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Robust Transmission in Downlink Multiuser MISO Systems: A Rate-Splitting Approach

Hamdi Joudeh, Bruno Clerckx

arXiv:1602.04345v3cs.IT

TL;DR

The paper addresses robust max-min fairness and QoS power minimization in downlink multiuser MISO systems with bounded CSIT uncertainty. It uses Rate-Splitting with a cutting-set/WMMSE-based robust algorithm and reports improved DoF, rates, and power-feasibility behavior over NoRS and conservative designs. The paper also identifies self-interference from the conservative approximation as a limitation of the earlier robust WMMSE approach.

  • Problem

    The paper studies robust max-min fairness and QoS-constrained transmit-power minimization when bounded CSIT uncertainty makes worst-case achievable rates and semi-infinite optimization difficult.

  • Method

    It splits each user message into common and private parts, transmits the resulting common and private streams with linear precoding, and solves the robust design using a cutting-set method with WMMSE optimization.

  • Results

    RS analytically outperforms NoRS in the interference-limited regime, achieves non-saturating rates for non-scaling CSIT, and improves simulations and QoS power minimization over NoRS and conservative designs.

  • Takeaways & Limitations

    Rate-Splitting provides a robust transmission strategy that improves fairness performance and addresses feasibility and power costs in the QoS problem under CSIT uncertainty.

  • Takeaways & Limitations

    The conservative robust WMMSE approximation introduces self-interference that can cause saturating performance and fail to match the optimum DoF, particularly at high SNR.

Abstract

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We consider a downlink multiuser MISO system with bounded errors in the Channel State Information at the Transmitter (CSIT). We first look at the robust design problem of achieving max-min fairness amongst users (in the worst-case sense). Contrary to the conventional approach adopted in literature, we propose a rather unorthodox design based on a Rate-Splitting (RS) strategy. Each user's message is split into two parts, a common part and a private part. All common parts are packed into one super common message encoded using a public codebook, while private parts are independently encoded. The resulting symbol streams are linearly precoded and simultaneously transmitted, and each receiver retrieves its intended message by decoding both the common stream and its corresponding private stream. For CSIT uncertainty regions that scale with SNR (e.g. by scaling the number of feedback bits), we prove that a RS-based design achieves higher max-min (symmetric) Degrees of Freedom (DoF) compared to conventional designs (NoRS). For the special case of non-scaling CSIT (e.g. fixed number of feedback bits), and contrary to NoRS, RS can achieve a non-saturating max-min rate. We propose a robust algorithm based on the cutting-set method coupled with the Weighted Minimum Mean Square Error (WMMSE) approach, and we demonstrate its performance gains over state-of-the art designs. Finally, we extend the RS strategy to address the Quality of Service (QoS) constrained power minimization problem, and we demonstrate significant gains over NoRS-based designs.

I. INTRODUCTION

The paper studies worst-case max-min fairness and QoS power minimization in downlink multiuser MISO systems with bounded CSIT errors. It proposes Rate-Splitting and a non-conservative cutting-set algorithm, showing analytical and simulated gains over conventional designs.

  • Problem setting: Robust max-min fairness designs maximize the minimum user QoS under a total transmit-power constraint despite bounded CSIT errors.The related QoS problem minimizes transmit power subject to user QoS constraints, and the two formulations are closely related.
  • Rate-Splitting strategy: Rate-Splitting divides each message into common and private parts, packs common parts into one super common message, and transmits K+1 linearly precoded streams.Receivers decode the common stream and their corresponding private stream to recover their intended messages.
  • Asymptotic performance: For CSIT uncertainty scaling as O(SNR^-αk), the paper characterizes optimum max-min DoFs for NoRS and RS, establishing RS performance gains.The scaling factors satisfy αk ∈ [0,1], with α1 ≤ α2 ≤ . . . ≤ αK.
  • Asymptotic performance: With non-scaling CSIT, NoRS can have saturating rates, whereas RS achieves non-saturating rates regardless of the CSIT scaling.This contrast is presented as a consequence of the derived max-min DoF results.
  • Algorithm: The cutting-set algorithm alternates finite-subset optimization with exact worst-case pessimization, avoiding conservative convexification and guaranteeing convergence to a robust solution and a KKT point.This addresses the infinitely many constraints in the worst-case formulation without the conservative approximation used by robust WMMSE.
  • Evaluation and QoS extension: Simulations show the cutting-set method outperforms the conservative method for both RS and NoRS, while RS provides significant gains over NoRS for a fixed robust design method.The RS QoS extension resolves a NoRS feasibility issue and requires less transmission power for the same QoS constraints.

