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

TL;DR

The paper addresses non-uniformly sampled, non-stationary signals by introducing the non-uniform S-transform (NUST), which applies a density-adaptive GLS within sliding time windows. It achieves density-aware time-frequency representations by estimating local sample density with kernel density estimation and adjusting the Gaussian window width σ_adaptive(τ,f) accordingly, then computing a localized GLS power p(τ,f) to populate the spectrogram S_NUST(τ,f). On synthetic benchmarks, NUST outperforms the time-integrated GLS and tracks ground-truth time-frequency structure with high fidelity; on HARPS RV data for HD 10180, it separates persistent planetary signals from time-variable stellar activity. The proposed approach offers a practical, robust tool for non-uniform time series analysis with improved localization and interpretability for astronomical and other real-world data.

Abstract

The analysis of non-stationary signals in non-uniformly sampled data is a challenging task. Time-integrated methods, such as the generalised Lomb-Scargle (GLS) periodogram, provide a robust statistical assessment of persistent periodicities but are insensitive to transient events. Conversely, existing time-frequency methods often rely on fixed-duration windows or interpolation, which can be suboptimal for non-uniform data. We introduce the non-uniform Stockwell-transform (NUST), a time-frequency framework that applies a localized density adaptive spectral analysis directly to non-uniformly sampled data. NUST employs a doubly adaptive window that adjusts its width based on both frequency and local data density, providing detailed time-frequency information for both transient and persistent signals. We validate the NUST on numerous non-uniformly sampled synthetic signals, demonstrating its superior time-localization performance compared to GLS. Furthermore, we apply NUST to HARPS radial velocity data of the multi-planetary system HD 10180, successfully distinguishing coherent planetary signals from stellar activity.

Figures (6)

• Figure 1: Adaptive window weights at two time instances, $\tau=30$ (solid) at dense and $\tau=110$ (dash) sparse regions, marked by dotted vertical lines. Panel (A) uses $h = 20$, $\alpha = 0.18$ with varying $\gamma$, while Panel (B) uses $h = 20$, $\gamma = 0.5$ with varying $\alpha$.
• Figure 2: Illustration of the four test signals used in our study. The continuous signal is in blue, and the non-uniformly sampled points are marked in black. (A) Multi-transient signal, (B) Transient chirp, (C) Transient burst, and (D) Central gap signal.
• Figure 3: (A) The S-transform spectrograms for the 4 evenly sampled signals. (B) GLS periodograms of the corresponding non-uniformly sampled signals, with dotted red lines indicating the maximum power recovered. (C) NUST spectrograms for the same. The colour bar represents the normalised power.
• Figure 4: NUST spectrograms of a transient frequency jump signal obtained over multiple runs with random non-uniform sampling, while the parameters $h$, $\alpha$, and $\gamma$ are kept constant. The colour bar represents the normalised power.
• Figure 5: NUST spectrograms of Signal 1 under different noise conditions with SNR values of $8$, $2.5$ and $0.8$. White boxes indicate the approximate time window in which the signal is expected. The colour bar represents the normalised power.