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Near-Field Sparse Channel Representation and Estimation in 6G Wireless Communications

Xing Zhang, Haiyang Zhang, Yonina C. Eldar

arXiv:2212.13527v1eess.SP

TL;DR

Near-field channel estimation requires reconsidering planar-wave-based models. The paper proposes a distance-parameterized angular-domain sparse representation with joint dictionary learning and sparse recovery, and reports effectiveness and superiority over the polar-domain method in multi-user communications.

  • Problem

    Planar-wave-based channel models and estimation methods need reconsideration for near-field communications.

  • Method

    The paper uses a distance-parameterized angular-domain sparse representation and develops joint dictionary learning and sparse recovery algorithms.

  • Results

    The proposed model and estimation algorithms demonstrate effectiveness and superiority over the existing polar-domain method in multi-user communications.

  • Takeaways & Limitations

    The representation distinguishes itself from the polar-domain two-dimensional method by treating distance as an unknown dictionary parameter.

  • Takeaways & Limitations

    The polar-domain approach uses an angle-distance two-dimensional dictionary, and dictionary design involves a tradeoff associated with sampling interval.

Abstract

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The employment of extremely large antenna arrays and high-frequency signaling makes future 6G wireless communications likely to operate in the near-field region. In this case, the spherical wave assumption which takes into account both the user angle and distance is more accurate than the conventional planar one that is only related to the user angle. Therefore, the conventional planar wave based far-field channel model as well as its associated estimation algorithms needs to be reconsidered. Here we first propose a distance-parameterized angular-domain sparse model to represent the near-field channel. In this model, the user distance is included in the dictionary as an unknown parameter, so that the number of dictionary columns depends only on the angular space division. This is different from the existing polar-domain near-field channel model where the dictionary is constructed on an angle-distance two-dimensional (2D) space. Next, based on this model, joint dictionary learning and sparse recovery based channel estimation methods are proposed for both line of sight (LoS) and multi-path settings. To further demonstrate the effectiveness of the suggested algorithms, recovery conditions and computational complexity are studied. Our analysis shows that with the decrease of distance estimation error in the dictionary, the angular-domain sparse vector can be exactly recovered after a few iterations. The high storage burden and dictionary coherence issues that arise in the polar-domain 2D representation are well addressed. Finally, simulations in multi-user communication scenarios support the superiority of the proposed near-field channel sparse representation and estimation over the existing polar-domain method in channel estimation error.

I. INTRODUCTION

The paper addresses near-field channel estimation, where spherical wavefronts make conventional planar-wave models and algorithms inadequate. It proposes a distance-parameterized angular-domain sparse representation with joint dictionary learning and sparse recovery, reducing polar-domain burdens while improving estimation accuracy.

  • Motivation: Near-field propagation requires reconsidering planar-wave channel models and estimation algorithms because spherical wavefronts depend on both user angle and distance.Most 6G users may lie in the radiating near-field, where the conventional planar wavefront assumption is not valid.
  • Representation: The proposed distance-parameterized angular-domain model makes dictionary size depend only on angular resolution, unlike the polar-domain angle-distance 2D dictionary.This design addresses the polar-domain method’s storage burden and high column coherence.
  • Estimation: Joint dictionary learning and sparse recovery algorithms, termed DL-OMP, estimate angle of arrival and distance for each path in both LoS and multi-path settings.The algorithms iteratively estimate the sparse vector and update the dictionary, while the recovered parameters can reconstruct the channel and provide user location information.
  • Analysis: Decreasing distance estimation error in the dictionary enables exact recovery of the angular-domain sparse vector after a few iterations.Theoretical analysis includes an RIP-based recovery condition and storage and computational complexity results.
  • Complexity: The proposed method requires lower storage and lower computational complexity than the polar-domain 2D approach, especially when the near-field range is large.Its smaller dictionary follows from dependence on angular resolution rather than joint angle-distance sampling.
  • Evaluation: Multi-user simulations show that the proposed representation and estimation outperform the polar-domain method in channel estimation NMSE.The improvement is attributed to lower dictionary coherence, which facilitates more accurate sparse recovery.

II. PRELIMINARIES ON CHANNEL MODELS

This section establishes far-field and near-field channel-model preliminaries for an uplink multiuser massive MIMO system. It introduces a general free-space spherical-wave model and explains that antenna-dependent channel differences arise primarily through phase delays.

