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Article
A new wavelet family based
on second-order LTI-systems
Tariq Abuhamdia
1
, Saied Taheri
1
and John Burns
2
Abstract
In this paper, a new family of wavelets derived from the underdamped response of second-order Linear-Time-Invariant
(LTI) systems is introduced. The most important criteria for a function or signal to be a wavelet is the ability to recover
the original signal back from its continuous wavelet transform. We show that it is possible to recover back the original
signal once the Second-Order Underdamped LTI (SOULTI) wavelet is applied to decompose the signal. It is found that
the SOULTI wavelet transform of a signal satisfies a linear differential equation called the reconstructing differential
equation, which is closely related to the differential equation that produces the wavelet. Moreover, a time-frequency
resolution is defined based on two different approaches. The new transform has useful properties; a direct relation
between the scale and the frequency, unique transform formulas that can be easily obtained for most elementary signals
such as unit step, sinusoids, polynomials, and decaying harmonic signals, and linear relations between the wavelet
transform of signals and the wavelet transform of their derivatives and integrals. The results obtained are presented
with analytical and numerical examples. Signals with constant harmonics and signals with time-varying frequencies are
analyzed, and their evolutionary spectrum is obtained. Contour mapping of the transform in the time-scale and the time-
frequency domains clearly detects the change of the frequency content of the analyzed signals with respect to time.
The results are compared with other wavelets results and with the short-time fourier analysis spectrograms. At the end,
we propose the method of reverse wavelet transform to mitigate the edge effect.
Keywords
SOULTI-wavelets, second-order systems wavelets, LTI-wavelets, second-order linear-time-invariant wavelets, time-fre-
quency analysis, spectrogram, scalogram, chirp analysis, frequency-identification, edge-effect, forward-wavelet-transform,
forward wavelet transform, reverse-wavelet-transform, reverse wavelet transform
1. Introduction
Wavelets provide a powerful tool to analyze signals
and extract information from them. They are capable
of extracting frequency, time, and nonharmonic infor-
mation. These potentials lured many scholars to use
them in the analysis of dynamic systems. Scholars
have used wavelets for system identification, system
modeling, system response solution, and even control
design. For broad and extensive survey on the use of
wavelets in systems and control, the reader is referred
to Abuhamdia and Taheri (2015).
Mathematically, the wavelet transform is an inner
product between a function and a set of basis functions
which are all derived from a single function called the
mother wavelet. It measures how much parallelism
exists between the analyzed function and the set of
basis functions. Therefore, if we seek to extract some
features from a signal, then the analyzing wavelet
family should also have these features. This is similar
to the way we measure the periodicity of a signal by
making inner product with the harmonic functions
because of their periodicity.
This idea also implies that if we want to use time-
frequency analysis on a dynamic system by analyzing
its response, a better understanding can be developed if
1
Center for Tire Research (CenTire), Virginia Polytechnic Institute and
State University, USA
2
Interdisciplinary Center for Applied Mathematics (ICAM), Virginia
Polytechnic Institute and State University, USA
Corresponding author:
Tariq Abuhamdia, Department of Mechanical Engineering, Virginia Tech,
100T Randolph Hall, 460 Old Turner St, Blacksburg, VA, 24060, USA.
Email: atariqm@vt.edu
Received: 18 May 2016; accepted: 4 September 2016
Journal of Vibration and Control
1–20
!The Author(s) 2016
Reprints and permissions:
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DOI: 10.1177/1077546316674089
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the analyzing wavelet is close in characteristics to sys-
tems responses and behaviors.
This observation led to investigating the character-
istics of the underdamped second-order response of
Linear Time-Invariant (LTI) systems to find if it can
serve as a mother wavelet. The underdamped second-
order impulse response of LTI systems is oscillatory
and decaying exponentially, and it dies out to effective
zero well within the defined period. Furthermore, its
frequency domain representation is a second-order
filter that can effectively extract certain frequency
bands from the signals.
It was intuitive to try to construct families of wave-
lets from the building blocks of systems responses, espe-
cially LTI systems. Such families of wavelets could be
useful in systems characterization and provide new per-
spective for understanding systems and how their
responses evolve. It was a remarkable coincidence
that Robinson (1962) called the response of a second-
order LTI systems a wavelet. However, the closest
point in this track was using the response of second-
order LTI-systems as pseudo wavelets (Freudinger
et al., 1998; Hou and Hera, 2001). They were con-
sidered pseudo wavelets because they failed to satisfy
the reconstruction conditions, namely the inverse
wavelet transform was not possible. In addition to
those efforts, Newland (1993) proposed the harmonic
wavelets which possess the important advantages
of being orthogonal and having excellent frequency
localization. Moreover, they can be viewed as per-
fect band-pass filters. Jezequel and Argoul (1986)
used a transfer function in the frequency domain
(ratio of zeros and poles) as a kernel for an integral
transform that transforms signals from the frequency
domain to another two-dimensional domain whose
axes represent some parameters of the model repre-
sented by kernel.
The response of second-order systems had been
used before to analyze signals for different purposes
and under different names but as a pseudo wavelet or
dictionary of wavelets. Freudinger et al. (1998) defined
the Laplace wavelet, by
ðf,,,tÞ¼ Ae
ffiffiffiffiffiffi
12
p2#ðtÞej2#ðtÞt2,þTs
½
0 else
(
ð1Þ
where represents the damping ratio, #is the fre-
quency, and T
s
is the effective duration of the wavelet
that defines the effective compact support. They formed
a dictionary of wavelets but not a basis or frame. Hou
and Hera (2001) used the the magnitude of second-
order LTI systems response in the frequency domain
as a pseudo wavelet and defined it by
ð!,!0,0Þ¼ !2
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð!2!2
0Þ2þð20!0!Þ2
p!0
0!50
8
<
:ð2Þ
and used the continuous wavelet transform but in the
frequency domain to identify the parameters of
dynamic systems by mapping the match between the
system frequency response and the pseudo wavelet.
