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Sum-Rate Maximization for Linearly Precoded Downlink Multiuser MISO Systems with Partial CSIT: A Rate-Splitting Approach

Hamdi Joudeh, Bruno Clerckx

arXiv:1602.09028v2cs.IT

TL;DR

The paper addresses ESR maximization for downlink MU-MISO systems with imperfect CSIT, where conventional interference suppression is constrained by transmitter-side uncertainty. It combines rate-splitting with linear precoder optimization using SAA and WMMSE, and reports gains over NoRS, including a high-SNR gap exceeding 4 dB for α = 0.6. The conservative approximation has a supported limitation: ignoring highly accurate CSIR in BS design may reduce performance and yield a loose lower bound.

  • Problem

    ESR maximization with linear precoding is difficult under partial CSIT, while optimized RS precoder design was less explored than simpler theoretical designs.

  • Method

    The paper combines RS with linear precoder optimization by maximizing conditional ASR using SAA and WMMSE, and also develops a conservative deterministic approximation.

  • Results

    At high SNR, the gap between RS-Opt and NoRS-Opt exceeds 4 dB for α = 0.6.

  • Takeaways & Limitations

    RS designs perform at least as well as corresponding NoRS designs across the entire SNR range and outperform them at high SNR through DoF analysis.

  • Takeaways & Limitations

    The conservative approach may lose performance because BS design ignores highly accurate CSIR, and its rate lower-bound may be loose.

Abstract

from arXiv · show

This paper considers the Sum-Rate (SR) maximization problem in downlink MU-MISO systems under imperfect Channel State Information at the Transmitter (CSIT). Contrary to existing works, we consider a rather unorthodox transmission scheme. In particular, the message intended to one of the users is split into two parts: a common part which can be recovered by all users, and a private part recovered by the corresponding user. On the other hand, the rest of users receive their information through private messages. This Rate-Splitting (RS) approach was shown to boost the achievable Degrees of Freedom (DoF) when CSIT errors decay with increased SNR. In this work, the RS strategy is married with linear precoder design and optimization techniques to achieve a maximized Ergodic SR (ESR) performance over the entire range of SNRs. Precoders are designed based on partial CSIT knowledge by solving a stochastic rate optimization problem using means of Sample Average Approximation (SAA) coupled with the Weighted Minimum Mean Square Error (WMMSE) approach. Numerical results show that in addition to the ESR gains, the benefits of RS also include relaxed CSIT quality requirements and enhanced achievable rate regions compared to conventional transmission with NoRS.

I. INTRODUCTION

The paper studies sum-rate maximization in downlink MU-MISO systems with imperfect CSIT, where interference management requires accurate transmitter-side channel knowledge. It combines rate-splitting with optimized linear precoding for ergodic performance across fading states and SNRs.

  • Motivation: Imperfect downlink CSIT constrains interference preprocessing at the base station, although multiple antennas provide multiplexing gains.Receivers can obtain accurate CSIR through downlink training, while timely, accurate CSIT is more difficult to provide.
  • Motivation: Under conventional transmission, linear strategies achieve optimal DoF with perfect CSIT, while imperfect-CSIT systems require errors to decay sufficiently with SNR.The cited condition includes an error decay rate of O(SNR^-1) for maintaining full DoF.
  • Rate-Splitting: Rate-splitting divides one user’s message into common and private parts, allowing the common part to be decoded by all users before private decoding.The approach was shown to achieve a DoF gain of 1 − α over NoRS in the MISO broadcast channel.
  • Contribution: Prior RS analyses largely emphasized information-theoretic performance with simple precoders, leaving optimized transceiver and precoder design less explored.This paper addresses that gap by marrying RS with linear precoder design and optimization.
  • Contribution: The paper allows channel states and BS estimates to vary throughout transmission according to a stationary process, with a precoder selected for each incoming estimate.The model includes fixed-estimate and fast-fading cases as special cases.
  • Contribution: The proposed RS design performs at least as well as the corresponding NoRS design across all SNRs because NoRS is a restricted RS case.High-SNR superiority is further established through DoF analysis.
  • Optimization: The paper evaluates long-sequence fading performance using ESR and solves conditional rate optimization through SAA and WMMSE methods.A simplified conservative approximation is also proposed.

B. Rate-Splitting and Transmit Signal Model

The RS transmit model sends one common stream alongside private user streams. Receivers decode the common stream first, remove it through SIC, and then decode their private streams.

