6G Multi-Domain Channel Extrapolation

• Multi-domain channel extrapolation is the inference of unobserved CSI across multiple domains (time, frequency, space, and more) from partial measurements to optimize communication performance.
• Recent methods employ AI-driven models, physics-based parameter estimation, and generative architectures, quantified by metrics like NMSE and SGCS, to effectively reconstruct missing CSI data.
• Implications include reduced pilot overhead, robust precoding, enhanced generalization across diverse 6G scenarios, and improved downstream communication efficiency.

Multi-domain channel extrapolation is the inference of unobserved channel state information (CSI) across more than one domain from partial CSI measurements. In the 6G setting emphasized by recent work, the principal domains are time, frequency, and space, so the full channel may be written as $H(t,f,a)\in\mathbb C^{N_t\times N_f\times N_a}$, while the practical task is to reconstruct missing entries from an observed subset $H_{\rm obs}$ through a mapping $\hat H=f_\theta(H_{\rm obs})$ trained by minimizing $$\mathbb E[\|f_\theta(H_{\rm obs})-H_{\rm true}\|_F^2]$$ (Gao et al., 1 Sep 2025). The same extrapolative idea has also been instantiated across radiation modes in pattern reconfigurable MIMO, across base stations through a position domain, across licensed and unlicensed bands through channel fingerprints, and across antenna, frequency, and spatial domains in near-field XL-MIMO, indicating that multi-domain extrapolation is a general CSI-acquisition paradigm rather than only a time–frequency–space problem (Liang et al., 2023, Guo et al., 23 Jul 2025, Xie et al., 2024, Li et al., 15 Jun 2026).

1. Formal definitions and problem variants

A common formalization treats CSI as a partially observed multidimensional object. In the time–frequency–space formulation, the observed tensor may include only past time steps $t=1\ldots T_o$, frequencies $f\in F_o$, or an antenna subset $a\in A_o$, and the extrapolation objective is to recover the missing entries of the full tensor $H(t,f,a)$ (Gao et al., 1 Sep 2025). A broader review uses $H(f,t,a)\in\mathbb C^{I\times J\times F\times T}$ and distinguishes single-domain mappings $\mathcal F_{\rm time}$, $H_{\rm obs}$0, and $H_{\rm obs}$1 from a joint mapping $H_{\rm obs}$2 that reconstructs unobserved entries from a partial tensor $H_{\rm obs}$3 (Gao et al., 1 Jan 2026).

The basic loss and accuracy measures are likewise standardized in recent formulations. The training objective is often an MSE or NMSE minimization, while 6G-oriented work also uses the squared generalized cosine similarity (SGCS), defined as

$H_{\rm obs}$4

These metrics make explicit that extrapolation is judged not only by entrywise error but also by directional consistency of the reconstructed channel (Gao et al., 1 Sep 2025).

The term “domain” is not restricted to time, frequency, and antenna indices. In PR-MIMO, the extrapolation target is the CSI of other radiation modes $H_{\rm obs}$5 from the estimated CSI of one reference mode, through a mapping $H_{\rm obs}$6 (Liang et al., 2023). In cell-free massive MIMO, the added domain is user position $H_{\rm obs}$7, which is invariant across base stations and can be used as a bridge between otherwise uncorrelated channels (Guo et al., 23 Jul 2025). In multi-band transmission, the extrapolated object can be a channel fingerprint map $H_{\rm obs}$8 rather than instantaneous CSI (Xie et al., 2024). In near-field XL-MIMO, the observed object is a masked complex tensor $H_{\rm obs}$9 over antenna, frequency, and spatial indices, and the task is infilling under severe masking (Li et al., 15 Jun 2026).

2. Performance criteria and 6G requirements

For 6G-oriented time–frequency–space extrapolation, explicit targets have been proposed: in in-distribution scenarios, $\hat H=f_\theta(H_{\rm obs})$0 dB and $\hat H=f_\theta(H_{\rm obs})$1; under SNR variation or Doppler variation, stable SGCS within $\hat H=f_\theta(H_{\rm obs})$2; under out-of-distribution testing such as a Doppler increase from $\hat H=f_\theta(H_{\rm obs})$3 kHz to $\hat H=f_\theta(H_{\rm obs})$4 kHz, performance degradation $\hat H=f_\theta(H_{\rm obs})$5 dB NMSE; and real-time inference $\hat H=f_\theta(H_{\rm obs})$6 samples/s on a GTX 1050 Ti, with model size $\hat H=f_\theta(H_{\rm obs})$7 M parameters and per-sample Flops $\hat H=f_\theta(H_{\rm obs})$8 (Gao et al., 1 Sep 2025). These requirements make adaptability and deployment cost first-class constraints, not secondary implementation details.

