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The Fourier Decomposition Method for nonlinear and nonstationary time series analysis
Pushpendra Singh, Shiv Dutt Joshi, Rakesh Kumar Patney, Kaushik Saha
TL;DR
Fourier methods were widely regarded as unsuitable for nonlinear and nonstationary data, motivating a Fourier-based alternative. The paper proposes FDM and zero-phase-filter-bank MFDM, which generate intrinsic band functions and time-frequency-energy representations; simulations and comparisons with EMD report effective analysis, including accurate localization for a unit sample sequence.
Problem
Fourier methods were widely regarded as unsuitable for analyzing nonlinear and nonstationary data, while existing time-frequency approaches did not represent nonstationary signals well.
Method
FDM decomposes data into Fourier intrinsic band functions, while MFDM uses zero-phase filtering and cutoff-frequency selection to generate band-limited multivariate FIBFs.
Results
The FDM time-frequency-energy plot for a unit sample sequence matched theoretical localization, whereas EEMD and CWT spread energy across frequencies and were less accurate.
Takeaways & Limitations
FDM and MFDM provide mathematically defined Fourier-based decompositions whose instantaneous frequencies yield time-frequency-energy distributions revealing embedded signal structures.
Abstract
from arXiv · showhide
Since many decades, there is a general perception in literature that the Fourier methods are not suitable for the analysis of nonlinear and nonstationary data. In this paper, we propose a Fourier Decomposition Method (FDM) and demonstrate its efficacy for the analysis of nonlinear (i.e. data generated by nonlinear systems) and nonstationary time series. The proposed FDM decomposes any data into a small number of `Fourier intrinsic band functions' (FIBFs). The FDM presents a generalized Fourier expansion with variable amplitudes and frequencies of a time series by the Fourier method itself. We propose an idea of zero-phase filter bank based multivariate FDM (MFDM) algorithm, for the analysis of multivariate nonlinear and nonstationary time series, from the FDM. We also present an algorithm to obtain cutoff frequencies for MFDM. The MFDM algorithm is generating finite number of band limited multivariate FIBFs (MFIBFs). The MFDM preserves some intrinsic physical properties of the multivariate data, such as scale alignment, trend and instantaneous frequency. The proposed methods produce the results in a time-frequency-energy distribution that reveal the intrinsic structures of a data. Simulations have been carried out and comparison is made with the Empirical Mode Decomposition (EMD) methods in the analysis of various simulated as well as real life time series, and results show that the proposed methods are powerful tools for analyzing and obtaining the time-frequency-energy representation of any data.
I. INTRODUCTION
The paper addresses limitations in existing time-frequency analysis and challenges in adaptive, basis-free decomposition of nonlinear and nonstationary signals. It motivates Fourier-based FDM and zero-phase-filter-bank MFDM as alternatives grounded in Fourier methods.
- Existing time-frequency methods provide insight into complex signals but do not represent nonstationary signals well.
- EMD decomposes data into finite-band intrinsic mode functions, whose extraction depends on local extrema and envelope interpolation choices.
- EMD is widely used for nonlinear and nonstationary analysis, but its empirical procedure can cause mode mixing, detrend uncertainty, aliasing, and end-effect artifacts.
- FDM uses the Fourier method to decompose nonlinear and nonstationary data into a small number of Fourier intrinsic band functions.
- MFDM uses zero-phase filtering to construct matched multivariate FIBFs and residue components for multivariate nonlinear and nonstationary data.
- Adaptive analysis lacks a general decomposition method that does not rely on a priori basis functions.
III. THE FOURIER DECOMPOSITION METHOD
FDM constructs a generalized Fourier expansion by decomposing a time series into Fourier intrinsic band functions with variable amplitudes and instantaneous frequencies. The resulting representation is complete, orthogonal, local, adaptive, and designed to accommodate nonstationary data.
- FIBF properties: FIBFs are zero-mean, mutually orthogonal functions whose analytic counterparts have positive amplitude and monotonically increasing phase.The analytic FIBFs are monocomponent signals with instantaneous frequency, while each FIBF combines zero-mean sinusoids from a consecutive frequency band.
- Signal representation: FDM represents a time series as a finite sum of analytic FIBFs and a residue or trend component.The framework models the signal using M single-component nonstationary signals together with a residue term.
