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Jonathan M. Lilly

  1. Generalized Morse Wavelets as a Superfamily of Analytic Wavelets.

    Пт, 03/16/2012 - 15:01 — SomNambul
    Authors: Sofia C. Olhede, Jonathan M. Lilly
    Subjects: Methodology
    Abstract

    The generalized Morse wavelets are shown to constitute a superfamily that
    essentially encompasses all other commonly used analytic wavelets. Details of
    the time/frequency concentration, frequency-domain symmetry, and Gaussianity of
    these wavelets are investigated. The generalized Morse wavelets are controlled
    by two parameters, one determining the Fourier-domain bandwidth, and the
    second, called $\gamma$, determining the lowest-order departure of the wavelet
    from a Gaussian form.

  2. Bivariate Instantaneous Frequency and Bandwidth.

    Втр, 10/18/2011 - 15:01 — SomNambul
    Authors: Sofia C. Olhede, Jonathan M. Lilly
    Subjects: Methodology
    Abstract

    The generalizations of instantaneous frequency and instantaneous bandwidth to
    a bivariate signal are derived. These are uniquely defined whether the signal
    is represented as a pair of real-valued signals, or as one analytic and one
    anti-analytic signal. A nonstationary but oscillatory bivariate signal has a
    natural representation as an ellipse whose properties evolve in time, and this
    representation provides a simple geometric interpretation for the bivariate
    instantaneous moments.

  3. Modulated oscillations in three dimensions.

    Ср, 04/13/2011 - 15:01 — SomNambul
    Authors: Jonathan M. Lilly
    Subjects: Methodology
    Abstract

    The analysis of the fully three-dimensional and time-varying polarization
    characteristics of a modulated trivariate, or three-component, oscillation is
    addressed. The use of the analytic operator enables the instantaneous
    three-dimensional polarization state of any square-integrable trivariate signal
    to be uniquely defined. Straightforward expressions are given which permit the
    ellipse parameters to be recovered from data. The notions of instantaneous
    frequency and instantaneous bandwidth, generalized to the trivariate case, are
    related to variations in the ellipse properties.

  4. Analysis of Modulated Multivariate Oscillations.

    Ср, 04/13/2011 - 15:01 — SomNambul
    Authors: Sofia C. Olhede, Jonathan M. Lilly
    Subjects: Methodology
    Abstract

    The concept of a common modulated oscillation spanning multiple time series
    is formalized, a method for the recovery of such a signal from potentially
    noisy observations is proposed, and the time-varying bias properties of the
    recovery method are derived. The method, an extension of wavelet ridge analysis
    to the multivariate case, identifies the common oscillation by seeking, at each
    point in time, a frequency for which a bandpassed version of the signal obtains
    a local maximum in power.

  5. On the Analytic Wavelet Transform.

    Чт, 10/29/2009 - 16:01 — SomNambul
    Authors: Sofia C. Olhede, Jonathan M. Lilly
    Subjects: Statistics
    Abstract

    An exact and general expression for the analytic wavelet transform of a
    real-valued signal is constructed, resolving the time-dependent effects of
    non-negligible amplitude and frequency modulation. The analytic signal is first
    locally represented as a modulated oscillation, demodulated by its own
    instantaneous frequency, and then Taylor-expanded at each point in time. The
    terms in this expansion, called the instantaneous modulation functions, are
    time-varying functions which quantify, at increasingly higher orders, the local
    departures of the signal from a uniform sinusoidal oscillation.

Math-arch

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