Three-Stage Unified Channel Estimation

Updated 29 November 2025

The paper introduces a three-stage unified approach that partitions channel estimation into sequential phases, effectively reducing pilot overhead and error propagation.
The methodology exploits common spatial features and sparsity using DFT, Bayesian inference, and deep learning to robustly extract CSI in mmWave, THz, and XL-MIMO scenarios.
Key results demonstrate up to 10 dB NMSE gains and significant runtime savings, highlighting the protocol’s efficacy in diverse multi-user and nonstationary settings.

A three-stage unified channel estimation strategy is a structured protocol for extracting channel state information (CSI) in multi-antenna wireless systems by partitioning the estimation process into three sequential phases, each targeting distinct subproblems. This approach has gained prominence in millimeter wave (mmWave) MIMO, reconfigurable intelligent surface (RIS)-aided massive MIMO, THz ultra-massive MIMO, multi-user uplink, and XL-MIMO settings. The unification refers both to the systematic progression—typically isolating direct channels, angles, and cascaded/sparse subchannels—and to computational synergies such as shared pilot designs or common spatial structures. Such frameworks are designed to minimize pilot/feedback overhead, suppress error propagation, enable robust parameter inference under nonstationarity, and maintain uniformly low estimation error across diverse scenarios (Zhang et al., 22 Nov 2025, Zhuo et al., 8 Feb 2025, Peng et al., 2022, Tang et al., 5 Mar 2024, Ma et al., 2020).

1. Channel and System Model Abstractions

Recent three-stage channel estimation protocols apply to MU-MIMO systems with uniform planar arrays (UPA), hybrid analog/digital architectures, RIS-aided configurations, and XL-MIMO infrastructures. A representative model considers:

BS: $N = N_1 \times N_2$ UPA
RIS: $M = M_1 \times M_2$ UPA
User: $Q_k = Q_{k1} \times Q_{k2}$ UPA, $K$ users
Channel: Cascaded links $G_k$ , direct links $H_{d,k}$ , and RIS subchannels $H_{r,k}$
Sparse Saleh-Valenzuela propagation (few dominant paths)—each characterized by AoA (BS), AoD (user/RIS), and complex gain (Zhang et al., 22 Nov 2025, Zhuo et al., 8 Feb 2025, Peng et al., 2022).

Protocols leverage the invariance of common channel components: all users typically share the same RIS–BS link, and cascaded links can be re-parametrized using a typical user’s estimated AoAs/AoDs. XL-MIMO settings additionally capture spherical wavefront effects and spatial nonstationarity via visibility regions and block sparsity (Tang et al., 5 Mar 2024).

2. Stage I: Direct Channel and Common Angle Estimation

The initial stage focuses on extracting direct channel components or common spatial frequencies.

RIS-aided MIMO: Cancels cascaded terms by designing RIS phase shifts with a $\pi$ difference—combining consecutive time slots leads to isolation of direct user–BS channels (Zhang et al., 22 Nov 2025). In hybrid architectures, all users transmit pilots simultaneously; the BS employs IDFT/DFT projections to recover AoAs, exploiting the low-rank structure and using high-resolution DFT peak/sparse search (Zhuo et al., 8 Feb 2025, Peng et al., 2022).
XL-MIMO: Atomic-norm SDP is applied to multi-pilot blocks for high-resolution AoD extraction under nonstationarity and spherical wavefront effects (Tang et al., 5 Mar 2024).
ESPRIT-based Massive MIMO: Overlapping subarrays and controlled pilot patterns with "OFF" antennas create the required shift-invariance for 2D/1D ESPRIT subspace methods (Ma et al., 2020).
Deep Learning ISAC: Direct sensing and communication channels are learned via CNNs using IRS-off input/output pilot blocks, with carefully designed DFT pilots (Liu et al., 29 Jan 2024).

This stage commonly exploits:

Sparse DFT-space projections, 1-D angle rotation or atomic-norm denoising to suppress leakage (Peng et al., 2022)
MMV/OMP or SOMP search to identify common AoAs/AoDs
In XL-MIMO, support estimation in the angular domain, prior to per-path inference (Tang et al., 5 Mar 2024)

Pilot overhead in Stage I is $O(\max\{I,J\} \log Q)$ , substantially smaller than simultaneous multi-user approaches (Zhang et al., 22 Nov 2025, Zhuo et al., 8 Feb 2025).