- Construction: The construction uses periodic extension and Fourier expansion of finite-duration data before grouping consecutive Fourier terms into analytic FIBFs.Forward and reverse frequency scans select band limits subject to a monotonically increasing phase condition.
- Generalized Fourier expansion: FDM obtains a variable-amplitude, variable-frequency representation using Fourier methods rather than the fixed-amplitude, fixed-frequency form of classical Fourier expansion.The authors state that this representation satisfies completeness, orthogonality, locality, and adaptiveness through the decomposition.
- Time-frequency representation: The Fourier-Hilbert spectrum records time-varying amplitude and instantaneous frequency, while its marginal form measures total amplitude or energy contribution by frequency.Frequency in the marginal Fourier-Hilbert spectrum indicates the likelihood that a wave appeared locally over the signal duration.
- Frequency scanning: FDM provides low-to-high and high-to-low frequency views, which can produce identical or different time-frequency-energy distributions depending on the signal.Both scan directions are summarized as alternative FDM algorithms for obtaining analytic FIBFs.
IV. MULTIVARIATE FOURIER DECOMPOSITION METHOD
MFDM extends FDM to multivariate signals through zero-phase filter-bank decomposition into matched multivariate FIBFs and residues. It preserves channel alignment and salient waveform features, while the authors note an approximate multivariate condition.
- Zero-phase filtering: FIBF generation is equivalent to Fourier-based zero-phase filtering, implemented by selecting a desired frequency band and applying the inverse DFT.The paper also describes convolution-based zero-phase filtering using a symmetric filter construction.
- MFDM decomposition: MFDM repeatedly applies zero-phase high-pass filters to each channel's residue, producing successive matched MFIBFs and a final residue.The residue after each stage is the previous residue minus the newly extracted MFIBF.
- Scope and limitation: For multivariate filtering, the first two FIBF conditions are fully satisfied, whereas the third is only approximately satisfied across all channels.The authors identify this as a limitation analogous to a condition not being imposed simultaneously in MEMD.
- Cutoff-frequency selection: Cutoff frequencies may be selected from the signal spectrum using dyadic, non-dyadic, uniform, or non-uniform strategies.Algorithm C selects m, initializes the highest band edge at Fs/2, and recursively computes cutoff frequencies; larger m produces narrower bands.
- Alternative ordering: The algorithm can instead use zero-phase low-pass filtering to decompose the signal from residue toward the first MFIBF.This reverses the ordering of the extracted components relative to the high-pass procedure.
- Multivariate properties: MFDM produces an equal number of scale-aligned MFIBFs across channels, supporting direct multichannel modeling while preserving joint channel properties.The paper also states that zero-phase filtering preserves salient maxima and minima at their original times.
V. SIMULATION RESULTS
The simulation-results passage identifies the use of existing MATLAB implementations of MEMD, EMD, and EEMD for the reported experiments.
- Simulation setup: The simulations use online MATLAB software implementations of MEMD, EMD, and EEMD.The passage names the software sources as the implementations used in the simulation results.
A. Multivariate data decomposition
MFDM decomposes multivariate signals into perfectly aligned intrinsic bands while substantially reducing computation time relative to MEMD in the reported simulation.
- A. Multivariate data decomposition: MFDM generated perfectly aligned intrinsic bands across all four channels, similarly to MEMD.The simulation used four channels containing shared and channel-specific sinusoids plus Gaussian white noise.
- A. Multivariate data decomposition: 0.45 sec. for MFDM versus 69.5 sec. for MEMD in the same simulation.
- A. Multivariate data decomposition: The comparison table covers Fourier, Wavelet, EMD-Hilbert, and proposed FDM methods in data analysis.
- A. Multivariate data decomposition: Figure 2 compares FDM, EMD, and EEMD decompositions of a sinusoid mixed with intermittent interference.
B. Intermittency and mode mixing
Intermittency in a time series is identified as a main cause of mode mixing and mode splitting in decomposition.
- B. Intermittency and mode mixing: Intermittency in a time series is a main cause of mode mixing and mode splitting.
C. Time-Frequency-Energy Analysis
The paper evaluates its method on signals with intrawave frequency modulation and highly nonlinear dynamics. These challenging cases are reported to be handled with good accuracy, demonstrating nonlinear decomposition by FDM.
- C. Time-Frequency-Energy Analysis: The first test signal has intrawave frequency modulation with very high-frequency modulation of its instantaneous frequency.
- C. Time-Frequency-Energy Analysis: The second test considers a model wave satisfying a highly nonlinear differential equation.
