_{l^{\prime},k}(j)}\mathbf{{P}}_{\pi_{l^{\prime},k}(j)}^{H}\mathbf{{H}}_{k}$
$\displaystyle+\sum_{\mathcal{A}^{\prime}\subseteq\mathcal{K},k\notin\mathcal{A}^{\prime}}\mathbf{{H}}_{k}^{H}\mathbf{{P}}_{{\mathcal{A}^{\prime}}}\mathbf{{P}}_{{\mathcal{A}^{\prime}}}^{H}\mathbf{{H}}_{k}.$

To guarantee the successful decoding of each stream vector
$\mathbf{s}_{\mathcal{A}}$, the achievable rates follow
$R_{\mathcal{A}}=\min_{k}\\{R_{{\mathcal{A}},k},k\in{\mathcal{A}}\\}$. As
$\mathbf{s}_{\mathcal{A}}$ contains sub-messages of users in ${\mathcal{A}}$,
we have $\sum_{k\in\mathcal{A}}C_{k}^{\mathcal{K}}=R_{\mathcal{A}}$, where
$C_{k}^{\mathcal{A}}$ denotes the portion of $R_{\mathcal{A}}$ allocated to
user-$k$ for the transmission of $\mathbf{w}_{k}^{{\mathcal{A}}}$. The overall
achievable rate of user-$k$ is
$R_{k,\mathrm{tot}}=\sum_{k\in\mathcal{A}^{\prime}}C_{k}^{\mathcal{A}^{\prime}}+R_{k,k}$.
GRS is a more general framework than 1-layer RS and HRS for MIMO BC. It boils
down to 1-layer RS when only the $K$-order stream vector
$\mathbf{s}_{\mathcal{K}}$ and the $1$-order stream vectors
$\mathbf{s}_{k},k\in\mathcal{K}$ are active. $\mathbf{s}_{\mathcal{K}}$ for
GRS is equivalent to $\mathbf{s}_{c}$ for 1-layer RS in such case. It reduces
to HRS when only $\mathbf{s}_{\mathcal{K}}$, $\mathbf{s}_{k},k\in\mathcal{K}$,
and $\mathbf{s}_{\mathcal{K}_{g}},g\in\mathcal{G}$ are active.

A toy $3$-user GRS example is illustrated in Fig. 17. The user message vector
of each user is split into 4 sub-message vectors. For user-1, $\mathbf{w}_{1}$
is split into $\\{\mathbf{w}_{1}^{123}$, $\mathbf{w}_{1}^{12}$,
$\mathbf{w}_{1}^{13}$, $\mathbf{w}_{1}^{1}\\}$, each is required to be decoded
by different groups of users.
$\\{\mathbf{w}_{1}^{123},\mathbf{w}_{2}^{123},\mathbf{w}_{3}^{123}\\}$ are
combined into $\mathbf{w}_{123}$, which is then encoded into the $3$-order
stream vector $\mathbf{s}_{123}$. Similarly,
$\\{\mathbf{w}_{1}^{12},\mathbf{w}_{2}^{12}\\}$ are combined into
$\mathbf{w}_{12}$ and encoded into the $2$-order stream vector
$\mathbf{s}_{12}$. Following this method, we obtain in total 7 transmit stream
vectors. 3 layers of SIC is adopted at each user to decode the intended $4$
stream vectors. In this example, user-$1$ decodes $\mathbf{s}_{123}$,
$\mathbf{s}_{12}$, $\mathbf{s}_{13}$, $\mathbf{s}_{1}$ sequentially with
$\mathbf{s}_{\pi_{2,1}{(1)}}=\mathbf{s}_{12}$ and
$\mathbf{s}_{\pi_{2,1}{(2)}}=\mathbf{s}_{13}$.

Figure 17: $3$-user downlink MIMO generalized RS.

####  IV-A4 Dirty paper coded RS

Dirty paper coded RS (DPCRS) is a non-linearly precoded RS framework built
upon DPC. It is proposed in [9] for MISO BC and is extended to MIMO BC in this
subsection. We first illustrate the transmission architecture of the simplest
DPCRS model 1-layer Dirty Paper Coded RS (1-DPCRS), and then briefly discuss
its extension to multi-layer dirty paper coded rate-splitting (M-DPCRS).

