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Rate-Splitting for Max-Min Fair Multigroup Multicast Beamforming in Overloaded Systems

Hamdi Joudeh, Bruno Clerckx

arXiv:1707.00304v1cs.IT

TL;DR

The paper addresses max-min-fair beamforming for overloaded multigroup multicasting, where inter-group interference cannot be fully neutralized. It analyzes classical and degraded strategies, then proposes rate-splitting with degraded and designated streams and solves its beamforming design using WMMSE. DoF analysis and simulations show that RS combines spatial multiplexing with non-saturating performance and achieves higher MMF rates in overloaded scenarios.

  • Problem

    The paper studies max-min fairness in overloaded multigroup multicasting, where insufficient transmit antennas prevent classical beamforming from neutralizing all inter-group interference.

  • Method

    The paper combines degraded beamforming with designated beamforming through rate-splitting, then solves the resulting MMF beamforming design using WMMSE.

  • Results

    RS combines partial spatial-multiplexing gains with non-saturating performance, and simulations show strictly higher MMF rates in partially and fully overloaded scenarios.

  • Takeaways & Limitations

    Rate-splitting provides a unified strategy that surpasses designated and degraded beamforming in some overloaded scenarios while retaining benefits of both.

Abstract

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In this paper, we consider the problem of achieving max-min fairness amongst multiple co-channel multicast groups through transmit beamforming. We explicitly focus on overloaded scenarios in which the number of transmitting antennas is insufficient to neutralize all inter-group interference. Such scenarios are becoming increasingly relevant in the light of growing low-latency content delivery demands, and also commonly appear in multibeam satellite systems. We derive performance limits of classical beamforming strategies using DoF analysis unveiling their limitations; for example, rates saturate in overloaded scenarios due to inter-group interference. To tackle interference, we propose a strategy based on degraded beamforming and successive interference cancellation. While the degraded strategy resolves the rate-saturation issue, this comes at a price of sacrificing all spatial multiplexing gains. This motivates the development of a unifying strategy that combines the benefits of the two previous strategies. We propose a beamforming strategy based on rate-splitting (RS) which divides the messages intended to each group into a degraded part and a designated part, and transmits a superposition of both degraded and designated beamformed streams. The superiority of the proposed strategy is demonstrated through DoF analysis. Finally, we solve the RS beamforming design problem and demonstrate significant performance gains through simulations.

I. INTRODUCTION

The paper studies max-min-fair beamforming for overloaded multigroup multicasting, where classical interference management is insufficient. It characterizes performance limits and proposes degraded and rate-splitting strategies to address the resulting trade-offs.

  • Motivation: Overloaded multigroup multicasting arises when classical beamforming cannot neutralize inter-group interference because data streams exceed available spatial DoF.The setting is relevant to ultralow-latency, ultra-high-connectivity wireless networks and multibeam satellite systems.
  • Motivation: Existing work largely studies multigroup multicast beamforming through simulations, leaving performance limits and explicit overloaded-interference treatment insufficiently analyzed.The paper focuses on max-min-fair designs in overloaded scenarios.
  • Contributions: DoF analysis characterizes classical beamforming limits and identifies high-SNR rate saturation in fully overloaded systems.MMF-DoF approximates high-SNR MMF-rate and captures interference-management capability as a function of antennas, groups, and users.
  • Contributions: Degraded beamforming uses successive interference cancellation to avoid saturating MMF rates, but at least one receiver decodes all messages, eliminating spatial multiplexing gains.Its sum-DoF is limited to unity, making it DoF-equivalent to a degraded single-beam strategy.
  • Contributions: Rate-splitting divides each group message into degraded and designated parts and superposes their beamformed streams to bridge classical and degraded strategies.The strategy exploits partial spatial-multiplexing gains while retaining non-saturating behavior through the degraded part.
  • Contributions: The paper derives RS MMF-DoF performance, solves the RS beamforming design problem with WMMSE, and evaluates gains through simulations.Unlike the preliminary version, the analysis allows unequal multicast-group sizes and derives an exact RS MMF-DoF characterization.

III. DESIGNATED (CLASSICAL) BEAMFORMING

Classical beamforming maps one message per multicast group to designated streams and optimizes the minimum group rate under a power constraint. Its MMF-DoF depends on whether enough antennas can null inter-group interference.

