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Untangling the Geometry and Speed for RF Sensing Spectrograms

Mert Torun, Darius Cuenca, Yasamin Mostofi

arXiv:2609.26960v1eess.SPcs.LG

TL;DR

Single-link Doppler observations mix reflector speed with sensing geometry, leaving their physical components ambiguous. The paper combines a compact spectrogram representation, a physics-informed autoencoder, and synthetic-to-real training to estimate them jointly. Across synthetic tests and 31 real WiFi experiments, the framework recovers speed, geometry, and ridge properties without real WiFi training data.

  • Problem

    A single-link Doppler observation entangles reflector speed and geometry, while practical spectrogram smearing further complicates their extraction.

    Source: B. The Product Ambiguity · pp. 3–4

  • Method

    The paper combines a compact parametric spectrogram representation with a physics-informed autoencoder and synthetic-to-real training to estimate physical ridge parameters.

    Source: §I · p. 2, B. Differentiable RF Forward Model · p. 7

  • Results

    Across synthetic tests and 31 real WiFi experiments, the framework robustly recovers speed, geometry, ridge amplitudes, and widths without real WiFi measurements for training.

    Source: §VII · p. 14

  • Takeaways & Limitations

    The framework provides a way to factor single-link Doppler observations into reflector speed and geometry while also estimating ridge amplitudes and widths.

    Source: §VII · pp. 13–14

Abstract

from arXiv · show

A fundamental challenge in RF sensing is that Doppler signatures observed by a link entangle the target's motion with the sensing geometry, resulting in limited applicability to unconstrained real-world settings. In this paper, we establish a new foundation for physically interpretable RF sensing that disentangles reflector speed from geometry, jointly recovering the speed, geometry factor, relative amplitude, and width of each dominant Doppler ridge. More specifically, we first develop a compact parametric representation of WiFi spectrograms and establish its low-dimensional structure through a systematic computer-vision analysis of a large and diverse human-activity dataset, thereby providing a tractable foundation for learning. Building on this representation, we then design a physics-informed autoencoder whose structured bottleneck and differentiable RF forward model enforce physically meaningful estimates of reflector speed and geometry. We further introduce a synthetic-to-real training framework, eliminating the need for real WiFi training data. We extensively validate the proposed framework under both known and time-varying geometries, using both independently generated synthetic test sets and 31 real WiFi experiments. The results demonstrate the superior performance in speed and geometry extraction, robustly recovering the underlying geometry, speeds, Doppler-ridge amplitudes, and ridge widths across all settings, while substantially outperforming the strongest baselines.

I. INTRODUCTION

Single-link Doppler signatures entangle reflector speed with sensing geometry, limiting systems that assume fixed paths or known geometry. The paper proposes a physically constrained approach to jointly estimate speed, geometry, and ridge properties, trained synthetically and evaluated on synthetic and real WiFi data.

I. INTRODUCTION

  • A single WiFi link’s Doppler signatures entangle target motion with geometry, while fixed-direction or known-geometry assumptions constrain use in unrestricted settings.The paper frames its problem as recovering reflector profiles while simultaneously estimating the geometry factor from one WiFi spectrogram.Source: §I · p. 1
  • A compact spectrogram representation models dominant motion ridges using each reflector’s speed, geometry factor, amplitude, and smear, with its low-dimensional structure established through human-activity analysis.This representation is presented as a tractable foundation for the subsequent learning design.Source: §I · p. 2
  • The proposed physics-informed autoencoder jointly estimates reflector speeds and geometry through a structured bottleneck and a fixed differentiable RF model that reconstructs spectrograms.Speeds are parameterized with Fourier coefficients, while a handful of parameters determine geometry.Source: §I · p. 2
  • Synthetic-to-real training supplies complete labels and noisy-clean spectrogram pairs, allowing the model and hyperparameters to be learned without real WiFi measurements.The framework is also designed to teach the model to work with and denoise real WiFi spectrograms.Source: §I · p. 2
  • The pipeline robustly estimates speed profiles, geometry, ridge amplitudes, and widths across 31 real WiFi experiments and synthetic tests, without real WiFi training or hyperparameter selection.For varying geometry, speed NMSE is 0.027 on the synthetic test set and 0.166 on the real test set, reducing strongest-baseline NMSE by 86.3% and 41.8%, respectively.Source: §I · p. 2

II. PROBLEM FORMULATION

  • Observed ridge frequencies depend on the product of reflector speed and an instantaneous geometry factor, and limited bandwidth and noise can further obscure their separation.Similar Doppler frequencies can arise from fast motion under weak geometric projection or slower motion under stronger projection.Source: §II · p. 2

A. Doppler Signatures in WiFi Spectrograms – A Primer

The primer models the received WiFi signal as static and moving-reflector contributions, then relates reflector motion to the Doppler signatures observed in spectrograms. The spectrogram convention considered retains nonnegative frequencies, so Doppler sign is not preserved.

