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A Rate Splitting Strategy for Massive MIMO with Imperfect CSIT

Mingbo Dai, Bruno Clerckx, David Gesbert, Giuseppe Caire

arXiv:1512.07221v1cs.IT

TL;DR

Imperfect CSIT limits multiuser MIMO through interference, motivating methods that improve transmission in massive-MIMO settings. This paper generalizes rate splitting to the large-scale array regime and proposes HRS using channel second-order statistics and two-tier precoding. It analyzes asymptotic rates, optimizes common-message precoders, derives power allocation, and reports significant gains over conventional baselines.

  • Problem

    Imperfect CSIT causes multiuser interference, while existing rate-splitting analyses and optimization methods are difficult to extend to multiuser massive MIMO.

  • Method

    The paper generalizes RS to massive MIMO and proposes HRS using channel second-order statistics, two-tier precoding, asymptotic rate analysis, and common-message precoder optimization.

  • Results

    RS and HRS provide significant sum-rate gains over conventional multiuser broadcasting and single-user transmission.

  • Takeaways & Limitations

    HRS is particularly suited to massive-MIMO deployments because it combines rate splitting with channel second-order statistics and two-tier precoding.

Abstract

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In a multiuser MIMO broadcast channel, the rate performance is affected by the multiuser interference when the Channel State Information at the Transmitter (CSIT) is imperfect. To tackle the detrimental effect of the multiuser interference, a Rate-Splitting (RS) approach has been proposed recently, which splits one selected user's message into a common and a private part, and superimposes the common message on top of the private messages. The common message is drawn from a public codebook and should be decoded by all users. In this paper, we generalize the idea of RS into the large-scale array regime with imperfect CSIT. By further exploiting the channel second-order statistics, we propose a novel and general framework Hierarchical-Rate-Splitting (HRS) that is particularly suited to massive MIMO systems. HRS simultaneously transmits private messages intended to each user and two kinds of common messages that can be decoded by all users and by a subset of users, respectively. We analyse the asymptotic sum rate of RS and HRS and optimize the precoders of the common messages. A closed-form power allocation is derived which provides insights into the effects of system parameters. Finally, simulation results validate the significant sum rate gain of RS and HRS over various baselines.

I. INTRODUCTION

Imperfect CSIT makes multiuser interference a major challenge for massive MIMO, motivating rate splitting and its hierarchical extension. The proposed schemes combine common and private messages to improve robustness and sum-rate performance.

  • Rate-Splitting: RS splits one selected user’s message into common and private parts, superimposing the common message on privately precoded messages.All users decode the common message first, then use successive interference cancellation before decoding their own private messages.
  • Rate-Splitting: With proper power allocation, RS achieves sum DoF 1 + δ, strictly exceeding the 2δ achieved by ZF.At high SNR, the common message supplies the multiplexing gain of single-user transmission while private messages retain multiuser benefits.
  • Motivation: Imperfect CSIT creates multiuser interference and can make conventional linear precoding interference-limited at high SNR.Accurate CSIT also becomes harder as the number of transmit antennas increases, especially with limited feedback.
  • Hierarchical-Rate-Splitting: The paper generalizes RS to massive MIMO and proposes HRS, which uses channel second-order statistics and a two-tier precoding structure.HRS transmits private messages together with common messages decoded by all users and by user subsets.
  • Results: Simulation results show significant sum-rate gains for RS and HRS over conventional multiuser broadcasting and single-user transmission.The earlier RS optimization and analysis methods are described as difficult to extend to multiuser massive MIMO systems.
  • Contributions: RS and HRS are analyzed in the large-scale array regime, with optimized common-message precoders and derived power allocation.The analysis quantifies gains over conventional one-tier and two-tier multiuser broadcasting across SNR, CSIT quality, spatial correlation, and user count.

II. SYSTEM MODEL

The system is a single-cell FDD downlink with a multi-antenna base station serving single-antenna users over spatially correlated Rayleigh fading. Channel statistics are modeled through covariance eigenstructures and the received signal includes linear precoding and AWGN.

