GSRF: Complex-Valued 3D Gaussian Splatting for Efficient Radio-Frequency Data Synthesis
Authors: Kang Yang
Gaofeng Dong
Sijie Ji
Wan Du
Mani Srivastava
Venue: NeurIPS 2025 spotlight ()
Keywords: 3D Gaussian Splatting
Complex-Valued Modeling
RF Data Synthesis
Primary area: applications
TL;DR: We introduce complex-valued 3D Gaussian Splatting for real-time, high-fidelity radio-frequency (RF) data synthesis

Abstract: Synthesizing radio-frequency (RF) data given the transmitter and receiver positions, e.g., received signal strength indicator (RSSI), is critical for wireless networking and sensing applications, such as indoor localization. However, it remains challenging due to complex propagation interactions, including reflection, diffraction, and scattering. State-of-the-art neural radiance field (NeRF)-based methods achieve high-fidelity RF data synthesis but are limited by long training times and high inference latency. We introduce GSRF, a framework that extends 3D Gaussian Splatting (3DGS) from the optical domain to the RF domain, enabling efficient RF data synthesis. GSRF realizes this adaptation through three key innovations: First, it introduces complex-valued 3D Gaussians with a hybrid Fourier–Legendre basis to model directional and phase-dependent radiance. Second, it employs orthographic splatting for efficient ray–Gaussian intersection identification. Third, it incorporates a complex-valued ray tracing algorithm, executed on RF-customized CUDA kernels and grounded in wavefront propagation principles, to synthesize RF data in real time. Evaluated across various RF technologies, GSRF preserves high-fidelity RF data synthesis while achieving significant improvements in training efficiency, shorter training time, and reduced inference latency.

GSRF: Complex-Valued Valued 3D Gaussian Splatting for Efficient Radio-Frequency Data Synthesis
Kang Yang
Gaofeng Dong'
Sijie Ji 1.2
Wan Du Du 3est
Mani Srivastava
Unniersitty of California, Los Angeles CCllifornia Institute of Technology University of California, Merced
[kyang73, gfdong wcla.clal.oo edu wdu3@ucmerced, edu
sijieji@caltech. edu mbs@ucla.edi edu
Abstract
Synthesizing fudiu-freyuency (RF) data given the transmitter and receiver posi- tions, e.g., received signal strength indicator (RSSI), is critical for wireless net- working and sensing applications, such as indoor localization. However, it remains challenging due complex propagation interactions, including reflection, diffracc tion, and scattering. State-of-the-art neural radiance field (NeRF)-based methods achieve high-fidelity RF data synthesis but are limited by long training times and high inference latency. We introduce GSRF, a framework that extends 3D Gaussian Splatting (3DGS) from the optical domain the RF domain, enabling efficient RF data synthesis. GSRF realizes this adaptation through three key innovations: First, it introduces complex-valued 3D Gaussians with hybrid Fourier-Legendre basis model directional and deenee-deendentt radiance. Second, it employs orthographic splatting for efficient Gay-saan intersection identification. Third, incorpo- rates a complex-valee valued ray tracing algorithm, executed on RF-customized CUDA kernels and grounded in wavefront propagation principles, to synthesize RF data in real time. Evaluated across various RF technologies, GSRF preserves high- fidelity RF data synthesis while achieving significant improvements in training efficiency, shorter training time, and reduced inference latency.
1 Introduction
Wireless networks, egg. WiFi and Fifth Generation (5G) cellular networks, are increasingly tasked with supporting both communication and sensing applications through deep learning (DL) models, including indoor localization [1, 2, 3]. However, training these DL. models requires large-scale cadio-frequency (RF) datasets, e.g.. received signal strength indicator (RSSI) measurements across different transmitter and receiver positions within a 3D space. which are typically collected through site surveys. These site surveys involve labor-intensive and time-consuming RF signal measurements across numerous transmitter- locations [4, 5, 6].
Inspired by the success of generative models in computer vision [7, 8, 9, 10], a natural alternative approach is synthesize RF data through propagation modeling, which computes the received RF signal at receiver given a transmitter emitting signals from specific position [11]. However, generating high-fidelity RF data is challenging due complex propagation interactions between RF signals and surrounding objects, including reflection, diffraction, and scattering.
This work partially done when Kang Yang was PhD student in Dr. Wan Du's group at UC Mereed. I Corresponding Author: wdu3 @ weummeceddcd.
8 Mani Srivastava holds concurrent appointments Professor of ECE and CS (joint) the University of California. Angeles, and as Amazon Scholar Amazon. This paper describes work performed at UCLA and is not associated with Amazon.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Neural Radiance Field (NeRF) [7]-based methods [12, 13] address these challenges by extend- ing NeRF the RF domain, achieving state-of-the-ar fidelity in RF data synthesis. These NeRF- based methods adopt continuous RF scene representations to effectively model complex RF interac- tions. However, their stochastic sampling process and Multi-layer Perceptron (MIP) optimization are computationally intensive and slow, limiting real-time applicability. Efficient training and inference in the RF domain are crucial for applications such as real-time localization and tracking [14, 6].
This paper proposes GSRF, an efficient RF data synthesis framework that extends 3D Gaussian Splatting (3DGS) [8, 15], developed for real-time novel view synthesis, to the RF domain. However, this adaptation introduces challenges due to inherent differences between visible light and RF signals:
(i) Directional and Phase Modeling. In 3DGS [8, 16]. the color attribute of Gaussian distribution is parameterized by spherical harmonics (SH) coefficients [17, 8] to capture directional variations caused by optical propagation effects such as reflections and shading. In contrast, RF signals with centimeter-scale wavelengths exhibit complex phenomena such as diffraction [18] and phase- dependent interference (constructive and destructive), which SH coefficients struggle capture [19].
(ii) Data Capture Mechanism. In visible light, images are captured by camera sensors (e.g. CMOS or CCD) on a 2D image plane, allowing splatting through classical transformation matrices that project 3D Gaussians onto the plane to identify muy-Gaussian intersections. In contrast, RF signals collected by antenna arrays difference makes splatting algorithms designed for visible light unsuitable for the RF domain.
spherical region centered at the RF antenna. This fundamental
(iii) Rendering Algorithm. In 3DGS, point-based rendering algorithm aggregates amplitude-bass based attributes, e.g., color, to compute pixel values along each ray. In contrast, RF signal synthesis needs to consider both amplitude and phase to model interference patterns. This necessitates a complex-valus rendering algorithm, along with CUDA kernels that jointly process amplitude and phase information.
By tackling the three challenges above, we make the following key contributions:
Fourier-Legendre Radiance Fields. A scene is represented using 3D Gaussian distributions, each characterized by four attributes: a mean and covariance matrix, along with two RF-specific attributes, which are complex- -alued RF radiance and transmittance. The directional radiance is modeled using a Fourier-l Legendre Expansion (FLF) [20]. FLE leverages Fourier basis functions for the azimuthal angle and Legendre polynomials for the elevation angle B, with complex coefficients Cnt E C encoding both amplitude and phase. Additionally, the complex-valued transmittance models signal amplitude attenuation and phase shifts as the RF signal propagates through Gaussian.
Orthographic Splatting. To determine ray-Gaussian intersections, GSRF introduces an orthographic splatting method for the RF domain. GSRF operates on the Ray Emitting Spherical Surface (RESS), a spherical region where RF signals are captured. Each 3D Gaussian is then splatted onto this region via orthographic projection, enabling identification of intersecting Gaussians for each ray.
Complex- Valued Ray Tracing. GSRF incorporates complex-valuee ray tracing algorithm for RF signals, executed on RF-customized CUDA kernels. Building on the Huygens- Fresnel principle [21], which states that each point on wavefront acts as a source of secondary wavelets, GSRF models each Gaussian as an RF source. GSRF emits rays from the RESS, identifies intersecting Gaussians through the adapted splatting method, and employs a complex-valuee ray tracing algorithm to jointly process amplitude and phase attributes along each ray, computing the received RF signal data.
GSRF is trained with an RF-customized loss function derived from both time and frequency domains using 2D Fourier transforms to capture the intricate propagation characteristics of RF signals.
We evaluate GSRF various RF technologies, including radio- frequency identification (RFID), Blue- tooth Low Energy (BLF), and 5G networks, to synthesize different types of RF data, including RSSI, spatial spectra, and complex-valued channel state information (CST). Results show that GSRF achieves significantly higher efficiency than existing methods, with improvements in training data efficiency, training time, and inference latency. We release our code this GitHub repository.
2 Preliminaries
RF Signal Propagation Characteristics. Wireless systems, such as WiFi, rely RF signals propagating between transmitters and receivers [22, 23). A transmitted signal can be represented as:
s(t) (22

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where A is the amplitude. fc is the carrier frequency (e.g., 2.4 GHz), and 0 is the initial phase. As the signal propagates through the scene, it encounters obstacles that cause reflections, diffraction, and scattering, resulting in multiple propagation paths. The received signal is the sum of these paths:
N
de
r(t)
A (t (0-m)) T(), o, = fcT4 + 0,,
(2)
where N is the number of paths, is the RF signal speed, A, is the attenuated amplitude. d; is the phase shift, T; is the time delay of path length d,, and 0, is the phase change from reflections.
The phase greatly affects the received signal, as illustrated in the following example. For two paths with lengths d1 3 and d2 3.0625 m at a carrier frequency of fe = 2.4 GHz, the corresponding delays are T1 = 10 ns and T2 = 10.208 ns. The phase shifts are Φ1 = 0 and Φ = т, resulting in a phase difference of AΦ = π, which causes destructive interference, i.e., the two signals cancel each other out, leading to reduction or complete loss of signal strength. Conversely, when AΦ x O, constructive interference occurs, amplifying the signal [22]. Therefore, synthesizing RF data requires modeling these amplitude and phase interactions across all paths.
3D Gaussian Splatting (3DGS). is a real-time rendering technique for novel view synthesis in 3D scenes [8]. It represents a 3D scene as a collection of 3D Gaussian ellipsoids {Si...., ,k],, where each Gaussian primitive Ç& is defined by a 3D Gaussian distribution:
G& (z; pk. E+) exp (-2 1 2
(x- (x-))) = Σn (x (x- ps))
(3)
where Uı E R$ is the center position and Σg E R3x3 is the covariance matrix. It is decomposed as: Σg S R where R3 and Sy are learnable rotation and scaling matrices that ensure positive semi femi--efiniteness [8]. Each Gaussian also includes an opacity term Pk E [0. 1] and SH coefficients shy E RRd, making each Gaussian primitive represented as: Ca (Mu, Rk, Sk, Pks sha).
To render an image, 3DGS projects these 3D Gaussians onto a 2D image plane, forming 2D Gaussians:
G (r: 2D EDD) with 2P = T2D (us),, Σ2D JWE,
where J is the Jacobian of the projective transformation, and W is the world-to-camera transformation matrix [8]. The pixel color Ó' (r) at location E IR² is computed via a-blending:
( Σ w (r) e (shggr),
(5)
KES,
where 8, C {1...., K} is the subset of indices of Gaussians that contribute to pixel The term w/th (r) represents the contribution of each Gaussian, computed as:
k-1 II (1 -1- P1 (r: ,zn)). ==1
w (r) = P (r: Pk
2
(6)
where the Gaussians are ordered by increasing depth (i.e., from front back) to ensure correct rendering. Finally, e (she, r) is the color decoded from the SH coefficients sha-
3 Related Work
Conventional RF data synthesis methods include simulations [24, 25, 26), empirical models [22, 27, 28), and physics-unaware DL models [29, 30, 31], but all suffer from low modeling fidelity due to inherent limitations. Simulations require accurate scene Computer-Are Aided Design (CAD) models, which are often unavailable. Empirical models oversimplify propagation with limited parameters, predicting only coarse signal power. Paysics-umaware DL models map inputs to labels but fail to capture the underlying physics of RF propagation. NeRF-based methods [12, 13, 32] introduce voxel- based scene representations to capture scene impact on RF signal propagation and employ ray tracing algorithms to achieve state-of-the- from low efficiency, requiring long training times and exhibiting high inference latency. This work proposes 3DGS-based method to achieve high training and inference efficiency.
fidelity in RF data synthesis. However, they suffer
Two recent works, RF-3DGS [33] and WRF-GS [34], propose 3DGS- -DGS-iispired techniques for RF data synthesis, yet both face limitations. RF-3DGS [33] employs two-stage training process to
3

