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results as another
and one function will give the same qualitative
then the choice of wavelet function is not critical,
is primarily interested in wavelet power spectra,
smooth function such as a damped cosine. If one
smoothly varying time series one would choose a
a boxcar-like function such as the Harr, while for
with sharp jumps or steps, one would choose
of features present in the time series. For time series
4) Shape. The wavelet function should reflect the type
good frequency resolution.
broad function will have poor time resolution, yet
resolution but poor frequency resolution, while a
A narrow (in time) function will have good time
width in real space and the width in Fourier space.
function is determined by the balance between the
wavelet amplitude. The resolution of a wavelet
function is defined here as the e-folding time of the
3) Width. For concreteness, the width of a wavelet
single component and can be used to isolate peaks
A real wavelet function returns only a
and is better adapted for capturing oscillatory behavior.
return information about both amplitude and phase
2) Complex or real. A complex wavelet function will
amplitude are expected.
where smooth, continuous variations in wavelet
transform is useful for time series analysis,
times is highly correlated. The nonorthogonal
at large scales, where the wavelet spectrum at adjacent
(such as used in this study) is highly redundant
spectrum. Conversely, a nonorthogonal analysis
shift in the time series produces a different wavelet
for time series analysis, an aperiodic
the most compact representation of the signal. Unfortunately
power and is useful for signal processing as it gives
that contains discrete “blocks” of wavelet
basis at that scale. This produces a wavelet spectrum
each scale is proportional to the width of the wavelet
wavelet analysis, the number of convolutions at
1) Orthogonal or nonorthogonal. In orthogonal
Bessel, Legendre, etc.) In choosing the wavelet function, there are several factors which should be considered (for more discussion see Farge 1992).
One criticism of wavelet analysis is the arbitrary choice of the wavelet function, y0(h). (It should be noted that the same arbitrary choice is made in using one of the more traditional transforms such as the Fourier,
Functions of Wavelet
transform of such capability is not given.
in frequency. The conventional two dimensional Fourier
accuracy simultaneously both in a time-domain, and
allows to conduct the analysis with the necessary
axis – shift state, another – parameter of scaling. It
researched signal appears to be two dimensional: one
and called as wavelet. The wavelet-spectrum of a
basis only one functions possessing special properties
the help of stretching-compressions and shifts on the
area. This basis is remarkable because it is built with
restricted both in temporary, and in frequency
decomposition of a parsed signal on basis of functions,
century 2,3. The essence of the method consists in
method was designed in middle 80s of the twentieth
The wavelet analysis as a new mathematical
wavelet analysis is represented.
techniques of such signals. Expedient usage of the
is the application of new mathematical research
One of possible paths of escaping of the given situation
Wavelet analysis is becoming a common tool for analyzing localized variations of power within a time series. By decomposing a time series into time–frequency space, one is able to determine both the dominant modes of variability and how those modes vary in time.
at infinity does not play any role.
analyzing wavelet before convolving it with the signal. The limited spatial support of wavelets is important because then the behavior of the signal
use analyzing functions, called wavelets, which are localized in space.1 The scale decomposition is obtained by dilating or contracting the chosen
Wavelets are mathematical techniques, based on group theory and square integrable representations, which allows one to unfold a signal, or a field, into both space and scale, and possibly directions. They
number of techniques which had been developed independently for various signal processing applications.
Wavelet theory based on unified framework for a