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163
ALGORITHMS FOR MODELING FUNCTION AND MEDICAL SIGNALS
IN WAVELET METHODS
Gofurjonov M.R.
Tashkent University of Information Technologies named
after Muhammadal-Khwarizmi,
gofurjonov13@mail.ru
Xuramov L.Ya.
Samarkand State University named after Sharof Rashidov
Tel:+99(893) 546-21-41,e-mail: latifxya@gmail.com
Sayidova Nigorabonu
Samarkand State University named after Sharof Rashidov
https://doi.org/10.5281/zenodo.15705932
Abstract.
Objective.
This article is devoted to the construction of wavelet models,
which are considered important in function processing, and the use of Taylor
series as a method of approximating analytical functions.
These models are built using Haar wavelets. It is important to reduce the
number of coefficients required to approximate the total number of Binary
segments (with a given accuracy) as a result of transformation of given signals
using Haar wavelets in the form of a known function and analytical function. The
process of calculating the coefficients using the Xarra wavelet is found without
long operations. Only addition, scaling and transformation operations are used.
Methods.
Haar wavelet, Taylor series, conversion wavelet, digital
processing error, relative error.
Results.
The obtained results show that it can be seen that the Haar wavelet
gives higher accuracy compared to the Taylor series in the digital processing of
the given signals in an analytical form.
Conclusion.
In the problems of image recognition from waves, in the
processing and synthesis of various signals such as speech, in the analysis of
various images in nature (the color of the retina, radiography of the kidney,
studying the surface properties of crystals and nano-objects, satellite .images of
clouds or planetary surfaces etc. can be used to study the properties of vortex
fields and in other cases.
Key words:
Haar wavelet, Taylor series absolute error, relative error,
scaling function
Introduction
. Currently, there are several types of wavelets. It is
convenient to use uncomplicated methods to determine the coefficients of the
function, one of the uncomplicated methods is the Haara wavelet, in which the
coefficients of the function are determined without many operations only by
THEORETICAL ASPECTS IN THE FORMATION OF
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International scientific-online conference
164
adding, subtracting and scaling. It can also be noted that the preference of the
type of wavelets also depends on the analysis of the input looking function or
signal, because the scaling function is interpreted differently depending on the
wavelet types. In medicine, wavelets are widely used when studying various
signals, radiography of the kidney, properties of the surface of crystals and
nano-objects, when studying the properties of gastroenterological signals and in
other cases, interpolation. The waveforms of the Haar wavelet extend along the
time axis along with the signal graph. A Haar wavelet plot often approximates a
signal as a one-way waveform across the signal, which is good for compressing
some signals. Its mathematical interpretation allows analysis of wave states at
different frequencies. The amplitude of the graph of the Haar-wavelet function
decreases to zero, forming oscillating waves.
Methods.
Construction of a Haar wavelet. There are fast Haar wavelet
transformation algorithms, and its orthogonal wavelets are widely used in
solving practical problems.
An orthogonal Haar wavelet is expressed as follows:
Considered as a wavelet in Haar bases. Haar wavelets attract the attention
of experts for two reasons:
Reducing the number of coefficients required for approximation (with a
given accuracy) compared to the total number of binary segments.
Absence of "long" operations in the process of calculating coefficients. Only
add, scale, and transform operations are used.
In the digital processing of signals, wavelet functions are used to separate
the details and local features of the signals, and scaling functions are used to
approximate the signals. When choosing wavelet functions, special attention is
paid to their characteristics, such as smoothness, carrier size, and the number of
cases where their values are equal to zero.
The process of wavelet transformation of signals relies on the use of two
types of functions: a wavelet function and a scaling function, which means that
they are the same wavelet
)
(
t
- to shift in time across the signal
b
and time
scale
a
is built by changing:
)
(
)
(
,
)
,
(
,
1
)
(
2
R
L
t
R
b
a
a
b
t
a
t
ab
0
V
- that's all
]
1
,
0
[
we define a set of invariant functions on the interval, that
is, a set of linear vectors.
