Communications in Mathematical Sciences

Volume 13 (2015)

Number 3

Special Issue in Honor of George Papanicolaou’s 70th Birthday

Guest Editors: Liliana Borcea, Jean-Pierre Fouque, Shi Jin, Lenya Ryzhik, and Jack Xin

Sparse time frequency representations and dynamical systems

Pages: 673 – 694

DOI: http://dx.doi.org/10.4310/CMS.2015.v13.n3.a4

Authors

Thomas Y. Hou (Applied and Computational Mathematics, Caltech, Pasadena, California, U.S.A.)

Zuoqiang Shi (Mathematical Sciences Center, Tsinghua University, Beijing, China)

Peyman Tavallali (Applied and Computational Mathematics, Caltech, Pasadena, California, U.S.A.)

Abstract

In this paper, we establish a connection between the recently developed data-driven time-frequency analysis [T.Y. Hou and Z. Shi, Advances in Adaptive Data Analysis, 3, 1–28, 2011], [T.Y. Hou and Z. Shi, Applied and Comput. Harmonic Analysis, 35, 284–308, 2013] and the classical second order differential equations. The main idea of the data-driven time-frequency analysis is to decompose a multiscale signal into the sparsest collection of Intrinsic Mode Functions (IMFs) over the largest possible dictionary via nonlinear optimization. These IMFs are of the form $a(t) \cos(\theta(t))$, where the amplitude $a(t)$ is positive and slowly varying. The non-decreasing phase function $\theta(t)$ is determined by the data and in general depends on the signal in a nonlinear fashion. One of the main results of this paper is that we show that each IMF can be associated with a solution of a second order ordinary differential equation of the form $\ddot{x} + p(x,t) \dot{x} + q(x,t) = 0$. Further, we propose a localized variational formulation for this problem and develop an effective $l^1$-based optimization method to recover $p(x,t)$ and $q(x,t)$ by looking for a sparse representation of $p$ and $q$ in terms of the polynomial basis. Depending on the form of nonlinearity in $p(x,t)$ and $q(x,t)$, we can define the order of nonlinearity for the associated IMF. This generalizes a concept recently introduced by Prof. N. E. Huang et al. [N.E. Huang, M.-T. Lo, Z. Wu, and Xianyao Chen, US Patent filling number 12/241.565, Sept. 2011]. Numerical examples will be provided to illustrate the robustness and stability of the proposed method for data with or without noise. This manuscript should be considered as a proof of concept.

Keywords

sparse time frequency representations, order of nonlinearity, intrinsic mode function, dynamical system

2010 Mathematics Subject Classification

37M10, 94A12

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