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The First NeRF for Wireless Propagation

NeRF $^2$ (Zhao et al., 2023) was among the first works to adapt neural radiance fields for radio-frequency propagation modelling. Rather than rendering images, NeRF $^2$ predicts received signal strength (RSS) and channel state information (CSI) between arbitrary transmitter--receiver pairs by learning a volumetric representation of the RF environment. The key question: can a single neural scene function, trained on sparse measurements, predict RF propagation more accurately than ray tracing or empirical models?

Definition:
NeRF $^2$ Architecture

NeRF $^2$ represents the RF environment as two MLPs:

Geometry MLP $f_\sigma$ : maps position $\mathbf{x} \in \mathbb{R}^3$ (with positional encoding $\gamma(\mathbf{x})$ ) to volume density $\sigma(\mathbf{x}) \geq 0$ and an intermediate feature vector $\mathbf{h} \in \mathbb{R}^{256}$ .
Signal MLP $f_s$ : maps the feature vector $\mathbf{h}$ , ray direction $\mathbf{d}$ , and frequency $f$ to a complex signal contribution $s(\mathbf{x}, \mathbf{d}, f) \in \mathbb{C}$ .

The predicted received power between a transmitter at $\mathbf{x}_{\mathrm{tx}}$ and receiver at $\mathbf{x}_{\mathrm{rx}}$ is:

$\hat{P}_{\mathrm{rx}} = \Bigl|\sum_{i=1}^{N} T_i \alpha_i\, s_i\,e^{-j 2\pi f \|\mathbf{x}_i - \mathbf{x}_{\mathrm{tx}}\|/c}\Bigr|^2,$

with the sum taken over $N$ samples along the direct ray from $\mathbf{x}_{\mathrm{tx}}$ to $\mathbf{x}_{\mathrm{rx}}$ .

The architecture mirrors optical NeRF: the geometry MLP captures where material exists (walls, furniture), while the signal MLP captures how that material interacts with RF waves (reflection/absorption).

Theorem: NeRF $^2$ Training Objective

The training loss combines received power prediction error and a regularisation term:

$\mathcal{L}(\theta) = \frac{1}{|\mathcal{D}|}\sum_{(\mathbf{x}_{\mathrm{tx}}, \mathbf{x}_{\mathrm{rx}}) \in \mathcal{D}} \bigl|\hat{P}_{\mathrm{rx}}^{(\mathrm{dB})} - P_{\mathrm{rx}}^{(\mathrm{dB})}\bigr|^2 + \lambda \|\theta\|_2^2,$

where $\mathcal{D}$ is the set of measured transmitter--receiver pairs, $\hat{P}_{\mathrm{rx}}^{(\mathrm{dB})} = 10\log_{10}\hat{P}_{\mathrm{rx}}$ , and $\lambda$ controls weight decay.

The dB-domain loss is essential because received power spans orders of magnitude ( $\sim 100$ dB variation from path loss). Operating in dB normalises gradient magnitudes across near and far measurements, preventing nearby Tx--Rx pairs from dominating the training.

Proof

Why dB-domain loss

In linear domain, a measurement at 10 m ( $P \sim -50$ dBm) and one at 1 m ( $P \sim -20$ dBm) differ by a factor of $1000$ . The $L_2$ loss $|\hat{P} - P|^2$ in linear scale would be dominated by the near measurement. In dB, both contribute comparably to the gradient.

Regularisation role

The $\ell_2$ regulariser prevents overfitting when the measurement set $\mathcal{D}$ is sparse. RF datasets have $10^2$ -- $10^3$ measurements, versus $10^6$ -- $10^7$ pixels in optical NeRF. Without regularisation, the MLP memorises training points and fails on novel positions. $\square$

Definition:
Positional Encoding for RF Scenes

NeRF $^2$ uses the standard positional encoding

$\gamma(\mathbf{x}) = \bigl[\sin(2^0 \pi \mathbf{x}),\; \cos(2^0 \pi \mathbf{x}),\;\ldots,\; \sin(2^{L-1} \pi \mathbf{x}),\; \cos(2^{L-1} \pi \mathbf{x})\bigr]$

with $L = 10$ for position and $L = 4$ for direction. The maximum encoded frequency is $2^{L-1}\pi \approx 1608$ rad/m, corresponding to a spatial scale of $\sim 4$ mm.

For indoor RF scenes, the relevant spatial scales range from $\sim 1$ cm (wall boundaries) to $\sim 10$ m (room dimensions). $L = 10$ provides adequate bandwidth. Over-parameterising $L$ causes the network to fit measurement noise; under-parameterising produces over-smooth reconstructions that miss shadow boundaries.

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Example: NeRF $^2$ for Indoor RSS Prediction

An indoor office ( $20 \times 15$ m) has 4 Wi-Fi access points at known positions. A site survey collects RSS at 500 receiver locations. Train NeRF $^2$ and predict RSS at a new receiver location $\mathbf{x}_{\mathrm{rx}} = (8, 6, 1.5)$ m.

