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Terahertz Massive MIMO with Holographic Reconfigurable Intelligent Surfaces

Ziwei Wan, Zhen Gao, Feifei Gao, Marco Di Renzo, Mohamed-Slim Alouini

arXiv:2009.10963v4cs.ITeess.SP

TL;DR

The paper addresses passive holographic RIS transmission design for THz massive MIMO, where prior work had not tackled nearly-passive surfaces with spatially continuous apertures. It derives holographic-RIS beam patterns and proposes closed-loop, compressive-sensing channel estimation; simulations show practical ultra-dense RISs can approach ideal holographic performance and improve ASE over non-holographic alternatives.

  • Problem

    Prior research had not addressed physical-layer transmission design for passive holographic communications using nearly-passive RISs with spatially continuous apertures, while critically spaced RISs suffer sidelobe leakage and reduced effective reflection area.

  • Method

    The paper derives holographic-RIS beam patterns, proposes closed-form beamforming designs, and develops a closed-loop broadband CE scheme with downlink grouping, uplink finer-grained estimation, and dual-domain compressive sensing.

  • Results

    Simulation results show holographic RISs outperform traditional non-holographic RISs and RIS-free schemes, while the proposed CS-based CE scheme outperforms LS estimation with fewer pilot signals.

  • Takeaways & Limitations

    An ultra-dense RIS with practical spacing d = λ/8 can sufficiently approximate an ideal holographic RIS, while grouping and dual sparsity support effective channel estimation with reduced pilot overhead.

Abstract

from arXiv · show

We propose a holographic version of a reconfigurable intelligent surface (RIS) and investigate its application to terahertz (THz) massive multiple-input multiple-output systems. Capitalizing on the miniaturization of THz electronic components, RISs can be implemented by densely packing sub-wavelength unit cells, so as to realize continuous or quasi-continuous apertures and to enable holographic communications. In this paper, in particular, we derive the beam pattern of a holographic RIS. Our analysis reveals that the beam pattern of an ideal holographic RIS can be well approximated by that of an ultra-dense RIS, which has a more practical hardware architecture. In addition, we propose a closed-loop channel estimation (CE) scheme to effectively estimate the broadband channels that characterize THz massive MIMO systems aided by holographic RISs. The proposed CE scheme includes a downlink coarse CE stage and an uplink finer-grained CE stage. The uplink pilot signals are judiciously designed for obtaining good CE performance. Moreover, to reduce the pilot overhead, we introduce a compressive sensing-based CE algorithm, which exploits the dual sparsity of THz MIMO channels in both the angular domain and delay domain. Simulation results demonstrate the superiority of holographic RISs over the non-holographic ones, and the effectiveness of the proposed CE scheme.

I. INTRODUCTION

The paper addresses reliable THz communication under severe propagation and blockage constraints by developing passive holographic RISs and associated beamforming and channel-estimation methods. It positions ultra-dense RISs as practical approximations to ideal continuous apertures and uses dual-domain sparsity to reduce channel-estimation overhead.

  • Motivation: THz systems offer abundant bandwidth and high data rates but suffer atmospheric attenuation, free-space loss, and severe LoS blockage.Massive or ultra-massive MIMO can provide beamforming gain, but may impose unaffordable power consumption and design burdens.
  • Prior Work: Holographic communications integrate many tiny reconfigurable elements into compact, spatially continuous or quasi-continuous apertures, enabled by THz component miniaturization.The paper focuses on nearly-passive holographic RISs for passive holographic communications, an area identified as insufficiently studied.
  • Contributions: The paper derives discrete-RIS beam patterns using Fourier analysis and formulates an angular-domain beamforming framework based on weighted Dirichlet kernels.Reflection coefficients are reconstructed from designed weighting factors through the Fourier transform.
  • Contributions: Closed-form narrow beam steering and spatial bandpass filtering designs are extended to holographic RISs, with ultra-dense RISs shown to approximate ideal continuous apertures.These designs address beamforming functions important to RIS-aided communication systems.
  • Contributions: A closed-loop broadband channel-estimation scheme combines downlink coarse angle estimation, grouped uplink refinement, and compressive sensing over angular and delay domains.Uplink pilots exploit coarse LoS-angle information, while the CS stage reduces pilot overhead by leveraging dual sparsity.

III. BEAMFORMING DESIGN FOR HOLOGRAPHIC RISS

This section derives RIS beam patterns from discrete planar arrays and extends the analysis to continuous holographic apertures. It also obtains closed-form narrow beam steering and spatial bandpass filtering solutions used in the proposed channel-estimation scheme.

