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Synchrosqueezing-based Recovery of Instantaneous Frequency from Nonuniform Samples
Gaurav Thakur, Hau-Tieng Wu
TL;DR
The paper addresses the lack of a rigorous instantaneous-frequency definition and the challenge of recovering it from uniform or nonuniform samples. It develops an STFT-based Synchrosqueezing method for multi-component AM-FM signals, reports accurate and noise-robust recovery, and compares it with Hilbert-based analysis in experiments and ECG analysis.
Problem
Instantaneous frequency lacks a fully satisfactory rigorous definition, while recovering it from nonuniform samples is important for signal analysis applications.
Method
The paper defines an STFT-based Synchrosqueezing variant and applies it to sample-weighted impulse trains to estimate instantaneous frequencies from discrete samples.
Results
The method recovers instantaneous frequencies with high accuracy, is robust to noise, and is evaluated against Hilbert-based analysis on test cases and ECG respiration data.
Takeaways & Limitations
STFT Synchrosqueezing provides a practical alternative for studying instantaneous frequencies from uniform or nonuniform samples, including noisy ECG-derived signals.
Takeaways & Limitations
The ideal instantaneous-frequency definition requires known component functions, and the method is limited by resolution through a tradeoff between α and γ.
Abstract
from arXiv · showhide
We propose a new approach for studying the notion of the instantaneous frequency of a signal. We build on ideas from the Synchrosqueezing theory of Daubechies, Lu and Wu and consider a variant of Synchrosqueezing, based on the short-time Fourier transform, to precisely define the instantaneous frequencies of a multi-component AM-FM signal. We describe an algorithm to recover these instantaneous frequencies from the uniform or nonuniform samples of the signal and show that our method is robust to noise. We also consider an alternative approach based on the conventional, Hilbert transform-based notion of instantaneous frequency to compare to our new method. We use these methods on several test cases and apply our results to a signal analysis problem in electrocardiography.
1 Introduction
The paper addresses the lack of a rigorous instantaneous-frequency definition and the challenge of analyzing nonuniformly sampled signals. It proposes an STFT-based Synchrosqueezing approach for accurate, noise-robust recovery and compares it with Hilbert-based analysis in test cases and ECG.
- Instantaneous frequency remains a heuristic concept lacking a mathematically rigorous and entirely satisfactory definition.
- Nonuniformly sampled signal analysis is important in applications including radar detection, audio processing, and seismology.
- The paper defines an STFT-based Synchrosqueezing variant for multi-component AM-FM signals.
- Applying the modified transform to sample-weighted impulse trains recovers instantaneous frequencies accurately and robustly to noise.
- The authors compare this method with Hilbert-based instantaneous frequency and apply both approaches to test cases and ECG respiration analysis.
2 Background Material
This section frames instantaneous frequency through AM-FM components and reviews the conventional Hilbert-transform approach. It emphasizes representation ambiguity and restrictive assumptions that can make Hilbert-based estimates difficult to verify or numerically sensitive.
- For an AM-FM signal f(t) = A(t) cos(2πφ(t)), the target instantaneous frequency is the local oscillation rate φ′(t).
- The ideal instantaneous frequencies are defined from the phase derivatives of the signal's AM-FM components.
- This ideal definition requires known component amplitudes and phases, while arbitrary signals may admit multiple AM-FM representations.
- The conventional approach forms an analytic signal using the Hilbert transform and estimates instantaneous frequency from it.
- Hilbert-based analysis relies on restrictive spectral-support conditions on amplitude and phase, especially nonnegative frequency support.
- Hilbert-based computation can be sensitive to noise and numerical roundoff because of transform errors and possible numerator-denominator zero cancellation.
3 Synchrosqueezing with the Short-Time Fourier Transform
The paper develops an STFT-based Synchrosqueezing framework for estimating instantaneous-frequency components from nonuniformly sampled multi-component AM-FM signals. Its theorem characterizes concentration near component frequencies under structured signal and sampling assumptions, including noisy samples.
- Method: STFT Synchrosqueezing provides an alternative to wavelet-based Synchrosqueezing for estimating instantaneous-frequency components from sampled signals.The method is developed independently and is intended to estimate instantaneous frequencies from discrete, nonuniform samples.
- Signal model: The signal model B_ε,d contains superpositions of intrinsic mode functions with slowly varying amplitudes and instantaneous frequencies separated by at least d.The class is restrictive but is presented as a reasonable model for signals arising in many applications.
- STFT Synchrosqueezing: The instantaneous-frequency information divides ∂_tV_g f̃ by V_g f̃, reducing window influence and sharpening the time-frequency representation.The resulting Synchrosqueezing transform concentrates content closer to instantaneous-frequency curves.
