Modulated Oscillations in Three Dimensions

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. Rates of change of the ellipse parameters are found to be intimately linked to the first few moments of the signal's spectrum, averaged over the three signal components. In particular, the trivariate instantaneous bandwidth---a measure of the instantaneous departure of the signal from a single pure sinusoidal oscillation---is found to contain five contributions: three essentially two-dimensional effects due to the motion of the ellipse within a fixed plane, and two effects due to the motion of the plane containing the ellipse. The resulting analysis method is an informative means of describing nonstationary trivariate signals, as is illustrated with an application to a seismic record.Comment: IEEE Transactions on Signal Processing, 201

Higher-Order Properties of Analytic Wavelets

The influence of higher-order wavelet properties on the analytic wavelet transform behavior is investigated, and wavelet functions offering advantageous performance are identified. This is accomplished through detailed investigation of the generalized Morse wavelets, a two-parameter family of exactly analytic continuous wavelets. The degree of time/frequency localization, the existence of a mapping between scale and frequency, and the bias involved in estimating properties of modulated oscillatory signals, are proposed as important considerations. Wavelet behavior is found to be strongly impacted by the degree of asymmetry of the wavelet in both the frequency and the time domain, as quantified by the third central moments. A particular subset of the generalized Morse wavelets, recognized as deriving from an inhomogeneous Airy function, emerge as having particularly desirable properties. These "Airy wavelets" substantially outperform the only approximately analytic Morlet wavelets for high time localization. Special cases of the generalized Morse wavelets are examined, revealing a broad range of behaviors which can be matched to the characteristics of a signal.Comment: 15 pages, 6 Postscript figure

Generalized Morse Wavelets as a Superfamily of Analytic Wavelets

The generalized Morse wavelets are shown to constitute a superfamily that essentially encompasses all other commonly used analytic wavelets, subsuming eight apparently distinct types of analysis filters into a single common form. This superfamily of analytic wavelets provides a framework for systematically investigating wavelet suitability for various applications. In addition to a parameter controlling the time-domain duration or Fourier-domain bandwidth, the wavelet {\em shape} with fixed bandwidth may be modified by varying a second parameter, called $\gamma$. For integer values of $\gamma$, the most symmetric, most nearly Gaussian, and generally most time-frequency concentrated member of the superfamily is found to occur for $\gamma=3$. These wavelets, known as "Airy wavelets," capture the essential idea of popular Morlet wavelet, while avoiding its deficiencies. They may be recommended as an ideal starting point for general purpose use

On the Analytic Wavelet Transform

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. Closed-form expressions for these functions are found in terms of Bell polynomials and derivatives of the signal's instantaneous frequency and bandwidth. The analytic wavelet transform is shown to depend upon the interaction between the signal's instantaneous modulation functions and frequency-domain derivatives of the wavelet, inducing a hierarchy of departures of the transform away from a perfect representation of the signal. The form of these deviation terms suggests a set of conditions for matching the wavelet properties to suit the variability of the signal, in which case our expressions simplify considerably. One may then quantify the time-varying bias associated with signal estimation via wavelet ridge analysis, and choose wavelets to minimize this bias

Analysis of Modulated Multivariate Oscillations

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. The lowest-order bias is shown to involve a quantity, termed the instantaneous curvature, which measures the strength of local quadratic modulation of the signal after demodulation by the common oscillation frequency. The bias can be made to be small if the analysis filter, or wavelet, can be chosen such that the signal's instantaneous curvature changes little over the filter time scale. An application is presented to the detection of vortex motions in a set of freely-drifting oceanographic instruments tracking the ocean currents

On the relation between sSFR and metallicity

In this paper we present an exact general analytic expression $Z(sSFR)=y/\Lambda(sSFR)+I(sSFR)$ linking the gas metallicity Z to the specific star formation rate (sSFR), that validates and extends the approximate relation put forward by Lilly et al. (2013, L13), where $y$ is the yield per stellar generation, $\Lambda(sSFR)$ is the instantaneous ratio between inflow and star formation rate expressed as a function of the sSFR, and $I$ is the integral of the past enrichment history, respectively. We then demonstrate that the instantaneous metallicity of a self-regulating system, such that its sSFR decreases with decreasing redshift, can be well approximated by the first term on the right-hand side in the above formula, which provides an upper bound to the metallicity. The metallicity is well approximated also by the L13 ideal regulator case, which provides a lower bound to the actual metallicity. We compare these approximate analytic formulae to numerical results and infer a discrepancy <0.1 dex in a range of metallicities and almost three orders of magnitude in the sSFR. We explore the consequences of the L13 model on the mass-weighted metallicity in the stellar component of the galaxies. We find that the stellar average metallicity lags 0.1-0.2 dex behind the gas-phase metallicity relation, in agreement with the data. (abridged)Comment: 14 pages, 6 figures, MNRAS accepte