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Is NOMA Efficient in Multi-Antenna Networks? A Critical Look at Next Generation Multiple Access Techniques

Bruno Clerckx, Yijie Mao, Robert Schober, Eduard Jorswieck, David J. Love, Jinhong Yuan, Lajos Hanzo, Geoffrey Ye Li, Erik G. Larsson, Giuseppe Caire

arXiv:2101.04802v1cs.ITeess.SP

TL;DR

The paper examines whether multi-antenna NOMA is efficient by comparing it with MU–LP and rate-splitting rather than primarily with OMA. It finds multiplexing-gain and rate losses, increased receiver complexity, and stronger overall trade-offs for NOMA than for RS in multi-antenna settings.

  • Problem

    Multi-antenna NOMA is often evaluated against OMA, while comparisons with MU–LP and other multi-antenna non-orthogonal baselines receive less emphasis.

  • Method

    The paper analyzes multiplexing gains under perfect and imperfect CSIT, then compares multi-antenna NOMA with conventional multiuser precoding and multi-antenna rate-splitting.

  • Results

    Multi-antenna NOMA loses multiplexing gain and high-SNR rate relative to conventional multiuser precoding while increasing receiver complexity; 1-layer RS achieves a better rate–SIC-complexity trade-off.

  • Takeaways & Limitations

    NOMA should not be applied blindly to multi-antenna networks, whereas rate-splitting is identified as a more powerful strategy that exploits multiplexing gain and SIC benefits.

  • Takeaways & Limitations

    The analysis and numerical evaluations are limited to sum and MMF multiplexing gain or rate metrics and the MISO broadcast channel.

Abstract

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In this paper, we take a critical and fresh look at the downlink multi-antenna NOMA literature. Instead of contrasting NOMA with OMA, we contrast NOMA with two other baselines. The first is conventional Multi-User Linear Precoding (MULP). The second is Rate-Splitting Multiple Access (RSMA) based on multi-antenna Rate-Splitting (RS) and SIC. We show that there is some confusion about the benefits of NOMA, and we dispel the associated misconceptions. First, we highlight why NOMA is inefficient in multi-antenna settings based on basic multiplexing gain analysis. We stress that the issue lies in how the NOMA literature has been hastily applied to multi-antenna setups, resulting in a misuse of spatial dimensions and therefore loss in multiplexing gains and rate. Second, we show that NOMA incurs a severe multiplexing gain loss despite an increased receiver complexity due to an inefficient use of SIC receivers. Third, we emphasize that much of the merits of NOMA are due to the constant comparison to OMA instead of comparing it to MULP and RS baselines. We then expose the pivotal design constraint that multi-antenna NOMA requires one user to fully decode the messages of the other users. This design constraint is responsible for the multiplexing gain erosion, rate loss, and inefficient use of SIC receivers in multi-antenna settings. Our results confirm that NOMA should not be applied blindly to multi-antenna settings, highlight the scenarios where MULP outperforms NOMA and vice versa, and demonstrate the inefficiency, performance loss and complexity disadvantages of NOMA compared to RS. The first takeaway message is that, while NOMA is not beneficial in most multi-antenna deployments. The second takeaway message is that other non-orthogonal transmission frameworks, such as RS, exist which fully exploit the multiplexing gain and the benefits of SIC to boost the rate in multi-antenna settings.

I. INTRODUCTION

The paper critically evaluates downlink multi-antenna NOMA against MU–LP and RS rather than OMA, finding that NOMA is generally inefficient because its SIC and decoding design waste spatial multiplexing gains. Analytical and numerical comparisons identify where NOMA can outperform MU–LP, while showing that RS offers better rate–complexity trade-offs.

