Amplitudes of mono-component signals and the generalized sampling functions

There is a recent trend to use mono-components to represent nonlinear and non-stationary signals rather than the usual Fourier basis with linear phase, such as the intrinsic mode functions used in Norden Huang's empirical mode decomposition [12]. A mono-component is a real-valued signal of finite energy that has non-negative instantaneous frequencies, which may be defined as the derivative of the phase function of the given real-valued signal through the approach of canonical amplitude-phase modulation. We study in this paper how the amplitude is determined by its phase for a class of signals, of which the instantaneous frequency is periodic and described by the Poisson kernel. Our finding is that such an amplitude can be perfectly represented by a sampling formula using the so-called generalized sampling functions that are related to the phase. The regularity of such an amplitude is identified to be at least continuous. Such characterization of mono-components provides the theory to adaptively decompose non-stationary signals. Meanwhile, we also make a very interesting and new characterization of the band-limited functions.

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

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

[3]  Qiuhui Chen,et al.  Analytic unit quadrature signals with nonlinear phase , 2005 .

[4]  A. Nuttall,et al.  On the quadrature approximation to the Hilbert transform of modulated signals , 1966 .

[5]  P. Ktonas,et al.  Instantaneous envelope and phase extraction from real signals: Theory, implementation, and an application to EEG analysis , 1980 .

[6]  R. L. Stens,et al.  The Shannon sampling series and the reconstruction of signals in terms of linear, quadratic and cubic splines , 1986 .

[7]  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).

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

[9]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[10]  E. Bedrosian.  A Product Theorem for Hilbert Transforms , 1963 .

[11]  U. Kähler,et al.  A necessary and sufficient condition for a Bedrosian identity , 2010 .

[12]  R. Kumaresan,et al.  Model-based approach to envelope and positive instantaneous frequency estimation of signals with speech applications , 1999 .

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

[14]  Joerg F. Hipp,et al.  Time-Frequency Analysis , 2014, Encyclopedia of Computational Neuroscience.

[15]  Stefan L. Hahn,et al.  On the uniqueness of the definition of the amplitude and phase of the analytic signal , 2003, Signal Process..

[16]  Yi Wang,et al.  A sampling theorem for non-bandlimited signals using generalized Sinc functions , 2008, Comput. Math. Appl..

[17]  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.

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

[19]  Boualem Boashash,et al.  Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals , 1992, Proc. IEEE.

[20]  S. Hahn,et al.  The History of Applications of Analytic Signals in Electrical and Radio Engineering , 2007, EUROCON 2007 - The International Conference on "Computer as a Tool".

[21]  C.E. Shannon,et al.  Communication in the Presence of Noise , 1949, Proceedings of the IRE.

[22]  William E. Sabin.  The Hilbert Transform , 2007 .