3. Stage II: Equivalent Channel Construction, Angle Estimation, and Sparsity Exploitation

The second stage leverages parameters from Stage I to reparameterize and estimate cascaded or equivalent channels:

RIS-aided protocols: Orthogonal subspace projection removes direct components before estimating AoDs of user–RIS links for all users. Typical users’ cascaded parameters (gains, angle offsets) enable equivalence in subsequent estimation (Zhang et al., 22 Nov 2025, Zhuo et al., 8 Feb 2025).
XL-MIMO: Each path—with steering towards an extracted AoD—is isolated; subchannel inference proceeds via a three-layer Bayesian scheme: block sparsity in angle domain, spatial nonstationarity in antenna domain (Markov VR chain), and bilinear GAMP decoupling in the measurement domain (Tang et al., 5 Mar 2024).
ESPRIT: Additional subarray pilot settings and cross-domain Hermitian processing permit angle pairing, extracting AoDs after AoAs, with permutation alignment to maximize diagonal path-gain concentration (Ma et al., 2020).
ISAC: IRS remains ON, and CNN-based estimation predicts reflected communication channels with pilot slots dedicated to IRS/UE transmissions (Liu et al., 29 Jan 2024).
AoSA THz: Variation-metric $\eta$ determines field regime; compressive observation with matched dictionaries enables spatially localized sparsity (Tarboush et al., 2023).

Key algorithmic themes:

MMV/OMP sparse recovery on projected blocks for AoD estimation
Subspace projections to orthogonal complements to decouple direct/cascaded contributions
Use of equivalence and invariance (gain and angle correlation) in cascaded multipath recovery (Zhang et al., 22 Nov 2025, Zhuo et al., 8 Feb 2025)
Layered Bayesian message passing for block-structured angular and antenna domain inference (Tang et al., 5 Mar 2024)

4. Stage III: Fast Cascaded Gain Update, Multiuser/Virtual User Expansion

The third stage focuses on re-estimating fast-varying gain parameters and extending inference to all users:

RIS protocols: Time slot signals with the same pilots are recombined, projected onto orthogonal complements of direct-channel spans, yielding reduced measurements with only cascaded components. Angular parameters and gains are evaluated via OMP and search over equivalent virtual-user channels (Zhang et al., 22 Nov 2025, Zhuo et al., 8 Feb 2025).
Multi-antenna users: Decomposition into virtual single-scatterer channels enables reuse of Stage I/II methods for each antenna/path, with pilot overhead and computational complexity scaling as $O(J \log Q + J \log M/L)$ (Peng et al., 2022).
XL-MIMO: TL-GAMP inference converges in 10–20 iterations, giving robust SnS/antenna region detection and angular support (Tang et al., 5 Mar 2024).
AoSA THz: Reduced dictionary sizes for non-reference subarrays sharply lower the per-SA computational burden (Tarboush et al., 2023).
Geometry-based estimation: Environment map and user location from Stage I, combined with profile-likelihood prediction, are fused with measured CSI in a Bayesian MMSE stage for robust beamforming (Deutschmann et al., 22 Mar 2025).

Stage III typically removes error propagation from previous stages (e.g., direct-to-cascaded) and achieves estimation complexity nearly independent of user/path number, with pilot efficiency enhanced by equivalence operations.

5. Algorithmic Structures, Pilot and Complexity Analysis

Three-stage strategies universally employ joint pilot/precoder/combiner design for noise suppression and power uniformity:

DFT-based combiners and combiner row assignments concentrate energy regionally, achieving uniform received power across AoA/AoD (Ma et al., 2020, Zhuo et al., 8 Feb 2025).
RIS phase shift patterns (e.g., π-shifted, "Type I" random Bernoulli) are selected to guarantee cascade cancellation or optimal spread (Zhang et al., 22 Nov 2025, Peng et al., 2022).
Hybrid analog/digital combining matrices are constructed to minimize noise covariance at the BS (Zhuo et al., 8 Feb 2025).