  • Conventional Transmission: Conventional linear transmission encodes each user’s message into a private stream and relies on precoder design to suppress multiuser interference.High CSIT uncertainty makes this interference suppression difficult.
  • Rate-Splitting: Rate-splitting broadcasts part of the interference as a common stream that all users decode and cancel before decoding private streams.The scheme combines private and common signaling but uses the common message to manage interference rather than to deliver identical requested information.
  • Rate-Splitting: The RS-user’s message is split into a common part and a private part, while the other users retain private messages.The common stream uses a public codebook and the private streams use conventional encoding.
  • Transmit Signal Model: The K + 1 streams are linearly precoded under a total transmit-power constraint.The common precoder is added to the private precoding matrix, and no splitting reduces the model to conventional transmission.
  • Receiver Processing: Each receiver first decodes the common stream while treating private signals as noise, then uses SIC before decoding its private stream.The private stream is decoded in the presence of the remaining interference.
  • Coding and Adaptation: The transmission assumes delay-unlimited coding over long channel-state sequences while precoders adapt to each instantaneous channel estimate.This supports average CSIT-error performance evaluation.

III. MOTIVATION, PROBLEM FORMULATION AND DOF ANALYSIS

The paper formulates ESR maximization under partial CSIT by optimizing precoders for conditional average rates, while comparing RS and NoRS through tractable power-constrained formulations and DoF analysis.

  • III. MOTIVATION, PROBLEM FORMULATION AND DOF ANALYSIS: With perfect CSIT, the BS adapts precoders to each channel state to maximize instantaneous sum rate under a long-term power constraint.A feasible solution is a set of precoding matrices indexed by channel state.
  • III. MOTIVATION, PROBLEM FORMULATION AND DOF ANALYSIS: Replacing the long-term constraint with a short-term power constraint decouples channel states and makes separate per-state optimization tractable.The paper adopts this peak-power formulation despite its potential for smaller ESR.
  • A. Average Sum Rate Maximization: Under partial CSIT, ESR maximization is achieved by optimizing the precoder so that conditional ASR is maximized for each channel estimate.This formulation reduces to the perfect-CSIT problem when the estimate is perfect.
  • A. Average Sum Rate Maximization: Under partial CSIT, robust precoding uses the channel estimate to improve receiver conditions and select rates that remain decodable despite estimation errors.Designing as if the estimate were perfect can cause interference and rate overestimation.
  • A. Average Sum Rate Maximization: The average rate for a given estimate is the conditional expectation of the instantaneous rate over the CSIT-error distribution.Ergodic rates are obtained by averaging these conditional average rates over channel-estimate variation.
  • A. Average Sum Rate Maximization: For RS, the common average rate is constrained by the weakest user, which couples common-rate decisions across channel states.Leaving the minimization outside the expectation creates an intractable formulation.
  • A. Average Sum Rate Maximization: DoF analysis compares how CSIT quality affects long-term NoRS and RS performance, including the transition between multiuser and single-user transmission.The analysis is used to answer both CSIT-quality and RS-versus-NoRS questions.

B. DoF Performance

The DoF analysis compares optimum NoRS and RS under CSIT errors characterized by exponent α. RS preserves a common-stream contribution and achieves strictly higher DoF than NoRS for α∈(0,1), while also guaranteeing no worse SR across all SNRs.

  • The DoF measures the number of interference-free streams supportable per channel use, highlighting the impact of multiuser interference and CSIT quality.
  • For optimum NoRS and RS, the paper derives DoF expressions as functions of the CSIT-quality exponent α.The analysis assumes isotropically distributed CSIT errors and full-column-rank estimated channels with probability one.
  • At α=1, both strategies attain the maximum DoF K; below α=1, NoRS loses DoF until switching to single-user transmission yields DoF 1 for α<1/K.With fixed and uniform power allocation, naive NoRS designs can instead have zero DoF at α=0.
  • RS scales private-stream powers as O(P_t^α), maintaining Kα private DoF, while allocating O(P_t) power to the common stream for an additional 1−α DoF.
  • RS DoF is strictly greater than NoRS DoF for every α∈(0,1), and optimized RS designs are reported as superior in rate performance despite unchanged achievable DoF in related work.
  • Across the entire SNR range, RS is guaranteed to achieve at least the NoRS SR because NoRS is obtained by restricting RS to zero common-stream power.

IV. SAMPLE AVERAGE APPROXIMATED WMMSE ALGORITHM

The stochastic ASR precoder problem is approximated with SAA and transformed through an augmented WMMSE formulation. Under finite-SNR and bounded-channel assumptions, the sampled problem converges to the stochastic problem as the sample count grows.