  • Beamforming Design: The beam-pattern analysis begins with discrete planar RIS arrays before extending the results to spatially continuous continuous metasurfaces.The extension supports the holographic-RIS model used for THz massive MIMO systems.
  • Beamforming Design: Closed-form narrow beam steering and spatial bandpass filtering solutions are derived as beamforming designs for RIS-aided THz massive MIMO.Both designs play an essential role in the proposed channel-estimation scheme.

A. RISs Based on Discrete Planar Arrays

The DPA-based RIS is modeled as an evenly spaced planar array whose programmable reflection coefficients determine its beam pattern. Fourier-domain analysis represents this pattern through weighted Dirichlet kernels and provides a two-step angular beamforming design.

  • Array model: A DPA-based RIS contains evenly spaced reflecting elements on the x-y plane, with Nx and Ny elements separated by distance d ≤ λ/2.Its physical dimensions are Ax = Nxd and Ay = Nyd.
  • Beam-pattern model: The incident and reflected signal directions are represented by angular variables ψin and ψout, and the beam pattern g(ψout, ψin) is the superposition of all reflected element signals.Each element has a software-programmable complex reflection coefficient Φ(m, n).
  • Fourier representation: The spatial reflection coefficients Φ(m, n) are treated as a two-dimensional discrete signal and represented through a discrete-time Fourier transform.The spatial sampling period is d in both azimuth and elevation directions.
  • Fourier representation: The beam pattern g(ψout, ψin) is formulated as a weighted integral of Dirichlet kernels Ξ(·), with weighting factors ω′(k, l).This formulation generalizes the far-field results cited in the paper.
  • Design procedure: Angular-domain beamforming first optimizes ω′(k, l) for the desired observation direction, then reconstructs Φ(m, n) through the inverse spatial relationship.The LoS angles of the BS-RIS channel are assumed fixed and known for this design.

B. Beamforming Design for DPA-Based RIS

The proposed DPA-based RIS framework designs angular coefficients for narrow beam steering and spatial bandpass filtering. Closed-form, physically realizable reflection coefficients produce focused regions suited to point-to-point transmission or broadcasting.

  • Narrow beam steering: Narrow beam steering maximizes |g(ψout, ψin)| at a desired direction ψopt while minimizing it elsewhere.The design uses angular-domain coefficients concentrated at the desired spatial frequencies.
  • Narrow beam steering: The resulting NBS reflection coefficients are ΦNBS(m, n; ψopt) ∝ ej(dmkopt+dnlopt), yielding the narrowest beam toward ψopt.The design provides optimal beamforming gain for point-to-point communications.
  • Spatial bandpass filtering: Spatial bandpass filtering designs |g(ψout, ψin)| to be quasi-constant within prescribed angular limits and approximately zero outside them.The limits are specified by the cut-off angles ψmin and ψmax.
  • Spatial bandpass filtering: The SBF angular-domain coefficients are constructed over one spatial-frequency period using intervals bounded by arbitrary values satisfying the stated cut-off constraints.The assumption d ≤ λ/2 guarantees the existence of the required interval parameters.
  • Validation and applications: The normalized NBS and SBF beam patterns fulfill their respective design goals, producing small and wide focused regions, respectively.The small region is useful for beamforming applications, whereas the wide region is useful for broadcasting applications.

C. Extension to Holographic RISs

The section extends RIS beamforming analysis to continuous holographic surfaces and shows that ultra-dense practical RISs can closely approximate their beam patterns. Reduced element spacing suppresses leakage while preserving spatial resolution determined by physical aperture size.

  • Practical approximation: d = {λ/4, λ/8} suppresses the power leakage observed for critically spaced RISs while retaining similar beam patterns.Increasing the number of elements at fixed physical size also increases the expected effective reflection area.
  • Continuous holographic RIS model: A CMS models the asymptotic regime d → 0 and Nx, Ny →∞ while keeping the physical dimensions Ax and Ay fixed.Its reflection coefficients are continuously distributed over the surface, and the discrete-time Fourier transform converges to a continuous-time Fourier transform.
  • Beam-pattern properties: CMS beam patterns replace Dirichlet kernel functions with sinc functions that have relatively small sidelobes and no periodicity.The comparison covers critically spaced and ultra-dense RIS implementations as well as CMSs.
  • Practical approximation: d ≤λ/4 makes ultra-dense RIS and CMS beam patterns almost indistinguishable in the reported examples.The ultra-dense RIS provides a practical approximation to the otherwise unrealizable infinite-element CMS.
  • Beam-pattern properties: The NBS mainlobe width is mainly determined by Ax and Ay, and spatial resolution is the same for critically spaced, ultra-dense, and CMS implementations at fixed physical size.This resolution characterizes the ability to distinguish closely located UEs.

