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Application of Wavelet Transform for Determining
Volodymyr Eremenko1[0000-0002-4330-7518], Artur Zaporozhets2[0000-0002-0704-4116],
Volodymyr Isaenko3[0000-0001-8010-8844], Kateryna Babikova3[0000-0002-5053-1999]
1 National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnical Institute”, Kyiv,
2 Institute of Engineering Thermophysics of NAS of Ukraine, Kyiv, Ukraine
3 National Aviation University, Kyiv, Ukraine
Abstract. It is proposed to apply the wavelet transform to localize in time the
frequency components of the information signals in this article. The wavelet
transform allows to fulfil time-frequency analysis of signals, which is very im-
portant for studying the structure of a composite material from the mode compo-
sition of free oscillations. The proposed approach to the development of infor-
mation signals using wavelet transform makes it possible to further study the na-
ture of the occurrence of free oscillations and the propagation of acoustic waves
in individual layers of composites and to study the change in the structure of
composites from the changes in the three-dimensional wavelet spectrum.
Keywords: wavelet transform, MHAT wavelet, Morlet wavelet, diagnostic
signs, information signal, composite material, free oscillation method
Information signals obtained in the process of diagnosing composite materials by low-
frequency acoustic methods belong to the class of single-pulse signals with locally con-
centrated features. These are, for example, signals of free oscillations, whose modes
have not only frequency, but also temporal distribution, signals of impulse impedance
method change frequency and current carrier phase for one radio pulse, signals of low-
speed impact method locally change their shape depending on material defectiveness.
For such signals, the task of identifying diagnostic signs is significantly more diffi-
cult than for signals in which the information component is evenly distributed over the
observation interval [1, 2]. This is explained by the fact that diagnostic signs are focused
on small time intervals or fragments of signal realization, and the signal itself has a
rather complex form that cannot be described by a formal constructive model. The most
common methods for isolating diagnostic features of such signals are [3, 4, 5, 6]:
methods for evaluating the integral characteristics ‒ the center of mass of the pulse,
the similarity coefficients, etc. These methods have high noise immunity, but have
very little sensitivity to local changes in signal parameters;
methods of decomposition of signals with an orthogonal basis. In general, they pro-
vide information about the shape of the pulsed signal, but provide only an integral
representation of its components throughout the entire domain of definition and are
not sensitive to local variations in characteristics;
methods of structural analysis using signal segmentation as a sequence of separate
fragments. Based on segmentation signal clustering is performed, chains of clusters
are used to structure the signal;
methods for representing signals in phase space ‒space, which is determined by a
finite set of state parameters. The disadvantage is the need for multiple repetition of
the pulse signal and the analysis of multidimensional data arrays;
heuristic methods, in particular, methods using neural network technologies, which
allow to select informative fragments of signals and to make a comparison with the
Classical methods for processing signals of low-frequency acoustic diagnostic meth-
ods, in particular, the method of decomposing signals with an orthogonal basis (gener-
alized Fourier transform), which give an integral representation of the signal compo-
nents in the entire domain of their definition, are ineffective . Therefore it was pro-
posed to apply the wavelet transform for localizing the frequency components of infor-
mation signals in time .
The signals of free oscillations are damped and consist of several modes, then each
mode, depending on which layer of the composite it is excited, will have its own am-
plitude, frequency and attenuation coefficient. That is, knowing the time behavior of
each mode, one can draw conclusions about the structure of the controlled zone of the
composite . The continuous wavelet transform allows to carry out a time-frequency
analysis . These studies can be used in many areas, including the monitoring of
complex technical systems [11, 12].
2 Main Part
The wavelet transform combines two types of transformations ‒ direct and inverse,
which, respectively, translate the studied function f(t) into a set of
( , )W a b f
coefficients and vice versa . The direct wavelet transform is performed according
to the rule:
( , ) ( )
W a b f f t dt
where a and b are the parameters that determine, respectively, the scale and offset of
the function ψ, which is called the analyzing wavelet; Cψ is normalization coefficient.
The basic, or maternal, wavelet ψ forms with the help of stretch marks and landslides
a family of functions ψ (t–b/a). Having a known set of coefficients Wψ(a, b)f, we can
restore the original form of the function f(t):
( ) ( , ) .
t b da db
f t W a b f
The direct (1) and the inverse (2) transforms depend on some function
( ) ( )t L R
which is called the basic wavelet. In practice, the only restriction on its choice is the
condition for the finiteness of the normalizing coefficient :
( ) ( )
2,C d d
is Fourier image of the
ˆ( ) ( ) .
t e dt
This condition satisfies many functions, so it is possible to choose the type of wavelet
that is most suitable for solving a specific problem. In particular, for analyzing damped
harmonic oscillations, it is more expedient to select wavelets, which are also damped
oscillations. The article deals with the MHAT-wavelet and Morlet wavelet.