- Theoretical result: Under suitable resolution and threshold conditions, the estimated instantaneous-frequency set matches the true set up to the prescribed resolution and is supported near the component frequencies.Away from component regions, the STFT magnitude is bounded by E_1 + E_2; with an appropriate threshold, the transform is supported in a 2K-point set.
- Noise robustness: With appropriately chosen α and γ, the instantaneous-frequency calculation remains robust to sample noise.The noisy case uses an increased threshold condition involving E_1 + E_2 + I_0 + 2I′_0 and retains support near the component frequencies.
4 A Bandlimited Reconstruction Approach
The section reconstructs bandlimited signals from nonuniform samples by least squares, then uses the recovered signal to approximate Hilbert-transform-based instantaneous frequency. It also establishes practical sampling assumptions and a sinc-basis implementation that improves reconstruction accuracy.
- Sampling assumptions: Stable reconstruction assumes sampling points are sufficiently dense, with average sampling rate above the Nyquist rate: 1/T > 2b.For admissible nonuniform points, sample and signal norms are bounded by constants depending on the sampling pattern and bandwidth.
- Nonuniform reconstruction: For DFT bases with appropriate weights, the approximation converges uniformly on compact sets, remains well-conditioned, and can be accelerated with FFTs.These properties hold as the truncation parameter N tends to infinity.
- Basis choice: The paper instead uses sinc(t − n − M) basis functions, obtaining convergence on the entire real line and better practical accuracy.M is an integer centering constant, while the weights remain those used for the DFT construction.
- Nonuniform reconstruction: Nonuniform samples can be converted into approximate uniform samples by solving a weighted least-squares problem over suitable basis functions.The reconstructed expansion is then used in a classical sampling series to recover the signal.
- Instantaneous-frequency estimation: After recovering f, the conventional approach computes IFHf, whereas the Synchrosqueezing method determines the signal’s individual IF components directly.This distinguishes whole-signal Hilbert-transform analysis from component-wise Synchrosqueezing analysis.
- Sampling assumptions: Although AM-FM signals are generally not strictly bandlimited, they can be closely approximated on finite intervals by sufficiently high-bandwidth bandlimited functions.Oversampling then allows the reconstruction method to approximately recover the finite-interval approximation.
5 Numerical Experiments and Applications
Numerical experiments on uniform and randomly perturbed nonuniform samples show that STFT Synchrosqueezing recovers instantaneous-frequency information across AM-FM, chirp, noisy, undersampled, multicomponent, and crossing-frequency signals. Applied to ECG-derived respiration analysis, IFS tracks the true respiration frequency more closely than the Hilbert-based alternative.
- Experimental setup: The experiments compare IFS from STFT Synchrosqueezing with IFH from bandlimited reconstruction using uniform and randomly perturbed nonuniform sampling times.Nonuniform samples use t_n = Tn + T′a_n with T′ < T; figures show uniform results alongside nonuniform results.
- Single-component signals: Both methods closely recover the known IIF for the AM-FM and chirp test signals under uniform and nonuniform sampling.The AM-FM IIF is 3−sin t, while the chirp IIF is 1 + 0.1t.
- Noise and sampling limitations: IFS remains fairly robust under Gaussian noise, whereas IFH produces very poor results and can amplify noise or numerical roundoff effects.For the noisy harmonic, the noise variance is σ^2 = 0.4; the broader experiments report the same contrast for Figures 4, 6, and 7.
- Noise and sampling limitations: Under heavy undersampling, STFT Synchrosqueezing produces a good IFS result, while bandlimited reconstruction cannot determine IFH.The signal has constant IIF 5 and a sampling rate below half its Nyquist rate.
- Multicomponent signals: For a two-component signal, IFS recovers both IIF curves, while IFH roughly returns their average and exhibits spurious singularities.The recovered IIF set is {2 − 0.2 sin t, 3 + 0.04t}; the approximate IFH average is 2.5 + 0.02t − 0.1 sin t.
- Multicomponent signals: When IIF curves cross, the synchrosqueezed time-frequency plot maintains sharp separation and distinguishes the two curves clearly.This extends the multicomponent test to interlacing IIF elements, a signal type noted as difficult and outside B_ε,d.
- ECG application: In ECG analysis, IFS computed from R peaks provides a good approximation to true respiration IFS, whereas IFH from the same peaks has little in common with it.The respiration IFH is often negative and has no obvious interpretation, while IFS also reflects closer respiration-cycle spacing through higher values.