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Beam Squint and Channel Estimation for Wideband mmWave Massive MIMO-OFDM Systems

Bolei Wang, Mengnan Jian, Feifei Gao, Geoffrey Ye Li, Shi Jin, Hai Lin

arXiv:1903.01340v1cs.IT

TL;DR

Large-array propagation delays cause beam squint, so conventional MIMO-based channel estimation and precoding do not apply directly. The paper develops a beam-squint-aware super-resolution estimation scheme for FDD hybrid-precoding systems, using reciprocal path parameters to simplify downlink estimation and covariance construction. Numerical results demonstrate superiority over conventional methods.

  • Problem

    Spatial-wideband effects make different OFDM subcarriers observe distinct AoAs for the same path, limiting conventional MIMO-based channel estimation and precoding.

  • Method

    A super-resolution compressive-sensing approach extracts uplink path parameters and uses their FDD reciprocity to estimate downlink gains and reconstruct channels with limited training.

  • Results

    Numerical results demonstrate superiority of the proposed channel model and estimation strategies over conventional algorithms under general mmWave system configurations.

  • Takeaways & Limitations

    Frequency-insensitive path parameters can support simplified FDD downlink estimation and covariance-based MMSE channel estimation.

Abstract

from arXiv · show

With the increasing scale of antenna arrays in wideband millimeter-wave (mmWave) communications, the physical propagation delays of electromagnetic waves traveling across the whole array will become large and comparable to the time-domain sample period, which is known as the spatial-wideband effect. In this case, different subcarriers in an orthogonal frequency division multiplexing (OFDM) system will "see" distinct angles of arrival (AoAs) for the same path. This effect is known as beam squint, resulting from the spatial-wideband effect, and makes the approaches based on the conventional multiple-input multiple-output (MIMO) model, such as channel estimation and precoding, inapplicable. After discussing the relationship between beam squint and the spatial-wideband effect, we propose a channel estimation scheme for frequency-division duplex (FDD) mmWave massive MIMO-OFDM systems with hybrid analog/digital precoding, which takes the beam squint effect into consideration. A super-resolution compressed sensing approach is developed to extract the frequency-insensitive parameters of each uplink channel path, i.e., the AoA and the time delay, and the frequency-sensitive parameter, i.e., the complex channel gain. With the help of the reciprocity of these frequency-insensitive parameters in FDD systems, the downlink channel estimation can be greatly simplified, where only limited pilots are needed to obtain downlink complex gains and reconstruct downlink channels. Furthermore, the uplink and downlink channel covariance matrices can be constructed from these frequency-insensitive channel parameters rather than through a long-term average, which enables the minimum mean-squared error (MMSE) channel estimation to further enhance performance. Numerical results demonstrate the superiority of the proposed scheme over the conventional methods in mmWave communications.

I. INTRODUCTION

Large antenna arrays and wide bandwidth create spatial-wideband effects that produce beam squint, making conventional MIMO-based estimation and precoding inadequate. The paper proposes beam-squint-aware uplink and downlink estimation for FDD hybrid-precoding systems using physical channel parameters and limited feedback.

  • Motivation: Large arrays can create antenna-dependent propagation delays, so conventional channel models and algorithms for estimation and precoding require revision.The spatial-wideband effect becomes relevant when propagation-delay differences across antennas are no longer negligible.
  • Motivation: Beam squint makes different OFDM subcarriers observe distinct AoAs for the same physical path and can steer them toward different directions.The effect is especially important for physical-angle-based approaches and broadband transmission.
  • Motivation: In FDD hybrid transceivers, fixed analog phase settings cannot generate frequency-dependent steering vectors across subcarriers.The paper addresses this by using multiple RF chains and subcarrier-specific digital precoding.
  • Proposed approach: The proposed scheme estimates frequency-insensitive path parameters and complex gains with super-resolution compressed sensing, then simplifies FDD downlink estimation through parameter reciprocity.Only a small amount of downlink training and user feedback is needed to obtain downlink gains and reconstruct channels.
  • Results: Covariance matrices are reconstructed from frequency-insensitive parameters, enabling MMSE estimation, while numerical results show superiority over existing algorithms under general mmWave configurations.The approach constructs both uplink and downlink covariances without relying on long-term averaging.