The broader literature evaluates multi-domain extrapolation with a larger metric set. Reviews identify NMSE, inference FLOPs, and adaptation to dynamic scenarios as central criteria (Gao et al., 1 Jan 2026). ChannelKAN reports NMSE, spectral efficiency, and bit error rate (Jiang et al., 11 May 2026). XL-ChannelDiff reports NMSE, $\hat H=f_\theta(H_{\rm obs})$9 distance, cosine distance, and achievable rate under MRT beamforming (Li et al., 15 Jun 2026). PCEnet evaluates pilot length, feedback bits, NMSE, and localization error (Guo et al., 23 Jul 2025). SEMRA evaluates an eCSI reconstruction metric,

$$\mathbb E[\|f_\theta(H_{\rm obs})-H_{\rm true}\|_F^2]$$0

along with spectral efficiency (Li et al., 22 May 2026).

A plausible implication is that “accuracy” in multi-domain extrapolation is increasingly operationalized through downstream communication utility. This is particularly clear in works that connect extrapolated CSI to robust precoding, beamforming, achievable rate, or sum rate rather than treating reconstruction as an isolated inverse problem (Xie et al., 2024, Li et al., 22 May 2026).

3. Model-driven and physics-grounded lineages

A major model-driven lineage treats extrapolation as parameter estimation followed by channel resynthesis. In wideband MIMO, one early formulation used double-directional models and proposed three predictors based on 4D, 3D, and 2D extensions of ESPRIT. The 4D model estimated receive spatial frequency, transmit spatial frequency, Doppler, and delay; the 3D and 2D models reduced the parameterization; and a Cramér–Rao lower bound on prediction error was derived through a vector formulation for functions of parameters (Adeogun et al., 2014). This line established a concrete multi-domain template: estimate latent propagation parameters jointly across several channel axes, then extrapolate to future states by re-evaluating the parametric model.

A related but more deployment-specific example appears in TDD 5G NR systems with hopping uplink pilot patterns. There, a two-stage 2D extrapolation scheme over frequency and time combines a multi-band and multi-timeslot high-resolution parameter estimation algorithm with a channel tracking stage based on a sparse Markov channel model and an expectation-maximization compressive tracking algorithm. The method explicitly estimates and compensates Doppler phase rotation, random phase noise, and time offset, and reports $$\mathbb E[\|f_\theta(H_{\rm obs})-H_{\rm true}\|_F^2]$$1–20 dB TNMSE gains over baselines depending on $$\mathbb E[\|f_\theta(H_{\rm obs})-H_{\rm true}\|_F^2]$$2 and SNR (Wan et al., 2023).

Other physics-grounded formulations move beyond path-parameter estimation. Roughness-calibrated CCM extrapolation infers scatterer surface roughness parameters from measured CCMs in one area and then reuses those calibrated parameters in a ray tracer to predict CCMs in other areas or other domains. Reported results include frequency-to-frequency relative error reduced by $$\mathbb E[\|f_\theta(H_{\rm obs})-H_{\rm true}\|_F^2]$$3, frequency-to-spatial CCM error $$\mathbb E[\|f_\theta(H_{\rm obs})-H_{\rm true}\|_F^2]$$4 of baseline, and spatial-to-frequency extrapolation with similar slope but slower convergence (Zhang et al., 2024). A modal-based multi-scatterer channel model instead builds source radiation, single-scatterer response, and inter-scatterer coupling in spherical-wave modes, learns scatterer responses and locations from sparse measurements, and extrapolates radiomaps in both the spatial domain and the beam domain. For beam extrapolation under novel beams, the reported optimum is $$\mathbb E[\|f_\theta(H_{\rm obs})-H_{\rm true}\|_F^2]$$5 with $$\mathbb E[\|f_\theta(H_{\rm obs})-H_{\rm true}\|_F^2]$$6, $$\mathbb E[\|f_\theta(H_{\rm obs})-H_{\rm true}\|_F^2]$$7, and $$\mathbb E[\|f_\theta(H_{\rm obs})-H_{\rm true}\|_F^2]$$8 (Li et al., 4 May 2026).