  • Designated beamforming: Each group’s message is mapped to one designated stream, beamformed with a group-specific vector, and decoded while other streams contribute interference.The transmit covariance is constrained by the total beamforming power.
  • Max-min fairness: The MMF objective maximizes the symmetric rate simultaneously achievable by all multicast groups, with each group limited by its weakest receiver.The formulation is equivalent to an SINR optimization because rate and SINR are monotonically related.
  • MMF-DoF performance: With sufficient antennas, each beam can null unintended groups and achieve interference-free streams; otherwise, transmission becomes interference-limited.The analysis assumes independently drawn continuous channel vectors, giving generic full-rank channel subspaces.
  • Degrees of freedom: MMF-DoF measures high-SNR rate growth relative to log2(P), while MMF-DoF is the minimum group DoF achieved simultaneously across groups.It captures the fraction of an interference-free single-stream transmission retained under inter-group interference.
  • MMF-DoF performance: Classical beamforming achieves MMF-DoF 1 for N ≥ N_M, 0.5 for N_M−1 + G_1 ≤ N < N_M, and 0 for N < N_M−1 + G_1.These regimes distinguish full, partially overloaded, and fully overloaded antenna configurations.

1) Achievability of Proposition 1:

In the partially overloaded regime, classical beamforming allocates spatial dimensions to preserve interference-free transmission for smaller groups and uses power scaling to equalize users’ DoF. This achieves MMF-DoF 0.5, although the required zero-forcing construction can be rate-suboptimal.

  • Achievability of Proposition 1: Excluding the largest group allows interference-free transmission among the remaining M−1 groups, while the largest group absorbs residual interference.The largest-group beam’s interference is nulled at the other groups.
  • Achievability of Proposition 1: Power allocations scale with P so all user SINRs have the same asymptotic power growth despite interference at the largest group.The resulting construction balances the users’ high-SNR rate scaling.
  • Achievability of Proposition 1: 0.5 MMF-DoF is achieved for every user and group through the proposed beamforming construction.All user rates scale as 0.5 log2(P) + O(1).
  • Achievability of Proposition 1: The DoF construction needs only simple zero-forcing precoders, which are generally suboptimal for finite-SNR rate performance.DoF depend on interference-free dimensions rather than O(1) power gains.

2) Insight:

Classical beamforming’s MMF-DoF reveals distinct overload regimes: partial overload retains multiplexing gain, whereas full overload causes rate saturation. The section also motivates degraded beamforming as an interference-cancellation alternative.

  • Insight: In fully overloaded systems, unavoidable mutual interference collapses the classical MMF-DoF to zero.When N < N_M−1 + G_1, one group’s gain becomes another group’s loss.
  • Insight: A scheme matching the DoF characterization need not maximize MMF rate, especially at medium and low SNR where fixed beam directions can be suboptimal.The converse guarantees DoF optimality only in the asymptotic sense.
  • Insight: Zero MMF-DoF appears at finite SNR as MMF-rate saturation because inter-group interference dominates additive noise.The three DoF regimes can be identified in the numerical rate curves.
  • Insight: For equal-size groups, MMF-DoF collapses to zero immediately when the interference-nulling antenna condition is violated.In that case, N_M equals N_M−1 + G_1.
  • Insight: Degraded beamforming orders group decoding and uses successive interference cancellation, removing decoded streams before later streams are decoded.This strategy is introduced to improve MMF-DoF when N < N_M−1 + G_1.

A. DoF Analysis

Degraded beamforming avoids the overloaded-system DoF collapse by having receivers decode all streams, but it reduces the system to a single shared DoF and sacrifices spatial multiplexing gains.

  • Degraded beamforming: Decoding all M streams at each receiver limits the sum group-DoF to one, which is equally divided among groups.The resulting channel behaves like a degraded channel with one transmitting antenna.
  • Degraded beamforming: 1/M is the MMF-DoF achieved by degraded beamforming.The result is established by matching an achievable construction with a single-antenna MAC upper bound.
  • Trade-off: Degraded beamforming prevents MMF-DoF collapse in fully overloaded systems, but sacrifices all spatial multiplexing gains available to classical beamforming.Classical spatial multiplexing gains are available when N ≥ N_{M−1} + G_1.
  • Simplified strategy: The simplified single-stream formulation packs all users into one multicast group and solves a single-group multicast beamforming problem.This formulation is easier to optimize than the general ordered degraded-beamforming problem.
  • Simplified strategy: The single-stream strategy has the same asymptotic DoF as degraded beamforming but incurs a finite-SNR rate gap because it is equivalent to OMA.The degraded strategy accesses the single DoF non-orthogonally, whereas the single-stream strategy uses orthogonal access.

