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Channel Estimation and Hybrid Precoding for Frequency Selective Multiuser mmWave MIMO Systems

J. P. González-Coma, J. Rodríguez-Fernández, N. González-Prelcic, L. Castedo, R. W. Heath

arXiv:2401.16241v1cs.ITeess.SPmath.IT

TL;DR

Frequency-selective multiuser mmWave MIMO makes channel estimation and hybrid precoder/combiner configuration challenging. The paper uses reciprocal uplink compressive estimation and proposes two hybrid-design approaches; its reported results favor uplink estimation over downlink at fixed overhead and show good performance for the hybrid designs.

  • Problem

    Hybrid precoder and combiner design and channel estimation are challenging in frequency-selective multiuser mmWave MIMO systems with constrained analog hardware and multiuser training requirements.

    Source: §I · p. 2

  • Method

    The BS jointly estimates users’ uplink channels using compressive methods that exploit common support and reciprocity, then designs hybrid filters by digital-solution factorization or iterative alternating minimization.

    Source: §I · p. 2, B. Contributions · p. 3

  • Results

    Uplink channel estimation at the BS outperforms downlink estimation at the MSs for fixed overhead, and the proposed hybrid solutions show good performance in simulations.

    Source: B. Contributions · p. 3, §VI · p. 13

  • Takeaways & Limitations

    Combiner performance depends on SNR, with low-complexity MRC described as an alternative at low and mid SNRs; digital-solution rank limits hybrid performance.

    Source: §VI · p. 13

    The rank of the digital solutions is a performance-limiting factor for the hybrid designs.

    Source: §VI · p. 13

Abstract

from arXiv · show

Configuring the hybrid precoders and combiners in a millimeter wave (mmWave) multiuser (MU) multiple-input multiple-output (MIMO) system is challenging in frequency selective channels. In this paper, we develop a system that uses compressive estimation on the uplink to configure precoders and combiners for the downlink (DL). In the first step, the base station (BS) simultaneously estimates the channels from all the mobile stations (MSs) on each subcarrier. To reduce the number of measurements required, compressed sensing techniques are developed that exploit common support on the different subcarriers. In the second step, exploiting reciprocity and the channel estimates, the base station designs hybrid precoders and combiners. Two algorithms are developed for this purpose, with different performance and complexity tradeoffs: 1) a factorization of the purely digital solution, and 2) an iterative hybrid design. Extensive numerical experiments evaluate the proposed solutions comparing to state-of-the-art strategies, and illustrating design tradeoffs in overhead, complexity, and performance.

I. INTRODUCTION

The paper addresses channel estimation and hybrid precoder/combiner design for frequency-selective multiuser mmWave MIMO, where hybrid hardware, limited RF chains, and multiuser training complicate both tasks. It proposes reciprocal uplink estimation and two hybrid-design approaches, against prior work that largely treats narrowband or single-user settings.

I. INTRODUCTION

  • Hybrid analog-digital hardware, frequency-flat analog filters, limited RF chains, and multiuser channel requirements make training and precoder/combiner design difficult.The analog network prevents direct access to antenna outputs, while wide bandwidth lowers SNR and channel dimensionality is high.Source: §I · p. 2

A. Prior Work

  • Prior work leaves a gap in joint frequency-selective multiuser estimation and hybrid design, with existing methods often limited by channel assumptions, user count, or design scope.Earlier estimation generally focused on narrowband channels; the few frequency-selective multiuser approaches had constraints such as strongest-path-only estimates or downlink CSI feedback.Source: A. Prior Work · pp. 2–3

II. SYSTEM MODEL

  • The system uses OFDM, assumes frequency-flat RF precoders and combiners, and relies on channel reciprocity and BS channel estimates shared with each MS.The BS simultaneously estimates channels for all MSs during training; analog phase-shifter entries have constant modulus.Source: B. Contributions · p. 3, §II · p. 4

B. Contributions

  • The proposed estimator jointly estimates multiuser frequency-selective channels by exploiting shared sparsity and TDD reciprocity.The paper also reports that uplink estimation at the BS outperforms downlink estimation at the MSs for fixed overhead, even neglecting feedback.Source: B. Contributions · p. 3
  • The paper proposes two hybrid-filter designs: factorizing a digital solution and directly designing hybrid precoders and combiners with alternating minimization.It also designs uplink MS precoders to maximize individual mutual information and BS combiners using criteria intended to mitigate inter-user interference.Source: B. Contributions · p. 3
  • The authors report that their designs perform well against state-of-the-art methods and that digital-solution rank limits hybrid performance.They also examine how precoder and combiner performance relates to the quality of the estimated channels.Source: B. Contributions · p. 3

III. HYBRID PRECODER/COMBINER DESIGN

The design objective is to maximize downlink achievable sum-rate using available CSI, but coupled precoders make the optimization difficult even before accounting for phase-shifter constraints.

