Wireless Radiance Field (WRF)

• Wireless Radiance Field (WRF) is a continuous, high-dimensional model that represents electromagnetic signal propagation across space, direction, and frequency.
• It employs both implicit NeRF-style MLPs and explicit Gaussian splatting to synthesize detailed channel maps from sparse measurements efficiently.
• Applications include channel knowledge map construction, beamforming, and integrated sensing, enabling real-time, environment-aware 6G communications.

Wireless Radiance Field (WRF) refers to a class of continuous, high-dimensional field representations for modeling site-specific electromagnetic (EM) signal propagation as a function of space, direction, and frequency. WRF methods translate radiance-field rendering—originating in computer vision—into wireless channel modeling, yielding neural or semi-explicit signal field approximations from sparse measurement data. These representations underpin high-fidelity channel knowledge map (CKM) construction, environment-aware communications, and enable real-time applications in next-generation wireless systems.

1. Mathematical Formulation and Representation

A Wireless Radiance Field is a continuous function

$$L(\mathbf{x}, \mathbf{u}, f):\; \mathbb{R}^3 \times S^2 \times \mathcal{F} \to \mathbb{R} \text{ or } \mathbb{C},$$

where $\mathbf{x}$ denotes a 3D spatial location, $\mathbf{u}$ a unit vector (typically angle of arrival, AoA, at the receiver), and $f$ the frequency (or subcarrier). $L(\mathbf{x}, \mathbf{u}, f)$ returns the attenuated, phase-shifted “radiance” or spatial spectrum received just beyond $\mathbf{x}$ along direction $\mathbf{u}$ at frequency $f$ (Zhang et al., 29 Nov 2024, Ren et al., 7 Nov 2025).

In parametric WRF-GS and RF-3DGS models, $L$ is represented as a sum over $N$ anisotropic 3D Gaussians

$$L(\mathbf{x}, \mathbf{u}, f) = \sum_{i=1}^N G_i(\mathbf{x}) \cdot R_i(\mathbf{u},f),$$

where $G_i(\mathbf{x})$ is the spatial density (Gaussian with mean $\mu_i$, covariance $\Sigma_i$, density/opacity $\alpha_i$), and $R_i(\mathbf{u},f)$ encodes view-dependent radiance, often via spherical harmonics (SH) (Zhang et al., 29 Nov 2024, Wen et al., 6 Dec 2024, Li et al., 27 May 2025). Frequency-dependence is modeled via either indexed SH coefficients or frequency-aware MLPs.

In NeRF-based architectures, $L$ is approximated implicitly via MLPs driven by high-dimensional positional encodings of $(\mathbf{x},\mathbf{u},f)$, regressing attenuation (“density”), in-phase, quadrature (I/Q), or amplitude/phase (Lu et al., 5 Mar 2024, Ren et al., 7 Nov 2025).

Generalizable transformer approaches such as GWRF further input transmitter location, neighbor spectra, and scene context, producing per-voxel latent vectors for ray integration (Yang et al., 8 Feb 2025).

2. Core Modeling and Rendering Algorithms

Two main computational paradigms predominate:

Volumetric and Implicit (NeRF-style) Rendering

NeRF-based WRFs cast parametric rays from a receiver, querying the MLP at stratified/interpolated positions, then integrating along the ray: $$\hat{C}(\mathbf{o},\mathbf{d}) = \int_{0}^{D}\!T(t)\,\sigma\bigl(\mathbf{r}(t)\bigr)\,c(\mathbf{r}(t),\mathbf{d})\,dt,$$ with transmittance $T(t) = \exp(-\int_0^t \sigma(\mathbf{r}(s))ds)$, rendering the received signal as a sum of “attenuated emissions” along the ray. Outputs include both amplitude and phase, matching the complex-valued channel response (Lu et al., 5 Mar 2024, Ren et al., 7 Nov 2025). Training is supervised with MSE or perceptual losses against measured spatial spectra or channel impulse responses.

