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Limited Feedback Hybrid Precoding for Multi-User Millimeter Wave Systems

Ahmed Alkhateeb, Geert Leus, Robert W. Heath

arXiv:1409.5162v2cs.IT

TL;DR

Conventional multi-user MIMO precoding is difficult at mmWave because hardware constraints and limited channel knowledge impede fully digital processing. The paper develops low-complexity hybrid analog/digital precoding with analog receiver combining, analyzes large-array and single-path regimes, and characterizes quantization loss. The proposed techniques achieve higher sum rates than analog-only beamforming while approaching unconstrained digital beamforming with relatively small codebooks.

  • Problem

    High-cost mixed-signal hardware and difficult channel-state acquisition limit conventional fully digital multi-user precoding for mmWave systems.

  • Method

    The paper develops low-complexity hybrid analog/digital precoding with analog-only receiver combining, and analyzes single-path, large-array, and quantized-codebook cases.

  • Results

    Hybrid precoding achieves higher sum rates than analog-only beamforming and approaches unconstrained digital beamforming with relatively small codebooks.

  • Takeaways & Limitations

    Hybrid precoding provides a low-overhead route to multi-user mmWave beamforming while retaining gains over analog-only solutions.

Abstract

from arXiv · show

Antenna arrays will be an important ingredient in millimeter wave (mmWave) cellular systems. A natural application of antenna arrays is simultaneous transmission to multiple users. Unfortunately, the hardware constraints in mmWave systems make it difficult to apply conventional lower frequency multiuser MIMO precoding techniques at mmWave. This paper develops low complexity hybrid analog/digital precoding for downlink multiuser mmWave systems. Hybrid precoding involves a combination of analog and digital processing that is inspired by the power consumption of complete radio frequency and mixed signal hardware. The proposed algorithm configures hybrid precoders at the transmitter and analog combiners at multiple receivers with a small training and feedback overhead. The performance of the proposed algorithm is analyzed in the large dimensional regime and in single path channels. When the analog and digital precoding vectors are selected from quantized codebooks, the rate loss due to the joint quantization is characterized and insights are given into the performance of hybrid beamforming compared with analog-only beamforming solutions. Analytical and simulation results show that the proposed techniques offer higher sum rates compared with analog-only beamforming solutions, and approach the performance of the unconstrained digital beamforming with relatively small codebooks.

I. INTRODUCTION

mmWave multi-user systems need precoding that respects costly hardware and limited channel knowledge. The paper develops low-complexity hybrid analog/digital precoding with reduced training and feedback overhead, analyzing its performance and quantization effects.

  • Motivation: mmWave’s large bandwidth makes it desirable for future cellular systems, but multi-user precoding must accommodate large antenna arrays and low pre-beamforming SNR.These conditions make complete channel-state information difficult to obtain because training overhead is large.
  • Motivation: Fully digital baseband precoding is difficult at mmWave because mixed-signal components have high cost and power consumption.The paper therefore targets algorithms that respect hardware constraints while reducing complexity.
  • Prior approaches: Analog-only beamforming is constrained by quantized phase shifters and limited gain control, restricting sophisticated processing such as multi-user interference management.Hybrid precoding divides processing between analog and digital domains to provide more freedom for multiplexing several data streams.
  • Prior approaches: Existing hybrid algorithms primarily addressed single-user diversity or spatial multiplexing, while prior general approaches did not specifically incorporate mmWave hardware constraints and channel characteristics.This motivates low-complexity hybrid precoding specifically for multi-user mmWave systems.
  • Contributions: The paper develops a low-complexity hybrid precoding and combining algorithm for downlink multi-user mmWave systems using hybrid base-station processing and analog-only mobile-station combining.The design assumes at least as many RF chains as mobile stations and targets limited training and feedback overhead.
  • Contributions: The paper analyzes single-path and large-array cases, characterizes rate loss from joint analog and digital codebook quantization, and compares hybrid gains with analog-only solutions.Simulations compare the proposed method with analog-only beamforming and unconstrained digital precoding under limited feedback and training overhead.

II. SYSTEM MODEL

The system models multi-user mmWave downlink transmission with hybrid BS precoding and analog-only mobile-station combining under sparse, quantized RF hardware constraints.