In the following section, we show that it is possible
to construct a family of wavelets from the response of
Second-Order Underdamped LTI (SOULTI) systems
that we call, for brevity, the SOULTI wavelets. We
show that their inverse continuous wavelet transform
exists and define the basic properties an analyst needs
to perform time-scale or time-frequency analysis.
In Section 2, we define the SOULTI wavelet families.
Section 3 constructs and proves the existence of the
inverse wavelet transform for the SOULTI wavelets.
Section 4 explores the basic properties of the
SOULTI wavelet and the associated transform and
lists the SOULTI wavelet transform for elementary sig-
nals. Section 5 defines the time and the frequency prop-
erties of the wavelet and derives different definitions
for the time-frequency resolution of the wavelet trans-
form. Section 6 presents an application with numerical
examples for analyzing signals with different frequency
characteristics, and Section 7 addresses the edge effect
and proposes a solution to reduce its influence on the
analysis.
2. Second-order underdamped LTI
wavelets
Second-order LTI systems are very common in most
dynamic fields of science. The mechanical mass-spring-
damper system, shown in Figure 1(a), and the
RLC-electrical circuit, shown in Figure 1(b), are typical
examples of such systems. The response of the SOULTI
system in Figure 1, for the impulse input !2
nðtÞis
given by
hðtÞ¼ !n
ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
p
!
e!ntsinð!nffiffiffiffiffiffiffiffiffiffiffiffiffi
12
ptÞuðtÞð3Þ
where <1 is the damping ratio, !
n
is the natural
frequency and u(t) is the Heaviside step function.
The impulse input is scaled by !2
nto simplify the deriv-
ation of the frequency properties in Sections 4 and 5.
The damped frequency !
d
of the underdamped system
is given by
!d¼!nffiffiffiffiffiffiffiffiffiffiffiffiffi
12
pð4Þ
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so equation (3) can be rewritten in terms of !
d
as
hðtÞ¼ !d
12
e!d
ffiffiffiffiffiffi
12
ptsinð!dtÞuðtÞð5Þ
Let the reciprocal of the damped frequency, defined in
equation (4), be the scaling parameter as
s¼1
!dð6Þ
Substitute equation (6) into equation (5) to get
hðtÞ¼ s1
12
e
t
s
ffiffiffiffiffiffi
12
psin t
s
uðtÞð7Þ
which is the impulse response in terms of the scaling
parameter swhere s2ð0, 1Þ. Now, the SOULTI wave-
let can be defined as
s,
¼
t
s
¼sp
12
e
ffiffiffiffiffiffi
12
pt
s
ðÞ
sin t
s
uðtÞ
ð8Þ
which represents the real part of the Laplace wavelet
defined by Freudinger et al. (1998). The parameter pis
used to give the wavelet a preservation property. For
example, when the wavelet is scaled, its energy content
is also scaled, so to preserve the energy of the L
2
norm
under scaling we use p¼1/2. However, to preserve the
L
1
norm of the wavelet, namely
Z1
1
t
s
dt¼Z1
1 j ðtÞjdtð9Þ
we use p ¼1. Figure 2 graphs the SOULTI wavelet
versus time showing its time properties. The wavelet
function defined in equation (8) represents more than
one family of wavelets, where each family is linked to a
single value of , where 0 <<1. It retains the LTI
second-order response characteristics completely.
Suppose that Jða,1Þ R, and let fðtÞ:J!R
and fðtÞ2L1and is exponentially bounded, see section
(2.1). The SOULTI wavelet transform of f(t) can be now
defined by the generic continuous wavelet transform
definition
WffðtÞg ¼ ~
fð,sÞ¼Z1
1
fðtÞ
t
s
dt,2 ð1,1Þ
ð10Þ
The SOULTI wavelet transform in equation (10) offers
a measurement of similarity between any signal and the
response of second-order LTI systems for characteriza-
tion and identification purposes. In addition, The
SOULTI wavelet gives a direct and simple relationship
between scale and frequency as shown in equation (6),
where the frequency is the reciprocal of the scale.
2.1. Region of convergence
The region of convergence of the SOULTI transform
defines the region ST, where s2S¼
ð0, 1Þ,2T¼ ð1,1Þ, in which the transform in
equation (10) converges to a finite value. Before explor-
ing such region, notice that f(t) has to be exponentially
bounded in order for the transform in equation (10) to
converge.
Exponential boundedness is defined in the following;
define Jða,1Þ R, and let fðtÞ:J!R,if9,k2R
such that jfðtÞj jketj8t2J, then f(t) is exponentially
bounded.
If f(t) is exponentially bounded, then the SOULTI
transform is convergent in the scale region defined by
05s5
ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
pð11Þ
When the time domain is considered for convergence,
i.e. considering the values of the time shift that ren-
ders the transform convergent, we have to be careful
about the uniqueness of the transform because
Figure 1. (a) Mass-spring-damper system; (b) RLC electrical circuit.
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the transform on finite time interval could be identical
for two different functions on a set of measure greater
than zero. Therefore, the time region has to be expli-
citly indicated on the transform, and uniqueness is not
achieved in this case. In the next section, we will define
the inverse continuous wavelet transform with respect
to the SOULTI wavelet.
