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Fil
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Fil
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m
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ly
em
p
lo
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,
s
u
ch
as
th
e
L
ap
lacia
n
f
ilter
[
1
]
,
B
ay
esian
o
n
e
[
2
]
,
an
d
th
e
Ga
u
s
s
ian
o
n
e
[
3
]
.
A
s
ig
n
if
ican
t
ar
ea
o
f
a
p
p
licatio
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s
is
f
ilter
in
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d
ata
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e
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I
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A
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231
s
ig
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als,
s
u
ch
as
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n
ce
p
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alo
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r
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m
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io
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am
(
E
C
G)
,
a
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d
elec
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m
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r
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E
MG
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Nu
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ap
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ac
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es a
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atu
r
e
u
s
in
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d
if
f
e
r
en
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p
es o
f
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[
4
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–
[
6
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.
Hea
r
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r
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d
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o
s
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to
o
l
f
o
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ass
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ciate
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illn
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[
7
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,
[
8
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ality
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en
tial
[
9
]
,
[
1
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.
Den
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iac
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s
ig
n
als
ar
e
o
f
ten
co
n
tam
in
ated
b
y
v
a
r
io
u
s
ty
p
es
o
f
n
o
is
e,
s
u
ch
as
b
aselin
e
wan
d
er
,
p
o
wer
lin
e
in
ter
f
e
r
en
ce
,
an
d
m
u
s
cle
ar
tifa
cts,
wh
ich
ca
n
d
is
to
r
t
th
e
o
r
ig
in
al
E
C
G
s
ig
n
al
[
1
2
]
.
T
h
e
n
o
is
e
m
ak
es
it
d
if
f
icu
lt
to
id
en
tify
im
p
o
r
tan
t
f
ea
tu
r
es,
s
u
ch
as th
e
P
-
wav
e,
QR
S c
o
m
p
lex
,
an
d
T
-
wav
e,
ess
en
tial f
o
r
d
iag
n
o
s
in
g
co
n
d
itio
n
s
lik
e
ar
r
h
y
th
m
ias
o
r
is
ch
em
ia.
Sev
er
al
t
y
p
es
o
f
f
ilter
s
f
o
r
b
io
m
ed
ical
s
ig
n
a
l
p
r
o
ce
s
s
in
g
h
av
e
b
ee
n
u
s
ed
:
Gau
s
s
ian
,
Mittag
L
ef
f
ler
,
Go
lley
,
ad
a
p
tiv
e,
an
d
m
ed
ian
f
ilter
s
.
T
h
ey
wer
e
a
n
aly
ze
d
in
ter
m
s
o
f
ef
f
ec
tiv
e
n
ess
in
s
u
p
p
r
ess
in
g
f
lu
ctu
atio
n
s
d
u
e
t
o
ar
tifa
cts,
p
o
wer
lin
e
n
o
is
e
an
d
o
th
e
r
n
o
i
s
e
s
o
u
r
ce
s
in
o
r
d
er
to
im
p
r
o
v
e
th
e
o
r
ig
in
al
E
C
G
s
ig
n
al’
s
q
u
ality
[
8
]
,
[
1
3
]
–
[
1
5
]
.
I
n
s
ig
n
al
f
ilter
in
g
,
a
k
ey
ad
v
a
n
ce
m
en
t
is
th
e
d
ev
el
o
p
m
en
t
o
f
g
en
e
r
alize
d
f
ilter
s
s
u
ch
as
th
e
Mittag
-
L
ef
f
ler
o
n
e,
wh
ic
h
p
r
o
v
i
d
es
ad
d
itio
n
al
u
s
er
-
ad
j
u
s
tab
le
p
ar
am
ete
r
s
,
en
s
u
r
in
g
h
ig
h
er
f
lex
ib
ilit
y
th
an
tr
a
d
itio
n
al
ex
p
o
n
en
tial
o
r
Gau
s
s
ian
f
ilter
s
.
T
h
is
f
ilter
ty
p
e
allo
ws
f
o
r
a
b
ett
er
b
alan
ce
b
etwe
en
n
o
is
e
r
ed
u
ctio
n
an
d
s
ig
n
al
r
eten
tio
n
b
y
tu
n
in
g
t
h
e
p
ar
am
ete
r
s
,
m
ak
in
g
it
p
ar
ticu
lar
l
y
ef
f
ec
tiv
e
f
o
r
n
o
is
y
E
C
G
s
ig
n
als.
T
h
is
ap
p
r
o
ac
h
en
h
an
ce
s
th
e
f
ilter
p
er
f
o
r
m
an
ce
b
y
ad
d
r
ess
in
g
ch
allen
g
es
s
u
ch
as
f
ilter
in
g
s
to
ch
asti
c
co
m
p
o
n
en
ts
wh
ile
r
ed
u
cin
g
co
m
p
u
tatio
n
al
c
o
m
p
lex
ity
an
d
o
u
tp
e
r
f
o
r
m
in
g
c
o
n
v
e
n
tio
n
a
l
f
ilter
s
[
1
6
]
.
Fo
r
in
s
tan
ce
,
in
[
1
7
]
,
th
e
au
th
o
r
s
in
tr
o
d
u
ce
d
a
Mittag
-
L
ef
f
ler
f
i
lter
,
an
e
x
ten
s
io
n
o
f
t
h
e
Gau
s
s
ian
o
n
e,
u
s
in
g
t
h
e
Mittag
-
L
ef
f
ler
f
u
n
ctio
n
in
its
p
r
o
b
ab
ilit
y
-
d
e
n
s
ity
f
u
n
ctio
n
an
d
co
n
v
o
lu
tio
n
k
er
n
el.
T
h
e
f
ilter
h
as
th
r
ee
ad
ju
s
tab
le
p
ar
am
eter
s
in
f
lu
e
n
ce
d
b
y
a
f
o
r
g
ettin
g
f
ac
to
r
,
o
f
f
er
in
g
ad
v
a
n
tag
es
o
v
e
r
class
ic
al
Gau
s
s
ian
f
ilter
in
g
in
task
s
s
u
ch
as
E
C
G
s
ig
n
als
d
en
o
is
in
g
.
T
h
e
im
p
lem
en
tati
o
n
d
etails
an
d
th
e
d
ev
elo
p
ed
MA
T
L
AB
f
u
n
ctio
n
ar
e
p
r
o
v
id
ed
.
I
n
[
1
8
]
,
th
e
au
t
h
o
r
s
p
r
o
p
o
s
ed
th
e
Alex
an
d
er
f
r
ac
tio
n
al
d
if
f
e
r
en
tial
win
d
o
w
(
AFDW
)
f
ilter
f
o
r
E
C
G
s
ig
n
al
d
en
o
is
in
g
b
ased
o
n
th
e
g
en
er
alize
d
Alex
an
d
er
p
o
ly
n
o
m
ial
an
d
f
r
ac
tio
n
al
ca
lcu
l
u
s
;
it
u
s
es
a
f
o
r
war
d
an
d
b
ac
k
war
d
f
ilter
in
g
a
p
p
r
o
ac
h
,
av
er
ag
i
n
g
th
e
c
o
ef
f
icien
t
s
o
f
b
o
th
f
ilter
s
.
B
ased
o
n
m
o
r
p
h
o
lo
g
ical
p
o
we
r
p
r
eser
v
atio
n
m
ea
s
u
r
e
(
MPPM)
,
th
e
r
esu
lts
d
em
o
n
s
tr
ated
th
at
th
e
f
ilter
p
r
eser
v
es
s
ig
n
al
p
o
wer
an
d
QR
S
m
o
r
p
h
o
lo
g
y
.
T
h
e
s
tu
d
y
in
[
1
9
]
p
r
o
p
o
s
ed
f
r
ac
tio
n
al
-
o
r
d
e
r
w
av
elet
f
ilter
s
f
o
r
E
C
G
s
ig
n
al
d
en
o
is
in
g
,
r
e
p
lacin
g
tr
ad
itio
n
al
lo
w
-
an
d
h
ig
h
-
p
ass
f
ilter
s
.
T
h
e
f
r
ac
tio
n
al
wav
ele
ts
wer
e
co
m
p
ar
ed
u
s
in
g
ap
p
r
o
p
r
iate
th
r
esh
o
ld
in
g
an
d
wav
elet
d
ec
o
m
p
o
s
itio
n
.
R
esu
lts
s
h
o
wed
s
u
p
er
io
r
p
er
f
o
r
m
a
n
ce
o
f
f
r
ac
tio
n
al
wa
v
elets
co
m
p
ar
ed
to
tr
ad
itio
n
al
wav
elets,
esp
ec
ially
in
r
em
o
v
i
n
g
h
i
g
h
-
f
r
eq
u
en
cy
n
o
is
e
with
o
u
t
r
eq
u
ir
in
g
p
r
io
r
f
r
eq
u
en
c
y
k
n
o
wled
g
e.
Fu
r
th
e
r
m
o
r
e
,
a
n
ew
alg
o
r
ith
m
f
o
r
d
en
o
is
in
g
E
C
G
d
ata
co
n
tam
in
ate
d
b
y
wi
d
e
-
b
an
d
n
o
is
e
was
p
r
o
p
o
s
ed
,
wh
er
e
th
e
E
C
G
s
ig
n
al
is
s
eg
m
en
ted
in
to
co
m
p
o
n
en
ts
with
d
is
jo
in
t
tim
e
an
d
o
v
er
lap
p
in
g
f
r
eq
u
e
n
cy
d
o
m
ain
s
[
2
0
]
.
