Time-Frequency Analysis of Non-Uniformly Sampled Signals via Sample Density Adaptation

Technical Summary: Time-Frequency Analysis of Non-Uniformly Sampled Signals via Sample Density Adaptation

Problem Statement
The analysis of non-stationary signals within non-uniformly sampled data presents a significant challenge in signal processing. Real-world datasets from fields such as astronomy, geophysics, and biomedicine often exhibit sparse sampling, gaps, or irregular time intervals ($t_{i+1} - t_i \neq \text{constant}$). Existing approaches face distinct limitations:

1. Interpolation: Converting non-uniform data to a uniform grid to apply standard techniques (e.g., Short-Time Fourier Transform, standard S-transform) can introduce significant artifacts.
2. Time-Integrated Methods: Techniques like the Generalized Lomb-Scargle (GLS) periodogram offer robust statistical assessment of persistent periodicities in non-uniform data but lack temporal resolution, rendering them blind to transient events or frequency drifts.
3. Fixed-Window Time-Frequency Methods: Existing time-frequency approaches applied to non-uniform data often rely on fixed-duration windows or ignore local sample density. This leads to unreliable estimates, particularly in sparsely sampled regions where a fixed narrow window may contain insufficient data points for a stable spectral estimate.

Methodology: The Non-Uniform S-Transform (NUST)
The authors introduce the Non-Uniform S-Transform (NUST), a time-frequency framework designed to synthesize the statistical robustness of the GLS periodogram with the adaptive multi-resolution principles of the S-transform. The core innovation is a doubly adaptive windowing scheme that operates directly on non-uniformly sampled data without interpolation.

The NUST methodology consists of two primary components:

1. Sample Density Estimation:
   To quantify local data density, the method employs Kernel Density Estimation (KDE). A continuous density function $\rho(t)$ is estimated from the discrete time samples $\{t_i\}$ using a Gaussian kernel and a bandwidth hyperparameter $h$. This allows the algorithm to identify regions of sparse versus dense sampling.

2. Doubly Adaptive Windowing and Localized GLS:
   Unlike the standard S-transform, where the Gaussian window width $\sigma$ scales only with frequency ($\sigma \propto 1/|f|$), the NUST defines an adaptive standard deviation $\sigma_{\text{adaptive}}(\tau, f)$ that depends on both frequency and local sample density:
   $$\sigma_{\text{adaptive}}(\tau, f) = \frac{\alpha}{|f| \cdot [\rho(\tau)]^\gamma}$$
   Here, $\alpha$ sets the baseline width, and $\gamma$ controls the sensitivity to density changes.

   • Frequency Adaptation: High frequencies utilize narrower windows for better temporal resolution; low frequencies utilize wider windows for better frequency resolution.
   • Density Adaptation: In sparse regions, the window expands to gather sufficient data points for a stable fit. In dense regions, it contracts to provide finer temporal detail.

   The spectral power at a specific time-frequency point $(\tau, f)$ is computed by applying a localized Generalized Lomb-Scargle (GLS) analysis. A sinusoidal model $y(t) = a \cos(2\pi ft) + b \sin(2\pi ft) + c$ is fitted to the data within the adaptive window. The fitting process uses combined weights $W_i$ that account for both the intrinsic measurement uncertainty ($1/\sigma_i^2$) and the adaptive Gaussian window value at each sample point. The output is the normalized power $p(\tau, f)$, representing the goodness-of-fit relative to a constant-offset model.

Key Contributions

• Framework Development: The formulation of NUST, which replaces the Fourier kernel of the S-transform with a localized GLS analysis, enabling direct processing of non-uniform data.
• Adaptive Mechanism: The introduction of a window width that dynamically adjusts based on local sample density, addressing the instability of spectral estimates in sparsely sampled regions.
• Validation: Comprehensive testing on synthetic signals and real-world astronomical data.

Experimental Results
The authors validated NUST against the GLS periodogram and the standard S-transform (applied to interpolated data as a ground truth) using four synthetic signal types: multi-transient signals, transient chirps, transient bursts, and signals with central gaps.

• Synthetic Data: NUST successfully localized transient and non-stationary features in both time and frequency, closely matching the ground-truth S-transform spectrograms. In contrast, the GLS periodogram produced ambiguous peaks by averaging power over the entire duration, failing to capture transient behavior.
• Robustness: Tests with random non-uniform sampling realizations and varying Signal-to-Noise Ratios (SNR) demonstrated that NUST consistently recovers time-frequency structures. Detection limits were observed at low SNR (0.8), where low-frequency components began to merge with the noise floor.
• Real-World Application (HD 10180): Applied to HARPS radial velocity data of the multi-planetary system HD 10180, NUST successfully distinguished coherent planetary signals from stellar activity.
  • For the high-frequency planet C, NUST revealed a stable horizontal track, resolving ambiguity caused by nearby noisy peaks in the GLS periodogram.
  • For the low-frequency planet D, NUST visualized the signal's temporal behavior, showing it as less consistent and intermittently weaker than planet C. The authors note that while wide windows are necessary for low-frequency resolution (approaching GLS behavior), the method still provides valuable diagnostic context for separating planetary signals from transient stellar noise.

Significance and Claims
The paper claims that NUST provides a powerful diagnostic tool for non-stationary signal analysis in non-uniform datasets by adding the temporal context absent in standard periodograms. It asserts that the method overcomes the limitations of time-integrated methods (blindness to transients) and fixed-window methods (instability in sparse regions). The authors position NUST as a superior alternative for characterizing transient, localized, or frequency-varying signals in fields like astronomy, where data gaps and irregular sampling are prevalent. The work concludes by suggesting future generalization to include wavelets for non-uniform data processing.