Near-Field Channel Estimation Techniques

• Near-field channel estimation is the process of recovering channel coefficients when the conventional planar-wave model fails, using spherical-wave geometry and quadratic phase corrections.
• Advanced methods integrate polar-domain dictionaries, tensor decompositions, and sparse Bayesian techniques to accurately capture both angular and distance dependencies.
• These techniques reduce pilot overhead and improve estimation accuracy in emerging systems like ELAA, XL-MIMO, THz, and RIS-assisted networks.

Near-field channel estimation is the recovery of channel coefficients or latent propagation parameters when the array aperture is large enough that the planar-wave assumption breaks down and the response depends jointly on angle and distance. In extremely large-scale MIMO, ELAA, XL-RIS, THz, and sub-THz systems, the Rayleigh distance can extend to tens or even hundreds of meters, so the phase across the array varies nonlinearly with antenna position, angular-domain sparsity weakens, and beam squint may arise in both angle and distance domains (Chen et al., 15 Mar 2026, Kosasih et al., 2024, Yang et al., 2023). Under these conditions, channel estimation has been formulated through exact spherical-wave geometry, Fresnel and polynomial approximations, polar- and wavenumber-domain representations, subspace and tensor methods, sparse Bayesian learning, gridless super-resolution, and sensing-assisted or distributed architectures (Dkhan et al., 14 May 2025, Wang et al., 26 Jan 2026).

1. Physical regime and channel models

A standard near-field approximation for a ULA writes the distance from path $l$ to antenna $n$ as

$$r_l^{(n)} \approx r_l - (n-1)d\sin\theta_l + \frac{(n-1)^2d^2\cos^2\theta_l}{2r_l},$$

which turns the steering vector into a complex exponential modulated by a chirp-like quadratic phase term (Xi et al., 28 Nov 2025). This quadratic phase is the basic signature that distinguishes radiative near-field propagation from far-field planar-wave propagation.

In mixed LoS/NLoS XL-MIMO, one physically motivated model separates exact LoS free-space propagation from NLoS near-field array responses. The LoS component is written antenna pair by antenna pair as

$${\bf H}(n_2,n_1)=\frac{1}{r_{n_2,n_1}}e^{-j\frac{2\pi}{\lambda}r_{n_2,n_1}},$$

whereas the NLoS component is expanded by near-field response vectors sampled over angle and distance (Lu et al., 2022). This distinction matters because the LoS part need not be well represented by the same response-vector product model used for NLoS paths.

Distance thresholds are correspondingly refined. In dual-band XL-MIMO, the Rayleigh distance is written as

$Z=\frac{2D^2}{\lambda},$

which explains why a higher-frequency band is more likely to operate in the near field than a lower-frequency co-located band (Wu et al., 2024). For mixed LoS/NLoS XL-MIMO, the MIMO Rayleigh distance and MIMO advanced Rayleigh distance are

$$\frac{2(D_1+D_2)^2}{\lambda} \quad\text{and}\quad \frac{4D_1D_2}{\lambda},$$

respectively, separating far-field, intermediate near-field, and strong near-field regimes (Lu et al., 2022).

At the lower end of the approximation range, a wavenumber-domain study defines the Fresnel distance as

$d_F = 0.5\left(\frac{D^3}{\lambda}\right)^{0.5},$

and argues that below this distance the usual Fresnel or parabolic approximation may become invalid, especially in the reactive near field (Guo et al., 2024). Near-field modeling is therefore not exhausted by the second-order Fresnel expansion.

The spherical-wavefront model is also not universal. For a single curved reflection path, the channel can be written as

$$\mathbf{h}=a\,\mathbf{c}(\bar{\mathbf{k}},\bar{\mathbf{Q}}),$$

where the quadratic phase term is governed by an effective anisotropic curvature matrix $\bar{\mathbf{Q}}$. In that formulation, the conventional spherical-wavefront channel is recovered as a special case when the reflector is planar or the point-scatterer limit applies (Zhang et al., 21 May 2026).

