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Adaptive Short-time Fourier Transform and Synchrosqueezing Transform for Non-stationary Signal Separation
Lin Li, Haiyan Cai, Hongxia Han, Qingtang Jiang, Hongbing Ji
TL;DR
The paper targets the inability of fixed-window STFT-based methods to jointly achieve high time and frequency resolution and reliably separate multicomponent non-stationary signals. It develops adaptive STFT/FSST methods using an LFM-based well-separated condition and localized automatic window selection, with reported effectiveness for frequency estimation, sharp time-frequency representation, and component separation, including fast-varying frequencies.
Problem
Fixed-window STFT cannot simultaneously provide high time and frequency resolution, and mixed component STFTs prevent FSST from separating components accurately.
Method
The paper introduces adaptive STFT and adaptive FSST with time-varying windows, derives an LFM-based well-separated condition, and uses localized optimization to estimate window width.
Results
The proposed adaptive FSST supports instantaneous-frequency estimation, sharp time-frequency representation, and separation of multicomponent non-stationary signals with fast-varying frequencies.
Takeaways & Limitations
The method provides a framework for sharper time-frequency analysis and more accurate component recovery in multicomponent non-stationary signals.
Takeaways & Limitations
The paper leaves theoretical analysis of adaptive FSST and recovery of components with crossover instantaneous-frequency curves for future work.
Abstract
from arXiv · showhide
The synchrosqueezing transform, a kind of reassignment method, aims to sharpen the time-frequency representation and to separate the components of a multicomponent non-stationary signal. In this paper, we consider the short-time Fourier transform (STFT) with a time-varying parameter, called the adaptive STFT. Based on the local approximation of linear frequency modulation mode, we analyze the well-separated condition of non-stationary multicomponent signals using the adaptive STFT with the Gaussian window function. We propose the STFT-based synchrosqueezing transform (FSST) with a time-varying parameter, named the adaptive FSST, to enhance the time-frequency concentration and resolution of a multicomponent signal, and to separate its components more accurately. In addition, we also propose the 2nd-order adaptive FSST to further improve the adaptive FSST for the non-stationary signals with fast-varying frequencies. Furthermore, we present a localized optimization algorithm based on our well-separated condition to estimate the time-varying parameter adaptively and automatically. Simulation results on synthetic signals and the bat echolocation signal are provided to demonstrate the effectiveness and robustness of the proposed method.
1 Introduction
The paper addresses limitations of fixed-window STFT-based synchrosqueezing for separating multicomponent non-stationary signals. It introduces adaptive STFT and adaptive FSST methods guided by an LFM-based separation condition and automatic window selection.
- Limitations of existing methods: EMD can suffer from mode mixing, making two close intrinsic mode functions difficult to distinguish.The cited issue arises when disparate scales occur within one IMF or similar scales occur across different IMFs.
- Limitations of existing methods: Synchrosqueezing sharpens time-frequency representations by reallocating transform coefficients, but frequency-varying signals can make its squeezing effect undesirable.The paper also notes that existing WSST and FSST methods are robust to noise and small perturbations.
- Motivation: Fixed-window STFT cannot provide high time and frequency resolution simultaneously, motivating a time-varying window parameter.Wide windows suit low-frequency broadband components, whereas narrow windows suit high-frequency components.
- Proposed methods: The paper proposes adaptive FSST, based on STFT with a time-varying window, to enhance time-frequency resolution, concentration, frequency estimation, and component separation.The adaptive STFT uses a positive time-varying parameter σ(t) controlling the window width.
- Proposed methods: An LFM-based well-separated condition is established so component STFTs lie in non-overlapping time-frequency regions, supporting more accurate recovery.The approach locally approximates non-stationary components by linear frequency modulation signals.
- Proposed methods: A localized optimization method estimates the time-varying window width, while the 2nd-order adaptive FSST further improves time-frequency energy concentration.The selected parameter prioritizes component separation and sharp associated adaptive FSSTs rather than necessarily optimizing overall STFT sharpness.