A. Doppler Signatures in WiFi Spectrograms – A Primer

  • The received narrowband complex baseband signal is modeled as a sum of a static component and reflections from moving reflectors.The static contribution includes the direct path and reflections from static objects; each moving reflector has an effective signal amplitude and path length.Source: A. Doppler Signatures in WiFi… · p. 3
  • Reflector motion changes the reflected path length and phase, and Doppler frequency follows from the rate of phase change.The primer derives the rate of path-length change before defining signed instantaneous Doppler frequency.Source: A. Doppler Signatures in WiFi… · p. 3
  • For real-valued WiFi motion signals derived from CSI power or receiver phase difference, the two-sided spectrum is conjugate symmetric and only nonnegative frequencies are considered, so Doppler sign is not preserved.The spectrogram representation therefore uses observed Doppler magnitude.Source: A. Doppler Signatures in WiFi… · p. 3
  • The geometry factor is defined as nonnegative, and the STFT concentrates a reflector’s energy near its Doppler frequency when that frequency varies sufficiently slowly within a window.These conditions connect the physical motion model to the ridge representation in the spectrogram.Source: A. Doppler Signatures in WiFi… · p. 3

B. The Product Ambiguity

A single ridge observation admits infinitely many speed–geometry pairs, making true speed unidentifiable without additional information or assumptions. The paper motivates a compact parametric spectrogram model and validates the low-dimensional structure of human-activity data as a tractable basis for learning.

B. The Product Ambiguity

  • Infinitely many speed–geometry pairs can explain the same ridge observation, so true speed cannot be recovered from a single component without assumptions or additional measurements.The ambiguity holds for a noiseless single-component spectrogram; practical bandwidth and sampling limits and extended-reflector contributions can further smear signatures.Source: B. The Product Ambiguity · p. 3
  • Limited bandwidth, sampling effects, and nonidentical Doppler contributions from points on an extended reflector smear signatures, complicating speed and geometry extraction.Real spectrogram ridges occupy frequency bands rather than appearing as infinitely narrow tracks.Source: B. The Product Ambiguity · p. 4

III. PARAMETRIC FORWARD MODELING FOR WIFI SPECTROGRAMS

  • The paper develops an efficient parametric WiFi-spectrogram model and tests whether human-motion redundancy supports a low-dimensional representation suitable for learning.The section describes this validation as establishing a tractable foundation for the subsequent machine-learning design.Source: §III · p. 4

A. Ridge Modeling of WiFi Spectrograms

The section represents WiFi spectrograms as a small superposition of parameterized motion ridges, then supports that compact representation with human-motion correlations and real spectrogram examples.

A. Ridge Modeling of WiFi Spectrograms

  • Each dominant motion component is a localized frequency-band kernel whose center depends on reflector speed and geometry, while amplitude and width represent reflection strength and Doppler smearing.The summed-ridge model is an approximation: exact spectrograms may also contain interference and cross-reflection terms.Source: A. Ridge Modeling of WiFi Spectrograms · p. 4

B. Establishing a Tractable Low-Dimensional Representation

  • The representation is tractable because human motion has structural redundancy, with many moving points sharing correlated motion.The authors establish this low-dimensional structure through a computer-vision study of diverse human activities.Source: B. Establishing a Tractable… · p. 4
  • Within body parts, sampled-point speed correlations averaged 0.967, while geometry-factor correlations averaged 0.898 across activities.Geometry-factor similarity was weaker for the left leg, which the authors attribute to the right-side video view reducing HMR reliability for that less-visible leg.Source: B. Establishing a Tractable… · pp. 4–5
  • Real WiFi spectrograms showed no more than three visually distinct ridges, motivating the modeling assumption M ∈{0, 1, 2, 3}.The study analyzed eight activities using HMR-derived points from five main body parts; the figure caption describes most Doppler energy as concentrated around no more than three motion components.Source: B. Establishing a Tractable… · pp. 4–5, Figure 1 · p. 5

IV. PHYSICS-INFORMED AUTOENCODER FOR WIFI SPECTROGRAMS

The physics-informed autoencoder predicts a structured set of physical parameters from WiFi spectrograms and reconstructs them with a fixed differentiable RF model. Its bottleneck uses compact parameterizations for reflector speed and geometry alongside ridge amplitude and width.