E
ac
h
s
eg
m
e
n
t
is
d
en
o
is
ed
u
s
in
g
id
ea
l
f
ilter
s
d
esig
n
ed
b
y
m
in
im
izin
g
a
p
e
n
alize
d
least
-
s
q
u
ar
es
f
u
n
ctio
n
;
t
h
e
m
eth
o
d
o
u
tp
er
f
o
r
m
s
ex
is
tin
g
tech
n
iq
u
es.
Als
o
,
Sav
itzk
y
-
Go
lay
f
ilter
s
wer
e
u
s
ef
u
l
f
o
r
ef
f
icien
tly
d
en
o
is
in
g
th
e
E
C
G
s
ig
n
als;
in
[
2
1
]
,
a
lo
w
-
d
is
to
r
tio
n
ad
ap
tiv
e
Sav
itzk
y
-
Go
lay
(
L
DASG)
f
ilter
in
g
m
eth
o
d
f
o
r
E
C
G
d
en
o
is
in
g
was
p
r
o
p
o
s
ed
,
wh
ich
,
u
n
lik
e
s
tan
d
a
r
d
Sav
itzk
y
-
Go
lay
f
ilter
,
u
s
es
d
is
cr
ete
cu
r
v
atu
r
e
esti
m
atio
n
to
ad
ju
s
t f
o
r
s
ig
n
al
v
ar
iatio
n
s
,
r
ed
u
cin
g
d
is
to
r
tio
n
wh
ile
m
ain
tain
in
g
ef
f
ec
tiv
e
d
a
ta
s
m
o
o
th
in
g
.
B
ased
o
n
th
e
tr
ad
itio
n
al
ex
p
o
n
en
tial
f
ilter
,
t
h
is
ar
ticle
in
v
esti
g
ates
a
n
o
v
el
s
tr
etch
ed
-
c
o
m
p
r
ess
ed
ex
p
o
n
e
n
tial
f
ilter
f
o
r
d
en
o
is
in
g
th
e
E
C
G
s
ig
n
als.
Giv
en
th
e
ef
f
ec
tiv
en
ess
o
f
th
e
e
x
p
o
n
en
ti
al
f
ilter
,
wh
ic
h
ca
n
lead
to
th
e
Gau
s
s
ian
o
n
e,
esp
ec
ially
in
E
C
G
s
ig
n
als
'
p
r
o
ce
s
s
in
g
,
a
n
o
v
el
id
ea
em
er
g
ed
f
o
r
a
s
tr
etch
ed
-
co
m
p
r
ess
ed
ex
p
o
n
en
tial
lo
w
-
p
ass
(
SC
E
L
P)
f
ilter
.
T
h
e
SC
E
L
P
f
ilter
r
etain
s
th
e
b
asic
s
tr
u
ctu
r
e
o
f
th
e
ex
p
o
n
e
n
tial
f
ilter
,
b
u
t
it
h
as
s
u
p
er
io
r
p
er
f
o
r
m
a
n
ce
b
y
ad
ap
tin
g
its
s
h
ap
e,
m
ak
in
g
it
m
o
r
e
ef
f
ec
tiv
e
in
h
an
d
lin
g
E
C
G
s
ig
n
als
wh
ile
m
ain
tain
in
g
s
im
p
licity
a
n
d
ef
f
icien
cy
.
T
h
e
SC
E
L
P
f
ilter
em
p
lo
y
s
a
n
e
x
p
o
n
e
n
tial
f
u
n
ctio
n
as
a
k
er
n
el
to
th
e
s
tan
d
ar
d
e
x
p
o
n
en
tial
Gau
s
s
ian
f
ilter
t
h
at
m
ath
em
atica
lly
m
o
d
if
ies
th
e
i
n
p
u
t
s
ig
n
al
th
r
o
u
g
h
co
n
v
o
l
u
tio
n
.
L
i
k
e
th
e
e
x
p
o
n
e
n
tial
f
ilter
,
f
ea
tu
r
e
d
b
y
an
ex
p
o
n
en
tial
f
u
n
ctio
n
as
im
p
u
ls
e
r
esp
o
n
s
e,
th
e
SC
E
L
P
f
ilter
h
as th
e
ad
v
a
n
tag
e
o
f
n
o
t
o
v
er
s
h
o
o
tin
g
th
e
s
ig
n
al,
m
in
i
m
izin
g
r
is
e
an
d
f
all
tim
es
[
2
2
]
–
[
2
7
]
.
A
MA
T
L
AB
im
p
lem
en
tatio
n
o
f
th
e
SC
E
L
P
f
ilter
h
as
b
ee
n
d
ev
elo
p
e
d
an
d
test
ed
,
d
en
o
is
in
g
p
u
b
licly
av
ailab
le
E
C
G
s
am
p
les
(
5
0
n
o
r
m
al
a
n
d
5
0
ab
n
o
r
m
al)
f
r
o
m
th
e
Ph
y
s
io
Net
d
atab
ase
b
y
v
a
r
y
in
g
th
e
β
p
ar
am
eter
.
T
h
e
c
h
ar
ac
ter
izati
o
n
r
esu
lts
d
em
o
n
s
tr
ated
t
h
at
a
β
v
alu
e
r
an
g
i
n
g
f
r
o
m
1
t
o
2
p
r
o
v
id
es
o
p
tim
al
p
er
f
o
r
m
an
ce
in
ter
m
s
o
f
m
ea
n
SNR
an
d
MSE
v
alu
es.
Fu
r
th
er
m
o
r
e,
th
e
ca
p
ab
ilit
y
o
f
SC
E
L
P
f
ilter
in
tr
ea
tin
g
s
ig
n
als
af
f
ec
ted
b
y
ad
d
itiv
e
wh
ite
n
o
is
e
was
test
ed
.
C
o
m
p
ar
ed
with
o
th
er
f
ilter
ty
p
o
lo
g
ies
(
i.e
.
,
Gau
s
s
ian
,
Mittag
–
L
ef
f
ler
,
a
n
d
Sav
itzk
y
-
Go
lay
)
,
th
e
p
r
o
p
o
s
ed
SC
E
L
P
f
ilter
p
r
o
v
id
es
b
etter
p
er
f
o
r
m
an
ce
,
as
d
etailed
i
n
s
ec
tio
n
3
.
T
h
e
m
ain
co
n
tr
ib
u
ti
o
n
s
to
th
e
p
r
o
p
o
s
ed
r
esear
ch
a
r
ticle
ar
e
th
e
f
o
llo
win
g
:
-
A
n
o
v
el
s
tr
etch
ed
-
c
o
m
p
r
ess
ed
ex
p
o
n
en
tial
lo
w
-
p
ass
f
ilter
is
p
r
esen
ted
,
wh
o
s
e
im
p
u
l
s
e
r
esp
o
n
s
e
is
o
p
tim
ized
as
a
f
u
n
ctio
n
o
f
th
e
β
p
ar
am
eter
to
m
ax
im
ize
th
e
SNR
an
d
m
in
im
ize
MSE
,
o
u
tp
er
f
o
r
m
in
g
tr
ad
itio
n
al
f
ilter
s
ty
p
ically
u
s
e
d
in
d
e
n
o
is
in
g
E
C
G
s
ig
n
als.
-
T
h
e
p
r
o
p
o
s
ed
SC
E
L
P
f
ilter
h
a
s
b
ee
n
ch
ar
ac
ter
ized
,
p
r
o
v
in
g
its
ef
f
ec
tiv
en
ess
in
d
en
o
is
in
g
th
e
E
C
G
s
ig
n
als
b
y
v
ar
y
in
g
t
h
e
h
id
d
en
ex
p
o
n
e
n
tial p
ar
am
eter
(
β)
t
o
d
eter
m
i
n
e
th
e
o
p
tim
al
f
ilter
s
ettin
g
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
1
6
,
No
.
1
,
Feb
r
u
ar
y
20
2
6
:
2
3
0
-
245
232
-
A
co
m
p
ar
ativ
e
an
aly
s
is
o
f
th
e
SC
E
L
P
f
ilter
,
in
ter
m
s
o
f
m
ea
n
SNR
an
d
MSE
,
with
d
if
f
er
en
t
f
ilter
ty
p
o
lo
g
ies
(
i.e
.
,
Gau
s
s
ian
,
Mittag
–
L
ef
f
ler
,
an
d
Sav
itzk
y
-
G
o
lay
f
ilter
s
)
is
p
r
esen
ted
,
d
e
m
o
n
s
tr
atin
g
th
e
s
u
p
er
io
r
ity
in
d
e
n
o
is
in
g
th
e
E
C
G
s
ig
n
als.
I
n
ad
d
itio
n
,
th
e
SC
E
L
P
f
ilter
’
s
p
er
f
o
r
m
an
ce
wa
s
ev
alu
ated
o
n
E
C
G
s
ig
n
als
af
f
ec
ted
b
y
ad
d
itiv
e
wh
ite
n
o
is
e
an
d
co
m
p
ar
ed
with
th
e
o
th
er
f
ilt
er
s
'
ty
p
o
lo
g
ies,
d
em
o
n
s
tr
atin
g
its
ef
f
ec
tiv
en
es
s
an
d
s
lig
h
t su
p
er
io
r
ity
in
ter
m
s
o
f
m
ea
n
SNR
an
d
MSE
.
T
h
e
r
esear
ch
ar
ticle
is
o
r
g
an
i
ze
d
as
f
o
llo
ws:
in
s
ec
tio
n
2
,
t
h
e
f
ilter
's
m
ath
e
m
atica
l
r
ep
r
esen
tatio
n
is
p
r
o
p
o
s
ed
,
an
d
th
e
E
C
G
s
ig
n
als'
d
ataset
u
s
ed
to
e
v
alu
ate
th
e
f
ilter
p
er
f
o
r
m
an
ce
is
d
es
cr
ib
ed
.