  • The stochastic ASR problem is first approximated deterministically using SAA and then transformed into an equivalent solvable augmented Average Weighted Sum MSE problem.
  • Sample Average Approximation: For a given channel estimate, M independent conditional channel realizations approximate average rates through sample average functions, producing the SAA optimization problem.
  • Assumptions: The SAA construction keeps the precoder sequence fixed across the M realizations, under finite SNR, positive noise variance, finite transmit power, and bounded channel realizations.
  • Convergence: The strong law of large numbers gives almost-sure convergence of the sampled average-rate functions to their conditional averages.
  • Convergence: Uniform convergence and continuity imply that global optima of the SAA problem converge almost surely to global optima of the stochastic problem as M→∞.The convergence result can also be extended to points satisfying first-order optimality conditions.
  • Augmented WMMSE formulation: The augmented WMMSE formulation introduces optimization weights and logarithmic weight terms, creating a block-wise convex structure through equalizers and weights.
  • Rate-WMMSE relationship: Common and private receiver MSEs are formed from successive decoding, then converted into weighted MSEs whose MMSE equalizers and weights establish the Rate-WMMSE relationship.
  • Sampled WMMSE problem: Sampled AWMSEs and per-realization equalizers and weights yield a deterministic augmented AWSMSE minimization problem equivalent to the sampled ASR problem under affine transformations.

C. Equivalence

The augmented WMMSE problem is equivalent to the sampled ASR formulation at the level of objective values and stationary solutions. This equivalence enables optimization despite joint non-convexity.

  • Because AWMSEs decouple across their corresponding equalizers and weights, minimizing each individually gives the MMSE solution for a fixed precoder.
  • Under the MMSE solution, the augmented AWSMSE problem reduces to the sampled ASR problem through affine transformations of the objective and common rate.
  • The Rate-WMMSE equivalence extends beyond global optima: any KKT point of the augmented WMMSE problem satisfies the KKT conditions of the sampled ASR problem.
  • The stationary-point correspondence applies because both objectives are non-smooth max-min expressions of sum-rate terms, despite differences from the referenced formulation.

D. Alternating Optimization Algorithm

The proposed alternating-optimization algorithm exploits convexity in each variable block and closed-form MMSE updates. It alternates receiver/weight updates with a convex precoder QCQP until convergence.

  • Although jointly non-convex, the augmented WMMSE problem is convex in each block of precoders, weights, and equalizers when the other two blocks are fixed.
  • Equalizer and weight update: Each iteration first updates equalizers and weights using their MMSE solutions for the current precoder.
  • Sample-average updates: The algorithm then constructs sample-average quantities, including MSE-related terms, weights, and auxiliary variables, from the updated equalizers and weights.
  • Precoder update: The precoder update substitutes the MMSE equalizers and weights into the augmented WMMSE formulation and solves the resulting optimization problem.
  • Iteration: Equalizer/weight and precoder updates repeat alternately until convergence, as summarized in Algorithm 1.
  • Precoder update: The precoder subproblem is a convex QCQP solvable with interior-point methods.

3) Algorithm and Convergence:

Algorithm 1 uses alternating optimization with convex approximations to solve the sampled ASR problem, and its iterates converge to KKT solutions; as the sample size grows, this convergence extends almost surely to the stochastic ASR problem.

  • Convergence: Algorithm 1 converges to the set of KKT solutions of the sampled ASR problem and, as M →∞, almost surely to KKT solutions of the stochastic ASR problem.The convergence result applies for a given channel sample H(M).
  • Convergence: The sampled ASR objective decreases monotonically across optimization steps and is bounded, yielding convergence of the objective sequence.The boundedness follows from the power-constrained feasible set.
  • Algorithm: The alternating-optimization step is an instance of Successive Convex Approximation because it solves a convex approximation around the previous precoder iterate.The approximation is formed for the sampled ASR objective at P[n−1].
  • Limitations: Global optimality is not guaranteed because the problem is non-convex, and different initializations may produce different limit points.The paper examines initialization effects and finite sample-size effects through simulations.
  • Conservative approximation: The conservative approximation removes SAA and reduces equalizer and weight updates, but it sacrifices achievable performance because it is a lower-bound approximation.Its relaxed variables remain unchanged across the M realizations, producing a restricted version of the sampled AWSMSE problem.

B. Conservative Performance Limitations

The conservative design simplifies optimization by relaxing channel-state dependencies, but its lower-bound performance can be limited by self-interference and by the information needed to estimate achievable rates.

  • Approximation: The conservative design restricts equalizers and weights to remain unchanged across channel realizations, thereby maximizing a lower bound on ASR.The resulting conservative ESR is achievable under the relaxed design.
  • CSIR assumption: The conservative approximation ignores highly accurate CSIR at the BS during design because its equalizers are updated using only partial CSIT.This ignorant approach may lose performance relative to SAA.
  • Performance limitation: Self-interference terms in the conservative SINR denominators can scale significantly with transmit power, depending on CSIT quality, limiting rate performance.The paper identifies these terms as the source of performance loss observed in simulations.
  • Rate evaluation: Receivers may still use perfect CSIR, but predicting their higher achievable rates at the BS requires numerical evaluation of the rate expressions.Thus, rate adjustment at the transmitter can reintroduce numerical calculations.