  • System architecture: The BS uses NBS antennas and NRF RF chains to serve U mobile stations, with U ≤ NRF and one stream per user.The number of simultaneously served users is limited by the available RF chains and spatial multiplexing gain.
  • Receivers: Each mobile station applies a single analog RF combiner, motivated by lower hardware cost and power consumption.The combiner follows constant-modulus and quantized-angle constraints similar to the RF precoders.
  • Transmitter: The transmitted signal combines a U × U baseband precoder with RF precoding vectors and equal power allocation across user streams.The symbol vector has average total power P, and the RF precoder entries have constant modulus.
  • Hardware constraints: RF phase-shifter angles are quantized to a finite set, with each RF precoder entry represented as 1/√NBS times a unit-modulus phase.The BS and mobile stations are assumed to know their antenna-array geometry; simulations use UPAs and ULAs.
  • Channel model: The channel follows a narrowband block-fading geometric model with Lu scatterers, each contributing one propagation path.The model represents each path using a complex gain, angles of arrival and departure, and BS/MS array-response vectors.

III. PROBLEM FORMULATION

The paper formulates sum-rate-maximizing joint analog/digital precoder and analog-combiner design under mmWave hardware, training, and feedback constraints.

  • Objective: The design objective is to choose BS RF and baseband precoders and MS analog combiners to maximize system sum-rate.The analog and digital layers are jointly involved in the optimization.
  • Codebooks: Beamsteering codebooks are adopted because their vectors match single-path array responses and require only single-parameter angle quantization.The formulation is general for any codebook, but the evaluated performance depends on the selected codebook.
  • Optimization challenge: The resulting optimization is a mixed integer programming problem requiring a search over all RF precoder and combiner codeword combinations.The digital precoder must be jointly designed with the analog beamforming and combining vectors.
  • Optimization challenge: Direct optimization is impractical because it can require full channel feedback, large training and feedback overhead, and an unknown optimal digital precoder.The paper characterizes direct sum-rate maximization as neither practical nor tractable.
  • Design target: The proposed mmWave-suitable algorithm targets performance close to the joint optimum while requiring low training overhead and small feedback overhead.It is motivated by practical difficulties affecting prior iterative and non-iterative multi-user precoding methods.

IV. TWO-STAGE MULTI-USER HYBRID PRECODING ALGORITHM

The proposed hybrid precoding algorithm separates analog signal acquisition from digital interference management. It uses low-dimensional effective-channel feedback before zero-forcing digital precoding.

  • First stage: The two-stage design first selects BS RF precoders and MS RF combiners to maximize each user’s desired signal power while neglecting inter-user interference.Single-user beam-training methods can perform this first-stage design without explicit channel estimation and with low training overhead.
  • Second stage: The second stage designs the BS digital precoder to manage multi-user interference.This separates the interference-management task from the initial analog beamforming and combining design.
  • Limited feedback: Each MS feeds back a quantized U × 1 effective channel rather than an NBS × 1 channel vector.The reduced dimension is a central overhead advantage over prior receiver-side designs with larger effective channels.
  • Digital precoding: The BS constructs a zero-forcing digital precoder from the quantized effective channels.Sparse mmWave channels and narrow beams are expected to make the effective channel well-conditioned, supporting near-optimal zero-forcing performance.
  • Scope and analysis: The multi-user setting requires interference management and therefore differs fundamentally from earlier single-user hybrid-precoding analyses.The paper analyzes the resulting multi-user problem using beamsteering RF codebooks and RVQ for effective-channel quantization.

V. PERFORMANCE ANALYSIS WITH INFINITE-RESOLUTION CODEBOOKS

The analysis studies the proposed hybrid precoding algorithm under single-path and large-antenna assumptions, deriving achievable-rate bounds and comparisons with analog-only beamsteering.