II. BEAM SQUINT IN WIDEBAND MASSIVE MIMO SYSTEMS

The wideband channel model accounts for antenna-dependent propagation delays in massive MIMO-OFDM systems. These delays make steering vectors frequency-dependent, producing beam squint when array size and bandwidth are sufficiently large.

  • System model: The system uses a BS ULA with M antennas, antenna spacing d, and OFDM with Nc subcarriers over bandwidth W.The subcarrier spacing is η = W/Nc.
  • Channel model: Each user-to-BS channel comprises Lk incident paths with path-specific delays, angles, and complex gains.The model indexes paths by k and l and antennas by m.
  • Beam squint: The resulting wideband channel model has frequency-dependent steering vectors, which constitute the beam squint effect.The model can also be extended to multidimensional antenna configurations.
  • Spatial-wideband effect: The spatial-wideband model retains antenna-dependent delay terms that conventional models neglect when delay differences are assumed negligible.The neglected terms become important when array size or transmission bandwidth increases.
  • Spatial-wideband effect: A 128-antenna half-wavelength ULA can produce a maximum delay of 1.36Ts at 28 GHz with 600 MHz bandwidth.This example illustrates a regime in which steering vectors become frequency-dependent.

B. Beam Squint over OFDM Subcarriers

The spatial-wideband effect induces beam squint: each path shifts across angular indices as the OFDM subcarrier changes. The maximum squint is approximately the path’s propagation delay across the antenna array measured in sample periods.

  • After DFT transformation, the path power concentrates near an angular index that varies with subcarrier index.
  • The spatial-wideband effect induces each path in the angle domain to squint along with subcarrier indices.
  • The maximum angular squint is approximately the propagation delay across the antenna array in sample periods.
  • Theorem 1 establishes the relationship between beam squint and the spatial-wideband effect, illustrated by different observed path angles across subcarriers.
  • The effect is less apparent in conventional small MIMO systems because propagation delay across antennas is small and can be neglected.

III. SYSTEM MODEL WITH SPATIAL-WIDEBAND EFFECT

The system model represents wideband mmWave channels with hybrid analog/digital precoding while retaining spatial-wideband effects. It uses pilot observations and a sparse angle-delay representation to support channel covariance construction and FDD reciprocity.

  • The model considers wideband mmWave massive MIMO channels whose frequency-domain manifestation of spatial-wideband effects is beam squint.
  • Hybrid precoding combines an analog phase-shifter combiner with a digital baseband combiner that varies by subcarrier and OFDM block.
  • The received pilot model assigns subcarriers to users and collects noisy antenna observations across successive OFDM blocks.
  • The wideband channel has a sparse representation using a basis that explicitly considers beam squint.
  • Under zero-mean independent multipath gains, the channel covariance is formed from path parameters, and AoA-delay reciprocity permits downlink covariance reconstruction.

IV. INITIAL UPLINK CHANNEL PARAMETER EXTRACTION

The initial uplink extraction estimates each path’s AoA, delay, and complex gain from pilot observations using gridless compressive sensing that accounts for beam squint. The iterative formulation avoids predefined angle-delay grids and reaches a stationary point through gradient-based updates.

  • Initial parameter extraction: Initial parameter extraction estimates each path’s AoA, time delay, and complex gain before subsequent multi-user uplink and downlink channel estimation.The extracted parameters are intended to support later channel-estimation stages.
  • Problem formulation: Because the physical parameter dimension is much smaller than the received-signal dimension, compressive sensing is used to formulate parameter extraction as sparse recovery.The stated condition is 3Lk ≪ NRFPTup.
  • Gridless extraction: The proposed gridless off-grid approach keeps the dictionary unknown during iterative extraction, avoiding the grid mismatch of discretized AoAs and delays.On-grid methods degrade when continuous parameters fall between predefined grid points.
  • Beam-squint-aware model: The channel basis uses frequency-dependent steering vectors, thereby incorporating beam squint into uplink channel expansion and parameter recovery.This basis is explicitly contrasted with approaches that ignore beam squint.
  • Iterative optimization: The optimization combines a sparsity objective with a data-fitting term and applies majorization-minimization followed by gradient descent over angle and delay parameters.The objective is non-increasing and the iterations reach a stationary point.

C. Parameters Selection

The parameter-selection procedure balances sparsity, fitting accuracy, convergence, and numerical stability through scheduled regularization and pruning choices. It gradually reduces the smoothing parameter while warning that excessively small values can slow convergence and cause ill-conditioning.

