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Ratesplitting multiple access for downlink communication systems: bridging, generalizing, and outperforming SDMA and NOMA
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 133 (2018)
Abstract
Spacedivision multiple access (SDMA) utilizes linear precoding to separate users in the spatial domain and relies on fully treating any residual multiuser interference as noise. Nonorthogonal multiple access (NOMA) uses linearly precoded superposition coding with successive interference cancellation (SIC) to superpose users in the power domain and relies on user grouping and ordering to enforce some users to fully decode and cancel interference created by other users.
In this paper, we argue that to efficiently cope with the high throughput, heterogeneity of quality of service (QoS), and massive connectivity requirements of future multiantenna wireless networks, multiple access design needs to depart from those two extreme interference management strategies, namely fully treat interference as noise (as in SDMA) and fully decode interference (as in NOMA).
Considering a multipleinput singleoutput broadcast channel, we develop a novel multiple access framework, called ratesplitting multiple access (RSMA). RSMA is a more general and more powerful multiple access for downlink multiantenna systems that contains SDMA and NOMA as special cases. RSMA relies on linearly precoded ratesplitting with SIC to decode part of the interference and treat the remaining part of the interference as noise. This capability of RSMA to partially decode interference and partially treat interference as noise enables to softly bridge the two extremes of fully decoding interference and treating interference as noise and provides room for rate and QoS enhancements and complexity reduction.
The three multiple access schemes are compared, and extensive numerical results show that RSMA provides a smooth transition between SDMA and NOMA and outperforms them both in a wide range of network loads (underloaded and overloaded regimes) and user deployments (with a diversity of channel directions, channel strengths, and qualities of channel state information at the transmitter). Moreover, RSMA provides rate and QoS enhancements over NOMA at a lower computational complexity for the transmit scheduler and the receivers (number of SIC layers).
Introduction
With the dramatic upsurge in the number of devices expected in 5G and beyond, wireless networks will be operated in a variety of regimes ranging from underloaded to overloaded (where the number of scheduled devices is smaller and larger than the number of transmit antennas at each access point, respectively). Moreover, due to the heterogeneity of devices (highend such as smartphones and lowend such as Internet of Things and MachineType Communications devices), deployments, and applications in 5G and beyond, the transmitter will need to serve simultaneously users with different capabilities, deployments, and qualities of channel state information at the transmitter (CSIT). This massive connectivity problem together with the demands for high throughput and heterogeneity of quality of service (QoS) has recently spurred interests in rethinking multiple access for the downlink of communication systems.
In this paper, we propose a new multiple access called ratesplitting multiple access (RSMA). In order to fully assess the novelty of the proposed multiple access paradigm and the design philosophy, we first review the state of the art of two major multiple accesses, namely nonorthogonal multiple access (NOMA) [1], also called MultiUser Superposition Transmission (MUST) in 3GPP LTE Rel13 [2] and spacedivision multiple access (SDMA). We identify their benefits and limitations and make critical observations, before motivating the introduction of the novel and more powerful RSMA.
SDMA and NOMA: the extremes
Contrary to orthogonal multiple access (OMA) that schedules users or groups of users in orthogonal dimensions, e.g., time (TDMA) and frequency (FDMA), NOMA superposes users in the same timefrequency resource via the power domain or the code domain, leading to the powerdomain NOMA (e.g., [1]) and codedomain NOMA (e.g., sparse code multiple access (SCMA) [3]). Powerdomain NOMA^{Footnote 1} relies on superposition coding (SC) at the transmitter and successive interference cancellation (SIC) at the receivers (denoted in short as SC–SIC) [1, 4–6]. Such a strategy is motivated by the wellknown result that SC–SIC achieves the capacity region of the singleinput singleoutput (SISO) (Gaussian) broadcast channel (BC) [7, 8]. It is also well known that the capacity region of the SISO BC is larger than the rate region achieved by OMA (e.g., TDMA) when users experience a disparity of channel strengths [8]. On the other hand, when users exhibit the same channel strengths, OMA based on TDMA is sufficient to achieve the capacity region [8].
The benefit of a singleantenna NOMA using SC–SIC is therefore to be able, despite the presence of a single transmit antenna in a SISO BC, to cope with an overloaded regime in a spectrally efficient manner where multiple users experience potentially very different channel strengths/path losses (e.g., cellcenter users and celledge users) on the same time/frequency resource.
The limitation of a singleantenna NOMA lies in its complexity as the number of users grows. Indeed, for a Kuser SISO BC, the strongest user needs to decode using SIC the K−1 messages of all coscheduled users and therefore peel off K−1 layers before accessing its intended stream. Though SIC of a small number of layers should be feasible in practice^{Footnote 2}, the complexity and likelihood of error propagation becomes quickly significant for a large number of users. This calls for ways to decrease the number of SIC layers at each user. One could divide users into small groups of users with disparate channels and apply SC–SIC in each group and schedule groups on orthogonal resources (using OMA), but that may lead to some performance loss and latency increase.
In nowadays wireless networks, access points are often equipped with more than one antenna. This spatial dimension opens the door to another wellknown type of multiple access, namely SDMA. SDMA superposes users in the same timefrequency resource and separates user via a proper use of the spatial dimensions. Contrary to the SISO BC, the multiantenna BC is nondegraded, i.e., users cannot be ordered based on their channel strengths in general settings. This is the reason why SC–SIC is not capacityachieving, and the complex dirty paper coding (DPC) is the only strategy that achieves the capacity region of the multipleinput singleoutput (MISO) (Gaussian) BC with perfect CSIT [9]. DPC, rather than performing interference cancellation at the receivers as in SC–SIC, can be viewed as a form of enhanced interference cancellation at the transmitter and relies on perfect CSIT to do so. Due to the high computational burden of DPC, linear precoding is often considered the most attractive alternative to simplify the transmitter design [10]. Interestingly, in a MISO BC, multiuser linear precoding (MU–LP), e.g., either in closed form or optimized using optimization methods, though suboptimal, is often very useful when users experience relatively similar channel strengths or longterm signaltonoise ratio (SNR) and have semiorthogonal to orthogonal channels [11]. SDMA is therefore commonly implemented using MU–LP. The linear precoders create different beams with each beam being allocated a fraction of the total transmit power. Hence, similarly to NOMA, SDMA can also be viewed as a superposition of users in the power domain, though users are separated at the transmitter side by spatial beamformers rather than by the use of SIC at the receivers.
SDMA based on MU–LP is a wellestablished multiple access that is nowadays the basic principle behind numerous techniques in 4G and 5G such as multiuser multipleinput multipleoutput (MU–MIMO), coordinated multipoint (CoMP) coordinated beamforming, network MIMO, millimeterwave MIMO, and massive MIMO.
The benefit of SDMA using MU–LP is therefore to reap all spatial multiplexing benefits of a MISO BC with perfect CSIT with a low precoder and receiver complexity.
The limitations of SDMA are threefold.
First, it is suited to the underloaded regime and performance of MU–LP in the overloaded regime quickly drops as it requires more transmit antennas than users to be able to efficiently manage the multiuser interference. When the MISO BC becomes overloaded, the current and popular approach for the transmitter is to schedule group of users over orthogonal dimensions (e.g., time/frequency) and perform linear precoding in each group, which may increase latency and decrease QoS depending on the application.
Second, its performance is sensitive to the user channel orthogonality and strengths and requires the scheduler to pair semiorthogonal users with similar channel strengths together. The complexity of the scheduler can quickly increase when an exhaustive search is performed, though lowcomplexity (suboptimal) scheduling and userpairing algorithms exist [10].
Third, it is optimal from a degrees of freedom^{Footnote 3} (DoF), also known as spatial multiplexing gain, perspective in the perfect CSIT setting but not in the presence of imperfect CSIT [12]. The problem of SDMA design in the presence of imperfect CSIT has been to strive to apply a framework motivated by perfect CSIT to scenarios with imperfect CSIT, not to design a framework motivated by imperfect CSIT from the beginning [12]. This leads to the wellknown severe performance loss of MU–LP in the presence of imperfect CSIT [13].
In view of SC–SIC benefits in a SISO BC, attempts have been made to study multiantenna NOMA. Two lines of research have emerged that both rely on linearly precoded SC–SIC.
The first strategy, which we simply denote as “SC–SIC,” is a direct application of SC–SIC to the MISO BC by degrading the multiantenna broadcast channel. It consists in ordering users based on their effective scalar channel (after precoding) strengths and enforce receivers to decode messages (and cancel interference) in a successive manner. This is advocated and exemplified for instance in [14–17]. This NOMA strategy converts the multiantenna nondegraded channel into an effective singleantenna degraded channel, as at least one receiver ends up decoding all messages. While such a strategy can cope with the deployment of users experiencing aligned channels and different path loss conditions, it comes at the expense of sacrificing and annihilating all spatial multiplexing gains in general settings. By forcing one receiver to decode all streams, the sum DoF is reduced to unity^{Footnote 4}. This is the same DoF as that achieved by TDMA/singleuser beamforming (or OMA). This is significantly smaller than the sum DoF achieved by DPC and MU–LP in a MISO BC with perfect CSIT, which is the minimum of the number of transmit antennas and the number of users^{Footnote 5}. Moreover, this loss in multiplexing gain comes with a significant increase in receiver complexity due to the multilayer SIC compared to the treat interference as noise strategy of MU–LP. As a remedy to recover the DoF loss, we could envision a dynamic switching between NOMA and SDMA, reminiscent of the dynamic switching between SU–MIMO and MU–MIMO in 4G [18]. One would dynamically choose the best option between NOMA and SDMA as a function of the channel states. A particular instance of this approach is taken in [19] where a dynamic switching between SC–SIC and zeroforcing beamforming (ZFBF) was investigated.
The second strategy, which we denote as “SC–SIC per group,” consists in grouping K users into G groups. Users within each group are served using SC–SIC, and users across groups are served using SDMA so as to mitigate the intergroup interference. Examples of such a strategy can be found in [1, 20–24]. This strategy can therefore be seen as a combination of SDMA and NOMA where the multiantenna system is effectively decomposed into G hopefully noninterfering singleantenna NOMA channels. For this “SC–SIC per group” approach to perform at its best, users within each group need to have their channels aligned and users across groups need to be orthogonal.
Similarly to SDMA, multiantenna NOMA designs also rely on accurate CSIT. In the practical scenario of imperfect CSIT, NOMA design relies on the same above two strategies but optimizes the precoder so as to cope with CSIT imperfection and resulting extra multiuser interference. As an example, the MISO BC channel is again degraded in [17] and precoder optimization with imperfect CSIT is studied.
The benefit of multiantenna NOMA, similarly to the singleantenna NOMA, is the potential to cope with an overloaded regime where multiple users experience different channel strengths/path losses and/or are closely aligned with each other.
The limitations of multiantenna NOMA are fourfold.
First, the use of SC–SIC in NOMA is fundamentally motivated by a degraded BC in which users can be ordered based on their channel strengths. This is the key property of the SISO BC that enables SC–SIC to achieve its capacity region. Unfortunately, motivated by the promising gains of SC–SIC in a SISO BC, the multiantenna NOMA literature strives to apply SC–SIC to a nondegraded MISO BC. This forces to degrade a nondegraded BC and therefore leads to an inefficient use of the spatial dimensions in general settings, leading to a DoF loss.
Second, NOMA is not suited for general user deployments since degrading a MISO BC is efficient when users are sufficiently aligned with each other and exhibit a disparity of channel strengths, not in general settings.
Third, multiantenna NOMA comes with an increase in complexity at both the transmitter and the receivers. Indeed, a multilayer SIC is needed at the receivers, similarly to the singleantenna NOMA. However, in addition, since there exists no natural order for the users’ channels in multiantenna NOMA (because we deal with vectors rather than scalars), the precoders, the groups, and the decoding orders have to be jointly optimized by the scheduler at the transmitter. Taking as an example, the application of NOMA based on “SC–SIC” to a threeuser MISO BC, we need to optimize three precoders, one for each user, along with the six possible decoding orders. Increasing the number of users leads to an exponential increase in the number of possible decoding orders. “SC–SIC per group” divides users into multiple groups but that approach leads to a joint design of user ordering and user grouping. To decrease the complexity in user ordering and user grouping, multiantenna NOMA (SC–SIC and SC–SIC per group) forces users belonging to the same group to share the same precoder (beamforming vector) [1]. Unfortunately, such a restriction can only further hurt the overall performance since it shrinks the overall optimization space.
Fourth, multiantenna NOMA is subject to the same drawback as SDMA in the presence of imperfect CSIT, namely its design is not motivated by any fundamental limits of a MISO BC with imperfect CSIT.
The key is to recognize that the limitations and drawbacks of SDMA and NOMA originate from the fact that those two multiple accesses fundamentally rely on two extreme interference management strategies, namely fully treat interference as noise and fully decode interference. Indeed, while NOMA relies on some users to fully decode and cancel interference created by other users, SDMA relies on fully treating any residual multiuser interference as noise. In the presence of imperfect CSIT, CSIT inaccuracy results in an additional multiuser interference that is treated as noise by both NOMA (SC–SIC per group) and SDMA.
RSMA: bridging the extremes
In contrast, with RSMA, we take a different route and depart from the SDMA and NOMA literature and those two extremes of fully decode interference and treat interference as noise. We introduce a more general and powerful multiple access framework based on linearly precoded rate splitting (RS) at the transmitter and SIC at the receivers. This enables to decode part of the interference and treat the remaining part of the interference as noise [12]. This capability of RSMA to partially decode interference and partially treat interference as noise enables to softly bridge the two extreme strategies of fully treating interference as noise and fully decoding interference. This contrasts sharply with SDMA and NOMA that exclusively rely on the two extremes or a combination thereof.
In order to partially decode interference and partially treat interference as noise, RS splits messages into common^{Footnote 6} and private messages and relies on a superimposed transmission of common messages decoded by multiple users and private messages decoded by their corresponding users (and treated as noise by coscheduled users). Users rely on SIC to first decode the common messages before accessing the private messages. By adjusting the message split and the power allocation to the common and private messages, RS has the ability to softly bridge the two extreme of fully treat interference as noise and fully decode interference.
The idea of RS dates back to Carleial’s work and the Han and Kobayashi (HK) scheme for the twouser singleantenna interference channel (IC) [25]. However, the use of RS as the building block of RSMA is motivated by recent works that have shown the benefit of RS in multiantenna BC and the recent progress on characterizing the fundamental limits of a multiantenna BC (and IC) with imperfect CSIT. Hence, importantly, in contrast with the conventional RS (HK scheme) used for the twouser SISO IC, we here use RS in a different setup, namely (1) in a BC and (2) with multiple antennas. The use and benefits of RS in a multiantenna BC only appeared in the last few years^{Footnote 7}.
