# Rate-Splitting Multiple Access: The First Prototype and Experimental
Validation of its Superiority over SDMA and NOMA

Xinze Lyu, , Sundar Aditya, , Junghoon Kim, and Bruno Clerckx This work was
supported in part by UKRI Impact Acceleration Account (IAA) grant
EP/X52556X/1.X. Lyu, S. Aditya and B. Clerckx are with the Dept. of Electrical
and Electronic Engg., Imperial College London, London SW7 2AZ, U.K. (e-mail:
{x.lyu21, s.aditya, b.clerckx}@imperial.ac.uk). B. Clerckx is also with
Silicon Austria Labs, 8010 Graz, Austria.J. Kim is with the College of Ocean
Science and Engineering, National Korea Maritime & Ocean University, Busan
49112, South Korea (e-mail: j.kim@kmou.ac.kr).

###### Abstract

In multi-user multi-antenna communications, it is well-known in theory that
Rate-Splitting Multiple Access (RSMA) can achieve a higher spectral efficiency
than both Space Division Multiple Access (SDMA) and Non-Orthogonal Multiple
Access (NOMA). However, an experimental evaluation of RSMA’s performance,
relative to SDMA and NOMA, is missing in the literature, which is essential to
address the ongoing debate between RSMA and NOMA over which is better suited
to handle most efficiently the available resources and interference in 6G. In
this paper, we address this critical knowledge gap by realizing the first-ever
RSMA prototype using software-defined radios. Through measurements using our
prototype, we empirically solve the modulation and coding scheme limited sum
throughput maximization problem for RSMA, SDMA and NOMA for the two-user
multiple-input single-output (MISO) scenario over (a) different pairs of line-
of-sight channels that vary in terms of their relative pathloss and spatial
correlation, and with (b) different channel state information quality. We
observe that RSMA achieves the highest sum throughput across all these cases,
whereas SDMA and NOMA are effective only in some cases. Furthermore, RSMA also
achieves better fairness at a higher sum throughput than both SDMA and NOMA.

###### Index Terms:

Rate-Splitting Multiple Access (RSMA), Space Division Multiple Access (SDMA),
Non-Orthogonal Multiple Access (NOMA), RSMA prototyping, RSMA measurements,
Software-defined Radio (SDR), RSMA for 6G.

##  I Introduction

The evolutionary trajectory over successive generations of wireless networks
has been towards achieving higher spectral efficiency, greater user fairness
and higher energy efficiency. For upcoming 6G in particular, improvements in
spectral and energy efficiency of at least 2x over 5G have been speculated
[1]. To meet these targets, the underlying multiple access technique must be
capable of offering higher efficiency, flexibility and coping with much higher
levels of multi-user interference than Space Division Multiple Access (SDMA) –
the state-of-the-art multiple access technique used in current wireless
standards such as 3GPP 5G NR and IEEE 802.11. The two candidates widely
believed to be capable of achieving this in future wireless networks are Non-
Orthogonal Multiple Access (NOMA) and Rate-Splitting Multiple Access (RSMA).
We provide a brief description of each below, specifically focusing on
practical implementations.

###  I-A NOMA

NOMA is inspired by the fact that the capacity region of the $K$-user single-
input-single-output (SISO) broadcast channel is achieved by superposition
coding at the transmitter (TX) and successive interference cancellation (SIC)
at the receivers (RXs) [2]. In SISO NOMA, users are ordered in decreasing
order of their channel strength and the $l$-th strongest user decodes and
subtracts the interference from the $K-l$ weaker users (i.e., SIC) before
decoding its desired signal. The interference from the $l-1$ stronger users is
treated as noise111This is commonly referred to as power-domain NOMA to
distinguish it from other variants, such as code-domain and frequency-domain
NOMA [3]. In this paper, NOMA refers only to power-domain NOMA..

The optimality of NOMA for the SISO broadcast channel (in terms of achieving
capacity) has been the driving force behind extensive investigations into NOMA
for multiple-input-multiple-output (MIMO) systems with the aim of improving
spectral efficiency [4, 5] and fairness [6] in 5G networks and beyond [7].
However, for the multi-antenna broadcast channel, NOMA is not optimal, in
general, in a capacity-achieving sense [8]. This is because with multiple
antennas at the TX, the spatial domain can be used to suppress interference
without SIC at the RXs through either multi-user linear precoding (MU-LP) or
non-linear precoding at the TX. MU-LP is the common approach adopted by SDMA
[9]. Essentially, SDMA and NOMA represent two distinct interference management
strategies; in SDMA, the interference is treated as noise, whereas in NOMA the
interference from the weaker users is decoded by the stronger users [10].
Hence, NOMA provides substantial spectral efficiency gains over SDMA only when
MU-LP cannot suppress the interference to near-noise levels. This occurs
either in overloaded systems (i.e., more users than the number of antennas at
the TX) or when the user channels in underloaded systems experience high
spatial correlation. On the other hand, in non-overloaded systems where user
channels do not experience high spatial correlation, NOMA can perform worse
than SDMA [8]. While it can be argued that under high user density, groups of
users with highly spatially correlated channels can be found with high
probability, it does mean that MIMO-NOMA requires a user grouping strategy
based on their channels, which has given rise to two approaches - beamforming-
based and cluster-based NOMA [11].

### NOMA Implementation

In terms of experimental evaluations, several papers have focussed on SISO-
NOMA [12, Table 1]. MIMO-NOMA implementations are relatively fewer [13, 14,
15]. Link and system-level simulations were carried out in [13], along with
measurements over emulated fading channels. Measurements involving over-the-
air transmissions were carried out in [14, 15]. However, in these papers, the
NOMA performance is compared with orthogonal multiple access instead of SDMA.
To the best of our knowledge, no experimental comparison between NOMA and SDMA
exists in the literature.

###  I-B RSMA

Unlike NOMA with its SISO roots, RSMA was designed for multi-antenna
communications from the outset. For the $K$-user multiple-input-single-output
(MISO) case, RSMA produces $K+1$ data streams (i.e., channel-coded and
modulated symbol streams) at the TX from the $K$ user messages [16] (details
in Section II-B). In contrast, both SDMA and NOMA generate $K$ data streams.
Similar to SDMA, each of the $K+1$ streams is linearly precoded (i.e., MU-LP)
and transmitted over the air. At each RX, SIC is used (like NOMA) to decode
_only two_ streams to retrieve the desired message, regardless of a users’s
channel strength relative to the other users. Hence, unlike NOMA, no user
grouping mechanism prior to data transmission is needed to realize spectral
efficiency gains over SDMA.

RSMA’s spectral efficiency gains over SDMA and NOMA stem from the fact that it
is degrees-of-freedom (DoF) optimal under perfect and imperfect channel state
information (CSI) [8, 16]. Furthermore, RSMA has also been shown, in theory,
to achieve spectral efficiency gains over SDMA and NOMA under varying
interference levels, network loads and user deployments [10]. Apart from
spectral efficiency, RSMA has also been shown to achieve – again, in theory –
performance improvements over SDMA and NOMA in terms of other metrics such as
energy efficiency, fairness, mixed-critical quality-of-service and tight
latency requirements, etc. (see [17, 18] and the references therein for a
comprehensive survey). These performance improvements stem from RSMA’s unique
interference mitigation strategy, wherein at each user, the interference is
_partially decoded and partially treated as noise_. In doing so, RSMA
generalizes both SDMA and NOMA [10, 19]. Taken together, all of these features
make RSMA an attractive candidate for physical layer multiple access in 6G,
provided the promised gains can be demonstrated in practice. In particular, an
experimental comparison between RSMA, SDMA and NOMA is needed as a starting
point to address the pressing question of which multiple access technique is
best equipped to deliver the spectral efficiency enhancements expected from
6G.

### RSMA Implementation

Despite its impressive benefits, the practical implementation of RSMA is still
in its infancy, with the state-of-the-art focusing on the closely related
issues of erroneous SIC, receiver design, finite constellation and adaptive
modulation and coding (AMC) for RSMA to minimize the block error rate due to
the combined effects of imperfect/outdated CSI at the TX (CSIT) and finite
blocklength [20, 21, 22, 23, 24]. Among these, RSMA link-level simulations
were conducted in [20, 21, 22], feedback-based RSMA link adaptation was
investigated in [23], while [24] considered a flexible variant of RSMA, where
only a subset of the receivers employ SIC. Though important, these practical
aspects have also been tackled only through theory/simulations, and no
experimental verification of RSMA’s benefits has been reported till date.

In this paper, we address this critical knowledge gap by realizing the first-
ever RSMA prototype using software-defined radios (SDRs). Using our prototype,
we experimentally compare the throughput and fairness performance of RSMA,
SDMA and NOMA. Our contributions are as follows:

  * •

We implement RSMA on SDRs using a two-stage protocol. In Stage 1, orthogonal
pilot signals are transmitted by the TX for CSI estimation at each RX. The
estimated CSI is then fed back to the TX, either unquantized or quantized to
emulate perfect and imperfect CSIT, respectively (Section II-A). The CSIT is
used to design the precoded RSMA signal that is transmitted in Stage 2
(Section II-B). We use OFDM signals largely based on IEEE 802.11g physical
layer frames to implement the data payload in the RSMA signal (Section III).
The above two-stage protocol can also be used to implement SDMA and NOMA in
our prototype with appropriate changes to Stage 2.