3. SOULTI wavelet inverse transform
Notice that equation (10) represents an inner product in
the time domain, which is equivalent to the inner prod-
uct in the frequency domain according to Plancherel’s
theorem (Yoshida, 1965). Applying Plancherel’s the-
orem to equation (10) yields
~
fð,sÞ¼hfðtÞ, ,s
i¼ 1
2hFð!Þ,,s
ð!Þi ð12Þ
where ,s
ð!Þis the Fourier transform of ,s
, and using
the Fourier transform shift and scale properties it can
be expressed in terms of the Fourier transform of the
mother wavelet, ð!Þ,as
F ,s
¼sej! ðs!Þð13Þ
Using Plancherels theorem, equation (10) becomes
~
fð,sÞ¼ s
2Z1
1
ej! ðs!ÞFð!Þd!ð14Þ
where the conjugate of sej! ðs!Þis substituted in the
inner product. Note that the integral in the right side of
equation (14) represents the inverse Fourier transform
of ðs!ÞFð!Þ. Applying the Fourier transform to equa-
tion (14) yields
Z1
1
ej! ~
fð,sÞd¼s
2ðs!ÞFð!Þð15Þ
In general, we cannot divide both sides by sðs!Þ
because it could vanish at some values of !or s.
However, in our case sðs!Þis given by
sðs!Þ¼
s1p
12
!2s2þ1
12j2!s
ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
p
ð16Þ
which does not vanish for any value of !or s2(0,1).
Figure 3 shows the wavelet spectrum magnitude,
which is equivalent to its conjugate spectrum
Figure 2. The SOULTI mother wavelet time function at ¼0.3, s¼1, and p¼1.
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magnitude. The curve never crosses the zero axis and
decays asymptotically to zero. Consequently dividing
by sðs!Þis a legitimate operation. Therefore, we
can solve for F(!) in equation (15) to get
Fð!Þ¼ 1
sðs!ÞZ1
1
ej! ~
fð,sÞdð17Þ
To retrieve f(t), take the inverse Fourier transform of
equation (17), so the inverse wavelet transform with
respect to the SOULTI wavelet becomes
W1
f~
fð,sÞg ¼ fðtÞ¼Z1
1 Z1
1
ej!t
sðs!Þej! ~
fð,sÞdd!
ð18Þ
Equation (18) forms the inverse wavelet transform with
respect to the SOULTI wavelet or the reconstruction
formula of the original wavelet definition shown in
equation (10). If f(t) is differentiable, we can use a sim-
pler and probably more practical inverse formula to
retrieve f(t) back from its wavelet transform.
Theorem 1. Let J ða,1Þ R, and let f(t): J!Rbe
differentiable and exponentially bounded, and let
the SOULTI wavelet transform of f(t)be given by equa-
tion (10), then the inverse wavelet transform satisfies the
identity
fðtÞ¼W
1
f~
fðt,sÞg ¼ sp1ð12Þs2d2~
fðt,sÞ
dt2
2
4
2ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
ps
d~
fðt,sÞ
dt þ~
fðt,sÞ3
5
ð19Þ
Proof. Substituting sðs!Þfrom equation (16) into
equation (18) gives
fðtÞ¼Z1
1
ej!t12
s1p!2s2þ1
12j2!s
ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
p
!
Z1
1
ej!t~
fðt,sÞdtd!ð20Þ
Note that is substituted by tinside the second inte-
gral. Using the operator notation for the Fourier trans-
form, equation (20) becomes
fðtÞ¼sp1F1!2s212
þ1jffiffiffiffiffiffiffiffiffiffiffiffiffi
12
p2!s
F~
fðt,sÞ
nono
ð21Þ
Figure 3. SOULTI wavelet in the frequency domain with the mean frequency !
CG
, the standard deviation-based frequency window
SD
, and the (half-power)-based frequency window
BW
.¼0.5, s¼1, and p¼1.
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Applying the linear operators properties (Naylor and
Sell, 2000) to equation (21) gives
fðtÞ¼sp1F1!2s212
F~
fðt,sÞ
nono
þsp1F1F~
fðt,sÞ
nono
sp1F1jffiffiffiffiffiffiffiffiffiffiffiffiffi
12
p2!sF~
fðt,sÞ
nono
ð22Þ
and after applying the Fourier transform differentiation
property to equation (22) we arrive at equation (19). #
This provides a simple and direct method in the time
domain to calculate the inverse wavelet transform with
respect to the SOULTI wavelet. However, in order for
the formula in equation (19) to apply, ~
fð,sÞhas to be
at least twice differentiable with respect to time. When
considering the transform that defines ~
fð,sÞin equa-
tion (10), we find that ~
fð,sÞis twice differentiable with
respect to time if f(t) is differentiable. So if f(t) is expo-
nentially bounded and fðtÞ2C1, then its SOULTI
wavelet transform is unique and f(t) can be retrieved
using equation (19).
Equation (19) also provides information about the
uniqueness of the SOULTI wavelet transform. The
inverse wavelet transform given by equation (19) is a
linear second-order differential equation, which we will
call the Reconstructing Differential Equation. The ori-
ginal function f(t) is the input function and its wavelet
transform at scale sis a solution or part of the response.
However, the other conditions must be satisfied in
order for equation (19) to server as inverse formula
for the SOULTI wavelet transform.
Corollary 2. Let Jða,1Þ R, and let fðtÞ:J!R
be differentiable and exponentially bounded, then the
SOULTI wavelet transform with respect to the wavelet
family t
s
of f(t) at scale sis a solution of the fol-
lowing nonhomogeneous differential equation
fðtÞ¼sp1ð12Þs2d2ðyðtÞÞ
dt22ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
psdðyðtÞÞ
dt þyðtÞ
ð23Þ
Proof. The proof follows by direct substitution. Fix s,
so it can be treated as a constant. Now, suppose that a
solution of equation (23) is given by
ypðtÞ¼~
fðt,sÞð24Þ
Substitute y
p
(t) back into the right side of equation (23)
to get
Gðt,sÞ¼sp1ð12Þs2d2
dt2
~
fðt,sÞ
2ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
psd
dt
~
fðt,sÞ
þ~
fðt,sÞð25Þ
but we just proved by Theorem 1 that G(t,s)¼f(t). #
4. SOULTI transform of elementary
signals and its properties
Let us examine the validity of equation (19) with an
example. Let fðtÞ¼et, then its SOULTI wavelet
transform is given by
~
fð,sÞ¼ sp
12Z1
ete
ffiffiffiffiffiffi
12
pt
s
ðÞ
sin t
s
dtð26Þ
Figure 4. SOULTI Wavelet transform surface of the decaying exponential function at ¼0.7 and ¼2.
