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Adaptive Local Iterative Filtering for Signal Decomposition and Instantaneous Frequency analysis
Antonio Cicone, Jingfang Liu, Haomin Zhou
TL;DR
Nonlinear and nonstationary signals require local, adaptive, and stable analysis. The paper proves convergence conditions for iterative filtering, introduces data-driven ALIF with compactly supported FP filters, and defines instantaneous frequency from local signal properties. Numerical examples demonstrate IF and ALIF stability, while theoretical bounds on ALIF accuracy and convergence for non-uniform mask lengths remain open.
Problem
Time-frequency analysis of nonlinear and nonstationary signals is challenging because conventional Fourier and wavelet methods are not designed to be data-adaptive, while convergence for IF on general signals had not been explored.
Method
The paper combines IF with pointwise, data-driven filter-length adaptation in ALIF, constructs smooth compactly supported FP filters, and defines instantaneous frequency from normalized IMFs and their derivatives.
Results
The paper derives sufficient filter-shape conditions guaranteeing IF inner-loop convergence for general nonstationary and nonperiodic signals, and numerical examples show IF and ALIF stability.
Takeaways & Limitations
IF and ALIF provide a local, adaptive, and stable iterative-filtering toolbox, including instantaneous phase and frequency definitions based only on local signal properties.
Takeaways & Limitations
Theoretical bounds on ALIF decomposition accuracy remain open, and convergence with non-uniform mask lengths requires a slowly varying mask length.
Abstract
from arXiv · showhide
Time-frequency analysis for non-linear and non-stationary signals is extraordinarily challenging. To capture features in these signals, it is necessary for the analysis methods to be local, adaptive and stable. In recent years, decomposition based analysis methods, such as the empirical mode decomposition (EMD) technique pioneered by Huang et al., were developed by different research groups. These methods decompose a signal into a finite number of components on which the time-frequency analysis can be applied more effectively. In this paper we consider the iterative filters (IFs) approach as an alternative to EMD. We provide sufficient conditions on the filters that ensure the convergence of IFs applied to any $L^2$ signal. Then we propose a new technique, the Adaptive Local Iterative Filtering (ALIF) method, which uses the IFs strategy together with an adaptive and data driven filter length selection to achieve the decomposition. Furthermore we design smooth filters with compact support from solutions of Fokker-Planck equations (FP filters) that can be used within both IFs and ALIF methods. These filters fulfill the derived sufficient conditions for the convergence of the IFs algorithm. Numerical examples are given to demonstrate the performance and stability of IFs and ALIF techniques with FP filters. In addition, in order to have a complete and truly local analysis toolbox for non-linear and non-stationary signals, we propose a new definition for the instantaneous frequency which depends exclusively on local properties of a signal.
1 Introduction
Nonlinear and nonstationary signals require local, adaptive, and stable analysis, but conventional Fourier and wavelet methods use predetermined, nonadaptive bases. The paper develops ALIF with smooth compactly supported FP filters, data-driven filter lengths, and local instantaneous-frequency definitions.
- Motivation: Fourier and wavelet methods are limited for nonlinear and nonstationary signals because they are linear, use predetermined bases, and are not data-adaptive.These limitations can prevent desirable results for the target signals.
- Related approaches: Decomposition methods first split signals into simpler components and then apply time-frequency analysis to each component.The paper situates decomposition as an iterative or optimization-based strategy.
- Related approaches: EMD iteratively decomposes a signal into intrinsic mode functions whose instantaneous frequencies are intended to be well behaved.Its sifting process repeatedly extracts components from the signal.
- Limitations of EMD: EMD computes moving averages from cubic-spline upper and lower envelopes, but repeated spline use makes it unstable under perturbations.EEMD addresses this by averaging IMFs from trials with artificially added random perturbations.
- Iterative filtering: IF replaces EMD’s envelope-based moving average with convolution using low-pass filters while retaining the iterative decomposition framework.The paper reviews IF as an alternative to EMD.
