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Rate-Splitting Multiple Access for Downlink Multiuser MIMO: Precoder Optimization and PHY-Layer Design

Anup Mishra, Yijie Mao, Onur Dizdar, Bruno Clerckx

arXiv:2105.07362v1cs.ITeess.SPmath.IT

TL;DR

The study addresses limited evidence on RS with multi-antenna receivers, especially the role of multiple common streams and rate regions under perfect and imperfect CSIT. It develops an RS precoder-optimization framework and reports that optimized RS ergodic-rate regions outperform MU-MIMO and MIMO NOMA in both CSIT settings.

  • Problem

    In MIMO broadcast channels, prior RS studies mostly use single-antenna receivers, leaving multiple-common-stream benefits and the achievable rate region under perfect or imperfect CSIT unresolved.

    Source: §I · p. 4

  • Method

    The paper develops an RS framework for MIMO broadcast channels with arbitrary common streams and uses vectorization- and WMMSE-based precoder optimization for perfect and imperfect CSIT.

    Source: A. Motivations and Contributions · p. 5, §VII · p. 24

  • Results

    Optimized RS ergodic-rate regions always outperform MU-MIMO and MIMO NOMA under perfect and imperfect CSIT, and under perfect CSIT come closer to DPC capacity.

    Source: A. Motivations and Contributions · p. 5

  • Takeaways & Limitations

    Link-level simulations show RS's finite-constellation, finite-length-code PHY architecture achieves significant throughput gains over MU-MIMO and MIMO NOMA.

    Source: A. Motivations and Contributions · p. 5

Abstract

from arXiv · show

Rate-Splitting Multiple Access (RSMA) has recently appeared as a powerful and robust multiple access and interference management strategy for downlink Multi-user (MU) multi-antenna communications. In this work, we study the precoder design problem for RSMA scheme in downlink MU systems with both perfect and imperfect Channel State Information at the Transmitter (CSIT) and assess the role and benefits of transmitting multiple common streams. Unlike existing works which have considered single-antenna receivers (Multiple-Input Single-Output--MISO), we propose and extend the RSMA framework for multi-antenna receivers (Multiple-Input Multiple-Output--MIMO) and formulate the precoder optimization problem with the aim of maximizing the Weighted Ergodic Sum-Rate (WESR). Precoder optimization is solved using Sample Average Approximation (SAA) together with the proposed vectorization and Weighted Minimum Mean Square Error (WMMSE) based approach. Achievable sum-Degree of Freedom (DoF) of RSMA is derived for the proposed framework as an increasing function of the number of transmitted common and private streams, which is further validated by the Ergodic Sum Rate (ESR) performance using Monte Carlo simulations. Conventional MU-MIMO based on linear precoders and Non-Orthogonal Multiple Access (NOMA) schemes are considered as baselines. Numerical results show that with imperfect CSIT, the sum-DoF and ESR performance of RSMA is superior than that of the two baselines, and is increasing with the number of transmitted common streams. Moreover, by better managing the interference, RSMA not only has significant ESR gains over baseline schemes but is more robust to CSIT inaccuracies, network loads and user deployments.

I. INTRODUCTION

The paper extends rate-splitting to multiuser MIMO with multi-antenna receivers, addressing gaps in precoder optimization and the role of multiple common streams. It reports rate and sum-DoF advantages over MU-MIMO and MIMO NOMA, including under imperfect CSIT.

