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Channel Estimation and Hybrid Combining for Wideband Terahertz Massive MIMO Systems

Konstantinos Dovelos, Michail Matthaiou, Hien Quoc Ngo, Boris Bellalta

arXiv:2102.06772v1cs.ITeess.SP

TL;DR

Wideband THz massive MIMO suffers beam squint and difficult channel estimation because array responses vary across OFDM subcarriers and training may lack combining gain. The paper proposes low-complexity TTD-based combining and GSOMP channel estimation, achieving near-digital combining performance and improved low-to-moderate-SNR estimation.

  • Problem

    Beam squint makes narrowband combining ineffective across OFDM subcarriers, while limited training gain complicates channel estimation in wideband THz massive MIMO.

  • Method

    The paper combines virtual-subarray TTD beam-squint mitigation with GSOMP, which exploits a common channel support across subcarriers.

  • Results

    The proposed combiner achieves 514 Gbps versus 303 Gbps for narrowband combining and 517 Gbps for digital combining, while GSOMP outperforms OMP at low and moderate SNR.

  • Takeaways & Limitations

    The design provides nearly beam-squint-free operation and accurate CSI acquisition even in the low-SNR regime.

Abstract

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Terahertz (THz) communication is widely considered as a key enabler for future 6G wireless systems. However, THz links are subject to high propagation losses and inter-symbol interference due to the frequency selectivity of the channel. Massive multiple-input multiple-output (MIMO) along with orthogonal frequency division multiplexing (OFDM) can be used to deal with these problems. Nevertheless, when the propagation delay across the base station (BS) antenna array exceeds the symbol period, the spatial response of the BS array varies across the OFDM subcarriers. This phenomenon, known as beam squint, renders narrowband combining approaches ineffective. Additionally, channel estimation becomes challenging in the absence of combining gain during the training stage. In this work, we address the channel estimation and hybrid combining problems in wideband THz massive MIMO with uniform planar arrays. Specifically, we first introduce a low-complexity beam squint mitigation scheme based on true-time-delay. Next, we propose a novel variant of the popular orthogonal matching pursuit (OMP) algorithm to accurately estimate the channel with low training overhead. Our channel estimation and hybrid combining schemes are analyzed both theoretically and numerically. Moreover, the proposed schemes are extended to the multi-antenna user case. Simulation results are provided showcasing the performance gains offered by our design compared to standard narrowband combining and OMP-based channel estimation.

I. INTRODUCTION

THz massive MIMO offers very large bandwidth and can offset severe path attenuation with dense antenna arrays, but wideband array effects create beam squint and make channel estimation difficult. The paper develops TTD-based hybrid combining and GSOMP-based channel estimation for these challenges.

  • Motivation: THz communication offers bandwidths at least 10 GHz, while dense massive-MIMO arrays help compensate for its severe path attenuation.The cited passage contrasts THz bandwidth with mmWave and links dense antenna packing to beamforming gain.
  • Problem: Large array apertures and wideband OFDM create spatial-frequency wideband effects, making array responses and signal directions vary across subcarriers.This frequency dependence is the source of beam squint and motivates frequency-dependent combining.
  • Problem: Beam squint can substantially reduce array gain when conventional narrowband combining is used across OFDM subcarriers.The paper identifies frequency-dependent combining as necessary because typical hybrid arrays cannot directly provide it.
  • Problem: Channel estimation is difficult without training-stage combining gain, while many antennas and few RF chains can make estimation overhead excessive.The challenge is especially pronounced in the low-SNR regime and under hybrid-array hardware constraints.
  • Contributions: The paper models SFW effects for BS UPAs and proposes virtual-subarray partitioning with TTDs to reduce the number of delay elements needed for beam-squint mitigation.The resulting wideband combiner is analyzed against fully digital combining through analysis and simulations.
  • Contributions: GSOMP uses a wideband dictionary and common support across OFDM subcarriers to improve channel recovery over OMP at low and moderate SNR.The estimator matches OMP accuracy at high SNR while exploiting multiple subcarriers during support detection.
  • Contributions: The paper analyzes GSOMP with a CRLB and achievable-rate evaluation, and extends the design to multi-antenna users.The reported analysis includes imperfect channel knowledge and the effects of deploying multiple user antennas.

B. Hybrid Transceiver Model

The model combines OFDM with a hybrid analog-digital architecture for wideband THz massive MIMO, where frequency-flat combining suffers from beam squint across subcarriers.