III. PROBLEM STATEMENT AND ASYMPTOTIC PERFORMANCE

The paper formulates worst-case max-min fairness and QoS design under bounded CSIT uncertainty, then analyzes how RS compares with NoRS as uncertainty scales with SNR.

  • Problem formulation: The BS models each channel as an estimate plus a bounded error and assumes perfect CSIR throughout.
  • Problem formulation: Worst-case design optimizes precoders against achievable rates guaranteed for every channel in each uncertainty region.
  • Max-min fairness: RS assigns each user a portion of a common rate, combines it with the private rate, and uses these total rates for max-min fairness.
  • Max-min fairness: NoRS is the restricted RS problem with zero common-rate portions and no common precoder, so its optimum cannot exceed RS.
  • Asymptotic performance: Fixed CSIT uncertainty makes residual interference dominate as private-stream powers grow, producing saturating NoRS rates at high SNR.
  • Asymptotic performance: The CSIT exponent α_k quantifies error decay with SNR: α_k=0 denotes a fixed uncertainty region, while α_k=1 is perfect CSIT in the DoF sense.

C. DoF Analysis

The DoF analysis defines worst-case high-SNR performance for RS and NoRS and characterizes their optimum max-min DoFs under CSIT uncertainty. RS strictly improves max-min DoF over NoRS and avoids the NoRS saturation associated with multiple users having non-scaling CSIT errors.

  • DoF definition: Worst-case DoF measures the high-SNR first-order rate growth of a feasible SNR-indexed precoding family.It is roughly interpreted as the number of interference-free streams simultaneously communicated in one channel use.
  • Assumptions: The analysis assumes actual channels remain SNR-independent with entries bounded away from zero and infinity, while channel estimates and errors may depend on SNR.The channel estimate matrix is additionally subject to a full-column-rank condition; this condition is not required for solving the later optimization problems.
  • Theorem 1: Theorem 1 derives optimum max-min DoFs for both NoRS and RS by first proving upper bounds and then constructing feasible precoding schemes that achieve them.The NoRS optimum is governed by the worst two CSIT scaling factors, whereas RS allocates private and common DoF components.
  • Comparison: RS provides a strict max-min DoF improvement over NoRS for all α1, …, αK ∈ [0,1), while increasing K can reduce RS max-min DoF by dividing common DoF among more users.The latter dependence is reflected by the minimization in the RS expression.
  • Comparison: When at least two users have αk = 0, NoRS has zero max-min DoF and a saturating max-min rate, whereas RS satisfies d̄*_RS ≥ 1/K regardless of CSIT scaling.Thus RS achieves an ever-growing max-min rate in this setting.
  • Interpretation: DoF-optimal precoders need not be rate-optimal, so the characterized DoF can be achieved by precoders that are suboptimal in finite-SNR rate.The paper connects this distinction to rate-suboptimal yet DoF-optimum ZF-BF strategies and later simulations.

IV. CONSERVATIVE APPROACH

The conservative approach converts the robust RS max-min problem into a tractable WMSE formulation while preserving a block-wise convex structure. It removes infinitely many uncertainty-dependent variables and constraints through fixed equalizer-weight abstractions and LMI reformulations.

