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arXiv:2105.07362v1 [cs.IT] 16 May 2021
1
Rate-Splitting Multiple Access for Downlink
Multiuser MIMO: Precoder Optimization and
PHY-Layer Design
Anup Mishra, Yijie Mao, Member, IEEE, Onur Dizdar, Member, IEEE, and
Bruno Clerckx, Senior Member, IEEE
Abstract
Rate-Splitting Multiple Access (RSMA) has recently appeared as a powerful and robust multiple access and
interference management strategy for downlink Multi-user (MU) multi-antenna communications. In this work, we
study the precoder design problem for RSMA scheme in downlink MU systems with both perfect and imperfect
Channel State Information at the Transmitter (CSIT) and assess the role and benefits of transmitting multiple
common streams. Unlike existing works which have considered single-antenna receivers (Multiple-Input Single-
Output–MISO), we propose and extend the RSMA framework for multi-antenna receivers (Multiple-Input Multiple-
Output–MIMO) and formulate the precoder optimization problem with the aim of maximizing the Weighted Ergodic
Sum-Rate (WESR). Precoder optimization is solved using Sample Average Approximation (SAA) together with the
proposed vectorization and Weighted Minimum Mean Square Error (WMMSE) based approach. Achievable sum-
Degree of Freedom (DoF) of RSMA is derived for the proposed framework as an increasing function of the number
of transmitted common and private streams, which is further validated by the Ergodic Sum Rate (ESR) performance
using Monte Carlo simulations. Conventional MU-MIMO based on linear precoders and Non-Orthogonal Multiple
Access (NOMA) schemes are considered as baselines. Numerical results show that with imperfect CSIT, the sum-
DoF and ESR performance of RSMA is superior than that of the two baselines, and is increasing with the number
of transmitted common streams. Moreover, by better managing the interference, RSMA not only has significant
ESR gains over baseline schemes but is more robust to CSIT inaccuracies, network loads and user deployments.
Index Terms
Rate-Splitting Multiple Access (RSMA), interference management, imperfect CSIT, Degree of Freedom (DoF),
MU–MIMO, MIMO, MIMO NOMA, Weighted Sum-Rate (WSR).
This work has been partially supported by the U.K. Engineering and Physical Sciences Research Council (EPSRC) under grant
EP/N015312/1, EP/R511547/1.
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I. INTROD UCTIO N
MUltiple-Input Multiple-Output (MIMO) communication networks are one of the key enabling
technologies for current and future wireless networks. By multiplexing signals in space, MIMO
networks are capable of providing remarkably higher spectral efficiency. For a point-to-point MIMO
channel, the channel capacity is known to scale linearly with the minimum number of transmit and receive
antennas at high Signal-to-Noise Ratio (SNR), regardless of the level of Channel State Information (CSI)
available to the Base Station (BS) [1]. However, such result does not hold in a Multi-user (MU) network
environment. When Channel State Information at the Transmitter (CSIT) is perfect, it is well known that
Dirty Paper Coding (DPC) achieves the capacity region of the MIMO Broadcast Channel (BC) [2], [3] but
its implementation is prohibitive due to high computational complexity. A more practical approach that
has attracted great attention is Space Division Multiple Access (SDMA) and MU–MIMO implemented
using MU Linear Precoders (LP)1[4]–[7]. However, this approach has many limitations. Among them,
one crucial limitation is that SDMA and MU-MIMO require accurate CSIT to design beamforming and
manage interference. Therefore, it is sensitive to the CSIT inaccuracy. In practical wireless communication
networks, the CSIT quality can deteriorate due to several reasons such as pilot reuse in Time Division
Duplex (TDD), mobility, latency, quantization errors in Frequency Division Duplex (FDD) and hardware
impairments. Poor CSIT quality leads to poor interference management in current MIMO BC based on
SDMA and MU–MIMO, and therefore, acts as a primary bottleneck in meeting demands of higher data
rates.
Besides SDMA, another multiple access scheme that has recently attracted great attention in multi-
antenna MU networks to manage interference is Non-Orthogonal Multiple Access (NOMA). Motivated
by the established result in Single-Input Single-Output (SISO) BC, where power-domain NOMA based
on Superposition Coding (SC) at the transmitter and Successive Interference Cancellation (SIC) at the
receivers is known as a capacity-achieving scheme (such scheme is also known as SC–SIC), NOMA has
been applied to multi-antenna BC [8], [9]. A typical MIMO NOMA strategy is to group the users into
different user groups and apply SIC within each group to decode the intra-group interference. The inter-
group interference is treated as noise [10]. However, it has been pointed out in [10]–[13] that NOMA,
which forces one user to decode the message of other users, causes spatial Degree of Freedom (DoF) loss
and is inefficient in multi-antenna settings. Hence, in multi-antenna (Gaussian) BC with perfect CSIT, the
only known capacity-achieving scheme is Dirty Paper Coding (DPC), rather than NOMA.
To overcome the limitations of SDMA and NOMA, a novel multiple access scheme, called Rate-
Splitting Multiple Access (RSMA) is introduced in [10] for Multiple-Input Single-Output (MISO) BC.
1MU–MIMO techniques based on LP will be referred to as MU–MIMO in the rest of the paper.
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RSMA is based on splitting user messages into common and private parts at the transmitter. The common
parts of the user messages are combined into common messages and encoded into common streams to be
decoded by multiple users (but not necessarily intended to all those users). The private parts of all users
are independently encoded into private streams to be decoded by the corresponding users only and treated
as noise by other users. The encoded common and private streams are superposed in a non-orthogonal
manner. Users rely on SIC to first decode the intended common streams before decoding the intended
private streams. By adjusting the message split and the power allocation to the common and private streams,
RSMA manages to partially decode the interference and treat the remaining interference as noise. This
capability allows RSMA to act as a bridge between the two extreme interference management schemes
of fully treating interference as noise (as in SDMA and MU–MIMO) and fully decoding interference (as
in NOMA), and creates the opportunity to enhance the Quality of Service (QoS) and reduce complexity
[14]. RSMA is shown to be a more general multiple access scheme embracing SDMA and NOMA as
special cases [10], [12]. It is built upon 1-layer Rate-Splitting (RS), a low-complexity strategy that relies
on one layer of SIC at each user [15]. For simplicity, 1-layer RS will be referred to by ‘RS’ for the rest
of the paper.
The concept of RS was first introduced in [16] for a two-user SISO Interference Channel (IC) and
has nowadays been further developed for multi-antenna BC [12], [17]–[24]. In multi-antenna BC, RS
has been studied from an information theoretic perspective [15], [20], [21], [25] demonstrating that it
achieves the optimal sum-DoF [20] and the entire DoF region [25] of the K-user underloaded MISO BC
with imperfect CSIT. [21] investigates the achievable DoF region of RS in asymmetric MIMO BC and
IC with imperfect CSIT and this achievable DoF region is shown in [26] to be optimal. Expanding the
scope from high SNR to finite SNR regime of MISO BC, the energy efficiency performance of RSMA
is investigated in [27] and its spectral efficiency performance is investigated in [10] with perfect CSIT
and in [18], [20] with imperfect CSIT. RSMA has been shown to achieve a rate region close to DPC in
MISO BC with perfect CSIT [10]. When CSIT is imperfect, linearly precoded RS is able to achieve a
larger rate region than DPC [28]. A novel non-linearly precoded RS scheme, namely, Dirty Paper Coded
RS is also proposed in [28], which is shown to outperform linearly precoded RS and DPC. [29] has
explored the benefits of RS using non-linear precoding technique named Tomlinson-Harashima Precoding
(THP) in MISO BC. [30] investigates precoder design and stream selection for RS in MISO BC. Apart
from the conventional MISO BC, performance benefits of RS have also been exploited in massive MIMO
[31], [32], millimeter wave systems [33], [34], multigroup multicasting [35], [36], multicarrier multigroup
multicast [37], joint unicast and multicast transmission [14], Cloud Radio Access Network (C-RAN)
[38], cooperative user relaying [39], secure transmission [40], etc. Furthermore, [41] investigates resource
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allocation for multicarrier RSMA systems, [42] studies RSMA in aerial networks and RSMA is shown
to provide better robustness, rate and QoS in multi-cell Coordinated Multipoint (CoMP) [43]. RSMA is
therefore a more promising strategy to manage interference in MIMO networks with both perfect and
imperfect CSIT.
RS has been fairly studied and analyzed in different aforementioned works. However, the number of
receive antennas at each user is limited to one in most of these works. While there are studies considering
multi-antenna receivers (in MIMO settings), the scope of such studies is limited. For example, [44]
proposes practical stream combining techniques together with Regularized Block Diagonalization (RBD)
precoding for RS in MIMO BC but with only a single common stream and without precoding optimization.
To the best of our knowledge, the role and benefits of multiple common streams in MIMO BC at finite
SNR have not been investigated and remains an open problem. Moreover, in both perfect and imperfect
CSIT settings, the achievable rate region of RS in MIMO BC is still unknown.
