Characterize highly oscillating frequency modulation using generalized Warblet transform

In this paper, in order to characterize highly oscillating time-frequency patterns of signals, whose instantaneous frequency (IF) is periodic or non-periodic, a generalized Warblet transform (GWT) is proposed. By replacing sine function kernel of conventional Warblet transform with Fourier series function, the GWT is able to generate a time-frequency representation (TFR) with satisfying energy concentration for such signals. As any oscillating function can be well approximated by a Fourier series, the GWT is guaranteed to provide an effective way to achieve accurate IF estimation. In addition, a signal-dependent iterative procedure for coefficients estimation is developed to enable the GWT to be applied in practice. Using the Fourier spectrum of the IF, the coefficients of the Fourier series kernel function of the GWT can be estimated and refined adaptively. The effectiveness of the proposed method is verified through comparing with other time-frequency analysis methods on several numerical examples and experimental vibration signal, which is collected from a rotor test rig undergoing speed-up and slow-down stages.

Research highlights
► We proposed a time frequency analysis method, named as generalized Warblet transform (GWT).
► We developed an iterative procedure to determine the coefficients of the kernel function for the GWT.
► We examined the effectiveness of the GWT.