  • System model: The uplink system uses an N-antenna uniform linear array, with orthogonal user pilots enabling independent channel estimation for each user.The antenna spacing is specified as d = λc, and the BS antenna coordinates are given along the array.
  • System model: A single-antenna user is represented by Cartesian coordinates (x, y) and polar coordinates (r, θ).The section considers the user location as the basis for describing its distances to the BS antennas.
  • General channel model: Free-space electromagnetic propagation is modeled with expanding spherical wavefronts, yielding a line-of-sight channel whose antenna distances determine the phase terms.The model includes carrier frequency, propagation speed, antenna-user distance, and approximately antenna-independent free-space path loss; extension to multipath is straightforward.
  • Channel simplification: Different antenna-user distances produce different delays relative to a reference antenna, so the channel vector can be simplified using correlated phase differences and a few parameters.The section subsequently distinguishes planar-wave far-field and spherical-wave near-field representations.

A. Far-Field Channel Model · B. Near-Field Channel Model

The far-field model treats the wavefront as planar and represents the channel with an angular-domain sparse Fourier model. In the near field, spherical propagation makes the steering vector depend nonlinearly on antenna index, while polar-domain representations face resolution, storage, and coherence tradeoffs.

  • A. Far-Field Channel Model: In the far field, the array’s small length relative to user distance makes spherical-wave curvature negligible, yielding an approximately planar wavefront with a common arrival angle.The distance difference across antennas is therefore modeled relative to a reference antenna.
  • A. Far-Field Channel Model: The far-field steering vector is a discrete Fourier vector because its phase varies linearly with antenna index, enabling an angular-domain sparse channel model.With the angle space divided into N parts, the dictionary is the discrete Fourier transform matrix F.
  • A. Far-Field Channel Model: In the LoS case the far-field sparse vector has one nonzero element, whereas in the L-path case it has L nonzero elements whose indices infer arrival angles.The nonzero values represent the corresponding channel coefficients, and compressed-sensing algorithms can estimate the sparse vector.
  • B. Near-Field Channel Model: In the radiating near field, planar-wave approximations fail, so the channel must be modeled under the spherical-wave assumption.The near-field steering vector depends on both user angle θ and distance r to the reference antenna.
  • B. Near-Field Channel Model: Near-field steering-vector phases are nonlinear in antenna index, preventing representation by a single far-field Fourier vector.This nonlinearity arises from antenna-dependent user distances under spherical propagation.
  • B. Near-Field Channel Model: The polar-domain representation samples both angle and distance, producing a dictionary with N·M columns over the angle and distance grids.The angle space is divided into N parts and the distance range [rmin, rmax] into M parts.
  • B. Near-Field Channel Model: The polar-domain sparse representation accurately describes spherical-wave near-field channels but uses an angle-distance two-dimensional dictionary.Its finer angular and distance sampling increases dictionary width and storage burden.
  • B. Near-Field Channel Model: Polar-domain dictionary design trades resolution against coherence: finer sampling widens the dictionary, whereas larger intervals reduce sampling-induced column coherence.Nonuniform distance sampling can limit coherence, but the resulting dictionary remains wide, especially over large distance ranges.

III. NEAR-FIELD CHANNEL SPARSE REPRESENTATION AND ESTIMATION · A. Distance-Parameterized Angular-domain Sparse Representation of Near-Field Channel

The paper introduces a distance-parameterized angular-domain sparse representation for near-field channels, then develops joint dictionary learning and sparse recovery methods for LoS and multi-path cases. The representation reduces storage burden and dictionary coherence compared with polar-domain 2D dictionaries while motivating simultaneous distance estimation and sparse-vector recovery.