The capacity region of the Kuser MISO BC with imperfect CSIT remains an open problem. As an alternative, recent progress has been made to characterize the DoF region of the underloaded and overloaded MISO BC with imperfect CSIT. In [26], a novel information theoretic upperbound on the sum DoF of the Kuser underloaded MISO BC with imperfect CSIT was derived. Interestingly, this sum DoF coincides with the sum DoF achieved by a linearly precoded RS strategy at the transmitter with SIC at the receivers [27, 28]. RS (with SIC) is therefore optimum to achieve the sum DoF of the Kuser underloaded MISO BC with imperfect CSIT, in contrast with MU–LP that is clearly suboptimum (and so is SC–SIC since it achieves a sum DoF of unity^{Footnote 8}) [28]. It turns out that RS with a flexible power allocation is not only optimum for the sum DoF but for the entire DoF region of an underloaded MISO BC with imperfect CSIT [29]. The DoF benefit of RS in imperfect CSIT settings were also shown in more complicated underloaded networks with multiple transmitters in [30] and multiantenna receivers [31]. Considering user fairness, the optimum symmetric DoF (or maxmin DoF), i.e., the DoF that can be achieved by all users simultaneously, of the underloaded MISO BC with imperfect CSIT with MU–LP and RS was studied in [32]. RS symmetric DoF was shown to outperform that of MU–LP. Finally, moving to the overloaded MISO BC with heterogeneous CSIT qualities, a multilayer power partitioning strategy that superimposes degraded symbols on top of linearly precoded ratesplitted symbols was shown in [33] to achieve the optimal DoF region.
The benefits of RS have also appeared in multiantenna settings with perfect CSIT. In an overloaded multigroup multicast setting with perfect CSIT, considering again fairness, the symmetric DoF achieved by RS, MU–LP, and degraded NOMA transmissions (where receivers decode messages and cancel interference in a successive manner as in SC–SIC) was studied in [34]. It was shown that RS here again outperforms both MU–LP and SC–SIC.
The DoF metric is insightful to identify the multiplexing gains of the MISO BC at high SNR but fails to capture the diversity of channel strengths among users. This limitation is countered by the generalized DoF (GDoF) framework, which inherits the tractability of the DoF framework while capturing the diversity in channel strengths [35]. In [36, 37], the GDoF of an underloaded MISO BC with imperfect CSIT is studied, and here again, RS is used as part of the achievability scheme.
The DoF (GDoF) superiority of RS over MU–LP and SC–SIC in all those multiantenna settings (with perfect and imperfect CSIT) comes from the ability of RS to better handle the multiuser interference by evolving in a regime in between the extremes of fully treating it as noise and fully decoding it.
Importantly, the rate enhancements of RS over MU–LP, as predicted by the DoF analysis, are reflected in the finite SNR regime as shown in a number of recent works. In [38], finite SNR rate analysis of RS in MISO BC in the presence of quantized feedback was analyzed and it was shown that RS benefits from a CSI feedback overhead reduction compared to MU–LP. Using optimization methods, the precoder design of RS at finite SNR was investigated in [28] for the sum rate and rate region maximization with imperfect CSIT, in [32] for maxmin fair transmission with imperfect CSIT, and in [34] for multigroup multicast with perfect CSIT. Moreover, the benefit of RS over MU–LP in the finite SNR regime was shown in massive MIMO [39], millimeterwave systems [40] and multiantenna deployments subject to hardware impairments [41]. Finally, the performance benefits of the powerpartitioning strategy relying on RS in the overloaded MISO BC with heterogeneous CSIT was confirmed using simulations at finite SNR in the presence of a diversity of channel strengths [33]. In particular, in contrast to the RS used in [12, 28, 29, 32–34, 38, 40, 41] that relies on a single common message, [39] (as well as [30]) showed the benefits in the finite SNR regime of a multilayer (hierarchical) RS relying on multiple common messages decoded by various groups of users.
In this paper, in view of the limitations of SDMA and NOMA and the above literature on RS in multiantenna BC, we design a novel multiple access, called ratesplitting multiple access (RSMA) for downlink communication system^{Footnote 9}. RSMA is a much more attractive solution (performance and complexitywise) that retains the benefits of SDMA and NOMA but tackles all the aforementioned limitations of SDMA and NOMA. Considering a MISO BC, we make the following contributions.
First, we show that RSMA is a more general class/framework of multiuser transmission that encompasses SDMA and NOMA as special cases. RSMA is shown to reduce to SDMA if channels are of similar strengths and sufficiently orthogonal with each other and to NOMA if channels exhibit sufficiently diverse strengths and are sufficiently aligned with each other. This is the first paper to explicitly recognize that SDMA and NOMA are both subsets of a more general transmission framework based on RS^{Footnote 10}.
Second, we provide a general framework of multilayer RS design that encompasses existing RS schemes as special cases. In particular, the singlelayer RS of [28, 29, 32–34, 38, 40, 41] and the multilayer (hierarchical and topological) RS of [30, 39] are special instances of the generalized RS strategy developed here. Moreover, the use of RS was primarily motivated by multiantenna deployments subject to multiuser interference due to imperfect CSIT in those works. The benefit of RS in the presence of perfect CSIT and/or a diversity of channel strengths in a multiantenna setup, as considered in this paper, is less investigated. RS was shown in [34] to boost the performance of overloaded multigroup multicast. However, no attempt has been made so far to identify the benefit of RS in multiantenna BC with perfect CSIT and/or a diversity of channel strengths.
Third, we show that the rate performance (rate region, weighted sumrate with and without QoS constraints) of RSMA is always equal to or larger than that of SDMA and NOMA. Considering a MISO BC with perfect CSIT and no QoS constraints, RSMA performance comes closer to the optimal DPC region than SDMA and NOMA. In scenarios with QoS constraints or imperfect CSIT, RSMA always outperforms SDMA and NOMA. Since it is motivated by fundamental DoF analysis, RSMA is also optimal from a DoF perspective in both perfect and imperfect CSIT and therefore optimally exploit the spatial dimensions and the availability of CSIT, in contrast with SDMA and NOMA that are suboptimal.
Fourth, we show that RSMA is much more robust than SDMA and NOMA to user deployments, CSIT inaccuracy, and network load. It can operate in a wide range of practical deployments involving scenarios where the user channels are neither orthogonal nor aligned and exhibit similar strengths or a diversity of strengths, where the CSI is perfectly or imperfectly known to the transmitter, and where the network load can vary between the underloaded and the overloaded regimes. In particular, in the overloaded regime, the RSMA framework is shown to be particularly suited to cope with a variety of device capabilities, e.g., highend devices along with cheap InternetofThings (IoT)/MachineType Communications (MTC) devices. Indeed, the RS framework can be used to pack the IoT/MTC traffic in the common message, while still delivering highquality service to highend devices.
Fifth, we show that the performance gain can come with a lower computational complexity than NOMA for both the transmit scheduler and the receivers. In contrast to NOMA that requires complicated user grouping and ordering and potential dynamic switching (between SDMA, SC–SIC and SC–SIC per group) at the transmit scheduler and multiple layers of SIC at the receivers, a simple onelayer RS that does not require any user ordering, grouping, or dynamic switching at the transmit scheduler and a single layer of SIC at the receivers still significantly outperforms NOMA. In contrast to SDMA, RSMA is less sensitive to user pairing and therefore does not require complex user scheduling and pairing^{Footnote 11}. However, RSMA comes with a slightly higher encoding complexity than SDMA and NOMA due to the encoding of the common streams on top of the private streams.
Sixth, though SC–SIC is optimal to achieve the capacity region of SISO BC, we show that a singlelayer RS is a lowcomplexity alternative that only requires a single layer of SIC at each receiver and achieves close to SC–SIC (with multilayer SIC) performance in a SISO BC deployment.
As a takeaway message, we note that the ability of a wireless network architecture to partially decode interference and partially treat interference as noise can lead to enhanced throughput and QoS, increased robustness, and lowered complexity compared to alternatives that are forced to operate in the extreme regimes of fully treating interference as noise and fully decoding interference.
It is also worth making the analogy with other types of channels where the ability to bridge the extremes of treating interference as noise and fully decoding interference has appeared. Considering a twouser SISO IC, interference is fully decoded in the strong interference regime and is treated as noise in the weak interference regime. Between those two extremes, interference is neither strong enough to be fully decoded nor weak enough to be treated as noise. The best known strategy for the twouser SISO IC is obtained using RS (socalled HK scheme). RS in this context is well known to be superior to strategies relying on fully treating interference as noise, fully decoding interference, or orthogonalization (TDMA, FDMA) [25, 35]. Limiting ourselves to those extremes strategies is suboptimal [25, 35].
The rest of the paper is organized as follows. The system model is described in Section 2. The existing multiple accesses are specified in Section 3. In Section 4, the proposed RSMA and its lowcomplexity structures are described and compared with existing multiple accesses. The corresponding weighted sum rate (WSR) problems are formulated, and the weighted MMSE (WMMSE) approach to solve the problem is discussed. Numerical results are illustrated in Section 5, followed by conclusions and future works in Section 6.
Notations: The boldface uppercase and lowercase letters are used to represent matrices and vectors. The superscripts (·)^{T} and (·)^{H} denote transpose and conjugatetranspose operators, respectively. tr(·) and diag(·) are the trace and diagonal entries, respectively. · is the absolute value, and ∥·∥ is the Euclidean norm. \(\mathbb {E}\{\cdot \}\) refers to the statistical expectation. \(\mathbb {C}\) denotes the complex space. I and 0 stand for an identity matrix and an allzero vector, respectively, with appropriate dimensions. \(\mathcal {CN}(\delta,\sigma ^{2})\) represents a complex Gaussian distribution with mean δ and variance σ^{2}. \(\mathcal {A}\) is the cardinality of the set \(\mathcal {A}\).
System model
Consider a system where a base station (BS) equipped with N_{ t } antennas serves K singleantenna users. The users are indexed by the set \(\mathcal {K}=\{1,\ldots,K\}\). Let \(\mathbf {{x}}\in \mathbb {C}^{N_{t}\times 1}\) denotes the signal vector transmitted in a given channel use. It is subject to the power constraint \(\mathbb {E}\{\left \Vert \mathbf {x}\right \Vert ^{2}\}\leq P_{t}\). The signal received at userk is
where \(\mathbf {{h}}_{k}\in \mathbb {C}^{N_{t}\times 1}\) is the channel between the BS and userk. \(n_{k}\sim \mathcal {CN}\left (0,\sigma _{n,k}^{2}\right)\) is the additive white Gaussian noise (AWGN) at the receiver. Without loss of generality, we assume the noise variances are equal to one for all users. The transmit SNR is equal to the total power consumption P_{ t }. We assume CSI of users is perfectly known at the BS in the following model. The imperfect CSIT scenario will be discussed in the proposed algorithm and the numerical results. Channel state information at the receivers (CSIR) is assumed to be perfect.
In this work, we are interested in beamforming designs for signal x at the BS. Specifically, the objective of beamforming designs is to maximize the WSR of users subject to a power constraint of the BS and QoS constraints of each user. We firstly state and compare two baseline multiantenna multiple accesses, namely SDMA and NOMA. Then, RSMA is explained. The WSR problem of each strategy will be formulated, and the algorithm adopted to solve the corresponding problem will be stated in the following sections.
SDMA and NOMA
In this section, we describe two baseline multiple accesses. The messages W_{1},…,W_{ K } intended for users 1 to K, respectively, are encoded into K independent data streams s=[s_{1},…,s_{ K }]^{T} independently. Symbols are mapped to the transmit antennas through a precoding matrix denoted by P=[p_{1},…,p_{ K }], where \(\mathbf {p}_{k}\in \mathbb {C}^{N_{t}\times 1}\) is the precoder for userk. The superposed signal is \(\mathbf {x}=\mathbf {P}\mathbf {{s}}=\sum _{k\in \mathcal {K}}\mathbf {p}_{k}s_{k}.\) Assuming that \(\mathbb {E}\{\mathbf {{s}}\mathbf {{s}}^{H}\}=\mathbf {I}\), the transmit power is constrained by tr(PP^{H})≤P_{ t }.
SDMA
SDMA based on MU–LP is a wellestablished multiple access. Each user only decodes its desired message by treating interference as noise. The signaltointerferenceplusnoise ratio (SINR) at userk is given by
For a given weight vector u=[u_{1},…,u_{ K }], the WSR achieved by MU–LP is
where R_{ k }= log2(1+γ_{ k }) is the achievable rate of userk. u_{ k } is a nonnegative constant which allows resource allocation to prioritize different users. \(R_{k}^{\text {th}}\) accounts for any potential individual rate constraint for userk. It ensures the QoS of each user. The WMMSE algorithm proposed in [42] is adopted to solve problem (3). The main idea of the WMMSE algorithm is to reformulate the WSR problem into its equivalent WMMSE problem and solve it using the alternating optimization (AO) approach. The rate region of the MU–LP strategy is approximated by R_{MU−LP}(u) for different rate weight vectors u. The resulting rate region R_{MU−LP} is the convex hull enclosing the resulting points. In general, solution to problem (3) would provide the optimal MU–LP beamforming strategy for any channel deployment (in between aligned and orthogonal channels and with similar or diverse channel strengths).
NOMA
NOMA relies on superposition coding at the transmitter and successive interference cancellation at the receiver. As discussed in the introduction, the two main strategies in multiantenna NOMA are the SC–SIC and SC–SIC per group. SC–SIC can be treated as a special case of SC–SIC per group where there is only one group of users.
SC–SIC
In SC–SIC, the precoders and decoding orders have to be optimized jointly. The decoding order is vital to the rate obtained at each user. To maximize the WSR, all possible decoding orders of users are required to be considered. Denote π as one of the decoding orders, the message of user π(k) is decoded before the message of user π(j),∀k≤j. The messages of user π(k),∀k≤i are decoded at user π(i) using SIC. The SINR experienced at user π(i) to decode the message of user π(k),k≤i is given by
For a given weight vector u=[u_{1},…,u_{ K }] and a fixed decoding order π, the WSR achieved by SC–SIC is
where \(R_{\pi (k)}=\min _{i\geq k,i\in \mathcal {K}} \{\log _{2}(1+\gamma _{\pi (i)\rightarrow \pi (k)})\}\). In [14], the problem (5) with equal weights is solved by the approximation technique minorizationmaximization algorithm (MMA). To keep a single and unified approach to solve the WSR problem of different beamforming strategies, we still use the WMMSE algorithm to solve it. By approximating the rate region with a set of rate weights, the rate region R_{SC−SIC}(π) with a certain decoding order π is attained. To achieve the rate region of SC–SIC, all decoding orders should be considered. The largest achievable rate region of SC–SIC is defined as the convex hull of the union over all decoding orders as R_{SC−SIC}=conv(∪_{ π }R_{SC−SIC}(π)).
SC–SIC per group
Assuming the K users are divided into G groups, denoted as \(\mathcal {G}=\{1,\ldots,G\}\). In each group, there is a subset of users \(\mathcal {K}_{g},g\in \mathcal {G}\). The user groups satisfy the following conditions: \(\mathcal {K}_{g}\cap \mathcal {K}_{g'}=\emptyset \), if g≠g^{′}, and \(\sum _{g\in \mathcal {G}}\mathcal {G}_{g}=K\). Denote π_{ g } as one of the decoding orders of the users in \(\mathcal {K}_{g}\), the message of user π_{ g }(k) is decoded before the message of user π_{ g }(j),∀k≤j. The messages of user π_{ g }(k),∀k≤i are decoded at user π_{ g }(i) using SIC. The SINR experienced at user π_{ g }(i) to decode the message of user π_{ g }(k),k≤i is given by
where \(I_{\pi _{g}(i)}=\sum _{g'\in \mathcal {G},g'\neq g}\sum _{j\in \mathcal {K}_{g'}}\mathbf {{h}}_{\pi _{g}(i)}^{H}\mathbf {{p}}_{j}^{2}\) is the intergroup interference suffered at user π_{ g }(i). For a given weight vector u=[u_{1},…,u_{ K }], a fixed grouping method \(\mathcal {G}\) and a fixed decoding order π={π_{1},…,π_{ G }}, the WSR achieved by SC–SIC per group is
where \(R_{\pi _{g}(k)}=\min _{i\geq k,i\in \mathcal {K}_{g}} \{\log _{2}(1+\gamma _{\pi _{g}(i)\rightarrow \pi _{g}(k)})\}\). Similarly to the SC–SIC strategy, the problem can be solved by using the WMMSE algorithm. To maximize the WSR, all possible grouping methods and decoding orders should be considered.
Remark 1
