CDCE: Cross-Domain Channel Estimation

• Cross-Domain Channel Estimation (CDCE) is a framework that bridges different channel technologies using statistical signal processing, machine learning, and domain adaptation to handle mismatches across diverse wireless regimes.
• CDCE employs regime detection, two-stage estimation, and adaptive sparse recovery to achieve significant NMSE gains, enhanced sample efficiency, and reduced pilot overhead in advanced MIMO systems.
• The methodology integrates fine-tuning from synthetic to realistic map-based channels, leveraging hybrid ML and model-based approaches to mitigate physical grid mismatches and beam-squint effects.

Cross-Domain Channel Estimation (CDCE) encompasses a collection of algorithmic frameworks designed to bridge fundamental modeling differences between channel regimes, domains, and environments in advanced wireless systems. These algorithms leverage statistical signal processing, ML, and domain adaptation principles to generalize channel estimators across mismatched or diverse scenarios, including: synthetic vs. map-based models, time-frequency vs. delay-Doppler domains, near-field vs. far-field propagation, and static vs. dynamic channels. CDCE is now central to high-precision channel estimation in next-generation MIMO-OFDM, terahertz (THz) UM-MIMO, and mmWave XL-MIMO architectures, delivering substantial NMSE gains, improved robustness with limited real measurements, and significant computational efficiency relative to conventional estimation pipelines (Hoang et al., 11 Jul 2025, Nie et al., 21 Jan 2026, Tarboush et al., 2023, Tarboush et al., 2024, Hou et al., 2023).

1. Problem Formulation and Domain Mismatch Scenarios

CDCE is motivated by marked distribution mismatches between source and target channel domains. Scenarios include:

• Synthetic vs. Map-Based Channels: Simulated (QSCM) channels, e.g., using simplified multipath models, diverge in delay, amplitude, and angular statistics from realistic map-based (MBCM) channels generated via CDL + ray-tracing. The Wasserstein-1 distance between datasets quantifies this domain gap (e.g., $W_1 \approx 0.46$) (Hoang et al., 11 Jul 2025).
• Near-/Far-Field Transitions: Ultra-massive array deployments induce spatial regimes where the planar (far-field) approximation fails, requiring channel estimators to adaptively select between spherical-wave (SWM), hybrid spherical-planar (HSPWM), or pure planar-wave (PWM) models as the user’s distance changes (Tarboush et al., 2023, Tarboush et al., 2024).
• Delay-Doppler vs. Time-Frequency Domain: High-mobility or doubly selective channels necessitate estimation in the delay-Doppler (DD) domain for sparsity, then translation back to the time-frequency (TF) domain of OFDM (Nie et al., 21 Jan 2026).
• Physical Grid Mismatch and Beam-Squint Effects: In mmWave XL-MIMO, spatial-frequency channels exhibit basis mismatch and frequency-dependent spatial signatures (beam-squint), which classic linear dispersive models cannot capture (Hou et al., 2023).

In each case, the mismatch prevents naive transfer of estimators: source-trained ML models or model-based solvers degrade or become ineffectual without adaptation mechanisms.

2. Core CDCE Algorithmic Principles

Across existing literature, CDCE implementations share several organizing principles:

• Regime Detection and Domain Partitioning: Algorithms typically start by measuring key channel statistics (e.g., cross-subarray power variation, delay-Doppler structure) to infer the active channel domain/regime (near/far-field, static/dynamic, synthetic/realistic).
• Two-Stage or Multiphase Estimation: Foundation or coarse-stage estimation is performed using computationally efficient techniques (e.g., LS, LMMSE, modified OMP/SOMP, CNNs pretrained on QSCM), building strong priors or support sets for subsequent domain-adaptive refinement (Hoang et al., 11 Jul 2025, Nie et al., 21 Jan 2026, Tarboush et al., 2023).
• Domain Adaptation via Lightweight Fine-Tuning: Transfer learning methods freeze most of the model (e.g., CNN/GAN backbone) pretrained on abundant synthetic data; only shallow layers are retrained on scarce target (real/map-based) samples, minimizing real measurement requirements (Hoang et al., 11 Jul 2025).
• Adaptive Sparse Recovery: In the presence of structure (e.g., delay/Doppler support, angular clusters, or polar grids), CDCE refines the dictionary or search space using outputs from the coarse stage and applies compressed sensing (CS) or $\ell_1$-regularized estimators for robust path coefficient estimation (Nie et al., 21 Jan 2026, Tarboush et al., 2023, Hou et al., 2023).
• Statistical Filtering for Regime Tracking: CDCE often employs Markovian or Bayesian filters (e.g., hidden Markov models (HMMs) over region estimates) to suppress regime misclassification due to SNR perturbations, especially in field-transitioning scenarios (Tarboush et al., 2024).

