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Channel Estimation for TDD/FDD Massive MIMO Systems with Channel Covariance Computing

Hongxiang Xie, Feifei Gao, Shi Jin, Jun Fang, Ying-Chang Liang

arXiv:1710.00704v1cs.IT

TL;DR

Massive-MIMO CSI acquisition is challenging because CCM-based estimation requires costly channel statistics, especially for FDD downlink systems. The paper reconstructs uplink and downlink CCMs from instantaneous uplink CSI using angle and PAS estimates, then uses them for channel estimation. The scheme improves uplink estimation without additional training and supports downlink CSI estimation in both TDD and FDD systems.

  • Problem

    Massive-MIMO CSI acquisition is challenging because obtaining CCMs requires costly channel estimates, while FDD systems lack direct channel reciprocity for downlink CCM acquisition.

  • Method

    The method extracts multipath angles and PAS from instantaneous uplink CSI, reconstructs the uplink CCM, infers the downlink CCM through angle and PAS reciprocity, and applies eigen-beamforming for downlink CSI estimation.

  • Results

    The reconstructed uplink CCM improves uplink channel estimation without additional training, while inferred downlink CCMs enable downlink CSI estimation even in FDD systems.

  • Takeaways & Limitations

    The strategy supports CCM-based channel estimation for both TDD and FDD massive MIMO using arbitrary PAS distributions and array geometries.

Abstract

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In this paper, we propose a new channel estimation scheme for TDD/FDD massive MIMO systems by reconstructing uplink/downlink channel covariance matrices (CCMs) with the aid of array signal processing techniques. Specifically, the angle information and power angular spectrum (PAS) of each multi-path channel is extracted from the instantaneous uplink channel state information (CSI). Then, the uplink CCM is reconstructed and can be used to improve the uplink channel estimation without any additional training cost. In virtue of angle reciprocity as well as PAS reciprocity between uplink and downlink channels, the downlink CCM could also be inferred with a similar approach even for FDD massive MIMO systems. Then, the downlink instantaneous CSI can be obtained by training towards the dominant eigen-directions of each user. The proposed strategy is applicable for any kind of PAS distributions and array geometries. Numerical results are provided to demonstrate the superiority of the proposed methods over the existing ones.

I. INTRODUCTION

The paper addresses difficult CSI acquisition in TDD/FDD massive MIMO by reconstructing channel covariance matrices from array-processing estimates of multipath angles and PAS. The reconstructed CCMs improve uplink estimation without extra training and enable downlink estimation, including in FDD systems.

  • Massive-MIMO CCM acquisition is difficult because the required number of channel estimates grows linearly with array size, making accuracy and feasibility questionable.
  • Downlink CCM acquisition is especially costly because training and feedback are hardly affordable, while channel reciprocity does not apply to FDD systems.
  • Existing CCM methods either require already-acquired uplink CCMs or impose restrictive assumptions such as uniform PAS, narrow angular spread, and uniform linear arrays.
  • The proposed scheme extracts multipath angles and PAS from one instantaneous uplink channel estimate, then reconstructs uplink CCMs using large-scale array structure.
  • Reconstructed uplink CCMs improve channel estimation without additional training, while angle and PAS reciprocity allow downlink CCM inference and eigen-beamforming for FDD downlink CSI estimation.
  • The method is designed for both TDD and FDD massive MIMO and for arbitrary PAS distributions and array geometries.

II. PROBLEM FORMULATION

The paper models massive-MIMO propagation as clustered multi-path channels, with each component formed by a continuum of rays over an angular region. The resulting channel covariance matrix is determined by the angular response, power distribution, and array geometry.

  • A. System Model: The base station uses M ≫ 1 antennas in a finite-scattering environment with P resolvable, independent multi-path components.Each component corresponds to a cluster of scatterers producing a continuum of simultaneous rays.
  • A. System Model: For the p-th component, the channel integrates incident rays across the angular region A_p = [¯ϑ_p − ∆_p, ¯ϑ_p + ∆_p].The ray coefficient contains amplitude attenuation and phase, while the array manifold vector describes the antenna response.
  • A. System Model: The array manifold vector depends on the deployed array geometry and, for a ULA, on antenna spacing, carrier frequency, and propagation speed.For a ULA, its entries form frequency-dependent phase progressions across the M antennas.
  • A. System Model: The user channel frequency response combines the multi-path components with their respective propagation delays.The p-th path delay is denoted by τ_p.
  • A. System Model: The p-th channel covariance matrix is obtained from the angular integral of the array response weighted by its power angular spectrum S_p(ϑ).S_p(ϑ) characterizes power distribution over angle, and the covariance depends on the central AOA, angular spread, PAS, and array response.