2. Structural representations of near-field channels

Many estimators are built on angle-distance dictionaries. For NLoS XL-MIMO, the polar-domain representation writes the channel as

${\bf H}_{\rm n\mbox{-}f} = {\bf D}_r {\bf H}^P_{\rm n\mbox{-}f} {\bf D}_t^H,$

with $n$0 and $n$1 sampled over angle-distance grids (Lu et al., 2022). In wideband XL-RIS systems, this idea becomes a wideband spherical-domain dictionary constructed by minimizing the coherence of two arbitrary beams, with common support across subcarriers to capture beam squint in both angle and distance domains (Yang et al., 2023). In XL-RIS-aided multi-user systems, the same logic appears in hybrid-field form: the BS-RIS link is modeled in the far field, while the RIS-user link is sparse in a polar dictionary (Dong et al., 7 May 2026).

Angular sparsity is often weakened rather than preserved. In dual-band XL-MIMO, a single near-field path spreads across adjacent DFT bins, so the angular-domain channel is described as weak sparsity and modeled as a Block Multiple-Measurement Vector structure with support influenced by out-of-band probabilities from a co-located lower-frequency band (Wu et al., 2024). For hybrid RIS-assisted systems, the transformed coefficient vector is likewise block sparse rather than single-spike sparse, which motivates boundary estimation and total-variation regularization instead of conventional far-field sparse recovery (Schroeder et al., 2024).

For extremely large-scale UPAs, the near-field steering matrix can be approximated as the outer product of two ULA-like near-field steering vectors. This yields a modified $n$2D-DFT dictionary,

$n$3

and a transformed coefficient matrix whose support forms multiple $n$4D rectangular blocks (Chen et al., 15 Mar 2026). The resulting $n$5D block-sparse structure is specific to UPA geometry and is not captured by direct extensions of $n$6D block-sparse models.

A separate representation abandons angle-distance atoms and parameterizes the wavefront itself. In near-field LoS propagation, the curved spherical wavefront can be approximated over a finite aperture by a multivariate polynomial in the antenna indices. In that hierarchy, $n$7 corresponds to the planar-wave or far-field model, $n$8 to a parabolic wavefront, and $n$9 to a higher-order refinement (Do et al., 5 May 2026).

3. Parametric, subspace, and tensor-based estimation

A classical parametric strategy estimates geometric quantities first and reconstructs the channel afterward. For extremely large aperture arrays, a two-step MUSIC estimator first evaluates the $$r_l^{(n)} \approx r_l - (n-1)d\sin\theta_l + \frac{(n-1)^2d^2\cos^2\theta_l}{2r_l},$$0D pseudospectrum

$$r_l^{(n)} \approx r_l - (n-1)d\sin\theta_l + \frac{(n-1)^2d^2\cos^2\theta_l}{2r_l},$$1

then performs a $$r_l^{(n)} \approx r_l - (n-1)d\sin\theta_l + \frac{(n-1)^2d^2\cos^2\theta_l}{2r_l},$$2D search in $$r_l^{(n)} \approx r_l - (n-1)d\sin\theta_l + \frac{(n-1)^2d^2\cos^2\theta_l}{2r_l},$$3 with the exact near-field steering vector, and finally refines the channel with a least-squares-based complex correction vector $$r_l^{(n)} \approx r_l - (n-1)d\sin\theta_l + \frac{(n-1)^2d^2\cos^2\theta_l}{2r_l},$$4. Spatial smoothing is used so that the effective number of snapshots becomes $$r_l^{(n)} \approx r_l - (n-1)d\sin\theta_l + \frac{(n-1)^2d^2\cos^2\theta_l}{2r_l},$$5 (Kosasih et al., 2024).

In RIS-aided sub-THz localization and channel estimation, NF-JCEL constructs a down-sampled Toeplitz covariance matrix from symmetric covariance entries for which the Fresnel quadratic term cancels. This decouples UE distance and AoAs, enables separate estimation of the vertical and azimuth AoAs with low complexities, reduces distance recovery to a simple one-dimensional search, and estimates channel attenuation coefficients through OMP (Pan et al., 2022).

For LoS ULA channels, the Joint Autocorrelation and Cross-correlation scheme decouples curvature and direction by using the identity

$$r_l^{(n)} \approx r_l - (n-1)d\sin\theta_l + \frac{(n-1)^2d^2\cos^2\theta_l}{2r_l},$$6