I. INTRODUCTION

  • The introduction identifies sensitivity to inaccurate CSIT as a key limitation of MU-MIMO and notes that multi-antenna RS studies had not resolved multiple-common-stream benefits or the achievable rate region.The system model serves K users, each with Q receive antennas, from a base station with M transmit antennas.Source: §I · pp. 2, 4, §II · p. 6

A. Motivations and Contributions

  • The proposed symmetric MIMO BC framework allows an arbitrary number of common streams from 1 through min(M, Q), with flexible common and private stream counts.The work also derives achievable sum-DoF for this framework at high SNR with imperfect CSIT.Source: A. Motivations and Contributions · p. 4
  • With imperfect CSIT, RS sum-DoF exceeds MU-MIMO and MIMO NOMA even with one common stream, and increases as more common streams are transmitted.Monte Carlo ESR results further test the sum-DoF findings.Source: A. Motivations and Contributions · p. 4
  • The paper proposes vectorization and WMMSE-based precoder optimization for RS in MIMO BC with perfect and imperfect CSIT, maximizing weighted sum-rate subject to a transmit-power constraint.The framework addresses optimization challenges introduced by matrix variables.Source: A. Motivations and Contributions · p. 5
  • Optimized RS ergodic-rate regions outperform MU-MIMO and MIMO NOMA under perfect and imperfect CSIT, and approach DPC more closely under perfect CSIT.The paper presents this as the first demonstration of these RS benefits in MIMO settings.Source: A. Motivations and Contributions · p. 5
  • Link-level simulations report significant RS throughput gains over MU-MIMO and MIMO NOMA in MIMO BC.The PHY architecture uses finite-constellation modulation, finite-length polar codes, and adaptive modulation and coding.Source: A. Motivations and Contributions · p. 5

A. Imperfect CSIT

The system models imperfect CSIT as an estimate plus channel error, then extends rate splitting to MIMO receivers using common and private stream vectors. Users decode and remove the common streams before decoding their own private streams, while rates account for the difference between estimated and actual channels.

A. Imperfect CSIT

  • The CSIT model writes the channel as an estimate plus an error, with error quality scaling with SNR; α=0 leaves quality unchanged with SNR, while α=1 corresponds to perfect CSIT from a DoF perspective.The channel-estimation error is modeled as i.i.d. zero-mean complex Gaussian, and its conditional distribution is assumed known at the BS.Source: A. Imperfect CSIT · p. 6

1) Transmitter:

  • Each user's message is split into common and private parts; common parts are encoded together into Qc streams decoded by all users, while private streams are intended for their corresponding users.The common-stream count satisfies Qc∈{1,…,min(M,Q)}.Source: 1) Transmitter · p. 7

2) MMSE Receiver and Rates:

  • Using SIC, each user first decodes and removes the common stream vector, then decodes its own private stream while treating other users' private streams as noise.Users reconstruct their messages by combining the common part intended for them with their decoded private part.Source: 2) MMSE Receiver and Rates · p. 7
  • Because the BS precodes from the channel estimate while users decode using the actual channel, imperfect CSIT can overestimate instantaneous rates; ergodic rates instead capture long-term performance across fading states.The common rate is shared among users, and each user's total ergodic rate combines its private rate with its allocated common-rate share.Source: 2) MMSE Receiver and Rates · pp. 7–9

C. Two User Example

The two-user example splits each user's messages into common and private sub-messages, combines common parts into a shared stream vector, and superposes the streams. Turning off common streams recovers MU–MIMO, while assigning one user's message to the common stream recovers a MIMO NOMA decoding order.

C. Two User Example

  • The transmitter combines the users' common sub-messages into a common stream vector and independently encodes their private sub-messages into user-specific stream vectors.The transmitted signal is the precoded superposition of the common and private stream vectors.Source: C. Two User Example · p. 9
  • With M=4, Q=2, Q1=Q2=2, and Qc=2 in the simulations, each user decodes the common stream first and then its own private stream using SIC.The other user's private stream is treated as noise during private-stream decoding.Source: C. Two User Example · p. 9
  • Turning off common-stream power reduces the two-user RS case to MU–MIMO; encoding user 2's message in the common stream and turning off its private stream gives MIMO NOMA with order 1→2.Table I maps messages to streams across schemes.Source: C. Two User Example · p. 9, Table I · p. 9

III. PROBLEM FORMULATION

The section formulates robust precoder design around weighted ergodic sum-rate and analyzes how common streams affect sum-DoF under imperfect CSIT. For α∈(0,1) and Qc=min(M,Q), RS achieves strictly greater sum-DoF than MU–MIMO and MIMO NOMA.