  * •

For the two-user MISO scenario, we formulate the practically relevant
_modulation and coding scheme (MCS)-limited sum throughput maximization
problem_ for RSMA, and establish its relationship to the canonical sum rate
maximization problem that is widely analyzed in theory (Section II-B). The
MCS-limited sum throughput maximization problem can also be formulated for
SDMA and NOMA (Table I).

  * •

Through measurements using our prototype, we empirically solve the MCS-limited
sum throughput maximization problem for RSMA, SDMA and NOMA for the two-user
MISO scenario over line-of-sight channels in a lab environment, where the pair
of channels differ in terms of their relative strength and spatial correlation
(Section IV). We empirically observe that

    * –

With unquantized CSI feedback, NOMA achieves significant throughput gains over
SDMA when channels have high spatial correlation. On the other hand, when the
channels have low spatial correlation, the SDMA throughput is higher than
NOMA. RSMA, however, achieves the highest throughput not only under either
extreme but also over channels that lie in between w.r.t their spatial
correlation. These outcomes are consistent with theoretical predictions in
[10, 19].

    * –

With quantized CSI feedback, NOMA achieves significant throughput gains over
SDMA even with channels having low spatial correlation, provided there is a
sufficiently large disparity between the channel strengths. The NOMA
throughput performance is also at par with that of RSMA under these
conditions. Essentially, the imperfect CSI disproportionately diminishes the
weaker user’s throughput for SDMA, whereas both RSMA and NOMA provide higher
throughput to the weaker user via the common stream (see Section II-B for
definition). However, when the channel strength disparity is not sufficiently
large, RSMA achieves throughput gains over NOMA as well. These outcomes are
consistent with theoretical predictions in [8, 17, 18].

    * –

With both quantized and unquantized CSI feedback, RSMA achieves better
fairness at higher throughputs than both SDMA and NOMA. This too is consistent
with theoretical predictions in [8, 17, 18].

###  I-C Organization

The rest of this paper is organized as follows. In Section II, we introduce
the RSMA system model for the two-user MISO scenario and formulate the
practically relevant MCS-limited sum throughput maximization problem. In
Section III, we provide a detailed description of our SDR-based RSMA
implementation. In Section IV, we empirically solve the MCS-limited sum
throughput maximization problem for RSMA, SDMA and NOMA through measurements
using our prototype and compare the sum throughput and fairness performance of
the three schemes. Finally, Section V concludes the paper.

###  I-D Notation

Column vectors are represented using lowercase bold letters (e.g.,
${\mathbf{x}}$). $(\cdot)^{H},\|\cdot\|,\cup$ and $\emptyset$ denote the
Hermitian operator, the Euclidean norm, the union operation and the empty set,
respectively. ${\rm Re}(\cdot)$ and ${\rm Im}(\cdot)$ denote the real and
imaginary parts of a complex number, $\lfloor\cdot\rfloor$ denotes the floor
function, and $\mathcal{CN}(0,\sigma^{2})$ denotes the circularly symmetric
complex Gaussian distribution with zero mean and variance $\sigma^{2}$.

##  II System Model

Consider a two-antenna TX communicating with two single-antenna users/RXs,
using OFDM222In principle, RSMA can be used with any waveform. Our choice of
OFDM is motivated by its prevalence in modern communications systems. However,
in the context of 6G, alternatives to OFDM, such as OTFS have received
considerable attention. RSMA for such waveforms – especially experimental
evaluations – is a relevant topic yet to be investigated. signals over $N_{c}$
subcarriers. The communications takes place over two stages.

  * •

Stage 1: First, the TX transmits orthogonal pilot signals for channel
estimation at the RXs. The estimated CSI is then fed back to the TX through a
separate link.

  * •

Stage 2: Using the CSI acquired in Stage 1, the TX designs the precoder and
the transmit signal to maximize the sum throughput.

We describe each stage, in turn, in Sections II-A and II-B.

###  II-A Stage 1: Channel Model and CSI feedback

Let $\mathbf{h}_{i}[k]:=[h_{i1}[k],~{}h_{i2}[k]]^{T}\in{\mathbb{C}}^{2\times
1}$ denote the true frequency flat, slowly varying channel between the TX and
RX $i~{}(=1,2)$ over the $k$-th subcarrier $(k=0,\cdots,N_{c}-1)$. Similarly,
let $\hat{{\mathbf{h}}}_{i}[k]$ denote the estimate of $\mathbf{h}_{i}[k]$,
obtained at RX $i$ through pilot signals transmitted by the TX. To reduce
feedback overhead, let $\hat{{\mathbf{h}}}_{i}$ denote the _wideband CSI_
evaluated at RX $i$ by averaging $\hat{{\mathbf{h}}}_{i}[k]$ over the
subcarriers, i.e.,

$\displaystyle\hat{{\mathbf{h}}}_{i}:=\frac{1}{N_{c}}\sum_{k=0}^{N_{c}-1}\hat{{\mathbf{h}}}_{i}[k],~{}(i=1,2).$
(1)

For CSI feedback of $\hat{{\mathbf{h}}}_{i}$ from RX $i$ to the TX in our
experiments, we consider two cases to emulate perfect and imperfect CSI at the
TX.

  * •

_Unquantized feedback_ : Let $\hat{{\mathbf{h}}}_{i}^{\rm
UQ}=[\hat{h}_{i1}^{\rm UQ}~{}\hat{h}_{i2}^{\rm UQ}]$ denote the unquantized
version of $\hat{{\mathbf{h}}}_{i}$, where each complex $\hat{h}_{il}^{\rm
UQ}~{}(i,l\in\\{1,2\\})$ is a 128-bit representation of $\hat{h}_{il}$ (i.e.,
a 64-bit double-precision floating-point number for the real and imaginary
parts). This is intended to emulate perfect CSI at the TX.

  * •

_Quantized feedback_ : Let $\hat{{\mathbf{h}}}_{i}^{\rm Q}=[\hat{h}_{i1}^{\rm
Q}~{}\hat{h}_{i2}^{\rm Q}]$ denote the quantized version of
$\hat{{\mathbf{h}}}_{i}$, where each complex $\hat{h}_{il}^{\rm
Q}~{}(i,l\in\\{1,2\\})$ is an 8-bit representation of $\hat{h}_{il}$ (i.e., 4
bits for the real and imaginary parts). The quantization mechanism is
described in Appendix A, and is based on based on IEEE 802.11n [25]. It
results in a more than $90\%$ reduction in the CSI feedback overhead, and is
intended to emulate imperfect CSI at the TX.

The fedback CSI is used by the TX to design the precoders, as described in the
following subsection.

###  II-B Stage 2: RSMA Signal Design

Figure 1: RSMA block diagram for the two-user MISO scenario.

For the two-user MISO scenario, Fig. 1 depicts RSMA in operation. Let $W_{1}$
and $W_{2}$ denote the messages corresponding to RX 1 and RX 2, respectively.
At the TX, each $W_{i}~{}(i=1,2)$ is split into common and private portions –
denoted by $W_{c,i}$ and $W_{p,i}$, respectively – by the message splitter.
The common portions of each RX’s message (i.e., $W_{c,1}$ and $W_{c,2}$) are
combined into a common message, which is then encoded and modulated to form a
common stream, $s_{c}[k]~{}(k=0,\cdots,N_{c}-1)$ over the subcarriers. On the
other hand, the private portions of RX $i$’s message are individually encoded
and modulated to form private streams, $s_{i}[k]$. All three streams (i.e.,
$s_{c}[k],s_{1}[k]$ and $s_{2}[k]$) are linearly precoded to form the transmit
OFDM frequency domain signal as follows:

$\mathbf{x}[k]=\mathbf{p}_{c}s_{c}[k]+\mathbf{p}_{1}s_{1}[k]+\mathbf{p}_{2}s_{2}[k]~{}(k=0,\cdots,N_{c}-1),$
(2)

where ${\mathbf{p}}_{c}$ is referred to as the common stream precoder, and
${\mathbf{p}}_{i}$ the private stream precoder of RX $i$.