  • Motivation and comparison: Multi-antenna NOMA is assessed against MU–LP and RS, exposing misconceptions that arise from the prevalent NOMA-versus-OMA comparison.The paper studies these alternatives across multiplexing gain, rate, and receiver-complexity dimensions.
  • Multiplexing-gain analysis: NOMA can gain or lose multiplexing gain relative to MU–LP, but always wastes multiplexing gain relative to RS across broad CSIT and loading regimes.The analysis covers perfect and imperfect CSIT, underloaded and overloaded systems, and both sum and max-min fair multiplexing gains.
  • Receiver complexity: More SIC operations in multi-antenna NOMA increase receiver complexity while reducing sum multiplexing gain and high-SNR sum-rate.MU–LP and RS can achieve higher multiplexing gains with lower receiver complexity and fewer SIC operations.
  • Design constraint: The central NOMA design constraint is forcing one user to fully decode other users’ messages, which can erode multiplexing gain and misuse SIC in multi-antenna networks.This approach is efficient for the single-antenna degraded broadcast channel but may be inefficient for multi-antenna networks.
  • Rate-splitting alternative: RS uses partial message decoding and one SIC layer to achieve better performance than MU–LP and NOMA with substantially lower receiver complexity than multi-antenna NOMA.The paper presents RS as avoiding user grouping and decoding-order optimization while benefiting from multi-antenna multiplexing gain.
  • Numerical evaluation: Numerical simulations confirm that NOMA may outperform MU–LP in some settings but underperform it in others despite higher receiver complexity, whereas RS and multi-antenna RS provide significant benefits.The simulations are designed to validate the predictive value of the multiplexing-gain analysis for sum-rate and max-min rate.

B. Definition of Multiplexing Gain

Multiplexing gain, or Degrees-of-Freedom, measures how effectively a strategy exploits spatial dimensions through high-SNR rate scaling. The paper uses user, sum, and max-min fair multiplexing gains to assess individual, aggregate, and symmetric performance.

  • Definitions: Multiplexing gain is the high-SNR pre-log factor describing how quickly a user’s rate increases with SNR.It can be interpreted as the number or fraction of interference-free streams simultaneously communicated to that user.
  • Definitions: The sum multiplexing gain is the high-SNR pre-log factor of the sum-rate and represents the total number of simultaneous interference-free data streams.A larger sum multiplexing gain means faster sum-rate growth with SNR.
  • Definitions: The MMF multiplexing gain is the maximum multiplexing gain simultaneously achievable by all users and reflects high-SNR symmetric-rate scaling.It captures the fairness-oriented rate growth common to every user.
  • Interpretation and scope: Multiplexing gain is an asymptotic high-SNR metric, so it does not precisely represent specific finite-SNR rates.Nevertheless, the paper notes that it can affect finite-SNR performance, provide theoretical insight, and guide strategy design.
  • Notation: The paper indexes multiplexing gain by communication strategy, including NOMA, MU–LP, Rate-Splitting, and the information-theoretic optimum.The notation uses j ∈ {N, M, R, ⋆}.

C. Discussions

In the two-user MISO setting, NOMA’s decoding structure limits its multiplexing gain, whereas MU–LP can exploit two spatial streams. Consequently, MU–LP strictly outperforms NOMA and OMA at high SNR, while NOMA’s rate is no higher than OMA’s for any SNR in this basic case.

  • NOMA multiplexing gain: 1 is the maximum sum multiplexing gain of two-user MISO NOMA, regardless of decoding order.The high-SNR NOMA sum-rate therefore scales at most as log2(P).
  • NOMA versus OMA: NOMA’s sum-rate is no higher than OMA’s for any SNR in the two-user MISO basic building block.This result motivates analyzing when, if ever, multi-antenna NOMA can outperform MU–LP despite its greater receiver complexity.
  • Fairness comparison: 1 is the two-user MISO NOMA MMF multiplexing gain, obtained by splitting its sum multiplexing gain equally between users.The corresponding MMF rate scales at most as 1/2 log2(P) at high SNR.
  • MU–LP comparison: 2 is the optimal sum multiplexing gain of the two-user MISO broadcast channel, achievable with conventional MU–LP.Zero-Forcing Beamforming provides two interference-free streams when the channel directions are not aligned.
  • Rate comparison: MU–LP achieves sum-rate scaling of 2 log2(P) and MMF-rate scaling of log2(P), strictly outperforming NOMA and OMA at high SNR.Each MU–LP user receives one full interference-free stream.
  • Architectural distinction: The NOMA and MU–LP transmit vectors may look similar, but their encoding, decoding, and rate expressions are different.NOMA requires one user to decode both streams, whereas MU–LP independently encodes and separately decodes the users’ streams.
  • Precoder implications: NOMA’s multiplexing-gain loss remains even with its best precoders because the analysis is based on an upper bound.More complicated precoders can improve rates but cannot improve the sum or MMF multiplexing gains.