: As described in the introduction, it is common in the multiantenna NOMA literature (SC–SIC and SC–SIC per group) to force users belonging to the same group to share the same precoder, so as to decrease the complexity in user ordering and user grouping. Note that, in the system model described for both SC–SIC and SC–SIC per group, we consider the most general framework where each message is precoded by its own precoder. Hence, we here do not constrain symbols to be superimposed on the same precoder as this would further reduce the performance of NOMA strategies and therefore leading to even lower performance. Hence, the performance obtained with NOMA in this work can be seen as the best possible performance achieved by NOMA.
Methods—proposed ratesplitting multiple access
In this section, we firstly introduce the idea of RS by introducing a twouser example (K=2) and a threeuser example (K=3). Then, we propose the generalized framework of RS and specify two lowcomplexity RS strategies. We further compare RSMA with SDMA and NOMA from the fundamental structure and complexity aspects. Finally, we discuss the general optimization framework to solve the WSR problem.
Twouser example
We first consider a twouser example. There are two messages W_{1} and W_{2} intended for user1 and user2, respectively. The message of each user is split into two parts, \(\left \{W_{1}^{12},W_{1}^{1}\right \}\) for user1 and \(\left \{W_{2}^{12},W_{2}^{2}\right \}\) for user2. The messages \(W_{1}^{12}, W_{2}^{12}\) are encoded together into a common stream s_{12} using a codebook shared by both users. Hence, s_{12} is a common stream required to be decoded by both users. The messages \(W_{1}^{1}\) and \(W_{2}^{2}\) are encoded into the private stream s_{1} for user1 and s_{2} for user2, respectively. The overall data streams to be transmitted based on RS is s=[s_{12},s_{1},s_{2}]^{T}. The data streams are linearly precoded via precoder P=[p_{12},p_{1},p_{2}], where \(\mathbf {{p}}_{12}\in \mathbb {C}^{N_{t}\times 1}\) is the precoder for the common stream s_{12}. The resulting transmit signal is x=Ps=p_{12}s_{12}+p_{1}s_{1}+p_{2}s_{2}.
We assume that tr(ss^{H})=I, and the total transmit power is constrained by tr(PP^{H})≤P_{ t }.
At user sides, both user1 and user2 firstly decode the data stream s_{12} by treating the interference from s_{1} and s_{2} as noise. Therefore, each user decodes part of the message of the other interfering user encoded in s_{12}. The interference is partially decoded at each user. The SINR of the common stream at userk is
Once s_{12} is successfully decoded, its contribution to the original received signal y_{ k } is subtracted. After that, userk decodes its private stream s_{ k } by treating the private stream of userj (j≠k) as noise. The twouser transmission model using RS is shown in Fig. 1. The SINR of decoding the private stream s_{ k } at userk is
The corresponding achievable rates of userk for the streams s_{12} and s_{ k } are \(R_{k}^{12}=\log _{2}\left (1+\gamma _{k}^{12}\right)\) and R_{ k }= log2(1+γ_{ k }). To ensure that s_{12} is successfully decoded by both users, the achievable common rate shall not exceed \(R_{12}=\min \left \{ R_{1}^{12},R_{2}^{12}\right \}\). All boundary points for the twouser RS rate region can be obtained by assuming that R_{12} is shared between users such that \(C_{k}^{12}\) is the kth user’s portion of the common rate with \(C_{1}^{12}+C_{2}^{12}=R_{12}\). Following the twouser RS structure described above, the total achievable rate of userk is \( R_{k,\text {tot}}=C_{k}^{12}+R_{k}\). For a given pair of weights u=[u_{1},u_{2}], the WSR achieved by the twouser RS approach is
where \(\mathbf {c}=\left [C_{1}^{12},C_{2}^{12}\right ]\) is the common rate vector required to be optimized in order to maximize the WSR. For a fixed pair of weights, problem (10) can be solved using the WMMSE approach in [28], except we have perfect CSIT here. By calculating \(R_{\text {RS}_{2}}(\mathbf {u})\phantom {\dot {i}\!}\) for a set of different rate weights u, we obtain the rate region.
In contrast to MU–LP and SC–SIC, the RS scheme described above offers a more flexible formulation. In particular, instead of hard switching between MU–LP and SC–SIC, it allows both to operate simultaneously if necessary, and hence smoothly bridges the two. In the extreme of treating multiuser interference as noise, RS boils down to MU–LP^{Footnote 12}by simply allocating no power to the common stream s_{12}. In the other extreme of fully decoding interference, RS boils down to SC–SIC by forcing one user, say user1, to fully decode the message of the other user, say user2. This is achieved by allocating no power to s_{2}, encoding W_{1} into s_{1} and encoding W_{2} into s_{12}, such that x=p_{12}s_{12}+p_{1}s_{1}. User1 and user2 decode s_{12} by treating s_{1} as noise and user1 decodes s_{1} after canceling s_{12}. A physicallayer multicasting strategy is obtained by encoding both W_{1} and W_{2} into s_{12} and allocating no power to s_{1} and s_{2}.
Remark 2
: It should be noted that while the RS transmit signal model resembles a broadcasting system with unicast (private) streams and a multicast stream, the role of the common message is fundamentally different. The common message in a unicastmulticast system carries public information intended as a whole to all users in the system, while the common message s_{12} in RS encapsulates parts of private messages, and is not entirely required by all users, although decoded by the two users for interference mitigation purposes [12].
Remark 3
: A general framework is adopted where potentially each user can split its message into common and private parts. Note however that depending on the objective function, it is sometimes not needed for all users to split their messages. For instance, for sumrate maximization subject to no individual rate constraint, it is sufficient to have only one user to split its message [28]. However, when it comes to satisfying some fairness (WSR, QoS constraint, maxmin fairness), splitting the message of multiple users appears necessary [28,32,34].
Threeuser example
We further consider a threeuser example. Different from the twouser case, the message of user1 is split into \(\left \{W_{1}^{123}, W_{1}^{12}, W_{1}^{13}, W_{1}^{1}\right \}\). Similarly, the message of user2 and user3 is split into \(\left \{W_{2}^{123}, W_{2}^{12},W_{2}^{23},W_{2}^{2}\right \}\) and \(\left \{W_{3}^{123}, W_{3}^{13},W_{3}^{23},W_{3}^{3}\right \}\), respectively. The superscript represents a specific group of users whose messages with the same superscript are going to be encoded together. For example, \(W_{1}^{123},W_{2}^{123},\text {and} W_{3}^{123}\) are encoded into the common stream s_{123} intended for all the three users. \(W_{1}^{12}\) and \(W_{1}^{13}\) are correspondingly encoded with the split messages of user2 \(W_{2}^{12}\) and user3 \(W_{3}^{13}\) into data streams s_{12} and s_{13}. s_{12} is the partial common stream intended for user1 and user2. Hence, user1 and user2 will decode s_{12} while user3 will decode its intended streams by treating s_{12} as noise. Similarly, we obtain s_{23} partially encoded for user2 and user3. \(W_{1}^{1}, W_{2}^{2}, \text {and} W_{3}^{3}\) are encoded into private streams s_{1},s_{2}, and s_{3}, respectively.
The vector of data streams to be transmitted is s=[s_{123},s_{12},s_{13},s_{23},s_{1},s_{2},s_{3}]^{T}. After linear precoding using precoder P = [p_{123},p_{12},p_{13},p_{23},p_{1},p_{2},p_{3}], the signals are superposed and broadcast. The decoding procedure when K = 3 is more complex comparing with that in the twouser example. The main difference lies in decoding partial common streams for twousers. Define the streams to be decoded by l users as lorder streams. The 2order streams to be decoded at user1 are s_{12}ands_{13}. The 2order streams to be decoded at user2 and user3 are s_{12}ands_{23} and s_{13}ands_{23}, respectively. As the 1order and 2order streams to be decoded at different users are not the same, we take user1 as an example. The decoding procedure is the same for other users. User1 decodes four streams s_{123},s_{12},s_{13},ands_{1} based on SIC while treating other streams as noise. The decoding procedure starts from the 3order stream (common stream) and progresses downwards to the 1order stream (private stream). Specifically, user1 first decodes s_{123} and subtracts its contribution from the received signal. The SINR of the stream s_{123} at user1 is
After that, user1 decodes two streams s_{12},s_{13} and treats interference of s_{23} as noise. Both decoding orders of decoding s_{12} followed by s_{13} and s_{13} followed by s_{12} should be considered in order to maximize the WSR. Denote π_{ l } as one of the decoding order to decode lorder streams. There is only one 1order stream and one 3order stream to be decoded at each user. Therefore, only one decoding order exists for both π_{1} and π_{3}. In contrast, each user is required to decode two 2order streams. Denote \(s_{\pi _{2,k}{(i)}}\) as the ith data stream to be decoded at userk based on the decoding order π_{2}. One instance of π_{2} is 12→13→23, where s_{12} is decoded before s_{13} and s_{13} is decoded before s_{23} at all users. Since only data streams s_{12} and s_{13} are decoded at user1, the decoding order at user1 based on π_{2} is π_{2,1}=12→13. Hence, \(s_{\pi _{2,1}{(1)}}=s_{12}\) and \(s_{\pi _{2,1}{(2)}}=s_{13}\). The data stream \(s_{\pi _{2,1}{(1)}}\) is decoded before \(s_{\pi _{2,1}{(2)}}\). The SINRs of decoding streams \(s_{\pi _{2,1}{(1)}}\) and \(s_{\pi _{2,1}{(2)}}\) at user1 are
User1 finally decodes s_{1} by treating other data streams as noise. The threeuser RS transmission model with the decoding order π_{2}=12→13→23 is shown in Fig. 2. The SINR of decoding s_{1} at user1 is
The corresponding rate of each data stream is calculated in the same way as in the twouser example. To ensure that s_{123} is successfully decoded by all users, the achievable common rate shall not exceed \(R_{123}=\min \left \{ R_{1}^{123},R_{2}^{123},R_{3}^{123}\right \}\). To ensure that s_{12} is successfully decoded by user1 and user2, the achievable common rate shall not exceed \(R_{12}=\min \left \{ R_{1}^{12},R_{2}^{12}\right \}\). Similarly, we have \(R_{13}=\min \left \{ R_{1}^{13},R_{3}^{13}\right \}\) and \(R_{23}=\min \left \{ R_{2}^{23},R_{3}^{23}\right \}\). All boundary points for the threeuser RS rate region can be obtained by assuming that R_{123}, R_{12}, R_{13}, and R_{23} are shared by the corresponding group of users. Denote the portion of the common rate allocated to userk for the message s_{123} as \(C_{k}^{123}\), we have \(C_{1}^{123}+C_{2}^{123}+C_{3}^{123}=R_{123}\). Similarly, we have \(C_{1}^{12}+C_{2}^{12}=R_{12}\), \(C_{1}^{13}+C_{3}^{13}=R_{13}\), \( \text {and}~ C_{2}^{23}+C_{3}^{23}=R_{23}\). Following the threeuser RS structure described above, the total achievable rate of each user is \( R_{1,\text {tot}}=C_{1}^{123}+C_{1}^{12}+C_{1}^{13}+R_{1}\), \( R_{2,\text {tot}}=C_{2}^{123}+C_{2}^{12}+C_{2}^{23}+R_{2},\) and \( R_{3,\text {tot}}=C_{3}^{123}+C_{3}^{13}+C_{3}^{23}+R_{3}.\) For a given weight vector u=[u_{1},u_{2},u_{3}] and a fixed decoding order π=[π_{1},π_{2},π_{3}], the WSR achieved by the threeuser RS approach is
where \(\mathbf {c}=\left [C_{1}^{123},C_{2}^{123},C_{3}^{123},C_{1}^{12},C_{2}^{12},C_{1}^{13},C_{3}^{13},C_{2}^{23},C_{3}^{23}\right ]\phantom {\dot {i}\!}\) is the common rate vector required to be optimized in order to maximize the WSR. By calculating \(R_{\text {RS}_{3}}(\mathbf {u},\pi)\phantom {\dot {i}\!}\) for a set of different rate weights u, we obtain the rate region \(R_{\text {RS}_{3}}(\pi)\phantom {\dot {i}\!}\) of a certain decoding order π. The rate region of the threeuser RS is achieved as the convex hull of the union over all decoding orders as \( R_{\text {RS}}=\text {conv}\left (\bigcup _{\pi }R_{\text {RS}}(\mathbf {\pi })\right).\)
Similar to the twouser case, SC–SIC and MU–LP are again easily identified as special substrategies of RS by switching off some of the streams. Problem (15) is nonconvex and nontrivial. We propose an optimization algorithm in Section 4.7 to solve it based on the WMMSE approach.
Generalized ratesplitting
We further propose a generalized RS framework for K users. The users are indexed by the set \(\mathcal {K}=\{1,\ldots,K\}\). For any subset \(\mathcal {A}\) of the users, \(\mathcal {A}\subseteq \mathcal {K}\), the BS transmits a data stream \(s_{\mathcal {A}}\) to be decoded by the users in the subset \(\mathcal {A}\) while treated as noise by other users. \(s_{\mathcal {A}}\) loads messages of all the users in the subset \(\mathcal {A}\). The message intended for userk (\(k\in \mathcal {K}\)) is split as \(\{ W_{k}^{\mathcal {A}'}  \mathcal {A}' \subseteq \mathcal {K}, k \in \mathcal {A}' \}\). The messages \(\{W_{k'}^{\mathcal {A}}k'\in \mathcal {A}\}\) of users with the same superscript \(\mathcal {A}\) are encoded together into the stream \(s_{\mathcal {A}}\).
The stream order defined in Section 4.2 is applied to the generalized RS. The stream order of data stream \(s_{\mathcal {A}}\) is \(\mathcal {A}\). For a given \(l\in \mathcal {K}\), there are \(K\choose l\) distinct lorder streams. For example, we have only one Korder stream (traditional common stream) while we have K 1order streams (private steams). Define \(\mathbf {s}_{l}\in \mathbb {C}^{{K\choose l}\times 1}\) as the lorder data stream vector formed by all lorder streams in \(\{s_{\mathcal {A}'}\mathcal {A}'\subseteq \mathcal {K},\mathcal {A}'=l\}\). Note that when l=K, there is a single Korder stream. s_{ K } reduces to \(s_{\mathcal {K}}\). For example, when K=3, the 3order stream vector is s_{3}=s_{123}. The 1order and the 2order stream vectors are s_{1}=[s_{1},s_{2},s_{3}]^{T} and s_{2}=[s_{12},s_{13},s_{23}]^{T}, respectively. The data streams are linearly precoded via the precoding matrix P_{ l } formed by \( \left \{\mathbf {p}_{\mathcal {A}'}\mathcal {A}'\subseteq \mathcal {K},\mathcal {A}'=l\right \}\). The precoded streams are superposed and the resulting transmit signal is
At user sides, each user is required to decode the intended streams based on SIC. The decoding procedure starts from the Korder stream and then goes down to the 1order stream. A given user is involved in multiple lorder streams with an exception of the Korder and 1order streams. Denote π_{ l } as one of the decoding orders to decode the lorder data streams s_{ l } for all users. The lorder stream vector to be decoded at userk based on a certain decoding order π_{ l } is \(\mathbf {s}_{\pi _{l,k}}=\left [s_{\pi _{l,k}{(1)}},\cdots,s_{\pi _{l,k}{(\mathcal {S}_{l,k})}}\right ]^{H}\), where \(\mathcal {S}_{l,k}=\{s_{\mathcal {A}'}\mathcal {A}'\subseteq \mathcal {K},\mathcal {A}'=l,k\in \mathcal {A}'\}\) is the set of lorder streams to be decoded at userk. We assume \(s_{\pi _{l,k}{(i)}}\) is decoded before \(s_{\pi _{l,k}{(j)}}\) if i<j. The SINR of userk to decode the lorder stream \({s}_{\pi _{l,k}{(i)}}\) with a certain decoding order π_{ l } is
where
is the interference at userk to decode \({s}_{\pi _{l,k}{(i)}}\). \(\sum _{j>i}\mathbf {{h}}_{k}^{H}\mathbf {{p}}_{\pi _{l,k}(j)}^{2}\) is the interference from the remaining nondecoded lorder streams in \(\mathbf {s}_{{\pi _{l,k}}}\). \(\sum _{l'=1}^{l1}\sum _{j=1}^{\mathcal {S}_{l',k}}\mathbf {{h}}_{k}^{H}\mathbf {{p}}_{\pi _{l',k}(j)}^{2}\) is the interference from lower order streams \(\mathbf {s}_{{\pi _{l',k}}},\forall l'<l\) to be decoded at userk. \(\sum _{\mathcal {A}'\subseteq \mathcal {K},k\notin \mathcal {A}'}\mathbf {{h}}_{k}^{H}\mathbf {{p}}_{{\mathcal {A}'}}^{2}\) is the interference from the streams that are not intended for userk. The corresponding achievable rate of userk for the data stream \({s}_{\pi _{l,k}{(i)}}\) is \( R_{k}^{\pi _{l,k}{(i)}}=\log _{2}\left (1+\gamma _{k}^{\pi _{l,k}{(i)}}\right).\) To ensure that the streams shared by more than two users are successfully decoded by all users, the achievable rate of each user in the subset \(\mathcal {A} (\mathcal {A}\in \mathcal {K},2\leq \mathcal {A}\leq K)\) to decode the \(\mathcal {A}\)order stream \(s_{\mathcal {A}}\) shall not exceed
For a given \(l\in \mathcal {K}\), the lorder streams to be decoded at different users are different. \(s_{\mathcal {A}}\) is decoded at userk\((k\in \mathcal {A})\) based on the decoding order \(\pi _{\mathcal {A},k}\). \(R_{\mathcal {A}}\) becomes the rate of receiving stream \(s_{\mathcal {A}}\) at all users in the user group \(\mathcal {A}\) with a certain decoding order \(\pi _{\mathcal {A}}\). All boundary points for the Kuser RS rate region can be obtained by assuming that \(R_{\mathcal {A}}\) is shared by all users in the user group \(\mathcal {A}\). Denote the portion of the common rate allocated to userk\((k\in \mathcal {A})\) as \(C_{k}^{\mathcal {A}}\), we have \(\sum _{k'\in \mathcal {A}}C_{k'}^{\mathcal {A}}=R_{\mathcal {A}}\). Following the RS structure described above, the total achievable rate of userk is
where R_{ k } is the rate of the 1order stream s_{ k }. It is intended for userk only. No common rate sharing is required for R_{ k }. For a given weight vector u=[u_{1},⋯,u_{ K }] and a certain decoding order π={π_{1},…,π_{ K }}, the WSR achieved by RS is
P = [P_{1},…,P_{ K }] is the precoding matrix of all order streams. c is the common rate vector formed by \(\big \{C_{k}^{\mathcal {A}} \mathcal {A}\subseteq \mathcal {K}, k\in \mathcal {A}\big \}\). For a fixed weight vector, problem (20) can be solved using the WMMSE approach discussed in Section 4.7 by establishing rateWMMSE relationships for all data streams. By calculating \(\phantom {\dot {i}\!}R_{\text {RS}}(\mathbf {u},\pi)\) for a set of different rate weights u, we obtain the rate region \(\phantom {\dot {i}\!}R_{\text {RS}}(\pi)\) of a certain decoding order π. To achieve the rate region, all decoding orders should be considered. The capacity region of RS is defined as the convex hull of the union over all decoding orders as