  • ϵ selection: ϵ must be chosen cautiously because excessively small values can cause very slow convergence and ill-conditioned objectives or matrices near zero-valued gains.The divide-by-zero problem is identified when [β]l is approximately zero.
  • λ selection: The regularization parameter λ trades sparsity against data-fitting deviation, with larger λ improving fit while increasing the possibility of overestimation.The method sets λ using the inverse noise variance and dynamically adjusts it toward a threshold.
  • LM selection: The initial path-count parameter LM is reduced by pruning gains below βmin, so the retained nonzero gains approach the number of actual channel paths.LM starts relatively large and the number of retained paths decreases during iteration.

ALGORITHM: ITERATIVE PARAMETERS EXTRACTION FOR UPLINK CHANNELS

The iterative uplink procedure extracts physical path parameters, groups users by angle-delay separability, and updates channel gains with pilot-efficient LS or MMSE estimation. Frequency-insensitive parameters can be reused, reducing later training requirements.

  • Iterative extraction: The algorithm uses a large initial path count, prunes weak gains below βmin, and stops when successive gain estimates differ by less than γT.The path-count setting should remain in the same order of magnitude as the actual number of paths.
  • Parameter reuse: For a user 500 meters from the base station moving at 80 km/h with W = 600 MHz, the maximum changes within 1 ms are 2.5×10^-3 degrees in AoA and 7.4×10^-2 ns in delay.This example supports treating the physical parameters as relatively stable over short intervals.
  • User grouping: Users with sufficiently distinct angle-delay channel signatures are assigned to the same training group and can transmit identical pilots without indistinguishable-path interference.The grouping criterion uses an orthogonality distance threshold ΩU.
  • Parameter reuse: The scheme reuses frequency-insensitive AoAs and path delays, so only channel gains need re-estimation or updating for new channel realizations.The approach relies on angles and delays varying more slowly than channel gains.
  • Pilot assignment: Pilot resources are reduced after initial extraction because users in a group share pilot resources across selected subcarriers and blocks.The text states that this saves many pilot resources compared with the initial parameter-extraction stage.

B. LS Estimator

The LS/MMSE stage updates uplink complex gains and reconstructs channels using the extracted physical parameters. Covariance matrices can be built efficiently from AoAs, delays, and gain statistics, while FDD downlink estimation exploits AoA-delay reciprocity with low training and feedback overhead.

  • LS estimation: The LS estimator updates uplink complex gains and reconstructs the user’s channel over all subcarriers using the extracted channel basis.The same physical parameters define the channel basis across subcarriers.
  • MMSE estimation: Channel covariance matrices are constructed efficiently from AoAs, path delays, and gain statistics rather than requiring the conventional long-term averaging process.The gain statistics can come from fewer samples or even a single initial estimate.
  • MMSE estimation: The constructed covariance matrices perform favorably against the true covariance matrices for channel estimation.The comparison is reported for the covariance-based estimation evaluation.
  • FDD downlink estimation: In FDD systems, downlink estimation exploits AoA-delay reciprocity and channel sparsity to require significantly low training overhead and limited user feedback.Only downlink complex gains need to be fed back for downlink-channel reconstruction.
  • Beam-squint mitigation: Ignoring beam squint causes severe performance degradation, whereas cooperative use of several RF chains generates frequency-dependent beamforming vectors across subcarriers.A single RF chain cannot realize the required frequency-dependent steering vectors.

A. Downlink Channel Model and User Grouping

The downlink design groups users by angular separation, reconstructs frequency-dependent channels from uplink physical parameters and updated gains, and uses frequency-dependent beamforming to address beam squint.

  • Channel formulation: Downlink channel parameters such as AoAs and path delays are obtained from uplink extraction and used to formulate the downlink channel.The downlink frequency-insensitive parameters are computed from the uplink version and the channel is reconstructed across subcarriers.
  • User grouping: Users are grouped according to angular separation because path delays are unknown and users need not be synchronized.The grouping uses an angular guard distance rather than delay information.
  • Downlink training: Successive OFDM blocks train grouped users on identical pilot subcarriers while combining their received signals across blocks.The received-signal model includes analog and digital precoders, pilot symbols, and additive Gaussian noise.
  • Hybrid precoding: The analog precoder stacks orthogonal steering vectors, while diagonal digital precoders form frequency-dependent spatial beams for each subcarrier.The digital design uses steering-vector combinations and orthogonal pilot rows to manage users within a group.
  • Beam-squint compensation: Different subcarriers use frequency-dependent beamforming vectors to address the beam squint effect.The construction assigns subcarrier-specific beamforming vectors while suppressing inter-user interference within groups.