III. K-USER MISO NOMA WITH PERFECT CSIT

This section defines grouped K-user MISO NOMA with perfect CSIT, where users decode messages within groups using SIC while treating inter-group interference as noise. Its multiplexing gains are constrained by the number of groups and by the requirement that strong users decode all messages in their groups.

  • System model: K users are partitioned into G groups of g=K/G users, with strong users decoding other users’ messages within each group.The architecture allows 1≤G<K; G=K corresponds to MU–LP rather than NOMA.
  • System model: The transmitter independently encodes K messages into linearly precoded streams under the total power constraint ΣP_k≤P.Some within-group messages use codebooks shared with decoding users so that they can be decoded and cancelled.
  • System model: Within each group, user-j decodes messages in descending user-index order, while treating interference from other groups as noise.Each message must be decodable by all earlier users required to decode it, constraining its rate accordingly.
  • System model: Additional constraints forcing one common precoder for all users in a group would further reduce the optimization space and rate performance.The paper notes that the channel-matrix rank is assumed to be min{M,K}.
  • Multiplexing gains: The sum multiplexing gain of perfect-CSIT MISO NOMA is min(M,G), because each group contributes at most one interference-free stream.Achievability uses ZFBF to transmit min(M,G) streams to strong users.
  • Multiplexing gains: The MMF multiplexing gain is 1/g when M≥K−g+1, but collapses to 0 when M<K−g+1 under one-shot transmission.The 1/g result follows from splitting each group’s multiplexing gain equally among its g users; insufficient antennas prevent eliminating inter-group interference.

IV. K-USER MISO NOMA WITH IMPERFECT CSIT

This section extends MISO NOMA multiplexing-gain analysis to imperfect CSIT using an ergodic channel model parameterized by CSIT quality α. The resulting gains depend differently on α for multiple groups and a single group.

  • CSIT model: Imperfect-CSIT rates and multiplexing gains are analyzed ergodically over stationary, ergodic joint channel states and their transmitter estimates.The channel estimate and error are represented through H=Ĥ+H̃, with the joint distribution known to the transmitter.
  • CSIT model: CSIT error variance follows σ_e^2=P^-α, where α=0 denotes non-improving CSIT and α=1 is perfect CSIT in the multiplexing-gain sense.Values 0<α<1 represent partial CSIT, with larger α indicating faster error decay as SNR increases.
  • Multiplexing gains: For G>1, NOMA’s sum and MMF multiplexing gains decay as α decreases, whereas for G=1 they are unaffected by α.The section identifies distinct sensitivity to CSIT quality between multi-group and single-group NOMA.
  • Multiplexing gains: The sum multiplexing gain of MISO NOMA is d_s^(N)=max(1,min(M,G)α).The achievable construction uses ZFBF for min(M,G) streams at power level P^α/min(M,G), or one stream when that quantity is below one.
  • Multiplexing gains: At α=1, the imperfect-CSIT sum-gain result reduces to the perfect-CSIT result.Thus the perfect-CSIT expression is recovered from the generalized CSIT model.

V. BASELINE SCHEME I:

This section introduces MU–LP as the first baseline for multi-antenna NOMA. Users independently decode their intended messages while treating residual interference as noise, and ZFBF determines its multiplexing gains.