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which can be evaluated using the integration by parts
technique to give
~
fð,sÞ¼e s1p
ð12Þs2þ2ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
psþ1ð27Þ
which represents an analytical formula in terms of the
scale s, the time shift , and the wavelet damping ratio
, in addition to the decay rate .
Note that the transform of etin equation (27) con-
sists of a multiplication of two functions, a function of
time and a function of scale. Note also, that the trans-
form is very similar to the Laplace Transform of a
delayed and scaled function. Figure 4 shows the wavelet
transform of the exponential function as described in
equation (27).
Let us now evaluate the SOULTI inverse transform
by using the formula in equation (19). Differentiating
equation (27) with respect to time twice and substitut-
ing the result into the right hand side of equation (27)
and substituting by tyields
sp1ð12Þs22etþetþ2ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
pset
h
s1p
ð12Þs2þ2ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
psþ1
!#¼et=#
ð28Þ
We can use the result in equation (27) to find the
SOULTI wavelet transform for the sinusoidal func-
tions. Table 1 lists the SOULTI wavelet transform for
some elementary signals. Figure 5 shows the wavelet
transform scalogram of the cos(!t) function.
Lemma 3. Let J ða,1Þ R, and let xðtÞ:J!Rbe
differentiable and exponentially bounded as defined in
Theorem 1, and the SOULTI wavelet transform of
x(t) be given by equation (10), then the SOULTI wave-
let transform of _
xðtÞis given by
~
_
xð,sÞ¼ e
ffiffiffiffiffiffi
12
p
sffiffiffiffiffiffiffiffiffiffiffiffiffi
12
p~
xðþs,sÞ
¼tan1ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
p
! ð29Þ
Proof. Substitute _
x(t) in equation (10) to have
~
_
xð,sÞ¼Z1
_
xðtÞ
t
s
dt,2 ð1,1Þ ð30Þ
which can be evaluated by the integration by parts tech-
nique to obtain
~
_
xð,sÞ¼ sp
12xðtÞe
ffiffiffiffiffiffi
12
pt
s
ðÞ
sin t
s
1
sp
12Z1
xðtÞ
sffiffiffiffiffiffiffiffiffiffiffiffiffi
12
pe
ffiffiffiffiffiffi
12
pt
s
ðÞ
sin t
s
dt
!
ð31Þ
Table 1. SOULTI wavelet transform for basic signals.
#f(t)~
fð,sÞ
1u(t)s1p
2tu(t)s1p2s
ffiffiffiffiffiffiffiffi
12
pþ1
12
3t2uðtÞs1p2ð221Þð12Þs2
þ4ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
psþð12Þ2
4ete s1p
ð12Þs2þ2ffiffiffiffiffiffiffiffi
12
psþ1
5 sin(!t)
s1p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2þB2
psinð! þÞ
¼tan1A
B
Aðs,!Þ¼1ð12Þs2!2
Bðs,!Þ¼2ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
ps!
6 cos(!t)
s1p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2þB2
pcosð! þÞ
¼tan1B
A
Aðs,!Þ¼1ð12Þs2!2
Bðs,!Þ¼2ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
ps!
7etcosð!tÞ
s1p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2þB2
pe cosð! þÞ
¼tan1B
A
Aðs,!Þ¼1þð12Þð2!2Þs2
þ2ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
ps
Bðs,!Þ¼2ð12Þ!s2þ2ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
p!s
8etsinð!tÞ
s1p
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
A2þB2
pe sinð! þ
¼tan1B
A
Aðs,!Þ¼1þð12Þð2!2Þs2
þ2ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
ps
Bðs,!Þ¼2ð12Þ!s2þ2ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
p!s
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where is given by
¼tan1ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
p
! ð32Þ
Since x(t) is exponentially bounded, the first term in
equation (31) vanishes, so equation (31) becomes
~
_
xð,sÞ¼ e
ffiffiffiffiffiffi
12
p
sffiffiffiffiffiffiffiffiffiffiffiffiffi
12
p
sp
12Z1
xðtÞe
ffiffiffiffiffiffi
12
ptðþsÞ
s
ðÞ
sin tðþsÞ
s
dt
ð33Þ
but the part inside the parenthesis is equal to
~
xðþs,sÞ, hence equation (33) is equivalent to equa-
tion (29). #
Lemma 4. Let J(a,1)R, and let x(t): J!Rbe
exponentially bounded as defined in Theorem 1, and
the SOULTI wavelet transform of x(t) be given by
equation (10), then the SOULTI wavelet transform of
(t), defined by
ðtÞ¼XðtÞXðaÞ¼Zt
a
xðrÞdrð34Þ
is given by
~
ð,sÞ¼sffiffiffiffiffiffiffiffiffiffiffiffiffi
12
pe
ffiffiffiffiffiffi
12
p~
xðs,sÞs1pXðaÞ
¼tan1ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
p
! ð35Þ
Proof. Since xðtÞ¼dXðtÞ
dt, substitute x(t) in place of _
x(t),
and X(t) in place of x(t) in equation (33), and the result
can be written as
Z1
xðtÞ
t
s
dt¼e
ffiffiffiffiffiffi
12
p
sffiffiffiffiffiffiffiffiffiffiffiffiffi
12
pZ1
XðtÞ
tðþsÞ
s
dt
ð36Þ
make the substitutions (t)þX(a)¼X(t) and ¼þs
into equation (36), then equation (35) follows.
5. The time-frequency resolution and
properties
The time-frequency resolution is an important property
of the wavelet transform. The time-frequency
Figure 5. SOULTI wavelet transform of f(t)¼cos(!t), at ¼0.7.
