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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

or discontinuities.

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



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




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