2 STFT and FSST with a time-varying parameter
The paper extends STFT and FSST by allowing a time-varying window parameter, enabling adaptive time-frequency representation and component recovery. It also introduces second-order adaptive FSST for LFM signals and faster-varying frequencies.
- FSST review: FSST reassigns the STFT frequency variable to sharpen time-frequency representations and can recover individual components under suitable conditions.For a pure tone, the phase transformation equals its instantaneous frequency.
- Adaptive STFT: Because overlapping component STFTs prevent FSST from separating those components, the paper seeks a parameter choice that avoids overlap.The paper introduces the adaptive STFT and establishes a separability condition for this purpose.
- Adaptive STFT: A time-varying parameter defines the adaptive STFT, which retains signal-recovery formulas under stated assumptions.The parameter varies with time while the transform remains invertible through the paper’s reconstruction theorem.
- Adaptive FSST: The adaptive FSST reassigns adaptive-STFT coefficients and provides formulas for reconstructing the signal and its components.Its phase transformation supplies a candidate instantaneous frequency at nonzero transform coefficients.
- Second-order adaptive FSST: The second-order adaptive FSST extends the construction with a second-order phase transformation that gives the instantaneous frequency exactly for LFM signals under Theorem 2.The method is designed for signals whose frequencies vary rapidly and uses a time-varying parameter.
3 Support zones of STFTs of LFM signals
This section characterizes how the Gaussian-window parameter controls STFT support zones for LFM signals. The resulting zones support analysis of sharpness and well-separated conditions for multicomponent signals.
- Support-zone framework: The parameter σ controls STFT sharpness, while support zones describe where an STFT is non-negligible in the time-frequency plane.The paper defines support using a threshold on the Fourier transform of the window.
- Gaussian support: For Gaussian windows, the effective Fourier-support interval is defined through α and the threshold ϵ.The construction treats the transform as supported where its magnitude remains above the chosen threshold.
- LFM support zones: For an LFM signal, the STFT ridge concentrates around η = c + rt within a corresponding time-frequency zone.The ridge follows the instantaneous frequency of the LFM signal.
- LFM support zones: The paper identifies a choice of σ that produces the sharpest Gaussian-window STFT representation of an LFM signal.The same time-varying choice is also described as optimal for representing a monocomponent signal.
- Multicomponent separability: These support zones are used to study when adaptive STFTs of multicomponent signals are well separated under a suitable σ(t).The constant-σ case reduces to the earlier condition with △ = α/σ.
4 Separability of multicomponent signals and selection of time-varying parameter
The paper derives separability conditions for adaptive STFTs of multicomponent signals, first under a sinusoidal model and then under a local LFM approximation. It uses these conditions to select time-varying parameters that sharpen adaptive FSST representations and support component separation.
- Problem: The section asks when adaptive-STFT representations of multicomponent signals occupy non-overlapping time-frequency regions.This provides the basis for separating components and obtaining sharper adaptive FSST representations.
- Sinusoidal signal model: For locally sinusoidal components, separation requires neighboring instantaneous frequencies to remain farther apart than the adaptive STFT zones determined by σ(t).With constant σ(t), the condition reduces to the previously stated fixed-window condition.
- LFM model: The LFM analysis replaces sinusoidal zones with larger zones accounting for local frequency modulation and derives an LFM model-based well-separated condition.The condition is formulated so that component STFTs lie in non-overlapping regions when σ(t) satisfies the derived bounds.
- Limitations: If the LFM separation inequality has no suitable σ solution, adjacent components cannot be separated in the time-frequency plane.The condition also requires boundedness of the relevant LFM quantities.
- Parameter selection: Any σ(t) between the derived bounds can separate components, but smaller σ(t) is preferred because it produces a sharper synchrosqueezing representation.The proposed choice therefore seeks the smallest admissible time-varying window parameter.
- Experiments: On a two-component LFM signal, adaptive FSST represents the components sharply, while 2nd-order adaptive FSST further improves time-frequency energy concentration.With σ = 0.057, conventional 2nd-order FSST is less sharp or clear than the 2nd-order adaptive FSST.