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resolution is defined by
¼T ð37Þ
where Tis the time resolution or the time window and
is the frequency resolution or frequency window.
The time window represents the time interval that a
frequency can be identified within, while represents
the range of frequencies within a time interval. There
are different ways to define the time and frequency reso-
lutions. One possible way is to use the standard devi-
ation definition, in which the resolutions are defined by
TSD
2
2
¼R1
1 ðttCGÞ2j ðtÞj2dt
R1
1 j ðtÞj2dtð38Þ
SD
2
2
¼R1
1 ð!!CGÞ2jð!Þj2d!
R1
1 jð!Þj2d!ð39Þ
where tCGand !CGrepresent the center of mass of the
signal in time and frequency respectively, and they are
given by
tCG¼R1
0tj ðtÞj2dt
R1
0j ðtÞj2dtð40Þ
!CG¼R1
0!jð!Þj2d!
R1
0jð!Þj2d!ð41Þ
Table 2 lists the results of calculating for some values
of 0 <<1atp¼1. The values of do not depend on
the scale value and they satisfy the Heisenberg principle
(Kaiser, 1994). Using the standard deviation, the reso-
lution satisfies the inequality >1/4(Kaiser, 1994).
!
CG
is proportional to the scale s, while t
CG
is inver-
sely proportional to s. However, the standard deviation
does not offer meaningful time and frequency windows
of resolution. The SOULTI wavelet is not symmetrical
neither in time nor in frequency. Moreover, it has no
compact support neither in time nor in frequency.
So we would question the significance of the standard
deviation window about the signal center in time
and the significance of the frequencies included in the
standard deviation window and weather that is really
what is accentuated in the time-scale or time-frequency
analysis.
We can attain an alternative definition for the
SOULTI wavelet time-frequency resolution based on
systems dynamics and control theory. The system
response is considered settled when it enters the 2%
margin of the final value and never leaves it again.
Therefore, we can use the 2% settling-time value to
define the time window, namely T2%¼2%tst.
In the frequency domain, the frequency correspond-
ing to attenuating the input power by a half is con-
sidered the frequency bandwidth of the system or the
cut-off frequency, so we can use the bandwidth to
define the frequency window.
The 2% settling time, t
st
is reached when the
enveloping function enters within 2% of the final
value. Therefor, for a scaled wavelet, it is given by
tst ¼sffiffiffiffiffiffiffiffiffiffiffiffiffi
12
plogð0:02Þ’4sffiffiffiffiffiffiffiffiffiffiffiffiffi
12
pð42Þ
On the other hand, the wavelet bandwidth is the fre-
quency at which the frequency spectrum of a scaled
wavelet satisfies
jsðs!Þj2¼
s22p
12
s2!2þ1
12
2
þ2!s
ffiffiffiffiffiffiffiffi
12
p
2¼1
2ð43Þ
Solving for !gives the bandwidth by
!jjsðs!Þj2¼1
2¼BW
¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffi
12
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
122þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2s22p42þ44
p
q
ð44Þ
Table 2. Time-frequency resolution based on the standard
deviation definition at p¼1.
zt
CGz
o
CGz
T
SD
SD
0.05 20.075 0.48531 39.85 0.24577 9.794
0.1 10.149 0.47298 19.704 0.33949 6.689
0.15 6.8878 0.46292 12.896 0.40724 5.252
0.2 5.2909 0.45506 9.4305 0.46191 4.356
0.25 4.3571 0.44939 7.3101 0.5088 3.719
0.3 3.7522 0.44595 5.8715 0.55084 3.234
0.35 3.3322 0.44484 4.8325 0.58997 2.851
0.4 3.0245 0.44625 4.0534 0.62763 2.544
0.45 2.7882 0.45046 3.4579 0.66502 3
0.5 2.5981 0.4579 3 0.7033 2.11
0.55 2.4372 0.46922 2.6493 0.74366 1.970
0.6 2.2933 0.48537 2.3828 0.78752 1.877
0.65 2.157 0.50783 2.1798 0.83668 1.824
0.7 2.02 0.53899 2.0203 0.89372 1.806
0.75 1.8741 0.58291 1.8838 0.96259 1.813
0.8 1.71 0.64715 1.7483 1.05 1.836
0.85 1.5153 0.74778 1.5888 1.1692 1.858
0.9 1.2689 0.927 1.3714 1.3512 1.853
0.95 0.92196 1.3553 1.0296 1.7057 1.756
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Substitute equation (42) and equation (44) into equa-
tion (37) gives the resolution as
¼4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
122þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2s22p42þ44
p
qð45Þ
Equation (45) gives us a way to determine an appro-
priate value for pbased on the time-frequency reso-
lution shape. In order for to be independent of s,
we must have p¼1, which yields
ðÞ¼4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
122þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
242þ44
p
qð46Þ
which indicates that the 2%tst BW resolution defin-
ition depends only on thus on the wavelet family
,
so we may write the resolution as (
).
Now, let us investigate the values of in range
0<<1. When !0, we have
lim
!0ðÞ¼1 ð47Þ
while when !1 we get
lim
!1ðÞ¼2:65 ð48Þ
Since d
d5082ð0, 1Þ, then
452:65 5ðÞ51ð49Þ
which means that the 2%tst BW definition satisfies
the Heisenberg principle when p¼1. Another advan-
tage of having p¼1, is preserving the wavelet frequency
function peak constant. This is sometimes useful since it
guarantees that all the frequency bands are amplified at
the same level, see Figure 6. This functions as a normal-
izing factor though it does not preserve the wavelet
energy. Figure 2 shows the standard deviation-based
and the 2%t
st
time windows, while Figure 3 illustrates
the standard deviation-based and the half-power band-
width-based frequency windows for the SOULTI
wavelet.
The 2%tst BW gives a better meaning for the time-
frequency resolution of the SOULTI wavelet, but when
is small, <0.4, the definition suffers from two prob-
lems. First, the variation in the frequency response
magnitude varies significantly within the bandwidth,
which requires better focus on the resonance range.