Then the following scaling function - belongs to the collection:
THEORETICAL ASPECTS IN THE FORMATION OF
PEDAGOGICAL SCIENCES
International scientific-online conference
165
if
t
t
t
,
0
1
0
,
1
)
(
)
(
0
,
0
(1)
(1)
0
i
zoom function when available.
if
j
t
j
t
n
n
j
n
,
0
2
1
2
,
1
)
(
,
,
1
2
,...,
1
,
0
n
j
(2)
(2)
n
i
zoom function when available, here,
n
n
n
j
t
j
j
t
2
1
2
,
1
2
0
is the interval of change of scaling functions,
)
(
,
t
j
n
-lar
n
V
are scaling
functions related to , in which there is a set of vectors with scalar multiplication,
so these sets form the Euclidean space. In our case as a scalar multiplication
1
0
)
(
)
(
)
,
(
dt
t
g
t
f
g
f
(3)
(3) we get the form using this formula
n
C
- coefficients of scaling functions
are defined.
In that case
1
2
...,
,
1
,
0
),
2
(
2
)
(
,
n
n
n
j
n
j
j
t
t
(4)
Using expressions (3) and (4), the coefficients of the Haar wavelet are
found:
1
0
)
(
)
(
dx
x
f
x
C
n
n
(5)
(5) Formula for finding coefficients of Haar wavelet.
0
)
(
)
(
n
n
n
x
C
x
f
Statement of the problem: Let's assume
t
x
f
)
(
function t=0,25 with step
[0,1) values are given, this function is required to be interpolated in the Haar
wavelet.
...
...
1
0
n
V
V
V
THEORETICAL ASPECTS IN THE FORMATION OF
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There is a way to solve the problem:
[𝑢, 𝑤] = {𝑡|𝑢 ≤ 𝑡 < 𝑤}
else
,
0
1
0
,
1
]
1
,
0
[
t
if
t
;
else
,
0
0
,
1
]
,
0
[
w
t
if
t
w
;
else
,
0
,
1
]
,
[
w
t
u
if
t
w
u
;
else
,
0
,
]
,
[
w
t
u
if
s
t
w
u
;
Generating the Haar wavelet function (6)
[
,
[
1
j
j
t
t
j
s
f
(6)
1
0
[
,
0
[
[
,
[
1
[
,
[
1
[
,
0
[
0
1
2
1
n
t
j
t
t
n
t
t
t
s
s
s
s
f
n
n
Generating the Haar wavelet wavefunction (7)
[
,
[
[
,
[
[
,
[
w
m
m
u
w
u
(7)
Results.
t
x
f
)
(
let the function be given in the interval [0,1) with a step of
t=0.25, the number of compressions when interpolating this function in the
Haara wavelet is equal to n=4.
else
,
0
4
1
0
,
1
2
)
(
4
t
t
;
else
,
0
2
1
4
1
,
1
4
)
(
t
t
;
else
,
0
4
3
2
1
,
1
4
)
(
t
t
;
else
,
0
1
4
3
,
1
4
)
(
t
t
;
To calculate the Haar wavelet coefficient, we calculate the following integral
n
i
i
t
C
t
f
0
)
(
)
(
;
8
1
)
(
)
(
0
0
0
t
C
t
f
;
8
3
)
(
)
(
1
1
1
t
C
t
f
;
8
7
)
(
)
(
2
2
2
t
C
t
f
;
8
7
)
(
)
(
3
3
3
t
C
t
f
Based on the given model, the initial experimental data of the function was
obtained and numerical processing was carried out on Haar wavelets (Fig.3).
Figure 3. f(x)=t Result of interpolation on Xaar wavelets.