which depends on the curvature parameter $$r_l^{(n)} \approx r_l - (n-1)d\sin\theta_l + \frac{(n-1)^2d^2\cos^2\theta_l}{2r_l},$$7 but not on the AoA parameter $$r_l^{(n)} \approx r_l - (n-1)d\sin\theta_l + \frac{(n-1)^2d^2\cos^2\theta_l}{2r_l},$$8. JAC-ISF estimates $$r_l^{(n)} \approx r_l - (n-1)d\sin\theta_l + \frac{(n-1)^2d^2\cos^2\theta_l}{2r_l},$$9 by inverse sinc on the main lobe, whereas JAC-GD minimizes the full autocorrelation mismatch by gradient descent; after quadratic-phase compensation, MUSIC estimates $${\bf H}(n_2,n_1)=\frac{1}{r_{n_2,n_1}}e^{-j\frac{2\pi}{\lambda}r_{n_2,n_1}},$$0. The paper states total complexity $${\bf H}(n_2,n_1)=\frac{1}{r_{n_2,n_1}}e^{-j\frac{2\pi}{\lambda}r_{n_2,n_1}},$$1 for JAC-ISF (Liu et al., 2024).

When multiple users share non-orthogonal pilots in near-field OFDM, the received data can be arranged as a third-order tensor. In LoS scenarios this tensor admits a canonical polyadic decomposition, while in NLoS scenarios it admits a block term decomposition. The corresponding factor matrices are recovered by alternating least squares and nonlinear least squares, and the uniqueness analysis shows that the framework remains effective even when the number of pilot symbols is much smaller than the number of users; under the stated conditions, $${\bf H}(n_2,n_1)=\frac{1}{r_{n_2,n_1}}e^{-j\frac{2\pi}{\lambda}r_{n_2,n_1}},$$2 and pairwise independent pilots are sufficient for identifiability (Wang et al., 26 Jan 2026).

Another two-stage XL-MIMO estimator separates the dominant LoS component from sparse NLoS components. The LoS parameters $${\bf H}(n_2,n_1)=\frac{1}{r_{n_2,n_1}}e^{-j\frac{2\pi}{\lambda}r_{n_2,n_1}},$$3 are obtained by coarse on-grid search followed by iterative refinement, and the residual NLoS term is recovered by OMP in the polar domain. The reported numerical results show improvements over far-field codebook OMP and earlier near-field codebook OMP on both a theoretical model and QuaDRiGa emulation (Lu et al., 2022).

4. Structured sparse, Bayesian, and gridless recovery

A wavefront-parameterization approach replaces direct geometric inversion by polynomial phase estimation. The spherical wavefront is approximated by a multivariate polynomial in the antenna indices, and the estimator then recovers polynomial coefficients from multidimensional finite differences, proceeding from highest-degree terms downward. The reported complexity is linear in the number of observations and subquadratic in the number of polynomial coefficients, and the numerical study identifies $${\bf H}(n_2,n_1)=\frac{1}{r_{n_2,n_1}}e^{-j\frac{2\pi}{\lambda}r_{n_2,n_1}},$$4 as the usual sweet spot: $${\bf H}(n_2,n_1)=\frac{1}{r_{n_2,n_1}}e^{-j\frac{2\pi}{\lambda}r_{n_2,n_1}},$$5 is too crude for true near-field operation, while $${\bf H}(n_2,n_1)=\frac{1}{r_{n_2,n_1}}e^{-j\frac{2\pi}{\lambda}r_{n_2,n_1}},$$6 reduces model mismatch but raises the SNR threshold (Do et al., 5 May 2026).

For extremely large-scale UPAs, the $${\bf H}(n_2,n_1)=\frac{1}{r_{n_2,n_1}}e^{-j\frac{2\pi}{\lambda}r_{n_2,n_1}},$$7D Pattern-Coupled Sparse Bayesian Learning algorithm uses a Gaussian prior whose precision couples each coefficient to its neighbors through

$${\bf H}(n_2,n_1)=\frac{1}{r_{n_2,n_1}}e^{-j\frac{2\pi}{\lambda}r_{n_2,n_1}},$$8

The posterior is approximated with GAMP inside an EM loop, yielding iterative complexity roughly $${\bf H}(n_2,n_1)=\frac{1}{r_{n_2,n_1}}e^{-j\frac{2\pi}{\lambda}r_{n_2,n_1}},$$9. In the reported $Z=\frac{2D^2}{\lambda},$0 UPA experiment with $Z=\frac{2D^2}{\lambda},$1 paths, the method achieves about $Z=\frac{2D^2}{\lambda},$2 dB NMSE at $Z=\frac{2D^2}{\lambda},$3 dB SNR, while BOMP needs at least $Z=\frac{2D^2}{\lambda},$4 dB to reach similar performance (Chen et al., 15 Mar 2026).