A. Motivations and Contributions
In light of the information theoretic results of [21], in a symmetric setup with Mtransmit antennas and
Qreceive antennas at each user, min(M, Q)common streams should be transmitted in RS to achieve the
information theoretical optimal DoF with imperfect CSIT. Motivated by this result and the performance
gain of RS over SDMA, NOMA and DPC in terms of rate and sum-DoF with imperfect CSIT in MISO
BC, we fill the aforementioned research gaps and make the following contributions:
•We introduce a general framework of RS in symmetric MIMO BC with the same number of receive
antennas at all users. The setting is general in the sense that RS can have arbitrary number of common
streams between 1and min(M, Q)inclusive. This is the first work that allows flexibility in the number
of common and private streams to be transmitted in RS.
•At high SNR, we derive the achievable sum-DoF for the proposed RS framework in MIMO BC
with imperfect CSIT and show the influence of multiple common streams on the sum-DoF of RS.
Even with a single common stream, the sum-DoF of RS is shown to be greater than the sum-DoF of
MU–MIMO and MIMO NOMA. This sum-DoF of RS increases as the number of common streams
increases. We show that by transmitting multiple common streams, the sum-DoF gain of RS over
MU–MIMO and MIMO NOMA increases. The assertions are further testified through the Ergodic
Sum Rate (ESR) performance using Monte-Carlo simulations. This is the first paper to compare the
DoF of MU-MIMO, MIMO NOMA and RS in a MIMO setting as opposed to current comparisons
as in [13] which are limited to MISO settings.
•We propose to utilize vectorization and Weighted Minimum Mean Square Error (WMMSE)-based
Alternative Optimization (AO) algorithm to optimize the precoders for RS in MIMO BC with the
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aim of maximizing the WSR subject to the transmit power constraint. The proposed optimization
framework addresses the challenge of intractable optimization introduced due to matrix variables. To
the best of our knowledge, this is the first work that studies the precoder optimization and the benefits
of transmitting multiple commons streams in RS-assisted MIMO BC with perfect and imperfect CSIT.
•Under the assumption of Gaussian signalling and infinite block lengths, we demonstrate that the
Ergodic Rate (ER) region of RS with optimized precoders always outperforms the ER regions of
MU–MIMO and MIMO NOMA in MIMO BC with both perfect and imperfect CSIT. When CSIT is
perfect, we also demonstrate that the ER region of RS comes closer to the capacity region achieved
by DPC than MU–MIMO and MIMO NOMA. This is the first work to demonstrate such benefits of
RS in MIMO settings.
•To demonstrate the performance of RS in practical systems, we design the Physical (PHY)-layer
architecture of RS with finite constellation modulation schemes, finite length polar codes and Adaptive
Modulation and Coding (AMC). We show via the Link Level Simulations (LLS) that RS achieves
significant throughput gain over MU–MIMO and MIMO NOMA in MIMO BC. This is the first work
to design the PHY-layer architecture and to provide the LLS of RS in MIMO settings.
B. Organisation
The rest of the paper is organized as follows. In Section II, the system model and CSIT assumptions
are delineated. Problem is formulated in Section III. Section IV contains the proposed methodology to
solve the optimization problem. Section V describes the PHY-layer architecture for RS. Simulation results
are illustrated in Section VI and Section VII concludes the paper. Appendix A contains the derivation of
the achievable sum-DoF for RS, MU-MIMO and MIMO NOMA schemes.
C. Notations
Matrices are denoted by boldface uppercase letters, column vectors are denoted by boldface lowercase
letters and scalars are denoted by standard letters. Trace and determinant of matrix Aare denoted by
tr(A)and det(A), respectively. diag(A)denotes the diagonal entries of the matrix. ATand AHdenote
the Transpose and Hermitian operators on the matrix A, respectively. Euclidean norm of a vector ais
denoted as kak.⊗denotes the kronecker product and vec(A)denotes vectorization of matrix A.EX{Y}
is expectation of Yw.r.t random variable X.CM×Nand RM×Ndenote the set of all M×Ndimensional
matrices with complex-valued and real-valued entries, respectively. The Circularly Symmetric Complex
Gaussian (CSCG) distribution with mean µand variance σ2is denoted as C N (µ, σ2).
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II. SYSTEM MO DE L
We consider a system model in which a BS consisting of Mtransmit antennas is serving Kusers indexed
by the set K={1,...,K}, each equipped with Qreceive antennas. The transmit signal x∈CM×1
is subject to a power constraint E{kxk2} ≤ Pt. The signal is transmitted through a MIMO BC with
Hk∈CM×Qdenoting the channel matrix between the BS and user-kand it is drawn from a continuous
distribution. The signal received at user-kis given by
yk=HH
kx+nk,(1)
where nk∼ CN (0, σ2
n,kIQ)is the Additive White Gaussian Noise (AWGN) vector and is independent of
the channel matrices. Without loss of generality, we assume the noise variances across users to be equal,
i.e., σ2
n,k =σ2
n,∀k∈ K. We assume that each user has complete knowledge of the channel information,
i.e., perfect CSI at the Receiver (CSIR). In contrast, the BS only has partial knowledge of users CSI. Next
we detail the channel acquisition at the BS.
A. Imperfect CSIT
The overall channel state can be denoted as H= [H1,H2,...,HK]∈CM×(QK ), where the fading
channel varies according to an ergodic stationary process during the time of transmission. The probability
density function of the stationary process is fH(H). Practical limitations in CSI acquisition such as
quantized feedback [45], feedback and processing delay [46], [47], hardware impairments [48] and channel
estimation [49] result in partial knowledge of the CSI at the BS given by b
H= [ b
H1,b
H2,..., b
HK]and is
modeled as H=b
H+e
H. We assume that the joint distribution of the channel state and its estimate {H,b
H}
is ergodic and stationary [20]. The conditional density fH|
b
H(H|b
H)is assumed to be known at the BS while
His unknown over the entire transmission. Error in the estimation is defined by the channel estimation
error matrix e
H= [ e
H1,e
H2,..., e
HK]in which each element of e
Hkis an independent and identically
distributed (i.i.d) complex Gaussian distribution variable with zero mean. Whereas, E{e
Hke
HH
k}=Re,k
is the covariance matrix of the error matrix, independent of b
Hk. Furthermore, the average CSIT error
power is defined as σ2
e,k ,Ee
Hkke
Hkk2=1
Mtr(Re,k).σ2
e,k is allowed to scale as O(P−α
t)felicitating
the scaling of the CSIT quality with SNR, where α∈[0,∞)is the quality scaling factor representing the
quality of CSI at the BS in the high SNR regime [20], [45]–[47]. Consequently, we write σ2
e,k =O(P−α
t)
such that the error variance is assumed to scale exponentially with SNR. For α=∞, the average CSIT
error power is equal to zero as σ2
e,k = 0,∀k∈ K, resulting in a perfect CSIT scenario. On the other
extreme, for α= 0, the CSIT quality remains invariant w.r.t SNR. Thus, a finite non-zero αleads to CSIT
quality improvement as SNR increases, e.g., increasing the number of feedback bits with SNR. Here we
truncate α∈[0,1]. From a DoF perspective α= 1 corresponds to perfect CSIT [20].
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B. MIMO Rate Splitting
Here we delineate the RS framework proposed for MIMO BC.
1) Transmitter: There are Qk≤min(M, Q)messages intended for user-k,∀k∈ K, such that PK
k=1 Qk=
Qp= min(M, K Q). These messages are expressed as wk={Wk
1, W k
2,...,Wk
Qk},k∈ K. Each message
of user-kis split into a common part and a private part as Wk
i={Wc,k
i, W p,k
i},∀i∈ {1,...,Qk}. The
common parts wc,1,...,wc,K of the messages of all users, with wc,k ={Wc,k
1, W c,k
2,...,Wc,k
Qk}denoting
common parts of user-k, are combined into Qc, Qc∈ {1,...,min(M, Q)}common messages denoted by
wc∈CQc×1, and encoded together into a common stream vector of size Qcdenoted by sc= [sc
1,...,sc
Qc]T.
scwill be decoded by all users. The private parts of user-k,wp,k ={Wp,k
1,...,Wp,k
Qk} ∈ CQk×1are
independently encoded into a private stream vector sk= [sp,k
1,...,sp,k
Qk]Tmeant to be decoded by the
corresponding user-konly. Therefore, the overall data stream vector to be transmitted is expressed as
s= [sc,s1, .., sK]T. We use linear precoders P= [Pc,P1, .., PK]to precode the data streams, where
Pc∈CM×Qcis the precoder for the common stream vector and Pk∈CM×Qkis the precoder for
the private stream vector of user-k. The resulting transmit signal is x=Ps. The assumption is that
E{ssH}=Ithereby making the transmit power constraint as Etr(PPH)≤Pt.