After appropriate DFT-based processing, the received OFDM frequency domain
signal, $y_{i}[k]$, at RX $i$ is given by:

$\displaystyle y_{i}[k]$
$\displaystyle=\mathbf{h}_{i}^{H}[k]\mathbf{x}[k]+n_{i}[k],~{}(k=0,\cdots,N_{c}-1)$
$\displaystyle={\mathbf{h}}_{i}^{H}[k]{\mathbf{p}}_{c}s_{c}[k]+{\mathbf{h}}_{i}^{H}[k]{\mathbf{p}}_{1}s_{1}[k]+{\mathbf{h}}_{i}^{H}[k]{\mathbf{p}}_{2}s_{2}[k]$
$\displaystyle~{}+n_{i}[k]~{}(i=1,2),$ (3)

where $n_{i}[k]\sim\mathcal{CN}(0,\sigma^{2})$ is the thermal noise. We assume
that precoded pilot signals have been interspersed in some of the subcarriers
of ${\mathbf{x}}[k]$ to help RX $i$ estimate
${\mathbf{h}}_{i}^{H}[k]{\mathbf{p}}_{c}$ and
${\mathbf{h}}_{i}^{H}[k]{\mathbf{p}}_{i}$ upon receiving $y_{i}[k]$333These
pilots play a role similar to the demodulation reference signals (DM-RS) used
in LTE and 5G NR.. Then, after suitable equalization (e.g., MMSE), RX $i$
first decodes $s_{c}[k]$ to recover $W_{c}$, while treating the interference
from the $s_{1}[k]$ and $s_{2}[k]$ as noise. The decoded estimate of $W_{c}$
at RX $i$, denoted by $\hat{W}_{c}^{(i)}$, is simultaneously sent to:

  * (a)

the message splitter to extract $\hat{W}_{c,i}^{(i)}$ – the portion of
$\hat{W}_{c}^{(i)}$ containing data meant for RX $i$’s, and

  * (b)

the SIC module, which generates an estimate of
${\mathbf{h}}_{i}^{H}[k]{\mathbf{p}}_{c}s_{c}[k]$ and subtracts it from
$y_{i}[k]$. The residue is once again subjected to equalization, this time to
decode $s_{i}[k]$ by treating the interference from the other private stream
as noise. Let $\hat{W}_{p,i}$ denote the decoded estimate of $W_{p,i}$ at RX
$i$.

Let ${\mathbf{P}}=[{\mathbf{p}}_{c}~{}{\mathbf{p}}_{1}~{}{\mathbf{p}}_{2}]$.
As a function of ${\mathbf{P}}$, from (II-B), the achievable rate (in
bits/s/Hz) for error-free decoding of $s_{c}[k]$ at RX $i$, denoted by
$R_{c,i}({\mathbf{P}})$, is given by:

$\displaystyle R_{c,i}({\mathbf{P}})$
$\displaystyle=\min_{k}\log_{2}\left(1+\frac{|\mathbf{h}_{i}^{H}[k]\mathbf{p}_{c}|^{2}}{\sigma^{2}+|\mathbf{h}^{H}_{i}[k]\mathbf{p}_{1}|^{2}+|\mathbf{h}^{H}_{i}[k]\mathbf{p}_{2}|^{2}}\right),$
(4)

where the minimum over $k$ arises because the encoded and modulated common
stream is embedded over subcarriers in the frequency domain. Additionally,
since both RXs must first decode $s_{c}[k]$, it follows that the achievable
rate for error-free decoding of $s_{c}[k]$ at _both_ RXs, denoted by
$R_{c}({\mathbf{P}})$, is given by:

$\displaystyle R_{c}({\mathbf{P}})$
$\displaystyle=\min(R_{c,1}({\mathbf{P}}),R_{c,2}({\mathbf{P}})).$ (5)

Assuming perfect SIC operation, the achievable rate for error-free decoding of
$s_{i}[k]$ at RX $i$, similar to (4), is given by:

$\displaystyle
R_{i}({\mathbf{P}})=\min_{k}\log_{2}\left(1+\frac{|\mathbf{h}_{i}^{H}[k]\mathbf{p}_{i}|^{2}}{\sigma^{2}+|\mathbf{h}_{i}^{H}[k]\mathbf{p}_{j}|^{2}}\right),~{}j\neq
i.$ (6)

Thus, the achievable rate at RX $i$ for error-free recovery of $W_{i}$,
denoted by $R_{i}^{\rm RSMA}({\mathbf{P}})$ is given by:

$\displaystyle R_{i}^{\rm RSMA}({\mathbf{P}})$
$\displaystyle=(W_{c,i}/W_{c})R_{c}({\mathbf{P}})+R_{i}({\mathbf{P}})$ (7)

where the first term captures the fraction of the common message, $W_{c}$,
intended for RX $i$. Consequently, the achievable RSMA sum rate, denoted by
$R^{\rm RSMA}({\mathbf{P}})$, is:

$\displaystyle R^{\rm RSMA}({\mathbf{P}})$ $\displaystyle=R_{1}^{\rm
RSMA}({\mathbf{P}})+R_{2}^{\rm RSMA}({\mathbf{P}})$
$\displaystyle=R_{c}({\mathbf{P}})+R_{1}({\mathbf{P}})+R_{2}({\mathbf{P}})$
(8)

The canonical _sum rate maximization problem_ , where the precoders at the TX
are optimized to maximize $R^{\rm RSMA}({\mathbf{P}})$ is formulated as
follows:

$\displaystyle OP_{\rm sr}^{\rm RSMA}:\max_{{\mathbf{P}}}$
$\displaystyle~{}R^{\rm RSMA}({\mathbf{P}})$ (9) s.t. $\displaystyle~{}{\rm
tr}({\mathbf{P}}{\mathbf{P}}^{H})\leq P_{t},$ (10)

where (10) restricts the transmit power to $P_{t}$. Through the well-known
Rate-WMMSE relationship [26], the non-convex problem in $OP_{\rm sr}^{\rm
RSMA}$ can be transformed into a sequence of tractable sub-problems that can
be iteratively solved till convergence to a local optimum. This is commonly
known in the literature as the _WMMSE method_ and despite only guaranteeing
local optimality, it is widely recognized as a benchmark solution for the
family of MIMO precoder design problems, of which $OP_{\rm sr}^{\rm RSMA}$ is
a specific instance corresponding to RSMA. Let ${\mathbf{P}}_{\rm wmmse}^{\rm
RSMA}$ denote the precoders obtained by applying the WMMSE method to solve
$OP_{\rm sr}^{\rm RSMA}$.

######  Remark 1 (Imperfect CSI at the TX).

$OP_{\rm sr}^{\rm RSMA}$ contains a slight abuse of notation for the sake of
simplicity. Let $\hat{{\mathbf{H}}}^{\rm TX}:=[\hat{{\mathbf{h}}}_{1}^{\rm
TX}~{}\hat{{\mathbf{h}}}_{2}^{\rm TX}]$ denote the CSIT, where
$\hat{{\mathbf{h}}}_{i}^{\rm TX}\in\\{\hat{{\mathbf{h}}}_{i}^{\rm
UQ},\hat{{\mathbf{h}}}_{i}^{\rm Q}\\}$, as described in Section II-A. Strictly
speaking, to obtain the objective function of $OP_{\rm sr}^{\rm RSMA}$ at the
TX, ${\mathbf{h}}_{i}[k]$ in (4) and (6) should be replaced by
$\hat{{\mathbf{h}}}_{i}^{\rm TX}$, so that the solution to $OP_{\rm sr}^{\rm
RSMA}$ becomes a function of the CSIT.

In this paper, the main performance metric in our measurements is the
throughput (measured in bits/s). Over an effective bandwidth444The effective
bandwidth is the portion of the total bandwidth available for data transfer,
after accounting for signaling overheads (e.g., cyclic prefix in the case of
OFDM)., $B$, the achievable RSMA sum throughput, denoted by $T^{\rm
RSMA}({\mathbf{P}})$, is related to $R^{\rm RSMA}({\mathbf{P}})$ in (II-B) via
$T^{\rm RSMA}({\mathbf{P}})=BR^{\rm RSMA}({\mathbf{P}})$. Hence, it follows
that $T^{\rm RSMA}({\mathbf{P}}^{\rm RSMA}_{\rm wmmse})$ is a good benchmark
in terms of the maximum achievable RSMA sum throughput. However, this
benchmark is hard to realize in practice because:

  * 1.

In state-of-the-art communications systems, PHY layer transmissions are
restricted to a finite collection of modulation and coding scheme (MCS) levels
(e.g., QPSK, rate $1/2$), which caps the maximum achievable sum throughput,
and

  * 2.

Error-free decoding of the transmitted messages at the RX cannot be guaranteed
due to finite block length effects, which further diminishes the realized
throughput.