T
h
e
r
esu
lts
r
elate
d
to
th
e
SC
E
L
P
f
ilter
ch
ar
ac
ter
izatio
n
ar
e
r
e
p
o
r
ted
in
s
ec
tio
n
3
,
d
em
o
n
s
tr
atin
g
t
h
e
b
est
p
er
f
o
r
m
an
ce
o
f
th
e
p
r
o
p
o
s
ed
f
ilter
co
m
p
ar
e
d
to
o
th
er
f
ilter
ty
p
o
lo
g
ies.
Sectio
n
3
also
r
ep
o
r
ts
o
n
th
e
d
is
cu
s
s
io
n
o
n
th
e
SC
E
L
P
f
ilter
’
s
p
er
f
o
r
m
a
n
ce
an
d
a
co
m
p
ar
ativ
e
an
al
y
s
is
with
o
th
er
co
m
m
o
n
E
C
G
-
d
en
o
is
in
g
f
ilter
s
.
Fin
ally
,
s
ec
tio
n
4
s
u
m
m
ar
izes th
e
m
ain
r
esu
lts
p
r
esen
ted
in
th
e
r
esear
c
h
wo
r
k
an
d
f
u
t
u
r
e
p
e
r
s
p
ec
tiv
es.
2.
P
RO
P
O
SE
D
M
E
T
H
O
D
T
h
is
s
ec
tio
n
in
tr
o
d
u
ce
s
th
e
f
u
n
d
am
e
n
tals
an
d
d
ef
in
itio
n
s
o
f
th
e
ex
p
o
n
e
n
tial
f
ilter
s
an
d
th
eir
m
ain
p
r
o
p
er
ties
;
th
en
,
th
e
s
tr
etch
ed
-
co
m
p
r
ess
ed
ex
p
o
n
en
tial
f
u
n
ctio
n
an
d
d
is
tr
ib
u
tio
n
a
r
e
p
r
esen
ted
,
wh
ich
,
in
teg
r
ated
in
to
th
e
k
e
r
n
el,
ar
e
u
s
ed
f
o
r
d
ef
i
n
in
g
th
e
n
o
v
el
s
tr
etch
ed
-
co
m
p
r
ess
ed
ex
p
o
n
en
ti
al
f
ilter
.
Fin
ally
,
th
e
MA
T
L
AB
im
p
lem
en
tatio
n
o
f
th
e
p
r
o
p
o
s
ed
cu
s
to
m
izab
le
ex
p
o
n
en
tial
f
ilter
an
d
m
etr
ics
to
ev
alu
ate
its
p
er
f
o
r
m
an
ce
in
d
en
o
is
in
g
th
e
E
C
G
s
ig
n
als ar
e
r
ep
o
r
ted
.
2
.
1
.
E
x
po
nentia
l f
ilte
r:
m
a
t
hem
a
t
ica
l f
o
rm
ula
t
io
n
a
nd
s
t
ep
re
s
po
ns
e
T
h
e
ex
p
o
n
e
n
tial
f
ilter
is
o
n
e
o
f
th
e
s
im
p
lest
f
o
r
m
s
o
f
lo
w
-
p
ass
f
ilter
,
with
h
ig
h
f
r
eq
u
en
cies
atten
u
ated
an
d
lo
w
f
r
e
q
u
en
cie
s
p
ass
ed
u
n
ch
an
g
e
d
.
T
h
e
s
am
p
lin
g
in
ter
v
al
is
th
e
o
n
ly
o
th
e
r
tu
n
in
g
p
ar
am
ete
r
av
ailab
le,
an
d
th
e
p
r
ev
io
u
s
o
u
tp
u
t
is
th
e
o
n
ly
v
ar
iab
le
th
at
n
ee
d
s
to
b
e
s
to
r
ed
.
I
t
is
a
n
in
f
in
ite
im
p
u
ls
e
r
esp
o
n
s
e
(
I
I
R
)
au
t
o
-
r
eg
r
ess
iv
e
f
ilter
,
m
ea
n
i
n
g
th
at
th
e
ef
f
ec
t
s
o
f
a
n
i
n
p
u
t
ch
an
g
e
o
n
th
e
f
ilter
o
u
t
p
u
t
d
im
in
is
h
ex
p
o
n
e
n
tially
,
tak
in
g
i
n
to
ac
c
o
u
n
t
th
e
co
m
p
u
tatio
n
al
lim
its
o
f
th
e
p
r
o
ce
s
s
in
g
a
n
d
d
is
p
lay
d
ev
ices.
T
h
is
f
ilter
is
also
k
n
o
wn
as
ex
p
o
n
en
tial
s
m
o
o
th
in
g
in
s
o
m
e
ac
ad
em
ic
f
ield
s
.
T
h
e
ex
p
o
n
en
tial
f
ilter
is
r
ef
er
r
ed
to
as
an
ex
p
o
n
e
n
tially
weig
h
ted
m
o
v
in
g
av
er
ag
e
(
E
W
MA
)
o
r
s
im
p
ly
an
ex
p
o
n
en
tial
m
o
v
i
n
g
av
e
r
a
g
e
(
E
MA
)
f
ilter
in
ce
r
tain
f
ield
s
,
s
u
ch
as
in
v
estme
n
t
r
esear
ch
;
f
o
r
t
h
is
f
ilter
ty
p
o
lo
g
y
is
im
p
r
o
p
e
r
ly
u
s
ed
th
e
ter
m
“
m
o
v
i
n
g
av
er
ag
e
”
r
ef
er
r
ed
t
o
th
e
tim
e
-
s
er
ies
an
aly
s
is
,
in
th
e
class
ic
au
to
r
eg
r
ess
iv
e
m
o
v
in
g
av
e
r
ag
e
(
AR
MA
)
m
o
d
el
s
in
ce
a
m
o
v
in
g
av
e
r
ag
e
f
ilter
ju
s
t
co
n
s
id
er
s
th
e
c
u
r
r
en
t
in
p
u
t
r
ath
er
t
h
an
th
e
in
p
u
t
h
is
to
r
y
.
T
h
e
ex
p
o
n
en
tial
f
ilter
is
th
e
an
alo
g
co
u
n
ter
p
ar
t
o
f
th
e
f
ir
s
t
-
o
r
d
er
lag
f
r
e
q
u
en
tly
em
p
lo
y
e
d
in
t
h
e
an
alo
g
m
o
d
elin
g
o
f
co
n
tin
u
o
u
s
-
tim
e
co
n
tr
o
l
s
y
s
tem
s
f
o
r
d
is
cr
ete
tim
e.
An
R
C
f
i
lter
,
co
n
s
titu
ted
b
y
a
s
in
g
le
r
es
is
to
r
an
d
ca
p
ac
ito
r
,
is
a
f
ir
s
t
-
o
r
d
er
lag
i
n
elec
tr
ic
al
cir
cu
its
.
W
h
en
h
ig
h
lig
h
tin
g
th
e
p
ar
allelis
m
with
an
alo
g
cir
cu
its
,
th
e
tim
e
co
n
s
tan
t,
r
ep
r
esen
ted
b
y
th
e
Gr
ee
k
s
y
m
b
o
l
tau
(
τ
)
,
is
t
h
e
o
n
ly
tu
n
in
g
p
ar
a
m
eter
;
g
iv
en
th
e
s
am
e
tim
e
co
n
s
tan
t,
th
e
v
alu
es a
t th
e
d
is
cr
ete
s
am
p
lin
g
p
er
i
o
d
s
p
er
f
ec
tly
m
atch
th
e
c
o
r
r
esp
o
n
d
in
g
co
n
tin
u
o
u
s
-
tim
e
lag
.
T
h
e
f
o
llo
win
g
eq
u
atio
n
s
illu
s
tr
ate
th
e
lin
k
b
etwe
en
th
e
d
ig
ital
im
p
lem
en
tatio
n
(
i.e
.
,
s
m
o
o
th
in
g
co
n
s
tan
t)
an
d
tim
e
co
n
s
tan
t
(
τ
)
.
T
o
en
s
u
r
e
th
at
th
e
o
u
t
p
u
t
an
d
in
p
u
t
s
ig
n
als
ar
e
id
en
tica
l
u
n
d
er
s
tead
y
-
s
tate
co
n
d
itio
n
s
,
th
e
ex
p
o
n
e
n
tial
f
ilter
co
m
b
in
es
th
e
m
o
s
t
r
ec
en
t
in
p
u
t
d
ata
with
a
weig
h
ted
co
m
b
in
atio
n
o
f
th
e
p
r
io
r
esti
m
ate
(
o
u
t
p
u
t)
,
with
t
h
e
to
tal
weig
h
ts
s
et
to
1
as
s
h
o
wn
in
Fig
u
r
e
1
.
T
h
e
o
u
tp
u
t
(
)
v
s
in
p
u
t
(
)
r
elatio
n
s
h
ip
o
f
a
n
ex
p
o
n
e
n
tial f
ilter
is
ex
p
r
ess
ed
as
(
1
)
:
(
)
=
∗
(
−
1
)
+
(
1
−
)
∗
(
)
(
1
)
w
h
er
e
(
)
is
th
e
r
aw
in
p
u
t
at
tim
e
s
tep
k
,
(
)
th
e
f
ilter
ed
o
u
tp
u
t
at
tim
e
s
tep
k
,
a
n
d
a
p
ar
a
m
eter
(
ca
lled
s
m
o
o
th
in
g
c
o
n
s
tan
t)
b
etwe
en
0
an
d
1
(
ty
p
ical
v
alu
es a
r
e
ch
o
s
en
in
th
e
r
an
g
e
0
.
8
÷
0
.