A further physics-assisted variant is movement-aided channel estimation in SEMRA. By moving antennas across four pilot slots to synthesize a $$\mathbb E[\|f_\theta(H_{\rm obs})-H_{\rm true}\|_F^2]$$9 virtual UPA, the method estimates delays and AoDs with higher resolution, reconstructs the EM-domain CSI, and then assembles eCSI at desired antenna positions without additional pilots. Under SNR $t=1\ldots T_o$0 dB, the reported NMSE-E falls below $t=1\ldots T_o$1 dB for SEMRA+ESPRIT versus only $t=1\ldots T_o$2 dB for EMRA+ESPRIT, while spectral efficiency increases from about $t=1\ldots T_o$3 bps/Hz for EMRA to about $t=1\ldots T_o$4 bps/Hz for SEMRA–WMMSE (Li et al., 22 May 2026).

4. AI-driven architectures and generative paradigms

Recent work increasingly treats multi-domain extrapolation as a generative or representation-learning problem. A 6G-oriented synthesis explicitly classifies VAEs, GANs, diffusion models, and Transformers as generative AI candidates. In that comparison, Transformers are marked as supporting both long-range dependency and hidden-feature capture, but not real-time processing; VAEs support hidden-feature capture and real-time processing; diffusion models support hidden-feature capture but are marked as training unstable and not real-time; and GANs support hidden-feature capture but neither long-range dependency nor real-time processing (Gao et al., 1 Sep 2025).

Within that framework, a specific encoder-only architecture modifies the standard Transformer in two ways: positional encoding is removed, and multi-head self-attention is replaced with an MLP mixer

$t=1\ldots T_o$5

with GELU nonlinearity. The rationale given is that CSI sequences in time, frequency, and space are real-valued signals without semantic reorder invariance, so positional encodings may corrupt local continuity (Gao et al., 1 Sep 2025). On a 3GPP CDL dataset at $t=1\ldots T_o$6 GHz with Tx $t=1\ldots T_o$7 Rx $t=1\ldots T_o$8, $t=1\ldots T_o$9 kHz SCS, observed $f\in F_o$0 frames, $f\in F_o$1 subcarriers, and $f\in F_o$2 antennas, the model uses about $f\in F_o$3 M parameters and about $f\in F_o$4 GFLOPs per sample, and reaches $f\in F_o$5 samples/s on GTX 1050 Ti with memory $f\in F_o$6 GB (Gao et al., 1 Sep 2025). In-distribution TFS-D performance is reported as SGCS $f\in F_o$7 and NMSE $f\in F_o$8 dB for the proposed model, compared with SGCS $f\in F_o$9 and NMSE $a\in A_o$0 dB for a standard Transformer; ablations report that removing positional encoding improves SGCS by $a\in A_o$1 and doubles speed, while replacing attention with an MLP yields $a\in A_o$2 dB NMSE gain and $a\in A_o$3 faster inference (Gao et al., 1 Sep 2025).

Cross-band extrapolation has produced a different AI design pattern. MDFCE extrapolates sub-6 GHz CSI to mmWave CSI through a Temporal Feature Extraction Module, a Multi-Domain Fusion Module with MHSA and sparse MoE, and a Deep Feature Interaction Module. The total loss is $a\in A_o$4 with $a\in A_o$5, and training uses AdamW for $a\in A_o$6 epochs (Xiao et al., 11 Jan 2026). Reported results include $a\in A_o$7 dB NMSE gain over direct LS plus linear interpolation at comparable pilot overhead, $a\in A_o$8 dB gain at $a\in A_o$9 pilot-overhead reduction, $H(t,f,a)$0–$H(t,f,a)$1 dB gain under low SNR, $H(t,f,a)$2 ms per sample on RTX 3090, and $H(t,f,a)$3 GFLOPs versus $H(t,f,a)$4 GFLOPs for the compared Transformer-based network (Xiao et al., 11 Jan 2026).

Another multi-domain AI strategy uses side information induced from CSI itself rather than from external modalities. A CSI-to-PDP autoencoder constrains the latent feature to represent CSI while reconstructing the PDP, then extracts total power and power-weighted delay for each antenna pair, and a masked autoencoder fuses masked CSI with these multipath features through cross-attention. With $H(t,f,a)$5 known CSI, the reported gain over a baseline MAE is $H(t,f,a)$6 dB NMSE; with $H(t,f,a)$7 known CSI, the gain is $H(t,f,a)$8 dB; cross-attention outperforms concatenation by $H(t,f,a)$9 dB; and inference on RTX 4090 increases from $H(f,t,a)\in\mathbb C^{I\times J\times F\times T}$0 ms to $H(f,t,a)\in\mathbb C^{I\times J\times F\times T}$1 ms, i.e. around $H(f,t,a)\in\mathbb C^{I\times J\times F\times T}$2 ms overhead (Gao et al., 29 Jan 2026).

