The Fourier Decomposition Method for nonlinear and nonstationary time series analysis

摘要

Since many decades, there is a general perception in literature that theFourier methods are not suitable for the analysis of nonlinear andnonstationary data. In this paper, we propose a Fourier Decomposition Method(FDM) and demonstrate its efficacy for the analysis of nonlinear (i.e. datagenerated by nonlinear systems) and nonstationary time series. The proposed FDMdecomposes any data into a small number of `Fourier intrinsic band functions'(FIBFs). The FDM presents a generalized Fourier expansion with variableamplitudes and frequencies of a time series by the Fourier method itself. Wepropose an idea of zero-phase filter bank based multivariate FDM (MFDM)algorithm, for the analysis of multivariate nonlinear and nonstationary timeseries, from the FDM. We also present an algorithm to obtain cutoff frequenciesfor MFDM. The MFDM algorithm is generating finite number of band limitedmultivariate FIBFs (MFIBFs). The MFDM preserves some intrinsic physicalproperties of the multivariate data, such as scale alignment, trend andinstantaneous frequency. The proposed methods produce the results in atime-frequency-energy distribution that reveal the intrinsic structures of adata. Simulations have been carried out and comparison is made with theEmpirical Mode Decomposition (EMD) methods in the analysis of various simulatedas well as real life time series, and results show that the proposed methodsare powerful tools for analyzing and obtaining the time-frequency-energyrepresentation of any data.