5 Selecting the time-varying parameter automatically
The paper proposes a localized algorithm that estimates a separability parameter σ(t) from STFT concentration and component-support separation. The estimate is then used to construct adaptive STFT, adaptive FSST, and 2nd-order adaptive FSST representations.
- Component identification: Local maxima of the adaptive STFT identify candidate components, while thresholding removes low-amplitude maxima treated as noise or interference.Non-overlapping support intervals are then used to validate candidate components.
- LFM estimation: The method estimates chirp rates by fitting a linear function to a local ridge in the time-frequency plane.The fitted chirp rate determines the support intervals used in the separability test.
- Localized separability search: The algorithm searches for the smallest σ whose estimated component support intervals remain non-overlapping.A candidate is accepted when adjacent intervals satisfy the separation criterion; otherwise σ is changed.
- Initialization and search: Rényi entropy supplies an initial upper-bound parameter because lower entropy indicates better time-frequency concentration.The procedure searches discretized σ values, tests separability, and smooths the resulting parameter trajectory with a low-pass filter.
- Constructing adaptive transforms: The estimated σest(t) defines adaptive STFT, adaptive FSST, and 2nd-order adaptive FSST representations.The low-pass smoothing step reflects the assumed continuity of the signal amplitudes and phases.
- Limitations: The method can fail when instantaneous frequencies of different components are too close for the separability procedure to distinguish them.This is presented as a limitation of the automatic parameter-selection procedure.
6 Further experiments and results
Experiments on synthetic and bat signals show that adaptive and higher-order adaptive FSSTs improve time-frequency representation, component separation, and robustness to noise.
- Three-component synthetic signal: The three-component signal has time-varying component durations and is sampled at 512 Hz.Its waveform and instantaneous frequencies are shown in Figure 3.
- Three-component synthetic signal: The 2nd-order adaptive FSST with σest2(t) provides sharp, clear representations when other adaptive FSST variants fail to separate closely spaced frequencies.The conventional and regular-PT variants do not separate the three components well in this case.
- Noise robustness: The proposed 2nd-order adaptive FSST works well across noisy conditions in comparisons with conventional 2nd-order FSST.Experiments use Gaussian noise at different signal-to-noise ratios.
- Bat echolocation signal: For the bat echolocation signal, the adaptive method improves representation of the highest-frequency component and signal ends relative to conventional 2nd-order FSST.The four bat-signal components are more separated than those in the synthetic example.
7 Conclusion
The paper introduces adaptive STFT-based synchrosqueezing methods and an automatic parameter-selection procedure for non-stationary multicomponent signals. Experiments support improved frequency estimation, sharper time-frequency representations, and component separation, while several theoretical and scope extensions remain open.
- Contributions: The paper introduces adaptive STFT, adaptive FSST, and 2nd-order adaptive FSST using a time-varying parameter.The analysis uses Gaussian-window STFTs of locally approximated linear frequency-modulation signals.
- Contributions: An automatic method selects the time-varying parameter using the proposed well-separated condition.The method is designed for adaptive parameter estimation.
- Conclusions: Experiments on synthetic and real data demonstrate efficient instantaneous-frequency estimation, sharp time-frequency representations, and multicomponent separation.The reported scope includes non-stationary signals with fast-varying frequencies.
- Limitations and future work: The paper does not yet provide a theoretical analysis of adaptive FSST or treat time-varying windows beyond the Gaussian function.These are identified as directions for future study.
- Limitations and future work: The experiments consider components without crossover instantaneous-frequency curves, leaving recovery with crossover curves for future work.This is an explicit scope boundary of the paper.
Appendix
The appendix establishes identities used in the analysis of the STFT and instantaneous-frequency estimation for linear frequency-modulation signals.
- Appendix derivations: The derivation permits exchanging the order of integration variables using Fubini’s theorem.This step establishes equation (16).
- Appendix derivations: For real-valued signals and real Gaussian windows, the STFT has symmetric frequency behavior used in the derivation.The appendix invokes this symmetry to derive a subsequent identity.
- Appendix derivations: For a linear instantaneous frequency φ′(t)=c+rt, the appendix identifies the signal’s instantaneous frequency through the stated STFT relation.The result is presented as Theorem 2.