IV. PHYSICS-INFORMED AUTOENCODER FOR WIFI SPECTROGRAMS

  • The autoencoder uses a structured physical bottleneck and a fixed differentiable RF forward model, with reconstruction loss coupling the estimated parameters to the target spectrogram.The framework also uses parameter-level supervision.Source: §IV · p. 5, Figure 2 · p. 6

A. Encoder Structure and Physical Bottleneck

  • A CNN encoder and prediction heads map the input spectrogram to bottleneck variables for speed, ridge amplitude, ridge width, and geometry factor.The encoder processes the spectrogram through convolutional blocks, adaptive pooling, and a fully connected layer before the prediction heads.Source: A. Encoder Structure and Physical… · p. 6
  • Reflector speeds are parameterized with truncated Fourier series, while four walking-path parameters generate a physically structured, time-varying geometry factor.These geometry parameters are initial position, heading angle, and mean translational speed; the primary reflector's relative amplitude is fixed at 1.Source: A. Encoder Structure and Physical… · p. 6

B. Differentiable RF Forward Model

The differentiable RF forward model maps the predicted physical bottleneck into spectrogram space without trainable parameters. It renders ridge locations from predicted speeds and geometry, and ridge strengths and smear from amplitudes and widths.

B. Differentiable RF Forward Model

  • A fixed, parameter-free differentiable RF forward model maps the predicted physical bottleneck to a spectrogram, so reconstruction loss jointly constrains the estimated parameters.Its analytical operations implement the spectrogram parameterization introduced earlier in the paper.Source: B. Differentiable RF Forward Model · p. 7
  • The geometry factor is generated from a walking-path model using predicted starting point, mean translational speed, and heading angle.The passage says the trajectory is calculated from these path quantities before the geometry factor is computed.Source: B. Differentiable RF Forward Model · p. 7
  • Predicted speeds and geometry determine ridge locations, while amplitudes and widths control relative strength and frequency smear.Differentiable rendering lets reconstruction gradients propagate through the ridge renderer, Doppler mapping, geometry model, and speed parameterization to the encoder and prediction heads.Source: B. Differentiable RF Forward Model · p. 7

C. Synthetic Training Data

The section builds synthetic WiFi spectrograms with known physical labels and uses them to train a model to recover ridge parameters from noisy inputs. Its training objective combines direct supervision with RF-model reconstruction and a penalty against persistent speed-component collapse.

C. Synthetic Training Data

  • Synthetic samples pair physical labels for speed, geometry, amplitude, and width with noisy encoder inputs and clean reconstruction targets.The generator draws parameters from predefined ranges, renders a clean spectrogram, then adds structured noise to form the input.Source: C. Synthetic Training Data · p. 7
  • The fixed-ψ training case sets ψ(t) = 2 and isolates recovery of component speeds from noisy, smeared, and potentially overlapping ridges.It samples reflector speeds and ridge amplitudes and widths.Source: C. Synthetic Training Data · p. 7
  • The varying-ψ case uses nine walking-path scenarios and models torso and leg motion, with torso-only samples included for cases where limbs may not be visible.The paths include centered and laterally offset horizontal, vertical, and diagonal motion.Source: C. Synthetic Training Data · p. 7

D. Loss Structure and Training

  • The reconstruction loss jointly constrains predicted bottleneck parameters by requiring the RF forward model to reproduce the clean target spectrogram.Only noisy spectrograms are fed to the network during training.Source: D. Loss Structure and Training · p. 8
  • Speed mismatch is weighted by ground-truth ridge amplitude, while direct ψ supervision teaches the geometry head to use its geometry-related bottleneck parameters.Weak or absent ridges therefore do not receive the same speed penalty as strong ridges.Source: D. Loss Structure and Training · p. 8
  • The model also directly supervises relative ridge amplitudes and widths, with amplitude-weighted smear loss scaled using a 10 Hz design parameter.The main reflector amplitude is fixed at 1; a zero predicted speed can still remove a component ridge.Source: D. Loss Structure and Training · p. 8
  • A coincidence loss discourages predicted speed components from collapsing together, averaging separation over one-second windows with 50% overlap so brief crossings remain possible.The penalty addresses a way the reconstruction objective could be exploited, especially when a secondary reflector is weak.Source: D. Loss Structure and Training · p. 9

V. EXPERIMENTAL SETUP

The experiments ask how well the pipeline recovers arbitrary reflector-speed trajectories with fixed geometry, jointly estimates speed and geometry during walking, and transfers from synthetic training to real WiFi measurements.

V. EXPERIMENTAL SETUP

  • With ψ fixed, the first question tests recovery of arbitrary reflector-speed trajectories under noisy, smeared, and partially overlapping ridges.For walking with time-varying ψ, the second question tests joint prediction of speed and ψ.Source: §V · p. 9
  • The study evaluates whether a model trained entirely on synthetic spectrograms performs on real WiFi measurements across motion types, subjects, and sensing conditions.This is the third stated experimental question.Source: §V · p. 9

A. Model Configurations and Hyperparameters

The paper configures separate fixed- and varying-geometry models, then generates paired synthetic training and validation spectrograms with physical labels and diverse motion, geometry, ridge, and noise conditions.