  • Assumptions and design: The analysis assumes perfect channel knowledge and continuous-angle RF beamsteering vectors for the single-path and large-dimensional cases.The proposed RF precoders and combiners are designed from users' array-response vectors before digital zero-forcing is applied.
  • Rate bounds: The proposed algorithm's rate admits a lower bound whose dependence separates channel gains from users' angles of departure.This separation supports optimality claims in some cases and insights into gains over analog-only solutions.
  • Rate bounds: At high SNR, each user's average achievable rate has the same slope as the single-user rate and remains within a constant gap.The gap depends only on the number of users and BS antennas.
  • Large arrays: As the number of BS antennas increases, the gap between the algorithm and the single-user rate decreases as the BS array response becomes better conditioned.The singular-value ratio approaches one, bringing the bound's factor closer to one.
  • Analog-only comparison: As the number of MS antennas tends to infinity, the expected rate advantage over analog-only beamsteering diverges under the single-path assumptions.By contrast, as BS antennas tend to infinity, RF beamsteering alone becomes optimal.

B. Large-dimensional Regime

The large-dimensional analysis uses a virtual channel model to obtain rate bounds for sparse multipath mmWave channels and characterize asymptotic behavior as antenna numbers grow.

  • Scope: The virtual-channel bound is an approximation that becomes accurate when the antenna count is very large and is valid only for uniform arrays.The stated array examples are ULAs and UPAs.
  • Virtual channel model: The virtual channel model expresses physical channels through fixed transmit and receive virtual directions shared by users with the same array sizes.For special virtual angles, the array-response matrices are DFT matrices.
  • Virtual channel model: Each virtual channel element aggregates the gains of physical paths associated with a transmit-receive direction pair.The model also incorporates angle spread through spatial-spreading functions.
  • Virtual channel model: The model provides a common transmit-eigenvector space across users and simplifies analysis for larger angle spreads.This common space is used to analyze the proposed multi-user hybrid precoding algorithm.
  • Rate bound: For arbitrary path counts, the analysis derives a lower bound on the algorithm's achievable rate and gives asymptotic insights for the multipath case.The bound uses a sparse-channel representation and a simplified effective-channel condition.
  • Asymptotic behavior: The bound approaches the single-user rate as antenna numbers grow and is tight when the number of paths is small relative to the number of antennas.It also indicates that paths beyond the strongest contribute relatively little when the channel is sufficiently sparse.

VI. RATE LOSS WITH LIMITED FEEDBACK

The limited-feedback analysis characterizes rate loss from jointly quantizing RF and baseband components, using single-path and large-dimensional cases to draw broader conclusions.

  • Scope and objective: The analysis considers finite RF and digital codebooks and quantifies the rate loss caused by their joint quantization.The special-case analysis is intended to provide conclusions about hybrid precoding over finite-rate feedback channels.

A. Single-Path Channels

For single-path channels with quantized beamsteering and effective-channel feedback, the analysis bounds rate loss and relates required feedback bits to system parameters.

  • Quantized design: The single-path analysis quantizes RF precoders, RF combiners, and each user's effective channel using finite codebooks.Each MS is assumed to know its channel perfectly.
  • Rate-loss bound: Theorem 7 provides an upper bound on the rate loss caused by quantization.The bound determines how RF and baseband quantization bits should scale to remain within a constant gap of the optimal rate.
  • Feedback scaling: To maintain a rate loss of log2 (b) bps/Hz per user, the required effective-channel quantization bits increase linearly with SNR in dB.They increase logarithmically with the number of antennas for a fixed number of users.
  • Feedback scaling: Poor RF beamsteering quantization increases the number of baseband quantization bits needed to control rate loss.This occurs when |µBS| and |µMS| are small.
  • Hybrid versus analog-only: When the effective channel is poorly quantized, analog-only beamforming can outperform hybrid precoding.The comparison is used to identify settings where the digital layer is useful for managing multi-user interference.

B. Large-dimensional Regime

In the large-dimensional regime, the proposed hybrid precoding approaches single-user and unconstrained digital performance while outperforming analog-only beamsteering. Simulations further show robustness to multipath and cellular interference, but quantization quality remains important.