2) MMSE Receiver and Rates: At user-k, first the common stream vector scis decoded into b
wcby
treating the interference from all private stream vectors as noise. Once the common stream vector is
decoded and removed successfully using SIC, the private stream vector skof user-kis decoded into
b
wp,k by treating interference from private stream vectors of other users as noise. User-kreconstructs its
original message by extracting b
wc,k from b
wcand combining it with b
wp,k to form b
wk. Fig. 1 shows the
K-user RS transmission model for MIMO BC. Next, we specify the instantaneous and ergodic rates for
the common and private stream vectors (which are respectively denoted as common rate and private rate
in the following).
Since the precoder design at the BS is dependent on the channel estimate b
Hwhile each user having
perfect CSIR decodes its intended streams based on the exact channel H, a joint fading state {b
H,H}
determines the instantaneous common and private rates of each user. Assuming the signalling to be
Gaussian, for a given channel realization, the instantaneous common and private rates Rz,k (H,b
H), z ∈
{c, p}2can be written as
Rc,k(H,b
H) = log2det(I+PH
cHk(Rc,k)−1HH
kPc),
Rp,k(H,b
H) = log2det(I+PH
kHk(Rp,k)−1HH
kPk).
(2)
2To avoid redundancy, wherever possible, subscript z, z ∈ {c, p}will be used throughout the paper to simultaneously represent entities
associated with the common and private stream vectors with crepresenting entities associated with the common stream vector and pwith
the private stream vector.
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Figure 1: Transmission model of RS in MIMO BC
The noise plus interference covariance matrices Rz,k(H,b
H), z ∈ {c, p}for the common and private stream
vectors at user-kare defined as
Rc,k(H,b
H) = IQ+
K
X
i=1
HH
kPiPH
iHk,Rp,k(H,b
H) = IQ+
K
X
i=1,i6=k
HH
kPiPH
iHk.(3)
Denote the receive filters for common and private stream vectors at user-kas Gc,k(H,b
H)∈CQc×Qand
Gp,k(H,b
H)∈CQk×Q, respectively. The estimated common stream vector is denoted as b
sc,k =Gc,kyk.
Assuming successful decoding and removal of the common stream vector, the private stream vector is
estimated as b
sk=Gp,kyk−HH
kPcsc. Therefore, Mean Square Error (MSE) matrices are written as
Ec,k(H,b
H) = E(b
sc,k −sc)(b
sc,k −sc)H,Ep,k(H,b
H) = E(b
sk−sk)(b
sk−sk)H.(4)
Minimizing the MSEs by solving ∂Ez,k
∂Gz,k = 0, z ∈ {c, p}leads to respective Minimum MSE (MMSE)
filters
GMMSE
c,k (H,b
H) = arg min
Gc,k
E[kGc,kyk−sck2] = PH
cHk(HH
kPcPH
cHk+Rc,k)−1,(5)
GMMSE
p,k (H,b
H) = arg min
Gp,k
E[kGp,k(yk−HH
kPcsc)−skk2] = PH
kHk(HH
kPkPH
kHk+Rp,k)−1.(6)
Substituting the MMSE filters (5) and (6) in (4), respectively, the MMSE matrices for common and private
stream vectors are calculated as
EMMSE
c,k (H,b
H) = (I+PH
cHk(Rc,k)−1HH
kPc)−1,EMMSE
p,k (H,b
H) = (I+PH
kHk(Rp,k)−1HH
kPk)−1.(7)
Using MMSE matrices in (7) and equation (2), we obtain the instantaneous rate expressions as Rz,k (H,b
H) =
log2det EMMSE
z,k (H,b
H)−1, z ∈ {c, p}.Partial knowledge of the CSI at the BS may result in overestima-
tion of the instantaneous rates, rendering them unachievable [20]. A robust method would be to design the
9
precoders based on the ER assuming that the transmission is delay unlimited. The ERs for the common
and private stream vectors of user-kare defined as
¯
Rz,k ,E{H,
b
H}{Rz,k (H,b
H)}, z ∈ {c, p}.(8)
ER characterizes the long-term performance of user-kover all possible joint fading states. To ensure
that each user is able to successfully decode the common stream vector, it needs to be sent at an ER
¯
Rc= minj¯
Rc,jj=K
j=1 . The common rate is shared by all users with ¯
Ckdenoting the share allocated to
user-ksuch that Pk∈K ¯
Ck=¯
Rc. Therefore, the total ER achieved by user-kis equal to ¯
Rk,tot =¯
Rp,k +¯
Ck.
C. Two User Example
To better illustrate RS, we consider a two-user case. At the BS, Qk, k ∈ {1,2}messages of both users
denoted by w1={W1
1,...,W1
Q1}and w2={W2
1,...,W2
Q2}are respectively split into sub-messages
as w1={Wc,1
1, W p,1
1},...,{Wc,1
Q1, W p,1
Q1}and w2={Wc,2
1, W p,2
1},...,{Wc,2
Q2, W p,2
Q2}. Sub-messages
{{Wc,k
1,...,Wc,k
Qk} | ∀k∈ {1,2}} are combined and jointly encoded into a common stream vector sc
of size Qc. Whereas, the private sub-messages {Wp,1
1,...,Wp,1
Q1}and {Wp,2
1,...,Wp,2
Q2}are respectively
encoded into streams s1of size Q1and s2of size Q2. For instance, in our simulations, we consider
M= 4,Q= 2,Q1=Q2= 2 and Qc= 2. The transmit signal is formed by precoding and superposing
the encoded data stream vectors sc,s1,s2, which is expressed as x=Pcsc+P1s1+P2s2. At the user
side, both users decode their intended streams using SIC. At user-1, first scis decoded by treating s1and
s2as noise. Assuming scis successfully decoded, user-1then removes it from the received signal and
decodes s1by treating s2as noise. Similarly, user-2decodes its intended common and private streams
sequentially. The two-user RS case reduces to the conventional MU–MIMO strategy by simply turning
off the common streams, i.e., allocating no power to the common stream vector. Considering the other
extreme of fully decoding the interference, we look at the MIMO NOMA strategy. By encoding the entire
w2into sc, allocating no power to s2and encoding w1into s1, the two-user RS reduces to MIMO NOMA
with decoding order 1→2, where the message of user-2is decoded by both users and the message of
user-1is decoded by user-1only. Table I illustrates the mapping of messages to streams.
Table I: Messages mapped to streams in different schemes.
s1s2sc
RS wp,1wp,2wc,1,wc,2
MU-MIMO w1w2-
MIMO NOMA w1-w2
decoded by its corresponding user decoded by
and treated as noise by the other user both users
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III. PROBLE M FORMULATIO N
In this section, our objective is to formulate the precoder optimization problem for the proposed system
model. A naive approach for precoder design is to assume the channel estimate b
Hto be perfect and
optimize the instantaneous precoder Pby maximizing the instantaneous WSR subject to the instantaneous
power constraint tr(PPH)≤Pt. However, this approach might not be able to cope with MU interference
and it may lead the BS to transmit at undecodable rates [20]. A more robust approach to designing
precoders is to maximize the Weighted Ergodic Sum-Rate (WESR) which captures the long-term WSR
performance of all users to ensure reliable transmission. We first define the WESR for RS as ¯
RRS (µ) =
PK
k=1 µk¯
Rk,tot, where µkis the weight allocated to user-kand µ={µ1, ..., µK}. Next, we consider
the Weighted Average SR (WASR) optimization approach to maximize WESR at the BS with imperfect
instantaneous CSIT. Though it is difficult to predict the instantaneous rates at the BS, the BS can instead
access the Average-Rate (AR).
Definition 1. For a given channel state estimate b
Hand precoder P(b
H), AR is defined as the expected
performance over the CSIT error distribution. The ARs for the common and private stream vectors at
user-kare given by
b
Rz,k (b
H) = EH|
b
HRz,k (H,b
H)|b
H, z ∈ {c, p}.(9)
AR should not be confused with ER. While ER captures the long-term performance over all channel
states, AR measures the short-term expected performance over CSIT error distribution for one channel
estimate. By using the law of total expectation and the definition of AR, the relation between ER and AR
of the common and private stream vectors at user-kis established as ¯
Rz,k =Eb
Hb
Rz,k (b
H), z ∈ {c, p}
[20]. The share of the AR allocated to user-kcorresponding to the common stream vector is defined as
b
Cksuch that PK
k=1 b
Ck=b
Rcand b
Rcmust not be greater than mink∈K{b
Rc,k}. For calculating AR of the
common stream vector b
Rc, we write
min
k∈K Eb
H{b
Rc,k}≥Eb
Hmin
k∈K {b
Rc,k},(10)
as interchanging minimization and expectation in (10) does not increase the value of left hand side.
Consequently, following Law of Large Numbers (LLN) we approximate the ER of each stream by
averaging its AR over all channel states and thereby remove dependencies among channel states. This
allows us to decompose the WESR maximization problem with short term3power constraints to a WASR
3For tractability, we replace the long-term power constraint EH{tr(PPH)} ≤ Ptwith short-term power constraints [20].
