# Data-driven atomic decomposition via frequency extraction of intrinsic mode functions

- 275 Downloads
- 1 Citations

## Abstract

Decomposition of functions in terms of their primary building blocks is one of the most fundamental problems in mathematical analysis and its applications. Indeed, atomic decomposition of functions in the Hardy space \(H^p\) for \(0<p\le 1\) as infinite series of “atoms” that have the property of vanishing moments with order at least up to 1 / *p* has significant impacts, not only to the advances of harmonic and functional analyses, but also to the birth of wavelet analysis, which in turn allows the construction of sufficiently large dictionaries of wavelet-like basis functions for the success of atomic decomposition of more general functions or signals, by such mathematical tools as “basis pursuit” and “nonlinear basis pursuit”. However, such dictionaries are necessarily huge for atomic decomposition of real-world signals. The spirit of the present paper is to construct the atoms directly from the data, without relying on a large dictionary. Following Gabor, the starting point of this line of thought is to observe that “any” signal *a* can be written as \(a(t) = A(t) \cos \phi (t)\) via complex extension using the Hilbert transform. Hence, if a given signal *f* has been decomposed by whatever available methods or schemes, as the sum of sub-signals \(f_k\), then each sub-signal can be written as \(f_k(t) = A_k(t) \cos \phi _k(t)\). Whether or not \(f_k\) is an atom of the given signal *f* depends on whether any of the sub-signals \(f_k\) can be further decomposed in a meaningful way. In this regard, the most popular decomposition scheme in the current literature is the sifting process of the empirical mode decomposition (EMD), where the sub-signals \(f_k\) are called intrinsic mode functions (IMF’s). The main contribution of our present paper is firstly to demonstrate that IMF’s may not be atoms, and secondly to give a computational scheme for decomposing such IMF’s into finer and meaningful signal building blocks. Our innovation is to apply the signal separation operator (SSO), introduced by the first two authors, with a clever choice of parameters, first to extract the instantaneous frequencies (IF’s) of each IMF obtained from the sifting process, and then (by using the same parameters for the SSO, with the IF’s as input) to construct finer signal building blocks of the IMF. In other words, we replace the Hilbert transform of the EMD scheme by the SSO in this present paper, first for frequency extraction, and then for constructing finer signal building blocks. As an example, we consider the problem in super-resolution of separating two Dirac delta functions that are arbitrarily close to each other. This problem is equivalent to finding the two cosine building blocks of a two-tone signal with frequencies that are arbitrarily close. While the sifting process can only yield one IMF when the frequencies are too close together, the SSO applied to this IMF extracts the two frequencies and recover the two cosine building blocks (or atoms). For this reason, we coin our scheme of sifting \(+\) SSO as “superEMD”, where “super” is used as an abbreviation of super-resolution.

## Keywords

Atomic decomposition Dictionaries for basis pursuit Data-driven atomic decomposition Local sifting process Instantaneous frequencies Empirical mode decomposition (EMD) Synchrosqueezing transform (SST) Hilbert transform Intrinsic mode functions Signal separation operator (SSO) Super-resolution SuperEMD## Mathematics Subject Classification

94A12## Notes

### Acknowledgments

The research of the first two authors is supported in part by Grant W911NF-15-1-0385 from the U. S. Army Research Office.

