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Published March 7, 2016 | public
Journal Article Open

Extracting a shape function for a signal with intra-wave frequency modulation


In this paper, we develop an effective and robust adaptive time-frequency analysis method for signals with intra-wave frequency modulation. To handle this kind of signals effectively, we generalize our data-driven time-frequency analysis by using a shape function to describe the intra-wave frequency modulation. The idea of using a shape function in time-frequency analysis was first proposed by Wu (Wu 2013 Appl. Comput. Harmon. Anal. 35, 181–199. (doi:10.1016/j.acha.2012.08.008)). A shape function could be any smooth 2π-periodic function. Based on this model, we propose to solve an optimization problem to extract the shape function. By exploring the fact that the shape function is a periodic function with respect to its phase function, we can identify certain low-rank structure of the signal. This low-rank structure enables us to extract the shape function from the signal. Once the shape function is obtained, the instantaneous frequency with intra-wave modulation can be recovered from the shape function. We demonstrate the robustness and efficiency of our method by applying it to several synthetic and real signals. One important observation is that this approach is very stable to noise perturbation. By using the shape function approach, we can capture the intra-wave frequency modulation very well even for noise-polluted signals. In comparison, existing methods such as empirical mode decomposition/ensemble empirical mode decomposition seem to have difficulty in capturing the intra-wave modulation when the signal is polluted by noise.

Additional Information

© 2016 The Author(s). Published by the Royal Society. Accepted: 29 September 2015; Published 7 March 2016. Data accessibility. Data in examples 5.1 and 5.2 are synthetic data. They can be reproduced following the descriptions in the paper. Data used in example 5.3 can be found in the electronic supplementary material. 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 ensured that a thorough study of the proposed method was carried out. Both authors gave final approval for publication. The authors declare that they have no competing interests. Funding. This work was supported by NSF FRG grant nos. DMS-1159138, DMS-1318377; AFOSR MURI grant no. FA9550-09-1-0613; and DOE grant no. DE-FG02-06ER25727. The research of Z.S. was supported by NSFC grant no. 11201257.

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August 20, 2023
August 20, 2023