Secondly, as sdecreases, the bandwidth of ðt
sÞcontains
all the bandwidths corresponding to larger s,
i.e.BWð,s2ÞBW ð,s1Þwhen s
1
<s
2
.
For <0.4, another definition of the time-frequency
resolution, that better reflects the data on the time-scale
or the time-frequency analysis domain can be provided
Figure 6. Wavelet amplitude in frequency domain for different values of the scale sat ¼0.2.
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based on the quality factor half-power bandwidth
definition. The quality factor is the peak value of the fre-
quency response. For small , the quality factor for LTI
second-order system can be approximated by
(Meirovitch, 1997)
Q¼1
2ð50Þ
The half power points, q
1
and q
2
are the points when
jðs!Þj ¼ Q
ffiffi2
p, see Figure 7. The bandwidth of the fre-
quency response is
Q¼!2!1ð51Þ
where !
1
is the corresponding frequency to q
1
, and !
2
is the corresponding frequency to q
2
, as shown in
Figure 7. To find !
1
and !
2
, we have to solve the wave-
let power in equation (52) for !where p¼1
jðs!Þj2¼1
ðs2!2ð12ÞÞ2þ2! ffiffiffiffiffiffiffiffiffiffiffiffiffi
12
ps
2¼1
82
ð52Þ
From equation (52) we find
!2
1¼12
s2ð12Þð53Þ
!2
2¼1þ2
s2ð12Þð54Þ
Using the approximation !1þ!2’2
sffiffiffiffiffiffiffiffi
12
p
, which is
valid for small values of , it is easy to show that
Q¼!2!1¼2
sffiffiffiffiffiffiffiffiffiffiffiffiffi
12
pð55Þ
Substituting equations (42) and (55) into equation (37),
the new time-frequency resolution definition becomes
¼T ’4sffiffiffiffiffiffiffiffiffiffiffiffiffi
12
p
2
sffiffiffiffiffiffiffiffiffiffiffiffiffi
12
p¼8ð56Þ
Equation (56) shows a very interesting result where the
time-frequency resolution is constant and does not
Figure 7. j(s!)jfor different values of showing the quality factor and the half-power bandwidth.
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depend on . Note that this approximation is valid for
values of <0.4, for larger values the quality factor is
smaller, hence the time-frequency resolution defined by
equations (45) and (46) would be more meaningful and
suitable to adopt. Of course one may require a wider
bandwidth than the half-power quality factor band-
width, which will make the time-frequency resolution
coarser. For example, if instead of the Q
ffiffi2
pbandwidth
limit we use Q/x, where x<Q, then the bandwidth
and the time-frequency resolution become
Q¼’ 2ffiffiffiffiffiffiffiffiffiffiffiffiffi
x21
p
sffiffiffiffiffiffiffiffiffiffiffiffiffi
12
pð57Þ
¼8ffiffiffiffiffiffiffiffiffiffiffiffiffi
x21
pð58Þ
The frequency at Qrepresents the frequency at which
the wavelet filter is centered at. Moreover, it is easy to
predict where the wavelet frequency is centered because
the scale is directly linked to frequency as stated by
equation (6). The peak occurs at (Meirovitch, 1997)
!Q¼!nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
122
pð59Þ
For small we have
!Q¼!nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
122
p’!nffiffiffiffiffiffiffiffiffiffiffiffiffi
12
p¼!dð60Þ
which with equation (6) shows that we can easily
approximate to a good accuracy the wavelet peak fre-
quency by the relation
!Q’1
sð61Þ
Figure 6 shows clearly the accuracy of equation (61) for
the 0:2ðt
sÞSOULTI family with different scaling values.
For larger values of , i.e. >¼0.4, the approximation
in equation (61) is not valid and we have to use the
exact relation
!Q¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
122
pffiffiffiffiffiffiffiffiffiffiffiffiffi
12
p
1
sð62Þ
The SOULTI wavelet has (0) ¼1 when p¼1. For
other values of p, the wavelet magnitude depends on
s1p. Moreover, we have
Z1
0
jð!Þj2
!d!¼1 ð63Þ
which implies that the SOULTI wavelet does not satisfy
the admissibility condition stated in equation (64) but
it has an inverse
05Z1
0
ð!Þ
2
!d!¼C51ð64Þ
6. Application: Frequency identification
and spectrogram
To validate the capability of the SOULTI wavelet in
detecting features of signals, we produce the frequency
evolution or the scale evolution with respect to time of
some signals using the SOULTI wavelet. We present
some examples of SOULTI wavelet analysis of finite
time signals with white noise added to them at different
Signal to Noise Ratio (SNR) levels.
It is important to emphasize that the continuous ver-
sion of the wavelet transform is performed in these
examples, where the transform integral is performed
numerically. In all the examples, ¼0.1 is used because
it gives the wavelet a large quality factor value as shown
in Figure 6.
6.1. Identifying constant frequencies in
time-invariant frequency signals
First, two noisy signals with the same frequency are tested.
The first has SNR ¼15 dB, and the second has SNR ¼
7.5dB. Figure 8 shows the two signals. Figure 9 shows
the contour map of the two signals wavelet transform.
We notice that in both cases the ridges and the
peaks are distinctly recognized at s¼1
2, which corres-
ponds to !¼2 rad/s by the scale-frequency relation in
equation (6). The ridge of the wavelet transform is
the set of points in the time-scale domain , where
the wavelet integral has stationary points (t,s)2
such that t
s
(t,s)¼s, where t
s
is a stationary point, i.e.
d~
fð,sÞ
ds ts¼0 (Tchamitchan and Torresani, 1992).
Notice also that at the end of the signal the transform
is distorted and the peaks diminish due to the edge
effect. Also notice that since the SOULTI wavelet is
causal the edge effect appears at the end of the time
scale of the signal only and the noisy signal with
SNR ¼7.5 dB has slightly worse edge effect.