Structured and lowcomplexity ratesplitting
The generalized RS described in Section 4.3 is able to provide more room for rate and QoS enhancements at the expense of more layers of SIC at receivers. Hence, though the generalized RS framework is very general and can be used to identify the best possible performance, its implementation can be complex due to the large number of SIC layers and common messages involved. To overcome the problem, we introduce two lowcomplexity RS strategies for K users, 1layer RS and 2layer hierarchical RS (HRS). Those two RS strategies require the implementation of one and two layers of SIC at each receiver, respectively.
1layer RS
Instead of transmitting all order streams, 1layer RS transmits the Korder common stream and 1order private streams. Only one SIC is required at each receiver. The message of each user is split into two parts \(\big \{W_{k}^{\mathcal {K}},W_{k}^{k}\big \},\forall k\in \mathcal {K}\). The messages \(W_{1}^{\mathcal {K}},\ldots,W_{K}^{\mathcal {K}}\) are jointly encoded into the Korder stream \({s}_{\mathcal {K}}\) intended to be decoded by all users. \(W_{k}^{k}\) is encoded into s_{ k } to be decoded by userk only. The overall data streams to be transmitted based on 1layer RS is \(\mathbf {{s}}=\left [ s_{\mathcal {K}},s_{1},\ldots, s_{K}\right ]^{T}\). The data streams are linearly precoded via precoder \(\mathbf {{P}}=\left [\mathbf {{p}}_{\mathcal {K}}, \mathbf {{p}}_{1},\ldots, \mathbf {{p}}_{K}\right ]\). The resulting transmit signal is \(\mathbf {{x}}=\mathbf {{P}}\mathbf {{s}}=\mathbf {{p}}_{\mathcal {K}}s_{\mathcal {K}}+\sum _{k\in \mathcal {\mathcal {K}}}\mathbf {{p}}_{k}s_{k}.\) Figure 3 shows a 1layer RS model. Readers are referred to Fig. 1 in [12] for a detailed illustration of the 1layer RS architecture.
At user sides, all users firstly decode the data stream \(s_{\mathcal {K}}\) by treating the interference from s_{1},…,s_{ K } as noise. The SINR of the Korder stream at userk is
Once \(s_{\mathcal {K}}\) is successfully decoded, its contribution to the original received signal y_{ k } is subtracted. After that, userk decodes its private stream s_{ k } by treating the 1order private streams of other users as noise. The SINR of decoding the private stream s_{ k } at userk is
The corresponding achievable rates of userk for the streams \(s_{\mathcal {K}}\) and s_{ k } are \( R_{k}^{\mathcal {K}}=\log _{2} \left (1+\gamma _{k}^{\mathcal {K}}\right)\) and R_{ k }= log2(1+γ_{ k }). To ensure that \(s_{\mathcal {K}}\) is successfully decoded by all users, the achievable common rate shall not exceed \(R_{\mathcal {K}}=\min \left \{ R_{1}^{\mathcal {K}},\ldots,R_{K}^{\mathcal {K}}\right \}\). \(R_{\mathcal {K}}\) is shared among users such that \(C_{k}^{\mathcal {K}}\) is the kth user’s portion of the common rate with \(\sum _{k\in \mathcal {K}}C_{k}^{\mathcal {K}}=R_{\mathcal {K}}\). Following the twouser RS structure described above, the total achievable rate of userk is \( R_{k,\text {tot}}=C_{k}^{\mathcal {K}}+R_{k}. \) For a given weight vector u=[u_{1},…,u_{ K }], the WSR achieved by the Kuser 1layer RS approach is
where \(\mathbf {c}=\left [C_{1}^{\mathcal {K}},\ldots,C_{K}^{\mathcal {K}}\right ]\). For a given weight vector, problem (24) can be solved using the WMMSE approach in [28].
In contrast to NOMA, this 1layer RS does not require any user ordering or grouping at the transmitter side since all users decode the common message (using single layer of SIC) before accessing their respective private messages. We also note that the 1layer RS is a subscheme of the generalized RS and is a superscheme of MU–LP (since by not allocating any power to the common message, the 1layer RS boils down to MU–LP). However, for K>2, SC–SIC and SC–SIC per group are not subschemes of 1layer RS (even though they were subschemes of the generalized RS). This explains why, in [12], the authors already contrasted 1layer RS and NOMA and expressed that the two strategies cannot be treated as extensions or subsets of each other. This 1layer RS appeared in many scenarios subject to imperfect CSIT in [28,29,32–34,38,40,41].
2layer HRS
The K users are divided into G groups \(\mathcal {G}=\{1,\ldots,G\}\) with \(\mathcal {K}_{g},g\in \mathcal {G}\) users in each group. The user groups satisfy the same conditions as in Section 3.2.2. Besides the Korder stream and 1order streams, 2layer HRS also allows the transmission of a \(\mathcal {K}_{g}\)order stream intended for users in \(\mathcal {K}_{g}\). The overall data streams to be transmitted based on 2layer RS is \(\mathbf {{s}}=\left [ s_{\mathcal {K}},s_{\mathcal {K}_{1}},\ldots,s_{\mathcal {K}_{G}},s_{1},\ldots, s_{K}\right ]^{T}\). The data streams are linearly precoded via precoder \(\mathbf {{P}}=\left [ \mathbf {{p}}_{\mathcal {K}}, \mathbf {{p}}_{\mathcal {K}_{1}},\ldots,\mathbf {{p}}_{\mathcal {K}_{G}},\mathbf {{p}}_{1},\ldots, \mathbf {{p}}_{K}\right ]\). The resulting transmit signal is \(\mathbf {{x}}=\mathbf {{P}}\mathbf {{s}}=\mathbf {{p}}_{\mathcal {K}}s_{\mathcal {K}}+\sum _{g\in \mathcal {G}}\mathbf {{p}}_{\mathcal {K}_{g}}s_{\mathcal {K}_{g}}+\sum _{k\in \mathcal {\mathcal {K}}}\mathbf {{p}}_{k}s_{k}.\)
Figure 4 shows an example of 2layer HRS. The users are divided into two groups, \(\mathcal {K}_{1}=\{1,2\}\), \(\mathcal {K}_{2}=\{3,4\}\). s_{1234} is a 4order stream intended for all the users while s_{12} and s_{34} are 2order streams for users in each group only.
Each user is required to decode three streams \(s_{\mathcal {K}}\), \(s_{\mathcal {K}_{g}}\), and s_{ k }. We assume \(k\in \mathcal {K}_{g}\). The data stream \(s_{\mathcal {K}}\) is decoded first by treating the interference from all other streams as noise. The SINR of the Korder stream at userk is
Once \(s_{\mathcal {K}}\) is successfully decoded, its contribution to the original received signal y_{ k } is subtracted. After that, userk decodes its group common stream \(s_{\mathcal {K}_{g}}\) by treating other group common streams and 1order private streams as noise. The SINR of decoding the \(\mathcal {K}_{g}\)order stream \(s_{\mathcal {K}_{g}}\) at userk is
After removing its contribution to the received signal, userk decodes its private stream s_{ k }. The SINR of decoding the private stream s_{ k } at userk is
The corresponding achievable rates of userk for the streams \(s_{\mathcal {K}}\), \(s_{\mathcal {K}_{g}}\), and s_{ k } are \( R_{k}^{\mathcal {K}}=\log _{2}\left (1+\gamma _{k}^{\mathcal {K}}\right)\), \( R_{k}^{\mathcal {K}_{g}}=\log _{2}\left (1+\gamma _{k}^{\mathcal {K}_{g}}\right) \) and R_{ k }= log2(1+γ_{ k }). The achievable common rate of \(s_{\mathcal {K}}\) and \(s_{\mathcal {K}_{g}}\) shall not exceed \(R_{\mathcal {K}}=\min \left \{ R_{1}^{\mathcal {K}},\ldots,R_{K}^{\mathcal {K}}\right \}\) and \(R_{\mathcal {K}_{g}}=\min _{k}\left \{ R_{k}^{\mathcal {K}_{g}}\mid k\in \mathcal {K}_{g}\right \}\), respectively. \(R_{\mathcal {K}}\) is shared among users such that \(C_{k}^{\mathcal {K}}\) is the kth user’s portion of the common rate with \(\sum _{k\in \mathcal {K}}C_{k}^{\mathcal {K}}=R_{\mathcal {K}}\). \(R_{\mathcal {K}_{g}}\) is shared among users in the group \(\mathcal {K}_{g}\) such that \(C_{k}^{\mathcal {K}_{g}}\) is the kth user’s portion of the common rate with \(\sum _{k\in \mathcal {K}_{g}}C_{k}^{\mathcal {K}_{g}}=R_{\mathcal {K}_{g}}\). Following the twouser RS structure described above, the total achievable rate of userk is \( R_{k,\text {tot}}=C_{k}^{\mathcal {K}}+C_{k}^{\mathcal {K}_{g}}+R_{k}, \) where \(k\in \mathcal {K}_{g}\). For a given weight vector u=[u_{1},…,u_{ K }], the WSR achieved by the Kuser 2layer HRS approach is
where c is the common rate vector formed by \(\big \{C_{k}^{\mathcal {K}},C_{k'}^{\mathcal {K}_{g}} k\in \mathcal {K}, {k'}\in \mathcal {K}_{g}, g\in \mathcal {G}\big \}\). For a given weight vector, problem (28) can be solved by simply modifying the WMMSE approach discussed in Section 4.7.
Comparing with SC–SIC per group where \(\mathcal {K}_{g}1\) layers of SIC are required at user sides, 2layer HRS only requires two layers of SIC at each user. Moreover, the user ordering issue in SC–SIC per group does not exist in 2layer HRS. The streams of a higher stream order will always be decoded before the streams of a lower stream order. Onelayer RS is the simplest architecture since only one SIC is needed at each user and it is a subscheme of the 2layer HRS. We also note that we can obtain a 1layer RS per group from the 2layer HRS by not allocating any power to \(s_{\mathcal {K}}\). Note that SC–SIC and SC–SIC per group are not necessarily subschemes of the 2layer HRS. The 2layer HRS strategy was first introduced in [39] in the massive MIMO context.
Encompassing existing NOMA and SDMA
A comparison of NOMA, SDMA and RSMA are shown in Table 1. Comparing with NOMA and SDMA, the most important characteristic of RSMA is that it partially decodes interference and partially treats interference as noise through the split into common and privates messages. This capability enables RSMA to maintain a good performance for all user deployment scenarios and all network loads, as it will appear clearer in the numerical results of Section 5.
Let us further discuss how the proposed framework of generalized RS in Section 4.3 contrasts and encompasses NOMA, SDMA, and RS strategies. We first compare the fouruser MIMO–NOMA scheme illustrated in Fig. 5 of [1] with the fouruser 2layer HRS strategy illustrated in Fig. 4. In Fig. 5 of [1], user1 and user2 are superposed in the same beam. User3 and user4 share another beam. The users are decoded based on SC–SIC within each beam. As for the fouruser 2layer HRS strategy in Fig. 4, the encoded streams are precoded and transmitted jointly to users. If we set the common message s_{12} to be encoded by the message of user2 only and decoded by both user1 and user2, the common message s_{34} to be encoded by the message of user4 and decoded by user3 and user4, we also set the precoders p_{12} and p_{1} to be equal, the precoders p_{34} and p_{3} to be equal, and the precoders of other streams to be 0, then the proposed RS scheme reduces to the scheme illustrated in Fig. 5 of [1]. Similarly, the Kuser RS model can be reduced to the Kuser MIMO–NOMA scheme. Therefore, the MIMO–NOMA scheme proposed in [1] is a particular case of our RS framework.
In view of the above discussions, it should now be clear that SDMA and the multiantenna NOMA strategies discussed in the introduction (relying on SC–SIC and SC–SIC per group) are all special instances of the generalized RS framework.
In the proposed generalized Kuser RS model, if we set P_{ l } = 0,∀l∈{2,⋯,K}, only 1order streams (private streams) are transmitted. Each user only decodes its intended private stream by treating others as noise. Problem (20) is then reduced to the SDMA problem (3). If the message of each user is encoded into one stream of distinct stream order, problem (20) is equivalent to the SC–SIC problem (5). By keeping 1order and Korder streams, we have the 1layer RS strategy whose performance benefit in the presence of imperfect CSIT was highlighted in various scenarios in [28,29,32–34,38,40,41]. There is only one common data stream to be transmitted and decoded by all users before each user decodes its private stream. By keeping 1order, Korder, and lorder streams, where l is selected from {2,⋯,K−1}, the problem becomes the 2layer HRS originally proposed in [39] with two layers of common messages to be transmitted. Another example of such a multilayer RS has also appeared in the topological RS for MISO networks of [30]. Therefore, the formulated Kuser RS problem is a more general problem. It encompasses SDMA, NOMA, and existing RS methods as special cases.
Though the current work focuses on MISO BC, the RS framework can be extended to multiantenna users and the general MIMO BC [31] as well as to a general network scenario with multiple transmitters [30]. Nevertheless, the optimization of the precoders in those scenarios remain interesting topics for future research. Applications of this RS framework to relay networks is also worth exploring. Preliminary ideas have appeared in [43], though joint encoding of the splitted common messages are not taken into account.
Complexity of RSMA
We further discuss the complexity of RSMA by comparing it with NOMA and SDMA. A qualitative comparison of NOMA, SDMA, and RSMA is shown in Table 2. In Table 2, RS refers to the generalized RS of Section 4.3.
As mentioned in the introduction, the complexity of NOMA in the multiantenna setup is increasing significantly at both the transmitter and the receivers. The optimal decoding order of NOMA is no longer fixed based on the channel gain as in the SISO BC. To maximize the WSR, the decoding order should be optimized together with precoders at the transmitter. Moreover, SC–SIC is suitable for aligned users with large channel gain difference. A proper user scheduling algorithm increases the scheduler complexity. At user sides, K−1 layers of SIC are required at each user for a Kuser SC–SIC system. Increasing the number of users leads to a dramatic increase of the scheduler and receiver complexity and is subject to more error propagation in the SICs.
SC–SIC per group reduces the complexity at user sides. Only \(\left \lceil \frac {K}{G}\right \rceil \) layers of SIC are required at each user if we uniformly group the K users into G groups. However, the complexity at the transmitter increases with the number of user groups. A joint design of user ordering and user grouping for all groups is necessary in order to maximize the WSR. For example, for a 4user system, if we divide the users into two groups with two users in each group, we should consider three different user grouping methods and four different decoding orders for each grouping method.
The complexity of MU–LP is much reduced as it does not require any SIC at user sides. However, as MU–LP is more suitable for users with (semi)orthogonal channels and similar channel strengths, the transmitter requires accurate CSIT and user scheduling should be carefully designed for interference coordination. The scheduler complexity at the transmitter is still high.
Comparing with NOMA and SDMA, RSMA is able to balance the performance and complexity better. All forms of RS are suitable for users with any channel gain difference and any channel angle in between, though a multilayer RS would have more flexibility. Considering the generalized RS, the decoding order of multiple streams with the same stream order should be optimized together with the precoders when there are multiple streams of the same stream order intended for each user (e.g., each user decodes two 2order streams in the 3user example of Section 4.2.). But its special case, 1layer RS, simplifies both the scheduler and receiver design, and it is still able to achieve a good performance in all user deployment scenarios. Onelayer RS requires only one SIC at each user. It does not rely on user grouping and user ordering for user scheduling. Therefore, the complexity of the scheduler is much simplified.
The cost of RSMA comes with a slightly higher encoding complexity since private and common streams need to be encoded. For the 1layer RS in a Kuser MISO BC, K+1 streams need to be encoded in contrast to K streams for NOMA and SDMA.
Optimization of RS
The WMMSE approach proposed in [42] is extended to solve the problem. The WMMSE algorithm to solve the sum rate maximization problem with 1layer RS (discussed in Section 4.4.1) is proposed in [28]. We further extend it to solve the generalized RS problem (20). To simplify the explanation, we focus on the 3user problem (15). It can be easily extended to solve the Kuser generalized RS problem.
As the 1order and 2order streams to be decoded at different users are not the same, we take user1 as an example. The procedure of the WMMSE algorithm is the same for other users. The signal received at user1 is \(\phantom {\dot {i}\!}y_{1}=\mathbf {h}_{1}^{H}\mathbf {{P}}\mathbf {{s}}+n_{1}\). It decodes four streams \(s_{123}, s_{\pi _{2,1}(1)}, s_{\pi _{2,1}(2)}, s_{1}\phantom {\dot {i}\!}\) sequentially using SICs. The 3order stream s_{123} is decoded first. It is estimated as \(\hat {s}_{123}=g_{1}^{123}y_{1}\), where \(g_{1}^{123}\) is the equalizer. After successfully decoding and removing s_{123} from y_{1}, the estimate of the 2order stream \( s_{\pi _{2,1}(1)}\) is \(\hat {s}_{\pi _{2,1}(1)}=g_{1}^{\pi _{2,1}(1)}\left (y_{1}\mathbf {h}_{1}^{H}\mathbf {{p}}_{123}s_{123}\right)\). Similarly, we calculate the estimates of \(\hat {s}_{\pi _{2,1}(2)}\) and \(\hat {s}_{1}\) as \(\hat {s}_{\pi _{2,1}(2)}=g_{1}^{\pi _{2,1}(2)}\left (y_{1}\mathbf {h}_{1}^{H}\mathbf {{p}}_{123}s_{123}\mathbf {h}_{1}^{H}\mathbf {{p}}_{\pi _{2,1}(1)}s_{\pi _{2,1}(1)}\right)\) and \(\hat {s}_{1}=g_{1}^{1}\left (y_{1}\,\,\mathbf {h}_{1}^{H}\mathbf {{p}}_{123}s_{123}\,\,\mathbf {h}_{1}^{H}\mathbf {{p}}_{\pi _{2,1}(1)}s_{\pi _{2,1}(1)}\mathbf {h}_{1}^{H}\mathbf {{p}}_{\pi _{2,1}(2)}s_{\pi _{2,1}(2)}\right)\), respectively. \(g_{1}^{\pi _{2,1}(1)},g_{1}^{\pi _{2,1}(2)}, g_{1}^{1}\) are the corresponding equalizers at user1. The mean square error (MSE) of each stream is defined as \(\varepsilon _{k}\triangleq \mathbb {E}\left \{s_{k}\hat {{s}_{k}}^{2}\right \}\). They are calculated as
