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   -> Volume 3, Issue 5


Answer: Wavelets and time series (WD 3.4 #2)
 
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Lonnie Hudgings (lonnie@golem.ps.uci.edu)
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PostPosted: Mon Dec 02, 2002 12:55 pm    
Subject: Answer: Wavelets and time series (WD 3.4 #2)
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Answer: Wavelets and time series (WD 3.4 #2)

Dear Esben,

In response to your inquiry regarding wavelet spectra for non-
stationary time series, I would like to offer a paper of mine:
"Wavelet Transforms and Spectral Estimation,"
by Lonnie H Hudgins.
This paper, which became part of my Ph.D. dissertation in 1992, explores the
relationships between wavelet spectra and Fourier spectra. Below is an
abstract:

"The wavelet transform is defined here in terms of the scale number
variable s instead of the more common scale length variable a where a = 1/s.
This convention permits the scale number for wavelets to compare directly
with wave number in Fourier, and the theory of wavelet spectral estimation
proceeds in parallel with that of Fourier transforms. After reviewing some
motivation for the usual Fourier power spectrum, definitions are given for
both one-sided and two-sided wavelet power spectra. These 'spectral densities'
are then interpreted in the context of standard Fourier theory and banks of
spectral estimation filters. Polarization then yields definitions for wavelet
cross spectrum, wavelet cospectrum, and wavelet quadrature spectrum, which
agree with their Fourier counterparts whenever we could reasonably expect
them to. These tools also provide us with the helpful notions of wavelet
transfer function, and wavelet coherence. Finally, the theory leads to a
bivariate wavelet cross transform: a useful new function for pairs of signals
which behaves like the Fourier cross transform, i.e., the magnitude
emphasizes signal similarities, and the phase contains relative timing
information between the two signals, while inheriting the 'zoom-in'
properties of wavelets."

The ideas developed in this paper have been successfully applied to
detect, identify, and locate intermittent large scale coherent structures in
turbulent fluid flow by analyzing the off-diagonal components in the
generalized Reynolds stress tensor (see Physical Review Letters, Nov 1993).

Good Luck,
Lonnie
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