Wavelet ridge estimation of jointly modulated multivariate oscillations

Wavelet ridge analysis is a technique for estimating the time-varying properties of a modulated oscillatory signal from a contaminated observation. Here this technique is extended to the multivariate case, that is, to a set of N real-valued time series. The bivariate case is illustrated with an application to a set of freely-drifting oceanographic floats. A freely distributed software package implementing this algorithm is available online at http://www.jmlilly.net.

[1]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[2]  Dennis Gabor,et al.  Theory of communication , 1946 .

[3]  Sofia C. Olhede,et al.  Bivariate Instantaneous Frequency and Bandwidth , 2009, IEEE Transactions on Signal Processing.

[4]  Sofia C. Olhede,et al.  Generalized Morse wavelets , 2002, IEEE Trans. Signal Process..

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

[6]  Jonathan M. Lilly,et al.  Multiwavelet spectral and polarization analyses of seismic records , 1995 .

[7]  A. T. Walden,et al.  Polarization phase relationships via multiple Morse wavelets. II. Data analysis , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[8]  Jonathan M. Lilly,et al.  Wavelet ridge diagnosis of time-varying elliptical signals with application to an oceanic eddy , 2006 .

[9]  Frank L. Vernon,et al.  Frequency dependent polarization analysis of high‐frequency seismograms , 1987 .

[10]  Sofia C. Olhede,et al.  Higher-Order Properties of Analytic Wavelets , 2008, IEEE Transactions on Signal Processing.

[11]  A. T. Walden,et al.  Polarization phase relationships via multiple Morse wavelets. I. Fundamentals , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[12]  David S. Stoffer,et al.  Wavelets: An Analysis Tool. , 1997 .

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

[14]  D. Dorson,et al.  The RAFOS System , 1986 .

[15]  Sofia C. Olhede,et al.  On the Analytic Wavelet Transform , 2007, IEEE Transactions on Information Theory.

[16]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[17]  Michael Schröder,et al.  Tracking three meddies with SOFAR floats , 1989 .

[18]  Matthias Holschneider,et al.  Wavelets - an analysis tool , 1995, Oxford mathematical monographs.

[19]  Boualem Boashash,et al.  Estimating and interpreting the instantaneous frequency of a signal. II. A/lgorithms and applications , 1992, Proc. IEEE.

[20]  Stéphane Mallat,et al.  A Wavelet Tour of Signal Processing, 2nd Edition , 1999 .

[21]  H. Voelcker.  Toward a unified theory of modulation part I: Phase-envelope relationships , 1966 .