where \(T_{1}^{123}\triangleq \mathbf {h}_{1}^{H}\mathbf {p}_{123}^{2}+\mathbf {h}_{1}^{H}\mathbf {p}_{12}^{2}+\mathbf {h}_{1}^{H}\mathbf {p}_{13}^{2}+\mathbf {h}_{1}^{H}\mathbf {p}_{23}^{2}+\mathbf {h}_{1}^{H}\mathbf {p}_{1}^{2}+\mathbf {h}_{1}^{H}\mathbf {p}_{2}^{2}+\mathbf {h}_{1}^{H}\mathbf {p}_{3}^{2}+1\) is the receive power at user1. \(T_{1}^{\pi _{2,1}(1)}\triangleq T_{1}^{123}\mathbf {h}_{1}^{H}\mathbf {p}_{123}^{2}\), \(T_{1}^{\pi _{2,1}(2)}\triangleq T_{1}^{\pi _{2,1}(1)}\mathbf {h}_{1}^{H}\mathbf {p}_{\pi _{2,1}(1)}^{2}\), \(T_{1}^{1}\triangleq T_{1}^{\pi _{2,1}(2)}\mathbf {h}_{1}^{H}\mathbf {p}_{\pi _{2,1}(2)}^{2}\). The optimum MMSE equalizers are
They are calculated by solving \(\frac {\partial \varepsilon _{1}^{123}}{\partial g_{1}^{123}}=0\), \(\frac {\partial \varepsilon _{1}^{\pi _{2,1}(1)}}{\partial g_{1}^{\pi _{2,1}(1)}}=0\), \(\frac {\partial \varepsilon _{1}^{\pi _{2,1}(2)}}{\partial g_{1}^{\pi _{2,1}(2)}}=0,\frac {\partial \varepsilon _{1}^{1}}{\partial g_{1}^{1}}=0\). Substituting (30) into (29), the MMSEs become
where \({I}_{1}^{123}=T_{1}^{\pi _{2,1}(1)}\), \({I}_{1}^{\pi _{2,1}(1)}=T_{1}^{\pi _{2,1}(2)}\), \({I}_{1}^{\pi _{2,1}(2)}=T_{1}^{1}\), and \(_{1}^{1}=T_{1}^{1}\mathbf {h}_{1}^{H}\mathbf {p}_{1}^{2}\). Based on (31), the SINRs of decoding the intended streams at user1 can be expressed as \(\gamma _{1}^{123}={1}/{\left (\varepsilon _{1}^{123}\right)^{\textrm {MMSE}}}1\), \(\gamma _{1}^{\pi _{2,1}(1)}={1}/{\left (\varepsilon _{1}^{\pi _{2,1}(1)}\right)^{\textrm {MMSE}}}1\), \(\gamma _{1}^{\pi _{2,1}(2)}={1}/{\left (\varepsilon _{1}^{\pi _{2,1}(2)}\right)^{\textrm {MMSE}}}1\), and \(\gamma _{1}^{1}={1}/{\left (\varepsilon _{1}^{1}\right)^{\textrm {MMSE}}}1\). The corresponding rates are rewritten as \(R_{1}^{123}=\log _{2} \left (\left (\varepsilon _{1}^{123}\right)^{\textrm {MMSE}}\right)\), \(R_{1}^{\pi _{2,1}(1)}=\log _{2}\left (\left (\varepsilon _{1}^{\pi _{2,1}(1)}\right)^{\textrm {MMSE}}\right)\), \(R_{1}^{\pi _{2,1}(2)}=\log _{2}\left (\left (\varepsilon _{1}^{\pi _{2,1}(2)}\right)^{\textrm {MMSE}}\right)\), and \(R_{1}^{1}=\log _{2} \left (\left (\varepsilon _{1}^{1}\right)^{\textrm {MMSE}}\right)\). The augmented WMSEs are
where \(u_{1}^{123},u_{1}^{\pi _{2,1}(1)},u_{1}^{\pi _{2,1}(2)},\text {and} u_{1}^{1}\) are weights associated with each stream at user1. By solving \(\frac {\partial \xi _{1}^{123}}{\partial g_{1}^{123}}=0,\frac {\partial \xi _{1}^{\pi _{2,1}(1)}}{\partial g_{1}^{\pi _{2,1}(1)}}=0, \frac {\partial \xi _{1}^{\pi _{2,1}(2)}}{\partial g_{1}^{\pi _{2,1}(2)}}=0, ~\text {and}~ \frac {\partial \xi _{1}^{1}}{\partial g_{1}^{1}}=0\), we derive the optimum equalizers as \(\left (g_{1}^{123}\right)^{*}=\left (g_{1}^{123}\right)^{\textrm {MMSE}}\), \(\left (g_{1}^{\pi _{2,1}(1)}\right)^{*}=\left (g_{1}^{\pi _{2,1}(1)}\right)^{\textrm {MMSE}}\), \(\left (g_{1}^{\pi _{2,1}(2)}\right)^{*}=\left (g_{1}^{\pi _{2,1}(2)}\right)^{\textrm {MMSE}}\), and \(\left (g_{1}^{1}\right)^{*}=\left (g_{1}^{1}\right)^{\textrm {MMSE}}\). Substituting the optimum equalizers into (32), we obtain
By further solving the equations \(\frac {\partial \xi _{1}^{123}\left (\left (g_{1}^{123}\right)^{\textrm {MMSE}}\right)}{\partial u_{1}^{123}}=0\), \(\frac {\partial \xi _{1}^{\pi _{2,1}(1)}\left (\left (g_{1}^{\pi _{2,1}(1)}\right)^{\textrm {MMSE}}\right)}{\partial u_{1}^{\pi _{2,1}(1)}}=0\), \(\frac {\partial \xi _{1}^{\pi _{2,1}(2)}\left (\left (g_{1}^{\pi _{2,1}(2)}\right)^{\textrm {MMSE}}\right)}{\partial u_{1}^{\pi _{2,1}(2)}}=0\), and \(\frac {\partial \xi _{1}^{1}\left (\left (g_{1}^{1}\right)^{\textrm {MMSE}}\right)}{\partial u_{1}^{1}}=0\), we obtain the optimum MMSE weights as
Substituting (34) into (33), we establish the rate WMMSE relationship as
Similarly, we can establish the rateWMMSE relationships for user2 and user3. Motivated by the rateWMMSE relationship in (35), we reformulate the optimization problem (15) as
where \(\mathbf {x}=\left [X_{1}^{123},X_{2}^{123},X_{3}^{123},X_{1}^{12},X_{2}^{12},X_{1}^{13},X_{3}^{13},X_{2}^{23},X_{3}^{23}\right ]\) is the transformation of the common rate c. \(\mathbf {u}=\left [u_{1}^{123},u_{2}^{123},u_{3}^{123},u_{1}^{12},u_{2}^{12},u_{1}^{13},u_{3}^{13},u_{2}^{23},u_{3}^{23},u_{1}^{1},u_{2}^{2},u_{3}^{2}\right ]\). \(\mathbf {g}=\left [g_{1}^{123},g_{2}^{123},g_{3}^{123},g_{1}^{12},g_{2}^{12},g_{1}^{13},g_{3}^{13},g_{2}^{23},g_{3}^{23},g_{1}^{1},g_{2}^{2},g_{3}^{2}\right ]\). \(\xi _{1,tot}=X_{1}^{123}+X_{1}^{12}+X_{1}^{13}+\xi _{1}^{1}\), \(\xi _{tot}=X_{2}^{123}+X_{2}^{12}+X_{2}^{23}+\xi _{2}^{2}\) and \(\xi _{3,tot}=X_{3}^{123}+X_{3}^{13}+X_{3}^{23}+\xi _{3}^{3}\) are individual WMSEs. \(\xi _{123}=\max \left \{ \xi _{1}^{123},\xi _{2}^{123},\xi _{3}^{123}\right \}\), \(\xi _{12}=\max \left \{ \xi _{1}^{12},\xi _{2}^{12}\right \}\), \(\xi _{13}=\max \left \{ \xi _{1}^{13},\xi _{3}^{13}\right \}\), \(\xi _{23}=\max \left \{ \xi _{2}^{23},\xi _{3}^{23}\right \}\) are the achievable WMSEs of the corresponding streams.
It can be easily shown that by minimizing (36a) with respect to u and g, respectively, we obtain the MMSE solutions (u^{MMSE},g^{MMSE}) formed by the corresponding MMSE equalizers and weights. They satisfy the KKT optimality conditions of (36) for P. Therefore, according to the rateWMMSE relationship (35) and the common rate transformation c=−x, problem (36) can be transformed to problem (15). For any point (x^{∗},P^{∗},u^{∗},g^{∗}) satisfying the KKT optimality conditions of (36), the solution given by (c^{∗}=−x^{∗},P^{∗}) satisfies the KKT optimality conditions of (15). The WSR problem (15) is then transformed into the WMMSE problem (36). The problem (36) is still nonconvex for the joint optimization of (x,P,u,g). We have derived that when (x,P,u) are fixed, the optimal equalizer is the MMSE equalizer g^{MMSE}. When (x,P,g) are fixed, the optimal weight is the MMSE weight u^{MMSE}. When (u,g) are fixed, (x,P) is coupled in the optimization problem (36), closedform solution cannot be derived. But it is a convex quadratically constrained quadratic program (QCQP) which can be solved using interiorpoint methods. These properties motivates us to use AO to solve the problem. In nth iteration of the AO algorithm, the equalizers and weights are firstly updated using the precoders obtained in the n−1th iteration (u,g)=(u^{MMSE}(P^{[n−1]}),g^{MMSE}(P^{[n−1]})). With the updated (u,g), (x,P) can then be updated by solving the problem (36). (u,g) and (x,P) are iteratively updated until the WSR converges. The details of the AO algorithm is shown in Algorithm 1, where WSR^{[n]} is the WSR calculated based on the updated (x,P) in nth iteration. ε is the tolerance of the algorithm. The AO algorithm is guaranteed to converge as the WSR is increasing in each iteration and it is bounded above for a given power constraint.
When considering imperfect CSIT, we follow the robust approach proposed in [28] for 1layer RS with imperfect CSIT. The precoders are optimized based on the available channel estimate to maximize a conditional averaged weighted sum rate (AWSR) metric, computed using partial CSIT knowledge. The stochastic AWSR problem was transformed into a deterministic counter part using the sample average approximated (SAA) method. Then, the rateWMMSE relationship is applied to transform the AWSR problem into a convex form and solved using an AO algorithm. The robust approach for 1layer RS in [28] can be easily extended to solve the Kuser generalized RS problem based on our proposed Algorithm 1, which will not be explained here.
Results and discussion
In this section, we evaluate the performance of SDMA, NOMA, and RSMA in a wide range of network loads (underloaded and overloaded regimes) and user deployments (with a diversity of channel directions, channel strengths, and qualities of channel state information at the transmitter). We first illustrate the rate region of different strategies in the twouser case followed by the WSR comparisons of the threeuser, fouruser, and tenuser cases.
Underloaded twouser deployment with perfect CSIT
When K=2, the rate region of all strategies can be explicitly compared in a twodimensional figure. As mentioned earlier, the rate region is the set of all achievable points. Its boundary is calculated by varying the weights assigned to users. In this work, the weight of user1 is fixed to u_{1}=1. The weight of user2 is varied as u_{2}=10^{[−3,−1,−0.95,⋯,0.95,1,3]}, which is the same as in [42]. To investigate the largest achievable rate region, the individual rate constraints are set to 0 in all strategies \(R_{k}^{\text {th}}=0,\forall k\in \{1,2\}\).
In the perfect CSIT scenario, the capacity region is achieved by DPC. Therefore, we compare the rate regions of different beamforming strategies with the DPC region. The DPC region is generated using the algorithm in [44]. Since the WSR problems for all beamforming strategies described earlier are nonconvex, the initialization of P is vital to the final result. It has been observed in [28] that maximum ratio transmission (MRT) combined with singular value decomposition (SVD) provides good overall performance over various channel realizations. It is used in this work for precoder initialization of RS. The precoders for the private message p_{ k } is initialized as \(\mathbf {p}_{k}=p_{k}\frac {\mathbf {h}_{k}}{\left \Vert \mathbf {h}_{k}\right \Vert }\), where \(p_{k}=\frac {\alpha P_{t}}{2}\) and 0≤α≤1. The precoder for the common message is initialized as p_{12}=p_{12}u_{12}, where p_{12}=(1−α)P_{ t } and u_{12} is the largest left singular vector of the channel matrix H=[h_{1},h_{2}]. It is calculated as u_{12}=U(:,1). U is derived based on the SVD of H, i.e., H=USV^{H}. To ensure a fair comparison, the precoders of MU–LP are initialized based on MRT. For SC–SIC, the precoder of the user decoded first is initialized based on SVD and that of the user decoded last is initialized based on MRT.
Random channel realizations
We firstly consider the scenarios when the channel of each user h_{ k } has independent and identically distributed (i.i.d.) complex Gaussian entries with a certain variance, i.e., \(\mathcal {CN}\left (0,\sigma _{k}^{2}\right)\). The BS is equipped with two or four antennas (N_{ t }=2,4) and serves two singleantenna users. Figure 5 shows the average rate regions of different strategies over 100 random channel realizations when \(\sigma _{1}^{2}=1, \sigma _{2}^{2}=1\), and N_{ t }=4. SNRs are 10 and 20 dB, respectively. When the number of transmit antenna is larger than the number of users, MU–LP achieves a good performance. The generated precoders of the users tend to be more orthogonal as the number of transmit antennas increases. In contrast, the average rate region achieved by SC–SIC is small. When \(\sigma _{1}^{2}=1 \text {and} \sigma _{2}^{2}=1\), there is no disparity of average channel strengths. SC–SIC is not able to achieve a good performance in such scenario. As the SC–SIC strategy is motivated by leveraging the channel strength difference among users, it achieves a good performance when the channels are degraded. Specifically, when the channels of users are close to alignment, SC–SIC works better than MU–LP if the users have asymmetric channel strengths. However, for the general nondegraded MISOBC, SC–SIC often yields a performance loss [19]. The simulation results when \(\sigma _{1}^{2}=1, \sigma _{2}^{2}=0.09\), and N_{ t }=2 is illustrated in Fig. 6. The average channel gain difference between the users increases to 5 dB, and the number of the transmit antenna reduces to two. In such scenario, the rate region gap between RS and MU–LP increases while the rate region gap between RS and SC–SIC decreases. It shows that SC–SIC is more suited to the scenarios where the users experience a large disparity in channel strengths. In both Figs. 5 and 6, the rate region gaps among different strategies increase with SNR. RS achieves a larger rate region than SC–SIC and MU–LP, and it is closer to the capacity region achieved by DPC.
Specific channel realizations
In order to have a better insight into the benefits of RS over MU–LP and SC–SIC, we investigate the influence of user angle and channel strength on the performance. When N_{ t }=4, the channels of users are realized as
In above channel realizations, γ and θ are control variables. γ controls the channel strength of user2. If γ=1, the channel strength of user1 is equal to that of user2. If γ=0.3, user2 suffers from an additional 5 dB path loss compared to user1. θ controls the angle between the channels of user1 and user2. It varies from 0 to \(\frac {\pi }{2}\). If θ=0, the channel of user1 is aligned with that of user2. If \(\theta =\frac {\pi }{2}\), the channels of user1 and user2 are orthogonal to each other. In the following results, γ=1,0.3, which corresponds to 0 dB, 5 dB channel strength difference, respectively. For each γ, θ adopts value from \(\theta =\left [\frac {\pi }{9},\frac {2\pi }{9},\frac {\pi }{3},\frac {4\pi }{9}\right ]\). Intuitively, when θ is less than \(\frac {\pi }{9}\), the channels of users are sufficiently aligned and SC–SIC performs well. When θ is larger than \(\frac {4\pi }{9}\), the channels of users are sufficiently orthogonal to each other and MU–LP is more suitable. Therefore, we consider angles within the range of \(\left [\frac {\pi }{9}, \frac {4\pi }{9}\right ]\). SNR is fixed to 20 dB. When N_{ t }=2, the channels of user1 and user2 are realized as h_{1}=[1,1]^{H} and h_{2}=γ×[1,e^{jθ}]^{H}, respectively. The same values of γ and θ are adopted in N_{ t }=2 as used in N_{ t }=4^{Footnote 13}.
Figure 7 shows the results when γ=1 and N_{ t }=4. In all subfigures, the rate region achieved by RS is equal to or larger than that of SC–SIC and MU–LP. When γ=1 and \( \theta =\frac {\pi }{9}\), the channels of user1 and user2 almost coincide. RS exhibits a clear rate region improvement over SC–SIC and MU–LP. SC–SIC cannot achieve a good performance due to the equal channel gain while the performance of MU–LP is poor when the user channels are closely aligned to each other. As θ increases, the gap between the rate regions of RS and MU–LP reduces as the performance of MU–LP is better when the channels of users are more orthogonal to each other while the gap between the rate regions of MU–LP and SC–SIC increases. The rate regions of RS and MU–LP tend to the capacity region achieved by DPC as θ increases. As shown in Fig. 7d, when the channels of users are sufficiently orthogonal to each other, the rate regions of DPC, RS, and MU–LP are almost identical. In such an orthogonal scenario, RS reduces to MU–LP.
Figure 8 shows the results when γ=1 and N_{ t }=2. In all subfigures, RS outperforms MU–LP and SC–SIC. Comparing with the results of N_{ t }=4, the rate region gap between RS and MU–LP is enlarged when N_{ t }=2. When the number of transmit antenna decreases, it becomes more difficult for MU–LP to design orthogonal precoders for users. MU–LP is more suited to underloaded scenarios (N_{ t }>K). In both Figs. 7 and 8, the rate region of SC–SIC is the worst due to the equal channel gain. In contrast, RS performs well for any angle between user channels.
Figure 9 shows the rate region comparison of DPC, RS, SC–SIC, and MU–LP transmission schemes with 5 dB channel strength difference between the two users, i.e., γ=0.3 and N_{ t }=4. RS and SC–SIC are much closer to the DPC region in the setting of Fig. 9 compared to Fig. 7 because of the 5 dB channel strength difference. Figure 9b, c are interesting as SC–SIC and MU–LP outperform each other at one part of the rate region. There is a crosspoint between the two schemes in each figure mentioned. The rate region of RS is equal to or larger than the convex hull of the rate regions of SC–SIC and MU–LP.
Figure 10 shows the rate region comparison when γ=0.3 and N_{ t }=2. Comparing Fig. 10 with Fig. 9, SC–SIC achieves a relatively better performance when the number of transmit antenna reduces. The WSRs of RS and SC–SIC are overlapped, and they almost achieve the capacity region when \(\theta =\frac {\pi }{9}\). However, as θ increases, the rate region gap between RS and SC–SIC increases despite the 5 dB channel gain difference. Both SC–SIC and RS rely on one SIC when there are two users in the system. Though the receiver complexity of SC–SIC and RS are the same, RS achieves explicit performance gain over SC–SIC in most investigated scenarios. Comparing with MU–LP and SC–SIC, RS is suited to any channel angles and channel gain difference.
More results of underloaded twouser deployments with perfect CSIT are given in Appendix 1. We further illustrate the rate regions of different strategies when SNR is 10 dB. Comparing the corresponding figures of 10 dB and 20 dB, we conclude that as SNR increases, the gaps among the rate regions of different schemes increase, with RS exhibiting further performance benefits. In all investigated scenarios, RS always outperforms MU–LP and SC–SIC.
Underloaded twouser deployment with imperfect CSIT
Next, we investigate the rate region of different transmission schemes in the presence of imperfect CSIT. We assume the users are able to estimate the channel perfectly while the instantaneous channel estimated at the BS is imperfect. We assume the estimated channel of user1 and user2 are \(\widehat {\mathbf {h}}_{1}=\left [ 1,1,1,1\right ]^{H}\) and \(\widehat {\mathbf {h}}_{2}=\gamma \times \left [ 1, e^{j\theta }, e^{j2\theta }, e^{j3\theta }\right ]^{H}\) when N_{ t }=4. For the given channel estimate at the BS, the channel realization is \(\mathbf {h}_{k}=\widehat {\mathbf {h}}_{k}+\widetilde {\mathbf {h}}_{k} ~\text {and}~ \forall k\in \{1,2\}\), where \(\widetilde {\mathbf {h}}_{k}\) is the estimation error of userk. \(\widetilde {\mathbf {h}}_{k}\) has i.i.d. complex Gaussian entries drawn from \(\mathcal {CN}\left (0,\sigma _{e,k}^{2}\right)\). The error covariance of user1 and user2 are \(\sigma _{e,1}^{2}=P_{t}^{0.6}\) and \(\sigma _{e,2}^{2}=\gamma P_{t}^{0.6}\), respectively. The precoders are initialized and designed using the estimated channels \(\widehat {\mathbf {h}}_{1} \text {and} \widehat {\mathbf {h}}_{2}\) and the same methods as stated in perfect CSIT scenarios. One thousand different channel error samples are generated for each user. Each point in the rate region is the average rate^{Footnote 14} over the generated 1000 channels. SNR is fixed to 20 dB.