B. Representation and Expansion of CCM

CCM reconstruction is reduced to estimating angular parameters and the PAS, then representing the covariance integral using either prior-based formulas or discrete expansions. Discrete angular sampling becomes more accurate as the number of terms increases, while closed-form approximations require narrow angular spread.

  • Angle and PAS estimation: Estimating a covariance matrix requires recovering both the channel angles and the PAS that weights them.The paper considers MUSIC, ESPRIT, compressive sensing, and DFT-based angle-acquisition methods.
  • Known PAS distribution: When the PAS is known, uniform or Laplacian models can yield simplified covariance expressions based on central angle and angular spread.For a uniform PAS, the covariance entries involve χ_mn and sinc(x), with approximations derived from a Taylor expansion.
  • Known PAS distribution: The closed-form covariance equations are approximations of the integral expressions under a very narrow angular-spread condition.If the angular spread is not small enough, the closed-form expressions may not accurately represent the covariance.
  • Unknown PAS distribution: Without a PAS prior, covariance reconstruction can use Monte Carlo integration or closed-form integral reconstruction when the corresponding approximation conditions apply.The paper names these procedures MC-iCCM and CF-iCCM, respectively.
  • Unknown PAS distribution: For a ULA, DFT expansion represents the covariance with M discrete PAS coefficients, making PAS estimation equivalent to estimating those coefficients.The coefficients are associated with the columns of the normalized M × M DFT matrix.
  • Unknown PAS distribution: An alternative uses non-orthogonal basis vectors or discrete samples within the actual angular region when the PAS distribution is unknown.The reconstructed covariance then requires the central angle, angular spread, and sampled PAS values.
  • Unknown PAS distribution: Increasing the number of discrete angular terms makes the sampled PAS a more accurate approximation of the continuous PAS and improves CCM approximation accuracy.The sampled values differ slightly from DFT expansion coefficients, but converge more accurately with more terms.

C. Inferring Downlink CCMs from Uplink CCMs

The method infers downlink CCMs from uplink angular information and PAS by exploiting approximate reciprocity, while accounting for carrier-frequency-dependent array responses. The reconstructed CCMs are then incorporated into limited-pilot channel estimation.

  • C. Inferring Downlink CCMs from Uplink CCMs: Downlink array responses differ from uplink responses because the array manifold depends on carrier frequency.The paper denotes the uplink and downlink carrier frequencies by f_u and f_d and introduces a frequency-dependent transformation.
  • C. Inferring Downlink CCMs from Uplink CCMs: The inference assumes reciprocal uplink and downlink angle parameters when the frequency discrepancy is not large.This links the uplink and downlink angular regions used in covariance reconstruction.
  • C. Inferring Downlink CCMs from Uplink CCMs: The method also assumes reciprocal PAS shapes between uplink and downlink channels, apart from a frequency-dependent scaling factor.These reciprocity assumptions are drawn from theoretical works and measurement tests.
  • C. Inferring Downlink CCMs from Uplink CCMs: Combining the uplink and downlink covariance relationships allows uplink angle parameters and PAS information to infer the downlink CCM, including in FDD systems.The discrete approximation follows by combining the covariance reconstruction expressions with the uplink/downlink transformation.
  • Transmission procedure: The described scheme focuses on one multi-path component at a time while using an additional user index for multi-user notation.The transmission phase is illustrated in Fig. 2.
  • Transmission procedure: The transmission procedure begins with an uplink preamble, updates instantaneous channel estimates using limited pilots, and dynamically updates users’ angle information.Reconstructed uplink and downlink CCMs are then used with MMSE estimators.

A. Preamble For Initial Angle Estimation and User Scheduling

The preamble phase obtains initial CSI and angular information for user grouping and scheduling, after which spatial signal-processing methods estimate the dominant angle regions. The selected spatial-rotation DFT method trades some angle-estimation accuracy for lower computational complexity than ML or CS.

