PaperlayerView exact revision on arXiv

Source-linked AI summary

Multi-user Precoding and Channel Estimation for Hybrid Millimeter Wave Systems

Lou Zhao, Derrick Wing Kwan Ng, Jinhong Yuan

arXiv:1703.04060v1cs.IT

TL;DR

Hybrid mmWave systems require channel estimation despite having far fewer RF chains than antennas, while existing methods may depend on sparse channels. The paper proposes strongest-AoA-based estimation with analog beamforming and orthogonal pilots, followed by digital ZF precoding. Simulations and analysis report performance approaching fully digital systems and robustness to several hardware imperfections.

  • Problem

    Channel estimation is challenging in hybrid mmWave systems because RF chains are much fewer than antennas, and existing methods may not suit non-sparse channels.

  • Method

    The scheme estimates strongest AoAs using frequency tones, designs analog beamformers, estimates the equivalent channel with orthogonal pilots, and applies digital ZF precoding.

  • Results

    The proposed scheme approaches fully digital rate performance and remains unbounded with increasing SNR under phase and transceiver beamforming errors.

  • Takeaways & Limitations

    The channel-estimation and multi-user downlink precoding scheme is applicable to sparse and non-sparse channels and is robust to the considered system imperfections.

Abstract

from arXiv · show

In this paper, we develop a low-complexity channel estimation for hybrid millimeter wave (mmWave) systems, where the number of radio frequency (RF) chains is much less than the number of antennas equipped at each transceiver. The proposed mmWave channel estimation algorithm first exploits multiple frequency tones to estimate the strongest angle-of-arrivals (AoAs) at both base station (BS) and user sides for the design of analog beamforming matrices. Then all the users transmit orthogonal pilot symbols to the BS along the directions of the estimated strongest AoAs in order to estimate the channel. The estimated channel will be adopted to design the digital zero-forcing (ZF) precoder at the BS for the multi-user downlink transmission. The proposed channel estimation algorithm is applicable to both nonsparse and sparse mmWave channel environments. Furthermore, we derive a tight achievable rate upper bound of the digital ZF precoding with the proposed channel estimation algorithm scheme. Our analytical and simulation results show that the proposed scheme obtains a considerable achievable rate of fully digital systems, where the number of RF chains equipped at each transceiver is equal to the number of antennas. Besides, by taking into account the effect of various types of errors, i.e., random phase errors, transceiver analog beamforming errors, and equivalent channel estimation errors, we derive a closed-form approximation for the achievable rate of the considered scheme. We illustrate the robustness of the proposed channel estimation and multi-user downlink precoding scheme against the system imperfection.

I. INTRODUCTION

The paper addresses channel estimation and precoding challenges in multi-user hybrid mmWave systems, where RF chains are fewer than antennas and existing methods may rely on sparsity. It proposes a non-feedback, non-iterative approach applicable to sparse and non-sparse channels, analyzes digital ZF rates and hardware imperfections, and reports near-fully-digital performance.

  • Motivation: Hybrid mmWave systems complicate conventional pilot-aided channel estimation because they use far fewer RF chains than antennas.Allocating analog and digital beams and estimating equivalent baseband channels remain open research problems.
  • Motivation: Existing mmWave channel-estimation methods often exploit sparsity, which may not hold in urban environments with substantial scattering.Field measurements and ray-tracing studies report non-negligible scattering clusters in urban propagation channels.
  • Contributions: The paper proposes a non-feedback, non-iterative channel-estimation algorithm applicable to both sparse and non-sparse multi-user mmWave channels.It also analyzes digital ZF precoding using the estimated channel and considers random phase, beamforming, and channel-estimation errors.
  • Contributions: The three-step scheme estimates strongest AoAs at the BS and users, designs analog beamformers, and uses orthogonal pilots for equivalent channel estimation.The estimated equivalent channel supports digital ZF precoder design for multi-user downlink transmission.
  • Results: The hybrid system’s average achievable-rate gap relative to the fully digital system is only  bits/s/Hz in the large-antenna regime.The supplied passage preserves the unit but not the numeric value of the reported gap.
  • Results: Phase and transceiver beamforming errors do not cause an achievable-rate ceiling, while the paper derives a closed-form approximation under multiple error sources.The analysis covers high receiver SNR and large antenna arrays.