  • System Model: OFDM divides bandwidth B into S subcarriers, while the hybrid combiner uses a frequency-flat RF stage and subcarrier-dependent baseband processing.The RF combiner has constant-amplitude, variable-phase elements, and each RF chain drives the antenna array through analog phase shifters.
  • System Model: The received subcarrier signal depends on the channel, transmitted data symbol, average subcarrier power, and additive Gaussian noise.
  • System Model: THz channels can have coherence bandwidths of hundreds of MHz, motivating relatively few OFDM subcarriers and making SC-FDE a considered alternative.
  • Beam Squint Problem: Propagation delay across the array can exceed the sampling period, making the direction of arrival and array gain frequency-dependent across OFDM subcarriers.
  • Beam Squint Problem: Under narrowband RF combining, normalized array gain can substantially decrease across subcarriers and may approach zero as the planar array grows.The paper therefore identifies beam-squint compensation as essential for THz massive MIMO deployment.

B. Proposed Combiner for Single-Path Channels

The proposed single-path combiner partitions the planar array into virtual subarrays and uses true-time-delay elements to compensate inter-subarray delay while retaining frequency-flat phase shifters.

  • Virtual Array Partition: The array is partitioned into Nsb×Msb virtual subarrays, each containing ˜NטM antennas, to analyze and mitigate wideband array-gain loss.
  • Virtual Array Partition: The array response is decomposed into x- and y-direction subarray responses, with each virtual subarray represented by its own response vector.
  • Virtual Array Partition: Each subarray response is related to the first subarray through a frequency-dependent phase term determined by its relative delay.
  • Virtual Array Partition: Choosing ˜N and ˜M sufficiently small makes the within-subarray Dirichlet-sinc factors approximately one, limiting intra-subarray frequency-dependent loss.
  • Virtual Array Partition: The subarray dimensions are selected so the maximum delay within a virtual subarray is below 1/B; with half-wavelength spacing and ˜N=˜M, this yields (˜N−1)<2fc/B.
  • TTD-Based Combiner: A single TTD element between virtual subarrays compensates their shared delay across all OFDM subcarriers, canceling the factor Ω(φ,θ,f).The compensated delay is Δmn(φ,θ)=(n−1)˜NΔx(φ,θ)+(m−1)˜MΔy(φ,θ).
  • TTD-Based Combiner: The resulting combiner approximately achieves array gain NB across the full bandwidth when the virtual subarrays are sufficiently small.The implementation uses (NsbMsb−1) TTD elements per RF chain.

C. Proposed Combiner for Multi-Path Channels

The TTD-based combiner extends to multipath channels by approximating each path’s wideband combiner with RF chains and baseband combining, while least-squares estimation exposes a severe training-overhead problem.

  • Multi-Path Combining: For a two-path NLoS channel, the fully digital optimum is the normalized maximum-ratio combiner, and two RF chains can represent the two path outputs.
  • Multi-Path Combining: The proposed architecture differs from prior ULA multi-beam TTD designs by assigning each frequency-flat phase shifter to one antenna and each TTD to a virtual subarray in a UPA.
  • Channel Estimation: Channel estimation is formulated as a compressive-sensing problem over subcarriers, with a wideband dictionary and an estimator that exploits cross-subcarrier information.
  • Training Model: Training uses pilot signals and hybrid combiners across Nslot slots, producing Nbeam=NslotNRF measurements and generally colored effective noise.
  • Least Squares Estimator: Least-squares estimation requires Nbeam≥NB, so its training overhead is prohibitively high when the number of RF chains is much smaller than the number of BS antennas.
  • Least Squares Estimator: The analysis uses least squares rather than MMSE because the estimators considered exploit only instantaneous channel state information.

C. Sparse Formulation and Orthogonal Matching Pursuit

The sparse formulation represents each wideband channel using a grid-based planar-array dictionary and solves the resulting sparse recovery problem with OMP, while super-resolution grids reduce angular mismatch.

  • Sparse Formulation and Orthogonal Matching Pursuit: The physical channel is represented by a wideband array-response matrix multiplied by a vector of path gains across subcarriers.
  • Sparse Formulation and Orthogonal Matching Pursuit: A predefined direction-of-arrival dictionary assigns nonzero coefficients to path directions and gains, producing an equivalent sensing matrix for channel recovery.
  • Sparse Formulation and Orthogonal Matching Pursuit: Because the channel gain vector has only L+1 nonzero entries among G dictionary elements, estimation becomes an (L+1)-sparse recovery problem.
  • Sparse Formulation and Orthogonal Matching Pursuit: The sparse optimization can be solved independently for each subcarrier using OMP, after which the channel estimate is reconstructed from the estimated dictionary coefficients.
  • 1) Wideband Dictionary for UPAs:: For half-wavelength spacing, the planar-array response is parameterized by spatial frequencies ωx=1/2 sinθ cosφ and ωy=1/2 sinθ sinφ.
  • 1) Wideband Dictionary for UPAs:: The dictionary samples both spatial-frequency dimensions over grids within [−1/2,1/2], with GxGy=G determining the total dictionary size.
  • 1) Wideband Dictionary for UPAs:: The wideband UPA dictionary is formed as ¯A[s]=¯Ax[s]⊗¯Ay[s], and super-resolution choices Gx>N and Gy>M reduce quantization mismatch.
  • 1) Wideband Dictionary for UPAs:: Quantization errors are reported as small enough not to significantly affect normalized array gain, supporting the assumption that path directions lie on the dictionary grid; Gx=N and Gy=M recover the narrowband VCR.