  • III. NEAR-FIELD CHANNEL SPARSE REPRESENTATION AND ESTIMATION: Section III proposes a distance-parameterized angular-domain sparse representation to address challenges from polar-domain representations.
  • III. NEAR-FIELD CHANNEL SPARSE REPRESENTATION AND ESTIMATION: The section develops joint dictionary learning and sparse recovery algorithms for both LoS and multi-path near-field channels.
  • III. NEAR-FIELD CHANNEL SPARSE REPRESENTATION AND ESTIMATION: It also analyzes RIP-based recovery conditions, storage burden, and computational complexity for the proposed approach.
  • A. Distance-Parameterized Angular-domain Sparse Representation of Near-Field Channel: Because angle and distance are coupled, the proposed model avoids sampling both dimensions in an extensive dictionary.
  • A. Distance-Parameterized Angular-domain Sparse Representation of Near-Field Channel: The model uses an angular-domain sparse vector s and a distance-parameterized dictionary W(r), whose columns pair sampled angles θi with distances ri that remain to be estimated.
  • A. Distance-Parameterized Angular-domain Sparse Representation of Near-Field Channel: The size of W(r) depends only on angular resolution, addressing the polar-domain 2D dictionary’s storage burden when distance sampling is extensive.
  • A. Distance-Parameterized Angular-domain Sparse Representation of Near-Field Channel: W(r) has lower coherence than the polar-domain 2D dictionary because each atom uses a different angle, with near-orthogonal columns when the distance is much larger than the array length.
  • A. Distance-Parameterized Angular-domain Sparse Representation of Near-Field Channel: The resulting estimation problem jointly estimates dictionary distances r and recovers the sparse vector s for LoS and multi-path channels.

B. Joint Dictionary Learning and Sparse Recovery based Channel Estimation · 1) Line of Sight Channel Estimation:

The proposed near-field LoS estimator combines sparse channel recovery with iterative distance-parameterized dictionary learning. It reduces pilot overhead for hybrid receivers by selecting antenna subsets and updates the dictionaries from angle and distance estimates until convergence conditions are met.

  • 1) Line of Sight Channel Estimation:: The LoS setup assumes straight-line propagation, mutually orthogonal user pilots, and a channel invariant over one or several transmission blocks.Orthogonal pilots allow channel estimation to be performed individually for each user.
  • B. Joint Dictionary Learning and Sparse Recovery based Channel Estimation: With NRF ≪ N RF chains, conventional channel estimation requires large pilot overhead because it requires NRFT ≥ N.The few-RF-chain architecture motivates compressed-sensing-based estimation.
  • B. Joint Dictionary Learning and Sparse Recovery based Channel Estimation: The method represents the near-field channel with the proposed parameterized sparse model and uses switching to select NRF antennas across T pilot times.A total of P = NRFT antennas are selected over time, generally with P < N, reducing pilot overhead.
  • 1) Line of Sight Channel Estimation:: Two antenna subsets, each of length P, provide measurements modeled as yj = Wj(r)sj + vj for j = 1, 2.The subsets use reference antennas separated by δ, with corresponding angles θ1, θ2 and distances r1, r2.
  • 1) Line of Sight Channel Estimation:: For fixed dictionaries, the angular-domain sparse vectors can be recovered with compressed-sensing algorithms, and the method adopts OMP for simplicity.The paper also identifies BP and ISTA as applicable alternatives.
  • 1) Line of Sight Channel Estimation:: The DL-OMP algorithm estimates angles, calculates distances using the law of sine, updates both dictionaries, and iterates before reconstructing the near-field channel.The estimated angle, distance, and channel are returned after channel-coefficient estimation and reconstruction.
  • 1) Line of Sight Channel Estimation:: The second angle estimate is accelerated by selecting a sub-dictionary using the first angle estimate and the user range [rmin, rmax].This narrows the search over W2 before applying OMP to y2.

2) Multi-path Channel Estimation:

The multi-path channel is represented by a sparse model with multiple nonzero components, but interference among paths reduces initial angular-estimation accuracy. The proposed approach addresses this by separately updating dictionary columns during iterative angle-distance estimation before support-based channel reconstruction.

  • Proposed method: Algorithm 2 iteratively estimates path angles, selects sub-dictionaries, calculates distances, updates dictionaries, and repeats across L paths and K_iter iterations.The procedure initializes angle and distance supports, then performs iterative path-wise refinement.
  • Channel reconstruction: After estimating each path, the method constructs the support B, solves for coefficient estimates, updates residual signals, and reconstructs the multi-path channel.The returned quantities are the angle support Λ_θ, distance support Λ_r, and reconstructed channel ĥ(x, y).
  • Challenges: Multi-path sparsity produces more than one nonzero element in s_j, creating interference from other paths during channel estimation.The interference reduces angular estimation accuracy at initial iterations.
  • Proposed method: The proposed method updates each dictionary column separately within every iteration to mitigate multi-path interference.This per-column dictionary update is the central modification for multi-path estimation.

