Semi-classical signal analysis

This study introduces a new signal analysis method, based on a semi-classical approach. The main idea in this method is to interpret a pulse-shaped signal as a potential of a Schrödinger operator and then to use the discrete spectrum of this operator for the analysis of the signal. We present some numerical examples and the first results obtained with this method on the analysis of arterial blood pressure waveforms.

[1]  L. Trefethen Spectral Methods in MATLAB , 2000 .

[2]  V. Marchenko Sturm-Liouville Operators and Applications , 1986 .

[3]  Stephanos Venakides,et al.  The Small Dispersion Limit of the Korteweg-De Vries Equation , 1987 .

[4]  P. Blanchard,et al.  Bound states for Schrodinger Hamiltonians: Phase space methods and applications , 1996 .

[5]  Ronald Smith The Inverse Scattering Transformation and the Theory of Solitons. By W. ECKHAUS and A. VAN HARTEN. North-Holland, 1981. 222pp. $31.75. , 1982, Journal of Fluid Mechanics.

[6]  Sergei Petrovich Novikov,et al.  NON-LINEAR EQUATIONS OF KORTEWEG-DE VRIES TYPE, FINITE-ZONE LINEAR OPERATORS, AND ABELIAN VARIETIES , 1976 .

[7]  P. Deift,et al.  Inverse scattering on the line , 1979 .

[8]  A. Messiah Quantum Mechanics , 1961 .

[9]  Taous-Meriem Laleg-Kirati,et al.  Separation of arterial pressure into a nonlinear superposition of solitary waves and a windkessel flow , 2007, Biomed. Signal Process. Control..

[10]  Some inverse spectral results for semi-classical Schr\ , 2005, math/0509290.

[11]  H. Holden,et al.  ERRATA: TRACE FORMULAS AND CONSERVATION LAWS FOR NONLINEAR EVOLUTION EQUATIONS , 1994 .

[12]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[13]  D. Robert,et al.  Riesz means of bound states and semiclassical limit connected with a Lieb–Thirring's conjecture , 1990 .

[14]  E. M. Lifshitz,et al.  Quantum mechanics: Non-relativistic theory, , 1959 .

[15]  C. David Levermore,et al.  The Small Dispersion Limit of the Korteweg-deVries Equation. I , 1982 .

[16]  S. Orszag,et al.  Theory and applications of spectral methods , 1984 .

[17]  Francesco Calogero,et al.  Spectral Transform and Solitons , 2012 .

[18]  C. Medigue,et al.  Arterial blood pressure analysis based on scattering transform II , 2007, 2007 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[19]  A new approach to the inverse scattering and spectral problems for the Sturm-Liouville equation , 1998 .

[20]  Validation of a New Method for Stroke Volume Variation Assessment: a Comparaison with the PiCCO Technique , 2009, 0911.0837.

[21]  Sharp Lieb-Thirring inequalities in high dimensions , 1999, math-ph/9903007.

[22]  Qinghua Zhang,et al.  Parsimonious Representation of Signals Based on Scattering Transform , 2008 .

[23]  Y. D. Verdière A semi-classical inverse problem II: reconstruction of the potential , 2008, 0802.1643.

[24]  I. Gel'fand,et al.  On the determination of a differential equation from its spectral function , 1955 .

[25]  Y. Papelier,et al.  Arterial blood pressure analysis based on scattering transform I , 2007, 2007 29th Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

[26]  T. Aktosun,et al.  Inverse Theory: Problem on the Line1 , 2002 .

[27]  B. Vainberg,et al.  First KdV Integrals¶and Absolutely Continuous Spectrum¶for 1-D Schrödinger Operator , 2001 .

[28]  M. Sorine,et al.  Identifiability of a reduced model of pulsatile flow in an arterial compartment , 2007, Proceedings of the 44th IEEE Conference on Decision and Control.

[29]  C. S. Gardner,et al.  Korteweg-devries equation and generalizations. VI. methods for exact solution , 1974 .

[30]  M. Sorine,et al.  Travelling-wave analysis and identification a scattering theory framework , 2007, 2007 European Control Conference (ECC).