III. PROPOSED CHANNEL ESTIMATION FOR HYBRID SYSTEM

The proposed estimator uses frequency tones and angular searches to identify strongest AoAs at the BS and users, then uses those directions for analog beamforming and equivalent-channel estimation. Orthogonal pilots produce equivalent CSI for digital ZF precoding in the downlink.

  • Challenge: Conventional fully digital pilot-aided estimation is not directly applicable because hybrid transceivers have fewer RF chains than antennas and unknown analog beamformers.This creates a fundamental channel-estimation challenge for the considered hybrid system.
  • AoA estimation: The first two steps transmit unique unmodulated frequency tones and linearly search angular directions to estimate strongest AoAs at the BS and users.The estimated AoAs determine analog transmit and receive beamforming matrices.
  • Step 1: The BS estimates uplink AoAs from simultaneously transmitted user-specific tones and uses the corresponding strongest directions to construct its analog beamforming matrix.Each RF chain downconverts its assigned tone before low-pass filtering removes other tones.
  • Step 3: The estimated equivalent channel is used to design the BS digital ZF precoder for downlink transmission.Algorithm 1 explicitly places equivalent-channel estimation before digital ZF precoder design.
  • Step 2: The BS retransmits frequency tones through its analog beamformer, allowing users to estimate downlink AoAs and construct their analog beamforming matrix.Users perform a similar search over candidate directions and select the maximum response.
  • Step 3: Users transmit orthogonal pilots through their analog beamformers, while the BS receives them through its analog beamforming matrix to estimate the equivalent channel.The equivalent channel combines the BS beamformer, mmWave channel, and user beamformer; all users’ equivalent CSI can be obtained simultaneously.

B. Performance Analysis of Proposed Channel Estimation

The proposed equivalent-channel estimation improves as pilot energy and antenna counts increase, with noise effects vanishing asymptotically. Simulations compare its normalized MSE against fully digital pilot-aided LS estimation.

  • Higher pilot energy and larger BS or user antenna arrays reduce the proposed equivalent-channel estimator’s normalized MSE.The reduction is attributed to lower effective noise and larger array gains from analog beamforming.
  • As BS and user antenna counts approach infinity, noise-induced estimation effects vanish for the proposed hybrid scheme.The analysis contrasts this asymptotic noise mitigation with fully digital massive-MIMO pilot-aided estimation.
  • Figure 4 compares normalized MSE for the proposed hybrid pilot-aided estimator with conventional pilot-aided LS in a fully digital system.The figure reports equivalent-channel MSE versus total BS and user antenna counts; simulations match the analytical result in Equation (18).
  • Increasing either BS or per-user antenna count can meet a required channel-estimation MSE.The analysis states that increasing BS antennas can improve performance even when user-side antenna counts are limited.

IV. ZF PRECODING AND PERFORMANCE ANALYSIS

The paper analyzes digital ZF downlink transmission using the estimated equivalent channel and derives a closed-form achievable-rate upper bound for the hybrid system.

  • Digital ZF precoding is based on the estimated equivalent channel, which incorporates both BS and user analog beamforming matrices.The analysis also compares the hybrid system’s achievable rate with relevant fully digital performance.

A. ZF Precoding

The estimated equivalent channel is used to construct a baseband digital ZF precoder for multi-user downlink transmission. Within the estimated AoA directions, digital ZF suppresses multi-user interference.

  • A. ZF Precoding: The equivalent channel is assumed perfectly estimated in the high-SNR analysis before designing the baseband digital ZF precoder.This assumption is motivated by the proposed estimator being affected only by noise in the stated model.
  • A. ZF Precoding: Each ZF precoder column w_eq,k is assigned to user k, whose hybrid transceiver has one RF chain.Each user consequently receives one superimposed signal at each time instant after beamforming.
  • A. ZF Precoding: The received signal model includes user symbol energy, a transmit-power normalization factor, and effective noise.These terms enter the post-beamforming signal expression for user k.
  • A. ZF Precoding: Digital ZF suppresses multi-user interference within the estimated AoA directions.The received post-beamforming signal is then used to formulate each user’s SINR.