11
maximization problem for each b
Hdefined as
b
RRS (Pt,µ) = max
P,b
c
K
X
k=1
µkb
Rk,tot (11a)
b
C1+b
C2+... b
CK≤b
Rc(11b)
tr(PPH)≤Pt(11c)
b
c≥0,(11d)
where b
c= [ b
C1,b
C2,..., b
CK]is the average common rate vector and b
Rk,tot =b
Rp,k +b
Ck. After formulating
the WASR problem for the RS scheme, we look at the effect of the instantaneous CSIT quality on its long-
term performance and observe how the RS scheme fares against conventional multiple access schemes.
We do that by looking at the sum-DoF analysis.
A. Common message and DoF Analysis
DoF is the total number of interference free streams that can be transmitted simultaneously in a single
channel use [15]. The sum-DoF for RS is defined as
dRS
s= lim
Pt→∞
Eb
H{b
RRS(Pt)}
log2(Pt),(12)
where b
RRS (Pt)is the Average SR (ASR) and is equal to b
RRS (Pt,µ)for equal user weights, i.e., µk=
1,∀k∈ K. We aim at establishing the sum-DoF achieved by the RS scheme for symmetric MIMO
BC transmission with imperfect CSIT under the assumption that the channel b
Hkis full rank and CSIT
error matrix e
Hkis isotropically distributed. It should be noted that these assumptions are not necessary
for optimization. With Qreceive antennas at each user, Qpbeing the total number of private streams
transmitted and Qcas the number of common streams, the sum-DoF achieved by the RS precoding
scheme is
dRS
s:
Qc(1 −α) + Qpα, M ∈ {2Q, 3Q, . . . , KQ}
M, M ≤Q.
(13)
For comparison, we consider the conventional MU–MIMO and MIMO NOMA schemes with sum-DoF
expressed as
dMU-MIMO
s= max min(M, Q), Qpα.(14)
dNOMA
s= min(M, Q).(15)
The procedure to obtain the sum-DoF achieved by all schemes is relegated to Appendix A. Following
the principle of SC-SIC, the ASR achieved by MIMO NOMA (SC–SIC) is limited by the decodability
of all users messages decoded in the last place. This restricts the sum-DoF to a maximum value of
12
min(M, Q). For MU–MIMO, the sum-DoF can attain a maximum value of Qpwhen α= 1. As αfalls
below 1, detrimental effects of interference lead to a decrease in its sum-DoF. Once αgoes further down,
CSIT quality deteriorates to a point where it is not conducive enough to support MU transmission and
transmitting to a single user yields a better sum-DoF. However, in the RS scheme, the presence of common
messages allows the transmitter to adjust the power allocated to the private stream vectors in a way that
the interference is always at the level of noise. Thus, the DoF of the private stream vectors is maintained
at Qpαby scaling down the power allocated to the private stream vectors to O(Pα
t). The remaining power
which scales as O(Pt)is allocated to the common stream vector. The DoF gain achieved by the common
streams is Qc(1 −α). For α∈(0,1) and Qc= min(M, Q), the sum-DoF of RS is strictly greater than
the sum-DoF of both MU–MIMO and MIMO NOMA.
Though optimization does not improve the achievable sum-DoF, it does play a significant role in
improving the rate performance. As b
RMU-MIMO(Pt)can be obtained by switching off the common streams,
the inequality b
RRS(Pt)≥b
RMU-MIMO(Pt)is guaranteed for the entire range of SNR. The results in Section
VI validate the theoretical assertions.
IV. OPTIMIZATION FRAMEWOR K
The optimization problem and sub-problems of (11) are non-deterministic in nature and thus solving
them becomes very difficult in their current form. We adopt a three-step approach to solve the optimization
problem (11). First, we use the Sample Average Approximation (SAA) method to obtain a deterministic
approximation of the problem, then we transform the WASR problem to a part-wise convex Weighted
Average MMSE (WAMMSE) problem making it solvable. Finally, we use vectorization to reduce the
matrix variables to their vectorized forms, thereby making WAMMSE problem tractable to solve. Using
AO, we obtain the precoders and consequently, the optimized rate for a given weight vector µ..
A. Sample Average Approximation
We first consider a set of Ni.i.d channel samples indexed N={1,2,...,N}drawn from a distribution
with density fH|
b
HH|b
H. Therefore, for a given channel estimate b
Hwe have Nchannel samples denoted
as H(N),H(n)=b
H+e
H(n)|b
H, n ∈ N . Using the channel realizations and Sample Average
Functions (SAFs) defined as: b
R(N)
z,k ,1
NPN
n=1 R(n)
z,k , z ∈ {c, p}, we approximate the average rates. Here,
R(n)
z,k ,Rz,k(H(n)), z ∈ {c, p}are the common and private rates associated with the nth realization. The
13
SAA of problem (11) is
b
R(N)
RS (Pt,µ) = max
P,b
c
K
X
k=1
µkb
R(N)
k,tot (16a)
b
C1+b
C2+... b
CK≤b
R(N)
c(16b)
tr(PPH)≤Pt(16c)
b
c≥0,(16d)
where b
R(N)
k,tot =b
R(N)
p,k +b
Ckand b
R(N)
c,minj{b
R(N)
c,j }K
j=1. The rates obtained here are bounded [20] and
therefore by applying LLN, it can be inferred that
lim
N→∞ b
R(N)
z,k (P) = b
Rz,k (P),∀P∈P, z ∈ {c, p}.(17)
The set Pdefined as {tr(PPH)≤Pt|P∈P}is the feasible set of precoders for which the rate functions
are bounded, continuous and differentiable in P, thereby making convergence in (17) uniform in P. The
ARs are also continuous and differentiable [20] and therefore using (17) we obtain
lim
N→∞ b
R(N)
k,tot =b
Rk,tot ∀P∈P.(18)
Based on (17), (18), we obtain that as N→ ∞, the optimum solutions of the SAA in problem (16)
converges to the solution of the stochastic problem in (11) [20], [50].
B. WASR →WAMMSE
In this subsection, we aim at solving the sample average approximated problem (16) by using the
methods adopted in [4] to transform problem (16) into an equivalent WAMMSE form.
First we define Augmented Weighted Mean Square Error (AWMSE) for common and private stream
vectors as
ξz,k (H,b
H) = tr(Uz,kEz,k)−log det (Uz ,k), z ∈ {c, p}.(19)
where Uz,k , z ∈ {c, p}are instantaneous weights introduced for common and private MSE matrices
of user-k. Next we aim to establish the Rate-WMMSE relationship by optimizing the AWMSEs w.r.t
equalizers (filters) and weights. By solving ∂ξz,k (H,
b
H)
∂Gz,k = 0, the optimum equalizers are obtained as G∗
z,k =
GMMSE
z,k ,z∈ {c, p}. Substituting the optimum equalizers in (19), we get
ξz,k GMMSE
z,k =tr(Uz,kEMMSE
z,k )−log det (Uz,k), z ∈ {c, p}.(20)
By solving ∂ξz, kGMMSE
z,k
∂Uz,k = 0, the optimum MMSE weights are obtained as U∗
z,k =UMMSE
z,k ,(EMMSE
z,k )−1,
z∈ {c, p}. Subsituting the obtained optimum weights for the weights in equation (20), the instantaneous
Rate-WMMSE relationship is established as
ξc,k(H,b
H),min
Gc,k,Uc,k
ξc,k =Qc−Rc,k(H,b
H), ξp,k(H,b
H),min
Gp,k,Up,k
ξp,k =Qk−Rp,k(H,b
H).(21)
14
Based on the principle of SAA, the AR-WAMMSE relationship is derived by taking the expectation over
the conditional distribution of channel Hfor a given channel estimate b
Hand is written as
b
ξc,k ,EH|
b
Hmin
Gc,k,Uc,k
ξc,k|b
H=Qc−b
Rc,k,b
ξp,k ,EH|
b
Hmin
Gp,k,Up,k
ξp,k|b
H=Qk−b
Rp,k.(22)
Next, we use the SAFs to obtain the deterministic equivalent relations of (22). Taking the Ni.i.d channel
samples, the average AWMSEs are b
ξ(N)
z,k ,1
NPN
n=1 ξ(n)
z,k ,z∈ {c, p}, where ξ(n)
z,k ,G(n)
z,k ,U(n)
z,k ,z∈ {c, p}
are all associated with the nth realization in H(N). For ease of notation, we use Gto represent the set of
equalizers for common and private stream vectors, i.e., G,{Gz,k | ∀ k∈ K}, where Gz,k ,{G(n)
z,k |
∀n∈ N , z ∈ {c, p}}. Following the same method, we obtain the set of weights for common and private
streams, denoted as U.