  • Channel Model: The base station has M antennas and serves K single-antenna users, with K ≤ M, over a spatially correlated Rayleigh-fading channel.The model uses a geometrical one-ring scattering assumption for spatial correlation.
  • Channel Model: The one-ring model characterizes each user through azimuth angle θk, angular spread Δk, wavelength λ, and antenna position vectors ri.The wave vector is defined as Ψ(α) = [cos(α), sin(α)].
  • Channel Statistics: The Karhunen-Loeve representation uses the non-zero eigenvalues Λk and associated eigenvectors Uk of the spatial correlation matrix Rk.These quantities describe the user channel in a reduced-dimensional eigen-subspace.
  • Signal Model: The base station employs linear precoding for simultaneous downlink transmission, subject to E[||x||2] ≤ P.The received vector includes the downlink channel matrix H and additive white Gaussian noise n.

III. RATE-SPLITTING

The RS transmission strategy combines a common message with private messages under imperfect CSIT, using linear precoding and successive decoding. The paper develops asymptotic rate and common-precoder analyses for the large-scale array regime.

  • Precoder Design: The private-message precoder is designed from channel estimates using ZF or RZF, while the common precoder is optimized for common-message rate.The asymptotically optimal common precoder equalizes the users’ common-message SINRs.
  • Motivation: With imperfect CSIT and fixed error variance, conventional BC with uniform power allocation is multiuser-interference-limited at high SNR.Adaptive scheduling can approach TDMA at high SNR but is computationally heavy for many users.
  • Transmission Scheme: RS splits one selected user’s message into a common part and a private part, while other users retain private messages.The common message is drawn from a public codebook and is decodable by all users.
  • Power Allocation: The common and private powers are Pc = P(1 − t) and Pk = Pt/K, where t is the fraction assigned to private messages.The paper uses uniform private-message power allocation for RS and conventional BC.
  • Transmission Scheme: Users decode the common message while treating private messages as noise, remove it by SIC, and then decode their own private messages.The common rate is constrained by the users that decode it.
  • Asymptotic Analysis: The paper derives asymptotic RS rates and a closed-form common-precoder solution in the large-scale array regime.The private-message precoders under imperfect CSIT are otherwise described as unknown in general and difficult to optimize directly.

D. Asymptotic Rate Analysis

The asymptotic rate analysis derives deterministic large-array approximations for RS SINRs and rates, then evaluates their accuracy for finite antenna dimensions. The approximations become increasingly accurate as the number of transmit antennas grows.

  • Asymptotic Rates: Random-matrix methods yield asymptotic SINR expressions for the RS common and private messages as M approaches infinity.The asymptotic common rate is expressed through the limiting common-message SINR.
  • Asymptotic Rates: The asymptotic sum rate of RS is obtained by combining the asymptotic common-message rate with the asymptotic private-message sum rate.The derivation uses the continuous mapping theorem after establishing asymptotic SINR convergence.
  • Approximation Accuracy: The asymptotic SINR and rate approximations become more accurate as the number of transmit antennas increases.This behavior follows the cited random-matrix-theory analysis.
  • Approximation Accuracy: The approximations are feasible and tight for large but finite systems such as a 64-antenna massive-MIMO prototype.Simulations also suggest effectiveness for smaller dimensions such as M = 16.

E. Power Allocation

The paper develops power allocation for RS and explains how the split between private and common messages affects sum-rate gains under imperfect CSIT.

  • E. Power Allocation: The optimal power split is characterized by a first-order condition and a closed-form expression for t.The resulting expression also yields a high-SNR lower bound on the RS sum-rate gain.
  • E. Power Allocation: A suboptimal power allocation allocates fraction t of total power to RS private messages and the remainder to a common message.The private messages approximately retain conventional multiuser-BC performance, while the common message uses residual power to enhance the sum rate.
  • E. Power Allocation: As CSIT quality worsens, t decreases, reducing private-message power and shifting more power toward the common message.At high SNR, private-message power is fixed to restore the non-interference-limited regime, while remaining power is assigned to the common message.
  • E. Power Allocation: As transmit power increases, t approaches zero but remains nonzero, so RS does not reduce to single-user transmission.The rate gain from power allocation is upper bounded by approximately 1.44 bps/Hz.
  • E. Power Allocation: The RS gain is affected by system parameters including the number of users, CSIT quality, spatial correlation, and channel-estimation requirements.A larger user count degrades RS rate benefits, while limited feedback makes high-quality CSIT difficult to obtain.
  • E. Power Allocation: HRS extends RS using spatial correlation statistics and two common-message types to enhance sum rate, reduce CSIT requirements, and mitigate large-user effects.Its outer and inner RS components address inter-group and intra-group interference, respectively.