Fig. 18 illustrates the proposed 1-layer MIMO DPCRS. It is an extension of
1-layer RS in Section IV-A1 by enabling dirty paper encoded private streams.
Following 1-layer RS, the $K$ user messages are split and combined into one
common message vector $\mathbf{w}_{\mathrm{c}}$ and $K$ private message
vectors $\mathbf{w}_{\mathrm{p},1},\ldots,\mathbf{w}_{\mathrm{p},K}$.
$\mathbf{w}_{\mathrm{c}}$ is encoded into a common stream vector
$\mathbf{s}_{\mathrm{c}}$ using a codebook shared by all users while
$\mathbf{w}_{\mathrm{p},1},\ldots,\mathbf{w}_{\mathrm{p},K}$ are encoded and
precoded by DPC for a certain encoding order $\pi$ into the private stream
vectors $\mathbf{s}_{\pi(1)},\ldots,\mathbf{s}_{\pi(K)}$. The resulting
transmit signal is

$\mathbf{x}={{\mathbf{P}_{\mathrm{c}}\mathbf{s}_{\mathrm{c}}}}+{{\sum_{k=1}^{K}\mathbf{P}_{\pi(k)}\mathbf{s}_{\pi(k)}}},$
(49)

where $\mathbf{P}_{\mathrm{c}},\mathbf{P}_{\pi(1)},\ldots,\mathbf{P}_{\pi(K)}$
are the precoders for the corresponding streams
$\mathbf{s}_{\mathrm{c}},\mathbf{s}_{\pi(1)},\ldots,\mathbf{s}_{\pi(K)}$.

Each user-$\pi(k)$ decodes $\mathbf{s}_{\mathrm{c}}$ and $\mathbf{s}_{\pi(k)}$
based on SIC or joint decoding. Assuming Gaussian signalling, the
instantaneous rates of decoding the common and private stream vectors based on
SIC are given by

$\displaystyle
R_{\mathrm{c},\pi(k)}=\log_{2}\det\left(\mathbf{I}+\mathbf{P}_{\mathrm{c}}^{H}\mathbf{H}_{\pi(k)}\left(\mathbf{R}_{\mathrm{c},\pi(k)}\right)^{-1}\mathbf{H}_{\pi(k)}^{H}\mathbf{P}_{\mathrm{c}}\right),$
(50) $\displaystyle
R_{\pi(k)}=\log_{2}\det\left(\mathbf{I}+\mathbf{P}_{\pi(k)}^{H}\mathbf{H}_{\pi(k)}\left(\mathbf{R}_{\pi(k)}\right)^{-1}\mathbf{H}_{\pi(k)}^{H}\mathbf{P}_{\pi(k)}\right),$

where $\mathbf{R}_{\mathrm{c},{\pi(k)}}$ and $\mathbf{R}_{\pi(k)}$ are noise
plus interference covariance matrices defined as

$\displaystyle\mathbf{R}_{\mathrm{c},{\pi(k)}}=\mathbf{I}+\sum_{j=1}^{K}\mathbf{H}_{\pi(k)}^{H}\mathbf{P}_{\pi(j)}\mathbf{P}_{\pi(j)}^{H}\mathbf{H}_{\pi(k)},$
(51)
$\displaystyle\mathbf{R}_{\pi(k)}=\mathbf{I}+\sum_{j>k}^{K}\mathbf{H}_{\pi(k)}^{H}\mathbf{P}_{\pi(j)}\mathbf{P}_{\pi(j)}^{H}\mathbf{H}_{\pi(k)}.$

The calculation of the overall achievable rate for each user follows 1-layer
RS, which is not detailed here. Besides 1-layer MIMO DPCRS, we can also
integrate DPC and different RS schemes to enhance the system performance
especially in imperfect CSIT. For instance, Fig. 19 illustrates another DPCRS
scheme, namely, multi-layer MIMO DPCRS, which marries DPC and GRS. In contrast
to GRS in Fig. 17 which relies on linear precoding to precode all streams,
multi-layer MIMO DPCRS considers dirty paper coded private streams.