  • Problem structure: The RS optimization is non-convex and semi-infinite because coupled sum-rate expressions and infinitely many channel-uncertainty constraints appear in each user’s rate.WMMSE reformulation is introduced to expose tractable structure.
  • Rate-WMMSE relationship: Under MMSE equalization, each common or private rate is related to its corresponding MMSE by Rc,k = −log2(εMMSEc,k) and Rk = −log2(εMMSEk).The construction proceeds from common and private MSEs, their optimum equalizers, and augmented WMSEs with positive weights.
  • WMSE reformulation: Introducing equalizers and weights yields an equivalent WMSE problem that is convex in each variable block when the remaining blocks are fixed.This block-wise convexity enables alternating optimization, also called block coordinate descent.
  • Conservative approximation: The conservative approximation swaps minimization and worst-case maximization so common and private equalizer-weight pairs no longer depend on perfect CSI realizations.The resulting quantities provide lower bounds on the worst-case common and private rates.
  • Optimization structure: Because equalizer-weight pairs decouple for fixed precoders, each pair can be optimized separately while maximizing the corresponding common or private rate constraints.This separability is used within the conservative WMSE counterpart.
  • Finite reformulation: The semi-infinite WMSE constraints are replaced by finite Linear Matrix Inequalities using the S-lemma, after Schur-complement reformulation.This converts uncertainty-region constraints into a finite optimization representation.

C. Alternating Optimization Algorithm

The alternating-optimization WMMSE approach solves a conservative robust formulation through block-wise updates, but its approximation can introduce self-interference that causes high-SNR rate saturation.

  • C. Alternating Optimization Algorithm: The WMMSE reformulation creates a block-wise convex structure exploited through alternating optimization over the involved variable blocks.The resulting subproblems are solved as semidefinite programs using interior-point methods.
  • C. Alternating Optimization Algorithm: Algorithm 1 converges monotonically and remains feasible for the original problem, but global optimality is not guaranteed because the problem is non-convex.The conservative approximations preserve feasibility while the alternating procedure may still yield a suboptimal solution.
  • D. Conservative Approach Limitations: The conservative max-min RS rate can saturate for non-scaling CSIT and fail to coincide with the optimum DoF.The observed saturation is attributed to self-interference introduced before alternating optimization.
  • D. Conservative Approach Limitations: The worst-case analysis uses an isotropic zero-mean error distribution over an origin-centered ball only to derive upper bounds, not to restrict actual errors.Actual CSIT errors may have nonisotropic distributions within the uncertainty region.
  • D. Conservative Approach Limitations: The approximation treats error-dependent desired-signal components as interference because equalizers and weights are made independent of the actual channel.This produces self-interference terms that undermine worst-case achievable rates, especially at high SNR.

V. CUTTING-SET METHOD

The cutting-set method alternates finite-subset optimization with exact worst-case channel analysis, progressively enforcing violated uncertainty-region constraints without conservative convexification.

  • V. CUTTING-SET METHOD: Each iteration alternates optimization over finite uncertainty subsets with pessimization that identifies and appends worst-case channels violating rate constraints.Private and common-rate constraints are sampled separately because the messages are independently decoded.
  • V. CUTTING-SET METHOD: The method avoids conservative approximations in the optimization step and solves the pessimization step exactly, targeting a robust solution of the original semi-infinite problem.This differs from approaches that convexify the optimization step through conservative approximations.
  • V. CUTTING-SET METHOD: The algorithm stops when the maximum rate violation falls below a specified tolerance.The violation-based stopping rule uses an arbitrary constant ǫV > 0.
  • V. CUTTING-SET METHOD: Because the sampled optimization is non-convex, global optimality is not generally guaranteed, although a stationary solution can be established.The cutting-set method reaches the stronger global-optimum guarantee only when both steps are globally solved.
  • V. CUTTING-SET METHOD: Under a KKT optimization step and exact pessimization, the iterates converge to KKT points of the semi-infinite robust problem.The guarantee is conditional on the stated properties of the two alternating steps.