III. HYBRID PRECODER/COMBINER DESIGN

  • The target is to maximize downlink achievable sum-rate using the available CSI, subject to transmit-power constraints.The stated power constraint applies per MS and subcarrier, with total transmit power Ptx.Source: §III · p. 5
  • The SNR is defined using total transmit power divided by the aggregate noise power across users.The passage specifies SNR = Ptx/(Uσ²_n) for E[αu,p] = 1.Source: §III · p. 5
  • The optimization is difficult because the precoders are coupled, even when phase-shifter constraints are ignored.The paper therefore develops alternative optimization problems and solves for all-digital filters as a step toward hybrid solutions.Source: §III · p. 5

A. Digital Design

The digital-design stage uses user-specific uplink precoders and several alternative BS combiners, including MSE-based and interference-nulling choices.

A. Digital Design

  • Uplink precoders are designed selfishly for each MS using its available CSI, with water-filling used for power allocation.The design neglects inter-user interference because an MS lacks the other users’ channels and the BS combiners.Source: A. Digital Design · p. 5
  • The MMSE uplink combiner can be designed independently for each MS and subcarrier when the channel estimate and corresponding precoder are known at the BS.The derivation begins from the uplink MSE for each user and subcarrier.Source: A. Digital Design · p. 5
  • The paper also considers MRC for low complexity and CB combiners that null inter-user interference, using the structure of the interference matrix.The CB combiner is directed toward the nullspace of the estimated interference matrix.Source: A. Digital Design · p. 5

B. Hybrid Design

The paper presents both factorization of digital filters into hybrid components and direct alternating-minimization design. The factorization couples users and subcarriers through the shared RF filters.

B. Hybrid Design

  • The hybrid-design section offers digital-solution factorization and direct alternating minimization to compute hybrid precoders and combiners.These are the two proposed routes for obtaining RF and baseband filters.Source: B. Hybrid Design · p. 6

1) Factorization of Digital Designs:

  • Factorization minimizes Frobenius-norm distance between unconstrained digital and hybrid solutions, with users and subcarriers coupled through the RF filters.The paper motivates this distance for precoding under unitary constraints and for MSE.Source: 1) Factorization of Digital Designs · p. 6
  • For fixed RF combiners, the baseband uplink combiner is obtained through a least-squares solution and must be scaled to satisfy power constraints.The factorization is performed separately for each MS for uplink precoders; the discussion focuses on hybrid uplink combiners.Source: 1) Factorization of Digital Designs · p. 6

MMSEFRF(F H

The section develops RF-only hybrid design strategies: least-squares factorization of digital filters and projected-gradient optimization that accounts for distortion and power normalization. It also gives a projected-gradient method for cases where factorization is inflexible.

MMSEFRF(F H

  • Least-squares baseband filters and power normalization define an RF-combiner design that avoids alternating minimization; factorization accuracy depends on the digital-solution rank and RF-chain count.When LBS = NBS and FRF is full rank, the projector is the identity and perfect factorization is ensured.Source: MMSEFRF(F H · p. 6

MMSE[k]FRF(F H

  • The projected-gradient method minimizes distortion under unit-modulus RF constraints, using gradient steps followed by projection onto the feasible set.A diminishing step size yields a local optimum; the algorithm incorporates the power constraint when computing the gradient.Source: MMSEFRF(F H · p. 6, MMSE[k]FRF(F H · pp. 6–7
  • The RF-only projected-gradient design makes the power constraint explicit during gradient computation, unlike approaches that normalize power only after convergence.The performance impact of those prior approaches’ power constraints had not been determined.Source: MMSE[k]FRF(F H · p. 7
  • Applying Algorithm 1 to uplink RF precoders requires designing them separately for each mobile station, and strategies assuming LMS,u ≥ K Ns,u are hardly applicable in frequency-selective settings.The number of subcarriers times the number of streams remains large even though the digital uplink precoder has fewer columns than the uplink combiner.Source: MMSE[k]FRF(F H · p. 7
  • The HD-PG method can factorize precoders and combiners for any number of users, streams, or subcarriers.The method projects updated RF precoders onto the feasible set by taking their phases.Source: MMSE[k]FRF(F H · pp. 6–7

2) Hybrid Design by Alternating Minimization:

The alternating-minimization design updates RF and baseband components in turn, avoiding the computationally complex factorization. The RF precoder is jointly designed across subcarriers, while uplink combiners use distinct criteria for RF and baseband components.