9
9
)
.
Fig
u
r
e
1
.
B
lo
ck
d
iag
r
am
o
f
a
g
en
er
ic
d
is
cr
ete
-
tim
e
f
ilter
T
h
e
s
m
o
o
th
in
g
co
n
s
tan
t
is
co
m
p
u
ted
an
d
s
av
ed
f
o
r
co
n
v
en
i
en
ce
o
n
ly
in
ca
s
es
wh
en
th
e
ap
p
licatio
n
d
ev
elo
p
er
s
p
ec
if
ies a
n
ew
v
alu
e
f
o
r
th
e
d
esire
d
tim
e
-
co
n
s
tan
t
in
s
y
s
tem
s
with
a
f
ix
ed
s
am
p
lin
g
p
er
io
d
T
:
=
(
−
)
(
2
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
A
n
o
ve
l str
etch
ed
-
co
mp
r
ess
ed
ex
p
o
n
e
n
tia
l lo
w
-
p
a
s
s
filt
er a
n
d
its
a
p
p
lica
tio
n
…
(
R
o
b
erto
d
e
F
a
z
io
)
233
wh
er
e
is
th
e
f
ilter
tim
e
co
n
s
ta
n
t
in
th
e
s
am
e
tim
e
u
n
its
as
T
.
T
h
e
ex
p
o
n
e
n
tial
f
u
n
ctio
n
in
(
2
)
m
u
s
t b
e
a
p
p
lied
to
ea
ch
tim
e
s
tep
in
s
y
s
tem
s
wh
er
e
d
ata
s
am
p
lin
g
o
cc
u
r
s
a
t
ir
r
eg
u
lar
in
ter
v
als
(
n
am
ely
,
T
is
n
o
t
co
n
s
tan
t)
.
Usu
ally
,
th
e
f
ir
s
t
in
p
u
t
is
u
s
ed
to
in
itialize
th
e
f
ilter
o
u
tp
u
t.
T
h
er
e
is
n
o
f
ilter
in
g
if
th
e
s
m
o
o
th
in
g
co
n
s
tan
t
(
)
g
o
es
to
ze
r
o
,
n
am
ely
wh
en
t
h
e
tim
e
co
n
s
tan
t
a
p
p
r
o
ac
h
es
ze
r
o
,
an
d
co
n
s
eq
u
en
tly
,
th
e
o
u
tp
u
t
(
(
)
)
eq
u
als
th
e
n
ew
in
p
u
t
(
(
)
)
.
Oth
er
wis
e,
if
th
e
s
m
o
o
th
in
g
co
n
s
tan
t
(
)
is
clo
s
e
to
1
,
an
d
th
er
e
f
o
r
e
τ
in
cr
ea
s
es,
th
e
n
ew
in
p
u
t is ess
en
tially
ig
n
o
r
ed
,
r
e
s
u
ltin
g
in
ex
tr
em
ely
s
tr
o
n
g
f
ilter
in
g
.
T
h
e
f
ilter
eq
u
ati
o
n
ca
n
b
e
r
ea
r
r
an
g
ed
i
n
to
th
e
f
o
llo
win
g
e
q
u
i
v
alen
t p
r
ed
icto
r
-
co
r
r
ec
to
r
r
elatio
n
s
h
ip
:
(
)
=
(
−
1
)
+
(
1
−
)
∗
(
(
)
−
(
−
1
)
)
(
3
)
T
h
er
ef
o
r
e,
t
h
e
f
ilter
o
u
tp
u
t
is
an
ticip
ated
to
r
e
m
ain
c
o
n
s
tan
t
f
r
o
m
t
h
e
p
r
io
r
esti
m
ate
(
−
1
)
p
lu
s
a
co
r
r
ec
tiv
e
ter
m
b
ased
o
n
u
n
e
x
p
ec
ted
c
o
n
tr
ib
u
tio
n
,
g
iv
en
b
y
th
e
d
if
f
er
en
ce
b
etwe
en
th
e
n
ew
in
p
u
t
(
)
an
d
f
o
r
ec
ast
(
−
1
)
;
th
e
(
3
)
m
ak
es
t
h
is
p
r
ed
ictio
n
clea
r
er
.
T
h
is
f
o
r
m
m
ay
also
b
e
o
b
tain
e
d
b
y
c
o
n
s
id
er
in
g
th
e
ex
p
o
n
e
n
tial
f
ilter
as
a
s
tr
aig
h
tf
o
r
war
d
s
p
ec
ial
ca
s
e
o
f
a
Kalm
an
f
ilter
,
wh
ich
is
th
e
b
est
s
o
lu
tio
n
f
o
r
a
n
esti
m
atio
n
p
r
o
b
lem
u
n
d
er
s
p
ec
if
ic
p
r
esu
m
p
tio
n
s
.
Fig
u
r
e
2
s
h
o
ws
th
e
u
n
it
-
s
tep
r
esp
o
n
s
e
o
f
an
ex
p
o
n
e
n
tial
f
ilter
,
o
b
tain
ed
b
y
ab
r
u
p
tly
c
h
an
g
in
g
t
h
e
in
p
u
t v
alu
e
to
1
f
r
o
m
ze
r
o
in
itial v
alu
e
.
Fig
u
r
e
2
.
Step
r
esp
o
n
s
e
o
f
th
e
ex
p
o
n
e
n
tial f
ilter
: in
p
u
t
(
b
lu
e
t
r
ac
e)
an
d
o
u
tp
u
t (
p
u
r
p
le
tr
ac
e
)
p
lo
ts
as a
f
u
n
ctio
n
o
f
t/
T
h
e
ex
p
o
n
en
tial
f
ilter
's
s
tep
r
esp
o
n
s
e
en
ab
les
m
o
r
e
r
ea
d
ily
p
r
ed
ictin
g
th
e
o
u
tc
o
m
es
f
o
r
an
y
tim
e
in
ter
v
al
an
d
v
al
u
e
o
f
th
e
f
ilter
tim
e
co
n
s
tan
t
(
)
b
y
d
iv
id
in
g
th
e
tim
e
b
y
.
T
h
e
f
ilter
o
u
t
p
u
t
cl
im
b
s
to
6
3
.
2
1
%
o
f
its
f
in
al
v
alu
e
af
ter
a
s
in
g
le
an
d
in
cr
ea
s
es
to
8
6
.
4
7
%
af
ter
two
-
tim
e
co
n
s
tan
ts
;
af
ter
3
,
4
,
an
d
5
τ
th
e
o
u
tp
u
t
r
ea
ch
es
9
5
.
0
2
%,
9
8
.
1
7
%,
an
d
9
9
.
3
3
%
o
f
th
e
f
in
al
v
alu
e,
r
esp
ec
tiv
ely
.
T
h
e
s
e
p
er
ce
n
tag
es
a
r
e
u
n
ch
an
g
ed
f
o
r
a
n
y
i
n
p
u
t
s
te
p
's
am
p
litu
d
e,
b
ec
au
s
e
th
e
f
i
lter
is
lin
ea
r
.
T
h
e
f
ilter
o
u
tp
u
t
tr
an
s
ien
t
ca
n
b
e
co
n
s
id
er
ed
ex
h
au
s
ted
af
ter
a
t
im
e
eq
u
al
to
4
o
r
5
tim
e
co
n
s
tan
ts
,
ev
en
if
th
e
s
tep
r
esp
o
n
s
e,
in
th
eo
r
y
,
tak
es
u
n
lim
ited
tim
e.
Oth
er
s
ettin
g
s
in
clu
d
e
=
0
.
9
0
a
n
d
=
0
.
9
9
8
,
r
e
s
p
ec
tiv
ely
,
co
r
r
esp
o
n
d
in
g
to
=
9
.
4
9
an
d
4
9
9
.
5
m
in
u
tes
f
o
r
a
s
am
p
lin
g
p
er
io
d
T
o
f
o
n
e
m
in
u
te.
On
e
wa
y
to
ar
r
an
g
e
th
e
ex
p
o
n
e
n
tial
f
ilter
s
is
in
s
er
ies,
r
esu
ltin
g
in
g
r
ea
ter
atten
u
atio
n
o
f
h
ig
h
-
f
r
e
q
u
en
c
y
n
o
is
e
b
u
t
also
ca
u
s
in
g
a
g
r
ea
ter
o
u
tp
u
t
d
elay
,
o
f
te
n
ex
ce
s
s
iv
e
f
o
r
c
o
n
tr
o
l
lo
o
p
s
o
r
d
iag
n
o
s
tic
ap
p
licatio
n
s
.
A
n
o
n
-
lin
ea
r
ex
p
o
n
e
n
tial
f
ilter
is
an
ex
p
o
n
e
n
tial
f
ilter
's
v
ar
ian
t
th
at
r
esp
o
n
d
s
m
o
r
e
q
u
ick
ly
to
g
r
ea
ter
in
p
u
t
ch
an
g
es
an
d
is
d
esig
n
ed
to
f
ilter
o
u
t
n
o
is
e
s
u
b
s
tan
tially
with
in
a
s
p
ec
if
ic
a
m
p
litu
d
e
r
a
n
g
e
[
2
8
]
.
I
n
ess
en
ce
,
th
e
ex
p
o
n
en
tial
f
ilter
s
o
p
er
ate
b
y
ass
u
m
in
g
th
at
th
e
s
ig
n
al
is
a
r
an
d
o
m
walk
o
r
B
r
o
wn
ian
m
o
tio
n
p
atter
n
,
with
r
a
n
d
o
m
p
r
o
ce
s
s
n
o
is
e
as
th
e
o
n
ly
v
ar
i
atio
n
s
o
u
r
ce
.