B. Optimization

The optimization procedure combines rate-WMMSE alternating updates with exact worst-case analysis, preserving channel-dependent equalizers and weights while obtaining KKT-point convergence.

  • B. Optimization: The sampled rate problem is transformed into an equivalent WMSE problem with a block-wise convex structure suitable for alternating optimization.The equalizers, weights, rates, common allocation, and precoders are updated in alternating blocks.
  • B. Optimization: Unlike the conservative approach, sampled equalizers and weights retain their dependencies on the actual channel and reflect perfect CSIR availability.For fixed equalizers and weights, the remaining update is convex and efficiently solvable by interior-point methods.
  • B. Optimization: The AO procedure converges to KKT points of each sampled rate problem, though non-convexity can make the resulting KKT point suboptimal.The paper demonstrates the algorithm’s effectiveness through simulations rather than claiming global optimality.
  • B. Optimization: Pessimization converts worst-case rate searches into MMSE-based non-convex QCQPs and solves them through relaxation into semidefinite programs.The relaxations are tight at optimality because the subproblems are single-constraint trust-region problems satisfying Slater’s condition.
  • B. Optimization: Dinkelbach’s algorithm solves the resulting fractional pessimization problems iteratively, independently for each user’s private and common-rate constraints.The auxiliary problems need not be convex if they can be solved globally for each parameter.
  • B. Optimization: Combining KKT-convergent sampled optimization with exact pessimization yields a KKT point for the original robust problem.This conclusion follows from the cutting-set convergence result.

VI. SIMULATION RESULTS

Simulations use a three-user, three-antenna system and compare conservative and cutting-set designs for both NoRS and RS strategies.

  • VI. SIMULATION RESULTS: The simulations consider a three-user system with three transmit antennas, unit noise variance, i.i.d. CN(0,1) channel entries, and uniformly distributed errors in uncertainty regions.The channel estimate is formed by subtracting the error matrix from the channel matrix.
  • VI. SIMULATION RESULTS: Four designs are compared: NoRS-con, NoRS-cs, RS-con, and RS-cs.The NoRS designs are obtained by discarding the common message, and NoRS-con is equivalent to the MSE-based design in.

A. Max-Min Fair Rate Performance

The cutting-set method improves robust max-min rate performance over conservative designs, while RS consistently outperforms NoRS and avoids rate saturation under non-scaling CSIT.

  • Non-scaling CSIT: RS-cs achieves approximately 0.31 and 0.33 DoF for δ = 0.05 and 0.15, while NoRS schemes saturate at zero DoF under non-scaling CSIT.The rates of NoRS and RS-con saturate, whereas RS-cs continues growing with SNR.
  • Method comparison: The cutting-set method outperforms the conservative method for both NoRS and RS, with the gap increasing at higher SNR and uncertainty.The conservative approximation introduces increasing self-interference effects in intermediate and high SNR regimes.
  • Scaling CSIT: For scaling CSIT, RS-cs achieves DoFs of 0.53 and 0.47 for δ = 0.05 and 0.15, compared with 0.26 and 0.24 for NoRS-cs.NoRS-con and RS-con fail to achieve the corresponding DoFs because of self-interference.
  • Larger systems: As the number of users and antennas increases, performance generally degrades, but RS retains significant gains over NoRS.The degradation reflects increased multiuser interference and the common message being shared among more users.
  • Complexity comparison: RS has longer running times than NoRS for a given robust design method because it involves more optimization variables.A rigorous analytic complexity comparison is unavailable because both algorithms are iterative and the cutting-set method has a nested structure.
  • QoS power minimization: Under the QoS constraint, RS yields feasible solutions for all realizations and exceeds NoRS feasibility by more than 100% at δ = 0.15.NoRS feasibility decreases as CSIT uncertainty increases, whereas RS remains feasible in the reported simulations.