3. Canonical Architectures and Algorithmic Workflows

(a) ML-driven CDCE for Map-Based MIMO-OFDM

• Phase 1: Foundation Model Training
  1. Generate QSCM data, apply LS to DM-RS pilots, output $\widehat H^{ls}$.
  2. Interpolate (LS-LI) for full grid $\widehat H^{li}$.
  3. Two ML modules: CNNs (for LS and LS-LI streams) and GANs (Pix2Pix variants), output $\widehat H^{ls,cnn}$, $\widehat H^{li,cnn}$, etc.
  4. Train using NMSE loss: $$\ell = \mathbb{E}[\|\widehat H - H\|_2^2 / \|H\|_2^2]$$.

• Phase 2: Domain Adaptive Fine-Tuning

  1. Freeze backbone layers; update only task-specific (last) layers.
  2. Fine-tune on small labeled set from map-based (MBCM) domain (e.g., 300/1000 samples).
  3. Optionally augment with domain alignment or adversarial losses (Hoang et al., 11 Jul 2025).

(b) Field-Adaptive CS CDCE

• AoSA Training: Apply pilot beams from reference subarrays; record per-SA received vectors.

• Field Regime Detection: Compute cross-SA metric $\eta = \max_{i<j} \|\chi^i - \chi^j\|_2^2$; classify into near, intermediate, or far-field via learned thresholds $\gamma_{S-H}$, $\gamma_{H-P}$.
• Domain-Specific CS Estimation:
  • SWM (polar-domain): Full-dictionary sparse recovery for reference SA, then reduced dictionaries for remaining SAs.
  • HSPWM/PWM: Similar, with search reduction commensurate with domain spatial coherence.
  • All regimes leverage OMP/SOMP with adaptive dictionary selection to minimize search and computation (Tarboush et al., 2023).
• Online HMM Filtering: Incorporate past regime observations into an HMM to address decision volatility at low SNR (Tarboush et al., 2024).

(c) Delay-Doppler Sparse CDCE for OFDM

• Twisted-Convolution Support Extraction: Transform TF domain pilot/data sequences to DD domain via SFFT; apply low-complexity $2$D twisted convolution to extract sparse support peaks $(\ell,k)$.
• Dictionary Construction and Sparse Recovery: Build a TF-domain dictionary tailored to detected DD taps; solve $\ell_1$-regularized least squares (LASSO) for complex tap amplitudes.
• Reconstruction: Map recovered support back to full TF domain, reconstructing the physical channel with low NMSE (Nie et al., 21 Jan 2026).

(d) Beam-Delay, MDGPP, and Bethe Free Energy Approaches

• Sparse Factor Graph Model: Model beam-delay channel coefficients $\beta_k$ as independent Bernoulli-Gaussian with learnable hyperparameters $\lambda_k$, $\gamma_k$.
• Hybrid Message Passing: Minimize a Bethe free energy functional under mean-variance consistency constraints using alternating measurement update and prior update steps.
• Grid Perturbation Refinement: The MDGPP model assigns per-path grid perturbations $(\Delta\psi, \Delta\eta, \Delta\tau)$, pruned by energy thresholding and updated via small fixed-point solves.
• Two-Stage Iterative Refinement: Coarse support detection via standard HMP; fine grid/parameter refinement on the pruned support (Hou et al., 2023).