Figure 18: $K$-user downlink 1-layer MIMO DPCRS. Figure 19: $3$-user downlink
multi-layer MIMO DPCRS.

###  IV-B Uplink

Uplink RSMA is first introduced in [23] for SISO MAC and further extended to
SIMO MAC in [25]. The main benefit of uplink RSMA discovered in existing works
is that the capacity region of the Gaussian MAC can be achieved by uplink RSMA
without time sharing among users. In the following, we introduce a generic
uplink RSMA transmission framework for MIMO MAC, which is simply denoted by
uplink MIMO RSMA.

Fig. 20 illustrates the proposed $K$-user uplink MIMO RSMA based on SC at the
transmitters and SIC at the receiver. A $Q_{k}$-dimensional message vector
$\mathbf{w}_{k}=[W_{k}^{(1)},\ldots,W_{k}^{(Q_{k})}]^{T}$ is transmitted from
user-$k$ to the receiver, where $Q_{k}\leq\min(M,N)$. At user-$k$, each user
message $W_{k}^{(i)}$ is split into two sub-messages $W_{k,1}^{(i)}$ and
$W_{k,2}^{(i)}$, resulting in two sub-message vectors $\mathbf{w}_{k,1}$ and
$\mathbf{w}_{k,2}$. Note that splitting $\sum_{k=1}^{K}Q_{k}-1$ messages is
sufficient to achieve the capacity region without time sharing in this
setting. For illustration simplicity, we here consider a more general model in
which the user messages of all users are split. At user-$k$, the sub-message
vectors $\mathbf{w}_{k,1}$ and $\mathbf{w}_{k,2}$ are independently encoded
into $\mathbf{s}_{k,1}$ and $\mathbf{s}_{k,2}$, precoded by $\mathbf{P}_{k,1}$
and $\mathbf{P}_{k,2}$, and superposed at the transmitter. The resulting
transmit signal is

$\mathbf{x}_{k}=\mathbf{P}_{k,1}\mathbf{s}_{k,1}+\mathbf{P}_{k,2}\mathbf{s}_{k,2},\forall
k\in\mathcal{K}.$ (52)

The signals of all users are transmitted via MIMO MAC, the receive signal is

$\mathbf{y}=\sum_{k\in\mathcal{K}}\mathbf{H}_{k}\mathbf{x}_{k}+\mathbf{n},$
(53)

where $\mathbf{H}_{k}\in\mathbb{C}^{M\times N}$ is the channel vector between
user-$k$ and the receiver, and
$\mathbf{n}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{M})$ is the AWGN vector.
The receiver then employs the receive filters
$\mathbf{W}_{k,1},\mathbf{W}_{k,2},k\in\mathcal{K}$ to detect the stream
vectors for all users. Denote the decoding order of the $2K$ received stream
vectors $\\{\mathbf{s}_{k,i}\mid k\in\\{1,\ldots,K\\},i\in\\{1,2\\}\\}$ by
$\pi$ such that $\pi_{k,i}<\pi_{k^{\prime},i^{\prime}}$ if $\mathbf{s}_{k,i}$
is decoded before $\mathbf{s}_{k^{\prime},i^{\prime}}$. $2K-1$ layers of SIC
is needed to decode all stream vectors. Assuming Gaussian signaling and
infinite blocklength, the rate of decoding $s_{ki}$ at the receiver is given
as follows

$\displaystyle
R_{k,i}=\log_{2}\det\left(\mathbf{I}+\mathbf{P}_{k,i}^{H}\mathbf{H}_{k}^{H}\mathbf{W}_{k,i}^{H}\left(\mathbf{R}_{k,i}\right)^{-1}\mathbf{W}_{k,i}\mathbf{H}_{k}\mathbf{P}_{k,i}\right),$
(54)

where $\mathbf{R}_{k,i}$ is defined as

$\displaystyle\mathbf{R}_{k,i}=\mathbf{I}+\sum\limits_{\pi_{k^{\prime},i^{\prime}}>\pi_{k,i}}\mathbf{W}_{k,i}\mathbf{H}_{k^{\prime}}\mathbf{P}_{k^{\prime},i^{\prime}}\mathbf{P}_{k^{\prime},i^{\prime}}^{H}\mathbf{H}_{k^{\prime}}^{H}\mathbf{W}_{k,i}^{H}.$
(55)

The proposed uplink MIMO RSMA model is generic. It embraces uplink SISO RSMA
in [23] and uplink SIMO RSMA in [25] as two sub-schemes. The study of uplink
MIMO RSMA is still in its infancy. Its SE/EE performance and applications in
different services such as eMBB, URLLC, and mMTC or their hybrid services are
worth more investigations.