Figures 11 and 12 show the results when γ=1 and γ=0.3, respectively. Similarly to the results in perfect CSIT, the gaps between the rate regions of RS and MU–LP reduce as θ increases in both figures. When \(\theta =\frac {4\pi }{9}\), the channels of the two users are sufficiently orthogonal. The rate regions of RS and MU–LP are almost identical. SC–SIC achieves a good performance when the channels of users are sufficiently aligned with enough channel gain difference, as shown in Fig. 12a.
Comparing Figs. 11 and 7, the rate region gap between RS and MU–LP increases in imperfect CSIT due to the residual interference introduced. The interferencenulling in MU–LP is distorted and yields residual interference at the receiver, which jeopardizes the achievable rate. In contrast, the rate region gap between RS and SC–SIC slightly reduces in imperfect CSIT, as observed by comparing Fig. 12 with Fig. 9. SC–SIC is less sensitive to CSIT inaccuracy comparing with MU–LP. However, the rate region gap between RS and SC–SIC is still obvious. In comparison, RS is more flexible and robust to multiuser interference originating from the imperfect CSIT, as evidenced by the recent literature on RS with imperfect CSIT [27–33,38–41]. With RS, the amount of interference decoded by both users (through the presence of common stream) is adjusted dynamically to the channel conditions (channel directions and strengths) and CSIT inaccuracy.
More results of underloaded twouser deployments with imperfect CSIT are given in Appendix 2. The rate regions of different strategies for varied SNR, N_{ t } and γ are illustrated. We further show that the performance of RS is stable in a wide range of parameters, namely number of transmit antennas, user deployments, and CSIT inaccuracy. RS achieves equal or better performance than MU–LP and SC–SIC in all simulated channels.
Underloaded threeuser deployment with perfect CSIT
When K = 3, the rate region of each strategy is a threedimensional surface. The gaps among rate regions of different strategies are difficult to display. As each point of the rate region is derived by solving the WSR problem with a fixed weight vector u, the WSRs instead of the rate regions of different transmission strategies are compared in the threeuser case.
Two RS schemes are investigated in threeuser deployments. RS refers to the generalized RS strategy of Section 4.2 and 1layer RS refers to the lowcomplexity RS strategy of Section 4.4.1. We compare the WSR of RS, 1layer RS, DPC, SC–SIC, and MU–LP. The beamforming initialization of different strategies is extended based on the methods adopted in the twouser case. There are three streams of distinct stream orders in RS (1/2/3order streams). The precoders of the streams are initialized differently. The transmit power P_{ t } is divided into three parts α_{1}P_{ t }, α_{2}P_{ t }, and α_{3}P_{ t } for streams of three distinct stream orders, where α_{1},α_{2},α_{3}∈[0,1] and α_{1}+α_{2}+α_{3}=1. The precoder p_{ k },∀k∈{1,2,3} of the 1order stream (private stream) s_{ k } is initialized as \(\mathbf {p}_{k}=p_{k}\frac {\mathbf {h}_{k}}{\left \Vert \mathbf {h}_{k}\right \Vert }\), where \(p_{k}=\frac {\alpha _{1} P_{t}}{3}\) is the allocated power. The precoders p_{12},p_{13},andp_{23} of the 2order streams are initialized as p_{12}=p_{12}u_{12},p_{13}=p_{13}u_{13},and p_{23}=p_{23}u_{23}, respectively, where \(p_{12}=p_{13}=p_{23}=\frac {\alpha _{2}P_{t}}{3}\) and u_{12} is the largest left singular vector of the channel matrix H_{12}=[h_{1},h_{2}]. Similarly, u_{13} and u_{23} are the largest left singular vectors of the channel matrices H_{13}=[h_{1},h_{3}] and H_{23}=[h_{2},h_{3}], respectively. The precoder p_{123} of the 3order stream (conventional common stream) s_{123} is initialized as p_{123}=p_{123}u_{123}, where p_{123}=α_{3}P_{ t } and u_{123} is the largest left singular vector of the channel matrix H_{123}=[h_{1},h_{2},h_{3}]. The beamforming initialization of 1layer RS is similar as RS except we have p_{123} and p_{ k },∀k∈{1,2,3} only. By setting α_{2}=0, the initialization of RS is applied to 1layer RS. To ensure a fair comparison, the precoders of MU–LP are initialized based on MRT. For SC–SIC, the precoder of the user decoded first p_{π(1)} is initialized as p_{π(1)}=p_{π(1)}u_{π(1)}, where p_{π(1)}=α_{3}P_{ t } and u_{π(1)} is the largest left singular vector of the channel matrix H_{123}=[h_{1},h_{2},h_{3}]. The precoder of the user decoded secondly p_{π(2)} is initialized as p_{π(2)}=p_{π(2)}u_{π(2)}, where p_{π(2)}=α_{2}P_{ t } and u_{π(2)} is the largest left singular vector of the channel matrix H_{π(23)}=[h_{π(2)},h_{π(3)}]. The user decoded last is initialized based on MRT.
We firstly consider an underloaded scenario. The BS is equipped with four transmit antennas (N_{ t }=4) and serves three singleantenna users in all simulations. The individual rate constraint is set to 0, \(R_{k}^{\text {th}}=0,\forall k\in \{1,2,3\}\). The channel of users are realized as
γ_{1}andγ_{2} and θ_{1}andθ_{2} are control variables as discussed in the twouser case. For a given set of γ_{1}andγ_{2}, θ_{1} adopts value from \(\theta _{1}=\left [\frac {\pi }{9},\frac {2\pi }{9},\frac {\pi }{3},\frac {4\pi }{9}\right ]\) and θ_{2}=2θ_{1}. When \(\theta _{1}=\frac {\pi }{9} \text {and} \theta _{2}=\frac {2\pi }{9}\), the channels of user1 and user2, and user2 and user3 are sufficiently aligned. When \(\theta _{1}=\frac {4\pi }{9} \text {and} \theta _{2}=\frac {8\pi }{9}\), the channels of user1 and user2 and user2 and user3 are sufficiently orthogonal. We consider SNRs within the range 0 to 30 dB. We assume the sum of the weights allocated to users is equal to one, i.e., u_{1}+u_{2}+u_{3}=1.
Figures 13 and 14 show the results when the weight vectors are u=[0.2,0.3,0.5] and u=[0.4,0.3,0.3], respectively. In both figures, γ_{1}=1andγ_{2}=0.3. There is a 5 dB channel gain difference between user1 and user3 as well as between user2 and user3. In all scenarios and SNRs, RS always outperforms MU–LP and SC–SIC. Comparing with Fig. 14, the WSR improvement of RS is more explicit in Fig. 13. It implies that RS provides better enhancement of system throughput and user fairness. The performance of SC–SIC is the worst in most subfigures. This is due to the underloaded user deployments where N_{ t }>K. One of the three users are required to decode all the messages, and all the spatial multiplexing gains are sacrificed. Therefore, the sum DoF of SC–SIC is reduced to 1, resulting in the deteriorated performance of SC–SIC in underloaded scenarios. In comparison, the performance of MU–LP is better than SC–SIC except in Fig. 14a. MU–LP is more likely to serve the users with higher weights and channel gains by turning off the users with poor weights and channel gains when there is no individual rate constraints. It cannot deal efficiently with user fairness when a higher weight is allocated to the user with weaker channel strength. In contrast, SC–SIC works better when user fairness is considered. The WSR achieved by lowcomplexity 1layer RS is equal to or larger than that of MU–LP and SC–SIC in most subfigures. Comparing with SC–SIC and MU–LP, 1layer RS is more robust to different user deployments and only a single SIC is required at each user. Moreover, the WSR of 1layer RS is approaching that of RS in all user deployments. Considering the tradeoff between performance and complexity, 1layer RS is a good alternative to RS.
In all threeuser deployments of SC–SIC, the decoding order is required to be optimized together with the precoder. To investigate the influence of different decoding orders, we compare the WSRs of SC–SIC using different decoding orders when u_{1}=0.2,u_{2}=0.3,and u_{3}=0.5. There are in total six different decoding orders:
In Fig. 15, the WSR of six different decoding orders are illustrated in the circumstance where there is a 5dB channel gain difference between user1/2 and user3. When γ_{1}=1andγ_{2}=0.3, it is typical to decode the message of user3 first as the channel gain of user3 is the worst. However, we notice that the optimal decoding order in Fig. 15 is order 3, user1 is decoded first. This is due to the smallest weight allocated to user1, u_{1}=0.2. It implies that the weights assigned to users will affect the optimal decoding order. The scheduler complexity of SC–SIC becomes extremely high in order to find the optimal decoding order. In contrast, 1layer RS has a much lower scheduling complexity and does not rely on any user ordering at the transmitter. Moreover, it only requires a single SIC at each receiver.
More results of underloaded threeuser deployments with perfect CSIT and imperfect CSIT are given in Appendices 3 and Appendix 5, respectively. The WSRs of different strategies for varied SNR, N_{ t }, γ_{1},γ_{2}, and u are illustrated. In all figures, RS outperforms SC–SIC and MU–LP. Though the scheduler and receiver complexity of 1layer RS is low, it achieves equal or better performance than SC–SIC and MU–LP in most figures of perfect CSIT and all figures of imperfect CSIT. All forms of RS are robust to a wide range of CSIT inaccuracy, channel gain difference, and channel angles among users.
Overloaded threeuser deployment with perfect CSIT
Two transmit antenna deployment
We first consider an overloaded scenario where the BS is equipped with two antennas (N_{ t }=2) and serves three singleantenna users. The channel realizations and beamforming initialization follows the methods used in the underloaded threeuser deployment. The channel of users are realized as h_{1}=[1,1]^{H}, \(\mathbf {h}_{2}=\gamma _{1}\times \left [ 1, e^{j\theta _{1}}\right ]^{H}\), and \( \mathbf {h}_{3}=\gamma _{2}\times \left [1, e^{j\theta _{2}}\right ]^{H}\). In overloaded scenarios, to guarantee some QoS, we add individual rate constraints to users as the system has otherwise a tendency to turn off some users. In all simulations of two transmit antenna deployment, we assume the rate threshold of each user is equal \(R_{1}^{\text {th}}\thinspace {=}\thinspace R_{2}^{\text {th}}\thinspace {=}\thinspace R_{3}^{\text {th}}\). Since the BS is able to serve users with higher QoS requirements as SNR increases, the rate threshold is assumed to increase with SNR. The rate threshold increases as r_{th} = [0.02,0.08,0.19,0.3,0.4,0.4,0.4] bit/s/Hz for SNR=[0,5,10,15,20,25,30] dBs.
We compare the performance of RS, 1layer RS, SC–SIC, MU–LP, and SC–SIC per group in the overloaded threeuser deployment. In SC–SIC per group, we consider a fixed grouping method. We assume user1 is in group 1 while user2 and user3 are in group 2. The decoding order will be optimized together with the precoder. The beamforming initialization of SC–SIC per group is different from SC–SIC. In group 1, the precoder of user1 is initialized based on MRT. In group 2, the precoder of the user decoded first p_{π(1)} is initialized as p_{π(1)}=p_{π(1)}u_{π(1)} and u_{π(1)} is the largest left singular vector of the channel matrix H_{23}=[h_{2},h_{3}]. The precoder of the user decoded secondly is initialized based on MRT.
RS exhibits a clear WSR gain over SC–SIC, SC–SIC per group, and MU–LP in Fig. 16, where γ_{1}=1,γ_{2}=0.3, and u=[0.4,0.3,0.3]. The WSR of MU–LP deteriorates in such overloaded scenario. When the individual rate constraints are not zero and N_{ t }<K, MU–LP cannot coordinate the multiuser interference coming from all the users served simultaneously. When the angles of channels are large enough (subfigure c and subfigure d of Fig. 16), the WSR of SC–SIC per group is better than SC–SIC. This is due to its ability to combine treating interference as noise (to tackle intergroup interference) with decoding interference (to tackle intragroup interference). However, as the angles of channels decrease, the performance of SC–SIC becomes better while that of SC–SIC per group is worse. Whether SC–SIC outperforms SC–SIC per group depends on SNR and user deployments. To ensure the WSR of the NOMA system is maximized, a joint optimization of NOMA strategies based on switching between SC–SIC and SC–SIC per group on top of deciding, the user grouping and user ordering is required. Such switching method has high scheduler and receiver complexity while its achieved performance is still lower than the simple 1layer RS in most user deployments.
Single transmit antenna deployment
In a SISO BC, there is no need to split the messages into common and private parts since the capacity region is achieved by SC–SIC. Nevertheless, in view of the benefit of 1layer RS in the MISO BC, we may wonder whether RS can be of any help in a SISO BC, especially when it comes to reducing the complexity of the receivers and the number of SIC needed.
We therefore compare the performance of 1layer RS with SC–SIC in a 3user SISO BC. We note that SC–SIC requires two layers of SIC while 1layer RS requires a single SIC for all users. The channel of each user h_{ k } has an i.i.d. complex Gaussian entry with a certain variance, i.e., \(\mathcal {CN}\left (0,\sigma _{k}^{2}\right)\). Figure 17 shows the average WSRs of different strategies over ten random channel realizations when \(\sigma _{1}^{2}=1, \sigma _{2}^{2}=0.3, \text {and} \sigma _{3}^{2}=0.1\). 1layer RS is able to achieve very close performance to SC–SIC. Comparing with SC–SIC, the complexity of 1layer RS is much reduced. There is no ordering issue at the BS, and only one SIC is required at each user. Jointly considering the performance and complexity of the system, 1layer RS is an attractive alternative to SC–SIC.
More results of overloaded threeuser deployments with perfect CSIT and imperfect CSIT are given in Appendices 4 and Appendix 6, respectively. The WSRs of different strategies for varied SNR, N_{ t }, γ_{1},γ_{2}, and u are illustrated. We further show that RS exhibits a clear WSR gain over SC–SIC, SC–SIC per group, and MU–LP in all simulated channels and weights. Onelayer RS outperforms SC–SIC, SC–SIC per group and MU–LP in most simulated scenarios. It is more robust and achieves a nearly equivalent WSR to that of RS in all user deployments. We also show that 1layer RS achieves near optimal performance in various channel conditions of SISO BC.
Overloaded fouruser deployment with perfect CSIT
We further investigate the fouruser system model shown in Fig. 4, where user1 and user2 are in group 1 while user3 and user4 are in group 2. We compare the 2layer HRS, 1layer RS per group, 1layer RS, SC–SIC per group, and MU–LP. In 2layer HRS, the intragroup interference is mitigated using the intragroup common streams s_{12} and s_{34}, and the intergroup interference is mitigated using the intergroup common stream s_{1234}. Onelayer RS and 1layer RS per group are two special strategies of 2layer HRS. All users in 1layer RS are treated as single group. Only the 4order common stream s_{1234} and 1order private streams are active. No power is allocated to s_{12} and s_{34}. In contrast, 1layer RS per group only allocate power to the intragroup common stream s_{12} and s_{34} and 1order private streams. No power is allocated to the intergroup common stream s_{1234}. Users within each group are served using RS and users across groups are served using SDMA so as to mitigate the intergroup interference.
We consider an overloaded scenario. The BS is equipped with two antennas and serves four singleantenna users. The channel of users are realized as
γ_{1},γ_{2},γ_{3} and θ_{1},θ_{2},θ_{3} are control variables. θ_{1} is the channel angle between user1 and user2. It is denoted as intragroup angle of group 1. θ_{2} is the channel angle between user1 and user2. θ_{2}−θ_{1} is the channel angle between user2 and user3, denoted as intergroup angle. θ_{3} is the channel angle between user1 and user3. θ_{3}−θ_{2} is the channel angle between user3 and user4. It is the intragroup angle of group 2. In the following, we assume the intragroup angle of group 1 is the same as that of group 2. We have θ_{3}=θ_{1}+θ_{2}. In each figure, the intragroup angle is varied as \(\theta _{1}=\left [0,\frac {\pi }{18},\frac {\pi }{9},\frac {\pi }{6}\right ]\). The individual rate constraint is set to r_{th}=[0.03,0.1,0.2,0.3,0.4,0.4,0.4] bit/s/Hz for SNR=[0,5,10,15,20,25,30] dBs. The weights of users are assumed to be equal, i.e., u_{1}=u_{2}=u_{3}=u_{4}=0.25. We also assume the channel gain difference within each group is equal. The channel gain of user3 is equal to that of user1 (γ_{2}=1), and the channel gain of user4 is equal to that of user2 (γ_{3}=γ_{1}).
Figures 18 and 19 show the results when γ_{1}=0.3. The intergroup angles are \(\frac {\pi }{9}\) and \(\frac {\pi }{3}\), respectively. The WSR achieved by 2layer HRS is equal to 1layer RS in both figures, which means that 2layer HRS reduces to 1layer RS in these user deployments. Twolayer HRS and 1layer RS outperform all other schemes. The intergroup and intragroup interference can be jointly mitigated by one layer common message. As the intergroup angle increases, the WSR gaps between 2layer HRS and 1layer RS per group reduces. The intergroup interference can be coordinated by SDMA when the intergroup angle is sufficiently large. Onelayer RS per group has the same WSR as SC–SIC per group in both figures. It reduces to SC–SIC per group because SC–SIC is more suitable when the intragroup angle is sufficiently small and the channel gain difference between users within each group is sufficiently large.