Dual-band side-information methods incorporate lower-frequency spatial information directly into support selection. In the resulting greedy rules, the correlation term is augmented by the logit prior

$Z=\frac{2D^2}{\lambda},$5

where $Z=\frac{2D^2}{\lambda},$6 is inferred from the lower-frequency channel. CSLW-BOMP is used for on-grid block-sparse recovery and CSLW-OMP for off-grid recovery; the reported pilot-overhead reduction for $Z=\frac{2D^2}{\lambda},$7 accurate estimation probability is about $Z=\frac{2D^2}{\lambda},$8 versus BOMP and $Z=\frac{2D^2}{\lambda},$9 versus OMP (Wu et al., 2024).

Gridless super-resolution removes explicit angle-range grids altogether. With a second-order spherical-wave approximation, the near-field steering vector is rewritten as a complex exponential modulated by a chirp-like waveform, and these waveforms are shown to lie in a common discrete chirp rate subspace whose dimension scales as $$\frac{2(D_1+D_2)^2}{\lambda} \quad\text{and}\quad \frac{4D_1D_2}{\lambda},$$0. After lifting, recovery is cast as regularized atomic norm minimization with an equivalent semidefinite program, followed by closed-form coarse range estimation and exact-model nonlinear least-squares refinement (Xi et al., 28 Nov 2025).

For hybrid mmWave MIMO with a UCA at the base station, Ring-Bayes uses a low-coherence concentric-ring codebook sampled jointly in angle and range, with spacings derived from the first zero of $$\frac{2(D_1+D_2)^2}{\lambda} \quad\text{and}\quad \frac{4D_1D_2}{\lambda},$$1, and recovers the sparse beamspace channel by EM-based Bayesian learning. In the reported simulations, Ring-Bayes yields lower NMSE than LS, OMP, and MFOCUSS/FOCUSS, and its BER approaches a Genie-aided detector (Garg et al., 7 May 2026).

5. Wideband, RIS, and hardware-constrained extensions

Wideband near-field estimation adds a frequency dimension to the geometric coupling. In XL-RIS-aided wideband mmWave systems, the near-field beam pattern exhibits squint in both angle and distance domains, a coherence-controlled wideband spherical dictionary is built across subcarriers, and the Multi-Measurement Parallelizable Subspace Recovery framework decomposes the large problem into smaller parallelizable subproblems. The reported time complexity is linear to the number of RIS elements (Yang et al., 2023).

In XL-RIS-aided multi-user mmWave MIMO, the cascaded channel is hybrid-field: the BS-RIS link is far-field, but the RIS-user links are near-field. A low-overhead two-stage estimator therefore decomposes each multi-antenna user into virtual single-antenna users, extracts the common BS-RIS parameters from a typical virtual user, initializes the RIS-user channels by compensated polar-domain sparse recovery, and refines the common BS-RIS operator and all user-specific RIS-side channels by alternating least squares. The paper states that the pilot overhead scales roughly logarithmically with dictionary size and is substantially lower than the cited near-field benchmark (Dong et al., 7 May 2026).

For U6G wideband XL-MIMO under beam squint, a distributed parametric symmetry-based algorithm works in the delay domain. Each subarray estimates local delays, neighboring subarrays extrapolate them over small search sets, and the CPU decouples angle, distance, and range by linear combinations of symmetric delay observations. Because the algorithm does not rely on scanning the polar-domain dictionary, only a single pilot is required even with hybrid architectures (Lu et al., 21 May 2026).

In THz-band RIS-aided links, physically faithful second-order statistics can dominate estimator quality. A Kronecker-correlated mixture-gamma model that includes spherical-wave propagation, molecular absorption, spatial correlation, and mutual coupling is combined with LS and LMMSE estimation. With $$\frac{2(D_1+D_2)^2}{\lambda} \quad\text{and}\quad \frac{4D_1D_2}{\lambda},$$2, near-field correlation improves accuracy by up to about $$\frac{2(D_1+D_2)^2}{\lambda} \quad\text{and}\quad \frac{4D_1D_2}{\lambda},$$3 dB at $$\frac{2(D_1+D_2)^2}{\lambda} \quad\text{and}\quad \frac{4D_1D_2}{\lambda},$$4 dB SNR, while ignoring both mutual coupling and correlation causes about $$\frac{2(D_1+D_2)^2}{\lambda} \quad\text{and}\quad \frac{4D_1D_2}{\lambda},$$5 dB degradation at $$\frac{2(D_1+D_2)^2}{\lambda} \quad\text{and}\quad \frac{4D_1D_2}{\lambda},$$6 dB SNR (Dkhan et al., 14 May 2025).