Figure 20: $K$-user uplink MIMO RSMA.

###  IV-C Multi-cell

In multi-cell networks, the transmission from multiple transmitters to
multiple receivers can be categorized into “coordinated transmission" and
“cooperative transmission" depending on whether the data is shared among the
transmitters [6]. RSMA has been investigated in both transmissions [59, 60,
61, 62]. It is shown to be a promising strategy to enhance SE by providing a
powerful inter-cell and intra-cell interference management capability. In the
following subsections, the transmission models of MIMO RSMA for “coordinated
transmission" and “cooperative transmission" are respectively delineated.

####  IV-C1 Coordinated transmission

A $K$-cell coordinated MIMO is illustrated in Fig.21(a). The message of each
user is sent from the serving cell (with one RSMA-enabled transmitter) while
the resource allocation and user scheduling are coordinated among cells. At
each transmitter-$k$, the message vector $\mathbf{w}_{k}$ for user-$k$ is
split into two sub-message vectors $\mathbf{w}_{\mathrm{c},k}$ and
$\mathbf{w}_{\mathrm{p},k}$, independently encoded into two stream vectors
$\mathbf{s}_{\mathrm{c},k}$ and $\mathbf{s}_{\mathrm{p},k}$, and linearly
precoded by the precoding matrices $\mathbf{P}_{\mathrm{c},k}$ and
$\mathbf{P}_{\mathrm{p},k}$. The transmit signal at transmitter-$k$ is given
as

$\mathbf{{x}}_{k}=\mathbf{{P}}_{\mathrm{c},k}\mathbf{s}_{\mathrm{c},k}+\mathbf{{P}}_{\mathrm{p},k}\mathbf{s}_{\mathrm{p},k}.$
(56)

The signal of all transmitters are transmitted to all users. The signal
received at user-$k$ is

$\mathbf{y}_{k}=\sum_{j=1}^{K}\mathbf{H}_{kj}^{H}(\mathbf{{P}}_{\mathrm{c},j}\mathbf{s}_{\mathrm{c},j}+\mathbf{P}_{\mathrm{p},j}\mathbf{s}_{\mathrm{p},j})+\mathbf{n}_{k},$
(57)

where $\mathbf{H}_{kj}^{H}\in\mathbb{C}^{N\times M}$ is the channel between
transmitter-$j$ and user-$k$ and
$\mathbf{n}_{k}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{N})$ is the AWGN vector
at user-$k$. Each user-$k$ is required to decode all common stream vectors
$\mathbf{s}_{\mathrm{c},1},\ldots,\mathbf{s}_{\mathrm{c},K}$ and the intended
private stream $\mathbf{s}_{\mathrm{p},k}$ based on SIC or joint decoding. For
a certain decoding order $\pi_{k}$ such that
$\mathbf{s}_{\mathrm{c},\pi_{k}(i)}$ is decoded before
$\mathbf{s}_{\mathrm{c},\pi_{k}(j)}$ at user-$k$ if $i<j$. Assuming Gaussian
signaling and infinite blocklength, the rate of decoding
$\mathbf{s}_{\mathrm{c},\pi_{k}(i)}$ and $\mathbf{s}_{\mathrm{p},k}$ at
user-$k$ are obtained as follows