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Geometry: Mean ji: Covariance
RF-related: Radiance W:: Transmittance
Splatting
Inttlalization
Equal-sized cube
Rendering
(1))
Primitive update
Ray
Transmitter
Groundtruth Prediction
Receiver
Figure I: Overview of GSRF architecture. The RF scene is represented by Gaussian primitives with mean μ., covariance Σ, and complex-vale valued radiance & and transmittance p. whose attributes are updated via gradient-based optimization with adaptive density control. For rendering, rays emitted from the receiver, Gaussians are splatted onto a 2D receiving RF plane, and the received data obtained by aggregating complex-valued contributions along each ray.
learn scene representations using Gaussian primitives defined by mean, covariance, opacity, and path loss. First, optical 3DGS optimizes mean, covariance, and opacity from visual images, then these parameters are fixed to train path loss with RF data. However, merging visible light and RF signals challenging due to their distinct properties. Moreover, visual data is often unavailable in RF domains.
WRF-GS [34] assigns each Gaussian four attributes: mean, covariance, radiance, and attenuation. adopts a NeRF-inspired approach to learn radiance and attenuation by optimizing a large MLP with each Gaussian's position as input. This dependence
computationally intensive MLP results in inefficiency and introduces NeRF-like bottlenecks: (i) dense querying of the MLPs for attribute prediction during training and inference, (ii) expensive backpropagation through deep networks for every Gaussian update, which scales poorly with scene complexity, and (iii) high inference latency due to per-query MLP evaluations (e.g., for novel transmitter positions). In contrast, GSRF eliminates MLP regressors entirely by directly optimizing per- ner-Gaussia attributes as learnable parameters. Combined with Nourier-Legente radiance fields, orthographic splatting, and complex-valued ray tracing, this design achieves faster training and inference compared to WRF-GS.
WRF-GS+ is extension of WRF-GS [34]. It introduces deformable Gaussians that decouple static components (e.g., path loss) and dynamic components (e.g., multipath) via learned offsets, thereby improving synthesis quality and mitigating the inefficiencies of WRF-GS's MLP-based attributes. While effective, this approach remains distinct from GSRF, which offers unified, complex- valued, MLP-free pipeline for RF propagation modeling; nevertheless, deformable mechanisms could be explored in future extensions of our framework.
4 Methodology
4.1 Problem Formulation
Given transmitter fixed position emitting RF signals (e.g., WiFi router) and receiver (e.g.. a smartphone) distributed throughout a scene, the objective is to synthesize the received RF data. Formally, for a transmitter located at t = (Ixx Mox 2xx and set of receiver positions {rr) where r; ( 3rx, 2x..), the goal is estimate model with parameters that synthesizes the received RF data S, at each receiver r;:
8" argmax ((S)) t, (rilien
where S; E C represents the received complex-valued RF data at receiver r., encapsulating both amplitude and phase. For specific RF technologies or applications, S, may represent a scalar signal power S, E E or spatial spectrum S, E Na over azimuth and clevation B angles. For example, with a one-degree angular resolution, we have N = 360 and Na = 180.
Model Overview. Figure 1 illustrates the GSRF, First, the scene is represented using 3D Gaussian distributions, each characterized by a mean A4 R3, covariance matrix Σ, E IR3x3 and two RF- specific complex-valued attributes: radiance 3/ E C and transmittance Pk: E C. To initialize the mean and covariance matrix, the scene partitioned into equal-sized cubes, deriving initial scene point clouds without Pructtur-frro-MMonn domain. Next, each 3D Gaussian is projected onto the receiver's receiving region, using orthographic projection efficiently identify ray Gaussian intersections. For each ray, intersecting Gaussians
(SfM) algorithms [35, 36), which are inapplicable to the RF

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sorted by depth, and complex-valuee ray tracing algorithm is applied to compute the received signal. Model optimization is performed by minimizing the loss function. Explicit gradients are computed to update the primitives via stochastic gradient descent, adjusting parameters (1u,, Ek; thk, Pk:) and refining primitive density through gradient-driven cloning, splitting, or removal.
4.2 Lourier-Legendre Radiance Fields
Each Gaussian primitive in GSRF is represented as tuple:
Sk = (pk: RRs Sky Vk Pk) with = = Rg S
(8)
The pair (Mx, Rg, S&)) defines 3D Gaussian distribution resembling ellipsoid, representing
probability distribution in 3D space. Its probability density function (PDF) is given by Equation (3).
The transmittance Pl: E C models the effect of an RF signal passing through the k-th Gaussian, resulting in an amplitude reduction |pk and phase shift Lpk. According to Maxwell's equations [37], transmittance depends on the material properties at the Gaussian's 's location μą. Therefore, Pk primarily captures the physical interaction of the RF signal with the medium.
The radiance 4ha E C represents the complex-valueee RF signal emitted by the k-th Gaussian. To model its directional dependency, the is defined using Fourier-Legendre Expansion (FLE) [20], which leverages Fourier basis functions for the azimuthal angle and Legendre polynomials for the elevation angle 3. This approach is physically grounded in the Huygens- Fresnel principle [21], which posits that each point on a wavefront, such as the k-th Gaussian at position μ(g, acts as source of secondary spherical wavelets. The emitted RF signal is modeled as a solution to the wave equation in spherical coordinates, where spherical harmonics and associated Legendre polynomials Plt. (cos B) form complete basis for representing directional wave fields on the unit sphere. Specifically, for direction (a, B), the radiance is expressed as:
cermmoncss-coppriiiii comprising Fourier functions e
dh (a, 3)
Σ Σ
e pmm (cos B), Smi
(K)
(9)
1=0 m==
where (1)
mmd E C are complex coefficients encoding the amplitude and phase of the radiance for the k-th Gaussian. This representation effectively captures phase-dependent interference crucial, as the separation of and B aligns with their geometric roles in spherical coordinates, while the complex coefficients model the interference effects stemming from the wave nature of RF signals.
4.3 Orthographic Splatting
For receiver positioned at r = (Iix. Vix: 2xx), rays are emitted to sample the RF signal across various directions around the receiver Each ray is parameterized as:
(d) = r dv, d 2 Tix:
(10)
where d is the distance along the ray from the receiver, and v is the unit direction vector. Therefore, rays are emitted from the Ray Emitting Spherical Surface (RESS), which is a sphere centered at r with radius Fix. and extend outward. For one-degree angular resolution, Nu. 360 and Na 180, resulting in a total of 360 x 180 rays being emitted, covering all directions around the receiver.
2D Receiving RF Plane. To enable splatting in the RF domain, where an image plane is absent, map the RESS onto a 2D RF plane. Consider a point p = (æ, y/, 2) E IR' on the RESS, satisfying ||p r|l Fix- We transform the Cartesian coordinates of into spherical coordinates (C, a, (). where 1-m/2, x x/2] is the elevation angle:
is the radial distance, a E [0, 2n) is the azimuthal angle, and
< = V x
+ y2 + z Tix
arctan 2(,,
(11)
8 arccos
Tix
We then project and 3 onto 2D grid with one one-degree resolution, defined as:
a 180
[ 180
90
(12)
π
π
where [.] denotes the floor function. The resulting coordinates (u, v) define the 2D RF plane.
5

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Splatting Process. Each 3D Gaussian, with mean μ* E R and covariance Σ3 E R is projected onto the 2D RF plane to identify ray-Gaussian intersections. The unit direction vector from the receiver position E R to the k-th Gaussian center 4A is given by:
1 r
VA
(13)
The projected center ni2 = (ut (x)) is computed from Equations (11) and (12), with the vector Va
as input. The 3D covariance Σg is projected onto the 2D plane as E JE where J is the Jacobian matrix, and the 2D spread is approximated by radius T. 33 Ammmx. with Amax the largest
eigenvalue of D. Rays at points (u, v) intersect the Gaussian if: / (u ux)² 2 + (v vx) Tk.
4.4 Complex-Valllle Valued Ray Tracing Algorithm
The received signal S C for a ray computed by aggregating the contributions from all inter- secting Gaussians, considering their geometric influence, radiance, and transmittance. The intersect- ing Gaussians are sorted in ascending order of their distance from the receiver along the ray path to ensure correct accumulation of transmittance effects. The received signal is computed as:
Kin
1-1
S =
Σ k=1
Su (Zrep.kil Mh: Mx) (1r2el ed e
II (1eml e m==1
(14)
Gaussian weight
Complex radiance
Curmlative transmittance
where Kint is the number of Gaussians intersecting the ray, and SR (2; pk; Σx) is the probability density function of the k-th Gaussian, evaluated at the representative intersection point Trep.k E R which is the midpoint of the intersection points between the ray trajectory and the ellipsoid defined by the Gaussian's 's mean μLk E Is and covariance Σn E R3xx3 The term 9% (Irep.k: ki μkx, Σx) weights the radiance based on the Gaussian's density at the intersection point, while the product term accumulates both amplitude attenuation and phase shifts from all preceding Gaussians, capturing both amplitude reduction and phase shifts during propagation. The detail of proor is provided in Appendix A.
Loss Function. The loss function is designed based on the receiver antenna type.
ANTENNA ARRAY. For a receiver equipped with antenna arrays, the signal power across all directions is represented as ground-truth spatial spectrum matrix S E R X.N, spanning N., azimuth and Na elevation angles. The predicted spatial spectrum is denoted $ E X.N,, The loss function C combines the & loss, the Structural Similarity Index Measure (SSIM) loss, and a Fourier-based loss:
L
A 22) Ln + X CSSIM + X2C
(15)
where &1 N. Ev E/e ((u,u) v) - S(u, w) measures the average absolute difference be- tween the predicted and soound-truth spectra. The term &SSIM captures spatial RF pattern similarity across directions. The term L.oourier quantifies the difference in the frequency domain:
Fourier
Σ Suifo
F(5) (S). - (f..
(16)
NauNel
where If (S)) (Fw, fo) and I (S) (Fas are the 2D Fourier transforms of the predicted and ground- truth spectra, respectively, with fa and fo representing the frequency indices in the azimuth and elevation dimensions. This term promotes consistency in the frequency domain, which is important for learning RF propagation behavior. The squared magnitude penalizes discrepancies in both amplitude and phase, enhancing the fidelity of synthesized signals.
SINGLE ANTENNA, For a receiver equipped with a single antenna, the ground-truth received signal S represents either real-valued power measurement or complex-valued signal encom- passing both amplitude and phase information. The synthesized signal Š is computed as Š EN=, INa Su,, where Su,v denotes the synthesized signal contribution from the ray at azimuth index u E (1,.... N and elevation index v E (1,..., , Na)- The loss function L is defined
as C $ s||, if S is real-valued (RF signal power), and L | -s(" if S complex- valued, penalizing both amplitude and phase errors.

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Gradient-Baser Gaussian Primitive Optimization. GSRF initializes the number of Gaussians and their primitives based on the scene's point clouds, which are obtained by partitioning the scene into equal-sized cubes. After calculating the loss function, the optimization of Gaussian primitives is performed through gradient- madieett-based strategies, as detailed in Appendix B.
Fast Differentiable RF Signal Renderer for Gaussians. In GSRF, we develop two CUDA ker- nels to enable efficient forward and backward computations for differentiable RF signal synthesis using Gaussian primitives. Implementation details of the CUDA kernels are provided in Appendix C. To reduce computational overhead, gradients are explicitly calculated as described in Appendix D.
5 Experiments
Our method is implemented in PyTorch with CUDA. Further implementation details and hyperparam- eter settings are provided in Appendix E. and additional experiments presented in Appendix F.
We evaluate GSRF across three RF technologies for various RF data synthesis tasks: (i) Radio- Frequency Identification (RFID) for spatial spectrum synthesis, (üi) Bluetooth Low Energy (BLE) for real-valued received signal strength indicator (RSSI) synthesis, (iii) 5G Cellular Network for complex-valued channel state information (CSI) [38] synthesis.
5.1 RFID Spatial Spectrum Synthesis
TASK. Given a transmitter sending RF signals at location (Ixx Mlx ax), the goal is to synthe- size the spatial spectrum received by the ceiver (equipped with an antenna array). The spatial spectrum, represented as a 360 x 90 ma- trix, captures the signal power from all directions around the receiver, covering azimuth and eleva- tion angles at a one-degree resolution. The eleva- tion angle is limited 90% as only the front hemi- sphere of the antenna array is considered [12].
Location
Location 2
Location 3
Location
Location
NeRF! GERF Groundtrath
Figure 2: Visualization comparison of synthesized from 2 [12], collected in real-world indoor spatial spectrum different positions.
DATASET,
The publicly released RFID dataset
environments, is employed. It contains 6, 6,123 transmitter (RFID tag) locations and their corresponding spatial spectra, received by receiver equipped with 4 x 4 antenna array operating at the 915 MHz frequency band. The dataset is randomly split by default into 70% for training and 30% for testing.
METRICS. We employ the two metrics:
(i) Mean Squared Error (MSE)J: This metric calculates the average of the squared differences in signal power between the synthesized spectrum and the ground truth for each entry. (ii) Peak Signal-to- to-oisee Ratio (PSNR, in dB)t: Treating the spatial spectrum image, PSNR measures structural similarity, with higher values indicating better quality.
BASELINES, We compare GSRF with NeRF2 [12] and WRF-GS [34]. Other simulation- -aaee or physics- unaware DL-based methods, such MATLAB simulation [26). DCGAN [39], and VAE [30]. perform worse on the same RFID dataset [12] compared to NeRF²
Overall Performance. To evaluate GSRF's per- formance in scenarios with insufficient data, we randomly select 220 instances from the training dataset instead of using the full training data. This creates a sparse dataset with a measurement den- sity of 0.8 measurements/ft". Figure 2 presents the real-collected spatial spectra for four ran- domly selected transmitter positions (first row), alongside those generated by baseline models. Visually, the spectra synthesized by GSRF more Figure 3: Metric comparison for a sparse measure- closely match the ground truth compared to those ment density of 0.8 mcasurements/
1.0
1.0
0.8
WRF G5
WRF
y
0.6
GSAF
GSAF
0.4
0.4
0.0
0.0
0.01
0.02
0.03
0.0
0
300
25
20
25
MSE
PSNR
(a) MSE score]
(b) PSNR score (dB)t
by NeRF² Figure 3 then shows the Cumulative Distribution Function (CDF) of the two metric scores on the testing data. GSRF achieves median improvements of 21.2% in PSNR and 56.4% in MSE over - while outperforming WRF-GS by 5.7% and 19.3%, respectively. This supe- riority stems from two advantages: First, our complex-value Gaussian representation explicitly models phase interactions throughout the architecture, which is critical for RF signal propagation,
7