A. Model Configurations and Hyperparameters

  • Inputs use six-second, single-channel spectrograms on a 0–100 Hz grid with 0.1 Hz frequency spacing and 0.01 s time spacing.Real test spectrograms use the same grid and column-wise normalization.Source: A. Model Configurations and… · p. 9
  • FIXED-ψ models up to two speed components with K = 20, whereas VARYING-ψ models up to three components with K = 30 and jointly estimates ψ.The fixed model uses ψ(t) = 2; the varying model uses nine walking-path scenarios.Source: A. Model Configurations and… · p. 9
  • Hyperparameters and model configurations are selected using validation data, without using synthetic or real WiFi test datasets for selection or optimization.The final checkpoint is selected using validation-set speed-loss improvement.Source: A. Model Configurations and… · p. 10

B. Synthetic Training and Validation Sets

  • For both variants, synthetic training and validation examples pair noisy encoder inputs with clean targets and labels for speeds, ψ, relative amplitudes, and widths.The fixed-ψ generator samples ten speed-shape families and varied component configurations; the varying-ψ generator samples walking trajectories and torso-only or torso-plus-limb cases.Source: B. Synthetic Training and Validation… · p. 10
  • The varying-ψ training set uses nine uniformly sampled walking-geometry scenarios with a shared transmitter–receiver pair.The scenario table is identified as the source of further geometry details.Source: B. Synthetic Training and Validation… · p. 10, Table II · p. 10
  • Training and validation spectrograms include varied ridge widths and structured noise, with 15,000 training and 1,500 validation samples generated for each case.Noise strengths are independently sampled across three ranges.Source: B. Synthetic Training and Validation… · p. 10

C. Testing with Real WiFi

Real WiFi testing covers controlled hand motion and human walking under fixed and varying geometries. The paper compares against tailored ridge-extraction baselines and evaluates speed error using per-experiment metrics averaged across recordings.

C. Testing with Real WiFi

  • Real WiFi tests include accelerometer-labeled hand motion, video-labeled walking at ψ = 2, and walking under varying ψ; test data are excluded from training and hyperparameter selection.Long recordings are evaluated in overlapping six-second windows, with predictions averaged across windows.Source: C. Testing with Real WiFi · p. 10

D. Baselines for Performance Comparison

  • With no existing method for jointly estimating multiple reflector speeds and ψ(t), the authors develop three tailored non-learning ridge-extraction baseline families.They adapt peak extraction, power-weighted frequency clustering, and dynamic-programming ridge tracking for the two- and three-component settings.Source: D. Baselines for Performance Comparison · pp. 10–11
  • The baselines assume ψ = 2 when converting extracted frequencies to speed, which is correct for fixed-ψ tests but affects performance when ψ is unknown.The resulting degradation measures the consequence of treating unknown geometry as ψ = 2.Source: D. Baselines for Performance Comparison · p. 11

E. Evaluation Metrics

  • Mean absolute error in m/s is the primary speed metric, with RMSE and NMSE reported as secondary metrics that penalize larger errors more heavily.Metrics are first computed per experiment and then averaged across experiments; unordered components are matched once per experiment by permutation search.Source: E. Evaluation Metrics · p. 11

VI. EXPERIMENTAL RESULTS

The experiments assess speed extraction first under fixed geometry, then joint speed–geometry recovery under varying geometry, using synthetic tests and real WiFi. The varying-ψ synthetic tests also assess ridge amplitude and width estimation.

A. Fixed-ψ – Parameter Estimation on Synthetic Data

B. Fixed-ψ Speed Estimation with Real WiFi

C. Varying-ψ Speed and ψ Estimation on Synthetic Data

D. Joint Speed–Geometry Estimation with Real WiFi

Real walking experiments evaluate joint speed and geometry estimation, while a separate timing evaluation measures runtime. The paper concludes that the pipeline recovers speed, geometry, ridge amplitudes, and widths across its evaluated settings.

D. Joint Speed–Geometry Estimation with Real WiFi

E. Computation Time

  • The overall pipeline processed input 43.54 times faster than real time, requiring 22.96 ms per second of input.Inference took 4.79±1.4 ms per second of input.Source: E. Computation Time · p. 13

VII. CONCLUSIONS

  • Across evaluated settings, the pipeline recovered underlying geometry, speeds, Doppler-ridge amplitudes, and widths while substantially outperforming the strongest baselines.Validation included independently generated synthetic tests and 31 real WiFi experiments.Source: §VII · p. 14

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