The Fourier decomposition method for nonlinear and non-stationary time series analysis

Abstract

for many decades, there has been a general perception in the literature that Fourier methods are not suitable for the analysis of nonlinear and non-stationary data. In this paper, we propose a novel and adaptive Fourier decomposition method (FDM), based on the Fourier theory, and demonstrate its efficacy for the analysis of nonlinear and non-stationary 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 variable frequencies of a time series by the Fourier method itself. We propose an idea of zero-phase filter bank-based multivariate FDM (MFDM), for the analysis of multivariate nonlinear and non-stationary time series, using the FDM. We also present an algorithm to obtain cut-off frequencies for MFDM. The proposed MFDM generates a 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 provide a time–frequency–energy (TFE) distribution that reveals the intrinsic structure of a data. Numerical computations and simulations have been carried out and comparison is made with the empirical mode decomposition algorithms.

Breaking the Limits: Redefining the Instantaneous Frequency

“…Thus, this study presents a new paradigm for nonlinear and nonstationary data analysis. Moreover, this work independently and uniquely resolves some misconceptions that have grown with regard to the significance of the Fourier theory and its usefulness in the representation and analysis of nonstationary signal, and also supports the results of other related studies [18,19]. This paper is organized as follows: The proposed methodology is presented in Section 2.…”

“…In order to avoid unnecessary variations in the IF, before evaluating the IF, the mean and dominating low frequency component like trend present in the signal can be removed. These can be easily removed by any zerophase low pass filtering operation [18,22]. Moreover, depending upon the requirements, signal can be decomposed into a set of desired frequency bands.…”

“…In this work, we use non-causal finite impulse response (FIR) and infinite impulse response (IIR) filter to decompose a signal into a set of LINOEP vectors by the filter mode decomposition (FMD) algorithm proposed in [13,18,22], as summarized in Algorithm 2. In this algorithm, for each iteration, ZP HP F i (ZP LP F i ) is zero-phase high (low) pass filter (e.g.…”

“…The IF is the basis of the TFD or time-frequency-energy (TFE) representation and analysis of a signal. The IF is a practically important parameter of a signal which can reveal the underlying process and provides explanations for physical phenomenon in many applications such as vibration, acoustic, speech signal analysis [18], meteorological and atmospheric applications [1], seismic [18], radar, sonar, solar physics, structural engineering, communications, health monitoring, biomedical and medical applications [20], cosmological gravity wave and financial market data analysis.…”

“…In order to avoid this problem, recently many nonlinear and nonstationary signal representation, decomposition and analysis methods, e.g. empirical mode decomposition (EMD) algorithms [1,[3][4][5][6][7][8]14], synchrosqueezed wavelet transforms (SSWT) [9], variational mode decomposition (VMD) [10], eigenvalue decomposition (EVD) [11], empirical wavelet transform (EWT) [12], sparse timefrequency representation [15], time-varying vibration decomposition [16], resonance-based signal decomposition [17] and Fourier decomposition methods (FDM) [13,18,19,21,22,24] based on the Fourier theory, are proposed. The Fourier theory is the only tool for spectrum analysis of a signal and the FDM has established that it is a superior tool for nonlinear and nonstationary time series analysis.…”