Haar's piece-invariant wavelet coefficients can also be determined by the
following method. We can express the approximation and dn-difference
coefficients of an-Haar through signal valuesin the following form
2
/
,.....,
3
,
2
,
1
,
2
2
1
2
N
n
f
f
a
n
n
n
(8)
THEORETICAL ASPECTS IN THE FORMATION OF
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International scientific-online conference
167
here
)
,....
,
(
2
/
2
1
N
i
a
a
a
a
- formula for determining average values.
The differential value representation of the signal is
2
/
,.....,
3
,
2
,
1
,
2
2
1
2
N
n
f
f
d
n
n
n
(9)
here
)
,....
,
(
2
/
2
1
N
i
d
d
d
d
- formula for determining difference values.
These values generate two new signals: one of which is the original signal
Z
n
a
a
n
},
{
and the second is to restore the initial signal.
Z
n
d
d
n
},
{
Haqiqatan
ham
n
n
n
d
a
f
1
2
;
n
n
n
d
a
f
2
One of the main features of wavelets is the fast calculation algorithms for
numerical processing of the calculated coefficients. Using the fast calculation
algorithm of numerical processing of coefficients in Haara wavelet
2
)
(
t
t
f
we
calculate the interpolation of the function in the interval [0,1) with a step of 0.1.
It is known that the number of compressions is equal to n=10.
obtained, and numerical processing was carried out on the Haar wavelet
(Fig. 4).
Figure 4. Using the algorithm of fast calculation of coefficients in the
Haar wavelet
2
)
(
t
t
f
the result of interpolating the function in the interval [0,1)
with a step of 0.1.
Taylor series using Haara wavelet
The Haara wavelet is a very versatile mathematical tool that can be used to
analyze, generate, and segment signals represented by functions or analytic
functions. The use of Taylor series as a method of approximating analytic
functions is one of the most common methods in applied mathematics. Using
Taylor series with wavelets is another way to approximate analytical functions.
let's say
)
(
x
f
The function
R
x
0
is one of the points
}
0
;
:
{
)
(
0
0
0
x
x
x
R
x
x
U
(10)
(10) Let have a derivative of any order around . This is the case
)
(
x
f
allows
us to write the Taylor formula of the function:
THEORETICAL ASPECTS IN THE FORMATION OF
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International scientific-online conference
168
)
(
)
(
!
)
(
...
)
(
!
2
)
(
)
(
!
1
)
(
)
(
)
(
0
)
(
2
0
0
''
0
0
'
0
x
r
x
x
n
x
f
x
x
x
f
x
x
x
f
x
f
x
f
n
n
n
(11)
(11) In this
)
(
x
r
n
residual term. As long as
)
(
x
f
function
)
(
0
x
U
has a
derivative of any order, then
...
)
(
!
)
(
...
)
(
!
2
)
(
)
(
!
1
)
(
)
(
0
)
(
2
0
0
'
'
0
0
'
0
n
n
n
x
x
n
x
f
x
x
x
f
x
x
x
f
x
f
(12)
(12) It is possible to look at the power series, (10) the coefficients of the
power series are numbers, which
)
(
x
f
function and its derivatives
0
x
expressed
by the values at the point, (11) rank series
)
(
x
f
is called the Taylor series of the
function.
In particular,
0
0
x
(8) is the degree series
1
)
(
)
(
2
''
!
)
0
(
...
!
)
0
(
...
!
2
)
0
(
!
1
)
0
(
)
0
(
n
n
n
n
x
n
f
n
f
x
f
x
f
f
(13)
(13) appears. Let's assume,
)
(
x
f
function is something
)
,
(
r
r
da
)
0
(
r
has a
derivative of any order, and its
0
0
x
point Taylor series
...
!
)
0
(
...
!
2
)
0
(
...
!
2
)
0
(
!
1
)
0
(
'
)
0
(
)
(
''
2
''
2
''
n
n
x
n
f
x
f
x
f
x
f
f
(14)
let it be This is the residual term of the series (14).