T
h
en
,
b
ef
o
r
e
d
is
p
lay
in
g
m
o
r
e
r
ec
e
n
t
d
ata,
th
e
p
r
e
v
io
u
s
v
alu
e
r
ep
r
esen
ts
th
e
b
est
esti
m
ate
o
f
th
e
s
u
b
s
eq
u
en
t
v
alu
e
.
All th
at
r
em
ain
s
o
f
th
e
f
in
al
esti
m
ate
is
a
weig
h
ted
av
er
ag
e
o
f
th
e
n
ew
o
b
s
er
v
ed
v
alu
e
an
d
t
h
e
ex
p
ec
ted
o
n
e.
C
r
ea
tin
g
a
r
ep
r
esen
tativ
e
d
ataset
o
f
th
e
s
y
s
tem
is
h
elp
f
u
l
f
o
r
s
o
m
e
ap
p
licatio
n
s
,
s
u
ch
as
s
o
p
h
is
ticated
co
n
tr
o
l
s
y
s
tem
s
as
well
as
f
au
lt
d
etec
tio
n
an
d
is
o
latio
n
alg
o
r
ith
m
s
,
b
ec
au
s
e
it
s
er
v
es
as
th
e
b
asis
f
o
r
u
n
d
er
s
tan
d
in
g
th
e
s
y
s
tem
’
s
b
eh
av
io
r
,
id
en
tify
in
g
a
n
o
m
alies,
an
d
d
esig
n
in
g
ef
f
ec
tiv
e
co
n
tr
o
l
s
tr
ateg
ies.
T
h
e
d
ataset
ca
p
tu
r
es
th
e
r
elatio
n
s
h
ip
s
b
etwe
en
th
e
s
y
s
tem
's
in
p
u
ts
,
o
u
tp
u
ts
,
an
d
in
ter
n
al
s
tates
u
n
d
er
n
o
r
m
al
an
d
ab
n
o
r
m
al
co
n
d
itio
n
s
,
m
a
k
in
g
it
an
in
v
alu
ab
le
r
eso
u
r
ce
f
o
r
an
aly
s
is
an
d
d
ec
is
io
n
-
m
a
k
in
g
.
So
m
e
co
n
t
r
o
l
tech
n
iq
u
es
in
ter
ac
t
d
ir
ec
tly
wi
th
th
at
d
ataset
with
o
u
t
th
e
n
ee
d
f
o
r
an
ex
p
licit
m
o
d
el
o
f
th
e
s
y
s
tem
,
in
clu
d
in
g
th
e
m
o
d
el
-
f
r
ee
b
in
a
r
y
d
ec
is
io
n
an
d
ac
tio
n
co
n
tr
o
l
(
B
DAC)
ap
p
r
o
ac
h
an
d
s
o
m
e
f
a
u
lt
is
o
latio
n
an
d
d
etec
tio
n
s
tr
ateg
ies
.
T
h
ese
m
o
d
el
-
f
r
ee
m
eth
o
d
s
ar
e
p
ar
ticu
lar
l
y
u
s
ef
u
l
in
s
y
s
tem
s
wh
er
e
d
ev
el
o
p
in
g
an
ac
cu
r
ate
m
ath
em
atica
l
m
o
d
el
is
ch
allen
g
in
g
d
u
e
to
co
m
p
lex
ity
,
n
o
n
l
in
ea
r
ity
,
o
r
u
n
ce
r
tai
n
ty
.
I
n
o
th
er
ca
s
es,
th
e
d
ataset
s
er
v
es
as
a
tr
ain
in
g
s
et
f
o
r
cr
e
atin
g
m
o
d
els
th
at
ar
e
s
u
b
s
eq
u
en
tly
u
s
ed
f
o
r
co
n
tr
o
l
o
r
d
iag
n
o
s
is
.
T
h
is
ap
p
r
o
ac
h
is
p
ar
ticu
lar
ly
r
elev
a
n
t
in
m
o
d
el
-
b
ased
ap
p
r
o
ac
h
es,
wh
e
r
e
a
m
ath
em
atica
l
o
r
d
ata
-
d
r
iv
e
n
m
o
d
el
o
f
th
e
s
y
s
tem
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
1
6
,
No
.
1
,
Feb
r
u
ar
y
20
2
6
:
2
3
0
-
245
234
is
co
n
s
tr
u
cted
a
n
d
u
tili
ze
d
f
o
r
d
ec
is
io
n
-
m
a
k
in
g
.
Neu
r
al
n
e
two
r
k
(
NN)
m
o
d
els,
m
o
d
el
-
b
ased
co
n
tr
o
l
(
lik
e
m
o
d
el
p
r
ed
ictiv
e
co
n
tr
o
l
(
MPC
)
)
,
an
d
tr
a
d
itio
n
al
r
e
g
r
ess
io
n
m
o
d
els
f
all
in
to
th
e
ca
teg
o
r
y
o
f
th
e
m
o
d
el
-
b
ased
ap
p
r
o
ac
h
.
A
co
m
p
lete
m
u
lti
-
v
ar
iab
le
m
eth
o
d
k
n
o
wn
as
r
ea
l
tim
e
ex
p
o
n
en
tial
f
ilter
clu
s
ter
in
g
(
R
T
E
FC
)
h
as
b
ee
n
cr
ea
ted
t
h
at
co
m
b
in
es e
x
p
o
n
en
tial f
ilter
in
g
with
r
ea
l
-
ti
m
e
clu
s
ter
in
g
[
2
9
]
.
2
.
2
.
Str
et
ched
-
co
m
press
ed
ex
po
nentia
l f
un
ct
io
n a
nd
dis
t
ributio
n
T
h
e
s
tr
etch
ed
ex
p
o
n
e
n
tial f
u
n
ctio
n
ca
n
b
e
ex
p
r
ess
ed
as
(
4
)
:
(
)
=
0
−
(
)
(
4
)
I
t
is
ac
h
iev
e
d
b
y
f
itti
n
g
th
e
e
x
p
o
n
e
n
tial
f
u
n
ctio
n
with
a
f
r
ac
tio
n
al
p
o
wer
law
,
wh
ich
is
o
n
ly
s
ig
n
if
ican
t
f
o
r
ar
g
u
m
en
t
t
b
etwe
en
0
an
d
+∞
in
m
o
s
t
ap
p
licatio
n
s
[
3
0
]
.
T
h
e
s
tan
d
ar
d
ex
p
o
n
e
n
tial
f
u
n
ctio
n
is
o
b
tain
ed
wh
en
=
1
.
T
h
e
f
u
n
ctio
n
g
ets
its
n
am
e
f
r
o
m
th
e
ch
a
r
ac
ter
is
tic
s
tr
etch
in
g
o
f
th
e
lo
g
ag
ain
s
t
g
r
ap
h
,
wh
ich
h
as
a
s
tr
etch
in
g
ex
p
o
n
e
n
t
b
etwe
e
n
0
an
d
1
.
An
o
t
h
er
p
r
ac
tica
l
s
ig
n
if
ican
ce
is
attac
h
ed
t
o
th
e
c
o
m
p
r
ess
ed
ex
p
o
n
e
n
tial
f
u
n
ctio
n
(
with
>
1
)
,
with
th
e
n
o
tab
le
ex
ce
p
tio
n
o
f
=
2
,
wh
ich
y
ield
s
to
th
e
n
o
r
m
al
d
is
tr
ib
u
tio
n
.
Fig
u
r
e
3
p
l
o
ts
th
e
(
)
f
u
n
ctio
n
r
e
p
o
r
ted
in
(
4
)
as a
f
u
n
ctio
n
o
f
t
h
e
/
r
atio
,
v
ar
y
in
g
th
e
p
ar
am
et
er
.
Fig
u
r
e
3
.
Plo
t o
f
I
(
t)
/
0
v
s
.
(
t⁄
τ
)
,
s
h
o
win
g
th
e
s
tr
etch
ed
e
x
p
o
n
e
n
tial f
u
n
ctio
n
f
o
r
s
ev
e
r
al
v
alu
es
I
n
m
ath
em
atics,
th
e
co
m
p
le
m
en
tar
y
cu
m
u
lativ
e
W
eib
u
ll
d
is
tr
ib
u
tio
n
is
an
o
th
er
n
a
m
e
f
o
r
th
e
s
tr
etch
ed
ex
p
o
n
en
tial
f
u
n
cti
o
n
[
3
1
]
.
T
h
e
ch
ar
ac
ter
is
tic
f
u
n
ctio
n
o
f
t
h
e
L
é
v
y
s
y
m
m
etr
ic
alp
h
a
-
s
tab
le
d
is
tr
ib
u
tio
n
is
also
th
e
s
tr
etc
h
ed
e
x
p
o
n
en
tial
o
n
e,
o
r
in
s
im
p
ler
ter
m
s
,
th
e
Fo
u
r
ier
tr
a
n
s
f
o
r
m
[
3
0
]
.
T
h
e
s
tr
etch
ed
ex
p
o
n
en
tial
f
u
n
ctio
n
is
a
p
h
en
o
m
en
o
lo
g
ical
ex
p
lan
atio
n
o
f
r
elax
atio
n
in
d
is
o
r
d
er
ed
s
y
s
tem
s
f
r
eq
u
e
n
tly
u
tili
ze
d
in
p
h
y
s
ics.
T
h
e
Ko
h
lr
au
s
ch
f
u
n
ctio
n
was
p
r
o
p
o
s
ed
f
r
o
m
R
.
Ko
h
lr
au
s
ch
to
ex
p
lain
h
o
w
a
ca
p
ac
ito
r
d
is
ch
ar
g
es
[
3
2
]
.