Hardware-constrained architectures introduce further structure. In modular ELAA uplinks with third-order LNA nonlinearities, closed-form estimators exploit the constant-modulus nature of LoS near-field channels and a reduced geometry-induced subspace. The strongest variant combines $$\frac{2(D_1+D_2)^2}{\lambda} \quad\text{and}\quad \frac{4D_1D_2}{\lambda},$$7D-DFT masking with constant-modulus reduced-subspace LS, and the retained fraction of nonzero $$\frac{2(D_1+D_2)^2}{\lambda} \quad\text{and}\quad \frac{4D_1D_2}{\lambda},$$8D-DFT coefficients is reported to drop from about $$\frac{2(D_1+D_2)^2}{\lambda} \quad\text{and}\quad \frac{4D_1D_2}{\lambda},$$9 for $d_F = 0.5\left(\frac{D^3}{\lambda}\right)^{0.5},$0 subarrays to about $d_F = 0.5\left(\frac{D^3}{\lambda}\right)^{0.5},$1 for $d_F = 0.5\left(\frac{D^3}{\lambda}\right)^{0.5},$2 subarrays. Under accurate distortion compensation, the impact of hardware impairments is reported as negligible (Demir et al., 20 Sep 2025).

A distinct overhead-reduction route is sensing-enhanced channel estimation. One framework inserts one sensing slot with embedded power sensors, reconstructs the propagation scene by time inversion, and then builds a location-aware DPSS-based eigen-dictionary of size $d_F = 0.5\left(\frac{D^3}{\lambda}\right)^{0.5},$3. For a target of $d_F = 0.5\left(\frac{D^3}{\lambda}\right)^{0.5},$4 dB NMSE, the reported required baseband samples are $d_F = 0.5\left(\frac{D^3}{\lambda}\right)^{0.5},$5 for DFT dictionaries, $d_F = 0.5\left(\frac{D^3}{\lambda}\right)^{0.5},$6 for spherical dictionaries, and $d_F = 0.5\left(\frac{D^3}{\lambda}\right)^{0.5},$7 for the proposed eigen-dictionary (Liu et al., 2024).

6. Limitations, model mismatch, and emerging directions

Several recurrent misconceptions are contradicted explicitly in the literature. Far-field methods are not reliable in ELAA near-field regimes, because the channel depends on both angle and distance and the transformed support can spread across multiple bins rather than collapsing to a single atom (Chen et al., 15 Mar 2026). Direct geometric maximum-likelihood estimation is statistically attractive, but the corresponding optimization is highly nonconvex, and brute-force or gradient-descent implementations often fail to converge globally as dimensionality grows (Do et al., 5 May 2026).

The Fresnel or parabolic approximation is likewise not universal. Below the Fresnel distance, a wavenumber-domain formulation shows that the spherical-wave spectrum of a scatterer is bounded by four ellipse-shaped curves determined jointly by the scatterer position and the rectangular array geometry. The resulting Wavenumber-Domain Ellipse Fitting method estimates geometry by thresholding the spectrum, clustering support regions, extracting boundary curves, and fitting ellipses, rather than by assuming Gaussian-like spectral blobs (Guo et al., 2024).

The spherical-wavefront channel is itself only one model class. Under wavefront anisotropy induced by curved reflecting surfaces, the usual angle-distance sparsity disappears: in the reported experiment, an anisotropic wavefront channel reaches only about $d_F = 0.5\left(\frac{D^3}{\lambda}\right)^{0.5},$8 maximum cosine similarity with any single spherical point-source model, and SWC-OMP-LM exhibits an error floor in anisotropic-wavefront environments because of model mismatch (Zhang et al., 21 May 2026).

This suggests that near-field channel estimation is best understood as a hierarchy of models and estimators rather than a single algorithmic template. Exact geometric LoS models, polar and spherical dictionaries, block-sparse and Bayesian formulations, wavefront parameterization, gridless atomic norms, wavenumber-domain geometry recovery, and sensing-assisted or distributed designs are each appropriate under different propagation assumptions, with the dominant tradeoffs being model mismatch, pilot overhead, and algorithmic complexity.