We address these two points by first characterizing the _MCS-limited_
throughput. An MCS level can be defined by a pair $(m,r)$, where positive
integer $m$ denotes the bits per constellation symbol (e.g., 2 for QPSK) and
$r\in(0,1]$ denotes the code rate. Hence, the achievable rate (in bits/s/Hz)
for MCS level $(m,r)$ equals $mr$, and the achievable throughput is $Bmr$. Let
$(m_{c},r_{c})$, $(m_{1},r_{1})$ and $(m_{2},r_{2})$ denote the MCS levels
chosen for $s_{c}[k]$, $s_{1}[k]$ and $s_{2}[k]$, respectively. Then, the
_MCS-limited_ RSMA sum throughput, denoted by $T^{\rm RSMA}_{\rm
mcs}({\mathbf{P}},{\mathcal{M}})$, is given by:

$\displaystyle T^{\rm RSMA}_{\rm mcs}({\mathbf{P}},{\mathcal{M}})$
$\displaystyle=Bm_{c}r_{c}({\mathbf{p}}_{c})\times{\mathbb{P}}(\hat{W}_{c}^{(1)}=W_{c};\hat{W}_{c}^{(2)}=W_{c})$
$\displaystyle~{}+Bm_{1}r_{1}({\mathbf{p}}_{1})\times{\mathbb{P}}(\hat{W}_{p,1}=W_{p,1})$
$\displaystyle~{}+Bm_{2}r_{2}({\mathbf{p}}_{2})\times{\mathbb{P}}(\hat{W}_{p,2}=W_{p,2}).$
(11)

The RHS of (II-B) indicates that the MCS levels of the streams are, in
general, a function of the corresponding precoders. The probability terms
capture the loss in throughput due to decoding errors – specifically,
${\mathbb{P}}(\hat{W}_{c}^{(1)}=W_{c};\hat{W}_{c}^{(2)}=W_{c})$ denotes the
probability that $W_{c}$, is correctly decoded at both RXs, while
${\mathbb{P}}(\hat{W}_{p,i}=W_{p,i})$ denotes the probability that $W_{p,i}$
is correctly decoded at RX $i$. Thus, similar to $OP_{\rm sr}^{\rm RSMA}$ in
(9)-(10), the _MCS-limited RSMA sum throughput maximization problem_ , which
is of considerable practical relevance, can be defined as follows:

$\displaystyle OP_{\rm mcs}^{\rm RSMA}:\max_{{\mathbf{P}},{\mathcal{M}}}$
$\displaystyle~{}T_{\rm mcs}^{\rm RSMA}({\mathbf{P}}^{\rm RSMA}_{\rm
wmmse},{\mathcal{M}})$ (12) $\displaystyle~{}{\mathcal{M}}\in{\mathbb{M}},$
(13)

where ${\mathbb{M}}$ denotes the collection of permissible MCS levels for the
three streams, which is typically pre-determined through standards (see Table
III for the ${\mathbb{M}}$ used in our measurements). $OP_{\rm mcs}^{\rm
RSMA}$ does not have a closed form solution, as the message decoding
probabilities in (II-B) are difficult to characterize in terms of
${\mathbf{P}}_{\rm wmmse}^{\rm RSMA}$ and ${\mathcal{M}}$. However, these
probabilities can be empirically evaluated through measurements. This
motivates us to empirically solve $OP_{\rm mcs}^{\rm RSMA}$ through a brute
force search555A more sophisticated empirical approach involves link
adaptation, where the most suitable MCS level is determined by an ARQ-based
mechanism [23]. This is left for future work. over ${\mathbb{M}}$, which is
the focus of our measurements in Section IV using our SDR-based RSMA
prototype. We conclude this subsection with the following remark:

######  Remark 2 (Wideband precoders).

In principle, narrowband precoders for each subcarrier (i.e.,
${\mathbf{p}}_{c}[k]$, ${\mathbf{p}}_{1}[k]$, ${\mathbf{p}}_{2}[k]$:
$k=0,\cdots,N_{c}-1$) could increase the achievable rate in (4)-(II-B) by
adapting to channel variations at subcarrier-level granularity. However, this
would involve huge signalling overhead in both directions, as: (a) narrowband
CSI (i.e., $\hat{{\mathbf{h}}}_{i}[k]$ for several, if not all, $k$) would
need to be fed back to the TX for precoder design, and (b) more demodulation
reference signals would have to be embedded in the transmit signal to estimate
${\mathbf{h}}_{i}^{H}[k]{\mathbf{p}}_{c}[k]$ and
${\mathbf{h}}_{i}^{H}[k]{\mathbf{p}}_{i}[k]$ at RX $i$ for all $k$. Thus, to
reduce the signalling overhead, we assume the same precoders, ${\mathbf{P}}$,
across all the subcarriers.

###  II-C RSMA v/s SDMA v/s NOMA

In general, the splitting of the user messages in Fig. 1 need not be equal
(i.e., 50-50) between the common and private portions, nor does the ratio
between the common and private portions need to be fixed across both messages.
In fact, SDMA and NOMA are special cases of RSMA corresponding to specific
choices of message splitting, as explained in the following remarks.

######  Remark 3 (SDMA as a special case of RSMA).

For the special case where the common stream is turned off – i.e.,
$W_{i}=W_{p,i}~{}(i=1,2)$ in Fig. 1 – RSMA reduces to SDMA [19].

######  Remark 4 (NOMA as a special case of RSMA).

Without loss of generality, suppose RX 2 has a weaker channel than RX 1. For
the special case where the private stream of RX 2 is turned off, and no
portion of $W_{1}$ is allocated to the common stream – i.e.,
$W_{1}=W_{p,1},W_{2}=W_{c,2}=W_{c}$ in Fig. 1 – RSMA reduces to NOMA [19].

This flexibility in message splitting provided by RSMA characterizes its
interference management philosophy, as follows:

  * •

SDMA: In the absence of a common stream, the interference experienced at RX
$i$ due to $W_{j}~{}(j\neq i)$ is treated as noise.

  * •

NOMA: The common stream allows the stronger RX 1 to fully decode and subtract
the interference it experiences (from $W_{2}$). The weaker RX 2 continues to
treat the interference it experiences (from $W_{1}$) as noise, like SDMA.

  * •

RSMA: The interference at RX $i$ due to $W_{j}~{}(j\neq i)$ is _partially
decoded_ ($W_{c,j}$) and _partially treated as noise_ ($W_{p,j}$). This
feature enables RSMA to softly bridge between SDMA and NOMA [10, 19]. In other
words, when the two channels are orthogonal, the optimal solution to $OP_{\rm
sr}^{\rm RSMA}$ in (9)-(10) coincides with SDMA. Similarly, when the channels
are aligned and one of them is much stronger than the other, the optimal
solution to $OP_{\rm sr}^{\rm RSMA}$ reduces to NOMA. Importantly, over the
vast majority of channels that do not fall in either extreme, RSMA is expected
to provide significant gains over both SDMA and NOMA, which we aim to
demonstrate experimentally in Section IV.

Based on Remarks 3 and 4, the expressions corresponding to SDMA and NOMA in
(2)-(13) can be obtained by discarding ${\mathbf{p}}_{c}$ (SDMA) and
${\mathbf{p}}_{2}$ (NOMA). These expressions are provided in Table I for
clarity. In Sections IV-B and IV-C, we provide an experimental RSMA v/s SDMA
v/s NOMA performance comparison by empirically solving $OP_{\rm mcs}^{\rm
ma},~{}{\rm ma}\in\\{{\rm RSMA,SDMA,NOMA}\\}$ using our SDR-based RSMA
prototype. In the following section, we present details of our prototype, as
well as how the system model described above in Sections II-A and II-B is
implemented.