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whereas WRF-GS relies on velled Gaussian primitives. Second, the Courier-Legendee radiance basis in GSRF provides directional resolution beyond the spherical harmonics used in WRF-GS, enabling finer capture of diffraction and scattering effects. These innovations allow GSRF to learn more physically accurate scene representations even from sparse measurements.
Training & Inference Efficiency. Training time measured by running each method on a com- puter equipped with GeForce RTX 3080Ti GPU. Inference time for each model is also recorded.
1.0
1.0
8
0.6
Figure 4 illustrates that our method achieves convergence in 0.27 hours, which is 18.56x faster than NeRF² (5.01 hours) and 5.96x faster than WRF-GS (1.61 hours). For inference latency, Figure 4: Training times as shown in Figure 5, our method synthesizes for spectrum synthesis.
0.2
0.2
WI--G5
63R
63R7
0.0
0.0
100
200
300
400
Training time (hours)
Inforence time (eea)
Figure 5: Test times for spectrum synthesis.
spatial spectra in 4.18 ms, yielding an 84.39x
speedup over NeRF (352.73 ms) and 1.81 speedup over WRF-GS (7.58 ms). This acceleration is due to: First, our explicit Gaussian representation with FLE eliminates the need for MLP queries, which are required in both NeRF and WRF-GS. Although WRF-GS also employs Gaussians, it still relies on a large MLP to query each Gaussian primitive's 's values using the Gaussian mean as input. Second, the hybrid LUDA-base' ray tracer optimizes complex-ver valuud operations through explicit gradient computation. These optimizations enable our method to support real-time applications, egg., sub-millin uub-millisccond tracking in 5G networks.
Measurement Density. Figure compares the MSE of GSRF (trained on the dataset with a den- sity of 0.8 measurements/ /ft)) to NeRF2 (varying densities ranging from 0.8 to 15.5). The densities are obtained by random sampling from the orig- inal 70% training set. GSRF achieves a compara- ble MSE to NeRF trained on the dataset with a density of 7.8. This indicates that GSRF requires 9.8x less training data to achieve comparable spectrum synthesis quality to NeRF The improvement arises from GSRF's 3D Gaussian- based scene representation, which focuses on object features rather than empty space, making more effi- cient than N voxel-based fields. More results for WRF-GS [34] are provided in Appendix F.4.
0.04
0.08
0.8-NF
1.6-MF
4.7-NI
7.8-NF
15.5-M
density
Figure 6: GSRF (GF) at 0.8 measurements/ and NeRF² (NF) across different densities.
5.2 5G Complex-Valune Valued CSI Synthesis
TASK. This task demonstrates GSRF's effectiveness in synthesising complex-valued signals. In 5G Orthogonal Trequency-Division Multiplexing (OFDM) modulation, downlink and uplink operate on different frequency bands [40]. Given uplink complex-vall -vauud CSI, the objective to predict the downlink CSI. The rationale for this task lies in the shared physical propagation environment, which correlates uplink and downlink CSI [41]. Furthermore, uplink CSI can serve as a transmitter position indicator due to its uniqueness across different positions [12, 42].
The public Argos dataset [43] is employed. is collected in outdoor environments, where a base station with 104 antennas measures CSI from signals sent by clients. Each CSI measurement includes 52 subcarriers. Following prior works [12, 30, 41], the first 26 subcarriers are treated as the uplink channel, and the remaining N 26 as the downlink channel. The dataset contains 100,000 measurements and is randomly split into 70% for training and 30% for testing.
DATASET.
METRICS. We adopt the Lignal-t-NNois Ratio (SNR) [30] to quantify synthesized CSI quality:
SNR = 100 (IS - --13 ||S|)=2).
(17)
where S N are the ground truth and synthesized CSI vectors, respectively.
BASELINES. We compare GSRF with NeRF² [12] and include two additional baselines:
R2F2 [41]: Extracts the number of propagation paths and each path's parameters to estimate CSI.
FIRE [30]: Uses the VAE [44] predict the downlink CSI by learning the latent distribution.
Overall Performance. To demonstrate GSRF's efficiency, it is trained on only 30% of the raw training data portion, while the basclines are trained on the full training set. All methods are evaluated
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on the same testing data. Since GSRF requires three- direesiimeiiin transmitter locations, we train an autoencoder [45] using 26 uplink subcarriers input reconstruct them. The autoencoder's hidden layer is set to three dimensions, representing the transmitter locations. Figure 7 illustrates a prediction example from GSRF, where the two eurves (blue and red) nearly overlap, demonstrating its high prediction accuracy. Figure 8 quantifies the SNR of the four methods. GSRF achieves mean SNR of 20.99 dB, outperforming R2F2 and FIRE. Additionally, GSRF achieves comparable CSI synthesis quality to NeR while using 3 less training data, highlighting its training efficiency.
It worth noting that NeRF also performs phase- aware modeling through an MLP that regresses amplitude and phase from voxel and transmitter coordinates. While this enables high-quality CSI
Phase (rad) Amolltude mana
25
20
30
synthesis, the volumetric ray-based querying of the MLP introduces significant computational cost during both training and inference. In con- trast, GSRF integrates phase modeling directly Figure 7: Channel am- Figure 8: Channel CSI into Gaussian primitives via Fourier-Legendre ba- plitude and phase trace. prediction SNR. sis expansion, avoiding MLP regressors entirely.
10 15 20
25
Subcariiers
ARE
GSRF
This explicit, complex-vale valued representation allows efficient gradient updates with lightweight CUDA operations, leading to faster convergence and inference without sacrificing accuracy. Thus, the comparable SNR to NeRF² does not diminish the contribution of GSRF, but instead underscores its ability to achieve phase-aware synthesis with substantially greater efficiency.
5.3 BLE Real-Valued RSSI Synthesis
TASK. This task verifies that GSRF supports single-antenna setups for capturing a single real- valued RSSI. Given a transmitter (BLE node) sending signals from location (Zu. Ulx 2x), the goal is to synthesize the RSSI (in dBm) received by a receiver (BLE gateway with a single antenna). The measured RSSI represents the aggregate signal power from all directions [12]. Additionally, we conduct a fingerprint-based localization application to demonstrate GSRF's sensing advantages.
The public BLE dataset [12), collected in an elderly nursing home, is employed. Twenty- one receivers operating at 2.4 GHz frequency band to capture RSSI. The dataset contains 6,000 transmitter positions, each paired with a 21 21-dimensional tuple of RSSI readings from the 21 receivers.
DATASET.
METRICS. RSSI synthesis error is the absolute difference between predictions and ground truth.
0.8
0.8
BASELINES. We compare GSRF with NeRF²
g g6
8
Other empirical and DL methods, e.g., MRI [46],
0.2
NeRF
0.4
0.2
NeRt?
are excluded because they perform worse
GSRE
GSRF
than NeRF on the same testing dataset [12].
(d8m)
(m)
Overall Performance. To evaluate the perfor- Figure 9: RSSI errors on
Figure 10: BLE-based
mance of GSRF in scenarios with sufficient data, the BLE dataset.
localization error.
both models (GSRF and NeRF²) are trained on the full training dataset. Figure indicates that GSRF achieves an average RSSI error 4.09 dBm, compared to ' 6.09 dBm. This represents 32.79% improvement, highlighting GSRF's effectiveness fo angle-antenna receivers. The performance gain stems from GSRF's flexible 3D Gaussian-based explicit scene representation, which efficiently utilizes training data by focusing on objects rather than large empty space and aligning with object geometry. Training and inference times are reported in Appendix F.2, Figures 11 and 12. These results demonstrate that GSRF achieves a 15.82-fold decrease in training time and 78.98-fold reduction in inference time for RSSI synthesis. Additional results for WRF-GS [34] are in Appendix F.2.
BLE-Based Localization. In fingerprinting- based localization, the RSSI value from an unknown transmitter queries fingerprint database containing pairs of transmitter positions and correspond- ing RSSI values. The K Nearest Neighbors (KNN) identifies the K nearest matches and estimates the unknown transmitter position as the average of these K positions [29]. We generate synthetic datasets using GSRF and F to build the fingerprint database for comparison. Figure 10 shows that GSRF outperforms 22 by 31. 11.40% on average. This improvement in localization accuracy demonstrates that high-fidelity synthesized databases generated by GSRF enhance localization applications, elimi- nating the need for time-consuming and labor-intensive manually collected fingerprinting databases.

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5,4 Ablation Study
We evaluate our design components using the RFID dataset introduced in Section 5.1. All versions trained the full training set, with results presented in Table 1.
FLE-Based Radiance. We employ FLE coeffi
Table 1: Effectivenees of components of GSRF.
cients to model the directional radiance of each
Metric
Radiance
Phase
Fourier loss GSRF
Gaussian in GSRF, unlike the SH coefficients used in 3DGS for visible light rendering [8]. Ex- perimental comparisons (first vs. last column) show that our FLE coefficients achieve better spectrum synthesis compared to SH coefficients. This improvement arises from FLE's ability to capture intricate RF signal interactions, e.g., phase-dependent interference, while SH is more suitable for smooth optical functions. Both FLE and SH are implemented with degree of L = 3, resulting in 16 coefficients each. Thus, GSRF enhances spatial spectrum synthesis by leveraging FLE's capability to handle complex- valued radiance fields.
PSNRT
20.51
20.89
21.30
22.64
Phase Information. Each Gaussian primitive represents radiance and transmittance as complex- valued attributes to capture RF signal propagation effects, such as constructive and destructive interference. Removing the phase channel while retaining only the amplitude results in 8.37% reduction in PSNR compared to the full model with phase inclusion (second vs. last column). This performance drop underscores the importance of phase information in synthesizing RF signal data.
Fourier Loss Fourier- We evaluate the impact of Fourier loss Pourier by comparing the model without this loss term (third column) against the full model (last column). Removing Fourier loss Fourier results in a 6.28% reduction in PSNR, indicating that trequency-oomal fidelity of the synthesized spatial spectra. Incorporating Fourier enables our model better preserve frequency-doman properties, enhancing overall RF data synthesis quality.
alignment enhances the
Discussion
Despite its advancements, our method has two main limitations. achieves efficient RF signal synthesis when training data is available for specific scene but lacks spatial generality for zero-shot inference in unseen environments. is also optimized for static settings: when the scene changes (e.g., moving obstacles or structural modifications), retraining or fine-tuning is required, limiting temporal adaptation. To address these issues, we outline two complementary directions: improving spatial generality across environments and enabling temporal adaptability to dynamic scenes.
For spatial generality, future work will explore pre-training GSRF large and diverse multi-scene RF datasets to learn transferable priors that capture common propagation patterns across environments. This could involve designing donain-general encoders that disentangle scene-invariant propagation features (e.g., free-space loss. reflection/diffie diffriito/dffaactiinnn signatures) from scene-specific geometry, and leveraging commin-adaptatiinn strategies to enable rapid adaptation to new environments with only few samples. Another promising direction is hierarchical Gaussian representations, where global Gaussians encode universal priors while local Gaussians specialize environment-speciff details.
For temporal adaptability, we propose a deformable 3DGS extension that supports dynamic RF scene rendering: shared set of complex-valued 3D Gaussians represents the baseline RF field, while a lightweight deformation module models time-varying changes without full retraining. spatiotemporal encoder factors the 4D space-time volume (I, y, 2,,)) t) into compact 2D planes (zy, y2, æt, yt, 2t), reducing parameter complexity from XC to 6R² and preserving locality for efficient CUDA querying; spatial planes capture multipath effects treflectionddiffraciion) and temporal planes capture motion-induced changes. A small multi-head decoder then predicts per- per-Gaussian deformations (position/rotatiionssalee)) while complex-value valued attributes (e.g., radi- ance thg transmittance Pı) are preserved for phase- aware modeling. Together, these directions aim to make GSRF both broadly generalizable and responsive to real- world dynamics.
7 Conclusion
This paper introduces GSRF, a novel complex-valued 3DGS-based framework for efficient RF signal data synthesis. We customize 3D Gaussian primitives with complex-vat valued attributes and integrate an RF--secificc CUDA-enabled ray tracing algorithm for efficient scene representation and received signal computation. Extensive experiments validate GSRF's efficiency, demonstrating significant improvements in training and inference speed while maintaining high- high-fidelity RF data synthesis.
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Acknowledgement
This research was funded part by the Air Force Office of Scientific Research under awards # FA95502210193 and FA95502310559, and the DEVCOM Army Research Laboratory under award Merced Fall 2023 Climate Action Seed Competition grant, and UC Merced Spring 2023 Climate Action Seed Competition grant. Kang Yang was partially supported at UC Merced by financial assistance award approved by the Economic Development Administration's 's Farms Food Future program. Sijie Ji's research is supported through a Schmidt Science Fellowship. Mani Srivastava was also partially supported by the Mukund Padmanabhan Term Chair UCLA.
Wan Du was partially supported by NSF Grant 2239458, UC
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NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: The abstract and introduction reflect the main contributions of GSRF, including efficient RF signal synthesis and improved inference speed, validated by experimental results.
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The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.
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It is fine to include aspirational goals as motivation long is clear that these goals are not attained by the paper.
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Answer: [Yes]
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Answer: [Yes]
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Justification:
The paper discusses the positive societal impacts of GSRF, including enhanced wireless coverage and support for smart agriculture.
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A Complex- Valued Ray Tracing Algorithm
This section derives Equation (14) for the received RF signal 8 C in GSRF, integrating wave propagation physics with Gaussian discretization.
A.1 Continuous Wave Propagation Model
Consider ray ((t) tv originating from the receiver position with direction V. The received signal modeled as an integral along the ray path:
S $ dl)
where:
(tt)) C: Emitted signal at position 7(t). defined by its amplitude |vb(t) and phase (t)().
.
T(t) E C: Cumulative transmittance from the receiver at (00) to position y(t,, given by:
T(t) exp (-/, a(s) ((s)ds)
where a(s) = ((() +s((s) is the complex attenuation coefficient, with:
a(s) 0: Amplitude attenuation.
- (s):: Phase propagation.
A.2 Discretization via 3D Gaussians
The scene is approximated using a set of 3D Gaussians
{CA }, each specified by:
(Jks Ex, the, ok),
a = ag + j3,,
where:
Pk I::: The mean position of the Gaussian.
Σg R3x3, The covariance matrix representing the spatial extent.
the C: The radiance, encoding amplitude and phase.
ak E C: The complex attenuation coefficient, where:
og 2 Governs amplitude attenuation.
Bx: Determines phase propagation.
Ray-Gaussian Intersection
For each Gaussian ks, we determine the intersection points of the ray >(t) r-+fV with the Gaussian's ellipsoid, as detailed in Section 4.3. We solve:
(7(1) n4) T (7(1) - /4) = 3,
which defines the ellipsoid boundary at the 3-sigma level. This yields the entry and exit times, finkk and fout,k, respectively. The path length through the Gaussian is then given by:
Aty
four,k
tmnk-
Midpoint Approximation
The contribution of Gaussian & is evaluated at the midpoint of the intersection segment:
fin,k + foulk
midd,
Xmid,k ((fiaa,x)
This approximation introduces error of order O(A()), which is negligible for small Ale. as is common in dense Gaussian representations.
Transmittance Derivation
21