)
(
x
r
n
let's say:
)
(
!
)
0
(
...
!
2
)
0
(
!
1
)
0
(
)
0
(
)
(
2
''
'
x
r
x
n
f
x
f
x
f
f
n
n
n
We find Taylor series of trigonometric functions.
let's say
x
x
f
sin
)
(
let it be Ravshanki
N
n
R
x
,
at
1
)
(
,
1
)
(
)
(
x
f
x
f
n
being
),
0
(
f
,
1
)
0
(
'
f
,
0
)
0
(
)
2
(
n
f
n
n
f
)
1
(
)
0
(
)
1
2
(
)
(
N
n
will
be.
So
x
x
f
sin
)
(
the function expands into a Taylor series and according to the
formula (10).
...
!
5
1
!
3
1
)!
1
2
(
)
1
(
sin
5
3
1
2
0
x
x
x
x
n
x
n
n
n
will be
)
2
(
0.179030
)
1
(
0.119712
)
(
059964
.
0
1
)
sin(
x
x
x
x
Discussions.
Aare the numerical performance errors of Haar
b
a
,
defined in
)
(
x
f
be a continuous function.
b
a
,
the segment
b
x
x
x
x
a
n
i
...
...
1
0
we can separate the nodes into points.
const
x
x
h
i
i
1
(15)
h
- distance between nodes.
THEORETICAL ASPECTS IN THE FORMATION OF
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International scientific-online conference
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There are formulas for determining methodical errors of interpolation for
polynomials of different degrees. For example, for polynomials of the zero
degree (for piece-invariant wavelets), the error estimation formula is expressed
as:
h
x
f
x
f
x
P
)
(
max
2
1
)
(
)
(
'
We present an estimate of the absolute and relative errors of digitizing the
geophysical signal in Haar piece-invariant wavelets.:
)
(
)
(
max
1
i
i
b
x
a
x
har
x
f
0.02%
1
- Absolute error of Haar's piece-invariant wavelets
Calculating the value of the function in the Taylor series and the Haar
wavelet and estimating the absolute error (Table 1)
Table
1
Calculating the value of the function in the Taylor series and the Haar wavelet and
estimating the absolute error
The first column of this table (Table 1) shows the initial values of the
function, the second column shows the value of the Taylor series distribution,
the third column shows the value of the coefficients of the square wavelet, and
the fourth column shows the absolute error value.
№
X[i]
)
6
sin(
)
(
x
t
f
Taylor series
Haar
Absolute
mistake
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
…
0.00000
0.01000
0.02000
0.03000
0.04000
0.05000
0.0600
0.0700
0.08000
0.09000
0.1000
0.1100
0.12000
0.13000
0.14000
0.1500
0.16000
...
0.97000
0.98000
0.99000
0.000000
0.059964
0.119712
0.179030
0.237703
0.295520
0.352274
0.407760
0.461779
0.514136
0.564642
0.613117
0.659385
0.703279
0.744643
0.783327
0.819192
...
-0.446800
-0.392350
-0.336488
0.00082
0.089838
0.149371
0.208366
0.266611
0.323897
0.380017
0.434770
0.487958
0.539389
0.588880
0.636251
0.681332
0.723961
0.763985
0.801259
0.835650
...
0.419575
0.364419
0.307952
0.04999
THEORETICAL ASPECTS IN THE FORMATION OF
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International scientific-online conference
170
Conclusion.
In this research work, the digital performance model of Haar's
wavelets of the given signal in the form of function and analytic function was
evaluated. It can be seen from Table 1 that the absolute error for the number of
signal nodes in the evaluation process is 0.0299944; was equal to the values. It
was found that the error of digital processing in Haara wavelets is small, it can
be concluded that the use of Haara wavelet in the process of digital processing of
signals gives good results. This method can also be used to solve problems such
as signal compression, medical signal filtering, and signal-to-noise separation.
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