2) Generalized Multiple Measurement Vector Problem:

The frequency-dependent channel dictionaries share a common support across subcarriers, enabling a GMMV formulation solved with simultaneous OMP. The pilot design whitens effective noise for reliable stopping and supports subsequent TTD-based combining.

  • Generalized Multiple Measurement Vector Problem: Frequency-dependent channel gain vectors share a common support across subcarriers, allowing the estimation problem to be formulated as GMMV.The formulation uses multiple sensing matrices, one for each pilot subcarrier.
  • Generalized Multiple Measurement Vector Problem: The proposed estimator applies simultaneous OMP to jointly recover the shared support from the multiple measurement vectors.The procedure is specified in Algorithm 2 as a GSOMP-based estimator.
  • Generalized Multiple Measurement Vector Problem: After spatial-frequency estimation, physical path angles are obtained and used in the TTD-based wideband combiner.This connects GSOMP channel estimation to the proposed beam-squint mitigation stage.
  • Generalized Multiple Measurement Vector Problem: Random RF pilot combiners are used because their low mutual coherence is expected to improve sparse-recovery probability.The corresponding baseband design compensates for the colored effective noise introduced by the RF pilots.
  • Generalized Multiple Measurement Vector Problem: Cholesky-based baseband combiners whiten the effective noise, yielding covariance σ^2I_Nbeam and enabling a noise-power stopping threshold.The combiners can be computed offline.

1) Lower Bound Error Analysis:

The error analysis derives a CRLB under semi-unitary combining and exact support recovery, then characterizes GSOMP complexity and extends the estimation model to multi-antenna users.

  • Lower Bound Error Analysis: For semi-unitary combiners, the effective-noise covariance is σ^2I_Nbeam, providing the noise model used in the CRLB analysis.The CRLB derivation assumes GSOMP recovers the exact support of the channel gain vector.
  • Lower Bound Error Analysis: Under exact support recovery, the resulting linear estimator is minimum-variance unbiased and attains the CRLB.The channel estimate is reconstructed from the dictionary columns indexed by the recovered support.
  • Lower Bound Error Analysis: GSOMP requires O(|S|Nbeam(G−l)) correlation work at iteration l and O(l^3 + 2l^2Nbeam) least-squares work per pilot subcarrier.Here, l is the iteration index and G is the dictionary size.
  • Lower Bound Error Analysis: GSOMP reduces online computation relative to OMP because OMP retains O(|S|G) correlation-search complexity.The reduction follows from examining only the remaining dictionary elements during each GSOMP iteration.
  • Lower Bound Error Analysis: For a multi-antenna user, the uplink channel is modeled as an N_B×N_U frequency response with a user-array response included in each path.The extension treats the user as an N_U-element ULA and incorporates hybrid transmit processing and a power constraint.

A. Hybrid Combining and Beamforming

The multi-antenna extension models user-side array responses and reconstructs the channel through an equivalent sensing dictionary, while hybrid transmission uses matched analog filters and digital SVD processing.

  • Hybrid Combining and Beamforming: TTD-based beamforming and combining provide approximately unit normalized array gain across frequency and add an N_U beamforming gain for multi-antenna users.The corresponding subcarrier SNR is approximately |β(f_s)|^2N_UN_BP_d/σ^2.
  • Hybrid Combining and Beamforming: For multipath channels, the hybrid design uses matched-filter RF processing and SVD-based baseband processing on the effective channel.Without inter-stream interference, the number of transmitted streams is at most min(L,N_RF).
  • Hybrid Combining and Beamforming: User-side training employs a codebook of N_Ubeam pilot RF beamformers collected over N_slot training slots.The received pilot vectors are assembled into a matrix for estimation.
  • Hybrid Combining and Beamforming: Vectorization converts the multi-antenna observation into a sensing problem involving vec(H[s]) and a noise vector.The identity vec(ABC)=(C^T⊗A)vec(B) produces the equivalent measurement representation.
  • Hybrid Combining and Beamforming: GSOMP is applied with an equivalent dictionary that combines the BS, user, and path-dependent dictionary factors, then reconstructs vec(H[s]) as Ā[s]β̄[s].The user-side dictionary has size G_u and is included through the Kronecker-product sensing model.

VI. NUMERICAL RESULTS

The simulations evaluate NMSE against receive SNR for the proposed estimator and baselines under wideband THz settings. GSOMP accurately recovers common support across the tested single-user SNR range and outperforms OMP-DFT.