---

Transmittance Pm for Gaussian m accounts for attenuation and phase shift over the path length Atm:
Pm = exp
Om dt
= mmmmm
am Aem
Cmm
farm
lpml
eicom
where:
|m = E amm Am < 1: Amplitude transmission factor, since O'm 2 0.
Lpm - Bm A bm Phase shift of the transmitted signal.
A.3 Discretized Rendering Equation
The Gaussians are sorted by their midpoint times midd (in increasing distance from the receiver) to ensure correct depth ordering. The discretized received signal is expressed as:
Kin
R
S
9z (Xmid,,)
II Pm
where:
S (X)i The Gaussian density at the midpoint, given by:
1
exp (-1 [ mn) EE (x mui)
Sn (x(x)= (2T) (2T)3/2(2>
4 = l ed The emitted radiance of Gaussian k.
[ Pm: This term represents the cumulative transmittance from the receiver through all preceding Gaussians up to k 1.
B Gaussian Primitive Optimization
The following two strategies are employed to update the number Gaussians and their attributes.
Attribute Updating Each 3D Gaussian stores its own attributes and updates them using SGD [47]:
mib+1 w(() The V (wrl))
(18)
where w represents any attribute of Gaussian, each with its own learning rate nw- The term V (w)(2) denotes the gradient of the loss function with respect iteration j. For radiance vt, the updated parameters are the FLE coefficients.
The covariance matrix Σ is physically meaningful only when positive semi-definite [48]. but the update equation above does not guarantee this property. To address this, we adopt the solution proposed in [8], which represents Σ = R. R where R is rotation matrix and S is scaling matrix. Updates are applied independently to and S, ensuring that remains positive semi-definite.
Number of Gaussian Updating. The initial number of Gaussians is set by cube-based initialization. However, this number suboptimal, as some areas require more Gaussians (e.g., object regions), while others need fewer (e.g., free space) to model RF signal propagation effectively. We observe that such cases lead to large gradients for the Gaussian's mean μ. as the cxisting 3D Gaussians do not adequately capture the area's 's effect on RF signal propagation. The mean exhibits larger gradients than other attributes because represents the position with the highest probability, making it crucial for modeling RF signal behavior.
To this end, we employ a madient-ttreshoole based strategy: First, every N₂ iterations, we compute the average gradient of the mean for all Gaussians and select those with mean gradient exceeding a threshold tu.- Second, we determine the radius of each selected Gaussian, approximated as average of the diagonal values of its covariance matrix. A radius threshold classifies them small or large Gaussians. Third, small Gaussians are cloned by duplicating them and shifting the copies in the direction of the gradient. Large Gaussians split into two Gaussians, reducing their scaling
22

---

Algorithm 1: Forward CUDA Kernel for Ray Tracing Algorithm
Input: w, h: numbers of rays in azimuth and elevation
Input: M, C: means & covariances of all Gaussians
Input: E,A: radiances & transmittances of all Gaussians
Input: L: positions of receiver and transmitter
Output: O: received signals for all rays
Function RayTracing (w, h, M, C, E, A, L): C, L)
M', C² appherialGaussian
Grids < buildGrid tc. h)
Idx, Kys f- sphericallplaatiinggM M' Grids)
Ranges <- compuneCrridRange(lan Kys)
0-00
forall grid G in Grids do
forall ray G do
< getGridRangeeRanges 9)
10
Oii + Blend(i, ldx, ra Kys, M', C², A)
11
end
end
13
return O
matrix R by a factor of ø and initializing their positions by sampling from the original Gaussian's 's PDF Additionally, every N, iterations, we remove Gaussians with attenuation p below a threshold cps as they minimally impact signal propagation, e,g, in free space. A single 3D Gaussian distribution can represent a large free space.
C CUDA Kernel
We develop two CUDA kernels for the forward and backward computations for RF signal renderer.
Forward Kernel. Algorithm 1 outlines the forward kernel. The inputs include the number of rays in azimuth and elevation, the means, covariance matrices, radiances, and attenuations of all 3D Gaussians, well the positions of the receiver and transmitter. The output is the received signal computed for all Naz x Nei rays.
Specifically, Line 2 projects the 3D Gaussians onto the 2D RF plane. Line 3 partitions all rays into multiple grids, each containing Nays rays in the azimuth and elevation directions, to accelerate processing. Line 4 applies the splatting process to identify which Gaussians influence each grid. Line 5 records the sorted Gaussians within each grid. Finally, Lines 7-12 compute the received signal for each ray in parallel using the complex-valued blending algorithm.
Backward Kernel. Since the Forward Kernel invoked ray tracing forward computation, PyTorch cannot automatically compute the corresponding computation graph gradients. After computing dc the received signal S and the loss &, PyTorch calculates the gradient as which is then passed to the Backward Kernel. This kernel reverses the computations of the Forward Kernel compute the gradients for each Gaussian attribute. We explicitly derive the gradients for all Gaussian attributes for gradient-based attribute learning, with detailed computations provided in Appendix D.
D Gradient Computation
After computing the received signal S and the loss L, PyTorch calculates the gradient ac We apply the chain rule [49] to compute the derivatives for each Gaussian in the Backward Kernel:
Kmn
k-1
II
Pimu,
Pm
k=1
m=1
where:
Pintr & Sx (Irep i M4. Ex): The Gaussian weight.
the l//1 v The radiance of the k-thh Gaussian.
Pm leml < The complex-valued transmittance.
23

---

D.1 Gradient for k-th Gaussian Radiance
Compute ac ac
Oc as
Pintr ks e
II
Pm
04 Compute
82
k-1
0c
aC PPun
|
II m=1
Pm
0Lyb
as
D.2
Gradient for k-th Gaussian Transmittance
The transmittance Pk influences all emissions from subsequent Gaussians j = k + L,.... Kiow- The cumulative transmittance for Gaussian j's radiance is defined as:
i-1
Pj
II
Pm
Compute
0C plpu1
Partial derivative P3 with respect to |pa|:
aP₁
0
if S k
Olppl
Pi loal
i > k
Diffcrentiating S with respect to lpx|:
Kut
as
1855le3200
Pj
alpk
Pintr,j
1pxl
j=k+1
Using the chain rule:
0C
aC
Kmn
W
0ppk|
0
0S
Pintr
Ipx
j=k+1
Oc Compute 8Z
The partial derivative of P3 with respect to Lpu is expressed as:
OP;
0
if S k
0/
(j,
if k
Differentiating the loss with respect to the phase of Pk:
Kmn
ac
OL
Wylletess P3
j Pintr.j
7Lpr
aS
j=h-1
D.3 Gradients for k-th Gaussian Mean and Covariance Matrix
For brevity, we omit the subscript Pintrs ft, and E, where implicitly denotes the k-th Gaussian. Before computing the derivatives for the mean μ and covariance matrix we first calculate a2
Sper
aC ac
|
II
Pm
0 as
m=1
To compute the gradients for the mean y and the covariance matrix Σ, we first examine the forward computation of the probability Piatr-

---

D.3.1
Defining 3-Sigma Ellipsoid
(x u2)²)> 20=11xxx - -9
where:
X: A point 3D space.
p The mean (center of the ellipsoid).
Σ: The covariance matrix defining the ellipsoid's shape and orientation.
D.3.2
Ray Parameterization
The parametric equation of ray, as given in Equation (10), is:
x(d) = 11x + dv, d 2 Trx
where:
Ix: Receiver position.
v: Unit direction vector of the ray.
d: Distance parameter along the ray.
Tix Radius of the Ray Emitting Spherical Surface (RESS), defining the starting point for each ray.
Substitute x(d) into the ellipsoid equation:
(ex + dv ---1 -1) E + dv =²=
Expanding the dot product:
(hx 1A-1)T=--0- 0x p)) 2dv 2 - (1x -1) -
D.3.3 Solving the Quadratic Equation
Rearrange the equation:
Ad + 2Bd (C =0
where:
A = V=
B = v - (1x -M), C = (In 11) [)) T E-1 4)
Using the quadratic formula:
d1,2 - B + / B2 A(C 9)
A
The two intersection points along the ray are:
x + V X2 1 d2f
The valid intersection is the one where 2 T'x- If no valid exists, the ray does not intersect the ellipsoid:
1 + d1 d2 4
X
2
The probability of intersection is then computed as:
exp - (xintr - 11) Σ (Xintr 11)
Pintr
( 1/2
25

---

D.3.4 Backward Computation for Mean
Compute
Opmn and
Oran 8y
direct
Since Pinur depends on both Xintr and jr, we compute the gradients as follows:
Opnnt
Vxintr
Piittr
Σ
(Xintr - (1)
Öptntr
Pintr
Σ-
Xintr (1)
direct
8Xurr Compute
By
(10dg
0d1
0d2
+
0p 2
0p
dp
0d. Compute and on 8y
The partial derivatives of di and d2 with respect to the mean μ given by:
0B
(B²
A(C 9)) 1/2 2B 0B
A AAC 3p
Od1.2
By
0p
where:
0B
ac
--
-22
2-2--11a 1b)
0p
0p
Compute Oraet: Op
The gradient of Pints with respect the mean j is computed using the chain rule follows:
Opint
0pmn
0xiui
Öμ
Ix
Op
direct
Compute
0c; 8p
The gradient of the loss & with respect the Gaussian mean is computed
ac
Oc
Opinir
0μ
Ppms
The gradient for the covariance matrix we omit it for brevity.
ac follows similar steps the mean gradient computation, so
ax.
E Implementation
Training. Table 2 presents the hyper- parameter settings, defined Section 4. These values are determined through ex-
Table 2: Hyperparameter settings.
Symbol
Meaning
Value
tensive empirical studies. The attributes of all 3D Gaussians are updated us-
Eg
Threshold for mean gradient
0,0002
Er
Threshold for radius
10.0
ing SGD [47]. The learning rates are set as follows: ?p 0.01 for attenuation,
Ep
Threshold for attenuation
0.004
N
Gradient check frequency
100
The = 0.0025 for emission, 7ss 0.01
No
Attenuation check frequency
100
for the scaling matrix, and π = 0.005
Tr
Radius of the RESS
1.0
for the rotation matrix. The learning rate for the mean, M/, starts at 0.00016 and de-
Scaling matrix reduction factor
1.6
creases exponentially to 1.6 x 10-6 over 30,000 iterations. For Gaussian count optimization, the number of Gaussians is optimized only during the first half the total iterations. After that, only the attributes of the Gaussians are updated.
26

---

CUDA Kernel. Each grid contains 16 rays in both azimuth and elevation angles, totaling Nays 16 16 rays per grid. Gaussians intersecting each grid are sorted using the CUDA built-in cub: Devii cotadixSrrt API [50]. Each splatting instance (a Gaussian intersecting a grid) assigned a 64-bit key: the lower 32 bits store the distance to the receiver, while the upper 32 bits encode the grid index. This structure enables efficient parallel sorting of all splats by distance with single invocation the cub: beviceRadixSort API.
To integrate PyTorch with CUDA execution, we implement a custom PyTorch extension using C++ and CUDA, enabling efficient GPU-accelerated computations. The forward and backward com- putations are encapsulated within subclass of torch. autograd. Function, ensuring seamless differentiation and gradient propagation within PyTorch's 's computational graph. The Python interface, implemented via Py PyTorc''s 's C++ API, facilitates interaction between PyTorch tensors and CUDA kernels, handling memory layout conversions and efficient CPU-GPU data transfers.
F
Experimental Results
F.1 Scenario Overview
This paper considers scenario where a receiver fixed at a position (e.g., 5G base station or LoRa gateway), while a transmitter (e.g., smartphone LoRa node) can be at any location in 3D space. Given a dataset of some transmitter locations and their corresponding received signals, the goal is to predict the received signal from transmitter at a new position.
Alternatively, the roles can be reversed: the transmitter is fixed (e.g.. WiFi router), while the receiver is placed at different locations (e.g., smartphone). According to reciprocity between the transmitter and receiver [30], these two scenarios are essentially equivalent. Consequently, this work focuses solely on the first scenario.
F.2 BLE RSSI Synthesis
Training and Inference Time. Both meth- ods are trained for 100,000 iterations. Train- ing time measured by running each method 10 times a computer equipped with an NVIDIA GeForce RTX 3080 Ti. In- ference time for each model is also recorded. Figure 11 shows that GSRF reduces train- ing time from 2.69 hours with NeRF² to 0.17 hours, achieving a 15ld decrease. Figure 11: Training times Similarly, Figure 12 illustrates that GSRF for RSSI synthesis.
1.0
1.0
8
0.4
0.4
Nult?
Nelf?
0.2
G5AF
0.2
ESRF
0.0
1
3
0.0
30
60
90
120
150
Training time (hours)
Inference time (ms)
Figure 12: Inference times for RSSI synthesis.
reduces inference time from 139.01 ms with NeRF² to 1.76 ms, 78.98-fold reduction. These short inference times enable GSRF support real-time applications [30].
BLE RSSI Prediction and Localization. We extend the evaluation to include WRF-GS [34), trained on the full dataset, for both RSSI prediction and localization tasks. GSRF reduces RSSI error by 3.92% compared to WRF-GS, demonstrating the benefit of its unified RF modeling. Localization errors remain similar across models, reflecting the inherent resilience of the KNN baseline: by selecting the h nearest neighbors in the RSSI fingerprints and averaging their positions, KNN effectively acts as a low-pass filter that mitigates synthesis noise.
Table 3: BLE RSSI prediction error and localization emor across models.
N
GSRF
WRF-GS 4.261#3.943
RSSI error (dBm) Localization error (m)
4091=5.427 0.699
4.09443.900 0.47910.692
0.481=0.685
F.3 Parameter Study
We further investigate the effect of several key hyperparameters in GSRF, including the Fourier-Legendre Expansion (FLE) degree, the angular resolution, and the number of Gaussians. These studies provide insights into the trade-offs between accuracy, efficiency, and stability, and guide the recommended default settings.
27