FRF (

The achievable-rate upper bound depends on channel characteristics and the BS analog beamformer, while large antenna arrays support asymptotic performance with RF chains equal to the number of users.

  • FRF (: The SINR expression is used to state an upper bound on per-user achievable rate for the proposed digital ZF precoding.The result is presented as Theorem 1 and proved by substitution into the preceding expressions.
  • FRF (: The per-user rate upper bound depends on the Rician K-factor and the BS analog beamforming matrix designed during channel estimation.The BS beamformer transmits each user’s signal through strong AoA directions.
  • FRF (: In the large-antenna regime, the hybrid system requires only as many RF chains as users.The asymptotic upper-bound analysis is stated for M →∞ and F_H^H F_H approaching I_N almost surely.
  • FRF (: Asymptotic proposed-precoding performance is mainly determined by the numbers of equipped antennas and RF chains.This observation follows from Equation (24).

C. Comparison with Fully Digital Systems

The paper compares hybrid and fully digital mmWave systems through achievable-rate bounds and simulations, examining how Rician K-factor and channel conditions affect their gap.

  • Analytical comparison: The fully digital reference assumes perfect CSI at users and the BS, with ZF precoding applied to the equivalent channel.This reference represents the maximal achievable-rate gap between the fully digital and hybrid systems.
  • Analytical comparison: The fully digital system’s achievable rate per user is bounded analytically, with a corresponding asymptotic bound in the large-antenna regime.The bound is used as a reference for evaluating the proposed hybrid system.
  • Analytical comparison: The large-antenna achievable-rate gap between the hybrid and fully digital systems is explicitly quantified and has a tight upper-bound characterization.The comparison is based on the respective bounds for the two systems.
  • Rician-channel effect: As the Rician K-factor increases, the hybrid and fully digital performance upper bounds converge because the line-of-sight component becomes dominant.The estimated strongest AoA then captures a larger portion of the relevant channel energy.
  • Simulation comparison: With M = 100, N = 10, and υ = 2, simulations verify tight upper bounds and show high hybrid sum-rate performance from ZF interference suppression.Figure 5 compares average achievable rate per user against SNR and rate against Rician K-factor under the stated configurations.

A. Transceiver Beamforming Errors and Random Phase Errors

The paper models transceiver beamforming and phase-shifter errors, derives achievable-rate approximations, and identifies power loss as the main rate penalty in the large-antenna regime.

  • Error model: The analysis considers high pilot transmit power so equivalent-channel estimation noise is negligible, focusing on phase-shifter and transceiver beamforming errors.The two error types arise respectively from phase-shifter noise or quantization and AoA estimation errors.
  • Error-aware estimation: The pilot-based equivalent-channel estimation incorporates the effects of phase and transceiver beamforming errors before digital ZF precoding.The resulting estimated equivalent channel is used to construct the downlink ZF precoder.
  • Transceiver beamforming errors: AoA estimation errors are represented by a power-loss coefficient ξ ∈ (0, 1] determined by the array beam pattern and error variance.When the error variance is no larger than half the HPBW, the half-power principle gives ξ ≈ 0.5.
  • Transceiver beamforming errors: The achievable-rate approximation under transceiver beamforming errors shows that the error matrices produce a finite power loss rather than an upper performance ceiling.The loss depends on ξ and can be compensated by increasing transmit power.
  • Joint impact: Joint random phase and transceiver beamforming errors degrade average achievable rate relative to ideal hardware, but additional transmission power can compensate the degradation.The joint effect is summarized through a large-antenna achievable-rate approximation.

B. Hardware Impairment and Imperfect Channel Estimation

The paper extends its rate analysis to imperfect equivalent CSI, deriving large-antenna approximations for ZF precoding under channel-estimation errors and hardware impairments.