Following the LLN as in (17) and the approach used to obtain (21), the AR-WAMMSE in (22) is
written as
b
ξMMSE
c,k (N),min
Gc,k,Uc,k b
ξ(N)
c,k =Qc−b
R(N)
c,k ,b
ξMMSE
p,k (N),min
Gp,k,Up,k b
ξ(N)
p,k =Qk−b
R(N)
p,k .(23)
The sets of optimum MMSE equalizers and weights associated with equation (23) are defined as GMMSE ,
GMMSE
z,k (n)|z∈ {c, p},∀n∈ N ,∀k∈ K,UMMSE ,UMMSE
z,k (n)|z∈ {c, p},∀n∈ N ,∀k∈ K
respectively. Motivated by the AR-WAMMSE relationship in (23), the deterministic WAMMSE optimiza-
tion problem is formulated as
min
P,b
x,U,G
K
X
k=1
µkb
ξ(N)
k,tot (24a)
b
X1+b
X2+... b
XK+Qc≥b
ξ(N)
c(24b)
tr(PPH)≤Pt(24c)
b
x≤0,(24d)
where b
ξ(N)
c= max{b
ξ(N)
c,k }K
k=1,b
ξ(N)
k,tot =b
ξ(N)
p,k +b
Xkand b
x={b
X1,b
X2,..., b
XK}=−b
c. (24) is optimized
w.r.t (U,G)by minimizing individual AWMSEs shown in (23) as the AWMSEs are decoupled in their
corresponding weights and equalizers. This can be validated by showing that for a given precoder P, the
KKT optimality conditions of (24) are satisfied by the MMSE solution {UMMSE ,GMMSE}. Consequently,
it can be shown that for the MMSE solution, (24) boils down to (16) and the Rate-WMMSE relationship
is not just limited to the global optimum solution but can be extended to the entire set of stationary points.
For any point {U∗,G∗,P∗,b
x∗}satisfying the KKT optimality conditions of (24), also satisfies the KKT
optimality conditions of (16), with b
c=−b
x[20]. Therefore, as N→ ∞, solving (24) yields a solution
for (16), which in turn, converges to a solution of the WASR problem in (11).
15
C. Vectorization and Alternate Optimization
Problem (24) is non-convex for the joint optimization of variables G,U,b
xand Pbut it is convex for
each block of variables if the other two are fixed. Therefore, we utilize the AO algorithm with 2 steps, 1)
updating the equalizers Gand weights Uby using (fixing) the precoders Pfrom the previous iteration
and 2) updating the precoders Pand the message split xby solving the optimization problem for a given
Gand U. Unlike the MISO case in [10], optimization in (24) encounters difficulties of optimizing matrix
variables. Furthermore, the presence of R−1
z,k, z ∈ {c, p}matrices in the AWMSE expressions makes the
optimization intractable. Bearing that in mind, we first use vectorization and deduce the objective function
into a tractable form. Let us consider the augmented AWMSE expression (19) for the common stream.
In step 2 of the AO, the weights are fixed. However, the calculation of the term tr(Uc,k Ec,k)introduces
difficulties due to the aforementioned reasons and thus we try to simplify the expression and consequently
the entire objective function into a solvable form. Expanding tr(Uc,k Ec,k)it follows,
tr(Uc,k Ec,k) = trUc,kE(Gc,kyk−sc)(Gc,kyk−sc)H=trUc,kGc,k HH
kPcPH
cHkGH
c,k
+Gc,kHH
k(
K
X
i=1
PiPH
i)HkGH
c,k −Gc,kHH
kPc−PH
cHkGH
c,k +σ2
nGc,kGH
c,k +I.
(25)
Simplifying4the expanded expression in (25), we get
tr(Uc,k Ec,k)−log det(Uc,k) = pH
cA′
c,kpc+
K
X
i=1
pH
iAc,kpi−aH
c,kpc−pH
cac,k + Φc,k.(26)
Similarly, the resultant expression for tr(Up,k Ep,k)is
tr(Up,k Ep,k)−log det(Up,k) = pH
kAp,kpk+
K
X
i6=k
pH
iAp,kpi−aH
p,kpk−pH
kap,k + Φp,k,(27)
where pc=vec(Pc),pk=vec(Pk),A′
c,k =IQc⊗HkGH
c,kUc,kGc,k HH
k,Az,k =IQk⊗HkGH
z,k Uz,k Gz,k HH
k,
az,k =vec(Uz,kHkGH
z,k )and Φz,k =σ2
ntr(Uz,kGz,kGH
z,k ) + tr(Uz,k)−log det(Uz,k),z∈ {c, p}. Next,
we calculate the SAFs of the AWMSEs following (26) and (27).
1) STEP 1: Let us denote the precoders from the previous iteration as P[i−1]. For each channel
realization, the equalizers G(P[i−1])(n)and weights U(P[i−1])(n)are calculated for both common
and private stream vectors. Precoders are fixed for each i.i.d realization. After obtaining equalizers and
weights, we consider SAFs of the following entities: b
A
′(N)
c,k =1
NPN
n=1(A′
c,k)(n),b
A(N)
z,k =1
NPN
n=1 A(n)
z,k ,
b
a(N)
z,k =1
NPN
n=1 a(n)
z,k ,b
Φ(N)
z,k =1
NPN
n=1 Φ(n)
z,k ,z∈ {c, p}.
4Applying matrix manipulation tr(AB) = tr(BA)and tr(ABC) = vec(AH)H(I⊗B)vec(C)we transform (25) to (26).
16
2) STEP 2: The next step is to update the precoders by substituting the updated equalizers, weights
and SAFs of dependent entities into equation (24). The problem is formulated as
min
P,b
x
K
X
k=1
µkb
Xk+pH
kb
Ap,kpk+
K
X
i6=k
pH
ib
Ap,kpi−b
aH
p,kpk−pH
kb
ap,k +b
Φp,k(28a)
K
X
i=1 b
Xi+Qc≥pH
cb
A′
c,kpc+
K
X
i=1
pH
ib
Ac,kpi−b
aH
c,kpc−pH
cb
ac,k +b
Φc,k,∀k∈ K (28b)
tr(PPH)≤Pt(28c)
b
x≤0.(28d)
Problem (28) is a convex Quadratically Constrained Quadratic Program (QCQP) and can be solved
using interior-point methods [51], [52]. Step 1 and 2 are repeated until the convergence is reached as
specified in Algorithm 1. Proposition 1 of [20] and its proof shows that for a given H(N), Algorithm 1
converges to a KKT solutions of the sampled WASR problem (16) and as N→ ∞, converges to a KKT
solution of the WASR in problem (11).
V. PHY-LAYER DE SI GN FO R MIMO CHANNE LS
In addition to the theoretical foundations, it is of high importance to demonstrate the improved perfor-
mance of RS in practical setups. In this section, we propose a practical transceiver architecture for RS in
MIMO settings. Fig. 2 illustrates the proposed transceiver architecture, which is build upon and generalizes
the design in [53] of RS in MISO channels. The transmitter employs finite alphabet modulation schemes
4-QAM, 16-QAM, 64-QAM and 256-QAM, finite-length polar coding [54] for Forward Error Correction
(FEC), and an Adaptive Modulation and Coding (AMC) algorithm.
The combined common messages are mapped to binary vectors wc
iof length Kc
i, for i∈ {1,2,...,Qc},
respectively. Similarly, the private messages are respectively mapped to binary vectors wp,k
jof length
Kp,k
j, for k∈ K and j∈ {1,2,...,Qk}. We assume the split messages are independent, such that, the
Algorithm 1 AO ALGORITHM
1: Initialize i= 0,P[0]
2: Iterate
3: i=i+ 1,G=G(P[i−1]),U=U(P[i−1]).
4: Compute b
A′
c,k,b
Ac,k,b
Ap,k,b
ac,k,b
ap,k,b
Φc,k,b
Φp,k,∀k∈ K.
5: Solve (28),update P[i],b
x.
6: until convergence
17
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Reconstructor
+-
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Figure 2: Proposed transceiver architecture
common and private information bit vectors are independent and uniformly distributed in FKc
i
2and FKp,k
j
2.
The information bit vectors are independently encoded and modulated into common and private symbol
streams sc
iand sp,k
j, each of length S. The AMC algorithm selects a suitable modulation-coding rate pair
based on the ARs. In this work, the transmit rate calculations for the AMC algorithm are performed
assuming the instantaneous channel is known at the AMC module. More details on the channel coding
procedure and the AMC algorithm are given in [53] for the interested reader. The precoders for the
common and private streams are obtained as described in Algorithm 1.
We note that the rate expressions in (2) are valid under the assumption of joint decoding of all Qc
common streams (and all Qkprivate streams, ∀k∈ K). This restricts the use of conventional and practical
point-to-point decoding methods for channel coding at the receiver side. Although there are studies on
joint decoding of several types of channel codes ( e.g., polar codes, Low-Density Parity Check codes),
such implementations have higher complexities than point-to-point decoding methods, especially when
the number of jointly decoded streams increase.