Explicit Gaussian Splatting

Gaussian-splatting WRFs use explicit Gaussians (each with position, orientation, and view-dependent radiance), which are projected onto a virtual receiver plane or angular grid. The “splatting” step accumulates complexes over sorted depth (by transparency or attenuation) and directly forms the spatial spectrum: $$S^{(i)} = \left(\prod_{j=1}^{i-1} \delta_j\right) S_i,$$

$R_k = \sum_{i=1}^{N_k} S^{(i)},$

where the $i$th Gaussian’s contribution at pixel $k$ is modulated by the accumulated attenuation and radiance (Wen et al., 6 Dec 2024). Differentiable rasterization and tile-based rendering enable efficient GPU parallelization, scaling to hundreds of FPS and beyond (Zhang et al., 29 Nov 2024, Liu et al., 15 Jun 2025).

Frequency-embedded or deformable variants allow the properties of each Gaussian to adapt as a function of frequency, mobility, or other context (Li et al., 27 May 2025, Liu et al., 15 Jun 2025).

3. Training Objectives, Data, and Pipeline

Data acquisition involves collecting spatial spectra or element-level channel impulse responses at a sparse set of transmitter-receiver placements. Antenna arrays may be used to measure angular spectra (e.g., 360x90 beams). No dense pilot sweeps are needed—sparse world-locked samples suffice (Jiang et al., 4 Sep 2024, Zhang et al., 29 Nov 2024, Ren et al., 7 Nov 2025).

Training objectives typically combine:

• L1 or MSE loss between predicted and ground-truth spectra or channels,
• Perceptual similarity indices (SSIM, LPIPS) to capture spatial structure,
• Regularization on model parameters (e.g., Gaussian scale, opacity, SH coefficient norms) to enforce compactness and avoid overfitting.

Some frameworks employ two-stage pipelines: an initial visual-based geometry fit (e.g., from photometric images), followed by RF-specific direction/radiance tuning (Song et al., 6 May 2025). Others use hierarchical coarse-to-fine MLPs or transformer encoders to accelerate convergence and improve generalization (Yang et al., 8 Feb 2025).

4. Empirical Results and Quantitative Benchmarks

The table below summarizes key performance metrics of representative WRF frameworks (verbatim from the source literature):

Method	Training Time	Rendering Latency	Median SSIM	Median LPIPS	AoA Error/Notes
RF-3DGS	~3 min	~2 ms	N/A	0.065	AoA error 5.94°
NeRF²	~3 hours	~1 s	N/A	0.420
WRF-GS	N/A	~5 ms	0.82	N/A
SwiftWRF	N/A	100,000 FPS	0.90 (avg.)	N/A	AoA error 1.2–1.8°
GWRF	N/A	~1.8 s	0.766	0.136	AoA error reduced 61.6%
Wideband 3DGS	~200k steps	~10–100 ms	0.72	N/A	SSIM drop 2.8% (zero-shot)

RF-3DGS achieves an 84.6% reduction in LPIPS error relative to NeRF² (0.065 vs. 0.420), with 3-minute training and millisecond-level rendering (Zhang et al., 29 Nov 2024). SwiftWRF achieves 100,000 FPS spectrum synthesis, with 0.904 average SSIM, outperforming both NeRF² and WRF-GS in speed and accuracy (Liu et al., 15 Jun 2025). GWRF, with a transformer geometry encoder, yields state-of-the-art cross-scene generalization (PSNR up to 21.94 dB, SSIM 0.766) (Yang et al., 8 Feb 2025).

Frequency-embedded WRF models (e.g., Wideband 3DGS) demonstrate robust cross-frequency prediction, achieving average SSIM 0.72 and retaining 97.2% performance even at unseen frequencies (Li et al., 27 May 2025). For THz fields, RF-3DGS+ yields PSNR 19.68 dB, SSIM 0.635 with just 3.4 ms inference (Song et al., 6 May 2025).

