Sparse Time-Frequency decomposition by adaptive basis pursuit

In our recent paper, we proposed a data-driven time-frequency analysis method. We also presented convincing numerical evidence to demonstrate the effectiveness of our method. Convergence analysis has been also carried out for periodic signals that satisfy certain scale separation property. In this paper, we propose an improved time-frequency analysis method that can be used to decompose signals that do not have good scale separation property. Our method is formulated as a nonlinear optimization problem using nonlinear basis pursuit. Unlike the classical basis pursuit where the basis is known a priori, the basis in our nonlinear basis pursuit is not known a priori. Instead, it is adapted to the signal and is determined as part of the nonlinear optimization problem. This nonlinear optimization problem is solved by using the Augmented Lagrangian Multiplier method (ALM) iteratively. We further accelerate the convergence of the ALM method in each iteration by using the fast wavelet transform. We apply our method to decompose a number of multiscale data without scale separation, including signals with noise and signals with outliers. Our results show that our method can give accurate recovery of both the instantaneous frequencies and the intrinsic mode functions even for signals that have poor or no scale separation.