Quantity |  | Expression |
---|---|---|---
Message |  | $W_{i}$ | $=W_{c,i}\cup W_{p,i}~{}(i=1,2)$
splitting |  | $W_{c}$ | $=W_{c,1}\cup W_{c,2}$
| RSMA: | $W_{c,i},W_{p,i}$ |  $\neq\emptyset$, in general
| SDMA: | $W_{c,i}$ | $=\emptyset,W_{p,i}=W_{i}$
| NOMA: | $W_{c,1}$ |  $=\emptyset,W_{p,1}=W_{1}$ and $W_{c,2}=W_{2},W_{p,2}=\emptyset$
Precoders | RSMA: | ${\mathbf{P}}$ | $=[{\mathbf{p}}_{c}~{}{\mathbf{p}}_{1}~{}{\mathbf{p}}_{2}]$
| SDMA: | ${\mathbf{P}}$ | $=[{\mathbf{p}}_{1}~{}{\mathbf{p}}_{2}]$
| NOMA: | ${\mathbf{P}}$ | $=[{\mathbf{p}}_{c}~{}{\mathbf{p}}_{1}]$
Achievable | RSMA: | $R_{c,i}({\mathbf{P}})$ | $=\displaystyle\min\limits_{k}\log_{2}\left(1+\frac{|\mathbf{h}_{i}^{H}[k]\mathbf{p}_{c}|^{2}}{\sigma^{2}+|\mathbf{h}^{H}_{i}[k]\mathbf{p}_{1}|^{2}+|\mathbf{h}^{H}_{i}[k]\mathbf{p}_{2}|^{2}}\right)$
common stream |  | $R_{c}({\mathbf{P}})$ | $=\displaystyle\min\limits_{i}R_{c,i}({\mathbf{P}})$
rate | SDMA: | $R_{c,i}({\mathbf{P}})$ | $=R_{c}({\mathbf{P}})=0$
at RX $i$ | NOMA: | $R_{c,i}({\mathbf{P}})$ | $=\displaystyle\min\limits_{k}\log_{2}\left(1+\frac{|\mathbf{h}_{i}^{H}[k]\mathbf{p}_{c}|^{2}}{\sigma^{2}+|\mathbf{h}^{H}_{i}[k]\mathbf{p}_{1}|^{2}}\right)$
|  | $R_{c}({\mathbf{P}})$ | $=\displaystyle\min\limits_{i}R_{c,i}({\mathbf{P}})$
Achievable pvt. | RSMA: | $R_{i}({\mathbf{P}})$ | $=\displaystyle\min\limits_{k}\log_{2}\left(1+\frac{|\mathbf{h}_{i}^{H}[k]\mathbf{p}_{i}|^{2}}{\sigma^{2}+|\mathbf{h}^{H}_{i}[k]\mathbf{p}_{j}|^{2}}\right)~{}(j\neq i)$
stream rate | SDMA: | $R_{i}({\mathbf{P}})$ | $=\displaystyle\min\limits_{k}\log_{2}\left(1+\frac{|\mathbf{h}_{i}^{H}[k]\mathbf{p}_{i}|^{2}}{\sigma^{2}+|\mathbf{h}^{H}_{i}[k]\mathbf{p}_{j}|^{2}}\right)~{}(j\neq i)$
at RX $i$ | NOMA: | $R_{1}({\mathbf{P}})$ | $=\displaystyle\min\limits_{k}\log\left(1+\frac{|\mathbf{h}_{1}^{H}[k]\mathbf{p}_{1}|^{2}}{\sigma^{2}}\right)$
|  | $R_{2}({\mathbf{P}})$ | $=0$
Achievable | RSMA: | $R^{\rm RSMA}({\mathbf{P}})$ | $=R_{c}({\mathbf{P}})+R_{1}({\mathbf{P}})+R_{2}({\mathbf{P}})$
sum rate | SDMA: | $R^{\rm SDMA}({\mathbf{P}})$ | $=R_{1}({\mathbf{P}})+R_{2}({\mathbf{P}})$
| NOMA: | $R^{\rm NOMA}({\mathbf{P}})$ | $=R_{c}({\mathbf{P}})+R_{1}({\mathbf{P}})$
Sum rate |  | $OP_{\rm sr}^{\rm ma}:~{}\displaystyle\max\limits_{\mathbf{P}}$ | $R^{\rm ma}({\mathbf{P}})$
max. problem |  | s.t. | ${\rm tr}({\mathbf{P}}{\mathbf{P}}^{H})\leq P_{t}$
Achievable |  | $T^{\rm ma}({\mathbf{P}})$ |  $=BR^{\rm ma}({\mathbf{P}})$,
sum throughput |  | where ${\rm ma}$ | $\in\\{\rm{RSMA,SDMA,NOMA}\\}$
MCS levels | RSMA: | ${\mathcal{M}}$ | $=\\{(m_{c},r_{c}),(m_{1},r_{1}),(m_{2},r_{2})\\}$
| SDMA: | ${\mathcal{M}}$ | $=\\{(m_{1},r_{1}),(m_{2},r_{2})\\}$
| NOMA: | ${\mathcal{M}}$ | $=\\{(m_{c},r_{c}),(m_{1},r_{1})\\}$
MCS-limited | RSMA: | $R_{\rm mcs}^{\rm RSMA}({\mathbf{P}},{\mathcal{M}})$ | $=m_{c}r_{c}({\mathbf{p}}_{c})+m_{1}r_{1}({\mathbf{p}}_{1})+m_{2}r_{2}({\mathbf{p}}_{2})$
sum rate | SDMA: | $R_{\rm mcs}^{\rm SDMA}({\mathbf{P}},{\mathcal{M}})$ | $=m_{1}r_{1}({\mathbf{p}}_{1})+m_{2}r_{2}({\mathbf{p}}_{2})$
| NOMA: | $R_{\rm mcs}^{\rm NOMA}({\mathbf{P}},{\mathcal{M}})$ | $=m_{c}r_{c}({\mathbf{p}}_{c})+m_{1}r_{1}({\mathbf{p}}_{1})$
MCS-limited | RSMA: | $T^{\rm RSMA}_{\rm mcs}({\mathbf{P}},{\mathcal{M}})$ | $=Bm_{c}r_{c}({\mathbf{p}}_{c})\times{\mathbb{P}}(\hat{W}_{c}^{(1)}=W_{c};\hat{W}_{c}^{(2)}=W_{c})$
sum throughput |  |  | $+Bm_{1}r_{1}({\mathbf{p}}_{1})\times{\mathbb{P}}(\hat{W}_{p,1}=W_{p,1})$
|  |  | $+Bm_{2}r_{2}({\mathbf{p}}_{2})\times{\mathbb{P}}(\hat{W}_{p,2}=W_{p,2})$
| SDMA: | $T^{\rm SDMA}_{\rm mcs}({\mathbf{P}},{\mathcal{M}})$ | $=Bm_{1}r_{1}({\mathbf{p}}_{1})\times{\mathbb{P}}(\hat{W}_{p,1}=W_{1})$
|  |  | $+Bm_{2}r_{2}({\mathbf{p}}_{2})\times{\mathbb{P}}(\hat{W}_{p,2}=W_{2})$
| NOMA: | $T^{\rm NOMA}_{\rm mcs}({\mathbf{P}},{\mathcal{M}})$ | $=Bm_{c}r_{c}({\mathbf{p}}_{c})\times{\mathbb{P}}(\hat{W}_{c}^{(1)}=W_{2};\hat{W}_{c}^{(2)}=W_{2})$
|  |  | $+Bm_{1}r_{1}({\mathbf{p}}_{1})\times{\mathbb{P}}(\hat{W}_{p,1}=W_{1})$
MCS-limited |  | $OP_{\rm mcs}^{\rm ma}:~{}\displaystyle\max\limits_{{\mathcal{M}}}$ | $T^{\rm ma}_{\rm mcs}({\mathbf{P}}^{\rm ma}_{\rm wmmse},{\mathcal{M}})$
sum throughput |  | s.t. | ${\mathcal{M}}\in{\mathbb{M}}$
max. problem |  |  |
(Experiments |  |  |
in this paper) |  |  |
TABLE I: System model differences between RSMA, SDMA and NOMA w.r.t the
terminology in Section II-B.

##  III RSMA Prototype using SDR

###  III-A Hardware setup

Figure 2: Block diagram of RSMA Prototype.

Our RSMA prototype is built using National Instruments’ (NI) Universal
Software Radio Peripherals (USRPs). As shown in Fig. 2, two USRP-2942 units
are used to realize the two-user MISO scenario. In particular, the RX antennas
(TP-LINK TL-ANT2405C) are connected to its USRP using coaxial cables, which
allows them to be moved to create different channel configurations (Fig. 4).
The USRPs share a common timing source (CDA-2990), and are connected to a
workstation through which they are controlled using LabVIEW NXG. All
connections are through PCIe cables, facilitated by a PCIe bus (CPS-8910). The
total transmit power, $P_{t}$, is $23{\rm dBm}$. A list of hardware components
is provided in Table II.

No. | Name | Units Description |
---|---|---|---
1. | Workstation | Running LabVIEW NXG |
2. | NI USRP-2942 | SDRs used to realize TX and RXs |
3. | NI CPS-8910 | Provides additional PCIe ports |
4. | NI CDA-2990 | 8 Channel, 10 MHz Clock Distribution Device |
5. | Linx Technologies ANT-W63WSx | TX antennas |
6. | TP-LINK TL-ANT2405C | RX antennas |
TABLE II: List of hardware components.

###  III-B RSMA Implementation Details

Figure 3: (Top and middle): The two-stage transmission protocol; (Bottom):
OFDM symbol structure within the payload. In conventional 802.11g, every DATA
symbol for an RX contains four pilot subcarriers for FPS. However, for RSMA,
this approach would cause the pilot subcarriers in the common and private
streams to interfere with each other. In particular, the inability to do FPS
effectively for the common stream leads to large phase noise. To avoid this,
the common (private) stream(s) carry non-zero pilots only in the odd (even)
DATA symbols [27].

The USRPs are capable of operating in the $2.4{\rm GHz}$ ISM band. Hence, we
choose a center frequency of $2.484{\rm GHz}$ for our transmissions666This
corresponds to channel no. $14$ in the IEEE 802.11 family of standards for the
$2.4$GHz band. We use this channel to avoid ambient WiFi interference, as it
is not commercially used in the UK.. For ease of practicalities like
synchronization and channel estimation, it is convenient to implement signals
based on well-known physical layer standards (e.g., IEEE 802.11 frames) in a
prototype. Hence, we adopt several features of the IEEE 802.11g physical layer
frames in our signal design. Next, we describe in detail how Stages 1 and 2
from Sections II-A and II-B, respectively, are implemented in our prototype:

#### Stage 1

Each TX antenna transmits a pilot signal orthogonally in time comprising a
Short Training Field (STF, $8\mu{\rm s}$ in duration) and a Long Training
Field (LTF, $8\mu{\rm s}$ in duration), as shown in the top portion of Fig. 3.
The STF is used for synchronization and coarse frequency offset estimation,
while the LTF is used to obtain a least squares estimate of the CSI at each RX
over all the subcarriers (i.e., $\hat{{\mathbf{h}}}_{i}[k]$ in Section II-A).

#### Stage 2

Here, the transmitted signal consists of a preamble followed by the data
payload, as shown in the middle portion of Fig. 3.