Instead of performing joint decoding, we perform interference nulling and interference cancellation
among the streams at the receiver in order to benefit from low-complexity decoding methods. Such
receiver design is originally proposed in [55], [56] for V-BLAST systems. The proposed design allows
to obtain a separate transmission rate for each common and private stream for Modulation and Coding
Scheme (MCS) selection, as opposed to assigning a single rate value calculated by (2) to all common and
private streams. The proposed receiver architecture is illustrated in Fig. 2 for a detection and decoding
ordering based on the natural ordering of the stream indexes. We note that the detection and decoding
18
ordering of the common (and private) streams in the figure and the following explanations is for the sake
of simplicity and the actual stream ordering criterion we use in our design is also explained in this section.
Consider the scenario where the common streams 1,2,...,l−1, for any l < Qc, have been correctly
decoded and removed from the received signal at user-kto obtain the resulting interference cancelled
received signal e
yl
k. We define the effective channel for the l-th common stream as ¯
h(l)
c,k ,HH
kPc(l)∈CQ×1,
where Pc(l)is the l-th column of the matrix Pc. We can write e
yl
kin terms of the real and effective channels
as
e
yc,l
k=
Qc
X
i=l
¯
h(i)
c,ksc
i+X
j∈K
HH
kPjsj+nk.(29)
The detection of the l-th common stream is performed by multiplying e
yc,l
kwith a linear nulling filter, gl
c,k.
The nulling filter is designed based on the MMSE criterion and expressed as
gl
c,k = (¯
h(l)
c,k)H IQ+
Qc
X
i=l
¯
h(l)
k(¯
h(l)
k)H+X
j∈K
HH
kPjPH
jHk!−1
.
The definitions above use the assumption that the detection order follows the natural indexing of the
streams for the brevity of the explanations. It is demonstrated in [55], [56] that ordering the streams
according to their post-processing SINR yields the best performance. Therefore, we follow such approach
by calculating the post-processing SINRs over the streams which are filtered by their corresponding linear
nulling filters. Consider the detection and decoding of the l-the common stream at user-k. The index i′
of such stream is determined as the solution of the problem
i′= arg max
i∈Sl
γi
c,k,
where Slis the index set of the undetected streams with a cardinality of |Sl|=Qc−l+1,γi
c,k = 1/ǫi
c,k −1
is the post-processing SINR of the undetected common stream iat user-kand ǫi
c,k ,E|gi
c,ke
yc,i
k−sc,i|2
is the MSE of the undetected common stream i.
The decoding of the common stream lis performed by a Soft Decision (SD) decoder, for which the Log-
Likelihood Ratios (LLRs) are calculated over the equalized symbol gl
c,ke
yc,l
k. We use the LLR calculation
method in [57]. Let λl
c,k,i denote the LLR of the i-th bit of the equalized common stream symbol lfor
the k-th user. We write
λc,l
k,i =γl
c,k "min
a∈θ(i)
1
ψ(a)−min
a∈θ(i)
0
ψ(a)#,
where θ(i)
bis the set of modulation symbols with the value b,b∈ {0,1}at the i-th bit location,
ψ(a) = |gl
c,k
e
yc,l
k
ρl
c,k
−a|2and ρl
c,k =γl
c,k/(1 + γl
c,k).
19
A similar procedure is applied to the private streams intended to user-kafter all common streams are
decoded. The decoding operation is performed using a Successive Cancellation List (SCL) decoder for
point-to-point polar codes [58].
Remark: As our aim is to verify the theoretical foundations in the paper, we propose a receiver
architecture which has a higher complexity than a receiver with a single interference cancellation (IC)
process. An example design for a receiver with single IC would employ linear equalizers to detect each
common (or private) stream by treating all other streams as interference and then decode all common (or
private) streams in parallel. Although such receiver design is expected to suffer a performance loss due
to its sub-optimality, it may be more suitable for practical systems due to its reduced complexity.
VI. NUME RI CA L RESULTS
In this section, we evaluate the performance of the RS in MIMO BC with perfect and imperfect CSIT.
In the following, we first illustrate the WSR performance of RS in MIMO BC followed by the sum-DoF
performance. At last, we illustrate the LLS results. For comparison, the following three strategies are
considered as baselines:
•DPC: Implemented based on the algorithm in [59]. With perfect CSIT, DPC is a capacity achieving
scheme which cancels interference at the transmitter.
•MU–MIMO: Results are produced by turning off the common stream and solving problem (11), i.e,
allocating no power to the common stream vector.
•MIMO NOMA: Implemented by extending the degraded beamforming methodology proposed in
[11] to the MIMO case, delineated in Appendix A. The precoders and the decoding order are jointly
optimized to achieve maximum performance, where precoders are optimized using the WMMSE and
AO algorithm.
Note that the comparison with DPC is limited to the perfect CSIT scenario (as DPC is capacity achieving
only with perfect CSIT) and MIMO NOMA is limited to the 2-user case because of the high complexity
in joint optimization of decoding order and precoders for more than 2users. User channels are randomly
generated in accordance with [18], [20]. The actual channel experienced at user-kHkand the channel
estimation error e
Hkboth have complex Gaussian entries drawn from the distribution C N (0, σ2
k)and
CN (0, σ2
e,k)respectively. The channel estimation error power is defined as σ2
e,k ,σ2
kP−α
tsuch that the
CSIT quality for user channels scale with both channel variance and transmit power. Consequently, the
channel estimate b
Hk=Hk−e
Hkalso follows Gaussian distribution with entries CN (0, σ2
k−σ2
e,k). By
averaging the WASR over 100 channel realizations, WESR is obtained. For each channel estimate b
H,
N= 1000 channel error samples are generated to form H(n)and then the SAA method is used to
20
approximate the AR. For a given channel estimate b
H, the nth channel estimation error e
H(n)sample is
generated randomly from the error distribution and forms the nth conditional channel H(n)=b
H(n)+e
H(n).
In the case of perfect CSIT, N= 0 and b
H=H.
Initialization of precoders of all three schemes is crucial and plays an important role at higher SNRs,
especially for convergence [20]. For MU–MIMO and MIMO NOMA, the precoders are initialized using
Maximum Ratio Transmission (MRT). Initial power allocation is uniform among the users. For RS, the
initialization of precoders is according to MRT-Singular Value Decomposition (SVD) in which the private
streams are initialized using MRT and the initial precoder for the common stream vector is the dominant
M×Qcsub-matrix of the left singular matrix of b
H. Power distribution is qc=Pt−Pα
tfor the common
stream vector and the remaining power Pα
tis uniformly distributed among the private stream vectors of
all users. The noise variance is assumed to be σ2
n= 1.
A. WESR Performance: Rate-Region
We first consider M= 4,K= 2,Q= 2,Qc= 2 and SNR=20 dB. The ER-regions achieved by
different strategies are illustrated for perfect and imperfect CSIT in Fig. 3 and Fig. 4, respectively, where
different channel strength disparities are considered for analysis. A boundary point for any transmission
strategy is realized by solving the WESR problem for a weight pair by averaging the WASR over 100
channel realizations such that for each channel realization, we use Algorithm 1to obtain the WASR for
that strategy. The entire rate region is calculated over a set of different weight pairs assigned to users.
To obtain the rate-regions, the weight of user-1is fixed at µ1= 1 and the weight of user-2is varied as
µ2∈10[−3,−1,−0.95,...,0.95,1,3].
Fig. 3 illustrates the ER-region of all the four transmission strategies in the perfect CSIT scenario.
In both subfigures (a) and (b), DPC achieves the highest rate-region, which is the capacity region. In
Fig. 3(a), we observe that with no disparity in the strength of user channels, MIMO NOMA achieves the
worst rate region. As the MIMO NOMA strategy is motivated to exploit disparities in channel strengths, it
is unable to properly manage the interference in this scenario. Whereas, MU–MIMO achieves a larger rate
region compared to MIMO NOMA as it depends on the precoder design at the transmitter. At the receiver
side, each user decodes its own streams by treating the streams of other users as noise. In contrast, when
user-2suffers an additional 10 dB path loss, MIMO NOMA achieves a larger rate-region as illustrated in
Fig. 3(b) compared to MU–MIMO when the weight of user-2is either more than or comparable to the
weight of user-1. When the weight of user-1is significantly larger than the weight of user-2, the effect
of disparities in channel strength fades as user-2is weighted significantly less. Hence, MU–MIMO starts
performing better than MIMO NOMA. In comparison, RS performs better than both MU–MIMO and
21
0 5 10 15
ER User-1 (bps/Hz)
0
5
10
15
ER User-2 (bps/Hz)
RS
MU-MIMO
NOMA
DPC
(a) σ2
2= 1
0 5 10 15
ER User-1 (bps/Hz)
0
2
4
6
8
ER User-2 (bps/Hz)
RS
MU-MIMO
NOMA
DPC
(b) σ2
2= 0.09
Figure 3: ER-region comparison of different strategies with perfect CSIT, averaged over 100 random channel realizations,
SNR= 20 dB, M= 4,K= 2,Q= 2,Qc= 2,σ2
1= 1.