  • Imperfect CSI model: The imperfect-CSI model expresses the estimated equivalent channel as the impaired equivalent channel plus normalized channel-estimation error.The normalized error variance is δ^2, defined through the equivalent-channel MSE.
  • Imperfect-CSI precoding: ZF precoding is constructed from the imperfect equivalent channel, and the received signal and user SINR include the resulting precoder error.The precoder normalization factor is tr(f_weq f_weq^H).
  • Rate approximation: Theorem 2 gives a high-SNR, large-antenna approximation for each user’s achievable rate under ZF precoding with imperfect hybrid CSI.The expression depends on the channel-estimation error variance and the diagonal element η_kk of K^-1.
  • Rate approximation: Corollary 4 provides the corresponding asymptotic average achievable rate per user for the hybrid system as M → ∞.This result summarizes the average-rate behavior under the imperfect-CSI model.
  • Hardware-impact comparison: The paper further summarizes the additional achievable-rate gap jointly caused by random phase errors and transceiver beamforming errors relative to ideal hardware.The gap is stated for the large-antenna regime and follows from the ideal- and impaired-hardware rate approximations.

VI. SIMULATION AND DISCUSSION

Simulations evaluate the proposed channel estimation and digital ZF precoding under sparse and non-sparse channels, hardware impairments, CSI errors, and varying antenna counts. The results show robustness to hardware imperfections and rate scaling with antenna resources.

  • The proposed algorithm achieves higher achievable rates than the compared method for non-sparse channels and outperforms analog-only beamforming through digital ZF precoding.For sparse single-path channels, its achievable rate matches the compared algorithm.
  • With random phase and transceiver beamforming errors, the achievable rate remains unbounded as SNR increases, with only a small gap from perfect hardware.The simulations also verify a 3 dB extra power consumption caused by these hardware impairments.
  • At normalized CSI MSE δ2 = 0.005, Figure 7 compares imperfect equivalent CSI under perfect hardware with imperfect CSI and hardware errors to validate Theorem 2.The corresponding high-SNR asymptotic approximations are represented by separate comparison curves.
  • ΔGap ≈ 1 bits/s/Hz characterizes the performance gaps between the hardware-error and perfect-hardware cases, as predicted by Equation (49).
  • With CSI estimation errors, the proposed system’s achievable rate increases with BS antennas and has a slope similar to the fully digital system.Similar behavior occurs when increasing user antennas, while larger antenna arrays provide higher array gains.

VII. CONCLUSIONS

The paper concludes that its low-complexity strongest-AoA channel estimation supports sparse and non-sparse MU hybrid mmWave channels. Analysis and simulations show near-fully-digital rates under suitable conditions and robustness to key hardware errors.

  • The proposed low-complexity channel estimation exploits the strongest AoA and applies to both sparse and non-sparse MU hybrid mmWave channels.Its achievable-rate performance is evaluated with analog beamforming and digital ZF precoding based on the estimated channel.
  • The proposed scheme can approach fully digital rate performance with sufficiently large Rician K-factors.The paper also derives and verifies a high-SNR closed-form achievable-rate approximation incorporating phase, beamforming, and CSI errors.
  • The results show robustness against random phase errors and transceiver beamforming errors.

APPENDIX

The appendix develops analytical rate bounds and approximations for the hybrid-system analysis. It uses eigenvalue decomposition, convexity, Jensen’s inequality, SINR expressions, and large-antenna asymptotics.

  • A. Proof of Theorem 1: The proof begins by decomposing the positive definite Hermitian equivalent-channel matrix through eigenvalue decomposition.The trace of its eigenvalues equals the trace of the matrix, enabling reformulation of the power-normalization factor.
  • A. Proof of Theorem 1: The derivation applies the strictly decreasing convex function f(x) = x−1 and Jensen’s inequality to obtain the required inequality and rewrite Equation (50).
  • The error analysis starts from the receive SINR and separates the interference term due to errors before applying asymptotic approximations.Negligible terms are omitted, and the large-antenna limit uses K = bHTeq ≈ M→∞ ξMPGL with ξ ∈ (0, 1].
Loading 1703.04060v1…