  * •

Preamble: The preamble consists of one STF and three LTFs. The function of the
STF is the same as the first stage, while the LTFs are precoded in order to
estimate the _precoded CSI_ for equalization at the RX; in particular, the
first LTF is used to estimate ${\mathbf{h}}_{i}^{H}{\mathbf{p}}_{c}$ at RX
$i~{}(=1,2)$ for decoding the common stream, the second to estimate
${\mathbf{h}}_{1}^{H}{\mathbf{p}}_{1}$ to decode the private stream meant for
RX 1, and the third to estimate ${\mathbf{h}}_{2}^{H}{\mathbf{p}}_{2}$ to
decode the private stream meant for RX 2.

  * •

Data Payload: For the payload, we use OFDM signals over a total bandwidth of
$20{\rm MHz}$ with $N_{c}=64$ subcarriers (labeled DATA in Fig. 3) and a
cyclic prefix (CP) of 16 samples per OFDM symbol. Aligned with IEEE 802.11
frames, $52$ subcarriers are used for communications while the rest serve as
guard bands. Among these $52$ subcarriers, $48$ are used to carry data
symbols, with the remaining used for fine phase shifting (FPS)777FPS is used
to correct the common phase error across all subcarriers in one OFDM symbol
[28].. This yields an effective bandwidth of:

$\displaystyle B=20{\rm MHz}\times\underbrace{(64/80)}_{\text{CP
overhead}}\times\underbrace{(48/64)}_{\begin{subarray}{c}\text{Guard band}\\\
\text{overhead}\end{subarray}}=12{\rm MHz}$ (14)

The payload consists of three superposed streams (one common, two private),
each comprising 50 OFDM symbols (labeled DATA 1 through DATA 50, as shown in
the bottom portion of Fig. 3).

  * •

MCS Implementation: Table III lists the MCS levels, ${\mathbb{M}}$,
implemented in our prototype. MCS indices 0 through 7 are identical to IEEE
802.11g, where 64-QAM is the largest supported constellation. To allow for the
possibility of higher data rates due to RSMA’s enhanced interference
suppression, we implement two more MCS levels supporting 256-QAM (MCS indices
8 and 9). After the preamble, the first OFDM symbol (labelled SERVICE in the
bottom portion of Fig. 3) contains the MCS information of each stream.

Unlike the IEEE 802.11 family of standards, which uses LDPC codes for error
correction, we implement Polar codes augmented with an 8-bit cyclic redundancy
check [29, 30], along with successive cancellation list decoding [31], with a
list depth of 2.

MCS Index | Modulation ($m$) | Code Rate, $r$ | Data Rate, $Bmr$ (Mbps)
---|---|---|---
$0$ | BPSK (1) | $1/2$ | $6$
$1$ | BPSK (1) | $3/4$ | $9$
$2$ | QPSK (2) | $1/2$ | $12$
$3$ | QPSK (2) | $3/4$ | $18$
$4$ | 16QAM (4) | $1/2$ | $24$
$5$ | 16QAM (4) | $3/4$ | $36$
$6$ | 64QAM (6) | $2/3$ | $48$
$7$ | 64QAM (6) | $3/4$ | $54$
$8$ | 256QAM (8) | $3/4$ | $72$
$9$ | 256QAM (8) | $5/6$ | $80$
TABLE III: MCS levels (largely based on IEEE 802.11g) implemented in our
prototype. The data rate in the last column is equal to $Bmr$, where $B$ is
the effective bandwidth given by (14).

An instance of Stage 1 and Stage 2, as described above and illustrated in Fig.
3, constitutes a single measurement run. To empirically solve $OP_{\rm
mcs}^{\rm ma}~{}({\rm ma}\in\\{{\rm RSMA,SDMA,NOMA}\\})$ from Table I, we
conduct $100$ measurement runs. Let $D_{s,c}^{\rm ma}$ denote the number of
runs in which the common stream is successfully decoded by both RXs.
Similarly, let $D_{s,i}^{\rm ma}~{}(i=1,2)$ denote the number of runs in which
RX $i$ successfully decodes its private stream. Consider the expressions for
the MCS-limited sum throughput in Table I (third last row). Replacing the
decoding probabilities therein with their empirical estimates, the _measured_
sum throughput for each multiple access scheme is given by:

$\displaystyle T_{\rm mcs}^{\rm RSMA}({\mathbf{P}}_{\rm wmmse}^{\rm
RSMA},{\mathcal{M}})$ $\displaystyle=\frac{D_{s,c}^{\rm
RSMA}}{100}Bm_{c}r_{c}+\sum_{i=1}^{2}\frac{D_{s,i}^{\rm
RSMA}}{100}Bm_{i}r_{i}$ (15) $\displaystyle T_{\rm mcs}^{\rm
SDMA}({\mathbf{P}}_{\rm wmmse}^{\rm SDMA},{\mathcal{M}})$
$\displaystyle=\frac{D_{s,1}^{\rm SDMA}}{100}Bm_{1}r_{1}+\frac{D_{s,2}^{\rm
SDMA}}{100}Bm_{1}r_{1}$ (16) $\displaystyle T_{\rm mcs}^{\rm
NOMA}({\mathbf{P}}_{\rm wmmse}^{\rm NOMA},{\mathcal{M}})$
$\displaystyle=\frac{D_{s,c}^{\rm NOMA}}{100}Bm_{c}r_{c}+\frac{D_{s,1}^{\rm
NOMA}}{100}Bm_{1}r_{1},$ (17)

where $B$ is given by (14) and the MCS levels are chosen from Table III.

##  IV Measurement Results

In this section, we first describe our measurement campaign in Section IV-A.
Then, in Section IV-B, we present measurement results pertaining to $OP_{\rm
mcs}^{\rm ma}$ with which we compare the measured sum throughput for RSMA,
SDMA and NOMA. Then, in Section IV-C, we present a fairness comparison between
RSMA, SDMA and NOMA.

Figure 4: Layout of measurement cases. The RX antenna positions in the above
snapshot correspond to Case 6.

###  IV-A Measurement campaign

Our measurements were carried out on a lab bench, as shown in Fig. 4. The TX
position is fixed along with the antenna of RX 1, which is placed $0.8{\rm m}$
away from the TX888As seen in Table IV, this is the closest TX-RX distance in
our experiments. Since it exceeds the Fraunhofer distance, we are operating in
the far-field.. The antenna position of RX 2 is varied as marked in Fig. 4 to
create pairs of LoS channels with varying relative strengths and spatial
correlation. For NOMA in particular, RX 1 is treated as the strong user and RX
2 the weak user. Every attempt was made to avoid any relative motion between
the TXs and RXs during the measurements. For easy reference, a full list of
all the parameters relevant to our measurements is provided in Table IV.

Parameter | Notation (where applicable) | Value
---|---|---
Center frequency | $f_{c}$ | $2.484{\rm GHz}$
Transmit power | $P_{t}$ | $23{\rm dBm}$
TX antenna length |  | $0.13{\rm m}$
Fraunhofer distance |  | $0.28{\rm m}$
Total bandwidth |  | $20{\rm MHz}$
Subcarriers | Total ($N_{c}$) | $64$
| Data | $48$
| Pilot (FPS) | $4$
| Guard band | $12$
CP length |  | $16$
Effective bandwidth | $B$ | $12{\rm MHz}$
OFDM symbols in payload |  | $50$
Experiment runs (per case) |  | $100$
Distances | RX 1 $\rightarrow$ TX | $0.8{\rm m}$
| RX 2 $\rightarrow$ TX (Cases 1-3) | $1.25{\rm m}$
| RX 2 $\rightarrow$ TX (Cases 4-6) | $2.10{\rm m}$
| RX 2 $\rightarrow$ TX (cases 7-9) | $3.30{\rm m}$
Channel coding |  | Polar codes
List depth |  | 2
Environment |  | LoS
TABLE IV: List of parameters used in our experiments.

To capture the relative pathloss difference between the two channels, we
introduce the channel strength disparity parameter, $\alpha$, defined in dB
scale as follows:

$\displaystyle\alpha~{}[{\rm
dB}]=10\log_{10}\frac{\|\hat{{\mathbf{h}}}_{2}\|}{\|\hat{{\mathbf{h}}}_{1}\|}.$
(18)

where $\hat{{\mathbf{h}}}_{i}~{}(i=1,2)$ is given by (1). Hence, RX 2 has a
weaker channel when $\alpha$ is negative.

Similarly, to measure the spatial correlation between the two channels, we
introduce the correlation parameter, $\rho\in[0,1]$, defined as follows:

$\displaystyle\rho=1-\frac{|\hat{{\mathbf{h}}}_{1}^{H}\hat{{\mathbf{h}}}_{2}|^{2}}{\|\hat{{\mathbf{h}}}_{1}\|.\|\hat{{\mathbf{h}}}_{2}\|}.$
(19)

Thus, $\rho=0$ when the channels are completely aligned, whereas $\rho=1$ when
they are orthogonal.