The condition (3) means that the Fourier transform of the wavelet is zero at zero
ˆ( ) 0
. In another case the denominator of the fraction in the in-
tegral (3) is equal to zero, while the numerator has a nonzero value, and the Сψ coeffi-
cient ceases to be finite.
In turn, this requirement can be presented in another form. Since the Fourier trans-
at zero frequency has the form
, it can write the following:
ˆ( ) ( ) .
t e dt
A characteristic feature of the analyzing wavelets is time-frequency localization. This
means that wavelets ψ(t) and their Fourier transforms
differ significantly from
zero only at small time intervals and frequencies, and differ very little from zero (or
simply equal to zero) outside these intervals.
The quantitative measure of the localization of a function
( ) ( )tL
can be its
1( ) ,t t t dt
1( ) .
tt t t dt
In this case, the effective wavelet width is assumed to be
Nowadays a large number of basic wavelet functions are known [15, 16]. As men-
tioned above, the members of any family of wavelets must satisfy condition (7) one of
such families, the Gauss and wavelets. The functions of this family are derived from
the Gaussian exponent:
( ) ( 1) n
g t e
The normalization coefficient takes the value
2 ( 1)!
The most widely used Gaussian wavelets of small orders. The properties of Gaussian
wavelets are discussed in detail in .
2.1 MHAT Wavelet
This wavelet is a second-order wavelet of the family of Gaussian wavelets (Fig. 1) and
is formed by a double differentiation of the Gauss function:
2( ) (1 ) .
g t t e
Fig. 1. Second order Gaussian wavelet (MHAT) (a) and its Fourier transform (b)
The Fourier transform of this wavelet is
The graph of this function is shown in Fig. 1 (b). MHAT wavelets are well localized in
both the time and frequency domains. The centers and localization radii in both areas
have the following meanings;
2.2 Morlet Wavelet
The analytical representation of the Morlet wavelet and its Fourier transform is given
by the following expressions:
4ik t k
t e e e
The Morlet wavelet is a plane wave modulated by a Gausian. The parameter α specifies
the width of the Gausian, the parameter k0 is the frequency of the plane wave. Usually
. With these values with sufficient accuracy it can be
222t i t
t e e
Graphs of these functions are shown in Fig. 2.
Fig. 2. The real part of the Morlet wavelet (a) and its Fourier transform (b)
The center and localization radius of the Morlet wavelet in the time domain are deter-
mined by the corresponding values;
. In the Morlet wavelet, only
zero moment is equal zero.
The definitions of the integral wavelet transform introduced above cannot be used
in practice, since in digital processing of results the main transformation objects are not
functions defined on the entire time axis, but discrete signals whose length are always
finite . For this reason, instead of the above theoretical concepts, their practical
counterparts (assessments) should be introduced.
We assume that the signal is given by the function values with a constant step Δt:
f f t
t t k
To estimate the wavelet transform of this sequence, we use the following expression:
( , ) ( , )
W a b f
n a b a
( , ) k
n a b e
where B=2 for MHAT wavelet and B=α2 for Morlet wavelet
In the transition from (1) to (17), the multiplier
from the denominator of the
formula (1) is replaced as follows
e dt a B
the discrete approximation of which is function (18).
This made it possible to eliminate the dependence of the amplitudes of the harmonic
components on the parameter a, which usually makes it difficult to correctly estimate
their relative intensities from the graphic image of the wavelet spectra. In addition, the
function n(a,b) as an approximation allows to “equalize” for different values of the
scale factor a the number of samples of the original function involved in the calculation
Calculating the wavelet transform in the scale-offset coordinates is somewhat incon-
venient for perception, since the scale ai specified with constant pitch compresses the
high-frequency region and the components of the signal under study that belong to this
region become difficult to distinguish. Therefore, it is proposed by replacing
to switch to νi value that is analogous to the frequency in the Fourier transform .
Then a pair of wavelet transforms of the function f(t) with (17) will look like this:
( , ) ( ) ( )
( , )
W b f t b f t dt
( ) ( ) ( , )
f t t b W b f d db
( , ) tb
n b e dt
The amplitude value of the discrete signal wavelet function will be calculated by the
( , ) ( )
( , )
i j k k j i
W b f t b
( , ) k j i
n b e
2.3 Discretization of Arguments
Each wavelet has its own shape and characteristic size, which for a fixed value of the
scale factor is determined by the value
where Δt is the wavelet radius.
The function W(a,b) (22) determines the correlation between the analyzing wavelet
located at a point b and at a certain part of a signal of da length with a center at a point
b. The module of this function takes the greatest value in the case when the size of the
wavelet coincides with the size of the "current" signal detail. In the case of polyhar-
monic functions, the natural measure of the scale of its details is the period of the har-
monic components, while the measure of the wavelet length da is determined by the
value a of the scale factor.