  • MU–LP architecture: MU–LP independently encodes each user’s message, linearly precodes the resulting streams, and has each user treat other-user interference as noise.The receiver directly decodes only its intended message.
  • Perfect CSIT: With perfect CSIT, MU–LP achieves sum multiplexing gain min(M,K) using min(M,K) interference-free ZFBF streams.This is also the information-theoretic optimal sum multiplexing gain of the K-user MISO BC.
  • Perfect CSIT: With perfect CSIT, MU–LP’s MMF multiplexing gain is positive when M≥K and collapses to zero when M<K.ZFBF fully eliminates interference only when the transmitter has at least K antennas.
  • Imperfect CSIT: When M<K, MU–LP experiences rate saturation at high SNR in the MMF setting.The corresponding MMF multiplexing gain is zero in this antenna-limited regime.

C. Multiplexing Gains with Imperfect CSIT

This section describes one-layer multi-antenna RS as a second baseline, splitting each message into common and private parts. A common stream is decoded by all users before private-stream decoding, while the scheme retains MU–LP’s sum multiplexing gain under perfect CSIT.

  • Baseline motivation: The section positions multi-antenna RS and RSMA as non-orthogonal transmission baselines for the multi-antenna broadcast channel.The approach extends rate-splitting to multi-antenna broadcast-channel transmission.
  • RS architecture: One-layer RS splits each message into common and private parts, combines all common parts into one common stream, and sends private streams separately.The common stream is decoded by every user using a shared codebook.
  • RS architecture: Each user first decodes and removes the common stream using SIC, then decodes its private stream while treating remaining private interference as noise.The user reconstructs its original message by combining its decoded common and private parts.
  • RS architecture: One-layer RS uses a single common stream and one SIC layer per user.This architecture is explicitly termed 1-layer RS.
  • Multiplexing gains: With perfect CSIT, one-layer RS achieves sum multiplexing gain min(M,K).MU–LP is a subscheme of one-layer RS, so ZFBF private precoding and zero common-stream power achieve this gain at high SNR.

C. Multiplexing Gains with Imperfect CSIT

Under imperfect CSIT, the paper derives sum and max-min fair multiplexing gains for NOMA, MU–LP, and RS across general multi-antenna regimes. NOMA can match or lose to MU–LP, but always wastes multiplexing gain relative to RS.

  • 1-layer RS achieves the optimal sum multiplexing gain in underloaded multi-antenna broadcast channels with imperfect CSIT.
  • MISO NOMA never achieves a higher sum multiplexing gain than MU–LP, although it can match MU–LP in low-antenna configurations.
  • For perfect CSIT with M > G, NOMA’s high-SNR sum-rate slope is strictly lower than MU–LP’s.
  • MISO NOMA can have either higher or lower MMF multiplexing gain than MU–LP, depending on M, K, G, and CSIT quality α.

B. NOMA vs. Baseline II (RS)

The RS comparison shows that 1-layer RS matches or exceeds the multiplexing gains of every considered MISO NOMA configuration while requiring only one SIC layer. Thus, NOMA brings only multiplexing loss and/or additional complexity relative to RS.

  • 1-layer RS achieves the same or higher sum and MMF multiplexing gains than the best MISO NOMA scheme for all M, K, and α.
  • 1-layer RS uses one SIC layer per receiver, whereas MISO NOMA can require more SIC operations.
  • MISO NOMA never achieves a higher sum multiplexing gain than 1-layer RS.
  • Neither single-group nor multi-group MISO NOMA achieves an MMF multiplexing gain larger than 1-layer RS.

C. Misconceptions of Multi-Antenna NOMA

The paper identifies misconceptions about multi-antenna NOMA involving spatial-dimension use, SIC complexity, comparisons with OMA, and overloaded operation. Its analyses show that NOMA often sacrifices multiplexing gain despite greater receiver complexity.

  • Spatial dimensions: NOMA with G = 1 achieves sum multiplexing gain 1 regardless of M, wasting the transmit antenna array.
  • Spatial dimensions: With perfect CSIT, MU–LP and 1-layer RS achieve min(M, K), whereas NOMA with G = K/2 achieves only min(M, K/2).
  • Overloaded operation: In the overloaded regime, NOMA may outperform MU–LP on MMF multiplexing gain, but its sum multiplexing gain still erodes in most scenarios.
  • RS comparison: 1-layer RS guarantees optimal sum multiplexing gain, enhances MMF performance, reduces receiver complexity versus NOMA, and is more robust to CSIT inaccuracy.
  • SIC complexity: In the two-user perfect-CSIT MISO broadcast channel, NOMA requires one SIC layer but provides only half MU–LP’s sum and MMF multiplexing gains.
  • SIC complexity: For fixed precoders, NOMA can increase user-1’s rate but decreases or preserves user-2’s rate relative to MU–LP.