$\displaystyle R_{k,\pi_{k}(i)}^{\mathrm{c}}=$ (58)
$\displaystyle\log_{2}\det\left(\mathbf{I}+\mathbf{P}_{\mathrm{c},\pi_{k}(i)}^{H}\mathbf{H}_{k\pi_{k}(i)}\left(\mathbf{R}_{\mathrm{c},{\pi_{k}(i)}}\right)^{-1}\mathbf{H}_{{k\pi_{k}(i)}}^{H}\mathbf{P}_{\mathrm{c},\pi_{k}(i)}\right),$
$\displaystyle
R_{k}^{\mathrm{p}}=\log_{2}\det\left(\mathbf{I}+\mathbf{P}_{\mathrm{p},k}^{H}\mathbf{H}_{kk}\left(\mathbf{R}_{k}\right)^{-1}\mathbf{H}_{kk}^{H}\mathbf{P}_{\mathrm{p},k}\right).$

where $\mathbf{R}_{\mathrm{c},{\pi_{k}(i)}}$ and $\mathbf{R}_{k}$ are defined
as

$\displaystyle\mathbf{R}_{\mathrm{c},{\pi_{k}(i)}}=\mathbf{I}+\sum_{j>i}\mathbf{H}_{k\pi_{k}(j)}^{H}\mathbf{P}_{\mathrm{c},\pi_{k}(j)}\mathbf{P}_{\mathrm{c},\pi_{k}(j)}^{H}\mathbf{H}_{k\pi_{k}(j)}$
(59)
$\displaystyle\quad\quad\quad\,\,+\sum_{j=1}^{K}\mathbf{H}_{kj}^{H}\mathbf{P}_{\mathrm{p},j}\mathbf{P}_{\mathrm{p},j}^{H}\mathbf{H}_{kj},$
$\displaystyle\mathbf{R}_{k}=\mathbf{I}+\sum_{j\in\mathcal{K},j\neq
k}\mathbf{H}_{kj}^{H}\mathbf{P}_{\mathrm{p},j}\mathbf{P}_{\mathrm{p},j}^{H}\mathbf{H}_{kj}.$

The achievable rate of user-$k$ is

$R_{k,\mathrm{tot}}=\min\left\\{R_{1,k}^{\mathrm{c}},\ldots,R_{K,k}^{\mathrm{c}}\right\\}+R_{k}^{\mathrm{p}}.$
(60)

As discussed in [59], The performance of coordinated MIMO RSMA could be
further boosted by splitting the message of each user into $N>2$ parts. Each
part is decoded by a different group of users with the same instantaneous CSIT
quality.

####  IV-C2 Cooperative transmission

As discussed in [25], cooperative MIMO requires all transmitters to be
connected as a virtual giant transmitter to serve all users in the downlink.
All downlink transmission frameworks of RSMA discussed in IV-A therefore can
be applied for cooperative MIMO. Compared with single-cell MIMO BC which is
subject to a sum-power constraint, multi-cell cooperative MIMO BC requires
per-cell transmit power constraint and the fronthaul capacity constraints
[62].

(a) Coordinated transmission

(b) Cooperative transmission

Figure 21: Multi-cell RSMA-enabled transmission [25].

###  IV-D Numerical Results

Next, we show numerically the benefits of RSMA in terms of the significant
wireless communication performance metrics, namely, spectral efficiency,
energy efficiency, generality, flexibility, robustness, reliability and
latency.

Unless stated otherwise, the elements of the channel matrix $\mathbf{H}_{k}$
are generated as i.i.d. complex Gaussian random variables
$\mathcal{CN}(0,\sigma_{k}^{2})$ and the noise variances are normalized. The
imperfect CSIT model is
$\mathbf{H}_{k}=\widehat{\mathbf{H}}_{k}+\widetilde{\mathbf{H}}_{k}$, where
both the channel estimate $\widehat{\mathbf{H}}_{k}$ and the channel error
$\widetilde{\mathbf{H}}_{k}$ have i.i.d. complex Gaussian random entries drawn
from $\mathcal{CN}(0,\sigma_{k}^{2}-\sigma_{e,k}^{2})$ and
$\mathcal{CN}(0,\sigma_{e,k}^{2})$, respectively. All results are averaged
over 100 channel instances.