C. The RIP-based Recovery Condition Analysis

The analysis establishes that exact angle recovery by DL-OMP follows when the estimated-distance dictionary satisfies an RIP condition of order L + 1. Reducing distance-estimation error relaxes this condition, while in the LoS case the induced interference remains below one-quarter of sparse-signal energy.

  • Recovery condition: If W(ˆr) satisfies the RIP of order L + 1 with the theorem’s specified isometry constant, DL-OMP exactly recovers the angle of arrival.The recovery condition accounts for dictionary perturbation caused by distance estimation error and observation noise.
  • Interference interpretation: µ represents the interference-to-signal ratio, combining interference from distance-estimation error and observation noise.The recovery bound is a monotonically decreasing function of µ.
  • Recovery condition: As distance-estimation error decreases, the bound on δL+1 increases, yielding a more relaxed RIP condition for angle recovery in the next iteration.When µ = 0, the condition becomes δL+1 < 1, matching the typical OMP recovery condition.
  • LoS case: 0 < µ < 0.25 in the LoS case, so channel-noise and distance-error-induced interference is at most around one-quarter of sparse-signal energy.For LoS, L = 1 and the general recovery condition simplifies accordingly.

D. Storage and Computational Complexity Analyses

The proposed DL-OMP uses an angular-domain dictionary, reducing storage relative to polar-domain 2D dictionaries, especially over large near-field ranges. Its complexity is approximately O(3LKiterPN), with Kiter = 3 sufficient for accurate channel estimation and lower complexity than both polar-domain alternatives under 3Kiter < ˜M < M.

  • Storage complexity: The proposed dictionary W(r) samples only the angular domain, so its storage requirement is much smaller than polar-domain 2D dictionaries, especially when the near-field range is large.The comparison includes uniform P-OMP and nonuniform P-OMP.
  • Computational complexity: The proposed method has overall computational complexity approximately O(3LKiterPN) for L paths and Kiter iterations.This accounts for angle estimation and dictionary updating across the iterations.
  • Computational complexity: Kiter = 3 is enough for an accurate channel estimate, while near-field channels have a small path number L.Consequently, complexity is mainly determined by pilot length P and antenna number N, corresponding to angular resolution.
  • Computational complexity: Uniform P-OMP and nonuniform P-OMP have approximate complexities O(LPNM) and O(LPN ˜M), respectively.Uniform P-OMP searches a 2D dictionary with NM columns, whereas nonuniform P-OMP uses ˜M distance samples.
  • Computational complexity: Under 3Kiter < ˜M < M, DL-OMP has lower computational complexity than both polar-domain algorithms.This condition is typical for near-field estimation with many antennas and high carrier frequency.

IV. SIMULATION RESULTS

Simulations show that DL-OMP consistently outperforms uniform and nonuniform P-OMP for LoS and multi-path near-field channel estimation. Its advantages persist across pilot lengths and varying near-field conditions, while multi-path interference and parameter errors limit performance.

  • IV. SIMULATION RESULTS: DL-OMP achieves superior NMSE performance over uniform and nonuniform P-OMP in both LoS and multi-path cases, before and after refinement.The advantage is attributed partly to the low column correlation of its distance-parameterized angular-domain dictionary, which supports sparse recovery.
  • IV. SIMULATION RESULTS: Multi-path estimation incurs about 2 ∼5 dB performance loss in NMSE for all evaluated algorithms because interference from other paths degrades accuracy.
  • IV. SIMULATION RESULTS: DL-OMP remains superior across the tested pilot-length range, although its comparatively low accuracy for P < 70 indicates better operation with more measurements.
  • IV. SIMULATION RESULTS: As near-field effects strengthen, uniform P-OMP suffers significant performance degradation, nonuniform P-OMP remains comparatively poor, and DL-OMP retains robust performance advantages.DL-OMP’s distance estimate becomes more accurate with stronger near-field effects, while uniform P-OMP’s higher angle errors cause sharp channel-NMSE degradation.