C. Downlink Channel Estimation with LS or MMSE estimator

Downlink estimation updates complex gains with limited pilots and reconstructs channels from those gains together with uplink-derived physical parameters, using LS or MMSE estimation.

  • Covariance construction: The downlink channel covariance matrix is constructed from the extracted physical parameters, with path-power terms estimated from gains or a single initial gain estimate.The covariance construction can use average previous gains or one complex-gain estimate from initial uplink extraction.
  • MMSE estimation: MMSE downlink complex-gain estimates are determined from the constructed covariance and the downlink observation model.The section presents the MMSE estimator after defining the covariance-based formulation.
  • Estimation procedure: The downlink strategy first extracts uplink physical parameters, then estimates downlink gains from pilots, and finally reconstructs downlink channels.The reconstruction combines AoAs and path delays from the initial stage with updated downlink gains.
  • Feedback and reconstruction: The complete strategy uses limited downlink training and feedback: users estimate gains by LS or MMSE and return them to the base station for reconstruction.Users in the same group can be trained efficiently in the same time-frequency band.

VII. SIMULATION RESULTS

Simulations show that accounting for beam squint preserves estimation quality as array size, bandwidth-related squint, RF resources, and frequency reuse vary, whereas conventional models develop severe error floors or degradation.

  • Initial parameter extraction: With increasing beam squint, conventional compressed sensing fails to extract path AoAs and delays, while the proposed algorithm maintains consistent performance.The conventional method assumes identical observed AoAs across subcarriers, causing unpredictable averaged estimates under frequency-dependent beam squint.
  • Initial parameter extraction: Conventional extraction can worsen with more antennas or larger bandwidths because beam squint increases model mismatch.The paper identifies this worsening as a counterproductive effect of ignoring frequency-dependent observed AoAs.
  • Uplink estimation: With more BS antennas, the proposed uplink MMSE approach maintains estimation performance, whereas the beam-squint-ignoring approach suffers increasing error floors.The comparison uses NMSE versus SNR with M = 16, 32, and 64 antennas and W = 600 MHz.
  • Uplink estimation: Increasing RF chains and OFDM blocks improves the proposed estimation, while the conventional approach cannot benefit because of high error floors.This comparison varies RF-chain and OFDM-block counts while holding the antenna configuration fixed.
  • Covariance-aided estimation: MMSE estimators using computed covariance matrices perform similarly to those using known covariance, with differences attributed to AoA and delay estimation errors.The computed covariance provides an excellent approximation because the relevant channel subspace depends on AoAs and path delays.
  • Multi-user estimation: MMSE outperforms LS under frequency reuse by reducing inter-user interference, especially when the number of users is large.The same pattern appears for downlink estimation, where MMSE maintains strong performance while LS degrades with many reused users.
  • Downlink beamforming: Conventional beamforming can create up to 18 dB of energy difference between subcarriers, producing large spectral-efficiency discrepancies.Beam-squinted subcarriers may direct energy into sidelobes or nulls instead of the intended users.

VIII. CONCLUSIONS

The paper develops a beam-squint-aware wideband channel model and FDD estimation strategies that recover physical parameters, update gains with limited training and feedback, and outperform conventional-model approaches in simulations.

  • Contribution: The proposed FDD strategy targets wideband mmWave massive MIMO systems with hybrid precoding and explicitly accounts for beam squint.The contribution covers both uplink and downlink channel estimation.
  • Channel model: The channel model uses physical parameters and frequency-dependent steering vectors to represent the beam squint effect.The paper reports that the model accurately depicts the wideband massive MIMO-OFDM channel.
  • Parameter extraction: A super-resolution compressive-sensing method extracts channel path parameters, separating frequency-sensitive AoAs and delays from frequency-insensitive complex gains.These extracted parameters support the subsequent uplink and downlink estimation strategies.
  • Channel estimation: The proposed uplink and downlink strategies estimate and update channels with a small amount of training and user feedback in FDD systems.The conclusion presents this as a central practical property of the proposed strategies.
  • Results: Numerical results demonstrate superiority of the proposed channel model and estimation strategies over algorithms based on conventional MIMO models.The comparison is stated for general mmWave system configurations.
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