A fast approach of implementing the Fourier decomposition method for nonlinear and non-stationary time series analysis

Abstract

The Fourier Decomposition Method (FDM) is an advanced tool to gather information about signals from nonlinear and/or non-stationary systems. It decomposes a signal into a finite set of zero-mean band-limited oscillation modes, so-called analytic Fourier intrinsic band functions (AFIBFs). Owing to its amplitude and frequency modulation properties, each AFIBF enables local analysis of signals. Thus, the determination of AFIBFs is of the key point in performing the FDM. In the traditional case, AFIBFs are obtained iteratively by evaluating numerous inverse discrete Fourier transforms (IDFTs). Also, phase calculations and unwrapping operations must be performed on IDFTs to examine the positivity of instantaneous frequencies. Hence, the classical FDM suffers from heavy computational burden, making it challenging to analyze large-size signals. This paper proposes a new approach to implement the FDM faster than its traditional one, without the need for phase calculation, unwrapping and derivation operations, and also exploits the computational efficiency of the inverse fast Fourier transform. Despite its low computational cost and thus improved computation speed, the new approach decomposes the signal into the same AFIBFs as the conventional one. Simulation results show that the proposed approach outperforms its traditional counterpart in terms of both computational complexity and computation time.