V. CONCLUSIONS

The work addresses near-field channel estimation for future 6G systems by introducing a distance-parameterized angular-domain sparse representation and corresponding dictionary-learning recovery algorithms. Compared with polar-domain two-dimensional modeling, the approach reduces dictionary dependence to angular resolution while achieving low storage and coherence, with theoretical analyses and multi-user simulations verifying its effectiveness.

  • Future 6G systems using large-scale antennas and high frequencies are likely to operate in the radiating near-field, requiring channel representation and estimation algorithms to be reconsidered.
  • The proposed distance-parameterized angular-domain sparse model represents near-field channels by treating distance as an unknown dictionary parameter.
  • Unlike the polar-domain two-dimensional method, the dictionary size depends only on angular resolution, providing low storage and dictionary coherence from the design stage.
  • Dictionary learning orthogonal matching pursuit algorithms were designed for line-of-sight and multi-path channels, with restricted-isometry-based recovery, storage, and complexity analyses.
  • Theoretical analyses and multi-user communication simulations verified the effectiveness and superiority of the proposed model and estimation algorithms.

APPENDIX A PROOF OF Proposition 1 · APPENDIX B PROOF OF Proposition 2

Appendix A proves dictionary-coherence expressions by analyzing maximum column correlations in two shared-parameter cases and distinguishing the distance-parameterized and polar-domain dictionaries. Appendix B proves that dictionary updates reduce distance mismatch by modifying selected-atom elements using updated distance estimates.

  • APPENDIX A PROOF OF Proposition 1: APPENDIX A: Dictionary coherence is defined as the largest correlation between any two columns.The proof begins from the column-correlation definition and identifies the maximum over distinct dictionary columns.
  • APPENDIX A PROOF OF Proposition 1: APPENDIX A: Maximum correlation may occur when two columns share either angle or distance while differing in the other parameter.The proof therefore examines two extreme cases separated by one sampling interval.
  • APPENDIX A PROOF OF Proposition 1: APPENDIX A: Dictionary W(r) permits only the shared-distance case, whereas dictionary D permits both cases and has generally larger correlation in case 2.The resulting coherence expressions are denoted εW(θp, θq, rp) and εD(θp, rp, rq), respectively.
  • APPENDIX A PROOF OF Proposition 1: APPENDIX A: The angular sampling interval follows from dividing cosθ ∈[−1, 1] into N parts.The proof uses this angular discretization together with an approximation to rewrite the coherence expressions.
  • APPENDIX A PROOF OF Proposition 1: APPENDIX A: For equal angles, the proof sets the distance sampling interval to 1 meter and re-expresses the resulting correlation.This is the second extreme case in the coherence analysis.
  • APPENDIX B PROOF OF Proposition 2: APPENDIX B: Dictionary updating aims to mitigate distance mismatch between dictionary atoms and the received signal.The proof measures matching through the absolute inner product between a selected atom and the received signal.
  • APPENDIX B PROOF OF Proposition 2: APPENDIX B: The mismatch can be eliminated by modifying the selected atom’s pth element with e−j2π fc.After obtaining a new distance estimate, Algorithm 1 updates dictionary W1 and similarly updates W2.

APPENDIX C PROOF OF Theorem 1

Theorem 1 is proved by showing that condition (37) enables recovery of the support set of s in L iterations. The argument establishes the first iteration and then extends the guarantee inductively to later iterations using residual reformulation and RIP properties.

  • Condition (37) guarantees successful support recovery in L iterations.The proof states this as its main objective and conclusion.
  • For the lth iteration, previously recovered indices imply Λl−1 ⊆ supp(s), enabling residual variables ˜s and ˜e to be defined.The residual is written as ˜s = s − s|Λl−1 and ˜e = −(∆W)˜s + v, with supp(˜s) ⊆ supp(s).
  • The later-iteration matching analysis repeats the first-iteration argument with L′ = L − l + 1.The proof defines ψl(i) and U for the residual problem before applying analogous inequalities.
  • The RIP relation δL′+1 < δL+1 supports the inequality used to extend the recovery guarantee to subsequent iterations.The passage attributes the second inequality directly to RIP properties.
  • With ||˜e||2^2 approximately equal to µ, condition (37) also guarantees success in the residual iteration.This connects the approximate residual-error norm to the theorem’s recovery condition.
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