2) Hybrid Design by Alternating Minimization:

  • Alternating minimization avoids computationally complex factorization by updating RF and baseband precoders or combiners in turn until convergence.The RF component is fixed while its baseband counterpart is computed, then the components are alternately updated.Source: 2) Hybrid Design by Alternating… · p. 7
  • The uplink RF precoder uses right singular vectors of the concatenated channel matrix to increase SNR across subcarriers, with entry-wise normalization for analog constraints.Baseband precoders maximize mutual information for the equivalent channel and can be computed at the base station from channel estimates.Source: 2) Hybrid Design by Alternating… · p. 7
  • For uplink combiners, the RF design maximizes received-signal SNR, while baseband combiners are independently designed for each user and subcarrier using an MSE criterion.The RF combiner has a closed-form expression when baseband combiners are given; MRC-like and CB-like baseband alternatives are also proposed.Source: 2) Hybrid Design by Alternating… · pp. 7–8
  • The alternating RF and baseband combiner updates have no theoretical convergence guarantee because they follow different criteria and may not reduce the cost at every step.The authors report good numerical results despite the lack of a guarantee.Source: 2) Hybrid Design by Alternating… · p. 7

IV. COMPRESSIVE CHANNEL ESTIMATION

The section formulates simultaneous uplink estimation of the frequency-selective channels of multiple users and proposes compressive recovery. Frequency-flat training permits a subcarrier-independent measurement matrix, and SW-OMP is used to recover the channels.

IV. COMPRESSIVE CHANNEL ESTIMATION

  • The proposed uplink framework simultaneously estimates the frequency-selective channel matrices for all users and subcarriers at the base station.The section also provides a multiuser channel-estimation CRLB and a criterion for fair uplink-downlink comparison.Source: §IV · p. 8

A. Uplink Compressive Channel Estimation

  • Frequency-flat training precoders and combiners reduce the complexity that otherwise grows with the number of subcarriers.Rotated versions of a baseline OFDM symbol make the measurement matrix frequency-independent while retaining spatial degrees of freedom.Source: A. Uplink Compressive Channel Estimation · p. 8
  • Different precoders across users in each training frame reduce measurement-matrix column correlation, supporting recovery of common channel support and subsequent channel gains.The subcarrier-dependent pilot effect can be eliminated at the receiver by multiplying by its inverse, or by its conjugate for equal-energy constellations.Source: A. Uplink Compressive Channel Estimation · pp. 8–9
  • The training model stacks measurements across frames and users to form a multiuser dictionary for sparse channel reconstruction.The model accounts for post-combining colored Gaussian noise.Source: A. Uplink Compressive Channel Estimation · p. 9
  • The sparse multiuser channel vectors are recovered by solving a relaxed optimization problem, using SW-OMP for illustration.SW-OMP and SS-SW-OMP+Th have been shown to achieve estimation performance within the CRLB when the angles of arrival and departure lie on the quantized spatial grid.Source: A. Uplink Compressive Channel Estimation · p. 9

B. Cram´er-Rao Lower Bound

The CRLB analysis derives a bound for multiuser channel estimation assuming the channel support is perfectly known. The overall variance sums subchannel variances because no prior relation between channel matrices is assumed.

B. Cram´er-Rao Lower Bound

  • The CRLB is derived assuming perfect estimation of the multiuser channel support, to assess support-estimator performance when channel-gain estimation is efficient for correctly identified support.The reduced model assumes the angles of arrival and departure have been found.Source: B. Cram´er-Rao Lower Bound · p. 9

HH[1] . . . , HH[K]

  • Without prior information about relations among channel matrices, the overall estimation variance sums the variances of the different subchannels.The passage attributes this summation to the lack of knowledge about possible correlations among channel matrices.Source: B. Cram´er-Rao Lower Bound · p. 9, HH[1] . . . , HH[K] · p. 10

C. Uplink vs. Downlink Multi-user Wideband Channel Estimation

The section compares uplink and downlink channel-estimation SNR on a fair basis. With uncorrelated user training signals, the base-station SNR sums the users’ contributions, so the comparison uses single-user SNR and accounts for the number of paths estimated.

C. Uplink vs. Downlink Multi-user Wideband Channel Estimation

  • Uncorrelated user training signals make the base-station receive SNR equal the sum of the individual users’ SNRs.The summed quantity corresponds to the downlink SNR for every user.Source: C. Uplink vs. Downlink Multi-user… · p. 10
  • The comparison uses single-user SNR because each mobile station measures SNR U times smaller than the base station.The base station estimates the combined sparse vectors for all users, while each mobile station estimates its own.Source: C. Uplink vs. Downlink Multi-user… · p. 10
  • The chosen SNR criterion represents the expected ratio of SNR to the number of multipath components estimated in either link direction.This is the stated rationale for comparing uplink and downlink channel estimation using the single-user SNR.Source: C. Uplink vs. Downlink Multi-user… · p. 10

V. NUMERICAL RESULTS

The numerical experiments compare uplink and downlink channel estimation, then evaluate digital and hybrid precoder/combiner designs with perfect CSI. They report stronger base-station uplink estimation than downlink estimation, and show that hybrid-design performance depends on the method and available RF chains.