Based on (18) and (19), we make the following remarks on the cases marked in
Fig. 4.

| Relative pathloss | Spatial correlation
---|---|---
| Cases |  $\alpha$ [dB] | Cases | $\rho$
Low | 1 | $-7.6$ | 1 | $0.15$
| 2 | $-6.4$ | 4 | $0.16$
| 3 | $-8.8$ | 7 | $0.24$
Medium | 4 | $-14.5$ | 2 | $0.48$
| 5 | $-13.0$ | 5 | $0.46$
| 6 | $-13.0$ | 8 | $0.35$
High | 7 | $-23.6$ | 3 | $0.95$
| 8 | $-24.7$ | 6 | $0.95$
| 9 | $-22.1$ | 9 | $0.85$
TABLE V: The cases marked in Fig. 4 correspond to pairs of channels that vary
in terms of their relative pathloss and spatial correlation, captured by
$\alpha$ and $\rho$ defined in (18) and (19), respectively. The above table
lists the empirical average of $\alpha$ and $\rho$ over all experiment
realizations for each case.

######  Remark 5.

For cases 1 through 3 in Fig. 4, we expect the relative pathloss difference
between the two channels to be low by design, as the two RX antennas are
fairly close together. In contrast, for cases 4 through 9, RX 2 is much
farther from the TX than RX 1; hence, we expect $\alpha$ to be more negative
by design. This is validated in Table V, which lists the empirical average of
$\alpha$ over all experiment realizations for each case.

######  Remark 6 (Channels with high spatial correlation).

In cases 1, 4 and 7, the channels are expected to be highly spatially
correlated (i.e., small $\rho$) by design, as the two RXs are nearly in a
straight line from the TX. This is validated in Table V, which lists the
empirical average of $\rho$ over all experiment realizations. RSMA and NOMA
are expected to provide large throughput gains over SDMA in these cases, as
the common stream allows for the decoding and elimination of interference.

######  Remark 7 (Channels with low spatial correlation).

Similarly, in cases 3, 6 and 9, the spatial correlation between the two
channels is expected to be relatively low (i.e., larger $\rho$) by design, due
to the larger angular separation between the RXs as seen by the TX. This is
validated in Table V. The gain of RSMA over SDMA is expected to be modest,
with the common stream being allocated lower power relative to the cases in
Remark 6. NOMA is also unlikely to yield significant gains over SDMA; in fact,
by forcibly imposing a common stream when the channels are not sufficiently
aligned, the NOMA throughput performance can even be worse than SDMA.

###  IV-B RSMA v/s SDMA v/s NOMA: Sum Throughput Comparison

Figure 5: For Case 1, the large spatial correlation (i.e., small $\rho$, see
Table V) between the RX channels induces high mutual interference. Under these
conditions, RSMA and NOMA can achieve significant throughput gains over SDMA
with the help of the common stream, provided its MCS level is not too
aggressive. The highest measured sum throughput above are: $66.48{\rm Mbps}$
for RSMA, $56.16{\rm Mbps}$ for NOMA, and $37.44{\rm Mbps}$ for SDMA.

As an example measurement, the sum throughput performance for Case 1 as a
function of the MCS levels is plotted in Fig. 5 with unquantized CSI feedback.
For RSMA/SDMA, the MCS levels for the two private streams are chosen to be
identical for ease of illustration999Since the two RXs are closely situated
for this case, this is also an intuitively good choice. However, in general,
the MCS levels of the private streams can be different depending on the
channel realizations.. For NOMA, the $x$-axis corresponds to the MCS of RX 1,
as only RX 1 has a private stream. We make the following observations in Fig.
5:

  * 1)

For a given common stream MCS level for RSMA/NOMA, the sum throughput
increases with the private stream MCS level up to a point as expected, before
dropping when the MCS level is too aggressive. The same behavior is observed
for SDMA as well, which does not have a common stream.

  * 2)

For a given private stream MCS level, a similar trend as above can also be
seen w.r.t common stream MCS level for RSMA/NOMA. Specifically, the RSMA/NOMA
sum throughput increases as the common stream MCS level is raised from QPSK
rate $1/2$ (MCS index 2 in Table III) to 16QAM rate $3/4$ (MCS index 5 in
Table III). However, the very next MCS level of 64QAM rate $3/4$ for the
common stream is too aggressive and due to the resulting error propagation,
the RSMA/NOMA sum throughput is lower than SDMA. To demonstrate this, Fig. 6
shows the superposed received symbol constellation for RSMA at RX 1,
corresponding to the MCS levels marked by points ‘C’ and ‘A’ in Fig. 5. The
MCS levels at point C being conservative, a QPSK constellation (private
stream) superposed on another QPSK constellation (common stream) is clearly
visible in Fig. 6a. In a similar vein, if the MCS levels at point A were _not_
aggressive, then we would expect to see 256 distinct symbol clusters in Fig.
6b, corresponding to a superposition of 64QAM (common stream) and QPSK
(private stream) constellations. However, we only observe 81 distinct symbol
clusters, which leads to high error rates for the common stream.

(a) Common stream: QPSK $1/2$, private stream: QPSK $1/2$

(b) Common stream: 64-QAM $2/3$, private stream: QPSK $1/2$

Figure 6: Superposed received symbol constellation for RSMA at RX 1 for the
MCS levels in Fig. 5 corresponding to: (a) point ‘C’, which is an example of a
conservative choice of MCS levels. Hence, the superposition of QPSK and QPSK
constellations is clearly visible; (b) point ‘A’, which is an aggressive
choice of MCS for the common stream. Instead of $256$ distinct symbol clusters
due to a superposition of 64-QAM and QPSK constellations, we only observe $81$
distinct symbol clusters, which leads to high error rates for the common
stream.

  * 3)

Points 1 and 2 above highlight the role of MCS selection in realizing sum
throughput gains for RSMA and NOMA over SDMA. The highest measured sum
throughput in Fig. 5 is the empirical solution to $OP_{\rm mcs}^{\rm ma}$ in
Table I, with the additional constraint that the MCS levels for the two
private streams in RSMA/SDMA are identical. For RSMA, this value is equal to
$66.48{\rm Mbps}$ (marked with a solid red circle); for NOMA, it is equal to
$56.16{\rm Mbps}$ (red square), and for SDMA it is only $37.44{\rm Mbps}$
(dashed red circle).

  * 4)

Finally, for all data points bar one, the RSMA sum throughput exceeds that of
NOMA. Since RX 2’s channel is not much weaker than RX 1 for Case 1 (see
$\alpha$ in Table V), RSMA provides additional throughput gain through the
private stream of RX 2.

Fig. 8 plots the empirical solution for $OP_{\rm mcs}^{\rm ma}$, broken down
in terms of the contributions of the common and private streams. We make the
following observations:

  * A.

Unquantized CSI feedback:

    * 1)

As expected for RSMA, the contribution of the common stream (green bars)
diminishes with increasing $\rho$ (Remarks 6 and 7); in particular, for cases
1, 4 and 7, the common stream constitutes $51\%$, $62\%$ and $62\%$ of the sum
throughput, respectively, whereas for cases 3, 6 and 9, this drops to $22\%$,
$34\%$ and $17\%$, respectively.

    * 2)

In cases 1, 4 and 7, where the channels have high spatial correlation (low
$\rho$, see Table V), SDMA’s sum throughput is significantly lower than those
of RSMA and NOMA due to the high mutual interference. The gains of RSMA and
NOMA in these cases stem from the decoding and elimination of interference
through the common stream (Remark 6). In particular, RSMA achieves gains of
$82\%$, $103\%$ and $134\%$, respectively, over SDMA as shown by the up arrows
in Fig. 8). On the other hand, RSMA’s gain over NOMA for these cases is quite
modest ($23\%$ for Case 1, none for Case 4 and $19\%$ for Case 7).

    * 3)

In cases 3, 6 and 9, where the channels have low spatial correlation (high
$\rho$, see Table V), RSMA’s gain over SDMA is modest due to the relatively
lower mutual interference. On the other hand, the NOMA sum throughput is
significantly worse than both RSMA and SDMA. In particular, RSMA achieves
gains of $38\%$, $45\%$, and $19\%$, respectively, for these three cases over
NOMA.

  * B.

Quantized CSI feedback:

    * 1)

Unsurprisingly, all three multiple access schemes experience a sum throughput
loss due to CSI quantization. However, SDMA and RSMA are more adversely
affected than NOMA, especially in cases 4 through 9, where there is a
significant pathloss difference between the two channels. For these cases,
most of the throughput loss for RSMA and SDMA can be attributed to the private
stream of RX 2 – the weaker user. In contrast, the robustness of NOMA to CSI
quantization stems from the fact that RX 2 does not have a private stream.
Furthermore, due to the very low private stream throughput for RX 2, the RSMA
throughput performance is similar to NOMA for these cases.

    * 2)

However, in the absence of a large pathloss difference between the two
channels – for instance, in cases 2 and 3 – the private stream throughput for
RX 2 is still quite high, despite the loss from CSI quantization. Under these
conditions, NOMA with its missing private stream for RX 2 continues to perform
worse than both RSMA and SDMA, mirroring the performance seen with unquantized
CSI feedback.