IV. CLOSED-LOOP CHANNEL ESTIMATION SCHEME

The closed-loop channel estimation framework uses short OFDM unique words as pilots and divides estimation into downlink and uplink phases. The two phases are applied successively to estimate THz massive-MIMO channels aided by holographic RISs.

  • Framework: Short-length OFDM symbols called unique words are transmitted as pilot signals throughout the channel-estimation stage.Each unique word contains NCP frequency-domain subcarriers and has duration 2NCPTs.
  • Framework: The proposed CS scheme has two phases applied to downlink and uplink transmissions, respectively.The downlink phase performs coarse estimation, while the uplink phase performs finer-grained estimation.

A. Downlink CE Stage

The downlink CE stage coarsely localizes the relevant RIS and UE angular groups using SBF and NBS beamforming. This grouping narrows the search space for the subsequent uplink finer-grained estimation.

  • Group search: The RIS angular AoD range is partitioned into Gx azimuth groups and Gy elevation groups.The azimuth group boundaries are separated using the RIS azimuth resolution λ/Ax.
  • Beam scanning: The RIS uses SBF over each angular group, while the UE uses NBS toward MU desired directions to coarsely estimate νLoS.The UE array is assumed relatively small, motivating the finite set of NBS directions.
  • Beam scanning: The UE collects pilot observations for the RIS and UE group combinations and selects the optimal indices by a search.The selected indices are (ĝx, ĝy, n̂x, n̂y).
  • Outcome: The NBS energy-focusing and SBF bandpass properties narrow the possible values of µLoS and νLoS after downlink CE.This significantly reduces the search space for uplink finer-grained CE.

B. Uplink CE stage

The uplink CE stage refines the angular estimate within downlink-selected groups while separating multiple UEs through dedicated subcarriers. A CS-based OMP estimator exploits angular-and-delay sparsity to recover underdetermined broadband channels.

  • Fine-grained estimation: The BS schedules UEs with the same downlink-selected RIS group together for uplink CE and payload transmission.The uplink search is restricted to the downlink-confined angular range.
  • Fine-grained estimation: Overlapped NBS beams cover all BxBy directions with different random phases instead of exhaustive beam scanning.The random phases are independently drawn from U[0, 2π).
  • Multiuser pilots: Dedicated subcarriers assign disjoint pilot subsets to different UEs, simplifying separation of their pilot signals at the BS.Each UE transmits on Nused of the NCP available subcarriers.
  • Sparse recovery: The uplink observation system is underdetermined because NP < BxBy and Nused < NCP, so conventional LS estimation cannot solve it.The constraints arise from limited channel coherence time and DSC allocation among multiple UEs.
  • Sparse recovery: Dual sparsity in the angular and delay domains enables CS-based OMP recovery of the effective uplink channel.Interpolation-based methods can reconstruct the channel on all K subcarriers after estimating HAdDd_u.

C. Pilot Overhead and Computational Complexity Analysis

The closed-loop CE scheme combines downlink grouping with uplink finer-grained estimation, while trading pilot overhead against computational complexity through RIS grouping. Its total pilot overhead is determined by the downlink and uplink UW requirements.

  • Trade-off analysis: Different group settings provide a trade-off between pilot overhead and computational complexity when NP = 40.Larger group counts increase downlink search complexity but can reduce the burden associated with broad channel estimation.
  • Uplink CE scheme: Algorithm 2 uses assigned DSCs for uplink UW transmission, followed by BS pilot reception and CS-based channel recovery.The procedure forms W, Yu, and Fu before iteratively recovering the channel estimate.
  • Pilot overhead: The proposed scheme requires GxGyM_U UWs for downlink CE and NP UWs for uplink CE.The total overhead combines these two stages, with each UW lasting 2N_CPT_s.
  • Computational complexity: Downlink search at each UE has complexity O(GxGyM_Uy), and this remains affordable because Gx and Gy are much smaller than the RIS and BS element counts.Uplink complexity varies with the sparse-recovery algorithm, including greedy, Bayesian, and deep-learning methods.