  • A. Preamble For Initial Angle Estimation and User Scheduling: During the preamble, users apply conventional uplink training to obtain initial CSI estimates.The paper uses least squares as an example and sends this orthogonal training only once at transmission start.
  • A. Preamble For Initial Angle Estimation and User Scheduling: Initial angle intervals are extracted from the estimated CSI and used for initial user grouping and scheduling.These intervals provide the angular information needed by the subsequent estimation procedure.
  • 1) Initial Angle Estimation: The estimated angular information supports subsequent PAS and covariance reconstruction for the channel-estimation pipeline.The flow chart identifies the proposed procedure as the IC-pCCM scheme.
  • 1) Initial Angle Estimation: The channel representation discretizes the angle domain with steering vectors and uses a sparse coefficient vector whose nonzero entries identify the actual angle interval.The unknown angular grid spans [−π, π], while the true angles lie inside the component’s incident angular region.
  • 1) Initial Angle Estimation: The nonzero sparse coefficients facilitate estimation of the angle-of-arrival interval and its associated center and spread parameters.The method determines a spatial rotation parameter and an index set before inferring the angular range.
  • 1) Initial Angle Estimation: Maximum likelihood and compressive-sensing methods can provide high angle-estimation accuracy but require exhaustive search or iterative nonlinear optimization.Both methods therefore incur relatively high computational complexity.
  • 1) Initial Angle Estimation: The proposed angle-acquisition step adopts spatial-rotation-enhanced DFT because it offers a tradeoff between angle-estimation accuracy and computational complexity.The approach uses the spatial rotation matrix and selects a limited index set, then infers the AOA range from those indices.

2) Initial User Scheduling:

The scheme groups users with non-overlapping angle-of-arrival intervals so they can reuse training sequences without pilot contamination, reducing subsequent training overhead. It estimates channel gains over selected angles using either least squares or a spatial-frequency approach, while updating scheduling as users move.

  • Angle-domain grouping: Users with non-overlapping AOA intervals are assigned to the same group for training reuse.Different groups use orthogonal training sequences to remove inter-group interference.
  • Angle-domain grouping: Users in the same group reuse one training sequence without causing pilot contamination, reducing subsequent training overhead after the preamble.This strategy is called angle-division multiple access (ADMA).
  • Subsequent channel updating: The SBEM-based update estimates each group’s uplink channels from received training signals, with inter-group interference absent because groups use orthogonal sequences.The method uses the previous coherence interval’s angle parameters because users’ AOA intervals are unlikely to change substantially in a short slot.
  • Subsequent channel updating: Updated instantaneous angle information monitors user motion and triggers rescheduling when two users’ angular distance falls below a threshold.The current update uses angle parameters from the preceding coherence time.
  • Channel-gain estimation: For L ≤ M selected angles, channel gains are obtained by least squares, whereas dense sampling with L ≥ M requires a spatial-frequency formulation because least squares is underdetermined.The selected angles lie inside the estimated AOA interval, and their gains approximate the continuous PAS.
  • Channel-gain estimation: The spatial-frequency method estimates channel gains by sampling the DTFT output at frequencies corresponding to the selected angles, but insufficient nonzero components can limit continuous-PAS accuracy.The desired gains are samples of β(ξ) at ξ_l = ψk,p(n) − χ cos(ϑ_l).

4) Uplink CCM Reconstruction:

The IC-pCCM scheme reconstructs uplink covariance matrices from instantaneous channel-derived angles and gains by generating auxiliary channels with independent random phases. The reconstructed covariance depends on angular information and PAS rather than channel phase information.

  • Auxiliary-channel construction: The method generates auxiliary instantaneous uplink channels by assigning independent uniformly distributed random phases to the estimated signal rays.These auxiliary realizations provide the random-phase model used in covariance reconstruction.
  • Covariance reconstruction: The reconstructed channel covariance is mainly determined by angular information and PAS, not by channel phase information.The random phases disappear under the expectation used in the reconstruction.
  • IC-pCCM procedure: IC-pCCM estimates channel gains at selected angles, then reconstructs the uplink CCM from the estimated angles and gains.Algorithm 1 obtains updated channel parameters, selects discrete angles, estimates their gains, and applies the CCM reconstruction formula.
  • Scheme designation: The scheme is termed Instantaneous Channel aided parametric CCM reconstruction (IC-pCCM), distinguishing it from MC-iCCM and CF-iCCM.The main reconstruction steps are summarized in Algorithm 1.

5) Improved Uplink Channel Estimation With Reconstructed CCMs:

The reconstructed uplink CCM is incorporated into an MMSE estimator to improve uplink channel estimation. Because narrow angular spread yields low-rank covariance matrices, training can use only dominant eigen-directions, and the reported improvement requires no additional training overhead.

  • MMSE estimation: The reconstructed CCM is used in an MMSE estimator to improve uplink channel estimation relative to SBEM.The covariance is decomposed into signal eigenvectors and eigenvalues for the estimator.
  • Low-rank training: Narrow angular spread makes the reconstructed CCMs low-rank, with ν ≪ M, enabling training along only ν dominant eigenvectors.The dominant eigenvectors form the signal subspace used for uplink training.
  • Training overhead: The uplink estimation enhancement and CCM construction are obtained without any additional training overhead.The reported comparison is against the SBEM method.