Diffusion-based generative extrapolation has also entered the field. XL-ChannelDiff formulates near-field antenna-, frequency-, and spatial-domain extrapolation as a conditional denoising-diffusion problem with a physics-aware CDDIM backbone, position-embedded patch tokenization, mask-guided multi-head attention, adversarial WGAN supervision, and RePaint-style refinement. Under antenna-domain random masks up to $H(f,t,a)\in\mathbb C^{I\times J\times F\times T}$3, the reported WGAN-enhanced CDDIM achieves NMSE $H(f,t,a)\in\mathbb C^{I\times J\times F\times T}$4 dB with $H(f,t,a)\in\mathbb C^{I\times J\times F\times T}$5 DDIM steps, versus $H(f,t,a)\in\mathbb C^{I\times J\times F\times T}$6 dB for conditional WGAN and $H(f,t,a)\in\mathbb C^{I\times J\times F\times T}$7 dB for a vanilla CDDPM with $H(f,t,a)\in\mathbb C^{I\times J\times F\times T}$8 steps; similar gains $H(f,t,a)\in\mathbb C^{I\times J\times F\times T}$9 dB are reported in frequency- and spatial-domain extrapolation, and inference latency is about $\mathcal F_{\rm time}$0 ms in 2D or $\mathcal F_{\rm time}$1 ms in 3D on RTX 4090 (Li et al., 15 Jun 2026).

5. Domain expansion beyond time–frequency–space

One prominent extension is the pattern or mode domain. In PR-MIMO, antennas are partitioned into $\mathcal F_{\rm time}$2 disjoint groups, each group transmits in a distinct radiation mode, and the pilot length drops from $\mathcal F_{\rm time}$3 to $\mathcal F_{\rm time}$4, i.e. by a factor of $\mathcal F_{\rm time}$5. For $\mathcal F_{\rm time}$6, the reported saving is $\mathcal F_{\rm time}$7. The complex-valued PR-Net uses three complex fully connected hidden layers with $\mathcal F_{\rm time}$8 neurons and CReLU, and at $\mathcal F_{\rm time}$9 dB SNR achieves NMSE $H_{\rm obs}$00 dB versus $H_{\rm obs}$01 dB for a real-valued DNN and $H_{\rm obs}$02 dB for the Duman Gram–Schmidt extrapolation benchmark (Liang et al., 2023).

Another extension is the position domain in cell-free massive MIMO. PCEnet first infers user position from a reconstructed main-BS channel, then uses the estimated position to guide pilot design and side-BS channel reconstruction. At SNR $H_{\rm obs}$03 dB, reported results are: E2E-AI4CSI with $H_{\rm obs}$04 gives $H_{\rm obs}$05 dB NMSE; full PCEnet with $H_{\rm obs}$06 gives $H_{\rm obs}$07 dB and a $H_{\rm obs}$08 overhead reduction; one-sided PCEnet with $H_{\rm obs}$09 gives $H_{\rm obs}$10 dB and a $H_{\rm obs}$11 overhead reduction; label-free PCEnet with $H_{\rm obs}$12 gives $H_{\rm obs}$13 dB; and direct black-box mapping $H_{\rm obs}$14 gives $H_{\rm obs}$15 dB, which the study attributes to spatial uncorrelation across base stations (Guo et al., 23 Jul 2025).

Multi-band statistical extrapolation appears in CF-CGN, which models channel fingerprints as multichannel images and learns bidirectional translation between bands with paired generative networks, variable-weight cycle consistency, and a refinement scheme based on channel-fingerprint resolution. The reported error reduction is $H_{\rm obs}$16–$H_{\rm obs}$17 dB relative to benchmarks across LOS and NLOS scenarios, and the resulting robust-precoding sum rate is within $H_{\rm obs}$18–$H_{\rm obs}$19 of the perfect-CSI upper bound (Xie et al., 2024).

Time–antenna multi-domain learning can also be embedded into a multi-task setting. MTCA uses a shared $H_{\rm obs}$20-layer GRU encoder with four heads for channel prediction, antenna-domain extrapolation, channel identification, and scenario classification. On a UAV-based multi-scenario dataset, the reported channel-prediction MSE is $H_{\rm obs}$21 versus $H_{\rm obs}$22 for Seq2Seq-attn-R, and the antenna-extrapolation MSE is $H_{\rm obs}$23 versus $H_{\rm obs}$24 for Seq2Seq-attn-R, corresponding to improvements of $H_{\rm obs}$25 and $H_{\rm obs}$26, while channel identification and scenario classification both reach $H_{\rm obs}$27 accuracy (Jiang et al., 26 Feb 2025). This suggests that semantic auxiliary tasks can regularize extrapolation features even when the primary targets remain CSI reconstruction.