V. NUMERICAL RESULTS

  • The experiments use a 128-antenna base station, four 16-antenna mobile stations, 32 subcarriers, and averages over 100 channel realizations.Channel-estimation tests use 100 training frames and four-bit phase-shifter quantization.Source: §V · p. 10, A. Channel Estimation Performance · p. 10

A. Channel Estimation Performance

  • Uplink estimation at the base station greatly outperforms downlink estimation at the mobile stations, and both lie close to their CRLBs.The base station’s larger dictionary and eight RF chains, versus two at each mobile station, are cited as reasons for the difference.Source: A. Channel Estimation Performance · pp. 10–11
  • The gap between them is approximately maintained; for SNR ranging from −10 up to 0 dB, NMSE performance with off-grid parameters is lower than −10 dB.Source: A. Channel Estimation Performance · p. 11

B. Performance Evaluation for Perfect CSI

  • Iterative hybrid methods perform better than factorization, especially with the MMSE combiner; factorization reaches digital-solution performance when enough RF chains are available.The factorization gap relative to alternating minimization remains positive, and MRC requires fewer RF chains than MMSE for accurate factorization.Source: B. Performance Evaluation for Perfect… · p. 11

C. Performance Evaluation for Imperfect CSI

The proposed methods combine uplink channel estimation with hybrid precoder and combiner design; simulations compare performance across imperfect-CSI conditions and design choices. The conclusions report uplink estimation gains, SNR-dependent combiner tradeoffs, and different complexity–performance tradeoffs for hybrid designs.

C. Performance Evaluation for Imperfect CSI

  • MRC performs similarly with perfect and imperfect CSI, whereas estimation error creates noticeable gaps for MMSE and CB combiners.Channel-gain and noise-variance estimation is more challenging in the multiuser setting, contributing to the perfect–imperfect CSI gap.Source: C. Performance Evaluation for… · pp. 11–12
  • As path count increases, frequency-domain channel rank increases, making low-rank hybrid approximation more difficult.The passage identifies this alongside other impairments contributing to performance gaps in the hybrid results.Source: C. Performance Evaluation for… · p. 11
  • Simultaneously estimating user subchannels without prior sparsity knowledge complicates path-count estimation; finite-resolution dictionaries add grid-quantization error.The passage describes this in the context of estimating a multiuser channel whose rank grows with the number of mobile stations.Source: C. Performance Evaluation for… · p. 11
  • Hybrid factorization is the system bottleneck because limited analog degrees of freedom must approximate multiple users’ digital combiners under stringent RF constraints.With the rank-limited Eckart–Young approximation and initialization described, the hybrid performance loss is mainly due to rank limitation rather than factorization itself.Source: C. Performance Evaluation for… · pp. 12–13
  • At low SNR, noise dominates the received covariance and alters MMSE combiner column spaces less, supporting MRC’s suitability in the expected mmWave regime.Across subcarriers, MRC column spaces remain related, while MMSE combiner rank grows with both user and subcarrier counts, contributing to a digital–hybrid performance gap.Source: C. Performance Evaluation for… · pp. 12–13

VI. CONCLUSIONS

  • Alternating minimization offers a complexity–performance tradeoff, with particular computational efficiency and good throughput for MRC.The conclusions contrast its lower computational complexity with the greater flexibility and performance of digital factorization methods.Source: C. Performance Evaluation for… · p. 13, §VI · p. 13

C. Performance Evaluation for Imperfect CSI

  • On-grid AoA/AoD avoids the cited quantization error and reduces the degrees-of-freedom difference between hybrid and digital solutions.The passage attributes this to possible shared path angles across users and on-grid delays, so multiuser channel rank need not increase with user and path counts.Source: C. Performance Evaluation for… · p. 13

VI. CONCLUSIONS

  • Uplink estimation outperforms downlink estimation for on-grid and off-grid AoA/AoD, while digital factorization is more flexible and performs better than lower-complexity alternating minimization.The conclusions also identify the rank of digital solutions as the performance-limiting factor.Source: §VI · p. 13

APPENDIX

The appendix describes gradient computation across mobile stations and subcarriers and relates a Frobenius-norm objective to an MSE lower bound.

A. Gradient Computation

  • The gradient computation separates contributions by mobile station and subcarrier, then sums them to obtain the stated expression.The passages say the separation enables derivative computation for each mobile station and subcarrier.Source: A. Gradient Computation · p. 13, MMSE[k]FRF(F H · p. 13

B. Frobenius Norm for MSE

+ MMSEUL

  • Table I is identified as a comparison of the performance of different approaches.Source: Table I · p. 14
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