On the Convergence of Iterative Filtering Empirical Mode Decomposition
Empirical mode decomposition (EMD), an adaptive technique for data and signal decomposition, is a valuable tool for many applications in data and signal processing. One approach to EMD is the iterative filtering EMD, which iterates certain banded Toeplitz
- PDF / 207,137 Bytes
- 16 Pages / 439.36 x 666.15 pts Page_size
- 83 Downloads / 231 Views
Abstract Empirical mode decomposition (EMD), an adaptive technique for data and signal decomposition, is a valuable tool for many applications in data and signal processing. One approach to EMD is the iterative filtering EMD, which iterates certain banded Toeplitz operators in l 1 .Z/. The convergence of iterative filtering is a challenging mathematical problem. In this chapter we study this problem, namely for a banded Toeplitz operator T and x 2 l 1 .Z/ we study the convergence of T n .x/. We also study some related spectral properties of these operators. Even though these operators don’t have any eigenvalue in Hilbert space l 2 .Z/, all eigenvalues and their associated eigenvectors are identified in l 1 .Z/ by using the Fourier transform on tempered distributions. The convergence of T n .x/ relies on a careful localization of the generating function for T around their maximal points and detailed estimates on the contribution from the tails of x. Keywords Finite impulse response filter • Toeplitz operator • Empirical mode decomposition • Intrinsic mode functions • Iterative filtering
1 Introduction Let a D .ak / 2 l 1 .Z/. We consider the operator Ta W l 1 .Z/!l 1 .Z/ associated with a, given by Ta .x/ D
X j 2Z
aj xkCj
k2Z
Y. Wang () • Z. Zhou Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA, e-mail: [email protected]; [email protected] T.D. Andrews et al. (eds.), Excursions in Harmonic Analysis, Volume 2, Applied and Numerical Harmonic Analysis, DOI 10.1007/978-0-8176-8379-5 8, © Springer Science+Business Media New York 2013
157
158
Y. Wang and Z. Zhou
where x D .xk / 2 l 1 .Z/. In the signal processing literature Ta is called a filter, and it is a finite impulse response (FIR) filter if ak ¤ 0 for only finitely many k 2 Z. Note that Ta is in fact a Toeplitz operator and an FIR filter simply means the Toeplitz operator Ta is banded. In this chapter we shall use the terms filter and Toeplitz operator interchangeably, and only FIR filters and banded Toeplitz operators will be considered. Toeplitz operators are classical operators that have been studied extensively, see [2] and the references therein. There is an even larger literature on filters, which we shall not divulge into. In this chapter our main focus is on the iteration of certain type of banded Toeplitz operators. More precisely, we consider the following question: Let Ta be banded and x 2 l 1 .Z/. When will Tan .x/ converge (in the sense that every entry converges) as n!1? This question arises from signal and data processing using empirical mode decomposition (EMD), which is an important tool for analyzing digital signals and data sets [8, 12]. Our study is motivated primarily by the desire to provide a mathematical framework for EMD. Signal and data analysis is an important and necessary part in both research and practical applications. Understanding large data set is particularly important and challenging given the explosion of data and numerous ways they are being collected today. Often the challenge
Data Loading...