For a polyharmonic function defined on a grid with a step Δt=const, the range of
periods of harmonics is determined by the quantities
max ( 1) t
accordance with this, the largest and smallest values of the scale factor are selected
from the condition of matching the size of the wavelet and the limiting periods of har-
where we get
max ( 1) 2 t
a N t
Note that these values are taken in cases where it is necessary to perform a signal
analysis in the full scale range. Often, however, it is advisable to examine the signal in
a narrower range of scales. In this case, the value amin and amax choose from other con-
We propose a discretization step
of the scales, after which we de-
fine the discrete values of the scale factors
a a a i
Since the width of the spectral line increases with increasing scale, sometimes the
value of the parameter is presented on a logarithmic scale.
If the calculation of the wavelet transform is carried out according to (22), (23), then
the minimum and maximum value of the quantity ν can be calculated using equations
(25). Values νi are calculated with a constant step
by the formula
In the simplest case, the boundaries of the landslide range are defined as follows
max ( 1)b N t
, and discrete values of the displacements can be calculated
by the following formulas:
b b b j
With this method of discretization of the parameter b near the boundaries bmin and
bmax of magnitude W(ai,bj) will be calculated with errors, since it is impossible to use
the entire length of the analyzing wavelet near the boundaries. To exclude marginal
effects, it is necessary to calculate the wavelet transform only for landslide values that
are remote from the boundaries by an amount equal to the current radius of the wavelet
aΔt. With this approach, formula (26) is transformed as follows
b b b j
j J Nb J
is the radius of the wavelet, expressed in units Δb, which corresponds to the
current scale value a (for the Morlet wavelet)
In the case of application instead of the scale of the formula we get:
In these formulas, the rounding operation to the nearest integer is indicated by square
The set of nodes of the discrete grid, which is defined by formulas (25) and (29), is
called the probability triangle. Note that very often the edge effects are ignored, and the
results of the wavelet analysis are simply represented in the rectangular area of the
nodes (25) and (27).
2.4 Construction of a Plurality of Diagnostic Features
The selection of each individual mode of oscillation is an important step in the study of
the properties and nature of the destruction of composite materials. Attenuation of the
components of free oscillations carries information about the quality factor of the con-
trolled zone of the composite. This allows to investigate defects that are not associated
with delamination, such as fatigue and impact damage to the surface.
The proposed approach with the use of wavelet transform makes it possible to further
study the nature of the propagation of acoustic waves in individual layers of composites
and to investigate the change in the structure of composites by the revealed changes in
the wavelet spectrum of free vibrations.
Next we consider the wavelet transform of signals received in the intact and damaged
area of a cellular panel with a thickness of 20 mm. The amplitude spectra of these sig-
nals are shown in Fig. 3, 4.
Fig. 3. Amplitude spectrum of the signal of free oscillations of the intact area of the cellular panel
Fig. 4. Amplitude spectrum of a signal of free oscillations of a zone with a defect of 20 mm radius
of a cellular panel
The estimated amplitude spectra preliminarily determine the frequency range within
which the wavelet transform will be performed — zone I in Fig. 3, 4.
Fig. 5 shows plots of the amplitude wavelet spectra of these signals, calculated by a
Morlet wavelet in the selected frequency range.
Fig. 5. Graphs of amplitude wavelet functions of signals of free oscillations: a ‒ intact zone, b ‒
zone with damage of 20 mm radius
From these figures it can be seen that, for example, the 1st, 2nd and 3rd modes of oscil-
lations change the nature of the attenuation with the appearance of a defect. In fig. 6
shows the restored third mode oscillations of the benign zone and the damaged zone.
Its attenuation coefficient changes with the appearance of a defect from 4.72 to 2.56.
For faster decision making on the presence or absence of a defect in the controlled
area of a composite material, it is proposed to compare the amplitude wavelet spectra
of free oscillations of the reference and controlled areas, calculated with the same shift.
For example, in fig. 7(a) amplitude wavelet spectra of a signal of free oscillations of a
benign zone with shifts
b b j
b b j
b b j
Fig. 7(b) shows similar spectra of free oscillations of a zone with a defect diameter of
20 mm. In other words, these spectra are actually a cross section of the graphs in Fig. 5.
Fig. 6. The third damping component of free vibrations of the benign zone (S1) and the damaged
Fig. 7. Plots of amplitude wavelet spectra of free vibrations of the intact zone (a) and zones with
a separation of 20 mm (b) at displacements b1<b2 <b3
The obtained wavelet spectra in fig. 7 are convenient for visual comparison, and also
allow you to more accurately determine the frequency range of each individual mode
in order to reduce errors in its recovery.
A constructive mathematical model of the information signal field of the process of
diagnosing composite materials was constructed, which made it possible to describe the
interaction of mechanical perturbation fields in composite materials with defects of var-
ious types. This allowed to use experimental results for statistical evaluation of field
characteristics, to conduct a wide range of mathematical and computer model experi-
Methods of primary processing of information signals of acoustic diagnostic meth-
ods in time-frequency coordinates have been improved and investigated, which made
it possible to carry out a structural analysis of single-pulse signals and signals with
locally concentrated parameter changes and to increase the probability of diagnosis by
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