III. PROBLEM FORMULATION

  • The precoder objective is to maximize WESR, which captures long-term weighted user performance, rather than instantaneous WSR based on an imperfect channel estimate.The instantaneous approach may fail to cope with multiuser interference and can lead to undecodable transmission rates.Source: §III · p. 10
  • For a given channel estimate, average rate is expected performance over CSIT error; the BS uses it to formulate the weighted average sum-rate problem for imperfect instantaneous CSIT.The common-rate shares sum to the average common rate, which cannot exceed the minimum users' average common-stream rate.Source: §III · p. 10
  • The law of large numbers approximates each stream's ergodic rate by averaging its average rate across channel states, enabling decomposition into per-estimate WASR problems.For tractability, the long-term power constraint is replaced with short-term power constraints.Source: §III · p. 10

A. Common message and DoF Analysis

  • RS scales private-stream power so interference remains at noise level, preserving private-stream DoF Qpα while allocating remaining power to common streams.The common streams contribute Qc(1−α) DoF.Source: A. Common message and DoF Analysis · p. 12
  • MU–MIMO can reach sum-DoF Qp at α=1, but its sum-DoF decreases as CSIT quality falls and may favor single-user transmission when quality is too poor for MU transmission.MIMO NOMA is limited to a maximum sum-DoF of min(M,Q).Source: A. Common message and DoF Analysis · pp. 11–12
  • For α∈(0,1) and Qc=min(M,Q), RS sum-DoF is strictly greater than that of MU–MIMO and MIMO NOMA.The common streams contribute a DoF gain of Qc(1−α).Source: A. Common message and DoF Analysis · p. 12
  • Although optimization does not improve achievable sum-DoF, RS sum-rate is at least MU–MIMO's across the SNR range because MU–MIMO is obtained by switching off common streams.The passage states this rate inequality for the entire range of SNR.Source: A. Common message and DoF Analysis · p. 12

IV. OPTIMIZATION FRAMEWORK

The optimization framework approximates the stochastic WASR problem with SAA, transforms it into a WAMMSE problem, and uses vectorization and alternating optimization to obtain precoders. As the sample count grows, the SAA optimum converges to the stochastic problem's solution.

IV. OPTIMIZATION FRAMEWORK

  • The framework uses SAA to obtain a deterministic approximation, transforms WASR into a part-wise convex WAMMSE problem, and vectorizes matrix variables before alternating optimization obtains precoders.The resulting precoders give the optimized rate for a specified user-weight vector.Source: §IV · p. 12

A. Sample Average Approximation

  • As the number of channel samples N tends to infinity, the optimum SAA solutions converge to the solution of the stochastic problem.The convergence argument uses bounded, continuous, and differentiable rate functions over the feasible precoder set.Source: A. Sample Average Approximation · p. 13

B. WASR →WAMMSE

  • The sample-averaged WASR problem is transformed into an equivalent deterministic WAMMSE optimization using the rate-WMMSE relationship.At MMSE equalizers and weights, the WAMMSE problem reduces to the sample-averaged problem; its stationary points satisfy the corresponding KKT conditions.Source: B. WASR →WAMMSE · pp. 13–14
  • The WAMMSE solution converges to a solution of the WASR problem as N→∞.The stated equivalence links KKT points of the WAMMSE and sample-averaged problems.Source: B. WASR →WAMMSE · p. 14

C. Vectorization and Alternate Optimization

The joint precoder and message-split optimization alternates between equalizer/weight updates and precoder/message-split updates. Vectorization makes the matrix-valued objective tractable for the precoder step.