5. Applications: CKM Construction, Channel Prediction, ISAC

Wireless Radiance Fields underpin channel knowledge map (CKM) reconstruction by modeling $L(\mathbf{x}, \mathbf{u}, f)$ over large spatial regions from minimal measurements. This enables site-specific, a priori channel prediction for environment-aware wireless communication, localization, and integrated sensing and communication (ISAC) (Ren et al., 7 Nov 2025, Zhang et al., 29 Nov 2024).

• Channel Prediction and Beamforming: WRFs synthesize fine-grained spatial channel state information (gain, delay, AoA, AoD) for arbitrary Rx positions and beam angles, supporting real-time beam alignment and predictive link adaptation (median AoA error sub-6°, accurate delay/AoD channel predictions) (Zhang et al., 29 Nov 2024, Wen et al., 6 Dec 2024, Jiang et al., 4 Sep 2024).
• Environment-aware ISAC: Explicit spatial representations enable multipath reconstruction, digital twin-assisted sensing, and robust performance in dynamic or obstructed environments (Jiang et al., 4 Sep 2024).
• Sample and Training Efficiency: Gaussian splatting models (WRF-GS, RF-3DGS, SwiftWRF) achieve comparable or better accuracy versus NeRF-based models while requiring 80–90% fewer measurements and providing orders-of-magnitude faster inference (Ren et al., 7 Nov 2025, Liu et al., 15 Jun 2025).
• Wideband and THz Regimes: Frequency-embedded WRF models generalize across GHz- to THz-scale bands, enabling scalable, low-cost spatial channel reconstruction for 6G and beyond networks (Li et al., 27 May 2025, Song et al., 6 May 2025).

6. Advantages, Limitations, and Future Directions

WRF models deliver:

• Real-time, site-specific spatial spectrum synthesis (as fast as 2 ms per spectrum, or 100k FPS for SwiftWRF) (Zhang et al., 29 Nov 2024, Liu et al., 15 Jun 2025).
• Explicit geometric interpretability (via splatted Gaussians) for downstream tasks—digital twins, cell-free MIMO, beam prediction (Zhang et al., 29 Nov 2024).
• Robustness to measurement sparsity, generalization across scenes, and multi-frequency support (Yang et al., 8 Feb 2025, Li et al., 27 May 2025).

Limitations and research frontiers include:

• High model size and inference cost for implicit-MLP and transformer methods (Yang et al., 8 Feb 2025).
• The need for periodic retraining or incremental updating in dynamic environments (Ren et al., 7 Nov 2025).
• Current frameworks address static scenes; modeling dynamic objects or temporal fading remains an open problem (Yang et al., 8 Feb 2025, Liu et al., 15 Jun 2025).
• Cross-domain generalization (across frequency, materials, or large geographic regions) is still a challenge; embedding physics-based priors or multi-modal fusion is a potential avenue (Ren et al., 7 Nov 2025).

7. Contextualization Within Channel Modeling

WRFs represent a transition from empirical interpolation and ray tracing toward hybrid physics/data-driven models. Unlike classical models, WRFs can infer fine angular-delay structure (multipath components) from sparse crowd-sourced samples, without requiring exhaustive pilot sweeps or detailed CAD models (Jiang et al., 4 Sep 2024, Wen et al., 6 Dec 2024). In comparative studies, WRF models reduce RMSE by up to 54% in non-line-of-sight conditions (BiWGS vs. MLP), achieve up to 0.90 SSIM on spatial spectra, and decrease AoA estimation error by up to 60% when augmenting ground-truth with synthesized spectra (Ren et al., 7 Nov 2025, Yang et al., 8 Feb 2025, Liu et al., 15 Jun 2025).

Within the CKM literature, WRF-based frameworks—using either implicit MLP or explicit Gaussian splatting representations—have established new standards for accuracy, speed, and sample efficiency, opening the door to real-time, environment-aware 6G communications and ISAC systems.