T
h
e
Fo
u
r
ier
tr
an
s
f
o
r
m
o
f
th
e
s
tr
etch
ed
ex
p
o
n
en
tial
f
u
n
ctio
n
is
also
k
n
o
wn
as
th
e
Ko
h
lr
au
s
ch
–
W
illi
am
s
–
W
atts
(
KW
W
)
f
u
n
ctio
n
;
it
wa
s
f
ir
s
t
ap
p
lied
in
1
9
7
0
b
y
W
illi
am
s
an
d
W
atts
to
ch
ar
ac
ter
ize
th
e
d
ielec
tr
ic
s
p
ec
tr
a
o
f
p
o
ly
m
e
r
s
[
3
3
]
.
Fo
r
s
m
all
-
tim
e
ar
g
u
m
en
ts
,
th
e
C
o
le
-
C
o
le
an
d
C
o
le
-
Dav
id
s
o
n
eq
u
atio
n
s
,
a
n
d
t
h
e
Hav
r
iliak
-
Neg
am
i
r
elax
atio
n
ar
e
e
x
am
p
les
o
f
th
e
p
r
im
a
r
y
d
ielec
tr
ic
m
o
d
els
wh
o
s
e
tim
e
-
d
o
m
ain
c
h
ar
g
e
r
esp
o
n
s
e
is
co
r
r
elate
d
with
t
h
e
KW
W
f
u
n
ctio
n
[
3
4
]
.
I
n
p
h
e
n
o
m
en
o
lo
g
ical
ap
p
licatio
n
s
,
it
is
o
f
ten
u
n
cl
ea
r
wh
eth
er
th
e
s
tr
etch
ed
e
x
p
o
n
en
tial
f
u
n
ctio
n
s
h
o
u
ld
d
e
s
cr
ib
e
th
e
in
te
g
r
al
d
is
tr
ib
u
tio
n
f
u
n
ctio
n
,
th
e
d
if
f
er
en
tial
o
n
e,
o
r
n
eith
er
.
T
h
e
a
s
y
m
p
to
tic
d
ec
ay
is
th
e
s
am
e
i
n
all
ca
s
es,
b
u
t
th
e
p
o
wer
law
p
r
e
-
f
ac
t
o
r
v
ar
ies,
l
ea
d
in
g
to
a
m
o
r
e
am
b
ig
u
o
u
s
f
i
t
th
an
s
im
p
le
ex
p
o
n
e
n
tials
.
T
h
e
asy
m
p
to
tic
d
ec
ay
h
as
b
ee
n
s
h
o
wn
to
b
e
a
s
tr
et
ch
ed
ex
p
o
n
en
tial
[
3
0
]
,
[
3
1
]
,
alth
o
u
g
h
th
e
p
r
e
-
f
ac
to
r
is
ty
p
ically
an
u
n
r
elate
d
p
o
wer
.
T
h
er
e
h
av
e
b
ee
n
attem
p
ts
to
e
x
p
lain
th
e
s
tr
etch
ed
ex
p
o
n
e
n
tial
b
eh
av
io
r
u
s
in
g
a
lin
ea
r
s
u
p
er
p
o
s
itio
n
o
f
s
im
p
le
e
x
p
o
n
en
tial
d
ec
ay
,
as
s
ee
n
in
th
e
d
is
tr
ib
u
tio
n
f
u
n
ctio
n
f
o
r
th
e
s
tr
etch
ed
ex
p
o
n
en
tial
f
u
n
ctio
n
.
T
h
er
ef
o
r
e,
a
n
o
n
tr
iv
ial
r
ela
x
atio
n
tim
e
d
is
tr
ib
u
tio
n
,
(
;
)
,
is
r
eq
u
ir
ed
,
wh
ich
is
im
p
licitly
d
ef
i
n
ed
b
y
(
5
)
:
−
=
∫
−
∞
0
(
;
)
(
5
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
A
n
o
ve
l str
etch
ed
-
co
mp
r
ess
ed
ex
p
o
n
e
n
tia
l lo
w
-
p
a
s
s
filt
er a
n
d
its
a
p
p
lica
tio
n
…
(
R
o
b
erto
d
e
F
a
z
io
)
235
Alter
n
ativ
ely
,
a
d
is
tr
ib
u
tio
n
r
e
lated
to
th
e
p
ar
a
m
eter
is
g
iv
en
:
(
;
)
=
(
;
)
(
6
)
wh
er
e
(
;
)
ca
n
b
e
ex
p
r
ess
ed
as
(
7
)
:
(
;
)
=
−
1
∑
(
−
1
)
!
∞
=
0
s
in
(
)
Γ
(
+
1
)
(
)
(
7
)
wh
er
e
Γ
is
th
e
Gam
m
a
f
u
n
ctio
n
.
Fo
r
r
atio
n
al
v
alu
es
o
f
,
(
;
)
ca
n
b
e
ca
lcu
lated
in
te
r
m
s
o
f
ele
m
en
tar
y
f
u
n
ctio
n
s
.
B
u
t
t
h
e
ex
p
r
ess
io
n
is
,
in
g
e
n
er
al,
t
o
o
c
o
m
p
lex
t
o
b
e
u
s
ef
u
l
e
x
ce
p
t
f
o
r
th
e
ca
s
e
β
=
1
/2
,
wh
er
e
th
e
d
is
tr
ib
u
tio
n
will b
e
as
(
8
)
:
(
;
1
2
⁄
)
=
(
;
1
2
⁄
)
=
1
2
√
√
−
4
(
8
)
Fig
u
r
e
4
s
h
o
ws th
e
f
u
n
ctio
n
in
(
8
)
f
o
r
d
if
f
e
r
en
t v
alu
es
o
f
th
e
.
Fig
u
r
e
4
.
Plo
t o
f
th
e
s
tr
etch
ed
ex
p
o
n
e
n
tial d
is
tr
ib
u
tio
n
f
u
n
cti
o
n
(
;
)
v
s
.
(
x
)
f
o
r
d
if
f
er
en
t
v
alu
es
T
h
e
Fo
u
r
ier
tr
an
s
f
o
r
m
o
f
th
e
s
tr
etch
ed
ex
p
o
n
e
n
tial
f
u
n
cti
o
n
h
as
to
b
e
co
m
p
u
ted
u
s
in
g
a
s
er
ies
ex
p
an
s
io
n
o
r
n
u
m
er
ical
in
teg
r
atio
n
;
th
e
Hav
r
iliak
–
Neg
a
m
i
f
u
n
ctio
n
ca
n
b
e
u
s
ed
to
ap
p
r
o
x
im
ate
th
e
Fo
u
r
ier
tr
an
s
f
o
r
m
[
1
3
]
.
Ho
wev
er
,
m
o
d
er
n
n
u
m
er
ical
c
o
m
p
u
tatio
n
i
s
s
o
ef
f
icien
t
th
at
th
e
Ko
h
l
r
au
s
ch
–
W
illi
am
s
–
W
att
s
f
u
n
ctio
n
s
h
o
u
ld
alwa
y
s
b
e
u
s
ed
in
th
e
f
r
eq
u
e
n
cy
d
o
m
ain
[
3
5
]
.
I
n
m
o
s
t
ca
s
es,
th
e
f
ilter
aim
s
to
s
ep
ar
ate
th
e
ac
tu
al
s
ig
n
al
f
r
o
m
th
e
n
o
is
y
m
ea
s
u
r
ed
s
ig
n
al:
(
)
=
(
)
+
(
)
(
9
)
wh
er
e
(
)
is
th
e
o
b
s
er
v
ed
(
m
ea
s
u
r
ed
)
s
ig
n
al
at
th
e
tim
e
,
(
)
th
e
tr
u
e,
d
eter
m
in
is
tic
p
ar
t
o
f
th
e
s
ig
n
al,
an
d
(
)
a
s
tatio
n
ar
y
n
o
is
e,
a
s
to
ch
asti
c
(
r
an
d
o
m
)
p
ar
t in
t
h
e
s
ig
n
al,
wh
ich
is
ass
u
m
ed
with
ze
r
o
m
ea
n
v
alu
e
.
An
ex
p
o
n
e
n
tial
lo
w
-
p
ass
f
ilte
r
is
f
ea
tu
r
ed
b
y
an
im
p
u
ls
e
r
e
s
p
o
n
s
e
eq
u
al
to
th
e
ex
p
o
n
e
n
tial
f
u
n
ctio
n
(
;
)
;
th
u
s
,
th
e
o
u
tp
u
t
o
f
th
e
e
x
p
o
n
en
tial
f
ilter
(
(
)
)
is
d
ef
in
ed
a
s
th
e
co
n
v
o
lu
tio
n
o
f
t
h
e
m
ea
s
u
r
ed
(
o
b
s
er
v
ed
)
s
ig
n
al
(
)
an
d
th
e
im
p
u
ls
e
r
esp
o
n
s
e
(
;
)
:
(
)
=
(
)
∗
(
;
)
=
∫
(
−
)
∞
−
∞
(
;
)
(
1
0
)
2
.
3
.
P
r
o
po
s
ed
co
m
press
ed
ex
po
nentia
l f
ilte
r
a
nd
perf
o
rm
a
nce
m
e
t
rics
T
h
e
n
o
v
el
e
x
p
o
n
en
tial
f
ilter
,
h
er
ein
af
ter
in
d
icate
d
as
th
e
s
t
r
etch
ed
e
x
p
o
n
en
tial
f
ilter
,
is
d
ef
in
ed
b
y
ex
ten
d
in
g
t
h
e
ex
p
o
n
en
tial
f
u
n
ctio
n
to
th
e
s
tr
etch
ed
e
x
p
o
n
e
n
tial
f
u
n
ctio
n
with
a
s
in
g
le
p
a
r
am
eter
.
