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II. MAIN PART
Construction of Haar's fragmentary vault
: Haar's orthogonal wavelets are widely used in
solving practical problems.Haar’s fragmentary wavelets attract the attention of experts for two
reasons:
1. Reduce the number of coefficients required to approximate (with a given accuracy) the total
number of binary segments.
2. Absence of "long" operations in the calculation of coefficients. Only add, zoom and edit
operations are used.
The process of changing the signal wavelet is based on the use of two types of functions: wavelet
function and scaling function, i.e. they are constructed by moving a single mother wavelet
)
(
t
along the signal in time b and changing
a
the time scale [3, 7, and 9]:
)
(
)
(
,
)
,
(
,
1
)
(
2
R
L
t
R
b
a
a
b
t
a
t
ab
]
1
,
0
[
i
0
2
q
2
2
1
-
q
if
2
1
2
2
1
-
q
2
1
-
q
if
2
1
)
(
p
p
2
/
p
p
2
/
x
f
x
N
x
N
t
p
p
(1)
Here,
)
0
(
2
q
1
1,
-
n
p
0
,
2
=
N
1,
-
N
,
…
0,
=
k
p
n
p
)
(
t
-wavelet function
ISSN: 2278-4853 Vol 10, Issue 9, September, 2021 Impact Factor: SJIF 2021 = 7.699
Asian Journal of Multidimensional Research (AJMR)
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132
AJMR
In digital signal processing, wavelet functions are used to distinguish the details and local
properties of signals, and a scaling function is used to approximate signals.
0
V
-we define a set of invariant functions in all intervals, i.e. a set of linear vectors [4,9,10,11].
In this case, the following scaling function belongs to the set
0
V
:
Otherwise
,
0
1
0
,
1
)
(
)
(
0
,
0
t
t
t
(2)
0
i
when scaling function
1
V
-collection
2
1
,
0
and
1
,
2
1
is a set of functions that do not change in the interval, it
forms linear vectors. The scaling function belongs to the
1
V
-set and is considered as its wavelet
function [5,12,17,18]:
Otherwise
,
0
2
1
0
,
1
)
2
(
)
(
0
,
1
t
t
t
and
Otherwise
,
0
1
2
1
,
1
)
1
2
(
)
(
1
,
1
t
t
t
(3)
1
i
when scaling function
This function is
]
1
,
0
[
in the range
2
1
,
0
and
1
,
2
1
does not change even at intervals. So
0
V
each element of the set
1
V
-is also an element of the set,
1
0
V
V
the attitude is reasonable.
2
V
we define the set in a similar way.
2
V
-
4
1
,
0
,
2
1
,
4
1
,
4
3
,
2
1
,
1
,
4
3
a set of interval-
dependent functions. A set of similar
n
V
scaling features, i.e.
1
2
,...,
1
,
0
),
2
(
)
(
,
n
n
j
n
j
j
t
t
n
n
n
j
t
j
j
t
2
1
2
,
1
2
0
Otherwise
,
0
2
1
2
,
1
)
(
,
n
n
j
n
j
t
j
t
,
1
2
,...,
1
,
0
n
j
(4)
...
...
1
0
n
V
V
V
ISSN: 2278-4853 Vol 10, Issue 9, September, 2021 Impact Factor: SJIF 2021 = 7.699
Asian Journal of Multidimensional Research (AJMR)
https://www.tarj.in
133
AJMR
n
i
The scaling function, where,
n
n
n
j
t
j
j
t
2
1
2
,
1
2
0
is the interval of change
scaling functions,
)
(
,
t
j
n
n
V
are scaling functions that contain a set of vectors that include a
scalar product, which means that these sets make up the Euclidean space. In our case as a scalar
product
1
0
)
(
)
(
)
,
(
dt
t
g
t
f
g
f
we obtain the view that the coefficients of the
n
C
-scale functions are
determined using this formula.
In that case
1
2
...,
,
1
,
0
),
2
(
2
)
(
,
n
n
n
j
n
j
j
t
t
Using Figures (3) and (4), find the coefficients of the Haar wavelet:
1
0
)
(
)
(
dx
x
f
x
C
n
n
(5)
The formula for finding the coefficients of the Haar wavelet.
0
1
)
(
)
(
)
(
n
n
n
i
i
x
C
x
f
x
f
(6)
Construction of Haar’s fragmentary wavelet:
The search for ways to reduce the number of
coefficients required for memory storage and improve “smoothness” performance requires a shift
to higher levels of wavelets. The simplest of these are the Xaar-line wavelets, they are formed as
a result of the integration of the fragment- invariant wavelets of the Haar- wavelet.
The disadvantage of Haar’s fragmentary wavelengths is the increase in errors when approaching
the signal, i.e., the need to memorize several coefficients of the signal to ensure 0.1% accuracy
[4,19,20].
In solving many practical problems, the capabilities of fragmentary wavelets are insufficient,
however, the interpolation error of fragmentary wavelets is greater than the interpolation error of
fragmentary wavelets, so it is advisable to switch to fragmentary wavelets.
An analysis of the available literature shows that Haar does not have an algorithm for
determining the fractional wavelet coefficients, so this type of wavelet is not widely used.
Sequence of approximation of the linear Haar wavelet:
x
hain
C
x
f
k
n
k
k
1
0
(7)
The disadvantage of this
k
C
sequence is the lack of an algorithm for quick calculation of
coefficients.
This defect can be remedied by applying a parabolic spline. If we take the second-order product
of the parabolic spline,
)
(
x
f
the function is interpolated in the interval [0,1], and with the
ISSN: 2278-4853 Vol 10, Issue 9, September, 2021 Impact Factor: SJIF 2021 = 7.699
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