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ConceFT: Concentration of Frequency and Time via a multitapered synchrosqueezed transform
Ingrid Daubechies, Yi Wang, Hau-tieng Wu
TL;DR
The paper addresses how to recover time-varying amplitudes and instantaneous frequencies from signals containing multiple oscillatory components, especially under noise. It introduces ConceFT, which combines synchrosqueezing, multi-taper estimation, and averaging over random projections. Theoretical analysis and numerical results report improved concentration and estimation under challenging noise conditions, subject to signal regularity and localization constraints.
Problem
Time-varying amplitudes and instantaneous frequencies are difficult to determine from Fourier representations, although they are important for describing and predicting underlying dynamics.
Method
ConceFT combines STFT- or CWT-based synchrosqueezing with multi-taper estimation and averaging over random projections.
Results
ConceFT provides improved estimation of time-varying signal characteristics, with numerical results confirming its effectiveness under significant and challenging noise.
Takeaways & Limitations
ConceFT yields better-concentrated time-frequency representations and can suppress spurious noise-induced concentrations more strongly than the reported MTSST comparison.
Takeaways & Limitations
The approach requires concentrated reference functions and is constrained by the limited number of orthonormal functions that can remain concentrated in a time-frequency region; the analysis also assumes smooth, positive, bounded amplitudes and instantaneous frequencies varying slowly relative to the signal.
Abstract
from arXiv · showhide
A new method is proposed to determine the time-frequency content of time-dependent signals consisting of multiple oscillatory components, with time-varying amplitudes and instantaneous frequencies. Numerical experiments as well as a theoretical analysis are presented to assess its effectiveness.
1. Introduction
Time-frequency analysis seeks to characterize multiple oscillatory components with time-varying amplitudes and instantaneous frequencies, while separating noise and recovering individual components. ConceFT addresses low-SNR artifacts by combining synchrosqueezing with multi-tapering and random projections, yielding more concentrated representations in challenging examples.
- Motivation: Time-frequency analysis aims to estimate component number, strength, oscillation rate, and time-varying behavior while separating noise from signal.These estimates help describe and quantify underlying dynamics in applications spanning geophysics, biology, medicine, finance, and social dynamics.
- Existing approaches: STFT and related time-frequency methods localize signal portions in time before measuring their oscillatory behavior.The STFT uses a user-chosen window function, commonly a Gaussian with bandwidth σ.
- Existing approaches: The synchrosqueezing transform reassigns STFT or CWT coefficients only in frequency, preserving causality and enabling component reconstruction.SST can extract instantaneous frequency and reconstruct oscillatory components in noise, but low SNR produces spurious TF concentrations from overcomplete analyses.
- Existing approaches: Multi-tapering reduces noise artifacts by averaging representations from multiple orthonormal windows, but increasing the taper count also linearly increases TF smearing.Thus, artifact suppression requires balancing readability at higher noise levels against loss of concentration.
- ConceFT: ConceFT averages synchrosqueezed representations from many projections in the vector space generated by a multilayered STFT- or CWT-based representation.The paper studies its theoretical properties and numerical performance under challenging SNR conditions.
- ConceFT: In a challenging noisy example, ConceFT suppresses spurious bubbles more dramatically than MTSST while also removing onset and cessation artifacts.MTSST uses two orthonormal windows in the reported comparison, whereas ConceFT uses the same vector space of windows with many projections.
2. The ConceFT algorithm
ConceFT constructs a time-frequency representation by combining synchrosqueezed transforms from multiple concentrated reference functions and averaging over random linear combinations. This reduces noise-induced artifacts while preserving dominant oscillatory components, though the number of usable orthonormal functions is limited by time-frequency concentration.
- Synchrosqueezed transform: The synchrosqueezed transform shifts spread CWT coefficients back toward frequency locations using phase-derived reassignment rules.A threshold Γ controls which coefficients are reassigned and can be adjusted to reduce numerical error and noise influence.
- Noise suppression: Different reference wavelets produce similar dominant time-frequency components, while noise-induced artifacts occur at differing locations across representations.Averaging across the resulting views therefore suppresses artificial concentrations caused by fortuitous correlations between noise and the wavelets.
- Design constraint: The method requires reference functions that are concentrated in time and frequency so reassignment does not mix distinct signal components.The number of orthonormal functions that can be mostly concentrated in a given time-frequency region is limited by its area.
- ConceFT construction: ConceFT averages synchrosqueezed transforms generated from random unit-norm linear combinations of J orthonormal reference wavelets.The algorithm selects J concentrated wavelets, samples N random vectors, forms each linear combination, computes its SST, and averages the results.
- Practical use: In practice, ConceFT can use as few as 2 reference functions, with N chosen as large as desired.The resulting representation can also be interpreted as an estimated time-varying power spectrum of the signal.
3. Theoretical Results
Theoretical analysis models multicomponent signals with smooth, slowly varying instantaneous frequencies under additive Gaussian noise and analyzes ConceFT using randomized combinations of orthonormal wavelets. The results show that increasing the representation dimension reduces frequency-estimation deviations, while the detailed bounds rely on restrictive signal and wavelet assumptions.
- Signal and noise model: The signal model consists of multiple intrinsic-mode components whose amplitudes and instantaneous frequencies are positive, bounded, smooth, and slowly varying relative to the oscillations.The model allows components with separated instantaneous frequencies and forms an adaptive harmonic model space.