V. RATE-SPLITTED BEAMFORMING

Rate-splitting combines degraded and designated beamforming by dividing each message into two parts and superposing their streams. The formulation includes both preceding strategies as special cases and represents each group-rate as degraded plus designated contributions.

  • RS construction: Rate-splitting divides each message into degraded and designated parts, then superposes the corresponding beamformed streams.The degraded stream is decoded first with designated streams treated as noise; receivers then cancel it before decoding designated streams.
  • Rate formulation: Each group-rate is the sum of a degraded contribution and a designated contribution.The degraded contribution is allocated through the common stream, while the designated contribution is determined by the group’s designated-stream rate.
  • Problem formulation: The RS beamforming problem jointly selects rates, splitting ratios, and beamforming vectors under common-stream decodability constraints.The rate-splitting variables are represented by the degraded-rate allocation vector and the common and designated beamformers.
  • Unifying formulation: Classical designated beamforming and degraded beamforming are both feasible special cases of the RS formulation.Setting the common-stream rate to zero recovers designated beamforming, while setting all designated beamformers to zero recovers degraded beamforming.
  • Partitioned beamforming: Partitioning groups between designated and degraded service changes the achievable rate performance and is used in the subsequent DoF analysis.Designated groups receive designated streams, while the remaining groups receive degraded-stream portions.

C. DoF Analysis

The RS MMF-DoF is achieved through partitioned beamforming: some groups receive interference-free designated streams while the rest share a degraded stream. Power partitioning balances the two service modes.

  • Proposition 3: M_D^⋆ is the maximum number of groups that can receive interference-free designated beamforming while the remaining groups are silenced.The threshold is determined by the minimum antenna requirement for serving a selected subset of groups without interference.
  • Achievability: The achievability construction serves M_D^⋆ groups with designated beamforming and the remaining groups with degraded beamforming.The designated directions are interference-free for the selected groups, while the degraded beam uses a randomly chosen direction and power scaling.
  • Achievability: The degraded super-symbol achieves DoF 1−α, while each designated group achieves DoF α after degraded-stream cancellation.Choosing α = 1/(1 + M − M_D^⋆) equalizes the group DoF and achieves the proposition’s MMF-DoF.
  • Signal-space interpretation: RS partitions signal power into bottom levels for interference-free designated beams and top levels for degraded transmission.The degraded DoF is divided among degraded groups, while designated beams retain their individual α DoF.

2) Insight: 

The RS DoF result can exceed both classical and degraded strategies in intermediate antenna regimes. At finite SNR, explicit message splitting can further improve rates, motivating WMMSE optimization of the nonconvex sum-rate formulation.

  • DoF insight: For N = N_{M−1}, RS achieves MMF-DoF 0.5 by transmitting the Mth stream in degraded form and cancelling it at receivers.Classical beamforming has zero MMF-DoF in this case because the Mth stream cannot be nulled.
  • Finite-SNR insight: The RS optimum DoF is achieved by partitioned beamforming without message splitting, although splitting is beneficial for finite-SNR achievable rates.Thus, the DoF-optimal structure and the finite-SNR rate-optimal structure need not coincide.
  • Optimization: Because each user-rate in the RS problem is a sum-rate rather than a single-SINR expression, the authors use the WMMSE approach.The method introduces equalizers, MSEs, weights, and a Rate-WMMSE relationship for optimization.

B. WMSE Reformulation and Algorithm

The RS beamforming design is reformulated as a WMSE optimization and solved by alternating optimization, with convergence to a stationary point but no guaranteed global optimum. Simulations compare designated, degraded, and RS strategies across partially and fully overloaded settings.