More results of overloaded fouruser deployments with perfect CSIT are given in Appendix 7. The WSRs of different strategies when there is no channel gain difference (γ_{1}=1) are illustrated. We further show that 2layer HRS, 1layer RS, and 1layer RS per group achieve equal or better performance than SC–SIC per group and MU–LP in all simulated channel conditions.
Overloaded tenuser deployment with perfect CSIT
We further consider an extremely overloaded scenario subject to QoS constraints. The BS is equipped with two antennas (N_{ t }=2) and serves ten users. The channel of each user h_{ k } has i.i.d. complex Gaussian entries with a certain variance, i.e., \(\mathcal {CN}(0,\sigma _{k}^{2})\). The rate of each user is averaged over the 10 randomly generated channels. We compare 1layer RS, MU–LP, multicast, and SC–SIC with a certain decoding order. There are 10! different decoding orders of SC–SIC in the tenuser case. The optimal decoding order of SC–SIC is intractable. In the following simulations, only the decoding order based on the ascending channel gain is considered for WSR calculation in SC–SIC. It is the optimal decoding order in SISO BC. Multicast can be regarded as a special scheme of 1layer RS with only the 10order stream to be transmitted to all users. The weight of each user is assumed to be equal to 1.
Figure 20 shows the WSRs of different strategies when \(\sigma _{1}^{2}=\sigma _{2}^{2}=\ldots =\sigma _{10}^{2}=1\), r_{th}=[0.01,0.03,0.05,0.1,0.1,0.1,0.1] bit/s/Hz. The WSR achieved by the multicast scheme is the worst. In such an overloaded user deployment, the spectral efficiency of multicast is low as it is difficult for a single beamformer to satisfy all users. Under the rate constraint r_{th}, the WSR of SC–SIC is better than that of MU–LP while the slopes of the WSRs are the same for large SNRs. It implies that SC–SIC and MU–LP achieve the same DoF of 1. In contrast, 1layer RS shows an obvious WSR improvement over all other strategies and exhibits a DoF of two. This highlights that RS exploits the maximum DoF of the considered deployments (that is limited by two, given the two transmit antennas). To further investigate the reason behind the results, we focus on one random channel realization. The WSRs achieved by all strategies when SNR = 30 dB are compared as shown in Fig. 21. The optimized common rate vector of onelayer RS is c=[0,0.1,0.1,0.1,0,0.1,0.1,0.1,0.1,0.1] bit/s/Hz. No common rate is allocated to user1 and user5. But in Fig. 21, we can observe that the rate allocated to user1 and user5 are the highest. It implies that RS uses the common message to pack messages from eight users and uses two transmit antennas to deliver private messages to user1 and user5. RS achieves a sumDoF of 2 in the overloaded regime. In contrast, MU–LP and SC–SIC allocate most of power to single user. The rate achieved by user5 when using MU–LP and the rate achieved by user10 when using SC–SIC are much higher than other users in Fig. 21. The DoFs achieved by MU–LP and SC–SIC are limited to 1 in such circumstance.
Note that results here show the usefulness of the RS framework for massive IoT or MTC services. Those devices are typically cheap. In the example above, user1 and user5 could be highend devices, for which RS would be implemented. Those devices would therefore perform SIC. All other devices could be IoT or MTC devices, who would not need to implement RS, nor SIC, but simply decode the common message. Hence, the RS framework can be used to pack the IoT/MTC traffic in the common message.
More results of overloaded tenuser deployments with perfect CSIT are given in Appendix 8. We further illustrate WSRs of different strategies when the rate threshold r_{th}, and channel gain difference are changed. We show that the when the rate threshold of each user is 0, MU–LP is able to achieve a DoF of 2. However, as the rate threshold increases, MU–LP cannot coordinate the interuser interference and its achieved DoF drops to 1. In the extremely overloaded scenario, the WSR gap between RS and SC–SIC is still large. SC–SIC makes an inefficient use of the transmit antennas and achieves a DoF of 1.
Conclusions
To conclude, we propose a new multiple access called ratesplitting multiple access (RSMA). We compare the proposed RSMA with SDMA and NOMA by solving the problem of maximizing WSR in MISOBC systems with QoS constraints. Both perfect and imperfect CSIT are investigated. WMMSE and its modified algorithms are adopted to solve the respective optimization problems. We show that SDMA and NOMA are subject to many limitations, including highsystem complexity and a lack of robustness to user deployments, network load, and CSIT inaccuracy. We propose a general multiple access framework based on rate splitting (RS), where the common symbols decoded by different groups of users are transmitted on top of private symbols decoded by the corresponding users only. Thanks to its ability of partially decoding interference and partially treating interference as noise, RSMA softly bridges and outperforms SDMA and NOMA in any user deployments, CSIT inaccuracy, and network load. The simplified RS forms, such as 1layer RS and 2layer HRS, show great potential to reduce the scheduler and receiver complexity but maintain good and robust performance in any user deployments, CSIT inaccuracy, and network load. Particularly, we show that 1layer RS is an attractive alternative to SC–SIC in a SISO BC deployment due to its near optimal performance and very low complexity. Therefore, RSMA is a more general and powerful multiple access for downlink multiantenna systems that encompasses SDMA and NOMA as special cases.
RSMA has the potential to change the design of the physical layer and MAC layer of nextgeneration communication systems by unifying existing approaches and relying on a superposed transmission of common and private messages. Many interesting problems are left for future research, including among others the role played by RSMA to achieve the fundamental limits of broadcast, interference and relay channels in the presence of imperfect CSIT and disparity of channel strengths, optimization (robust design, sumrate maximization, maxmin fairness, QoS constraints) of RSMA, performance analysis of RSMA, RSMA design for multiuser/massive/millimeterwave/multicell/network MIMO, modulation and coding for RSMA, RSMA with multicarrier transmissions, RSMA with linear versus nonlinear precoding, resource allocation and crosslayer design of RSMA, security provisioning in RSMA, RSMA design for cellular and satellite communication networks, prototyping and experimentation of RSMA, and standardization issues (link/systemlevel evaluations, receiver implementation, transmission schemes/modes, CSI feedback mechanisms, and downlink and uplink signaling) of RSMA.
Appendix 1
Underloaded twouser deployment with perfect CSIT
To further investigate the influence of SNR, we illustrate the rate region of different strategies when SNR is 10 dB in Figs. 22, 23, 24, and 25 and compare with the results when SNR is 20 dB in Figs. 7, 8, 9, and 10. Comparing the corresponding figures of 10 and 20 dB, we observe that the rate region gaps among different schemes grow with SNR. As SNR increases, the performance improvement of RS becomes more obvious. Specifically, SC–SIC and MU–LP outperform each other at one part of the rate region in Figs. 9b and 10d and the rate region of RS encompasses the convex hull of the rate regions of SC–SIC and MU–LP. However, as SNR decreases to 10 dB, the crosspoints disappear in Figs. 24b and 25d. The rate regions of SC–SIC overlap with that of RS. RS reduces to SC–SIC, and they outperform MU–LP in the whole rate region.
Appendix 2
Underloaded twouser deployment with imperfect CSIT
To further study the influence of CSIT inaccuracy, SNR, number of transmit antennas, and user deployments, we illustrate the rate region of different strategies when SNR, N_{ t }, and γ are varied in Figs. 26, 27, 28, 29, 30, and 31.
Figures 26 and 27 show the corresponding results of Figs. 11 and 12 when SNR decreases to 10 dB. The rate region gaps among users decreases when SNR decreases.
Figures 28 and 29 show the results when γ=1 and N_{ t }=2. When SNR is 10 dB, the rate regions of the three schemes are very close to each other. When SNR is 20 dB, the rate region of RS shows explicit improvement over the rate regions of MU–LP and SC–SIC. Comparing Fig. 29 with Fig. 8, the performance of MU–LP is worse when CSIT is imperfect. It shows that MU–LP requires accurate CSIT to design precoders. There is no crosspoint between SC–SIC and MU–LP in Figs. 27c and 12b compared, respectively, with Figs. 24c and 9b.
Figures 30 and 31 show the results when γ=0.3. SNR is 10 and 20 dB, respectively. The rate region gap between RS and SC–SIC reduces in imperfect CSIT, as observed by comparing Fig. 31 with Fig. 10. Comparing with MU–LP, SC–SIC is less sensitive to CSIT inaccuracy.
Appendix 3
Underloaded threeuser deployment with perfect CSIT
We consider three different sets of γ_{1},γ_{2}. When γ_{1}=γ_{2}=1, the three users have no channel strength difference. When γ_{1}=1,γ_{2}=0.3, there is a 5dB channel strength difference between user1 and user3 as well as between user2 and user3. When γ_{1}=0.3,γ_{2}=0.1, there is a 5dB channel strength difference between user1 and user2 as well as user2 and user3. The channel strength difference between user1 and user3 is 10 dB. We consider three different weight vectors for each set of γ_{1},γ_{2}, i.e., u=[0.2,0.3,0.5], u=[0.4,0.3,0.3], and u=[0.6,0.3,0.1].
In all figures (Figs. 32, 33, 34, 35, 36, 37, and 38), the WSR of RS is equal to or better than that of MU–LP and SC–SIC. Considering a specific scenario where \(\theta _{1}=\frac {2\pi }{9},\theta _{2}=\frac {4\pi }{9}\), and u=[0.6,0.3,0.1], the WSR of RS is better than that of MU–LP and SC–SIC as shown in Figs. 34b, 35b, and 38b. As SNR increases, the WSR improvement of RS is generally more obvious. For a fixed weight vector, the WSR of SC–SIC becomes closer to that of RS as the channel gain differences among users increase. For example, we compare Figs. 13, 32, and 36 for a fixed u=[0.2,0.3,0.5]. When u=[0.4,0.3,0.3], the WSR of RS and MU–LP are almost identical. In such scenario, RS reduces to MU–LP. In subfigure d of each figure, \(\theta _{1}=\frac {4\pi }{9} \text { and } \theta _{2}=\frac {8\pi }{9}\), the channels of user1 and user2, and the channels of user2 and user3 are sufficiently orthogonal while the channels of user1 and user3 are almost in opposite directions. In such circumstance, the WSRs of RS and MU–LP strategies overlap with the optimal WSR achieved by DPC.
Appendix 4
Overloaded threeuser deployment with perfect CSIT
(1) Two transmit antenna deployment
Figures 39, 40, 41, 42, and 43 show the results when γ_{1},γ_{2}, and u are varied as discussed in Appendix C.
RS exhibits a clear WSR gain over SC–SIC, SC–SIC per group, and MU–LP in all figures (Figs. 39, 40, 41, 42, and 43). Onelayer RS outperforms SC–SIC, SC–SIC per group, and MU–LP in most figures. It further shows that 1layer RS outperforms the joint switching between SC–SIC and SC–SIC per group in most user deployments while the complexity of 1layer RS is much reduced. In Figs. 39a–c and 40a–c, 1layer RS achieves the same WSR as RS. It implies that RS reduces to 1layer RS in these user deployments. Both of RS and 1layer RS achieve higher WSRs than all other strategies.
(2) Single transmit antenna deployment
Figures 44 and 45 show the average rate regions of different strategies over 10 random channel realizations when \(\sigma _{1}^{2}=\sigma _{2}^{2}=\sigma _{3}^{2}=1\) and \(\sigma _{1}^{2}=\sigma _{2}^{2}=1, \sigma _{3}^{2}=0.3\), respectively. We further show that 1layer RS is an attractive alternative to SC–SIC.
Appendix 5
Underloaded threeuser deployment with imperfect CSIT
We consider the imperfect CSIT scenarios. The channel model in the twouser deployment with imperfect CSIT is extended here. The estimated channel of user1, user2, and user3 are initialized using Eq. (38). For the given channel estimate at the BS, the channel realization is \(\mathbf {h}_{k}=\widehat {\mathbf {h}}_{k}+\widetilde {\mathbf {h}}_{k},\forall k\in \{1,2,3\}\), where \(\widetilde {\mathbf {h}}_{k}\) is the estimated error of userk. \(\widetilde {\mathbf {h}}_{k}\) has i.i.d. complex Gaussian entries drawn from \(\mathcal {CN}\left (0,\sigma _{e,k}^{2}\right)\). The error covariance of user1, user2, and user3 are \(\sigma _{e,1}^{2}=P_{t}^{0.6}\), \(\sigma _{e,2}^{2}=\gamma _{1} P_{t}^{0.6}\), and \(\sigma _{e,3}^{2}=\gamma _{2} P_{t}^{0.6}\), respectively. The precoders are initialized and designed using the estimated channels \(\widehat {\mathbf {h}}_{1},\widehat {\mathbf {h}}_{2}, \text {and} \widehat {\mathbf {h}}_{3}\) and the same methods as stated in perfect CSIT scenarios. One thousand different channel error samples are generated for each user. Each point in the rate region is the average rate over the generated 1000 channels.
Comparing with the simulation results in perfect CSIT, the WSR gap between RS and MU–LP increases in imperfect CSIT. In contrast, the WSR gap between RS and 1layer RS decreases in imperfect CSIT. Onelayer RS achieves equal or better WSRs than SC–SIC, SC–SIC per group, and MU–LP in all figures (Figs. 46, 47, 48, 49, 50, and 51). As mentioned earlier, all forms of RS are suited to any network load and channel circumstances of users. Moreover, all forms of RS are robust to imperfect CSIT.
Appendix 6
Overloaded threeuser deployment with imperfect CSIT
We further investigate the overloaded threeuser deployment with imperfect CSIT. The BS is equipped with two antennas (N_{ t }=2). Figures 52, 53, 54, 55, 56, and 57 show the simulation results when the rate threshold is r_{th}=[0.02,0.08,0.19,0.3,0.4,0.4,0.4] bit/s/Hz. Comparing Fig. 52 with Fig. 39, the WSR gaps between RS and SC–SIC per group, RS and MU–LP are increasing dramatically while the WSR gap between RS and SC–SIC is decreasing. The intergroup interference of SC–SIC per group becomes difficult to coordinate due to the limited number of transmit antenna and imperfect CSIT. RS is able to overcome the limitations of SC–SIC per group and MU–LP by dynamically determining the level of multiuser interference to decode and treat as noise.
Appendix 7
Overloaded fouruser deployment with perfect CSIT
Figures 58 and 59 show the results when γ_{1}=1. Comparing with SC–SIC per group, 1layer RS per group always achieves equal or better WSR. Onelayer RS per group is more general than SC–SIC per group. It enables the capability of partially decoding interference and partially treating interference as noise in each user group. When there is a sufficient channel gain difference between users within each group and a sufficient intergroup angle, the WSR of SC–SIC per group becomes closer to the WSR of RS comparing Figs. 59 and 19.
Appendix 8
Overloaded tenuser deployment with perfect CSIT
Figure 60 shows the simulation results when \(\sigma _{1}^{2}=\sigma _{2}^{2}=\ldots =\sigma _{10}^{2}=1\), r_{th}=[0,0.001,0.004,0.01,0.03,0.06,0.1] bit/s/Hz. Comparing with Fig. 21, the rate threshold of each SNR is reduced in Fig. 60. The WSR achieved by MU–LP is approaching RS when SNR is 0 or 5 dB in Fig. 60. This is because the rate threshold is set to 0 when SNR is 0 dB or 5 dB. When the rate threshold is 0, MU–LP could deliver two interference free streams since there are two transmit antennas. It achieves a DoF of 2 while SC–SIC is always limited by a DoF of 1.
Figure 61 shows the simulation results when \(\sigma _{1}^{2}=1, \sigma _{2}^{2}=0.9, \ldots \sigma _{10}^{2}=0.1\). The rate threshold is the same as in Fig. 60. In the extremely overloaded scenario, the WSR gap between RS and SC–SIC is still large despite the diversity in channel strengths. Here again, SC–SIC makes an inefficient use of the transmit antennas and achieves a DoF of 1. In contrast, 1layer RS, with a low scheduler and receiver complexity, achieves a good performance in all network loads.
Notes
In the sequel, powerdomain NOMA will be referred simply by NOMA.
Recall that SU–MIMO in LTE Rel. 8 was designed with minimum mean square error–SIC (MMSE–SIC) in mind [45].
The DoF characterizes the number of interferencefree streams that can be transmitted or equivalently the prelog factor of the rate at high SNR.
This can be easily seen since, for the receiver forced to decode all streams, the model reduces to a multiple access channel (MAC) with a singleantenna receiver, which has a sumDoF of 1. This was discussed in length in [34].
Recall that this spatial multiplexing gain is the main driver for using multiple antennas in a multiuser setup and the introduction of MU–MIMO in 4G [18].
“Common” is sometimes referred to as “public.”
Note that in the specific case where we have finite precision CSIT, the sum DoF collapses to 1 [26], and RS, SC–SIC,and TDMA all achieve the same optimal DoF.
It is worth noting that RateSplitting Multiple Access (RSMA) also exists in the uplink for the SISO Multiple Access Channel [46]. Though they share the same name and the splitting of the messages, they have different motivations and structures.
As already explained in [12], RS can also be seen as a form of nonorthogonal multiuser transmission. Indeed, in its simplest form, the common message in RS can be seen as a nonorthogonal layer added onto the private layers.
This benefit of RS was briefly pointed out in [39].
Note that OMA (singleuser beamforming) is a subset of MU–LP and is obtained by allocating power exclusively to s_{1} or s_{2}.
Note that for a given θ, the users’ direction of arrival (DoA) are the same for N_{ t }=2 and N_{ t }=4 scenarios while the channel angle is more orthogonal when N_{ t }=4 comparing with that when N_{ t }=2.
The readers are referred to [28] for a rigorous discussion about the notion of average rate.
Abbreviations
 AO:

Alternating optimization
 AWGN:

Additive white gaussian noise
 AWSR:

Averaged weighted sum rate
 CoMP:

Coordinated multipoint
 CSIR:

Channel state information at the receivers
 CSIT:

Channel state information at the transmitter
 DoA:

Direction of arrival
 DoF:

Degrees of freedom
 DPC:

Dirty paper coding
 FDMA:

Frequencydivision multiple access
 GDoF:

Generalized degrees of freedom
 HK:

Han and Kobayashi
 HRS:

Hierarchical rate splitting
 IC:

Interference channel
 IoT:

Internet of Things
 MISO:

Multipleinput singleoutput
 MRT:

Maximum ratio transmission
 MSE:

Mean square error
 MTC:

Machinetype communications
 MU–LP:

Multiuser linear precoding
 MU–MIMO:

Multiuser multipleinput multipleoutput
 MUST:

Multiuser superposition transmission
 NOMA:

Nonorthogonal multiple access
 OMA:

Orthogonal multiple access
 QCQP:

Quadratically constrained quadratic program
 QoS:

Quality of service
 RS:

Rate splitting
 RSMA:

Ratesplitting multiple access
 SAA:

Sample average approximated
 SC:

Superposition coding
 SC–SIC:

Superposition coding with successive interference cancellation
 SCMA:

Sparse code multiple access
 SDMA:

Spacedivision multiple access
 SIC:

Successive interference cancellation
 SISO:

Singleinput singleoutput
 SISO BC:

Singleinput singleoutput broadcast channel
 SNR:

Signaltonoise ratio
 SVD:

Singular value decomposition
 TDMA:

Timedivision multiple access
 WMMSE:

Weighted minimum mean square error
 WSR:

Weighted sum rate
 ZFBF:

Zeroforcing beamforming
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Acknowledgements
The authors are deeply indebted to Dr. Hamdi Joudeh for his helpful comments and suggestions.
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This work is partially supported by the UK Engineering and Physical Sciences Research Council (EPSRC) under grant EP/N015312/1.
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Authors’ contributions BC proposed the research idea. The coauthors discussed the model design and experiments together. YM performed the experiments. YM and BC cowrote the first draft of the manuscript. BC and VOKL gave advice on writing and revised the manuscript. All authors read and approved the final manuscript.
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Mao, Y., Clerckx, B. & Li, V.O. Ratesplitting multiple access for downlink communication systems: bridging, generalizing, and outperforming SDMA and NOMA. J Wireless Com Network 2018, 133 (2018). https://doi.org/10.1186/s1363801811047
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DOI: https://doi.org/10.1186/s1363801811047
Keywords
 RSMA
 NOMA
 SDMA
 MISO BC
 Linear precoding
 Rate region
 Weighted sum rate
 Rate splitting