C. Vectorization and Alternate Optimization

  • The non-convex joint problem is handled by alternating between equalizer and weight updates, then optimizing precoders and message split with those variables fixed.Each block is convex when the other variables are fixed.Source: C. Vectorization and Alternate… · p. 15
  • Vectorization reformulates the common-stream AWMSE trace term into a tractable objective for the precoder update.The trace term is identified as an obstacle when weights are fixed.Source: C. Vectorization and Alternate… · p. 15
  • The same simplification is applied to private-stream AWMSEs, whose sample-average functions are then calculated for the optimization.The resulting sample-average functions are used in the alternating procedure.Source: C. Vectorization and Alternate… · p. 15

1) STEP 1:

The design alternates between receiver-variable and precoder updates, then specifies a practical coded transceiver for MIMO RS. At the receiver, interference nulling and cancellation support stream-specific rate selection, with a complexity trade-off against simpler decoding.

2) STEP 2:

  • The precoder update solves a convex QCQP, and alternating updates converge to a KKT solution of the sampled WASR problem.As the number of channel samples grows, the algorithm converges to a KKT solution of the WASR problem.Source: 2) STEP 2 · p. 16

V. PHY-LAYER DESIGN FOR MIMO CHANNELS

  • The transmitter uses 4-, 16-, 64- and 256-QAM, finite-length polar coding, and adaptive modulation and coding.The architecture generalizes an RS design for MISO channels.Source: §V · p. 16
  • Rather than jointly decoding all common and private streams, the receiver uses interference nulling and cancellation to select a separate rate for each stream.Joint decoding is the assumption underlying the rate expressions; it can be more complex than point-to-point decoding, especially as the number of streams increases.Source: §V · p. 17
  • The proposed receiver has higher complexity than a single-interference-cancellation design, which may be more practical but is expected to lose performance.The stated aim of the more complex architecture is to verify the paper’s theoretical foundations.Source: §V · p. 19

VI. NUMERICAL RESULTS

The numerical evaluation compares RS with DPC, MU-MIMO, and MIMO NOMA using perfect and imperfect CSIT settings. It uses sample-average approximations and specified baseline implementations and initialization procedures.

VI. NUMERICAL RESULTS

  • The evaluation compares RS with DPC, MU-MIMO, and MIMO NOMA; DPC is included only with perfect CSIT, and NOMA is restricted to two users.The NOMA restriction reflects the high complexity of jointly optimizing decoding order and precoders for more than two users.Source: §VI · p. 19
  • For each channel estimate, 1,000 channel-error samples approximate the average rate, while WESR is averaged over 100 channel realizations.In the perfect-CSIT case, the error-sample count is zero and the estimate equals the channel.Source: §VI · pp. 19–20
  • MU-MIMO and MIMO NOMA precoders are initialized with MRT, while RS uses MRT for private streams and an SVD-based common-stream initialization.Initial power allocation is uniform among users for the first two schemes; RS distributes power between common and private streams.Source: §VI · p. 20

A. WESR Performance: Rate-Region

In the rate-region comparisons, RS outperforms MU-MIMO and MIMO NOMA and approaches the DPC capacity region with perfect CSIT. With imperfect CSIT, RS shows explicit gains and greater robustness as channel quality and user deployment vary.

A. WESR Performance: Rate-Region

  • With a 10 dB additional path loss for user 2, NOMA exceeds MU-MIMO when user 2’s weight is comparable to or greater than user 1’s; MU-MIMO does better when user 1’s weight is much larger.The rate-region comparison uses two users and averages over 100 channel realizations.Source: A. WESR Performance: Rate-Region · p. 20
  • RS outperforms MU-MIMO and MIMO NOMA and achieves a rate region closer to capacity-achieving DPC in both perfect-CSIT subfigures.DPC has the highest rate region; the relative performance of MU-MIMO and NOMA varies with channel-strength disparity.Source: A. WESR Performance: Rate-Region · pp. 20–21, Figure 3 · p. 21
  • With imperfect CSIT, RS shows explicit performance gains over MU-MIMO and MIMO NOMA as CSIT quality degrades.The comparison also reports robustness to different user deployments, attributed to better interference management.Source: Figure 4 · p. 21, A. WESR Performance: Rate-Region · p. 21