  • WMSE Reformulation: The WMSE reformulation introduces common-rate allocations, group rates, equalizers, and weights as optimization variables.The constraints allocate common rate across groups while enforcing group-rate and nonnegative-allocation conditions.
  • Algorithm: Alternating optimization updates MMSE equalizers and weights, then solves the remaining convex problem for fixed equalizers and weights.The convex subproblem can be solved with interior-point methods.
  • Algorithm: The algorithm increases the objective until convergence, but non-convexity prevents guaranteeing global optimality; stationarity can nevertheless be established.The convergence objective is bounded above under a fixed power constraint.
  • Numerical Results: For N = 6, designated and RS beamforming achieve full DoF and nearly identical rates, whereas degraded beamforming loses DoF.For N = 4, designated and RS both achieve MMF-DoF 0.5, with RS marginally higher rates; for N = 2, designated rates saturate while degraded and RS achieve MMF-DoF 1/3.
  • Numerical Results: In fully overloaded cases, RS achieves MMF-DoFs of 1/2 for M = 3 and 1/3 for M = 4, outperforming designated and degraded alternatives.The comparison uses N = 4 antennas and groups of two users each.
  • Numerical Results: RS rate contributions vary by group: designated transmission for the largest group can contribute substantially when its beam lies in the null space of the other groups.The considered configuration has N = 4 and group sizes 1, 2, and 3.

A. Converse of Proposition 1

The converse analysis bounds achievable MMF-DoF by relating interference patterns, power exponents, and beamforming dimensions. It shows how unavoidable interference constrains group DoF in overloaded systems.

  • DoF Framework: The generalized DoF framework allows beamforming directions and power exponents to vary with transmit power.The exponents are restricted to [0, 1] without changing the DoF result, and their limits are assumed well-defined.
  • Interference Bounds: For each group, the dominant interfering beam determines an upper bound on the group’s achievable DoF.Interference is characterized through the sets of groups affected by each beamformer and the largest interfering exponent.
  • Converse Argument: The MMF-DoF is bounded by 1 and by averages of group-DoFs selected from groups sharing unavoidable interference.The converse uses the fact that the minimum group-DoF cannot exceed the average of any subset of group-DoFs.
  • Converse Argument: When antennas are insufficient to eliminate interference, at least one group receives interference from a relevant beam, producing the converse bounds for partially overloaded regimes.For N = NM − 1, the analysis establishes the d ≤ 0.5 bound; further antenna reductions cannot increase DoF.
  • Converse Argument: The proof also establishes zero MMF-DoF when every beamformer necessarily interferes with at least one group under the stated antenna condition.The argument identifies a group affected by a beam with the maximum power exponent and derives d ≤ 0.
  • DoF Framework: The DoF definitions use lim sup and compactness assumptions to ensure the relevant asymptotic quantities exist.These assumptions support selecting exponent and power-scaling values that attain the limit superior.

B. Converse of Proposition 3

The converse for RS beamforming bounds its common and designated DoF contributions using unavoidable interference from designated beams. The resulting bounds depend on how many groups each beam must interfere with and how common DoF is allocated.

  • RS DoF Structure: RS assigns each group-DoF as a common contribution plus a designated contribution, with the common stream decoded by all receivers.The common DoF is divided among groups, while each group also receives its designated DoF.
  • Designated-Beam Interference: The number of groups that can receive interference-free designated beams is limited by the antenna-dependent quantity M⋆D.At least M⋆D groups receive non-zero interference from designated beams for any feasible precoding scheme.
  • Designated-Beam Interference: Lemma 1 shows that the first designated beam interferes with at least M⋆c groups, while each remaining designated beam also has a lower-bounded interference footprint.These bounds follow from the antenna requirements for placing beams in null spaces of unintended groups.
  • Null-Space Limits: A designated beam cannot generally be nulled at an arbitrary subset of groups when the antenna count is below the required sum of group sizes.Under this condition, the beam interferes with at least M − L − 1 groups after excluding its intended group.
  • Converse Bound: The converse selects groups affected by the highest-power designated beam and by the first beam to upper-bound the minimum group-DoF.The proof assumes the smallest relevant interference set for the upper bound and uses average group-DoF inequalities.
  • Converse Bound: The resulting upper bound applies across the considered cases because adding further interfered groups does not increase the achievable DoF bound.The proof completes the bound by treating the selected interfered groups as having zero designated DoF in the relevant step.
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