    * 3)

In all cases bar one, RSMA’s gain over SDMA is much larger when compared to
unquantized CSI feedback. This is because imperfect CSI has a more adverse
effect on private stream throughputs due to the higher residual interference.
Thus, due to the throughput contribution of the common stream, which stays
pretty steady even with quantized CSI, RSMA’s throughput loss is lower than
that of SDMA.

(a)

(b)

Figure 7: The maximum measured sum throughput under unquantized (UQ) and
quantized (Q) CSI feedback for RSMA, SDMA and NOMA, which correspond to the
empirical solutions of $OP_{\rm mcs}^{\rm RSMA}$, $OP_{\rm mcs}^{\rm SDMA}$
and $OP_{\rm mcs}^{\rm NOMA}$, respectively. The arrows indicate RSMA’s gain
w.r.t. SDMA and NOMA. (a) Figure 8: (contd.)

In summary, RSMA achieves the best throughput performance across (a) different
RX channels that vary in their relative strength and spatial correlation, and
(b) different CSI quality. In some cases, SDMA is effective while in others
NOMA is effective. RSMA is effective in all the cases, and thus effectively
bridges between both SDMA and NOMA.

###  IV-C RSMA v/s SDMA v/s NOMA: Fairness Comparison

In the previous subsection, we saw that for RSMA, the private stream
throughput of RX 2 is severely diminished when there is a large pathloss
difference between the two channels and imperfect CSI (bullet point B1). To
increase RX 2’s total throughput in such cases, a larger fraction ($>0.5$) of
the common stream can be allocated to it. This would not alter the RSMA sum
throughput101010For precisely this reason, the allocation of the common stream
between the two RXs is not mentioned when discussing the sum throughput
results in Section IV-B., but would achieve better (max-min) fairness [17].

In this subsection, we experimentally verify this possibility of improved
fairness afforded by RSMA. This is done in the following manner for each case
in Fig. 8:

  * S1)

if $R_{1}({\mathbf{P}}_{\rm wmmse}^{\rm RSMA})-R_{2}({\mathbf{P}}_{\rm
wmmse}^{\rm RSMA})\leq R_{c}({\mathbf{P}}_{\rm wmmse}^{\rm RSMA})$, then the
common stream is partitioned such that $R_{1}({\mathbf{P}}_{\rm wmmse}^{\rm
RSMA})+(W_{c,1}/W_{c})R_{c}({\mathbf{P}}_{\rm wmmse}^{\rm
RSMA})=R_{2}({\mathbf{P}}_{\rm wmmse}^{\rm
RSMA})+(W_{c,2}/W_{c})R_{c}({\mathbf{P}}_{\rm wmmse}^{\rm RSMA})$ to ensure
that both RXs have the same throughput;

  * S2)

otherwise, when $R_{1}({\mathbf{P}}_{\rm wmmse}^{\rm
RSMA})-R_{2}({\mathbf{P}}_{\rm wmmse}^{\rm RSMA})>R_{c}({\mathbf{P}}_{\rm
wmmse}^{\rm RSMA})$, the entire common stream is allocated to RX 2. This is
similar to NOMA, albeit RX 2 has a private stream as well.

######  Remark 8 (Fairness while maximizing throughput).

User fairness in RSMA/SDMA/NOMA is often framed as an optimization problem,
where the objective is to maximize the minimum user throughput through
precoder design. Hence, the optimal precoders for fairness are, in general,
different from the optimal precoders for sum throughput maximization. The
fairness strategy in S1 and S2 differs from this viewpoint by trying to
achieve fairness without changing the precoders. In other words, we are trying
to achieve fairness while still maximizing the sum throughput. This ability to
realize both objectives simultaneously is a unique feature of RSMA. S1 and S2
cannot be realized for NOMA and SDMA because the former does not have a
private stream for RX 2 and the latter does not have a common stream.

Fig. 9 compares the relationship between the sum throughput and the minimum
throughput for RSMA (subject to S1 and S2), SDMA and NOMA. The case number is
indicated beside each data point in the same color. The dashed $y=2x$ line
represents max-min fairness and the region above it is feasible for all three
multiple access schemes. Thus, points that are closer (in terms of Euclidean
distance) to the dashed line represent _fairer_ outcomes. Based on this
insight, we make the following remarks:

  * •

When S1 is true, the corresponding RSMA point lies on the dashed line (i.e.,
achieves max-min fairness), as seen for all cases bar one in Figs. 9a and b.
Otherwise, for S2, the RSMA point lies _as close as possible_ above the dashed
line, as seen for case 8 in Figs. 9a and b.

  * •

In Fig. 9a, SDMA also achieves approximate fairness for some cases (namely 1,
2, 3 and 4), as does NOMA (case 7). However, the RSMA points achieve fairness
at a higher sum for these cases lie to the _northeast_ of the corresponding
SDMA/NOMA points, which indicates fairness at a higher sum throughput.

In summary, RSMA is capable of achieving better fairness at a higher sum
throughput than both SDMA and NOMA, regardless of CSI quality, through a
judicious allocation of the common stream to the weaker RX.

(a) Unquantized CSI feedback

(b) Quantized CSI feedback

Figure 9: Fairness comparison between RSMA (subject to S1 and S2), SDMA and
NOMA. The number beside each data point indicates the measurement case from
Fig. 4. The dashed line ($y=2x$) corresponds to max-min fairness and points
that are closer (in terms of Euclidean distance) to this line represent fairer
outcomes.

##  V Summary

In this paper, we realized the first-ever RSMA prototype using SDRs in order
to experimentally validate its performance gains promised in the theoretical
literature. To this effect, we measured the throughput and fairness
performance of RSMA, SDMA and NOMA using our prototype for the two-user MISO
scenario over (a) different pairs of channels varying in terms of their
relative strength and spatial correlation and (b) different CSI quality.
Across these different cases, we observed that RSMA achieved the highest sum
throughput, whereas SDMA and NOMA were each effective only in some cases. We
also observed that RSMA achieved better fairness at a higher sum throughput
than both SDMA and NOMA. As the first-ever experimental comparison between
RSMA, SDMA and NOMA, our efforts in this paper address the pressing question
of which multiple access technique is best equipped to deliver the performance
enhancements expected from next-generation networks like 6G. Thus, the
superior performance of RSMA in these experiments provides the impetus for
further experimental efforts at a larger scale and in more challenging
operating environments.

## Acknowledgments

We thank Onur Dizdar for his valuable inputs on Polar code implementation.

##  Appendix A CSI Quantization Technique

The quantized wideband CSI, $\hat{\mathbf{h}}^{\rm Q}_{i}:=[\hat{h}_{i1}^{\rm
Q}~{}\hat{h}_{i2}^{\rm Q}]$, is obtained from
$\hat{\mathbf{h}}_{i}=[\hat{h}_{i1}~{}\hat{h}_{i2}]$, as follows:

  * (a)

The _linear scaler_ is evaluated as follows:

$\displaystyle M_{h}^{\mathrm{lin}}$
$\displaystyle:=\frac{m_{h}}{10^{M_{h}/20}}$ (20)
$\displaystyle\mbox{where}~{}m_{h}$
$\displaystyle:=\max_{i,l}\\{|\mathrm{Re}(\hat{h}_{il})|,|\mathrm{Im}(\hat{h}_{il})|\\}~{}(i,l\in\\{1,2\\}),$
(21) $\displaystyle\mbox{and}~{}M_{h}$
$\displaystyle:=\min\biggl{\\{}7,\Bigl{\lfloor}20\log_{10}(m_{h})\Bigr{\rfloor}\biggr{\\}}.$
(22)

$M_{h}$ in (22) is known as the scaling ratio and is quantized to three bits
for feedback.

  * (b)

Using (20), the real and imaginary parts $\hat{h}_{ij}$ are quantized using
$N_{b}=4$ bits in our prototype as follows.

$\displaystyle{\rm Re}(\hat{h}_{ij}^{\rm Q})$
$\displaystyle:=\mathrm{round}\biggl{(}\frac{\mathrm{Re}(\hat{h}_{ij})}{M_{h}^{\mathrm{lin}}}(2^{N_{b}-1}-1)\biggr{)}$
(23) $\displaystyle{\rm Im}(\hat{h}_{ij}^{\rm Q})$
$\displaystyle:=\mathrm{round}\biggl{(}\frac{\mathrm{Im}(\hat{h}_{ij})}{M_{h}^{\mathrm{lin}}}(2^{N_{b}-1}-1)\biggr{)}$
(24)

The feedback overhead for the entire CSI equals $3+2N_{b}il$ bits, where the
first term corresponds to the scaling ratio, and the factor of two in the
second term accounts for the real and imaginary parts of $\hat{h}_{ij}^{\rm
Q}$. For our prototype, this value equals $35$ bits, as opposed to the
$128il=512$ bits without quantization.

At the TX, the CSI estimates, denoted by $\hat{h}_{il}^{\rm TX}$, are
recovered as follows:

$\displaystyle\hat{h}_{il}^{\rm TX}:=\frac{\hat{h}_{il}^{\rm
Q}}{10^{M_{h}/20}}~{}(i,l\in\\{1,2\\}).$ (25)

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