B. Precoder Design

The HRS precoder design uses long-term spatial-correlation information for outer precoding and effective channel information for inner and common-message precoding.

  • B. Precoder Design: HRS uses channel covariance matrices and effective channel information, unlike RS, which accesses more detailed channel information.The outer precoder is designed from long-term CSIT based on spatial correlation matrices.
  • B. Precoder Design: The outer precoder selects dominant eigenmodes orthogonal to other groups’ dominant eigenspaces to reduce inter-group interference.Its dimension must satisfy K_g ≤ b_g ≤ M − P.
  • B. Precoder Design: The inner precoder is designed using regularized zero forcing with regularization parameter ε = K/bP.This design operates on each group’s reduced-dimensional effective channel.
  • B. Precoder Design: The outer common-message precoder is designed in the span of the reconstructed effective channels after resolving the dimension mismatch.Its columns become orthogonal asymptotically, enabling an equally weighted precoder.
  • B. Precoder Design: The optimal inner common-message precoder is generally NP-hard to obtain efficiently, so the paper uses an equally weighted matched-beamforming design.Under additional large-user assumptions, matched beamforming based on the inner precoder is equivalent.

C. Asymptotic Rate Analysis

The paper derives asymptotic SINRs and sum-rate approximations for HRS and conventional two-tier precoding, then quantifies HRS’s sum-rate advantage.

  • C. Asymptotic Rate Analysis: The HRS SINRs asymptotically converge to deterministic expressions as the massive-MIMO dimensions grow.The derivation follows an established asymptotic approach.
  • C. Asymptotic Rate Analysis: The asymptotic analysis yields approximations for the outer common, inner common, and private-message rates of HRS.These terms are combined to obtain the HRS sum-rate approximation.
  • C. Asymptotic Rate Analysis: The conventional two-tier precoding scheme is analyzed analogously through its asymptotic sum rate.The resulting expressions include inter-group and intra-group interference contributions.
  • C. Asymptotic Rate Analysis: HRS’s sum-rate gain over conventional two-tier precoding broadcast transmission is explicitly quantified.The paper compares the two asymptotic rate expressions through a dedicated gain formula.

D. Power Allocation

The paper derives closed-form power allocation for HRS in weak and strong inter-group interference regimes, showing how power shifts between private and common messages with CSIT quality and interference. HRS reduces to parallel inner RS under negligible inter-group interference and to RS under dominant inter-group interference, while exploiting excess high-SNR power for common transmission.

  • Closed-form optimal power-splitting ratios α and β are derived separately for weak and strong inter-group interference regimes.The general case does not yield a simple closed form guaranteeing an HRS gain over two-tier precoding BC.
  • When inter-group interference is negligible, HRS becomes parallel inner RS and the outer common message is unnecessary.
  • When inter-group interference dominates, HRS reduces to RS with reduced-dimensional CSIT because inner common and private transmissions are inter-group-interference limited.
  • At low SNR, α = β = 1, so HRS becomes conventional two-tier precoding BC and its sum-rate gain is zero.
  • As CSIT worsens or inter-group interference increases, HRS allocates less power to private messages and more power to common messages at high SNR.The private-message power remains invariant with total power at high SNR, while common-message power grows linearly with total power.
  • At high SNR, HRS sum-rate gain increases by G bps/Hz per 3 dB in weak inter-group interference and by 1 bps/Hz per 3 dB in strong interference.

V. SIMULATION RESULTS

Simulations validate the asymptotic rate analyses and closed-form power allocations for RS and HRS, showing rate gains and reduced precoding complexity over broadcasting baselines.

  • The asymptotic approximation RRS,◦sum properly characterizes RS sum rates across CSIT qualities and SNRs.
  • RS_CLF achieves almost the same sum rate as simulation-based exhaustive-search RS_EXS, validating the proposed power allocation.
  • At high SNR, RS avoids the BC_RZF rate ceiling caused by imperfect CSIT and achieves a multiplexing gain approaching 1.
  • The RS sum-rate gain over BC_RZF decreases as the number of users increases, because the common-message rate benefit becomes smaller.
  • RS with MBF reaches the same rate performance as BC with RZF at SNR = 30 dB, while reducing precoding complexity but increasing encoding and decoding complexity.
  • HRS simulations compare exhaustive-search and closed-form power allocation under clustered massive-MIMO configurations and multiple two-tier-precoding baselines.