- C. Time-Frequency-Energy Analysis: FDM handled these challenging cases with good accuracy in the reported figures.
- C. Time-Frequency-Energy Analysis: The examples demonstrate that FDM can analyze nonlinear signals and functions as a nonlinear decomposition method.
E. Analysis of a white Gaussian noise
The paper evaluates FDM on white Gaussian noise and related signals, comparing its time-frequency-energy and power-spectral-density representations with EMD and other methods. The reported analyses emphasize narrowband, orthogonal decomposition and enhanced tracking by FDM.
- FDM and EMD are compared on white Gaussian noise with zero mean, unit variance, 1024 samples, and Fs = 100 Hz.
- The mixture of a linear chirp and FM sinusoid is analyzed using FDM and EMD time-frequency-energy representations.
- FDM decomposes the signal into DC, FIBF1, FIBF2, and a highest-frequency component, whose summed components exactly synthesize x(t).
- The white-noise PSD analysis reports narrowband and orthogonal FIBFs, with enhanced TFE and PSD tracking using FDM.
- For the unit sample sequence, theoretical energy concentration occurs at t = 1.99 sec and f = 25 Hz, matching the FDM TFE result.
- The unit-sample example is presented as evidence that FDM is not limited by the uncertainty principle in simultaneous time-frequency concentration.
G. Application to a earthquake signal analysis
The paper applies its Fourier-based decomposition to earthquake and speech signals while presenting the proposed FDM, MFDM, and FIBF framework. These applications are used to show signal decomposition, time-frequency-energy analysis, and physically meaningful component extraction.
- Earthquake signal analysis: Elcentro earthquake data sampled at Fs = 50 Hz has nearly all its energy below the structural-design critical range of 10 Hz.
- Earthquake signal analysis: Earthquake arrivals include higher-frequency compression waves at 10 to 20 Hz and stronger lower-frequency shear and surface waves below 5 Hz.
- Proposed methods: FDM decomposes data into a small number of Fourier intrinsic band functions and generalizes Fourier expansion with variable amplitudes and frequencies.
- Proposed methods: MFDM uses a zero-phase filter bank to generate finite band-limited multivariate FIBFs and includes an algorithm for obtaining cutoff frequencies.
- Proposed methods: FIBFs are described as complete, orthogonal, local, and adaptive, with instantaneous frequencies producing a time-frequency-energy distribution.
- Conclusion: The proposed FDM and MFDM are mathematically defined through filter and Fourier-transform theory rather than extrema-based extraction procedures.
APPENDIX
The appendix formalizes Fourier, analytic-signal, instantaneous-frequency, group-delay, and energy relationships underlying the paper’s decomposition and time-frequency-energy representations. It also describes frequency scanning for constructing analytic FIBFs.
- The Fourier transform and inverse transform provide the continuous-time representation used for real, non-periodic functions satisfying Dirichlet conditions.
- For real signals, the analytic representation combines a signal with its Hilbert-transform component and uses the real part to recover the original signal.
- Low-to-high and high-to-low frequency scans select band boundaries to obtain a minimum number of analytic FIBFs.
- The analytic signal has twice the energy of the original signal under the stated relationship.
- Instantaneous frequency describes local frequency behavior over time, while group delay describes average time arrival as a function of frequency.
B. The FDM for discrete time real function
The discrete-time formulation extends the Fourier and analytic-signal construction to DTFT-based decomposition. Applications include vowel analysis, where FDM and EEMD are compared through components and time-frequency-energy representations.
- Discrete-time formulation: The discrete-time formulation defines the DTFT and inverse DTFT for real, non-periodic signals satisfying Dirichlet conditions.
- Speech signal analysis: For the small-cap I vowel, FDM generates a highest-frequency component, FIBFs y2 to y11, and a DC term y12.
- Speech signal analysis: FDM captures the fundamental frequency F0 accurately in FIBF y8, whereas EMD does not capture F0 in any mode.
- Speech signal analysis: EEMD captures F0 accurately in IMF y5 but introduces reconstruction error and substantial computation complexity at ensemble size 300.
- Discrete-time formulation: The discrete-time analytic representation uses conjugate symmetry and the real part of z[n] to represent a real sequence.
- Speech signal analysis: The FDM, EMD, and EEMD time-frequency-energy plots are compared, with enhanced tracking reported for FDM.