---

Fourier-Legendre Expansion (FLE) Degree. The FLE basis degree L controls the expressiveness of GSRF 's complex-valued 3D Gaussians in modeling phase-aware RF propagation effects such as interference and diffraction. Low degrees (I. = 1-2) capture only coarse angular components, which leads to underfitting Moderate degrees (T. = 3-4) capture essential variations efficiently, while high degrees (I. 2 5) risk overfitting and add computation. We find that I. = 3 provides the best balance between accuracy and efficiency, Experimental results are summarized Table 4.
Table 4: Effect of FLE degree on PSNR, training time, and inference speed.
Degree L
PSNR (dB)
Training time (min) Inference time (ms)
16.49
13.84
2.96
2
17.73
15,09
3.38
3
18.67
16.21
4.18
4
18.78
19.26
6.27
5
18.21
24.72
8.62
Angular Resolution. The angular resolution in GSRF a rigid hyperparameter but is instead governed by the antenna configuration and the measurement setup. Its role is to balance fidelity, coverage, and computational efficiency in modeling RF propagation.
For multi-antenna arrays, the effective angular resolution follows the spatial sampling theorem and scales with the number of array elements. For example, in RFID dataset, rectangular array supports ~~1 resolution using classical algorithms such as MUSIC (Multiple Signal Classification). We align the resolution with the measurement data: since the RFID dataset was collected at 1 intervals over azimuth and elevation, we preserve this resolution to avoid interpolation artifacts. If the measurement data had coarser sampling (e-g., 2 intervals due to a smaller array), GSRF could operate at that resolution without modification, since both the orthographic splatting process and the loss functions are resolution-agnosie and work with arbitrary ray grids.
4 uniform
For single-antenna configurations, the received signal is inherently scalar, with no native angular resolution per antenna theory. In such cases (e.g., RSSI synthesis), rendering at 1 to ensure dense coverage of propagation paths. This choice is flexible: finer bins (e.g.. 0.5%) increase the ray count without proportional fidelity gains at centimeter wavelengths, while coarser bins (e.g., 59) reduce runtime but risk missing important multipath effects. Our 1 setting thus represents practical trade-off, and is consistent with conventions in RF Computer-Aided Design (CAD) simulation tools such as Wireless InSite and the MATLAB Ray Tracing toolbox.
discretize the spherical
Experimental Validation. Table 5 reports the effect of angular resolution in the single-antenne RSSI synthesis task (Section 5.3). At 1" with 360 = 32,400 rays, GSRF achieves the lowest RSSI error due to dense angular sampling. As the resolution coarsens, accuracy degrades: 2 resolution produces slightly higher error (with 180x45 = 8,100 rays), and 5 resolution degrades further (72x 18 1,296 rays). Training and inference times scale proportionally, demonstrating tunable trade-off: high-resolution settings suit precision-cre critical applications, whereas coarser settings may be preferable when computational efficiency is paramount.
Table 5: Effect of angular resolution on RSSI synthesis.
Resolution
RSSI error (dBm) Training time (min)
Inference time (ms)
1
4.094
10.23
1.76
4.493
8.56
0.94
5
6.518
4.92
0.17
Number of Gaussians. The number of Gaussians is not a manually fixed hyperparameter. As de- tailed Appendix and following the original 3DGS [8], it dynamically optimized through densifi- cation and pruning during training. This process automatically adds Gaussians in mnder-reconstrueet regions and prunes redundant ones, ensuring the model adaptively balances representation capacity and efficiency without manual tuning.
Cube- Based Initialization. The cube- initialization in GSRF refers to a uniform strategy for placing Gaussian primitives at the start of training. This choice is motivated by the need for comprehensive coverage of the 3D scene volume, which accelerates convergence. By distributing Gaussians uniformly across the bounding box that encloses the transmitter, receiver, and environment, the model begins with a balanced representation of potential RF propagation paths. This avoids
28