  • NUMERICAL RESULTS: The simulations use 400 subcarriers for the NLoS multipath case and approximately 18 subcarriers for a 100×100 UPA LoS case with 40 GHz bandwidth.The NLoS setting follows a 5 ns delay spread, while the LoS setting is governed by the maximum spatial-wideband delay.
  • NUMERICAL RESULTS: The channel model includes directional BS antenna patterns based on 3GPP parameters and omnidirectional user antennas.The BS maximum gain is selected as 50 dBi.
  • NUMERICAL RESULTS: GSOMP accurately detects the common support from −15 dB to 10 dB and attains the CRLB, whereas NBOMP and OMP show poor or significant low-SNR errors.LS has prohibitively high NMSE because its error scales linearly with the number of BS antennas.
  • NUMERICAL RESULTS: GSOMP outperforms OMP-DFT because DFT-based pilot combiners and dictionaries become highly correlated, producing near-zero sensing-matrix columns.This correlation destroys the incoherence needed for reliable recovery, particularly with many BS antennas and high SNR.

2) Multi-Antenna User:

For multi-antenna users, the proposed estimator remains effective at low and moderate SNR, while the proposed true-time-delay combiner maintains near-maximum array gain and substantially improves achievable rate over narrowband combining.

  • Multi-Antenna User: The multi-antenna-user evaluation fixes the total number of antennas at NBNU = 160 and uses a 20 × 20-element BS UPA with a 4-element user ULA.The corresponding partial-training overhead is also kept fixed.
  • Multi-Antenna User: The GSOMP estimator has slightly higher NMSE than in the single-antenna case, with degradation becoming significant at high SNR, but it outperforms OMP at low and moderate SNR.The increase is attributed to higher total coherence of the equivalent sensing matrices for the multi-antenna user.
  • Subcarrier Selection: GSOMP-SS accurately estimates the uplink channel in the moderate-SNR regime using only a small number of pilot subcarriers for common-support detection.Using one subcarrier per 50 pilot subcarriers slightly increases NMSE in the low-SNR regime.
  • Achievable Rate with Perfect CSI: The proposed combiner attains approximately maximum normalized array gain across the entire B = 40 GHz signal bandwidth for the 100 × 100-element UPA.The evaluated configuration uses 10 × 10 virtual subarrays and 99 TTD elements.
  • Achievable Rate with Perfect CSI: 517 Gbps, 514 Gbps, and 303 Gbps are achieved by digital, proposed, and narrowband combiners, respectively, giving the proposed combiner a 40% gain over narrowband combining.The proposed scheme uses only 99 TTD elements for the 100 × 100-element UPA.
  • Achievable Rate with Imperfect CSI: Under imperfect CSI in the NLoS multi-path case, the achievable rate of the proposed channel estimator approaches the perfect-CSI rate.The imperfect-CSI analysis uses the GSOMP-based common-support estimate and accounts for effective noise associated with channel-estimation error.

C. Hybrid SVD Transmission for Multi-Antenna Users

For multi-antenna users, the proposed TTD-based wideband combiner uses virtual array partitioning and SVD-based baseband processing, achieving rates close to fully digital transmission. The analysis also considers near-field operation and identifies hardware impairments and mobility as future boundaries.

  • System and comparison: The multi-antenna-user evaluation fixes 100 × 100 total antennas, using a 100 × 50 BS UPA and a 2-element user ULA.
  • Proposed transmission: The proposed scheme implements a wideband RF combiner with TTD and virtual array partition, followed by SVD-based baseband processing.
  • Baselines: The narrowband baseline uses a frequency-flat RF combiner based on the array response at the carrier frequency.
  • Results: The proposed TTD-based method performs close to fully digital transmission in average achievable rate, while multiple user antennas improve rate relative to the single-antenna case.The comparison uses waterfilling power allocation and SVD-based processing of the effective channel.
  • Near-field considerations: The combiner remains closely matched under plane-wave and spherical-wave models at distances below the Fraunhofer distance, although broader near-field studies remain future work.For a 100 × 100-element UPA at 300 GHz, the cited Fraunhofer distance is approximately 9.8 meters.
  • Scope and future work: Future work includes evaluating hardware impairments, beam tracking in high-mobility scenarios, SC-FDE versus OFDM, and analytical PAPR expressions.

APPENDIX A

The appendix derives the normalized array-gain expression from the planar-array response relationships. These relationships yield the result stated in Proposition 1.

  • These relationships are used to obtain equation (23) in Proposition 1 for the normalized array gain.
  • The appendix expands the planar-array response into entries indexed by virtual-subarray positions.
  • The response entries factor into a reference element multiplied by phase terms depending on subarray indices, carrier frequency, and directional delays.
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