In the second test, a signal carrying two different fre-
quencies is analyzed. The signal has SNR ¼15 dB and is
graphed in Figure 10. Figure 11 shows two mappings. The
first maps the contours on the time-scale domain and it
showsclearlytworidgesthatstretchalongtwolinesof
constant scale s¼0.125 and s¼0.5, parallel to the time
axis. The second plots the contours on the time-frequency
domain. The scale-frequency conversion is performed
using equation (6). The ridges stretch along the constant
frequency values !¼2 rad/s and !¼8rad/s.
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6.2. Identifying the instantaneous frequency
in time-varying frequency signal
The advantage of the time-scale or time-frequency ana-
lysis over classic frequency analysis is that it is more
useful in analyzing time-varying and nonlinear oscilla-
tions. In this example, we analyze a signal consisting of
a combination of constant harmonics with linear chirp
as a time-varying frequency component. White noise is
added to the signal with SNR ¼20 dB. The signal is
Figure 9. Contour mapping of the SOULTI wavelet Transform of a harmonic signal of frequency ¼2 rad/s; (a) SNR ¼15 dB,
(b) SNR ¼7.5 dB.
Figure 8. Top: single harmonic with white noise of SNR ¼15 dB. Bottom: single harmonic with white noise of SNR ¼7.5 dB.
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given by equation (65)
xðtÞ¼10 sinð0:4t2Þþ5 cosð2tÞþ8 sin tþ
7
þDðtÞ
ð65Þ
where D(t) represents the white noise or the disturbance
term. Figure 12 plots the signal in the time domain.
Performing Fourier analysis to the signal does not
reveal the instantaneous frequency change in the
signal. Figure 13 shows the Fast Fourier Transform
(FFT) and the Welch averaging of the frequency spec-
trum. While the FFT identifies the constant harmonics
with peaks at !¼1 and !¼2, it is not possible to dis-
tinguish the instantaneous frequency change from the
FFT. The Welch averaging does not identify the con-
stant harmonics because of the interference from the
frequency-changing component.
Figure 14 shows the SOULTI wavelet transform of
the signal. The transform distinctly traces the instant-
aneous frequency with respect to time, where the
dashed lines represents ridgelines that trace this
Figure 11. SOULTI wavelet transform for the two harmonics signal in Figure 10. (a) Time-scale contour mapping. (b) Time-
frequency contour mapping.
Figure 10. Sum of two harmonics with white noise. !
1
¼2 rad/s, !
2
¼8 rad/s, and the SNR ¼15 dB.
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frequency along time. When examining the time vary-
ing component in equation (65) we find that the instant-
aneous frequency is given by !(t)¼0.8t, which is the
equation of the dashed line on the time-frequency
wavelet mapping shown in Figure 14(b). The dashed
curve in Figure 14(a) is the inverse of the line
!¼0.8t, namely s(t)¼1/0.8t, which conforms to the
scale-frequency relation in equation (6). Notice also,
that the other two constant frequencies are identified
along ridgelines of almost constant scales at s¼0.5 and
s¼1 in Figure 14(a) and along ridgelines of almost
constant frequency at !¼1 and !¼2 in Figure 14(b).
On 14(a), the parabolic dashed line, which traces the
instantaneous change of the chirp frequency, intersects
the s¼1 and the s¼0.5 lines at times t¼0.26 s and
t¼1.6 s respectively.
As a comparison between the SOULTI wavelet and
other wavelets in resolving frequencies with respect to
Figure 13. Frequency spectrum of the signal described in Figure 12. (Dashed line) FFT. (Solid line) Welsh spectrum averaging.
Figure 12. Two constant harmonics with a time varying frequency component signal.
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Figure 15. Scalograms of the signal described in Figure 12 using different wavelets. (a) By Morlet wavelet; (b) by Complex Shannon
wavelet (f
b
¼1, f
c
¼1); (c) by Mexican hat wavelet; (d) by Frequency B-Spline wavelet (order ¼2, f
b
¼1, f
c
¼1). f
b
: bandwidth fre-
quency. f
c
: wavelet center frequency.
Figure 14. SOULTI wavelet transform for the chirp signal described in Figure 12 and shown in Figure 12. (a) Scalogram (Time-scale)
contour mapping (b) Spectrogram (Time-frequency) contour mapping.
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time, the same chirp signal is analyzed with four differ-
ent wavelets, Morlet, complex Shannon, Mexican hat
and the frequency B-Spline. The four wavelets scalo-
gram graphs are shown in Figure 15. Notice that the
four are able to resolve the chirp into curves parabolic
in shape with different ridge widths. However, the
curves are not the reciprocal of the instantaneous fre-
quency. In addition, the constant scale ridges are not at
scales that can be easily matched to frequencies.
The Mexican hat wavelet gives the best match to the
parabolic curve, but when resolving the constant fre-
quencies in the signal it shows large shifts. It is difficult
to infer the accurate frequencies in the signal from
these wavelets scalogram maps, though one can infer
qualitative information about the shape of the instant-
aneous frequency change with respect to time. For each
wavelet, the relation between the scale and the fre-
quency along the ridgeline is different, but one can
argue that it is the reciprocal of some function of the
instantaneous frequency.
From the previous discussion, we conclude that it is
difficult to construct a spectrogram for each scalogram
shown in Figure 15. However the SOULTI wavelet
scalogram can be directly transformed into spectro-
gram by applying the scale-frequency change in equa-
tion (6).
To evaluate the SOULTI wavelet spectrogram, we
compare it to the Short Time Fourier Transform
Figure 16. Spectrograms of the linear chirp signal in Figure 12 at different Window widths (samples). (a) W¼8 (b) W¼16
(c) W¼32 (d) W¼64 (e) W¼128 (f) W¼256.