4. Theoretical Performance and Computational Characteristics

CDCE universally improves estimation robustness and efficiency under cross-domain mismatches. Key metrics:

• NMSE Gains: Foundation-model ML CDCE recovers $3-8$ dB NMSE improvement over classical LS/LI (MBCM at low SNR) post-fine-tuning; delay-Doppler LASSO approaches achieve $4-5$ dB gain over full-matrix LMMSE at moderate SNR (Hoang et al., 11 Jul 2025, Nie et al., 21 Jan 2026).
• Sample Efficiency: ML CDCE adapts effectively with $300$ target-domain samples, a substantial reduction compared to direct supervised training (Hoang et al., 11 Jul 2025). MDGPP-based two-stage HMP achieves $25\%$ pilot overhead reduction for –10 dB NMSE (Hou et al., 2023).
• Complexity Reduction: Field-adaptive CDCE in UM-MIMO can reduce SOMP solve complexity by up to $75\%$ (far-field, via support reuse), lowering average runtime by $40-70\%$ while preserving or improving NMSE and throughput (Tarboush et al., 2023).
• Robustness to SNR and Dynamics: HMM-based regime selection outperforms single-shot classification by $15-20$ points in correct detection under low SNR; DD-domain LASSO remains effective in high-mobility cases (Tarboush et al., 2024, Nie et al., 21 Jan 2026).

5. Limitations, Pitfalls, and Data Requirements

Several limitations are endemic across current CDCE frameworks:

• Residual Domain Gaps: Frozen-backbone fine-tuning does not eliminate all statistical discrepancies; a nontrivial number of real or map-based samples (typically $>100$) are needed for strong transfer (Hoang et al., 11 Jul 2025).
• Region Misclassification: In near/far-field discriminators, SNR-dependent errors can degrade overall estimation, though statistical filters (HMMs) significantly alleviate this (Tarboush et al., 2024).
• Pilot Overhead: Compressive sensing and sparse recovery approaches, while more efficient, retain some lower bound on pilot density per subdomain to ensure grid coverage and avoid severe basis mismatch (Hou et al., 2023).
• Static Scenario Emphasis: Most pipelines focus on block-fading or slowly-varying settings; real-time adaptation and Doppler-resilient design require further development (Hoang et al., 11 Jul 2025).

6. Future Directions and Open Questions

Research in CDCE is converging on several significant extensions:

• Unsupervised/Semi-supervised Domain Adaptation: Reducing or eliminating the dependence on labeled target-domain samples remains a critical need for scalable, real-world deployment (Hoang et al., 11 Jul 2025).
• Time-Varying Channel Tracking: Dynamic extension to channels with substantial Doppler (intra-slot or slot-to-slot), integrating real-time ML adaptation or model-based motion tracking (Hoang et al., 11 Jul 2025).
• Joint Pilot and Estimator Optimization: Co-designing pilots with CDCE, to maximize information-to-support mapping or minimize ambiguity under regime switching, is open (Hoang et al., 11 Jul 2025, Hou et al., 2023).
• Hierarchical/Recursive Filtering: Advanced regime-tracking beyond basic HMMs (e.g., particle filters or variational Bayes) may further suppress regime misclassification (Tarboush et al., 2024).
• Physical Model Generalization: Incorporation of more complex effects—material scatter, mutual coupling, array curvature, and hardware impairments—poses theoretical and computational challenges for CDCE’s foundational assumptions.

7. Comparative Overview of Recent CDCE Approaches

Reference	Model Domains	CDCE Mechanism	Key NMSE/Postprocessing Gains
(Hoang et al., 11 Jul 2025)	QSCM ↔ MBCM	Foundation CNN/GAN fine-tuning	$3$–$8$ dB NMSE gain (MBCM, low SNR)
(Nie et al., 21 Jan 2026)	TF ↔ Delay-Doppler	Twisted-conv + LASSO sparse reco.	$4$–$5$ dB vs. FS-LMMSE at $20$ dB SNR
(Tarboush et al., 2023)	Near/mid/far-field	Field detection + OMP/SOMP + RD	$2$–$5$ dB NMSE, $40$–$75\%$ complexity cut
(Tarboush et al., 2024)	Field regime tracking	HMM regime classifier + CS	$15$–$20$ pt gain in classification at low SNR
(Hou et al., 2023)	Beam-delay (XL-MIMO)	HMP (Bethe) + MDGPP 2-stage	$5$–$10$ dB vs. OMP, $3$–$5$ dB via MDGPP

Methodologically, CDCE unifies model-driven and data-driven paradigms by utilizing domain-specific priors, regime-adaptive dictionary strategies, and transfer learning from large source to scarce target regimes, thereby providing a robust foundation for future high-mobility, high-frequency, and geometrically complex wireless systems.