A. Convergence

The proposed optimization converges to a limit point across tested initializations, while initialization affects convergence speed and can affect the limit point. Optimized RS outperforms NoRS particularly at high SNR under moderate CSIT quality, and SAA performs well with modest sample sizes.

  • Convergence: Algorithm 1 converges to a limit point regardless of initialization, but convergence speed and possibly the limit point depend on the starting precoder.The initialization effect becomes more visible at high SNR.
  • Convergence: SVD initialization of the common precoder improves convergence at high SNRs, motivating the use of MRC-SVD for subsequent results.MRC-SVD was selected for good overall performance across channel realizations and a wide SNR range.
  • RS vs. NoRS: RS-Opt and NoRS-Opt perform similarly at low SNRs, but RS-Opt activates the common message as multiuser transmission becomes beneficial.At low SNRs, both schemes reduce to single-user transmission by switching off the weaker user.
  • RS vs. NoRS: The high-SNR RS-Opt versus NoRS-Opt gap exceeds 4 dB for α = 0.6, while RS remains less instrumental for α = 0.9 but still yields high-SNR rate gains.The common message primarily contributes at high SNRs.
  • RS vs. NoRS: With SNR-independent CSIT errors, NoRS-ZF saturates, whereas NoRS-Opt and RS-Opt substantially outperform their corresponding baselines across the SNR range.Residual interference caps NoRS-ZF at high SNR, while RS-ZF-SVD reduces to multicast transmission with DoF 1.

2) Increased Number of Users:

The SAA design outperforms the conservative design, especially as CSIT quality decreases, while RS enlarges achievable ergodic-rate regions relative to NoRS at higher SNR.

  • RS versus NoRS rate regions: RS provides relaxed CSIT quality requirements compared with NoRS while achieving closely related performance in the considered comparison.The corresponding NoRS system requires a higher CSIT quality exponent, α2 > α for α ∈ (0, 1).
  • SAA sample size: M = 10 samples already gives strong SAA performance, while M = 100 and M = 1000 perform almost identically.Using M = 1 significantly degrades performance because the design lacks statistical knowledge of CSIT uncertainty.
  • SAA versus conservative design: The SAA scheme achieves significant ESR gains over the conservative scheme, with gains increasing as CSIT quality decreases.The conservative lower bound becomes looser because self-interference terms become dominant at lower CSIT qualities.
  • RS versus NoRS rate regions: At 30 dB SNR, RS achieves almost 4 bps/Hz for user-2 while guaranteeing 10 bps/Hz for user-1, compared with just over 1.5 bps/Hz using NoRS.The gap between RS and NoRS grows with SNR, and RS significantly enlarges the whole ergodic-rate region.

APPENDIX A PROOF OF THEOREM 1

The appendix derives DoF upper bounds for RS under partial CSIT and shows that these bounds are achievable using feasible precoding schemes.

  • Proof of Theorem 1: The proof bounds conditional average DoFs using the interference scaling exponent āk = maxj≠k{aj} and derives d̄k ≤ min{(α + ak − āk)+, ak}.The bound follows by combining upper and lower bounds on terms in the achievable-rate expression.
  • Proof of Theorem 1: For J > 1 users with positive DoFs, the sum DoF satisfies d̄(J) ≤ Jα, while d̄(1) ≤ 1.The maximum upper bound is obtained with J = K for α ≥ K−1 and with J = 1 otherwise.
  • Proof of Theorem 1: A DoF of 1 is achieved through single-user TDMA, whereas Kα is achieved with ZF-BF vectors based on the channel estimate and private powers qk = P^α/K.These feasible constructions establish achievability of the relevant upper bounds.
  • Proof of Theorem 1: The RS sum DoF upper bound is 1 + (J−1)α for J users and reaches 1 + (K−1)α when all K users have positive DoFs.This maximum holds regardless of α and is achieved by the DoF-motivated design.

APPENDIX B GENERATING CHANNEL STATES

Channel states are generated conditionally from normalized channel estimates and normalized estimation errors, then averaged across estimate realizations to produce ESR curves.

  • Channel-state generation: For each normalized channel estimate bHn, M normalized estimation errors are generated and combined into M conditional channel realizations H(m).The normalized estimates and errors have entries drawn from CN(0, 1).
  • Channel-state generation: When α > 0, the conditional channel-state set varies with SNR because the CSIT error variance depends on SNR.The construction preserves the fading assumptions used in the simulations.
  • ESR evaluation: ESR curves are averaged over 100 normalized channel-estimate realizations, reusing the same normalized channels across SNRs.This reuse produces the smooth ESR curves shown in the figures.
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