####  IV-D1 Spectrally and Energy Efficient

Consider a downlink MU-MIMO with one transmitter equipped with $M=4$ antennas
and $K=2$ users, each is equipped with $N=2$ antennas. Each user requires
$Q_{1}=Q_{2}=2$ streams, which are split, combined, and encoded into a common
stream vector of length $Q_{\mathrm{c}}=2$ and two private stream vectors of
length 2. Readers are referred to [35] for more details on the simulation
setting and the rate region optimization algorithm. Fig. 22 illustrates the
ergodic rate region of RS, MU-MIMO, NOMA, and DPC in perfect CSIT with equal
channel variances or 10 dB channel variance disparity between the users. Note
that in the two-user case, HRS and GRS are equivalent to 1-layer RS, which are
therefore simply denoted by “RS". We observe that in both subfigures, the rate
region of RS is larger than that of MU-MIMO and NOMA, and is much closer to
the capacity region of MIMO BC achieved by DPC. When the users have equal
channel variances as in Fig. 22(a), NOMA achieves the worst performance as no
channel strength disparity can be leveraged to manage interference. Although
MU-MIMO is capable of exploiting the full DoF in this setting (i.e.,
underloaded and perfect CSIT), there is still a rate region gap between RS and
MU-MIMO. As MU-MIMO always treats interference as noise, it only works well
when user channels are (semi-)orthogonal [8]. When the users have a 10 dB
channel variance disparity, the rate region gap between NOMA and RS reduces.
Interestingly, the rate region of NOMA and MU-MIMO outperform each other in
part while RS always outperforms both. Although NOMA can utilize the channel
variance disparity to improve its rate region, it incurs a theoretical DoF
loss and therefore rate loss due to its interference management principle of
forcing one user to fully decode the streams of all users [7]. Owing to its
powerful interference management capability of partially decoding the
interference and partially treating the interference as noise, RS enhances the
spectral efficiency in various user deployments.

(a) $\sigma_{1}^{2}=1,\sigma_{2}^{2}=1$

(b) $\sigma_{1}^{2}=1,\sigma_{2}^{2}=0.09$

Figure 22: Ergodic rate region comparison in MIMO BC with perfect CSIT, SNR =
20 dB, $M=4$, $K=2,N=2$, and $Q_{\mathrm{c}}=Q_{1}=Q_{2}=2$ [35].

(a) $\sigma_{1}^{2}=1,\sigma_{2}^{2}=1$

(b) $\sigma_{1}^{2}=1,\sigma_{2}^{2}=0.09$

Figure 23: Ergodic rate region comparison in MISO BC with imperfect CSIT, SNR
= 20 dB, $\alpha=0.6$, $M=4,K=2$ [9].

In Fig. 23, we further compare SE of different strategies in imperfect CSIT.
Consider a downlink MU-MISO with $M=4$ and $K=2$. The power of the channel
error is defined as $\sigma_{e,k}^{2}=\sigma_{k}^{2}P^{\alpha}$, where
$P=\mathrm{tr}(\mathbf{P}\mathbf{P}^{H})$ and $\alpha$ is fixed to 0.6. More
details of the simulation setting and the rate region optimization algorithm
can be found in [9]. Besides linearly precoded RS, the ergodic rate region of
1-layer DPCRS is also illustrated. In both subfigures, the ergodic rate region
of DPC drops significantly as it is sensitive to CSIT uncertainty.
Surprisingly, linearly precoded RS achieves a larger rate region than DPC
while maintaining a much lower transceiver complexity. As RS has been proved
to achieve the optimal multiplexting gain/DoF of MISO BC with imperfect CSIT
[19], such gain is reflected in its spectral efficiency gain over MU-MIMO and
NOMA. DPCRS, which marries the advantages of DPC and RS, achieves the largest
achievable rate region in both subfigures at the sacrifice of a higher
transceiver complexity than DPC and RS.

Fig. 24 compares EE of different strategies for two-user MU-MISO with perfect
CSIT, $M=2,N=2$. EE is defined as
$\frac{\sum_{k=1}^{K}R_{k,\mathrm{tot}}}{\frac{1}{\eta}\mathrm{tr}(\mathbf{P}\mathbf{P}^{H})+P_{\textrm{cir}}}$
[63, 64], where $\eta=0.35$ is the power amplifier efficiency,
$P_{\textrm{cir}}=MP_{\textrm{dyn}}+P_{\textrm{sta}}$ is the static circit