- ALIF contribution: ALIF generalizes IF to non-uniform filters by adapting filter length point by point according to the signal and constructing smooth compactly supported FP filters.The adaptive strategy is data driven, while compact support supports local analysis.
- Local instantaneous frequency: The paper defines instantaneous frequency and phase from an IMF’s normalized signal, derivative, rotation speed, and rotation angle rather than the global Hilbert transform.The stated goal is to localize time-frequency analysis and better capture frequency changes in nonlinear signals.
2 Iterative Filtering Algorithm
Iterative Filtering extracts IMFs by repeatedly subtracting a filter-based moving average, and the paper derives filter conditions guaranteeing inner-loop convergence for L2 signals. Convolution-based filter constructions provide a broad sufficient class, including self-convolutions of symmetric, nonnegative, finitely supported L2 filters.
- Algorithm structure: IF decomposes a signal into finitely many intrinsic mode functions through nested inner and outer loops.The inner loop computes each IMF, while the outer loop derives all IMFs.
- Algorithm structure: Each IF iteration subtracts a convolution-based moving average from the current signal to retain its fluctuation part.The moving average uses a low-pass filter and a mask length.
- Algorithm structure: The first IMF is the limiting sequence produced by repeatedly applying the fluctuation operator, with subsequent IMFs extracted from the remainder.The process stops when the remainder becomes a trend signal.
- Convergence problem: Convergence for general signals with uniform or non-uniform filters had not previously been explored, motivating explicit IMF formulas and sufficient filter conditions.Earlier guarantees covered periodic signals and some l∞ signals with uniform filters.
- Convergence theorem: For an L2 signal, if |1 − F(w)(ξ)| < 1 or F(w)(ξ) = 0, the IF inner-loop sequence converges.The theorem also supplies an explicit formula for the resulting IMF.
- Admissible filters: The sufficient conditions are fulfilled by filters formed by convolving a symmetric, nonnegative, finitely supported L2 filter with itself.The resulting Fourier transform satisfies 0 ≤ F(u)(ξ) < 1.
3 Adaptive Local Iterative Filtering Techniques
ALIF extends iterative filtering with signal-dependent, spatially varying mask lengths and FP-based filters, while establishing convergence under stated conditions and identifying an a posteriori convergence criterion.
- ALIF generalizes iterative filtering to non-uniform filters for general oscillatory signals.
- The algorithm uses two loops: the inner loop extracts one IMF, while the outer loop extracts all IMFs from the signal.
- ALIF computes a positive mask length ln(x) that may be uniform or vary point by point with the signal.
- The non-uniform mask length is estimated from distances between consecutive extrema, interpolated smoothly, and then low-pass filtered by removing high-frequency oscillations.
- Under Theorem 2's assumptions, the inner-loop sequence converges almost uniformly and its limit is an IMF.
- The theorem provides only an a posteriori convergence criterion, not sufficient conditions on the filter and mask length guaranteeing ALIF convergence in advance.
4 Local Filters developed from a PDE model
The paper constructs smooth, compactly supported FP filters through diffusion and transport, then efficiently adapts their support lengths by interpolation while preserving filter shape.
- FP filters are constructed from Fokker–Planck equations to provide smooth filters with compact support for iterative filtering.
- Diffusion spreads solution mass toward interval boundaries while transport moves it toward the center until a steady state forms.
- The resulting steady-state filter is nonnegative, supported on [a, b], and has no leakage outside that interval.
- The filter shape can be adjusted by selecting h(x), g(x), and coefficients α and β.
- A filter for a new support interval can be obtained either by resolving the PDE or by spatially interpolating a previously computed steady state.
- The interpolation method preserves filter shape across lengths and supports any positive real, including non-integer, filter length.