Solution

Data preparation

Training set: 500 receiver locations $\times$ 4 APs $= 2{,}000$ (Tx, Rx, RSS) triples. Positional encoding: $L = 10$ (position), $L = 4$ (direction). Normalise RSS to $[0, 1]$ range.

Training

MLP: 8 layers, 256 hidden units, ReLU activations. Optimiser: Adam, learning rate $5 \times 10^{-4}$ , batch size 256 rays. Train for 50,000 iterations ( $\sim 30$ min on a single GPU). Validation: hold out 100 measurements for early stopping.

Prediction

For each AP, cast a ray from $\mathbf{x}_{\mathrm{tx}}$ to $\mathbf{x}_{\mathrm{rx}}$ , sample 128 points, evaluate the MLPs, and render received power via the discrete volume rendering sum. Typical accuracy: 3--5 dB RMSE, which is competitive with site-specific ray tracing. $\square$

The Multipath Limitation

NeRF $^2$ integrates along the direct ray only. This captures line-of-sight attenuation and through-wall propagation but misses reflected and diffracted paths. In indoor environments, multipath contributes 30--50% of received power.

Why this matters: In a room with a concrete wall blocking the direct path, NeRF $^2$ predicts near-zero received power. In reality, reflections off the ceiling and floor may provide usable signal strength.

Extensions address this via:

Casting multiple rays (reflected rays via learned or geometric reflection models) --- see WiNeRT in Section 24.4;
Adding a learned multipath correction network;
Using the geometry MLP to identify reflective surfaces and trace secondary rays.

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Common Mistake: Overfitting with Sparse RF Measurements

Mistake:

Training a large NeRF $^2$ model (8 layers $\times$ 256 units $\approx$ 530k parameters) on a few hundred RSS measurements without regularisation, and expecting it to generalise.

Correction:

RF datasets are orders of magnitude smaller than optical datasets. Mitigation strategies:

Weight decay ( $\lambda = 10^{-4}$ -- $10^{-3}$ );
Smaller MLPs (4 layers $\times$ 128 units);
Hash encoding (Instant-NGP) with smaller tables ( $T = 2^{16}$ instead of $2^{19}$ );
Early stopping on a validation set;
Data augmentation: add noise to Tx/Rx positions.

RF NeRF Volume Rendering

Visualise the RF NeRF volume rendering process along a single ray. The top panel shows the volume density $\sigma(\mathbf{x})$ profile (walls appear as density peaks). The bottom panel shows the accumulated complex signal magnitude and the final rendered received power. Observe how increasing the number of samples improves accuracy near sharp density transitions (walls), and how changing the frequency shifts the interference pattern.

Parameters

Samples per Ray128

Scene Type

Frequency (GHz)5

🔧Engineering Note

NeRF $^2$ for Network Planning

NeRF $^2$ is most practical for offline network planning where a site survey collects RSS measurements, the model trains overnight, and the trained model predicts coverage maps for AP placement optimisation. Real-time channel prediction (e.g., for beam management) requires faster inference methods.

Practical numbers: Training on 2,000 measurements takes $\sim 30$ minutes. Inference for 10,000 grid points takes $\sim 10$ seconds (batched GPU evaluation). Accuracy: 3--5 dB RMSE, comparable to calibrated ray tracing but without requiring a detailed 3D building model.

Practical Constraints

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Requires known Tx/Rx positions (GPS or floor plan coordinates)
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Per-scene training: a new model is needed for each environment
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Single-frequency training; multi-frequency requires architecture changes

Quick Check

Why does NeRF $^2$ use a dB-domain loss function rather than a linear-domain MSE?

dB is more intuitive for engineers

To normalise gradients across measurements with widely varying power levels

To make the loss function convex

To enable closed-form optimisation

Correction:

To normalise gradients across measurements with widely varying power levels

Correct. Received power varies by orders of magnitude ( $\sim 100$ dB range). The dB-domain loss ensures that measurements at all distances contribute comparably to the gradient.

🎓CommIT Contribution(2023)

NeRF $^2$ for RF Propagation Modelling

Y. Zhao, X. Zhu, Y. C. Eldar — arXiv:2305.06118

Zhao et al. introduced the first neural radiance field for RF propagation modelling. By replacing the colour output with a complex signal MLP and training on sparse RSS measurements, NeRF $^2$ achieves 3--5 dB RSS prediction accuracy in indoor environments without requiring a detailed 3D building model. The key insight is that the volume density MLP implicitly learns the scene geometry (walls, furniture) from signal measurements alone. Limitations include single-ray integration (no multipath) and per-scene training.

nerfrf-propagationrss-predictionneural-fields

Key Takeaway

NeRF $^2$ learns a volumetric RF scene representation from sparse RSS measurements, predicting received power between arbitrary Tx--Rx pairs. The dB-domain loss normalises gradients across the large dynamic range of RF signals. The primary limitation is single-ray integration, which misses multipath contributions significant in indoor environments. Extensions addressing multipath, channel estimation, and material properties are surveyed in Section 24.4.

NeRF-Squared (Zhao et al.)