T
h
er
ef
o
r
e
,
th
e
o
u
tp
u
t o
f
th
e
s
tr
etch
ed
ex
p
o
n
en
tial f
ilter
(
)
is
g
iv
en
b
y
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
1
6
,
No
.
1
,
Feb
r
u
ar
y
20
2
6
:
2
3
0
-
245
236
(
)
=
(
)
∗
(
;
)
=
∫
(
−
)
∞
−
∞
(
;
)
(
1
1
)
wh
er
e:
(
;
)
=
1
√
2
−
(
1
2
(
−
)
2
)
(
1
2
)
T
h
er
ef
o
r
e,
th
e
f
ilter
o
u
tp
u
t is ex
p
r
ess
ed
as
(
1
3
)
:
(
)
=
(
)
∗
(
;
)
=
1
√
2
∫
(
−
)
∞
−
∞
−
(
1
2
(
−
)
2
)
(
1
3
)
W
h
en
=
1
,
a
class
ical
Gau
s
s
ian
f
ilter
is
o
b
tain
ed
,
ad
d
itio
n
ally
,
v
ar
io
u
s
f
ilter
t
y
p
o
lo
g
ies
ar
e
ac
h
iev
ed
b
y
tu
n
in
g
th
e
h
id
d
en
f
ac
to
r
,
all
o
win
g
to
m
o
d
if
y
th
e
c
u
r
v
e
o
f
th
e
p
r
o
b
a
b
ilit
y
-
d
en
s
ity
f
u
n
c
tio
n
.
C
o
n
v
er
s
ely
,
d
if
f
er
en
t
d
is
tr
ib
u
tio
n
s
h
ap
es
n
ec
ess
itate
th
e
u
s
ag
e
o
f
d
is
tin
ct
d
is
tr
ib
u
tio
n
f
u
n
ctio
n
s
,
s
u
ch
as
th
e
n
o
r
m
al
d
is
tr
ib
u
tio
n
,
th
e
C
au
ch
y
o
n
e
,
an
d
o
th
er
s
o
f
th
is
f
am
ily
.
C
o
m
p
ar
ed
to
a
class
ical
o
n
e
-
p
ar
am
eter
f
ilter
,
th
e
p
r
o
p
o
s
ed
o
n
e
is
m
o
r
e
v
e
r
s
atile
g
iv
en
th
e
ex
tr
a
tu
n
in
g
p
ar
am
e
ter
s
,
en
ab
lin
g
m
o
d
if
y
in
g
t
h
e
d
is
tr
ib
u
tio
n
cu
r
v
e
an
d
th
e
e
x
p
o
n
en
tial stre
tch
an
d
,
th
u
s
,
p
r
o
v
id
i
n
g
m
o
r
e
d
e
g
r
ee
s
o
f
f
r
ee
d
o
m
.
T
h
e
s
ig
n
al
-
to
-
n
o
is
e
r
atio
(
SN
R
)
an
d
m
ea
n
s
q
u
ar
e
er
r
o
r
(
M
SE)
ar
e
u
s
u
ally
u
s
ed
in
th
e
li
ter
atu
r
e
to
ev
alu
ate
th
e
f
ilter
's
ab
ilit
y
to
r
ed
u
ce
th
e
n
o
is
e
[
3
6
]
.
T
h
is
ar
ticle
u
s
es
th
e
f
o
llo
win
g
d
ef
i
n
ed
ev
alu
atio
n
m
ar
k
e
r
s
to
co
m
p
a
r
e
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
with
th
e
e
x
is
tin
g
m
et
h
o
d
s
an
d
ass
ess
its
ef
f
ec
tiv
en
ess
in
r
ed
u
cin
g
th
e
n
o
is
e.
T
h
e
d
ef
in
itio
n
o
f
t
h
e
o
u
tp
u
t sig
n
al
-
to
-
n
o
is
e
r
atio
is
as
(
1
4
)
:
=
10
∗
10
(
∑
[
(
)
]
2
=
=
1
∑
[
(
)
−
(
)
̂
]
2
=
=
1
)
(
1
4
)
wh
er
e
(
)
ar
e
t
h
e
o
r
ig
in
al
(
id
ea
l
)
s
ig
n
al's
s
am
p
les,
wh
ile
̂
(
)
ar
e
th
e
s
am
p
les
o
f
th
e
f
ilter
ed
E
C
G
s
ig
n
al
[
3
7
]
.
T
h
e
m
ea
n
s
q
u
a
r
e
er
r
o
r
is
d
ef
in
ed
as in
[
1
5
]
; b
o
th
,
th
e
m
ea
n
SR
N
an
d
MSE
v
alu
es c
alcu
lated
o
n
all
E
C
G
s
ig
n
als in
th
e
d
ataset
wer
e
co
n
s
id
er
ed
f
o
r
test
in
g
t
h
e
ef
f
ec
tiv
en
ess
o
f
th
e
p
r
o
p
o
s
ed
SC
E
L
P
f
ilter
.
=
1
∑
[
(
)
−
̂
(
)
]
2
=
=
1
(
1
5
)
A
tech
n
iq
u
e
to
im
p
lem
en
t
th
e
Gau
s
s
ian
f
ilter
in
th
e
d
is
cr
ete
-
tim
e
d
o
m
ain
is
to
f
o
llo
w
th
e
in
s
tr
u
ctio
n
s
in
[
3
8
]
,
[
3
9
]
.
Sin
ce
th
e
Gau
s
s
ian
f
ilter
is
n
o
t
ca
u
s
ativ
e,
th
e
tim
e
-
d
o
m
ai
n
f
ilter
win
d
o
w
is
s
y
m
m
etr
ic.
B
ec
au
s
e
th
e
Gau
s
s
ian
f
u
n
ctio
n
f
o
r
x
∈
(
-
∞
,
∞
)
wo
u
ld
th
e
o
r
etica
lly
r
e
q
u
ir
e
an
in
f
in
ite
win
d
o
w
len
g
th
,
th
e
Gau
s
s
ian
f
ilter
is
p
h
y
s
ically
u
n
f
ea
s
ib
le.
I
n
p
r
ac
tice,
it
m
ak
es
s
en
s
e
to
r
ed
u
ce
th
e
f
ilter
win
d
o
w's
len
g
th
an
d
ap
p
l
y
it
s
tr
aig
h
t
to
n
ar
r
o
w
win
d
o
ws;
h
o
wev
er
,
o
cc
asio
n
ally
,
th
is
s
h
o
r
ten
in
g
ca
n
r
esu
lt
in
s
er
io
u
s
m
is
tak
es.
T
h
e
f
ilter
ca
n
n
o
t
b
e
ap
p
lied
to
th
e
s
ig
n
al
b
ein
g
p
r
o
ce
s
s
ed
u
n
til
th
e
i
n
co
m
in
g
s
am
p
les
o
cc
u
p
y
th
e
f
ilter
win
d
o
w
,
r
esu
ltin
g
i
n
a
laten
cy
in
r
ea
l
-
tim
e
s
y
s
tem
s
.
I
n
co
n
v
o
lu
tio
n
,
th
e
Gau
s
s
ian
f
ilter
k
er
n
el
is
co
n
tin
u
o
u
s
,
b
u
t
it
is
co
m
m
o
n
ly
ap
p
r
o
x
im
ated
b
y
a
d
is
cr
ete
s
am
p
led
Gau
s
s
ian
k
er
n
el
cr
ea
ted
b
y
s
am
p
lin
g
p
o
in
ts
f
r
o
m
th
e
co
n
tin
u
o
u
s
k
e
r
n
el.
T
h
is
d
is
cr
ete
v
er
s
io
n
is
t
h
e
m
o
s
t
wid
ely
u
s
ed
s
u
b
s
titu
te
f
o
r
th
e
co
n
tin
u
o
u
s
Gau
s
s
ian
k
er
n
el.
T
h
e
s
u
m
m
in
g
p
r
o
ce
s
s
ac
r
o
s
s
all
s
am
p
les
ca
n
b
e
u
s
ed
in
p
lace
o
f
an
i
n
teg
r
atio
n
o
p
er
atio
n
in
co
n
v
o
l
u
tio
n
[
4
0
]
.
I
t
is
also
co
m
m
o
n
ly
r
ec
o
g
n
ize
d
th
at
tr
ad
itio
n
al
m
o
v
in
g
av
er
ag
e
f
ilter
s
,
o
r
weig
h
ted
m
o
v
i
n
g
av
er
ag
e
f
ilter
s
with
s
tr
etch
ed
ex
p
o
n
e
n
tial,
ar
e
n
o
t
alwa
y
s
ap
p
r
o
p
r
iate
f
o
r
allo
ca
tin
g
weig
h
ts
to
p
r
ec
ed
in
g
f
ilter
ed
s
ig
n
al
s
am
p
les
[
4
1
]
.
A
co
m
m
o
n
r
e
q
u
ir
em
e
n
t
is
th
at
s
am
p
les
with
a
h
ig
h
p
r
o
p
o
r
tio
n
o
f
s
to
ch
asti
c
(
n
o
is
y
)
co
m
p
o
n
en
ts
s
h
o
u
ld
b
e
ass
ig
n
ed
lo
wer
weig
h
ts
r
ath
er
th
an
s
im
p
ly
ass
ig
n
in
g
lo
wer
weig
h
ts
to
o
ld
er
s
am
p
les.
B
y
p
r
io
r
itizin
g
m
o
r
e
r
ec
e
n
t
s
a
m
p
les,
s
u
ch
f
ilter
s
ca
n
r
esp
o
n
d
f
aster
to
ch
a
n
g
es
in
d
eter
m
i
n
is
tic
o
r
s
to
ch
asti
c
co
m
p
o
n
en
ts
.