5 A Different View of Instantaneous frequency and phase
The paper replaces Hilbert-transform-based instantaneous frequency with a local definition based on the normalized IMF and its derivative. The proposed phase and frequency use the rotation angle and rotation speed of this normalized representation, with ENO handling sudden amplitude changes.
- Motivation: The Hilbert transform is global, making it unsuitable for fully local time-frequency analysis of signals with transient features.The paper therefore seeks a definition based only on local information.
- Local definition: The proposed instantaneous phase is the rotation angle of the normalized IMF and its derivative, while instantaneous frequency is the corresponding rotation speed.Normalization uses envelope functions for the IMF and its derivative, producing a unit circle or a perturbation of one.
- Local definition: The local definition generally depends only weakly on the choice of the envelope functions.The envelopes can, for example, be cubic splines through local extrema of the IMF and its derivative.
- Handling amplitude changes: Sudden short-time changes in an IMF or its derivative can make the envelopes inaccurate and cause unexpected instantaneous phase and frequency errors.The paper uses ENO to detect such changes and construct separate envelopes on either side.
- Numerical tests: For slowly changing amplitude, both methods capture gradual frequency change, but the proposed method has almost no oscillations compared with the Hilbert-transform result.Figure 4 reports this comparison for a signal whose oscillation gradually accelerates and whose amplitude changes mildly.
- Numerical tests: For sudden amplitude changes, Hilbert-transform effects spread to distant positions, whereas the proposed method with ENO yields an almost constant instantaneous frequency away from the changes.Test 2 uses a constant-frequency signal with amplitude changes at times 3 and 6.
6 Numerical Experiments
Numerical experiments show that IF and ALIF can decompose synthetic, noisy, and real signals, while ALIF handles overlapping instantaneous-frequency ranges that defeat fixed-mask IF. The experiments also indicate sensitivity to intra-wave modulation, robustness to white noise, and unresolved theoretical accuracy bounds.
- IF decomposes the test signal into a frequency-modulated component and the trend 4(x − 0.5)^2.
- IF separates a signal into two IMFs that reproduce the ground-truth components with good accuracy and recover varying and constant instantaneous frequencies.
- IF fails when component instantaneous-frequency ranges overlap, because its fixed mask length groups components within one frequency interval.
- ALIF separates the same non-stationary signal using a signal-derived adaptive mask, without prior knowledge of its components.
- The new instantaneous-frequency definition is more sensitive to intra-wave modulation than the traditional definition and identifies intervals where that phenomenon occurs.
- Theoretical bounds on the expected accuracy of IF and ALIF decompositions remain an open problem.
- IF and ALIF remain effective under white noise, automatically isolating noise in early IMFs and recovering signal components in later IMFs, including at SNR around 0.
- Real-data experiments produced stable decompositions for similar signals and revealed periodic components in length-of-day and tsunami water-level records.
7 Conclusion
The paper establishes convergence results for IF and ALIF inner loops, proposes local instantaneous-frequency definitions, and reports numerical evidence of stability and outer-loop convergence. It also identifies unresolved theoretical questions about ALIF convergence, identifiability, and higher-dimensional extensions.
- Theorem 1 provides sufficient filter-shape conditions ensuring IF inner-loop convergence for general non-stationary and non-periodical signals.
- ALIF uses a pointwise adaptive filter length and compact-support FP filters to provide adaptive and local iterative filtering.The filter length changes according to the signal, while FP filters are designed from a PDE model.
- Theorem 2 proves ALIF inner-loop convergence through an a posteriori criterion, while a priori conditions for non-uniform mask lengths remain open.
- Numerical tests suggest outer-loop convergence under mild filter conditions, and examples demonstrate stability for both IF and ALIF.
- Theoretical accuracy bounds, identifiability of generic-signal decompositions, and convergence of higher-dimensional extensions remain future research problems.
- The paper defines instantaneous phase and frequency using only local signal properties, enabling completely local time-frequency analysis.