E. Shortcomings of Multi-Antenna NOMA

The paper attributes multi-antenna NOMA’s shortcomings to forcing one user to fully decode all co-scheduled users’ messages. RS instead preserves MU–LP’s spatial benefits while using SIC selectively.

  • Design constraint: A strong NOMA user acts as a single-antenna effective MAC receiver, whose sum multiplexing gain is limited to one.
  • Design constraint: Forcing one user to fully decode all streams in a group causes multiplexing losses relative to MU–LP and 1-layer RS.
  • RS relationship: In the two-user case, 1-layer RS contains MU–LP, NOMA, and multicasting as particular instances.
  • RS relationship: 1-layer RS converges to MU–LP when common-stream power is reduced to zero, whereas multi-user NOMA cannot reduce to full K-user MU–LP.
  • RS design: RS adjusts the common-stream contribution to the interference that receivers can cancel, covering a wider strategy set with multiplexing and complexity benefits.
  • Receiver design: Joint decoding instead of SIC does not improve multiplexing gain because the strong user remains an effective single-antenna MAC receiver.

A. Perfect CSIT

Under perfect CSIT, multi-antenna NOMA can lose multiplexing gain and rate despite higher receiver complexity. Its performance depends on grouping and channel disparities, while RS and MU–LP provide stronger baselines.

  • Sum-rate performance: At high SNR, the simulated multiplexing gains match the theoretical sum multiplexing gains for the evaluated strategies.The comparison considers K = 6 users, M = 3 or M = 6 antennas, and equal or randomly varied channel variances.
  • Sum-rate performance: MISO NOMA (G = 1) and OMA have sum multiplexing gain 1, while MISO NOMA (G = 3) remains at 3.For M = 3 and K = 6, the sum multiplexing gains follow these strategy-specific limits across the evaluated subfigures.
  • Sum-rate performance: MISO NOMA (G = 1) has the worst sum-rate performance and incurs significant rate loss at medium and high SNRs.Its inefficient use of multiple antennas is reflected in the reduced sum multiplexing gain.
  • Complexity: MISO NOMA has the highest complexity, requiring joint optimization of grouping, decoding order, and precoders plus multiple SIC layers per receiver.The number of SIC layers increases with the number of users K.
  • Max-min fairness: In overloaded regimes, MISO NOMA has poor max-min fairness performance and performs worse than 1-layer RS.The low MMF multiplexing gains of MISO NOMA translate into poor MMF rates in the evaluated settings.
  • Scope: The simulations validate the theoretical multiplexing-gain analysis, while larger antenna regimes are not directly evaluated here.The analysis applies to any antenna configuration, but the simulations use small MIMO systems.

B. Imperfect CSIT

With imperfect CSIT, the reported rate and multiplexing-gain results continue to favor 1-layer RS over multi-antenna NOMA. NOMA remains inefficient for both sum-rate and fairness, while RS is more robust to CSIT inaccuracy.