---

early coverage gaps that could arise from sparse or random initialization, enabling the subsequent densification and pruning process to refine the representation more effectively. While random initialization can achieve similar fidelity after sufficient optimization, it typically requires longer training time (0.59 hours for random initialization vs. 0.27 hours for uniform initialization on the RFID spatial spectrum synthesis task).
F.4 Measurement Density
We extend the measurement density in Section 5.1 analysis include WRF-GS [34]. When trained with 0.8 measurements/tt WRF-GS and GSRF achieve MSEs of 0.002659 + 0.003560 and 0.002147 0.003343, respectively, both comparable to NeRF 0.002405 0.003623 despite the latter being trained with a substantially higher density of 7.8 measurements/ft?" This advantage arises from the explicit 3DGS representation adopted by WRF-GS and GSRF, where Gaussian primitives provide greater representational power and flexibility than NeRF-based volumetric sampling, thereby improving efficiency under sparse measurement conditions.
E.5 Practical Benefits
RFID. An angular artificial neural network (AANN) identifies the Angle of Arrival (AoA) of line- sigh path from received spatial spectra, enabling spectrum- based localization [51]. The AANN is trained on pairs of spectra and their corresponding AoA labels. Both GSRF and NeRF can synthesize spectra for AANN training. Compared to NeRF adopting GSRF can significantly reduce real-world resource consumption. For example, in a conference room (26.2 x 16.4ft x 9.8 [52] with a mcasurement time of one minute per measurement, reducing the measurement density from 7.8 to 0.8 measurements /ft² saves approximately 200 hours of data collection time. Additionally, GSRF reduces computing time by 5.71 hours, including 4.74 hours for training and 0.97 hours for inference, both of which greatly save computational resources.
BLE. Similar to the previous field study, GSRF eliminates the need for site surveys, significantly reducing data collection time. Its fast training GPU-hours and low inference latency save server computation resources, accelerating the construction of the fingerprint database.
5G. The current method for obtaining downlink CSI requires client feedback. causing significant transmission overhead [30]. GSRF eliminates this overhead. Furthermore, GSRF's low inference latency makes it suitable for 5G networks. In contrast, NeRF²'s inference latency of over 300 ms exceeds the coherence time in dynamic scenarios [30], making it impractical for 5G applications.
G Design Discussion
Why Fourier-Legendre Basis Instead Spherical Harmonics. We adopt Lourier-Legendre Expansion (FLE) over the more common Spherical Harmonics (SH) to model directional radiance in GSRF. This choice is motivated by the fundamental differences between RF propagation and visible light rendering, well the mathematical properties of the two bases. RF signals at centimeter-scall wavelengths exhibit pronounced phase-dependent interference and diffraction, which SH is ill-suited to capture efficiently.
Limitations of SH in RF. SH provides an orthogonal basis on the sphere and is widely used in 3DGS for representing smooth, low- oow-freueeccy view view-dependent effects in the visible domain (e.g., shading, reflections). However, SH suffers from two limitations in RF applications. First, its low-frequency bias makes it converge slowly for oscillatory patterns: waveeengt RF fields often exhibit sharp constructive and destructive interference, leading to high-frequency angular variations that SH requires high degrees to approximate. Second, standard SH relies on real coefficients and is therefore chass--nsenssitvve making it poorly suited for modeling complex-valuee RF fields where phase differences govern interference outcomes.
Advantages of FLE. FLE combines Fourier series (for azimuthal periodicity) with Legendre poly- nomials (for elevation dependency), and is better aligned with the properties of RF propagation. Its Fourier component naturally captures periodic phase shifts and oscillatory interference patterns, while Legendre polynomials provide orthogonal support over elevation. Unlike SH, FLE employs complex coefficients, allowing direct encoding of both amplitude and phase, which essential for accurate RF modeling. Moreover, the aolmuuazimuthal decomposition of FLE matches the geometry of antenna measurements over spherical regions, providing better locality and efficiency for multipath effects compared to the global harmonics of SH.
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Overall, FLE offers more compact, phase-aware and physically aligned basis for directional RF radiance, enabling GSRF to efficiently capture the interference-rien characteristics of RF propagation.
Role of SSIM Loss. While the blob-like primitives in 3DGS representations can produce smoother outputs, the Structural Similarity Index (SSIM) remains valuable as a complementary loss. Unlike L1, which emphasizes pixel-wise accuracy, SSIM emphasizes structural and perceptual similarity, making particularly suitable for RF synthesis tasks where outputs such as spatial spectra are image-like data (e.g., directional signal power). This follows standard practice in prior 3DGS methods (e.g., 3DGS [8] and WRF-GS [34]), which incorporate SSIM to enhance perceptual quality. Our ablation confirms its effectiveness: removing SSIM and relying solely on LI and Fourier loss reduces RFID spatial spectrum synthesis PSNR by 0.73 dB (from 22.64 to 21.91 dB), indicating that SSIM helps refine structural details in the synthesized spectra.
Equirectangular Projection vs. Cube-Map. We adopt equirectangular projection for representing RF spatial spectra, despite its known polar stretching. due its simplicity and compatibility with parameterizations commonly used in antenna array data. This choice enables uniform angular sampling without additional remapping and aligns directly with the matitudellogitte grids of collected datasets. The distortion near high elevations has limited practical impact in RF scenarios, since paths above 60" typically correspond ceilings, floors, or skyward directions where signals are heavily attenuated or yield few useful multipath components. Although cube-maps could mitigate polar distortion, they introduce seam artifacts and gradient discontinuities across faces, destabilizing backpropagation in a differentiable rendering pipeline. Moreover, cube-maps exhibit non-uniform sampling density across faces, which conflicts with the uniform angular resolution of RF spatial spectrum measurements. Thus, equirectangular projection provides a more stable and dataset-aligned choice for our framework.
Multipath Effects. GSRF is designed to capture multipath effects by representing the RF scene as a collection of complex-valued 3D Gaussians, where each Gaussian acts as primitive that approximates a propagation path or interaction point. Multipath propagation introduces amplitude attenuation and phase shifts across different paths, which are modeled through complex- valued radiance and transmittance attributes encoded via the critical role: rays are emitted from the receiver across a spherical surface, and the contributions of intersecting Gaussians are aggregated. Transmittance encodes wath-length-deeendeer phase shifts and attenuation, and the summation of complex contributions enables both constructive and destructive interference. This effectively discretizes the continuous wave propagation integral, ensuring that path-specific interactions are preserved. Without ray tracing, aggregation would reduce to simple amplitude blending, which is insufficient for centimeter-scale RF modeling.
basis. Ray tracing plays
Empirical validation is performed using real-collected datasets, where explicit ground truth for multipath components is not directly observable due to measurement aggregation. Instead, multipath fidelity validated implicitly: interference effects were not captured, synthesized RF data would deviate significantly from real signals. leading to degraded quality metrics. The strong alignment of synthesized and measured signals thus confirms that multipath effects are effectively represented.
Antenna Beam Patterns. Directional antenna effects, including side lobes and attenuation, are in- corporated in GSRF through data-driven learning rather than explicit physics- based parameterization. The -ourier-Legendrr basis provides the representational capacity: Legendre polynomials capture polar variations such as main beam gain and off-sxis attenuation, while Fourier components represent azimuthal phase shifts and side lobe structures. During rendering, these directional dependencies are aggregated via spherical ray tracing, and orthographic splatting ensures that heam-induced modu- lations are preserved in the synthesized RF field. If training data reflects beam-specific effects, the optimization naturally adapts Gaussian attributes to encode them.
In our datasets, the antennas used are omnidirectional, so side lobes and beam shaping are not observed. Nevertheless, ablation studies of the Bourier-Legenden basis confirm its benefit, showing improved performance even under isotropic conditions. We anticipate that with directional antenna data (e.g., beamformed phased arrays), GSRF would capture and reproduce beam patterns faithfully, as the framework is agnostic to antenna type and adapts to observed propagation characteristics.
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Reviews:
[{"review_id": "DzDV8ITRJI", "reviewer": "Reviewer_c9tJ", "summary": "GSRF extends 3D Gaussian Splatting to the RF domain by modeling each Gaussian with complex radiance and transmittance parameters to capture signal amplitude and phase. It replaces spherical harmonics with a Fourier-Legendre expansion to represent directional, phase-dependent radiance. Ray-surface intersections are identified by orthographically splatting Gaussians onto a Ray Emitting Spherical Surface, reducing per-ray computation. A CUDA-based ray tracing algorithm aggregates complex-valued contributions along each ray using the Huygens–Fresnel principle in a differentiable framework. Evaluated on RFID, BLE, LoRa, and 5G datasets, GSRF speeds up training and inference while matching or outperforming NeRF-based and 3DGS-based RF synthesis methods.", "questions": "How sensitive are results to the Fourier-Legendre degree L or the angular resolution?", "limitations": "The authors discussed the limitations of the paper at a high-level but didn't show more detailed analysis such as presenting failure cases.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "+++ The paper proposes a solid method to model a novel application of neural rendering. Specifically on using 3D gaussian splitting for radio frequency data. The paper addresses relevant challenges like complex-value ray tracing, orthographic splitting and directional and phase modeling. It is a novel framework for a significant problem.\n\n+++ The method is very well-written and clear to readers. The high-level logic is very consistent and the paragraphs are well-connected, following a clear logical order.\n\n+++ The experiments are pretty solid, as the authors use run extensive experiments on several datasets and presented both qualitative and quantitative results. The authors included sufficient details but also keep the writing relatively concise and presented the experimental results very clearly.\n\n+++ The experiments contain detailed analysis and ablation studies to show statistical significance and the efficacy of individual technical choices.\n\n+++ The paper already included a well-documented github repo to reproduce the experiments in the paper.\n\n--- All experiments assume static geometry; moving obstacles or dynamic environments require additional modeling or retraining and I imagine extending dynamic environments are pressing for real-world applications.\n\n--- Performance depends on choices like number of Gaussians, basis degree and grid resolution, but the paper provides limited guidance on tuning these. It'll be great if the authors could provide some insights into whether the performance is sensitive to these values and if there are failure cases where the training is not stable.", "was_revised": true, "final_justification": "I appreciate the authors for the detailed response. I will maintain my positive rating.", "rebuttal": "## **To Reviewer c9tJ**\nThank you for your thoughtful assessment and for recognizing the strengths of our paper. Below, we provide responses to your questions and concerns.\n\n\n### **Q1 – Assumption of Static Scenarios**\nThank you for your comment and for highlighting this limitation. We fully agree that extending the method to support moving obstacles or time-varying scenes is critical for enhancing real-world applicability. As noted in Section 6 (Discussion), the current approach is optimized for static environments. When the scene changes, such as due to moving obstacles, the model would require retraining. This limits its temporal adaptation.\n\nTo address this in future work, we propose extending GSRF with a deformable 3DGS mechanism that supports dynamic RF scene rendering. The preliminary idea is to maintain a shared set of complex-valued 3D Gaussians that represent the baseline RF field and to introduce a lightweight deformation module that models time-varying changes efficiently. This would leverage GSRF's existing efficiency in rendering to enable real-time adaptation without full retraining.\n\n**Spatiotemporal Structure Encoder.** A spatiotemporal structure encoder captures both spatial RF interactions (such as reflection and diffraction) and temporal dynamics (such as obstacle motion) by decomposing the 4D space–time volume (x, y, z, t) into six compact 2D planes: (xy, yz, xz, xt, yt, zt). This decomposition is motivated by computational efficiency and locality preservation. Modeling the full 4D tensor directly would require R⁴ × C parameters, which is prohibitive for real-time applications. In contrast, the six-plane strategy reduces this to 6 × R² × C, significantly lowering memory and compute demands. It enables localized modeling of spatial features (e.g., reflections in xy, xz, yz) and temporal changes (e.g., motion in xt, yt, zt), allowing scalable and efficient querying on CUDA while preserving the multipath and interference effects crucial to RF modeling.\n\n**Multi-Head Decoder for Deformation.** A lightweight multi-head decoder, composed of small MLPs, would then predict per-Gaussian deformations in position, rotation, and scaling based on the features extracted by the encoder. The complex-valued attributes (e.g., radiance) remain unchanged to preserve phase-aware RF modeling. This setup enables each Gaussian to deform dynamically over time, allowing GSRF to simulate evolving RF paths and interference patterns without requiring a new set of Gaussians for each time step.\n\n\n### **Q2 – Hyperparameter: Fourier–Legendre Expansion (FLE) Degree**\nThank you for your comment on the FLE basis degree `L` in our GSRF framework. Below, we provide a brief analysis of its impact, supported by experimental results.\n\n**Analysis.** The FLE basis degree `L` controls the expressiveness of GSRF’s complex-valued 3D Gaussians in modeling phase-aware RF propagation effects such as interference and diffraction. Low degrees (`L = 1–2`) use fewer coefficients and capture only low-frequency angular components, which leads to underfitting of complex RF patterns. Moderate degrees (`L = 3–4`) better capture essential phase variations with a compact number of Fourier–Legendre coefficients. High degrees (`L ≥ 5`) risk overfitting to noise and introduce increased computational cost and numerical instability. This behavior is similar to how the degree of spherical harmonics (SH) controls fidelity in other 3DGS-based systems. We find that `L = 3` provides a good balance between modeling fidelity and efficiency.\n\n**Experimental Validation.** We use the sparse training data with 0.8 measurements/ft³ for spatial spectrum synthesis, as described in Section 5.1 (RFID Spatial Spectrum Synthesis). Our experiments (see table below) show that `L = 3` closely approaches the peak PSNR of `L = 4` while providing faster training and inference. Lower degrees (`L = 1` and `2`) underfit and yield lower PSNR, whereas a higher degree (`L = 5`) offers no clear accuracy gain and increases computational cost. We recommend `L = 3` as the default setting for its balanced trade-off between accuracy and efficiency.\n| Degree `L`              | 1     | 2     | 3     | 4     | 5     |\n|-------------------------|-------|-------|-------|-------|-------|\n| PSNR (dB)               | 16.49 | 17.73 | 18.67 | 18.78 | 18.21 |\n| Training time (minutes) | 13.84 | 15.09 | 16.21 | 19.26 | 24.72 |\n| Inference time (ms)     | 2.96  | 3.38  | 4.18  | 6.27  | 8.62  |\n\n\n### **Q3 – Hyperparameter: Angular Resolution**\nThank you for your comment on the angular resolution parameter. The one-degree resolution in our setting is not a rigid hyperparameter but is selected to match the inherent resolution of the collected data and to align with standard RF practices, balancing fidelity, coverage, and computational efficiency. Below, we elaborate based on antenna types, as these directly influence the effective angular resolution.\n\n**Multi-Antenna Arrays.** Our method does not impose a fixed angular resolution; instead, it adapts to the antenna array's capabilities and the training data's granularity. In multi-antenna setups, the resolution is governed by the array’s spatial sampling theorem, which approximately scales with the number of antenna elements. For instance, in our RFID dataset, a 4×4 uniform rectangular array supports around 1° precision using algorithms such as MUSIC (Multiple Signal Classification). We align the synthesis resolution with the measurement setup: since the RFID data was collected at 1° intervals over azimuth and elevation, we maintain this resolution for consistency and to avoid interpolation artifacts. If the measurement data had coarser resolution (e.g., 2° due to a smaller array), GSRF could be trained and evaluated at that resolution without modification, as the orthographic splatting and loss functions are resolution-agnostic and operate on arbitrary ray grids.\n\n**Single-Antenna Configurations.** In single-antenna cases (e.g., for RSSI synthesis), the received signal is a scalar power aggregated from all directions, per the antenna theory (no inherent angular resolution). Here, we discretize the spherical rendering at 1° to ensure coverage of propagation paths. Essentially, we could reduce the resolution to lower computation cost while preserving sufficient accuracy. Finer bins (e.g., 0.5°) increase ray count without proportional gains in fidelity for centimeter-wavelength RF. Coarser bins (e.g., 5°) risk missing details of propagation effects. Our 1° choice aligns with conventions in RF Computer-Aided Design (CAD) scene model-based simulation tools such as Wireless InSite and the MATLAB Ray Tracing toolbox.\n\n**Experimental Validation.** The table below shows the experiments for angular resolution in single-antenna RSSI synthesis, as described in Section 5.3 (BLE Real-Valued RSSI Synthesis). At 1°, with 360 × 90 = 32,400 rays, GSRF achieves the lowest RSSI error due to dense angular sampling. As the resolution coarsens, fidelity drops: 2° resolution yields slightly worse error with 180 × 45 = 8,100 rays, and 5° degrades further with 72 × 18 = 1,296 rays, due to the loss of directional detail. Training and inference time scale accordingly. Coarser resolutions reduce runtime but increase RSSI prediction errors. This tradeoff enables flexible tuning: the use of 1° resolution is suitable for precision-demanding tasks, while coarser settings are preferable in deployment scenarios where speed is critical.