Sparse Representation with Nonlinear Fourier Atoms

In this paper, we study the sparse representation of a finite energy signal with intrinsic mode functions in a dictionary consisting of nonlinear Fourier atoms. Each nonlinear Fourier atom is a mono-component with a physically meaningful nonnegative instantaneous frequency. The sparse representation is obtained adaptively by an orthogonal matching pursuit using a two-level greedy search. It is demonstrated that the representation has efficient energy decay in error compared to the Fourier expansion and wavelet expansion.

[1]  Nelly Pustelnik,et al.  Empirical Mode Decomposition revisited by multicomponent non smooth convex optimization 1 , 2014 .

[2]  Stéphane Mallat,et al.  A Wavelet Tour of Signal Processing - The Sparse Way, 3rd Edition , 2008 .

[3]  Boualem Boashash,et al.  The relationship between instantaneous frequency and time-frequency representations , 1993, IEEE Trans. Signal Process..

[4]  Khaled H. Hamed,et al.  Time-frequency analysis , 2003 .

[5]  S. Qian,et al.  Joint time-frequency analysis : methods and applications , 1996 .

[6]  T. Hou,et al.  Data-driven time-frequency analysis , 2012, 1202.5621.

[7]  Tao Qian,et al.  Adaptive Fourier decompositions and rational approximations, part I: Theory , 2014, Int. J. Wavelets Multiresolution Inf. Process..

[8]  Y. C. Pati,et al.  Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition , 1993, Proceedings of 27th Asilomar Conference on Signals, Systems and Computers.

[9]  Patrick Flandrin,et al.  Time-Frequency/Time-Scale Analysis , 1998 .

[10]  Richard Kronland-Martinet,et al.  Asymptotic wavelet and Gabor analysis: Extraction of instantaneous frequencies , 1992, IEEE Trans. Inf. Theory.

[11]  Qiuhui Chen,et al.  Chirp transforms and Chirp series , 2011 .

[12]  QIUHUI CHEN,et al.  Functions with spline Spectra and their Applications , 2010, Int. J. Wavelets Multiresolution Inf. Process..

[13]  Thomas F. Coleman,et al.  An Interior Trust Region Approach for Nonlinear Minimization Subject to Bounds , 1993, SIAM J. Optim..

[14]  S. Mallat,et al.  Adaptive greedy approximations , 1997 .

[15]  Bernard C. Picinbono,et al.  On instantaneous amplitude and phase of signals , 1997, IEEE Trans. Signal Process..

[16]  N. Huang,et al.  The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[17]  I. Daubechies,et al.  Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool , 2011 .

[18]  Qiuhui Chen,et al.  Time-Frequency Aspects of Nonlinear Fourier Atoms , 2006 .

[19]  Gabriel Peyré,et al.  The Numerical Tours of Signal Processing , 2011, Computing in Science & Engineering.

[20]  P. Loughlin,et al.  On the amplitude‐ and frequency‐modulation decomposition of signals , 1996 .

[21]  Qiuhui Chen,et al.  Sampling theorem and multi-scale spectrum based on non-linear Fourier atoms , 2009 .

[22]  Leon Cohen,et al.  On analytic signals with nonnegative instantaneous frequencies , 1999, 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258).

[23]  Milos Doroslovacki,et al.  On nontrivial analytic signals with positive instantaneous frequency , 2003, Signal Process..

[24]  Stéphane Mallat,et al.  Matching pursuits with time-frequency dictionaries , 1993, IEEE Trans. Signal Process..

[25]  W. K. Melville,et al.  Wave modulation and breakdown , 1983, Journal of Fluid Mechanics.

[26]  X. Rodet.  Time — Domain Formant — Wave — Function Synthesis , 1984 .

[27]  L. Carleson.  On convergence and growth of partial sums of Fourier series , 1966 .

[28]  Boualem Boashash,et al.  Time-Frequency Signal Analysis: Methods and Applications. , 1993 .