Complexity summaries from the data include: | Method | Dominant Complexity Term | Pilot Overhead / Scaling | |----------------|-----------------------------------------------------------------|-----------------------------------------------------------| | TDE ESPRIT (Ma et al., 2020) | $O(8(T_3^2+T_1^2)(T_1+T_3)+2L_u^2(10L_u+5T_3+6T_1)+2L_uT_1T_3)$ | 3 pilot stages; order-of-magnitude reduction over CS | | XL-MIMO TL-GAMP (Tang et al., 5 Mar 2024) | $O(N_RQ + MN_R)$ per iteration | Typically 10–20 iters; robust to VR size/path number | | RIS-MU Hybrid (Zhuo et al., 8 Feb 2025) | DFT: $O(N_{\mathrm{bs}}\log N_{\mathrm{bs}})$ , CS: $O(JM\tau)$ | $>5\times$ lower pilots vs. AoD-SOMP/CS multiuser | | AoSA THz Cross-field (Tarboush et al., 2023) | $O(G_RG_TK M_TM_RL)$ (oracle LS) | Reduced dictionary for non-ref SAs drastically lowers cost| | Multi-User UPA (Peng et al., 2022) | $O(J K \log Q + J\log M + (K-1)J\log M/L)$ | Supports fast gain update; scales to $K>10$ | | Geometry-based fusion (Deutschmann et al., 22 Mar 2025) | Each PF step: profile likelihood + amplitude update | Robust to low SNR, mobility; fusion for unreliable CSI |

6. Simulation Performance, Robustness, and Field Applicability

Three-stage unified methods realize substantial improvements in normalized mean squared error (NMSE), pilot overhead, and spectral efficiency across diverse empirical regimes:

RIS-aided MU-MIMO: Proposed methods achieve near-oracle NMSE for direct links once SNR $>0$ dB; cascaded NMSE outperforms conventional ON-OFF/OMP/SBL by 5–10 dB, with pilot overhead reduced from $O(KI\log Q)$ to $O(K\max\{I,J\}\log Q)$ (Zhang et al., 22 Nov 2025, Zhuo et al., 8 Feb 2025).
Multiuser scaling: Pilot overhead and NMSE remain stable for $K$ up to 10; error propagation from direct channel estimation is eliminated, yielding independence from path numbers (Zhang et al., 22 Nov 2025, Peng et al., 2022).
XL-MIMO: TL-GAMP provides 5–10 dB NMSE gain over OMP or antenna-only GAMP, consistently robust against increasing visibility region $\phi_l$ and path count, effective in near/far-field transition (Tang et al., 5 Mar 2024).
AoSA THz: Cross-field reduced-dictionary approach matches or outperforms benchmarks with up to 70% runtime savings and 1 bit/s/Hz AR gain in near/intermediate field (Tarboush et al., 2023).
Deep Learning ISAC: CNN-based pipeline yields up to 12.5 dB SNR gain over LS, exceeds generalization across $-10$ dB to $20$ dB test range, and remains computationally competitive (Liu et al., 29 Jan 2024).
Geometry-based fusion: Bayesian MMSE combination of uplink-inferred and predicted CSI enhances beamforming robustness in mobile and low-SNR conditions; gains stem from accurate user/environment modeling (Deutschmann et al., 22 Mar 2025).

7. Extensions, Unification Themes, and Limitations

The unification of three-stage strategies extends to: geometry-based SLAM with CSI prediction and fusion (Deutschmann et al., 22 Mar 2025); spatially nonstationary XL-MIMO via layered Bayesian inference (Tang et al., 5 Mar 2024); ML/DL frameworks for IRS/ISAC (Liu et al., 29 Jan 2024); multi-antenna user decomposition for RIS-aided systems (Peng et al., 2022). The universal themes are:

Joint exploitation of sparsity, block-angle invariance, and pilot reuse
Decoupling direct and cascaded channel estimation, propagating typical-user parameters to other users via reparameterization
Structured pilot and hybrid matrix design for uniform SNR and noise reduction
Use of equivalence, error-suppressing projections, and dictionary reduction
Robustness to mobility, nonstationarity, and high-user/path-count scenarios

Limitations include the need for prior knowledge of RIS/BS/UE array geometries, staticity of angle parameters within blocks, and, in DL-based protocols, retraining for new SNR/mobility regimes. A plausible implication is that further reductions in pilot overhead and more adaptive inference schemes—e.g., jointly learning the array manifold or applying online Bayesian hyperparameter updates—could further improve performance in time-varying and heterogeneous systems.