\n| Angular Resolution   | 1°     | 2°     | 5°     |\n|----------------------|--------|--------|--------|\n| RSSI error (dBm)     | 4.094  | 4.493  | 6.518  |\n| Training time (minutes) | 10.23  | 8.56   | 4.92   |\n| Inference time (ms)  | 1.76   | 0.94   | 0.17   |\n\n\n### **Q4 – Hyperparameter: Number of Gaussians**\nThank you for your comment on the number of Gaussians parameter. The number of Gaussians is not a fixed hyperparameter; as noted in Appendix B, which details the training procedure following the original 3DGS [7], it is dynamically optimized via densification and pruning during training. This adaptive process iteratively adds (densifies) new Gaussians in under-reconstructed regions while pruning redundant or low-contribution ones. This ensures the model efficiently captures the scene's RF propagation complexity without manual tuning of the number of Gaussians, balancing representation power and efficiency."}, {"review_id": "RdPUShOvod", "reviewer": "Reviewer_PLyQ", "summary": "The authors introduce GSRF which combines 3D Gaussian splitting for the RF domain to synthesize new signals. While NerF-based methods can achieve high performance, they are computationally expensive and have high inference latency. GSRF uses a Fourier-Legendre basis to model directional radiance and orthographic splitting for intersections with spherical surfaces. They also customize CUDa kernels for complex-valued ray tracing. The authors evaluate their method on a variety of domains and demonstrate significant improvements in training, inference times, and sample efficiency compared to NeRF methods.", "questions": "- Could the authors explain a bit more about why the Fourier-Legendre basis was chosen over standard spherical harmonics? What does it contribute?\n- Why was cube-based initialization chosen and how well does it work for spherical surfaces like RF signals?\n- I can understand how GSRF is much faster to train and inference than NeRF^2, but how is it so much faster than WRF-GS? Is this mostly due to the custom CUDA kernels?", "limitations": "Yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "## Strengths\n- Although I am only somewhat familiar with RF, Gaussian splats, and NeRF, the paper is easy enough to follow overall. Some of the experiments were difficult to understand.\n- Novel contribution of adapting 3DGS to RF domain and proposes methods to overcome limitations of current methods (RF-3DGS and WRF-GS).\n- Empirical results seem quite strong: GSRF not only achieves lower MSE and higher PSNR over baselines, it is also much faster to train and inference than both a NeRF-based method and 3DGS-based method.\n\n## Weaknesses:\n- The authors mention this in their limitations and I agree: there is almost no spatial generalization for novel scenes. This method seems to require retraining (although it is fast to train) for each scene separately (please correct me if I'm wrong).", "was_revised": false, "final_justification": "", "rebuttal": "## **To Reviewer PLyQ**\nThank you for your thoughtful assessment and for recognizing the strengths of our paper. Below, we provide responses to your questions and concerns.\n\n\n### **Q1 – Spatial Generalization**\nThank you for your comment and for highlighting this aspect in our limitations discussion. You are correct that GSRF, like many radiance field-based methods (including 3DGS and NeRF variants), is designed as a scene-specific model and does not inherently generalize to novel scenes without retraining. This stems from its core architecture: the 3D Gaussians are optimized to overfit the specific propagation environment captured in the training data, modeling unique multipath effects, interference patterns, and scene geometry (e.g., reflections off walls or objects in a given room). As a result, applying a trained model to a new scene (e.g., a different room layout) would yield poor synthesis fidelity, as the Gaussians would not align with the unseen propagation dynamics.\n\nThat said, the fast training time you noted mitigates this limitation in practice. Retraining on new data for each scene is feasible and aligns with RF applications, where environments often change incrementally (e.g., furniture rearrangements) rather than requiring broad generalization across vastly different scenes. For scenarios demanding stronger generalization, future extensions could incorporate hybrid approaches, such as integrating scene-agnostic priors (e.g., physics-based simulations) or meta-learning, but these are beyond our current scope focused on high-fidelity, efficient per-scene synthesis.\n\nTo enable spatial generalization while building on GSRF's framework, we propose actionable solutions: pretraining GSRF on diverse RF datasets and fine-tuning on target scenes, initializing Gaussians to preserve general priors. Incorporating conditional inputs like scene geometry embeddings via a lightweight MLP modulating attributes. Hybridizing with physics simulators (e.g., Wireless InSite [19]) to seed Gaussians along simulated paths for zero-shot inference on similar layouts.\n\n\n### **Q2 – Why Fourier–Legendre Basis Instead of Spherical Harmonics**\nThank you for your question on the choice of Fourier–Legendre Expansion (FLE) over Spherical Harmonics (SH) for modeling directional radiance in GSRF. Below, we provide a theoretical explanation for this design decision, grounded in the fundamental differences between RF signal propagation and visible light rendering, as well as the mathematical properties of the bases. This choice stems from the need to handle the unique challenges of RF domains—such as centimeter-scale wavelengths that cause pronounced phase-dependent interference and diffraction—which SH is ill-suited to capture efficiently.\n\n**Limitations of SH in the RF Domain.** SH forms an orthogonal basis on the sphere, commonly used in 3DGS for visible light to represent smooth, low-frequency view-dependent effects like shading and reflections. While effective for nanoscale visible light wavelengths, SH struggles in RF due to:\n\n*(i) Low-Frequency Bias.* SH excels at band-limited, diffuse functions but converges slowly for high-frequency or oscillatory patterns. RF signals, with wavelengths around centimeters (e.g., 915 MHz in our RFID dataset), exhibit sharp phase-dependent interference (constructive or destructive) and diffraction over propagation paths. These create high-frequency, non-smooth variations in the radiance field.  \n\n*(ii) Phase Insensitivity.* Standard SH uses real coefficients, poorly modeling complex-valued RF fields where phase drives interference (e.g., destructive cancellation). RF's longer wavelengths amplify these effects over distances, unlike light's smoother amplitude-based aggregation.\n\n**Why FLE.** FLE is a hybrid basis combining Fourier series (for azimuthal/phase periodicity) with Legendre polynomials (for elevation dependency), tailored for complex-valued, directional RF radiance. We chose FLE for its theoretical advantages in RF:\n\n*(i) Frequency-Domain Suitability.* The Fourier component naturally models periodic phase shifts and interference patterns inherent to wave propagation. This aligns with RF's wave-like nature (diffraction, scattering), allowing efficient capture of oscillatory behaviors without requiring high degrees. Legendre polynomials provide orthogonal support on the sphere, and the hybrid enables better representation of anisotropic, phase-sensitive fields.\n\n*(ii) Complex-Valued Support.* Unlike real SH, FLE's complex coefficients directly encode amplitude and phase, crucial for RF's interference modeling (e.g., constructive amplification). This integrates seamlessly with our complex-valued Gaussians and ray-tracing.\n\n*(iii) Physical Alignment.* RF antennas collect over spherical regions, and FLE's polar–azimuthal decomposition matches this geometry, better than SH's global harmonics for localized multipath effects.\n\n\n### **Q3 – Cube-Based Initialization**\nThank you for your question on the cube-based initialization strategy in GSRF. The term may be slightly misleading—this is essentially a uniform strategy for initializing Gaussian primitives. Below, we explain the rationale for this choice and its effectiveness.\n\n**Why Uniform Initial Distribution.** The uniform initial distribution is selected to ensure comprehensive coverage of the entire 3D scene from the outset, which accelerates model convergence during training. In GSRF, the scene is represented by 3D Gaussians that model RF propagation paths. Starting with Gaussians distributed uniformly across the bounding scene volume (encompassing the transmitter, receiver, and environment) provides a structured, balanced initial representation. This avoids the pitfalls of sparse or uneven starting points, allowing the densification and pruning process to refine the model more efficiently. Compared to alternatives like random initialization, this strategy reduces early imbalances and speeds up convergence toward stable, high-fidelity synthesis. While gradient-based updating allows random initialization to achieve comparable spectrum synthesis fidelity, it requires longer training time: 0.59 hours for random initialization compared to 0.27 hours for uniform initial distribution for RFID spatial spectrum synthesis.\n\nThis approach is inspired by efficient scene representations in 3DGS but adapted for RF's volumetric, wave-like nature, where paths span the full space rather than just surfaces. It prioritizes coverage over randomness to handle RF's complex, multipath environments without relying on prior geometric data, e.g., no Structure-from-Motion (SFM) points as in visual 3DGS.\n\n**Effectiveness for Spherical Surfaces Like RF Signals.** RF signals are inherently spherical, as they are collected over a spherical region centered at the antenna, with propagation effects radiating omnidirectionally. Despite the initial distribution's volumetric structure, it performs well by fully enclosing the spherical domain, ensuring Gaussians are initialized to capture radial paths without gaps. This initialization is only for the scene representation process, not related to the rendering process. The orthographic splatting and spherical rendering handle projection onto spherical coordinates during training and inference. The Gaussian shapes are then adapted by the adaptive density control (densification and pruning), which refines ellipsoid orientations and positions to fit the spherical, multipath nature of RF data effectively. This ensures robust performance without geometry-specific biases.\n\n\n### **Q4 – Why GSRF (Ours) Is Faster Than WRF-GS [28]**\nThank you for your question on the training and inference time comparison between GSRF and WRF-GS. While GSRF indeed benefits from custom CUDA kernels for acceleration, the speed gains over WRF-GS stem from a combination of architectural differences and optimizations tailored to RF synthesis. Below, we elaborate on the key factors.\n\nWRF-GS employs two NeRF-like MLPs to regress Gaussian attributes, using the Gaussian centers and transmitter coordinates as inputs. This MLP-centric design inherits NeRF's computational bottlenecks:\n\n(i) Dense querying of the MLPs for attribute prediction introduces inefficiency in both training, due to high parameter count and redundancy, and inference, as each Gaussian requires MLP evaluations.\n\n(ii) During optimization, backpropagation through the MLPs scales poorly with scene complexity, as gradients must flow through deep networks for every Gaussian update, leading to longer convergence times.\n\n(iii) Inference in WRF-GS involves re-evaluating MLPs per query (e.g., for novel transmitter positions), adding latency that increases with the number of Gaussians or rays.\n\nIn contrast, GSRF eliminates MLPs entirely by directly optimizing per-Gaussian attributes as learnable parameters. This explicit representation enables direct gradient updates via simple operations like complex-valued blending in the rendering equation, reducing both parameter overhead and computation per iteration."}, {"review_id": "bpApyuMl8v", "reviewer": "Reviewer_v8Xu", "summary": "The paper proposes a novel framework that extends 3D Gaussian Splatting (3DGS) to the domain of radio-frequency (RF) signal synthesis. Incorporating complex-valued 3D Gaussians, orthographic splatting, and an RF-customized CUDA-based complex ray tracing algorithm, the proposed model seems successfully develop RF-specific 3DGS, leading to efficient RF data synthesis. The paper addresses the limitations of previous NeRF-based models, particularly in terms of inefficiency in training and phase modeling deficiencies. The model is validated across diverse RF technologies (RFID, BLE, 5G, LoRa), and demonstrates superior synthesis accuracy and performance efficiency.", "questions": "1. Why is a one-degree angular resolution assumed, despite it being predefined by the number and configuration of Tx/Rx antennas in typical RF systems?\n2. How does the model capture and validate the influence of multipath effects? Is it dependent on the explicit ray tracing process utilized in the model?\n3. What mechanisms are in place to simulate or incorporate antenna beam patterns, including side lobes and directional attenuation, which are significant in RF synthesis accuracy?\n4. Given the comparable performance to NeRF on the 5G CSI dataset as discussed in weaknesses of the paper, results can suggest inefficiencies in phase estimation, or limitations of complex rendering in capturing RF-specific nuances during simulation. Needs more discussion on it.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 4, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n- The method introduces a domain-specific adaptation of 3DGS to RF environments using complex-valued Gaussian primitives, integrating amplitude and phase characteristics through a Fourier-Legendre Expansion.\n- The orthographic splatting technique and wavefront-based CUDA kernel design enable high-resolution angular modeling and real-time synthesis capabilities.\n- Extensive benchmarks show that GSRF outperforms RF NeRF and other 3DGS-based models (e.g., WRF-GS) in both quality and efficiency across multiple RF modalities.\n- The model significantly reduces training data requirements and inference latency, making it well-suited for practical applications in real-time wireless sensing.\n\n**Weaknesses**\n- While the paper highlights GSRF’s high fidelity in train-test settings, it does not sufficiently discuss or empirically validate its potential for downstream semi-supervised learning tasks in RF tasks, which is a major motivation of RF data synthesis outlined in the introduction.\n- In the 5G CSI synthesis experiment, GSRF achieves similar performance to NeRF-based methods despite the use of complex rendering. This raises concerns about the practical impact of complex-valued modeling, i.e., the efficacy of phase estimation.\n- The assumption of one-degree angular resolution, which is not always feasible given predefined Tx/Rx configurations, is not clearly justified.\n- Important RF characteristics such as the modeling of multipath propagation, side-lobe effects, and antenna beam pattern-induced attenuation are not thoroughly analyzed or empirically validated.", "was_revised": true, "final_justification": "Based on the novelty of the work in the context of integrating 3DGS into RF modality as well as the authors' detailed feedback to resolve my previous issues (regarding downstream tasks and multipath effects), I believe the paper can be accepted to the conference.", "rebuttal": "## **To Reviewer v8Xu**\nThank you for your thoughtful assessment and for recognizing the strengths of our paper. Below, we provide responses to your questions and concerns.\n\n\n### **Q1 – Downstream Tasks**\nThank you for your comment on the need for further discussion and empirical validation of GSRF's potential in downstream RF tasks, as motivated in the introduction. While our primary focus is high-fidelity synthesis, we do evaluate GSRF’s practical impact through application-specific experiments that go beyond standard metrics. Below, we summarize key results from relevant sections to highlight these benefits.\n\n**Section 5.3: BLE-Based Localization.** We apply GSRF-synthesized BLE fingerprints (RSSI readings per location) to a downstream localization task using a k-NN regressor. The database is constructed entirely from GSRF-synthesized signals, instead of real fingerprints. During testing, each real-collected fingerprint is used to query the synthesized database, retrieving the *k*-nearest locations and averaging them to estimate the position. This demonstrates that GSRF-generated fingerprints can directly support indoor positioning tasks, especially when exhaustive real-world data collection is impractical. The low localization error confirms that GSRF outputs are usable and accurate, reducing reliance on dense site surveys.\n\n**Appendix F.3: LoRa Gateway Coverage Estimation.** We use GSRF to synthesize LoRa RSSI maps for gateway coverage estimation. The resulting predictions achieve near-zero bit error rate (BER) across tested areas. Visual comparisons (Figure 13) show strong alignment with ground truth, enabling informed decisions on gateway placement. This validates GSRF’s role in practical deployment planning. For example, in domains such as precision agriculture or smart cities, sparse measurements can be extended into complete, actionable coverage maps.\n\n**Appendix F.5: Practical Benefits.** We summarize broader utility such as fast retraining (~10 minutes) for adapting to environmental changes (e.g., furniture rearrangement), and effective data augmentation for downstream models. For instance, using GSRF-synthesized fingerprints reduces training time for localization models by up to 50% while maintaining comparable accuracy to those trained on physics-based simulators. This shows how GSRF supports dynamic RF environments, enabling efficient updates without full re-measurement.\n\n\n### **Q2 – Similar Performance of GSRF to NeRF² for CSI Synthesis**\nWe appreciate your observation regarding the practical impact of our complex-valued modeling. While GSRF achieves comparable SNR to NeRF² [11], this does not undermine the efficacy of phase estimation in GSRF; rather, it highlights how our approach enables phase-aware synthesis more efficiently.\n\nTo clarify, NeRF² [11] also models phase explicitly. It uses an MLP network to output complex-valued signals, where the MLP takes voxel coordinates and transmitter coordinates as input and regresses both amplitude and phase components of the RF field. This allows NeRF² to capture interference effects, but its volumetric querying along rays incurs high computational costs, limiting speed and scalability. As a result, it inherits the computation overhead typical of NeRF-based methods.\n\nIn contrast, GSRF's complex-valued rendering integrates phase directly into Gaussian primitives via a Fourier–Legendre basis expansion, enabling amplitude–phase interactions without MLP overhead. This results in effective phase estimation while achieving faster training and inference times. The efficiency gain stems from our direct optimization of Gaussian attributes and custom CUDA kernels, which avoid the redundant computations of NeRF²'s MLP approach.\n\n\n### **Q3 – One Degree Angular Resolution**\nThank you for your comment on the angular resolution assumption. We agree that clearer justification is needed, particularly regarding feasibility with predefined Tx/Rx configurations, where hardware limits (e.g., antenna array size) may dictate coarser sampling. The one-degree resolution is not a rigid assumption but is chosen to align with the collected data's inherent resolution and standard RF practices, balancing fidelity, coverage, and efficiency. Below, we elaborate based on antenna types, since this directly influences the effective resolution.\n\n**For Multi-Antenna Configurations.