  • Sum-rate performance: For α = 0.5 and M = 3/6, 1-layer RS achieves sum multiplexing gains 2/3.5, versus 1.5/3 for MU–LP and 1/1 for MISO NOMA (G = 1) and OMA.These imperfect-CSIT results correspond to the evaluated sum-rate comparisons.
  • Sum-rate performance: MISO NOMA (G = 1) has the worst ergodic sum-rate performance despite the highest receiver complexity, including in its preferred overloaded setting.It performs worse than MU–LP regardless of whether CSIT is perfect or imperfect.
  • Sum-rate performance: 1-layer RS achieves explicit sum multiplexing gains and sum-rate improvement over the other evaluated strategies.Its advantage is reported under imperfect CSIT and across the corresponding perfect-CSIT comparisons.
  • Max-min fairness: 1-layer RS achieves significantly higher MMF multiplexing gains and much better MMF ergodic rates than MISO NOMA.This holds in both perfect and imperfect CSIT settings, so MISO NOMA does not improve user fairness in the evaluated scenarios.
  • CSIT robustness: Partial interference decoding and treating the remainder as noise make 1-layer RS more robust to CSIT inaccuracy.The performance gaps between RS and MU–LP or multi-antenna NOMA generally widen under imperfect CSIT.
  • Complexity: 1-layer RS requires no user-grouping or decoding-order optimization and only one SIC layer per receiver.Its rate gains over MISO NOMA come with reduced transmitter and receiver complexity.
  • Scope: The evaluations are limited to 1-layer RS, while multi-layer RS may provide further rate enhancements.The paper also limits multiplexing-gain and numerical evaluations to sum-rate/MMF metrics and the MISO BC.

APPENDIX A PROOF OF PROPOSITION 4

The proposition proof establishes achievable MMF multiplexing gains under imperfect CSIT through group-wise precoding and power allocation. It also shows collapse to zero in one antenna-limited regime.

  • Upper bounds: For G > 1 and M ≥ K − g + 1, equal group and user splitting yields an upper bound of α/g on MMF multiplexing gain.The total gain Gα is divided equally among G groups and then among g users per group.
  • Achievability: ZFBF precoding and power allocation achieve MMF multiplexing gain α/g for each user in the grouped construction.The resulting user SINR scales as O(P^α/g).
  • Illustrative construction: For K = 4, G = 2, g = 2, and M ≥ 3, inter-group zero forcing and power scaling O(P^(1−α/2)) support the construction.The example assigns lower-order power to users 1 and 3 and O(P) power to users 2 and 4.
  • Illustrative construction: All four streams in the example have SINR scaling P^α/2, achieving MMF multiplexing gain α/2.SIC first decodes and cancels s2 before decoding s1 in group 1; analogous reasoning applies to group 2.
  • Antenna-limited regime: When G > 1 and M < K − g + 1, the MMF multiplexing gain collapses to 0 under imperfect CSIT.The proof carries over the collapse already established for perfect CSIT.
  • Single-group case: For G = 1, the single-group sum multiplexing gain of 1 is divided equally among K users, yielding an upper bound of 1/K.Power allocation O(P^k/K) achieves the corresponding user SINR scaling O(P^1/K).

APPENDIX B WMMSE OPTIMIZATION FRAMWORK

The WMMSE appendix reformulates NOMA sum-rate and max-min problems using equalizers and weights, then solves the resulting block-wise convex formulation by alternating optimization.

  • Signal estimation: Each receiver uses an equalizer to estimate the streams it decodes after previously decoded streams are removed.The estimate of stream s_k at user j is formed as ĝ_j,k = g_j,k y_j,k.
  • Rate transformation: The MSE and MMSE equalizer yield the rate expression for user j decoding user k’s message.The rate is equivalently written using the minimum MMSE as R_j,k = −log2(ε_j,k^MMSE).
  • Rate transformation: The WMSE formulation introduces positive weights and establishes a rate–WMMSE relationship through joint minimization over equalizers and weights.The relationship is stated as min_u_j,k,g_j,k ξ_j,k = 1 − R_j,k.
  • Optimization formulation: The sum-rate WMMSE problem is formulated over the precoding matrix, equalizers, and weights for all decoded streams.The corresponding equalizer and weight sets include every user-stream decoding pair within each group.
  • Optimization formulation: Substituting MMSE equalizers and weights into the transformed problem gives KKT-consistent solutions for the original sum-rate problem.The transformed formulation remains non-convex but is block-wise convex in the precoder and equalizer-weight blocks.
  • Alternating optimization: Alternating optimization updates equalizers and weights first, then solves the convex precoder subproblem with fixed equalizers and weights.The precoder update can be solved using interior-point methods.
  • Max-min optimization: The same WMMSE transformation and alternating-optimization procedure is applied to the max-min rate problem.Replacing the sum-rate subproblem with the max-min formulation yields the corresponding AO algorithm.
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