## References

- Ahlfors, L.V.: Complex Analysis, An Introduction to the Theory of Analytic Functions of One Complex Variable. McGraw-Hill Book Company, New York (1966)Google Scholar
- Candès, E.J., Fernandez-Granda, C.: Super-resolution from noisy data. J. Fourier Anal. Appl.
**19**(6), 1229–1254 (2013)MathSciNetCrossRefMATHGoogle Scholar - Candès, E.J., Fernandez-Granda, C.: Towards a mathematical theory of super-resolution. Commun. Pure Appl. Math.
**67**(6), 906–956 (2014)MathSciNetCrossRefMATHGoogle Scholar - Chen, S.S., Donoho, D.L., Saunders, M.A.: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput.
**20**(1), 33–61 (1998)MathSciNetCrossRefMATHGoogle Scholar - Chui, C.K., Diamond, H.: A general framework for local interpolation. Numer. Math.
**58**(1), 569–581 (1990)MathSciNetCrossRefMATHGoogle Scholar - Chui, C.K., Mhaskar, H.N.: Signal decomposition and analysis via extraction of frequencies. Appl. Comput. Harmon. Anal. (2015). doi: 10.1016/j.acha.2015.01.003. (in press)
- Chui, C.K., van der Walt, M.D.: Signal analysis via instantaneous frequency estimation of signal components. Int. J. Geomath.
**6**(1), 1–42 (2015)MathSciNetCrossRefMATHGoogle Scholar - Chui, C.K., Lin, Y.-T., Wu, H.-T.: Real-time dynamics acquisition from irregular samples—with application to anesthesia evaluation. Anal. Appl. (2015). doi: 10.1142/S0219530515500165
- Coifman, R.R.: A real variable characterization of \({H}^p\). Stud. Math.
**3**(51), 269–274 (1974)MathSciNetMATHGoogle Scholar - Coifman, R.R., Meyer, Y., Stein, E.M.: Some new function spaces and their applications to harmonic analysis. J. Funct. Anal.
**62**(2), 304–335 (1985)MathSciNetCrossRefMATHGoogle Scholar - Coifman, R.R., Weiss, G., et al.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc.
**83**(4), 569–645 (1977)MathSciNetCrossRefMATHGoogle Scholar - Daubechies, I., Maes, S.: A nonlinear squeezing of the continuous wavelet transform based on auditory nerve models. In: Aldroubi, A., Unser, M.A. (eds.) Wavelets in Medicine and Biology, pp. 527–546. CRC Press, Boca Raton (1996)Google Scholar
- Daubechies, I., Lu, J., Wu, H.-T.: Synchrosqueezed wavelet transforms: an empirical mode decomposition-like tool. Appl. Comput. Harmon. Anal.
**30**, 243–261 (2011)MathSciNetCrossRefMATHGoogle Scholar - Demanet, L., Needell, D., Nguyen, N.: Super-resolution via superset selection and pruning. arXiv:1302.6288 (2013). (arXiv preprint)
- Donoho, D.L.: Superresolution via sparsity constraints. SIAM J. Math. Anal.
**23**(5), 1309–1331 (1992)MathSciNetCrossRefMATHGoogle Scholar - Gabor, D.: Theory of communication. J. Inst. Elec. Eng. Part III: Radio Commun. Eng.
**93**(26), 429–441 (1946)Google Scholar - Grafakos, L.: Modern Fourier Analysis, vol. 250. Springer, New York (2009)MATHGoogle Scholar
- Hou, T.Y., Shi, Z.: Data-driven time-frequency analysis. Appl. Comput. Harmon. Anal.
**35**(2), 284–308 (2013a)Google Scholar - Hou, T.Y., Shi, Z.: Sparse time-frequency decomposition by adaptive basis pursuit. arXiv:1311.1163 (2013b). (preprint)
- Huang, N.E., Shen, Z., Long, S.R., Wu, M.C., Shih, H.H., Zheng, Q., Yen, N.-C., Tung, C.C., Liu, H.H.: The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis. Proc. R. Soc. Lond. Ser. A: Math. Phys. Eng. Sci.
**454**(1971), 903–995 (1998)MathSciNetCrossRefMATHGoogle Scholar - Kahane, J.-P.: Turán’s new method and compressive sampling. In: Number Theory, Analysis, and Combinatorics: Proceedings of the Paul Turan Memorial Conference held 22–26 August 2011 in Budapest, p. 155. Walter de Gruyter (2013a)Google Scholar
- Kahane, J.-P.: Variantes sur un théoreme de candes, romberg et tao. Ann. Inst. Fourier (Grenoble)
**63**(6), 2081–2096 (2013b)Google Scholar - Latter, R.: A characterization of \( {H}^p({R}^n)\) in terms of atoms. Stud. Math.
**1**(62), 93–101 (1978)MathSciNetMATHGoogle Scholar - Stein, E.M., Murphy, T.S.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 3. Princeton University Press, Princeton (1993)Google Scholar
- Tang, G., Bhaskar, B.N., Recht, B.: Near minimax line spectral estimation. Inf. Theory IEEE Trans.
**61**(1), 499–512 (2015)MathSciNetCrossRefGoogle Scholar - van der Walt, M.D.: Wavelet analysis of non-stationary signals with applications. Ph.D. thesis, University of Missouri, St. Louis (2015)Google Scholar