** Our method does not impose a fixed prediction resolution; instead, it adapts to the antenna array's capabilities and the training data's angular granularity. In multi-antenna setups, the resolution is determined by the array's spatial sampling theorem, where the angular resolution scales with the number of elements. For example, in a 4×4 uniform rectangular array used in our RFID dataset, the system supports approximately 1° resolution using algorithms like MUSIC (Multiple Signal Classification). We align our synthesis resolution with the dataset's acquisition: the RFID spatial spectrum is measured at 1° intervals over azimuth and elevation, so we adopt this setting to maintain consistency with training data and avoid interpolation artifacts.\n\nHowever, we acknowledge that predefined Tx/Rx configurations may have lower resolution due to fewer antenna elements. For instance, a 2×2 array might only achieve 2–3° resolution. GSRF can flexibly accommodate such settings by training and evaluating at the available resolution without modification, since the orthographic splatting and loss functions operate on arbitrary ray grids. This makes our method hardware-agnostic and adaptable.\n\n**For Single-Antenna Configurations.** In single-antenna cases, the received signal is a scalar (power) aggregated omnidirectionally, per antenna theory and has no inherent angular resolution. Here, we discretize the spherical rendering at 1° to ensure thorough coverage of propagation paths while maintaining computational efficiency. Finer bins (e.g., 0.5°) increase the ray count without significant fidelity gains, while coarser bins (e.g., 5°) may miss interference patterns. Our 1° choice aligns with conventions in RF Computer-Aided Design (CAD) scene model-based simulation tools such as Wireless InSite and the MATLAB Ray Tracing toolbox.\n\n\n### **Q4 – Multipath Effects**\nThank you for your comment on how GSRF captures and validates multipath effects, and its dependency on ray tracing. We'll address this step by step, grounding our explanation in the model's design and empirical approach.\n\n**Design.** GSRF is designed to capture multipath effects by representing the RF scene as a collection of complex-valued 3D Gaussians, where each Gaussian serves as a primitive that approximates a propagation path or interaction point. This design is grounded in RF physics: multipath requires modeling multiple signal paths, each introducing amplitude attenuation and phase shifts. GSRF handles this using complex-valued radiance and transmittance attributes, encoded via a Fourier–Legendre basis that models directional and frequency-sensitive variations.\n\nThe capture of multipath is inherently tied to our ray tracing process, which emits rays from the receiver across a spherical surface and aggregates the contributions of intersecting Gaussians. Transmittance encodes phase shifts and attenuation due to path length. Summing complex contributions allows for phase-based interference—constructive when the phase difference is near zero and destructive when near π. This process effectively discretizes the continuous wave propagation integral. Without ray tracing, these path-specific interactions would be lost, and aggregation would reduce to a naive amplitude blending, which is insufficient for RF modeling at centimeter-scale wavelengths.\n\n**Empirical Validation.** For validation, we rely on real-collected datasets, where explicit ground-truth for multipath effects is unavailable due to the aggregated nature of measurements. Instead, we implicitly validate through the synthesized RF data's overall quality: if multipath were not captured, interference patterns would be inaccurate, leading to poor fidelity metrics.\n\n\n### **Q5 – Antenna Beam Patterns**\nThank you for your comment on the mechanisms for incorporating antenna beam patterns, including side lobes and directional attenuation, in GSRF. As a data-driven method, GSRF learns these characteristics implicitly from the training data rather than modeling them via explicit physics-based rules, allowing flexibility across antenna types.\n\n**Incorporating Beam Patterns.** GSRF models directional antenna effects through the Fourier–Legendre basis for complex-valued radiance and transmittance. This basis encodes angular dependencies: Legendre polynomials handle polar variations in amplitude (e.g., main beam gain and off-axis attenuation), while Fourier components capture azimuthal phase shifts and side lobe structures. During rendering, these directional effects are aggregated via ray tracing on the spherical surface, where rays simulate beam orientations, and the orthographic splatting projects Gaussians to preserve pattern-induced modulations. If the training data reflects beam-specific effects, such as side lobes from directional antennas, the model’s optimization adapts the Gaussian attributes to capture them naturally.\n\n**Datasets.** In our datasets, the antennas used are omnidirectional or isotropic, and thus side lobes and directional attenuation are not prominent. As a result, GSRF does not explicitly exhibit such effects in results. However, ablations of the Fourier–Legendre basis show that directional encoding improves performance, confirming its role in capturing any present beam patterns. We anticipate that if directional antenna data (e.g., beamformed phased arrays) were used, GSRF would learn and express those patterns, as the framework is agnostic to antenna type and adapts to observed propagation."}, {"review_id": "p35kH7Dqg6", "reviewer": "Reviewer_w6X4", "summary": "This work presents a physically grounded extension of 3D Gaussian splatting (3DGS) for reconstructing RF signals, leveraging the Huygens-Fresnel principle to treat each Gaussian blob as a secondary RF source. The method introduces complex-valued 3D Gaussians with Fourier-Legendre radiance fields models to represent both amplitude and phase-dependent interactions. The approach achieves state-of-the-art accuracy across multiple RF synthesis tasks (e.g., spatial spectrum, CSI, RSSI) while offering significant gains in training and inference speed over NeRF-based baselines.", "questions": "- Why are there no comparisons to RF-3DGS? \n\n- Are the comparisons to WRF-GS or WRF-GS+, how does the proposed deformability in WRF-GS+ affect the results?\n\n- Given how \"blobby\" the reconstructed content is, I'm not sure that SSIM is a useful loss function here? Are there any ablations on its purpose / improvement?\n\n- Eq 10-12, the 2D recieving plane is modeled with equirectangular projection, is there a strong motivation here for this (as it will inevitably stretch points close to the poles)? Would something like a cube-map (see: Greene, Ned. \"Environment mapping and other applications of world projections.\") be useful here?", "limitations": "The limitations of static scene reconstruction are addressed in the work, I do not see any major potential negative societal impacts of the work.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 4, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths:**\nForemost I’d like to commend the writing and formatting of this work, as the sections are laid out in clear and logical order, with consistent mathematical notation following established conventions. Overall the work is easy to follow, with physically intuitive reasons given behind most model choices (e.g., Fourier-Legendre Expansion, orthographic splatting). It addresses a real and relevant signal modeling problem, as simulating dense, realistic RF signal distributions remains a highly non-trivial task. The qualitative and quantitative results are compelling, with improvements demonstrate over both NeRF2 and WRF-GS in reconstruction and training speed. \n\n**Weaknesses:**\nAs this is not the first work using radiance fields for RF modeling, nor the first work to use 3DGS for RF, the niche for novelty is limited to specifically the importance of phase information for reconstruction (as expressed via the presented complex ray tracing algorithm). Given this I’m somewhat surprised the comparisons focus more on NeRF2 than the prior 3DGS works (with no results for RF-3DGS? and no WRF-GS results in Figure 2,6,9,10). This does not inspire confidence that the proposed approach is a strict improvement over other 3DGS approaches.\n\nWhile the writing in the paper is clear, I can’t fully follow what is being illustrated in Figure 1. Expanding the figure description might help, as would incorporating mathematical notation from the work directly into the elements of the figure.", "was_revised": true, "final_justification": "In light of the additional comparisons and clarifying details provided in the rebuttal, I believe that this work should be accepted into the conference proceedings. It advanced the state-of-the-art in modeling radio-frequency signal distributions in a computationally tractable way, and provides clear avenues for follow-on domain-specific applications.", "rebuttal": "## **To Reviewer w6X4**\nThank you for your thoughtful assessment and for recognizing the strengths of our paper. Below, we provide responses to your questions and concerns.\n\n### **Q1 – Contribution**\nGSRF advances 3DGS into the RF domain via a unified, complex-valued modeling pipeline:\n\n**Scene Representation.** GSRF introduces a compact hybrid Fourier–Legendre Expansion (FLE) basis to jointly encode angular dependence and frequency-sensitive phase shifts. This representation preserves phase coherence, which is essential for capturing phase-driven constructive and destructive interference patterns modulating signal amplitude.\n\n**Rendering.** GSRF projects Gaussians onto a spherical domain using orthographic splatting and applies ray-tracing-based blending to integrate complex-valued radiance and transmittance along propagation paths. This design aligns with spherical RF antenna reception and models RF propagation through path integrals, with fully differentiable gradients derived for all parameters.\n\n**Optimization.** GSRF optimizes Gaussian parameters using RF-customized CUDA kernels for both forward and backward passes. Explicit gradient derivations ensure efficient updates through phase-aware components. A Fourier-domain loss is applied to preserve frequency-domain fidelity.\n\n### **Q2 – Comparison to RF-3DGS[27]**\nRF-3DGS adopts a two-stage pipeline: first, an optical 3DGS is trained on co-located visual data (e.g., RGB images or LiDAR) to estimate Gaussian means, covariances, and opacities. These parameters are then fixed, and RF attributes (e.g., path loss) are learned using RF measurements, and uses spherical harmonics (SH) to encode directional information. Rendering relies on perspective projection and rasterization inherited from camera-based NeRFs, assuming camera-like intrinsics and extrinsics.\n\nIn contrast, in GSRF: (i) learns all Gaussian attributes are learned from RF measurements via optimization and adaptive density control; (ii) handles directional encoding is handled by the FLE, which preserves amplitude–phase coupling effectively; (iii) performs rendering is performed through orthographic splatting and ray-tracing-based blending aligned with spherical RF reception, enabling fully differentiable propagation paths.\n\n**Exclusion of Comparison.** RF-3DGS is not included in our evaluations due to its reliance on co-located visual inputs. This makes it incompatible with our RF-only benchmarks, which lack RGB or LiDAR data, and renders it infeasible for real-world scenarios such as indoor sensing, 5G CSI prediction, or IoT deployments. Its dependence on visual access limits its practicality as a general-purpose RF modeling solution.\n\n### **Q3 – Comparison to WRF-GS[28]**\nWRF-GS employs two NeRF-like MLPs to regress Gaussian attributes—the radiated signal and attenuation—using the Gaussian centers and transmitter coordinates as inputs. This MLP-centric design inherits NeRF's computational bottlenecks: \n\n(i) Dense querying of the MLPs for attribute prediction introduces inefficiency in both training, due to high parameter count and redundancy, and inference, as each Gaussian requires MLP evaluations.\n\n(ii) WRF-GS handles radiated signals of each Gaussian by separating them into real and imaginary components. While this approach maintains complex operations, it may not capture angular dependencies and frequency-sensitive phase shifts as directly, potentially limiting modeling of phase-driven interference that modulates RF signal amplitude.\n\nIn contrast, GSRF represents complex-valued Gaussian primitives using the FLE, which jointly encodes directional dependencies and frequency-sensitive phase shifts in a compact, orthogonal representation. This structure preserves amplitude–phase coupling without the need for decomposition. GSRF optimizes these primitives end-to-end using RF-customized forward and backward CUDA kernels, enabling efficient gradient flow through the entire pipeline and avoiding extensive per-Gaussian MLP querying for attribute learning.\n\n### **Q4 – Contemporaneous Work: WRF-GS+**\nWRF-GS+ is an extension of WRF-GS, released on arXiv after the NeurIPS 2025 contemporaneous work definition date (March 1st, 2025). It improves upon WRF-GS by introducing deformable Gaussians that decouple static components (such as path loss) from dynamic components (such as multipath), using learned offsets in signal strength, rotation, and scaling. This design enhances synthesis quality by adaptively reshaping Gaussians to better capture high-frequency multipath effects. It also reduces parameter redundancy and mitigates the inefficiencies of WRF-GS’s MLP-based attribute learning.\n\nWe acknowledge the strength of this design, which offers a principled mechanism for modeling environmental variation without changing the core architecture. GSRF could potentially incorporate these deformable mechanisms in future work.\n\nHowever, the two approaches remain methodologically distinct. GSRF emphasizes a unified, complex-valued RF modeling pipeline. This results in a fully differentiable, MLP-free framework tailored for RF propagation modeling.\n\n### **Q5 – Figures 2, 6, 9, 10 for WRF-GS**\n**Figure 2 (Qualitative Spectra Synthesis).** We present only NeRF² as a visual baseline in the manuscript. However, we have uploaded extensive qualitative results for WRF-GS, NeRF², and GSRF in the supplementary materials. We will include WRF-GS in Figure 2.\n\n**Figure 6 (Measurement Density).** We extend the measurement density analysis to include WRF-GS. WRF-GS and GSRF, trained with 0.8 measurements/ft³, achieve MSEs of 0.002659 ± 0.003560 and 0.002147 ± 0.003343, respectively. Both are comparable to NeRF²’s 0.002405 ± 0.003623, despite being trained with a higher density of 7.8 measurements/ft³. This benefit stems from WRF-GS’s use of 3DGS’s explicit representation, where Gaussian primitives offer greater representational power and flexibility, improving efficiency over NeRF-based volumetric sampling.\n\n**Figures 9 and 10 (BLE RSSI Prediction and Localization).** We add WRF-GS results to both tasks using the full training set. GSRF reduces RSSI error by 3.92% compared to WRF-GS due to its unified RF modeling. Localization errors remain similar across models due to the resilience of KNN. KNN selects the *k* nearest neighbors in the RSSI fingerprints and averages their positions. This averaging acts as a low-pass filter, mitigating the effects of synthesis noise.\n||NeRF²|GSRF|WRF-GS|\n|-|-|-|-|\n|RSSI error (dBm)|6.091±5.427|4.094±3.908|4.261±3.943|\n|Localization error (m)|0.699±0.804|0.479±0.692|0.481±0.685|\n\n### **Q6 – SSIM Loss**\nWe agree that the inherent blob-like primitives in 3DGS can lead to smoother, less pixel-sharp outputs. However, SSIM remains valuable as it emphasizes structural and perceptual similarity across spatial patterns, rather than focusing solely on the pixel-wise errors emphasized by L1.\n\nIn RF synthesis, where outputs like spatial spectrum are treated as image-like data (e.g., directional signal power), SSIM helps preserve meaningful directional spatial spectrum structures. This aligns with standard practice in 3DGS work (e.g., 3DGS and WRF-GS), which incorporate SSIM in their losses to enhance perceptual quality.\n\nOur new ablation experiments show that removing SSIM (relying on L1 plus Fourier loss) degrades RFID spatial spectrum synthesis PSNR by 0.73 dB (from 22.64 to 21.91 dB), confirming that SSIM refines structural details.\n\n### **Q7 – Equirectangular Projection and Cube-Map**\nWe appreciate the suggestion of cube-maps and agree that equirectangular projection introduces stretching near the poles, a known artifact in spherical mappings. However, our choice is motivated by several key factors:\n\n**Simplicity and Compatibility.** RF spatial spectra from antenna arrays are naturally parameterized by azimuth and elevation, which aligns with the latitude–longitude grid of equirectangular projection. This enables uniform angular sampling without additional remapping.\n\n**Limited Impact of Polar Distortion.** In practical RF antenna systems, high elevations (greater than 60°–70°) correspond to paths directed toward ceilings, floors, or the sky, where signals typically experience the following:\n\n*(i) Higher attenuation.* Materials like concrete and soil absorb or reflect signals with significant loss. Skyward paths (e.g., satellite) are line-of-sight but rare in indoor or occluded outdoor environments.\n\n*(ii) Fewer multipath components (MPCs).* Most useful MPCs arise from reflections and scattering at mid-elevations (10°–50°), typically from walls and objects at human height.\n\nWe did consider cube-map projections, but we ultimately opted against them for two main reasons:\n\n**Seam Artifacts and Gradient Discontinuities.** Cube-maps split the sphere into six faces, introducing seams that disrupt gradient continuity. In our differentiable rendering pipeline, these seams can cause instability in backpropagation.\n\n**Non-uniform Sampling Misalignment.** Cube-maps exhibit higher sampling density at face centers and lower at edges, which conflicts with the uniform angular resolution used in RF datasets (e.g., spatial spectrum).\n### **Q8 – Figure 1**\nThe confusion mainly arises because the figure lacks an explicit workflow and does not follow a typical left-to-right layout. We will improve this in the revision.\n\nThe scene is represented as a collection of Gaussian primitives. Each primitive has a mean position μ, covariance Σ, and complex-valued radiance ψ and transmittance ρ.\n\nThe set of Gaussians evolves during training. The model uses gradient-based optimization to update their attributes and adapt density based on the loss computed between predictions and ground truth.\n\nOn the right, rays γ are emitted from the receiver and traced through the scene. Gaussians are splatted onto a 2D angular grid using orthographic projection, and the received complex-valued signal is obtained by aggregating contributions along each ray."}]

Author rebuttal:

Meta-review: The paper introduces GSRF, a complex-valued extension of 3D Gaussian Splatting for RF signal synthesis. Reviewers broadly praised the novelty of extending 3DGS to the RF domain, the contributions (Fourier-Legendre basis, orthographic splatting, CUDA kernels for complex ray tracing), and the strong empirical results across diverse RF tasks (RFID, BLE, LoRa, 5G). The clarity of presentation, reproducibility via a public repository, and detailed ablations were also noted as strong points.

Questions on Fourier–Legendre vs. spherical harmonics, cube-based initialization, angular resolution assumptions, multipath handling, and beam patterns were addressed thoroughly in rebuttal. Reviewers generally found these explanations satisfactory.

After rebuttal and discussion, all reviewers confirmed their positive recommendations, leading to consensus for acceptance.
Overall, reviewers agreed that the work is technically solid, addresses a significant problem, and advances the state of the art in RF signal modeling with a clear, efficient, and reproducible approach. The AC concurs with the consensus and recommends acceptance.
Decision: Accept (spotlight)