Sparse time-frequency decomposition based on dictionary adaptation
In this paper, we propose a time-frequency analysis method to obtain instantaneous frequencies and the corresponding decomposition by solving an optimization problem. In this optimization problem, the basis that is used to decompose the signal is not known a priori. Instead, it is adapted to the signal and is determined as part of the optimization problem. In this sense, this optimization problem can be seen as a dictionary adaptation problem, in which the dictionary is adaptive to one signal rather than a training set in dictionary learning. This dictionary adaptation problem is solved by using the augmented Lagrangian multiplier (ALM) method iteratively. We further accelerate the ALM method in each iteration by using the fast wavelet transform. We apply our method to decompose several signals, including signals with poor scale separation, signals with outliers and polluted by noise and a real signal. The results show that this method can give accurate recovery of both the instantaneous frequencies and the intrinsic mode functions.
© 2016 The Author(s). Published by the Royal Society. Accepted: 29 September 2015; Published 7 March 2016. Data accessibility. Data in examples 6.1, 6.2 and 6.4 are synthetic data. They can be reproduced following the descriptions in the paper. Data used in example 6.3 can be downloaded at http://www.ece.rice.edu/dsp/software/bat.shtml. Authors' contributions. Both authors, T.Y.H. and Z.S., worked together to propose the proper formulation for this problem. Z.S. carried out most of the work in the design and the implementation of the numerical algorithms. He also drafted the manuscript. T.Y.H. revised the manuscript and made sure that we have done a thorough study to demonstrate the accuracy and robustness of the proposed method. Both authors gave final approval for publication. We declare we have no competing interests. This research was supported in part by NSF grants nos. DMS-318377 and DMS-1159138, DOE grant no. DE-FG02-06ER25727, and AFOSR MURI grant no. FA9550-09-1-0613. The research of Z.S. was supported by a NSFC grant no. 11201257.
Submitted - 1311.1163v2.pdf