  • Large-dimensional analysis: Large antenna arrays make the derived bound tight, and considering only the maximum-gain path gives very good performance.The marginal impact of additional paths becomes small when L−1 NBSNMS ≪1.
  • Large-dimensional analysis: Hybrid precoding approaches the single-user rate as the number of BS antennas grows and remains above beamsteering.The hybrid–single-user gap decreases at large BS-array sizes, while the hybrid–beamsteering gap remains considerable.
  • Rate comparisons: Hybrid precoding achieves almost the same performance as unconstrained block diagonalization and gains over analog-only beamsteering, especially at higher SNR.Beamsteering becomes interference limited as SNR increases.
  • Rate comparisons: The hybrid–beamsteering performance gap increases with the number of MS antennas, giving hybrid precoding greater gains for large receiver arrays.This behavior coincides with Corollary 5.
  • Quantization: Quantization bits must increase with antenna numbers to avoid significant degradation, although the hybrid gain stays almost constant for equal antenna counts.The result is shown for RF quantization with multipath channels.
  • Quantization: Hybrid precoding retains its gain over analog-only beamsteering with reasonable effective-channel quantization, whereas poor quantization can reverse the comparison.With poorly quantized effective channels, interference-management loss exceeds the hybrid gain.
  • Cellular evaluation: In cellular simulations, hybrid precoding provides a coverage gain over analog-only beamsteering, especially when many users are served simultaneously.The coverage metric is per-user P(Ru ≥ η) under inter-cell interference.

VIII. CONCLUSIONS

The paper proposes low-complexity hybrid analog/digital precoding for multi-user mmWave downlink systems using sparse channels and large antenna arrays. Its analyses and simulations show asymptotic gains over beamsteering, while limited feedback makes quantization quality a practical boundary.

  • Contribution: The paper proposes a low-complexity hybrid analog/digital precoding algorithm for downlink multi-user mmWave systems.The design leverages sparse channels and the large number of deployed antennas.
  • Analysis: The algorithm is analyzed for single-path channels and very large system dimensions, where its asymptotic optimality and gain over beamsteering are illustrated.These are the paper’s principal analytical regimes.
  • Implication: Interference management remains necessary in multi-user mmWave systems even when the number of antennas is very large.The conclusion is stated for the analyzed large-array setting.
  • Limited feedback: With limited feedback, the paper analyzes and simulates average rate loss from joint analog/digital codebook quantization.The quantization analysis concerns finite feedback through analog and digital codebooks.
  • Limited feedback: The simulations indicate that the hybrid precoding gain is not very sensitive to RF-angle quantization when quantization is sufficiently effective.Good RF-angle quantization is important for maintaining a reasonable gain over analog-only solutions.
  • Future work: Efficient multi-user cellular precoding and channel-estimation algorithms accounting for out-of-cell interference remain future work.The paper identifies this as an open direction rather than a result of the proposed method.

APPENDIX A

The appendix derives large-array consequences for hybrid precoding relative to RF-only beamsteering. As BS dimensions grow, steering-vector orthogonality simplifies the effective channel and supports the resulting rate bounds.

  • Large-array limit: As NBS→∞, the off-diagonal channel terms vanish with probability one, yielding BSABS = IU.The resulting matrix ABS is semi-unitary with σmax(ABS) = σmin(ABS) = 1.
  • Rate comparison: The appendix bounds the gain of Algorithm 1 over RF-only beamsteering using the hybrid-precoding rate lower bound.The comparison uses FRF = ABS for beamsteering alone.
  • Rate comparison: As NMS→∞, the first term in the gain bound diverges while the remaining terms stay constant.This establishes an asymptotically increasing gain over beamsteering under the stated assumptions.

APPENDIX D

The appendix analyzes rate loss from quantized analog and effective-channel codebooks. It models the receiver’s quantized effective channel, constructs digital zero-forcing from it, and bounds the resulting loss.

  • Quantized feedback: The receiver quantizes its effective channel with an RVQ codebook and feeds the quantized representation back to the BS.The effective channel includes the effects of quantized beamsteering directions.
  • Digital precoding: Using the quantized effective channel, the BS constructs and column-normalizes a digital zero-forcing precoder.The zero-forcing vectors lie in the null space of the other users’ channel vectors.
  • Rate-loss bound: The average per-user rate loss from joint RF and baseband quantization is bounded relative to the unquantized case.The derivation compares rates after quantization with rates without RF and baseband quantization.
  • Rate-loss bound: The effective channel occupies a subspace, so its average quantization error is no greater than the full-space case, which is upper bounded by 2−BBB.This subspace observation supplies the final quantization-error bound.
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