0 5 10 15
ER User-1 (bps/Hz)
0
5
10
15
ER User-2 (bps/Hz)
RS
MU-MIMO
NOMA
(a) σ2
2= 1
0 5 10 15
ER User-1 (bps/Hz)
0
2
4
6
8
ER User-2 (bps/Hz)
RS
MU-MIMO
NOMA
(b) σ2
2= 0.09
Figure 4: ER-region comparison of different strategies with imperfect CSIT, averaged over 100 random channel realizations,
SNR= 20 dB, M= 4,K= 2,Q= 2,Qc= 2,α= 0.6,σ2
1= 1.
MIMO NOMA and achieves a closer rate region to the capacity achieving DPC in both subfigures. This
is due to the fact that it allows the users to exploit the common streams thereby enabling them to partially
decode the interference and partially treat the interference as noise.
Fig. 4 illustrates the ER-regions obtained for the imperfect CSIT scenario. As the CSIT quality degrades,
the performance of MU-MIMO deteriorates significantly while RS, because of the presence of common
streams, exhibits robustness and shows explicit performance gain over MU-MIMO and MIMO NOMA.
In the case of different channel strengths, we observe that the results are consistent with the perfect CSIT
scenario. Comparing Fig. 3 and Fig. 4, we obtain that better management of interference makes RS robust
to CSIT inaccuracy and different user deployments.
22
B. Sum-DoF: Effect of Common Message
1) CSIT Quality: From Fig. 5(a), we see that at high SNR, RS with Qc= 2 always achieves a higher
ESR performance compared to RS with Qc= 1, and both perform better than MU–MIMO and MIMO
NOMA5. For α= 0.6, the sum-DoF obtained for RS with Qc= 2, RS with Qc= 1, MU-MIMO and MIMO
NOMA are [3.08,2.67,2.10,1.93], respectively. These values are close to the theoretical sum-DoF values
[3.2,2.8,2.4,2] calculated using (13)–(15). Thus, transmitting multiple common streams yields better ESR
and sum-DoF performance than transmitting a single common stream. Furthermore, as αdecreases from
0.9to 0.3, the ESR performance and the sum-DoF gain gaps between RS and the other two schemes
increase, as seen in Fig. 5(b) and Fig. 5(c). The results are inline with equations (13)–(15). Since α= 0.9
is closer to the perfect CSIT case, at higher SNRs, the MU–MIMO curve is nearly parallel to the RS curve
though with lower ESR. It is because the contribution of the common streams decreases with increase in
αand w.r.t sum-DoF, MU–MIMO gets closer to RS. However, optimization and contribution of common
stream still make the ESR performance of RS better than MU–MIMO. MIMO NOMA on the other hand
is observed to have the same sum-DoF irrespective of α. The observation is consistent with the theoretical
sum-DoF expression in (15) which clearly implies that the sum-DoF of MIMO NOMA is independent
of αin our scenario6. Therefore, RS is more suited in MU multi-antenna networks than MU–MIMO and
MIMO NOMA in MIMO BC, especially with the deteriorating CSIT quality.
2) Number of Users: Fig. 6(a) and Fig. 6(b) illustrate the ESR performances of the schemes in
the underloaded and overloaded regime, respectively. In the underloaded regime, for all the three user
configurations, i.e., K= 2,K= 3 and K= 4, RS with Qc= 2 has better ESR than RS with Qc= 1 which
in turn performs better than MU–MIMO and MIMO NOMA. Fig. 6(b) illustrates that in the overloaded
regime also, RS outperforms MU–MIMO and MIMO NOMA in ESR and sum-DoF performances. The
sum-DoF achieved by all the three schemes are inline with equations (13)–(15) for the underloaded and
overloaded scenarios. Fig. 6(c) illustrates the ESR performance of all the three schemes with higher
dimensions, i.e., M= 9, Q = 3. RS achieves ESR performance gain over MU–MIMO and MIMO
NOMA in all scenarios.
C. Link-Level Simulation Results
In this section, we aim to verify the theoretical foundations in the paper under realistic and practical
setups. We perform Link-Level Simulations (LLS) to analyze the throughput performance of RS and
compare it with those of MU-MIMO and MIMO NOMA. We employ the practical transceiver architecture
5Note that, the performance gain of RS over MIMO NOMA with Qc= 1 has implications on the receiver complexity with RS requiring
less number of SICs compared to MIMO NOMA.
6The results are consistent with the findings in [13] that NOMA is unable to exploit the available CSIT efficiently.
23
5 10 15 20 25 30 35
SNR(dB)
0
10
20
30
40
50
ESR(bps/Hz)
RS, Qc=2
RS, Qc=1
MU-MIMO
NOMA
(a) α= 0.9
5 10 15 20 25 30 35
SNR(dB)
0
10
20
30
40
ESR(bps/Hz)
RS, Qc=2
RS, Qc=1
MU-MIMO
NOMA
(b) α= 0.6
5 10 15 20 25 30 35
SNR(dB)
0
5
10
15
20
25
30
35
ESR(bps/Hz)
RS, Qc=2
RS, Qc=1
MU-MIMO
NOMA
(c) α= 0.3
Figure 5: ESR versus SNR comparison of different strategies with different imperfect CSIT inaccuracies, averaged over 100
random channel realizations, M= 4,K= 2,Q= 2,σ2
1= 1,σ2
2= 1.
5 10 15 20 25 30 35
SNR(dB)
0
10
20
30
40
50
60
ESR(bps/Hz)
RS, Qc=2
RS, Qc=1
MU-MIMO
NOMA
K=4
K=3
K=2
(a) Q= 2,M=KQ
5 10 15 20 25 30 35
SNR(dB)
0
10
20
30
40
ESR(bps/Hz)
RS
MU-MIMO
NOMA K=3
K=2
(b) Q= 2,M= (K−1)Q
5 10 15 20 25 30 35
SNR(dB)
0
20
40
60
80
ESR(bps/Hz)
RS
MU-MIMO
NOMA K=3
K=2
(c) Q= 3,M= 9
Figure 6: ESR versus SNR comparison of different strategies for different network loads, averaged over 100 random channel
realizations, α= 0.6,σ2
k= 1,∀k∈[1,4].
described in Section V for RS, MU-MIMO and MIMO NOMA. Note that in the proposed architecture,
the common signal is turned off to simulate MU-MIMO and one out of two private signals is turned off to
simulate MIMO NOMA. We assume that the instantaneous channel is perfectly known at the transmitter
for MCS selection. Let S(l)denote the number of channel uses in the l-th Monte-Carlo realization and D(l)
s,k
denote the number of successfully recovered information bits by user-kin the common stream (excluding
the common part of the message intended for the other user) and its private stream. Then, we calculate
the throughput as
Throughput[bps/Hz] = Pl(D(l)
s,1+D(l)
s,2)
PlS(l).(30)
Fig. 7 shows the Shannon Bound (ESR obtained with Gaussian signalling and infinite block length) and
throughput levels achieved by RS, MU-MIMO and MIMO NOMA in both underloaded and overloaded
scenario, for M= 4, Q = 2, Qc= 2 and α= 0.6. The throughput performance is consistent with the ESR
performance for all three schemes in both the underloaded and overloaded regime.
24
5 10 15 20 25 30 35
SNR(dB)
0
10
20
30
40
Throughput (bps/Hz)
RS - Shannon Bound
MU-MIMO - Shannon Bound
NOMA - Shannon Bound
RS - LLS
MU-MIMO - LLS
NOMA - LLS
(a) K= 2
5 10 15 20 25 30 35
SNR(dB)
0
10
20
30
40
Throughput (bps/Hz)
RS - Shannon Bound
MU-MIMO - Shannon Bound
RS - LLS
MU-MIMO - LLS
(b) K= 3
Figure 7: Throughput versus SNR comparison of different strategies averaged over 100 random channel realizations.
VII. CONCLUS ION
To conclude, we introduce a general framework for RSMA in MIMO BC with both perfect and imperfect
CSIT, where RSMA is able to transmit an arbitrary number of common streams. We study the proposed
framework in both finite and high SNR regimes. In the finite SNR regime, we propose the vectorization
and WMMSE-based approach for precoder optimization to analyze the rate performance with perfect
and imperfect CSIT. In the high SNR regime, we derive the sum-DoF achieved by RS, MU–MIMO and
MIMO NOMA in a symmetric MIMO BC setup with imperfect CSIT. We demonstrate via the theoretical
results that by transmitting multiple common streams, RSMA achieves a higher sum-DoF with imperfect
CSIT compared to the conventional multiple access schemes. Numerical results show that RSMA achieves
a higher rate performance than MU–MIMO and MIMO NOMA irrespective of the network load, user
deployments or the CSIT quality. Moreover, numerical results validate the theoretical sum-DoF expressions
derived for all the three transmission strategies and the sum-DoF gain of RS over MU–MIMO and MIMO
MOMA. Moving beyond the assumptions of Gaussian signalling and infinite block lengths, we design the
PHY-layer architecture of RS and analyze its performance in practical systems. Through LLS simulations,
we demonstrate that by better managing the interference, RS achieves a significant throughput performance
gain over MU-MIMO and MIMO NOMA in MIMO BC. Therefore, we conclude that RSMA is a powerful
and promising physical-layer strategy for multi-antenna networks.