C. Downlink CCM Inference and Downlink Channel Estimation

The downlink CCM is inferred from uplink angle and PAS estimates using reciprocity, then used for eigen-beamforming-based downlink training. Training only along dominant eigen-directions reduces overhead relative to full orthogonal downlink training, while unknown CCM amplitude scaling does not affect the eigenvectors.

  • Downlink CCM inference: The downlink CCM is inferred from uplink measurements together with estimated uplink angles and PAS.This inference follows the downlink CCM relation derived from uplink measurements.
  • Downlink CCM inference: An unknown amplitude scaling factor may separate the estimated and real downlink CCMs, but it does not affect their eigenvectors or subsequent eigen-beamforming.The limitation concerns covariance amplitude rather than the signal subspace used for training.
  • Downlink training: Downlink channel training uses the dominant eigenvectors of each user’s low-rank inferred CCM as eigen-beamforming directions.The same low-rank principle used for uplink training is applied to the downlink.
  • Downlink training: Training along ν dominant eigenvectors significantly reduces overhead compared with the M × M orthogonal training matrix required by conventional downlink training.The beamforming matrix is formed from the corresponding eigen-directions for each user.
  • Downlink estimation: The downlink receiver applies an MMSE estimator after receiving training through the eigen-beamforming directions.As M grows, asymptotic orthogonality suppresses inter-user interference and promotes downlink training performance.

IV. SIMULATIONS

The simulations evaluate CCM reconstruction efficiency, CCM rank accuracy, and MMSE channel-estimation performance for uplink and downlink settings. Across PAS distributions, angular spreads, and SNRs, IC-pCCM generally provides efficient, robust CCMs and improved estimation with reduced eigenvector overhead.

  • CCM reconstruction efficiency: IC-pCCM achieves high uplink and downlink CCM reconstruction efficiency for uniform and Laplacian PAS distributions and for narrow and wide angular spreads.The simulations state that IC-pCCM outperforms CF-iCCM and MC-iCCM for both uplink and downlink CCM construction.
  • CCM reconstruction efficiency: CF-iCCM performance deteriorates significantly for larger angular spreads and uniform PAS because it relies heavily on accurate angle estimation.IC-pCCM is described as more robust to angle-estimation errors because contributions outside the real angle-of-arrival interval receive small weights.
  • CCM rank comparison: IC-pCCM produces estimated CCM ranks close to the real CCM ranks, whereas CF-iCCM and MC-iCCM produce much larger ranks under uniform PAS.The rank gap between the integral methods increases with angular spread, and MC-iCCM is more accurate than CF-iCCM for large angular spreads.
  • Uplink channel estimation: MMSE estimation with IC-pCCM-reconstructed CCMs substantially improves uplink performance and remains relatively robust as angular spread increases.The simulations attribute this robustness to CCM eigenvectors concentrating more signal power in a lower-dimensional subspace.
  • Uplink channel estimation: Uplink MSE curves exhibit error floors because training uses only ν = 16 expansion vectors and instantaneous channel estimates contain truncation errors that the enhancement scheme cannot remove.Even the real-CCM curve has an error floor under this truncation limitation.
  • Eigenvector overhead: With ν about 20, IC-pCCM and real-CCM MSE curves become flat, while SBEM, CF-iCCM, MC-iCCM, and CS require ν = 32 or larger.This supports accurate CCM estimation with lower channel-estimation overhead for IC-pCCM.

V. CONCLUSIONS

The paper proposes reconstructing uplink and downlink CCMs for TDD/FDD massive MIMO from array-signal-processing information extracted from one instantaneous uplink channel estimate. The resulting method improves channel estimation, supports downlink CCM inference in FDD, enables dynamic angle-based scheduling, and applies across PAS distributions and array geometries.

  • CCM reconstruction: The proposed scheme reconstructs uplink and downlink CCMs using angle and PAS estimates extracted from one instantaneous uplink channel estimate.The reconstructed uplink CCM improves uplink training, while the downlink CCM can be inferred even in FDD systems.
  • CCM reconstruction: The method infers downlink CCMs in FDD systems using reconstructed uplink CCM information.The paper explicitly separates uplink CCM reconstruction from subsequent downlink CCM inference.
  • User scheduling: The proposed strategy includes dynamic angle division multiple access scheduling based on users’ real-time angle information.This scheduling strategy is presented as part of the overall enhanced channel-estimation scheme.
  • Scope and practical setting: Compared with existing methods, the proposed method does not require long-term uplink CCM acquisition and handles practical propagation environments with larger angular spreads.The conclusions also state applicability to arbitrary PAS distributions and array geometries.
  • Simulation conclusion: Numerical simulations corroborate the effectiveness of the proposed scheme.The conclusion states this outcome without specifying a single quantitative metric.
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