APPENDIX A PROOF OF THEOREM 1

The proof of Theorem 1 derives upper bounds on optimal max-min DoF and constructs feasible RS precoding schemes that attain the relevant bounds.

  • Achievability: A feasible RS scheme achieves private DoF ˆd_k = min{(α_k + a_k − ā_k)+, a_k} and common DoF ˆd_c = 1 − ā.These expressions characterize the private and common DoF contributions used in the construction.
  • Precoding construction: The RS construction uses zero-forcing-based private precoders from the imperfect channel estimate and a common precoder carrying the super-common stream.Private powers and common DoF allocation are selected to attain the derived upper bound.
  • Proof strategy: The proof first establishes upper bounds for the NoRS and RS max-min DoF problems, then proves achievability through feasible precoding constructions.The argument combines converse bounds with explicit power allocation and common-DoF splitting.
  • Upper-bound argument: The proof bounds private-stream performance by selecting worst-case channels that maximize interference from another user.The selected interferer has the largest private-power exponent among users other than k.
  • DoF allocation: Power allocations with equal private exponents can achieve private DoFs min{α_k, ā} and common DoF 1 − ā for all users.The common DoF is split among users so that each user reaches the target max-min DoF.

APPENDIX B PROOF OF LEMMA 1

The lemma’s proof replaces distributional average-MSE expressions with conservative worst-case bounds, yielding the conservative MMSE formulation used by the robust method.

  • MSE formulation: Averaging the common and private MSEs over the error distribution produces corresponding average WMSE expressions for fixed equalizers and channel estimates.The equalizers are then optimized through the resulting MMSE formulations.
  • MMSE reduction: Substituting the closed-form equalizer solutions into the averaged expressions yields the upper bounds defining the conservative MMSEs.The proof concludes after this substitution.
  • Conservative bound: The maximum worst-case quantity is lower-bounded by its average under any distribution supported on the uncertainty region.This relation connects averaged MSE analysis to conservative robust constraints.

APPENDIX C PROOF OF PROPOSITION 1

The cutting-set proof shows that sampled feasible iterates converge to a feasible point of the semi-infinite problem and, under regularity, to a KKT point.

  • Cutting-set algorithm: The cutting-set algorithm solves the semi-infinite problem through sampled subproblems whose finite constraint sets are updated by exact pessimization.Each iteration alternates optimization over the current subset with worst-case constraint analysis.
  • Feasibility: Under compactness, differentiability, and exact pessimization assumptions, the generated iterates converge to a feasible point of the original problem.The result does not require every sampled optimization step to be globally optimal.
  • KKT convergence: If each sampled iterate satisfies the KKT conditions and the stated regularity condition holds, every limit point satisfies the KKT conditions of the semi-infinite problem.The proof uses convergence of subsequences and associated multiplier measures.
  • Active constraints: At KKT points, the semi-infinite problem has finitely supported multiplier measures because only finitely many constraints are active.This supports the finite representation used in the KKT argument.
  • Regularity and compactness: The compactness of the precoder and rate-variable feasible sets completes the convergence argument for the robust optimization problems.The optimization variables and rate variables belong to compact feasible regions.

APPENDIX D PROOF OF PROPOSITION 2

The AO procedure is an instance of Successive Convex Approximation, updating key variables by solving convex approximations of problem (25). The WMSEs approximate rates around the previous iteration's precoder, under conditions satisfying the cited assumptions and Slater's condition.

  • The AO procedure is an instance of the Successive Convex Approximation method.The passage identifies the procedure in Section V-B as an SCA instance.
  • Each iteration updates ( ¯Rt, ¯c, P) by solving a convex approximation of problem (25).
  • The WMSEs in (27) approximate rates around the precoder P(n−1) from the previous iteration.
  • The cited SCA assumptions are satisfied, and Slater’s condition holds for the convex-approximated problem.
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