B. Sum-DoF: Effect of Common Message

Across CSIT qualities and network loads, RS with multiple common streams achieves higher ESR than the compared MU–MIMO and MIMO NOMA baselines. Link-level simulations show throughput consistent with ESR, and the conclusion reports RS rate gains across loads, deployments, and CSIT quality.

1) CSIT Quality:

  • At high SNR, RS with Qc = 2 has higher ESR than RS with Qc = 1, and both outperform MU–MIMO and MIMO NOMA; at α = 0.6, their sum-DoF values are 3.08, 2.67, 2.10, and 1.93, respectively.The reported values are close to the corresponding theoretical values of 3.2, 2.8, 2.4, and 2.Source: 1) CSIT Quality · p. 22, Figure 5 · p. 23

2) Number of Users:

  • For underloaded K = 2, 3, and 4 configurations, RS with Qc = 2 has better ESR than RS with Qc = 1, which outperforms MU–MIMO and MIMO NOMA; RS also outperforms both baselines in overloaded and higher-dimensional cases.The higher-dimensional case uses M = 9 and Q = 3.Source: 2) Number of Users · p. 22, Figure 6 · p. 23

C. Link-Level Simulation Results

VII. CONCLUSION

  • RS achieves higher rate performance than MU–MIMO and MIMO NOMA across network loads, user deployments, and CSIT quality, with link-level throughput consistent with ESR.The conclusion also reports significant throughput gains in practical link-level simulations.Source: C. Link-Level Simulation Results · p. 23, §VII · p. 24

APPENDIX A ACHIEVABLE SUM-DOF

The appendix derives achievable sum-DoF for RS and MU–MIMO in a homogeneous MIMO broadcast channel with imperfect CSIT. Its RS construction assigns private streams to a subset of users and multicasts common streams to all users.

A. RS and MU-MIMO

  • The analysis assumes equal Q receive antennas per user and focuses on M > Q, with J ≤ K private-stream users chosen so M = JQ; common streams are multicast to all K users.Users outside J form J* and receive common streams only; private-stream power is P^α/J and common-stream power is P − P^α.Source: A. RS and MU-MIMO · p. 25, Appendix A · p. 25

1) User-k in J

  • Each user in J receives Qα private DoF, while the common-stream vector provides Qc(1 − α) DoF shared among all users.Users in J* have zero private DoF; the derivation uses the common and private rates under the stated high-SNR setup.Source: 2) User-k in J · p. 26
  • The appendix obtains MU–MIMO’s sum-DoF expression by switching off the common stream in the RS construction.The resulting MU–MIMO expression is referenced as equation (14).Source: 2) User-k in J · p. 26

B. MIMO NOMA

The MIMO NOMA analysis models ordered decoding with successive interference cancellation and derives a sum-rate bound through user 1. The stated high-SNR scaling uses the minimum of the transmit and receive antenna counts.

B. MIMO NOMA

  • With decoding order 1 → K, user 1 successively decodes the other users’ messages, while each later user performs fewer SIC layers.User K decodes its own message while treating the other users’ messages as noise.Source: B. MIMO NOMA · p. 27
  • Each user’s achievable rate is limited by the weakest decoding rate among the users required to decode that message.The rate of user j is bounded by the minimum decoding rate over users k ≤ j.Source: B. MIMO NOMA · p. 27
  • The high-SNR bound scales as min(M, Q) log2(Pt) + O(1), using user 1’s optimal covariance in a single-user setup.The cited expression identifies the minimum of the transmit and receive antenna counts as the pre-log factor.Source: B. MIMO NOMA · p. 27

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