V. SIMULATION RESULTS

The simulation section evaluates different RIS types and the proposed channel-estimation scheme.

  • Evaluation scope: Numerical results assess different RIS types and the performance of the proposed CE scheme.The evaluation covers the RIS configurations and channel-estimation procedure introduced in the paper.

A. Experimental Setting

The simulations model RIS-aided THz massive MIMO with specified geometry, hardware, channel, and noise parameters, then evaluate beamforming, CE, spectral efficiency, and quantization effects. Results examine practical RIS density, grouping, pilot allocation, sparsity exploitation, and phase quantization.

  • Experimental Setting: The simulated BS and RIS serve UEs within a 20 m sector, using fc = 0.15 THz, λ ≈ 2 mm, 500 MHz bandwidth, and NCP = 64.The BS and RIS heights are 10 m, the UE height is 1.5 m, and the sector angle is 120°.
  • Experimental Setting: The channel model uses L = 1 NLoS path and Kf = 30 dB, with random NLoS angles and uniformly distributed delay offsets.Molecular absorption and array gains are included in the channel-coefficient model.
  • Pilot overhead and complexity: At NP = 40, grouping creates a pilot-overhead/computational-complexity trade-off; exhaustive downlink scanning with Gx = Gy = 200 yields TDL about 0.6 sec.The open-loop Gx = Gy = 1 case requires a 40,000 × 64 uplink channel estimate.
  • Downlink CE and RIS density: The CMS gives the best downlink grouping performance, while a practical ultra-dense RIS approaches the ideal CMS and smaller element spacing improves performance.The paper attributes grouping gains to narrower SBF passbands and higher passband array gain.
  • Uplink CE: Random DSC allocation outperforms uniform allocation and maintains high CE accuracy at compression ratios from 0.0063 to 0.0688.The random strategy is therefore used for the remaining results.
  • Uplink CE: The proposed CE outperforms SW-OMP and LS benchmarks by exploiting THz channel sparsity in both angular and delay domains, using NP = 40 or 80 versus LS NP = 1600.The open-loop scheme performs poorly because random UE pilots disperse transmit energy across directions.
  • ASE: The proposed CE achieves ASE close to the oracle upper bound, while practical d = λ/8 approaches ideal CMS performance and the RIS virtual LoS link significantly increases ASE.The well-determined LS estimator has worse ASE because of its longer pilot-transmission duration.
  • Quantization effects: ASE degradation from phase quantization is not significant for B = 2 or 3 across all d ≤ λ/2.The study compares quantized reflection phases with the ideal B = ∞ case.

VI. CONCLUSIONS

The paper develops holographic RIS beamforming and a closed-loop broadband CE scheme for THz massive MIMO. Simulations show that holographic and ultra-dense RIS designs can outperform non-holographic and RIS-free alternatives, while practical hardware and propagation issues remain future work.

  • VI. CONCLUSIONS: The paper derives holographic RIS beam patterns and closed-form NBS and SBF beamforming designs for closely spaced elements approaching a continuous aperture.The ideal holographic beam pattern can be approximated by an ultra-dense RIS.
  • VI. CONCLUSIONS: The proposed closed-loop broadband CE scheme combines downlink grouping with finer-grained uplink estimation and uses CS to exploit angular- and delay-domain sparsity.The uplink pilots are designed using coarsely estimated LoS-angle information.
  • VI. CONCLUSIONS: Simulation results show holographic RISs outperform traditional non-holographic RISs and communication schemes without RISs.The conclusion summarizes the reported superiority across the evaluated designs.
  • VI. CONCLUSIONS: Future work includes practical reflecting-element amplitude and phase models, hardware impairments, robust signal processing, near-field communications, and field validation.The paper identifies proof-of-concept validation and experiments as additional directions.

APPENDIX A PROOF OF Corollary 1

The appendix proves Corollary 1 by taking limits of the array-related function Ξ_Nx as the element spacing d approaches zero, using symmetry and separate cases for k_opt relative to Δx.

  • The result in (27) follows by substituting d_m = x and d_n = y into (17).
  • By symmetry, it suffices to establish the limiting expression for Ξ_Nx[d(k_opt − Δx)] as d approaches zero.
  • The proof treats separately the cases k_opt ≠ Δx and k_opt = Δx when evaluating the limit.
  • The derivation uses N_xd = A_x and De l’Hôpital’s rule to obtain the needed limiting result.
  • The resulting expression establishes (62), completing the proof of Corollary 1.
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