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to increase the resolution of the approach of the signal. The approaches
of
f
for the scales
and
2
j
-
are respectively equal for its orthogonal
j
2
V
W
projections
V
and
with
VV
. Be
and
the orthogonal complement
j
-
j
j
j
-1
of
V
in
. The orthogonal projection of
f
in
V
can be written as the
V
j
-
j
-
P
f
=
P
+
P
sum of orthogonal projections
P
in
V
and
W
. Then,
. The
V
V
W
j
−
1
j
j
åå
function
()
( ) ()
,
ft
=
A
W jn
,
y
t
can be reconstructed from the discrete
f
j n
j
n
(
)
W
f
,
where
j
is the scaling factor and
n
is the
movement factor [13, 14, 15, 17, 18].
The wavelets
j
,
n
wavelets coeffi cients
generated of the same wavelet mother function
ψ
(
t
)
j
,
n
ψ
have different scale
j
and place
n
, but they have the same form. Scale
factor
j
> 0 is always used. The wavelet is dilated when the scale
j
> 1, and
it is contracted when
j
< 1. This way, changing the value of
j
the different
range from frequencies is covered. Big values of the parameter
j
corre-
spond to frequencies of minor range, or a big scale of
(
t
)
ψ
.
Small values
of
j
correspond to frequencies of minor range or a very small scale of
)
(
t
)
j
,
n
ø
,j
[13- 18]. The continuous wavelet functions with discrete factors
of scale and movement are named discrete wavelets. Finally, the signal
)
(
t
(
f
can be compressed or expand in the time, this will have a few certain
after effects in the plane of frequencies,
æ
÷
1
t
()
() ()
ç
f
t compression by a factor
2
j
s
f
t
=
f
ç
÷
ç
èø
s
s
j
2
(1)
1
1
ˆ
ˆ
ˆ
( )
ˆ
( )
()
()
j
j
j
j
f
w compression by a factor
f
w
=
22
f
w
=
22
f
w
j
j
2
2
j
2
The coeffi cient of the decomposition of a function
f
in an orthogo-
nal base of wavelets is calculated by a subsequent algorithm of dis-
crete convolution with
h
and
g
, and realizes a sampling of the low-
pass fi lter (LPF)
=
å
[] [][ ]
xk
xnhkn
2-
and the high-pass fi lter (HPF)
low
=
å
[][ ]
n
xk
[]
xngkn
2 -
kh
are the impulse
responses of HPF and LPF, respectively, subsampled by a factor of 2
[17-20]. These coeffi cients are calculated by cascades of discrete fi lters
through of convolution and sampling.
DWT decomposes a discrete signal into two subsignals of half of the
original length. This subsignal is known as the approaches and the other one
is known as the details [19]. The fi rst subsignal
, where
and
g
[
2
k
-
n
]
[
2
-
n
]
high
n
(
)
, for the sig-
nal
x
is obtained making the average of the signal as follows: the fi rst value
a
is calculated by taking the fi rst set of values vector
aaa a
1
=
,
,
,
12
m
/ 2
[](
)
x
m
:
x
+
x
/
2
and
1
2
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