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(STFT) spectrograms. Six spectrograms based on the
STFT were computed for different window sizes (W),
where the size is measured by the number of samples.
To make the comparison compatible with the continu-
ous wavelet transform, the window overlap is set as
(W1) to perform window sweep over the time
vector of the signal. Figure 16 plots the spectrograms
with the three straight lines that represent the instant-
aneous frequencies imposed on it. Note that the narrow
windows (W64) are better in resolving the linear
chirp than resolving the constant frequencies, while
the wider windows (W128) are better in resolving
the constant frequencies.
However, notice that when applying the STFT the
wider the window the less time resolution is obtained,
the more end effect occurs and the more time trunca-
tion from both sides of the signal is taken. For example,
Figure 16(f) only shows frequency information for the
time period 12.8t17.2 s of the signal. Frequency
information for periods 0 t<12.8 s and 17.2<t30 s
is not available, while the SOULTI wavelet spec-
trogram provides information for the duration of the
signal as shown in Figure 14(b). Moreover, the
SOULTI spectrogram resolves both the linear chirp
and the constant harmonics, and its direct link between
frequency (spectrograms) and scale (scalograms) allows
checking the results for small or close frequencies.
7. Edge effect mitigation
Edge effect in harmonic and wavelet analysis of finite
duration signals is caused by many factors. First, the
measured signals are finite in duration and we do not
have information about the signal after or before the
times of recording. Second, many wavelets do not have
compact support rather they have an effective window.
Third, at the beginning of the analysis, (¼0), the
wavelet window is defined for negative and positive
range of time, t50 and t40, but the analyzed
signal is defined only for t40, so the inner product is
computed between the signal and part of the wavelet.
Similarly, at the end of the analysis, the wavelet effect-
ive window will move out of the signal range and only
part of it will take part in the inner product with the
signal. This partial inner product gives inaccurate
results at both edges.
The SOULTI wavelet is a right sided wavelet or
signal, i.e. the mother wavelet is zero for t<0.
Therefore, when performing the wavelet transform, the
effective wavelet window sets fully inside the range of the
signal at the beginning of the analysis when ¼0.
However, at the end of the analysis, the effective
window moves out of the signal range and the inner
product is performed between the signal and part of
the effective window. Therefore, though the SOULTI
wavelet solves naturally the edge effect at the beginning
it does not solve the problem at the end, which makes
the analysis at the end inaccurate and distorted.
As a solution for the end edge effect, we propose in
this section performing a Reverse Wavelet Transform
(RWT) analysis starting from the end of the signal. So
the mother wavelet is reflected about t¼0, then it is
shifted to the end of the signal, and the wavelet analysis
is performed end-to-start. Then, we reflect the results
Figure 17. Reverse wavelet transform of the signal in Figure 10. (a) Scalogram, time-scale contour. (b) Spectrogram, time-frequency
contour mapping.
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back with respect to time. The same result can be
obtained by just reflecting the signal, performing the
wavelet analysis as usual and then reflecting the
results back.
Figure 17 shows the RWT of the signal in Figure 10
and Figure 18 shows the RWT of the signal in
Figure 12. Note that at the end of the analysis there
are clear ridges in both the scalograms and the spectro-
grams whereas the beginning shows distortions. This
result gives an indication that the distortion of
the ridges at the end of the studied signals is due to
the edge effect.
Figure 18. Reverse wavelet transform of the signal in Figure 12. (a) Scalogram, time-scale contour mapping. (b) Spectrogram, time-
frequency contour mapping.
Figure 19. Average of FWT and RWT of the linear chirp in Figure 12. (a) Scalogram, time-scale contour mapping. (b) Spectrogram,
time-frequency contour mapping.
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Instead of producing two scalograms or spectro-
grams for each signal, the average of the Forward
Wavelet Transform (FWT) and the RWT can be com-
puted and mapped to give refined results at both ends
of the signal. Figure 19 shows the average of the FWT
and RWT. The edges are better resolved and the dis-
tortion at the edges almost disappeared. However, a
slight reduction in the ridges amplitude is notices.
8. Remarks and conclusions
It is shown that the impulse response of SOULTI sys-
tems can be used as a wavelet to obtain time-scale and
time-frequency analysis directly. We also proved that
the transform can be reversed to obtain the original
signal, hence an inverse wavelet transform for the
SOULTI wavelet exists. A region of convergence can
be defined for the transform on the scale domain. This
region defines in which range of scales the SOULTI
wavelet transform converges.
Moreover, it is shown that the original signal can be
retrieved back by substituting the transform into the
conjugate differential equation. The SOULTI wavelet
can be evaluated for most elementary functions and
basic signals. In addition, there is a direct relation between
the SOULTI wavelet transform of a signal and the trans-
form of its derivative or integral. For a wavelet scaling
power p¼1, we found that the time-frequency resolution
is preserved constant using the three definitions for com-
puting the time-frequency resolution, the standard devia-
tion based, the -3dB bandwidth based, and the Q-factor
bandwidth based.
The important result that the reconstruction differ-
ential equation shows is extending the notion that the
wavelet transform is the output of a filter bank from
digital wavelets to crude noncompactly supported
wavelets. The reconstruction differential equation in
equation (19) shows that the SOULTI wavelet trans-
form at scale sis part of the output (particular solution)
of the second-order system modeled by the differential
equation itself.
The RWT can reveal whether the distortion at the end
of the time range on scalograms and spectrograms is due
to the edge effect or not. Moreover, taking the average
between the FWT and RWT is a practical and simple
method to eliminate the edge effect on both edges.
The SOULTI wavelet transform formula provides
an analytical tool for time-frequency or time-scale
representation of basic signals. It also preserves all
the important characteristics and parameters that
exist in the time domain to the time-scale or time-fre-
quency domain.
Declaration of Conflicting Interests
The author(s) declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of this
article.
Funding
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