ChannelKAN represents yet another domain pairing: frequency-domain and delay-domain CSI branches are expanded, enhanced by a multi-scale frequency information module, processed by parallel CNN and Chebyshev-KAN blocks, and fused adaptively. The reported outcome is that ChannelKAN outperforms RNN, LSTM, GRU, CNN, and Transformer baselines in NMSE, spectral efficiency, and bit error rate across velocities and SNRs on 3GPP-compliant QuaDRiGa datasets (Jiang et al., 11 May 2026).

6. Generalization, datasets, and open problems

A central technical issue is distribution shift. One frequency-domain line analyzes shift as the combination of multipath-structure shift and single-path-response shift, then proposes a physics-based progressive distribution alignment strategy consisting of path-oriented design and path alignment. In unseen environments, the reported PO-DLE+PA achieves $H_{\rm obs}$28 dB NMSE in env-1 and $H_{\rm obs}$29 dB in env-2 when trained in env-3, i.e. more than $H_{\rm obs}$30 dB improvement over state of the art; with $H_{\rm obs}$31 dB SNR noise, degradation is $H_{\rm obs}$32 dB and the gain over baselines remains $H_{\rm obs}$33 dB; and about $H_{\rm obs}$34 samples suffice to reach $H_{\rm obs}$35 dB generalization NMSE (Wang et al., 20 May 2025).

A second line targets regime shift between far-field and near-field channels. UNiFi-DLE disentangles each measured channel into angular and delay features by SVD, aligns the delay features with oversampled DFT masks, learns only the aligned delay extrapolation, and reuses the angular features unchanged. Reported results state that UNiFi-DLE outperforms baselines by $H_{\rm obs}$36–$H_{\rm obs}$37 dB NMSE on unseen far-field and unseen near-field datasets, improves as the user moves into deep near-field down to $H_{\rm obs}$38 dB at $H_{\rm obs}$39 m, remains stable down to $H_{\rm obs}$40 dB SNR, generalizes from $H_{\rm obs}$41 antennas to $H_{\rm obs}$42 without retraining, and exceeds PO-DLE+PA by $H_{\rm obs}$43–$H_{\rm obs}$44 dB in sim-to-real experiments on RENEW and ESPARGOS (Wang et al., 27 Jun 2026). A related zero-shot result appears in the CSI-to-PDP framework, which remains $H_{\rm obs}$45–$H_{\rm obs}$46 dB better than a baseline at $H_{\rm obs}$47 GHz after training at $H_{\rm obs}$48 GHz (Gao et al., 29 Jan 2026). XL-ChannelDiff additionally reports robust generalization across carrier frequencies $H_{\rm obs}$49 GHz, array sizes $H_{\rm obs}$50 and $H_{\rm obs}$51, and far-field versus near-field regimes (Li et al., 15 Jun 2026).

The data infrastructure remains mixed. Reviews list measured datasets such as Industrial Radio, DICHASUS, Wireless Intelligence, and RENEW, RT-based datasets such as WARI-D, DeepMIMO, and DataAI-6G, and simulators such as SEU-PML-6GPCS, BUPTCMG-6G, NYUSIM, Sionna RT, QuaDRiGa, WiThRay, NirvaWave, and KUCG (Gao et al., 1 Jan 2026). Yet recent 6G-focused work still identifies dataset collection as an open challenge, stating that public CSI measurements for ultra-massive MIMO, high-mobility, and multi-band settings are scarce (Gao et al., 1 Sep 2025).

Open questions are correspondingly stable across the literature. One 6G report highlights explainability, generalization, and dataset collection as core unresolved issues, and suggests interpretability techniques such as attention visualization and feature attribution, domain-adaptive training and meta-learning, and generative augmentation through GANs, while noting that stable training is required (Gao et al., 1 Sep 2025). Review work adds data scarcity and domain gaps, joint domain coupling, computational constraints, and non-stationarity and dynamics as the main technical obstacles (Gao et al., 1 Jan 2026). A plausible implication is that the field’s next stage will be decided less by isolated in-distribution NMSE gains than by whether extrapolators can remain accurate under distribution shift, operate within strict inference budgets, and connect reconstructed CSI to robust downstream communication performance.