4,010 research outputs found
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 . For integer values of , the most symmetric,
most nearly Gaussian, and generally most time-frequency concentrated member of
the superfamily is found to occur for . 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
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 is the yield per stellar
generation, is the instantaneous ratio between inflow and star
formation rate expressed as a function of the sSFR, and 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
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