APPENDI X A
ACHIE VABLE SUM -DOF
Since the precoders for each channel estimate are decoupled with each other, we consider precoding
scheme for a given channel estimate b
Hwith precoders defined by {P}Pt. We begin with a general case
by denoting the CSIT quality of the user-kas αk=α, ∀k∈ K. Here, αassumes a non-negative value
25
and is in the range [0,1]. Also, replacing negative value of αwith zero does not alter the sum-DoF results
derived next. Therefore, one has E[|hH
k,ipk|2]∼P−α, with hk,i as the ith column of channel Hkand pk
being the vector of unit norm in the null space of b
Hk, i.e., ZF precoder. The entity |hk,ipk|2represents the
residual interference power at the unintended receiver. The power exponent of the private stream vector
of each user is taken as α= mink∈K{αk},∀k∈ K, as this power exponent maximizes the sum-DoF [21].
As aforementioned, here we consider a homogeneous network with equal number of receive antennas Q
at every user.
A. RS and MU-MIMO
We begin by first deriving the achievable sum-DoF of RS. We consider two possible scenarios, M≤Q
and M > Q. In the former scenario, the sum-DoF is restricted to Mand can be achieved by switching
to single user transmission. Therefore, we focus on the latter scenario where M > Q. This scenario
encompasses both network load regimes, i.e., underloaded when M=K Q7and overloaded when M=
LQ, 1< L < K. Consequently, we assume the number of users that will receive the private streams to
be J≤Ksuch that M=JQ. Whereas, the common streams will be multicast to all the Kusers. Let us
denote the set of users receiving the private streams to be J ⊆ K. Remaining K−Jusers will form a
set J∗such that J ∪ J ∗=K. Therefore, we have Qk=Q, ∀k∈ J and Qk= 0,∀k∈ J ∗. We assume
that the power is uniformly distributed among users for the private streams, i.e., Pα/J and consequently
the power allocated to the common streams is P−Pα, which simply boils down to P−Pα∼Pin high
SNR regimes. Hence, the transmission block is constructed as,
•A private stream vector denoted by sk∈CQ×1is transmitted to each user in the set Jusing a ZF
precoder Pk∈CM×Q,∀k∈ J .
•A common stream vector denoted by sc∈CQc×1is multicast to all users using a precoder Pc∈
CM×Qc, where Pcconstitutes the first Qcleft singular vectors of b
H.
As per the system model, the transmitted signal from the BS and the received signal at user-kwrites as
x=Pcsc
|{z}
P
+X
k∈J
Pksk
|{z}
Pα/J
,(31)
yk,k∈J =HH
kPcsc
|{z }
P
+HH
kPksk
| {z }
Pα
+X
j∈J ,j6=k
HH
kPjsj
|{z }
Pα−αk
+nk
|{z}
P0
.(32)
yk,k∈J ∗=HH
kPcsc
|{z }
P
+X
j∈J
HH
kPjsj
|{z }
Pα
+nk
|{z}
P0
.(33)
7In terms of the achievable sum-DoF, M > KQ is equivalent to the M=KQ scenario [60].
26
1) User-kin J:First, we write the common and private rates achieved by user-k,
b
Rc,k = log2det(Qc+Qk+Qηk)−log2det(Qk+Qηk),(34)
b
Rp,k = log2det(Qk+Qηk)−log2det(Qηk),(35)
where Qc=HH
kPcPH
cHk,Qk=HH
kPkPH
kHkand Qηk=Pj∈J ,j6=kHH
kPjPH
jHk+σ2
nIQare re-
spective covariance matrices of HH
kPcsc,HH
kPkskand Pj∈J ,j6=kHH
kPjsj+nk. Furthermore, we con-
sider the eigenvalue decomposition of Qc,Qkand Qηkas WcDcWH
c,WkDkWH
kand WηkDηkWH
ηk,
respectively, with Dc∼diag(PIQc,0Q−Qc),Dk∼diag(PαIQ)and Dηk∼diag(P(α−αk)+IQ). Here,
(x)+,max(x, 0). Then it follows,
log2det(Qc+Qk+Qηk) = Qc+ (Q−Qc)αlog2(Pt) + O(log2(Pt)),(36)
log2det(Qk+Qηk) = (Qα) log2(Pt) + O(log2(Pt)),(37)
log2det(Qηk) = O(log2(Pt)).(38)
2) User-kin J∗:For user-kin the the set J∗, we only have the common stream vector and the
common rate is written as
b
Rc,k = log2det(Qc+Qηk)−log2det(Qηk),(39)
where Qηk=Pj∈J HH
kPjPH
jHk+σ2
nIQwith the eigenvalue decomposition as Dηk∼
diag(PαIQ). It follows
log2det(Qc+Qηk) = Qc+ (Q−Qc)αlog2(Pt) + O(log2(Pt)),(40)
log2det(Qηk) = (Qα) log2(Pt).(41)
The term O(log2(Pt)) dies with Pt→ ∞ because of the negative exponent. We proceed to obtaining
the common and private DoF for user-kby using the expression in (12). From (34), (39), (36)–(37) and
(40)–(41), the DoF for the common stream vector is dc,k =Qc(1 −α)and is shared by all users in K.
Similarly, from (35) and (37)–(38), the DoF for the private stream vector at each user in Jis dp,k =Qα
and dp,k = 0 for users in J∗. Considering the total rate achieved by user-k,∀k∈ K, it can be written that
b
Rc+b
Rp,k ≤b
Rc,k +b
Rp,k. Hence, the sum-DoF achieved by RS is dRS
s≤dc,k +PK
k=1 dp,k and is expressed
in (13).
Switching off the common stream and using equations (35)-(38), the expression for the sum-DoF
achieved by the conventional MU–MIMO scheme is obtained and is expressed in (14).
27
B. MIMO NOMA
We now consider multi-antenna MIMO NOMA. Without loss of generality we assume the decoding
order81→Ksuch that user-1performs K−1layers of SIC to fully decode the messages (and therefore
remove interference) from the other K−1users. Similarly, the next user, i.e., user-2performs K−2
layers of SIC to fully decode messages from other K−2users and so on. Thus, user-Kdecodes its
own message vector treating messages of the rest K−1users as noise and the message vector of user-1
will be decoded by user-1after it decodes the messages of all the other K−1users and removes them.
Following the NOMA principle, the transmit signal vector xis generated such that the messages intended
for user-kare encoded using a shared codebook such that user-k,k∈ K is able to decode the message
of user-j, j ∈ K for j > k. After encoding the messages, linear precoders Pk∈CM×Qk,∀k∈ K can be
used to construct the transmit signal vector denoted by
x=X
k∈K
Pksk
|{z}
P/K
,(42)
and the received signal at user-kis given by yk=Hkx+nk. Using SIC, user-kdecodes the message
vector of user-j, j ≥kwhile treating the interference from users {i|i < j, i ∈ K} as noise. Under the
assumption of Gaussian signalling and perfect SIC, the rate at user-k, to decode the message vector of
user-jis given by
b
Rk,j = log2det IQ+Qk,j IQ+Q(in)
k,j −1,(43)
where Qk,j =HH
kPjPH
jHkand Q(in)
k,j =Pi<j,i∈K HH
kPiPH
iHk. In order to ensure decodability, message
vector of user-jis expected to be decoded at user-k|j≥k, k ∈ K and thus the rate of user-
jshould not exceed b
Rj= mink≤j, k∈K b
Rk,j . Therefore, the achievable rates of K-users is given by
b
R1=b
R1,1,b
R2= min( b
R1,2,b
R2,2),b
R3= min( b
R1,3,b
R2,3,b
R3,3),..., b
RK= min( b
R1,K ,b
R2,K ,..., b
RK,K ).
Since the message vector of each user is required to be decoded by user-1, the SR b
RNOMA of the K-user
MIMO NOMA can then be upper bounded as
b
RNOMA ≤
K
X
k=1 b
R1,k = log2det(IQ+
K
X
k=1
Q1,k).(44)
The SR bound achieved with this K-user MIMO NOMA strategy is further upper bounded as
b
RNOMA ≤log2det IQ+HH
1Q⋆
1H1Ptր
= min(M, Q) log2(Pt) + O(1),
where Q⋆
1refers to the optimal covariance matrix for user-1 in a single-user (OMA) setup with tr(P1PH
1) =
Pt, i.e., obtained by transmitting along the dominant eigenvector of H1HH
1and allocating power Pt
according to the water-filling solution. The sum-DoF obtained for MIMO NOMA is described in equation
(15).
8The sum-DoF analysis is independent of the basis of ordering and will hold for any ordering of users.
28
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