1) Validation of the Asymptotic Rate Analysis:

The simulations validate the asymptotic analyses of HRS across eigen-subspace configurations and CSIT qualities. HRS uses common messages to address interference and maintains gains over two-tier precoding baselines.

  • Disjoint eigen-subspaces: With disjoint eigen-subspaces, inter-group interference is negligible, making the outer common message unnecessary; with perfect CSIT, HRS reduces to two-tier precoding BC.
  • Imperfect CSIT: As CSIT quality decreases, HRS exploits inner common messages to mitigate intra-group interference, while approximation (48) remains valid for asymptotic sum rate.
  • Overlapping eigen-subspaces: With overlapping eigen-subspaces, approximation (48) approximately characterizes HRS sum rate, including low-to-medium-SNR behavior resembling two-tier precoding BC.
  • HRS performance: At SNR = 30 dB, HRS improves sum rate over two-tier precoding BC by 1.5 bps/Hz under strong eigen-subspace overlap.
  • HRS performance: Closed-form HRS power allocation achieves almost the same sum rate as simulation-based exhaustive search in both eigen-subspace settings.
  • HRS performance: At high SNR, HRS sum-rate gain over two-tier precoding BC increases by nearly G or 1 bps/Hz for each 3 dB power increment, depending on the setting.
  • Overall findings: HRS is robust to CSIT error and eigen-subspace overlap, achieves the multiplexing gain of two-tier BC with perfect user scheduling, and remains competitive without scheduling.
  • Overall findings: HRS reduces scheduler and precoder-design burden, but increases encoding and decoding complexity.

C. RS vs. HRS

RS and HRS mitigate interference from imperfect CSIT, with HRS exploiting channel second-order statistics and two-tier precoding for massive MIMO. Simulations report especially large HRS gains when users occupy disjoint eigen-subspaces and under reduced-dimensional CSIT.

  • CSIT quality: Lower CSIT quality, represented by larger τ^2, degrades the rate performance of the evaluated schemes.The paper also notes that conventional broadcasting schemes saturate at high SNR because of imperfect CSIT, whereas RS gains increase with transmit power.
  • Simulation comparison: 15.5 bps/Hz is the reported sum rate gain of HRS over BC_TTP at τ^2 = 0.48 and SNR = 30 dB.At the same example setting, RS only slightly outperforms BC_RZF by around 1 bps/Hz.
  • Simulation comparison: HRS is better suited than simple RS when the transmitter has channel second-order statistics available.The gain is enabled by multiple inner common messages whose rates are constrained by fewer users.
  • Simulation comparison: With reduced-dimensional CSIT, BC_TTP and HRS achieve much higher rates than BC_RZF and RS because users are partitioned into less-interfering groups.The outer precoder uses long-term CSIT, and each user experiences interference from fewer users.
  • Framework: HRS combines outer and multiple inner common messages with private messages, targeting inter-group and intra-group interference, respectively.The outer common message is decoded by all users, while inner common messages are decoded by subsets of users.
  • Complexity and scope: RS reduces precoder-design and scheduling complexity for a given sum-rate requirement while increasing encoding and decoding complexity.The paper concludes that RS and HRS are robust to CSIT errors and eigen-subspace overlap, with HRS particularly suited to massive MIMO.

APPENDIX

The appendix develops asymptotic power-allocation insights for RS and HRS across SNR and interference regimes. It explains how common-message power and hierarchical allocation depend on user count, intra-group interference, and inter-group interference.

  • RS power allocation: RS achieves approximately the same sum rate as conventional multiuser broadcasting with full power under the stated approximation.The appendix derives bounds on RS rate loss and uses them to characterize the power split.
  • RS power allocation: The common-message power should decrease as the number of users K increases because its achievable rate is constrained by the minimum user rate.The appendix motivates reducing P(1 − t) as K grows.
  • RS power allocation: At low SNR, t = 1 turns RS into conventional broadcasting because private-message transmission is not interference-limited.At high SNR, t < 1 allocates residual power to a common message after private transmission saturates.
  • HRS power allocation: For HRS, the outer common message suffers more interference and contributes less rate than inner common messages because its pre-log factor is 1 while G > 1.The optimal inter-group power split is reported as β = 1.
  • Asymptotic analysis: The appendix analyzes HRS rate loss and sum-rate bounds in weak and strong inter-group-interference regimes using high-SNR asymptotics.The analysis uses RZF-to-ZF convergence and approximations involving the channel correlation structure.
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