- Signal and noise model: The theoretical noise model adds Gaussian white noise with standard deviation 1 scaled by a positive noise level σ to signals in the adaptive model space.The analysis restricts attention to this additive noise setting, although broader noise models are discussed as possible extensions.
- ConceFT analysis: ConceFT constructs a larger time-frequency representation space from unit-norm linear combinations of an orthonormal wavelet family and averages synchrosqueezing transforms over random combinations.The analysis uses smooth, rapidly decaying wavelets with compactly supported Fourier transforms and orthonormality conditions.
- Error behavior: High-dimensional random projections reduce the relevant error terms because unrelated vectors are more likely to be nearly orthogonal as the dimension J increases.Other terms in the error expression also carry a factor J^-1 in the denominator.
- Error behavior: ConceFT yields sharper instantaneous-frequency estimates under noise, and even for modest J it reduces potential deviation between the time-varying power spectrum and instantaneous time-varying power spectrum.These conclusions are stated for the theoretical model and its associated bounds.
- Scope of the theory: The detailed estimates require restrictive conditions on the signals and wavelets, although the authors state that these conditions can be relaxed with more intricate estimates.Numerical examples reportedly show similar behavior in more complex situations and under more challenging noise models.
4. Numerical Experiments
The experiments evaluate ConceFT on simulated two-component signals with known time-varying amplitudes and instantaneous frequencies under Gaussian, ARMA(1,1), and Poisson noise. ConceFT achieves accurate time-frequency estimates, with performance improving over standard SST and multi-taper SST across tested conditions.
- Data simulation: The experiments use simulated two-component signals with explicit ground truth for time-varying amplitudes and instantaneous frequencies.Signals are sampled uniformly, and the analysis compares estimated time-varying power spectra with ideal spectra.
- Noise models: The noisy-signal tests add Gaussian white, ARMA(1,1), or Poisson noise, including experiments at 0 dB SNR.The ARMA model uses Student t4 innovations, producing time-dependent and potentially spiky noise.
- Parameter selection: N = 20 random projections were selected because the OT-distance decrease shows an elbow near 20 with small standard deviation.This setting was used in subsequent experiments for all three noise types.
- Noisy-signal results: The estimated time-varying power spectra remain highly accurate on a new signal not used to calibrate ConceFT parameters.The same qualitative accuracy is reported for the new example across the noisy conditions shown.
- Method comparison: ConceFT outperforms standard SST and multi-taper SST across all tested noise types and SNR conditions.The comparison uses OT-distance between the estimated and ideal time-varying power spectra over 20 noise realizations.
5. Conclusion
The paper concludes that ConceFT combines multi-taper estimation, random-projection averaging, and synchrosqueezing to improve time-frequency estimation for noisy multicomponent signals. It also introduces benchmark signals with known time-varying structure and an OT-distance for evaluating estimates against ground truth.
- Conclusion: ConceFT targets signals composed of a small number of intrinsic-mode functions with time-varying amplitudes and instantaneous frequencies.The method is intended to estimate these time-varying characteristics even under significant and challenging noise.
- Conclusion: The method combines multi-taper estimation, random-projection averaging, and synchrosqueezing for time-frequency representation.The approach can be based on STFT- or CWT-based synchrosqueezing.
- Conclusion: Theoretical analysis predicts improved estimation, while numerical experiments confirm the method’s effectiveness under significant noise.The conclusion reports this agreement between theory and experiments without restricting the claim to one noise model.
- Evaluation tools: The paper introduces explicit simulated signals with known time-varying characteristics and an OT-distance for comparison with ground-truth time-frequency representations.These tools are presented as applicable to ConceFT and other time-frequency methods.
Electronic Supplementary Materials for “ConceFT: Concentration of frequency and time via a multi-tapered synchrosqueezing transform”
The supplementary materials provide mathematical definitions, theoretical results, implementation remarks, parameter studies, and numerical evaluations supporting ConceFT for time-frequency analysis. They describe its randomized multi-taper construction, stability properties, localization choices, and performance assessment using optimal transport.
- Algorithm: The ConceFT algorithm averages many nonlinear SST estimates, each formed from a transform using a randomly selected reference wavelet.The supplementary discussion contrasts this with defining one master reassignment rule.
- Amplitude estimation: ConceFT can estimate amplitudes by identifying instantaneous-frequency curves and integrating the time-varying power spectrum around each curve.The supplementary materials state that the curves can be identified more stably under large noise than with simple SST or MTSST.
- Theoretical results: The theoretical analysis characterizes intrinsic-mode-type behavior by concentration near component instantaneous-frequency curves, with errors of order ϵ^(1/3) under the stated conditions.The supplementary statements specify that the relevant quantity is small away from those curves.
- Theoretical results: For the randomized noise analysis, projected complex Gaussian quantities have variances determined by the projection vector norm, while high-dimensional random vectors are typically nearly perpendicular to a fixed vector.The latter geometric effect becomes increasingly likely as J increases.
- Reference functions: CWT- and STFT-based ConceFT use orthogonal reference functions derived from time-frequency localization operators, with CWT wavelets and STFT windows selected as corresponding eigenfunctions.The materials identify Morse functions for CWT and scaled or chirped Hermite functions for STFT.
- Numerical evaluation: The parameter exploration selected β = 30, γ = 9, and J = 2 for CWT-based ConceFT on signal class C, while a separate STFT study used four window functions and an effective width of 600 samples.The CWT choice gave the best performance in the reported exploration, and the selected values were retained for later experiments.