T
o
b
e
u
s
ed
ev
e
n
in
d
ig
ital
co
n
tr
o
ller
s
with
l
im
ited
co
m
p
u
tin
g
p
o
wer
,
th
e
f
ilter
in
g
alg
o
r
ith
m
m
u
s
t
m
ee
t
two
r
eq
u
ir
em
en
ts
:
r
ea
s
o
n
ab
ly
s
im
p
le
to
im
p
l
em
en
t
an
d
ef
f
ec
tiv
e
ev
e
n
w
h
en
th
e
m
ea
s
u
r
ed
wav
ef
o
r
m
s
c
o
n
tain
s
ig
n
if
ican
t
s
to
ch
asti
c
n
o
is
e.
Similar
c
h
al
len
g
es
to
t
h
o
s
e
d
is
cu
s
s
ed
ea
r
li
er
m
ay
ar
is
e
wh
e
n
im
p
lem
en
tin
g
th
e
s
tr
etch
ed
e
x
p
o
n
en
tial
f
ilter
.
Sp
ec
if
ically
,
wh
ile
f
lex
ib
le
in
its
weig
h
tin
g
o
f
p
ast
s
am
p
les,
th
e
s
tr
etch
ed
ex
p
o
n
en
tial
f
u
n
ctio
n
ca
n
b
e
co
m
p
u
tatio
n
ally
d
em
a
n
d
in
g
a
n
d
s
en
s
itiv
e
to
p
ar
am
et
er
ch
o
ices.
I
n
r
ea
l
-
tim
e
ap
p
licatio
n
s
,
ev
alu
atin
g
th
e
f
u
n
ctio
n
o
v
er
ex
ten
d
e
d
tim
e
r
an
g
es
ca
n
p
o
s
e
p
r
ac
tical
d
if
f
icu
lties
,
p
ar
ticu
lar
ly
in
s
y
s
tem
s
with
lim
ited
co
m
p
u
tatio
n
al
r
eso
u
r
ce
s
.
Ad
d
itio
n
ally
,
th
e
ab
s
en
ce
o
f
a
n
atu
r
al
cu
to
f
f
in
th
e
s
tr
etch
ed
ex
p
o
n
en
tial
f
u
n
ctio
n
n
ec
ess
itates
ca
r
ef
u
l
tr
u
n
ca
tio
n
to
b
alan
ce
ac
c
u
r
ac
y
an
d
co
m
p
u
tatio
n
al
ef
f
icien
cy
,
f
u
r
th
er
co
m
p
licati
n
g
its
u
s
e
in
r
ea
l
-
tim
e
s
ce
n
ar
io
s
.
A
MA
T
L
AB
f
u
n
ctio
n
f
o
r
th
e
s
tr
etch
ed
ex
p
o
n
e
n
tial f
u
n
ctio
n
is
p
r
o
p
o
s
ed
in
th
is
r
esear
ch
wo
r
k
; its
h
e
ad
er
is
g
iv
en
b
elo
w:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2088
-
8
7
0
8
A
n
o
ve
l str
etch
ed
-
co
mp
r
ess
ed
ex
p
o
n
e
n
tia
l lo
w
-
p
a
s
s
filt
er a
n
d
its
a
p
p
lica
tio
n
…
(
R
o
b
erto
d
e
F
a
z
io
)
237
function [y]=SEF (beta, x)
% SEF (beta, x) is the stretched exponential function
% for each element of x and beta are scalars,
% array. The output is of the same size as x.
T
h
e
f
ilter
o
u
tp
u
t
ex
p
r
ess
io
n
r
ep
o
r
ted
in
th
e
(
1
3
)
ca
n
b
e
im
p
lem
en
ted
b
y
a
MA
T
L
AB
f
u
n
ctio
n
[
y
]
=SE
F (
b
eta,
x
)
.
T
h
e
MA
T
L
AB
f
u
n
ctio
n
o
f
d
ev
elo
p
ed
s
tr
e
tch
ed
ex
p
o
n
en
tial f
ilter
h
as th
e
f
o
llo
win
g
h
ea
d
er
:
function [y_filt]
=
SE_filter (t, y, beta, sigma)
% function [y_filt]
=
SE_filter (t, y, beta, sigma)
% Stretched Exponential filter
% Inputs: t
=
independent variable
% y
=
noisy data to be filtered at the points t
% beta
=
parameters of the Stretched Exponential function
% Output: y_filt
=
filtered data given in variable
y
2
.
4
.
E
CG
s
ig
na
ls
’
da
t
a
s
et
T
h
e
E
C
G
s
ig
n
als
wer
e
tak
en
f
r
o
m
t
h
e
p
u
b
lic
Ph
y
s
io
n
et
ar
ch
i
v
e,
a
v
ailab
le
at
th
e
web
s
ite
h
ttp
s
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ch
iv
e.
p
h
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ataset
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tain
s
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tal
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f
6
4
8
.
0
0
0
s
am
p
les.
Fo
r
test
in
g
th
e
p
r
o
p
o
s
ed
s
tr
etch
ed
f
ilter
,
th
r
ee
QR
S
co
m
p
lex
es
wer
e
co
n
s
id
er
e
d
f
o
r
n
o
r
m
al
E
C
G
s
ig
n
als
an
d
two
QR
S
co
m
p
lex
es
f
o
r
ab
n
o
r
m
al
E
C
Gs
w
ith
2
-
an
d
1
-
s
ec
o
n
d
d
u
r
atio
n
s
,
r
esp
ec
tiv
ely
.
T
ab
le
1
s
u
m
m
ar
izes
th
e
f
ea
tu
r
es o
f
E
C
G
s
ig
n
als u
s
ed
f
o
r
test
in
g
th
e
p
r
o
p
o
s
ed
s
tr
etch
ed
ex
p
o
n
en
tial f
ilter
.
T
ab
le
1
.
No
r
m
al
an
d
a
b
n
o
r
m
al
E
C
G
s
ig
n
al
p
ar
am
eter
s
s
elec
ted
f
o
r
test
in
g
t
h
e
s
tr
etch
ed
ex
p
o
n
en
tial f
ilter
N
o
r
mal
EC
G
A
b
n
o
r
ma
l
E
C
G
(
sl
e
e
p
a
p
n
e
a
)
N
u
mb
e
r
o
f
EC
G
R
e
c
o
r
d
s
50
50
EC
G
T
i
m
e
L
e
n
g
t
h
3
0
mi
n
3
0
mi
n
S
a
mp
l
i
n
g
R
a
t
e
3
6
0
H
z
3
6
0
H
z
P
r
o
c
e
ss
e
d
E
C
G
si
g
n
a
l
s'
d
u
r
a
t
i
o
n
2
se
c
1
se
c
3.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
Fig
u
r
es
5
(
a)
-
5
(
d
)
an
d
6
(
a)
-
6
(
d
)
s
h
o
w
th
e
ef
f
ec
t
o
f
th
e
d
esig
n
ed
SC
E
L
P
f
ilter
ap
p
lied
to
ab
n
o
r
m
al
an
d
n
o
r
m
al
E
C
G
n
o
is
y
s
ig
n
als,
r
esp
ec
tiv
ely
,
f
o
r
d
if
f
er
en
t
p
ar
am
eter
v
alu
es.
T
h
e
p
u
r
p
o
s
e
o
f
f
ilter
in
g
p
r
o
ce
s
s
is
to
d
en
o
is
e
th
e
r
aw
E
C
G
s
i
g
n
al
(
in
p
u
t,
r
ed
p
lo
ts
)
an
d
p
r
o
v
id
e
a
f
ilter
ed
o
u
tp
u
t
(
b
lu
e
p
lo
ts
)
s
u
itab
le
f
o
r
f
u
r
th
er
p
r
o
ce
s
s
in
g
.
T
ab
le
2
an
d
3
p
r
esen
t
th
e
o
b
t
ain
ed
m
ea
n
SNR
an
d
MSE
v
alu
es
af
ter
ap
p
ly
in
g
th
e
SC
E
L
P
f
ilter
f
o
r
d
if
f
er
en
t
v
alu
es
in
t
h
e
r
a
n
g
e
[
0
.
2
÷
1
0
.
0
]
to
th
e
d
ataset's
a
b
n
o
r
m
al
an
d
n
o
r
m
al
E
C
G
s
ig
n
als.
T
h
e
SNR
an
d
MSE
v
alu
es we
r
e
ca
lcu
lated
f
r
o
m
all
E
C
G
s
ig
n
als f
o
r
ea
ch
v
alu
e;
th
en
,
th
e
m
ea
n
v
alu
es
h
av
e
b
ee
n
r
ep
o
r
ted
in
T
ab
le
2
f
o
r
ab
n
o
r
m
al
E
C
G
s
ig
n
als
an
d
in
T
ab
le
3
f
o
r
th
e
n
o
r
m
al
o
n
es.
Fo
r
b
o
th
s
ig
n
al
ty
p
es,
th
e
f
ilter
o
u
tp
er
f
o
r
m
s
f
o
r
β
b
etwe
en
1
.
2
an
d
2
,
p
r
o
v
i
d
in
g
th
e
h
ig
h
est
SNR
an
d
lo
west
M
SE
v
alu
es
f
o
r
1
.
6
.
T
a
b
le
4
s
h
o
ws th
e
o
v
er
all
p
er
f
o
r
m
an
c
e
in
ter
m
s
o
f
av
er
ag
e
SNR
an
d
MSE
f
o
r
h
i
g
h
er
th
a
n
2
,
b
e
twee
n
1
.
2
an
d
2
,
an
d
eq
u
al
to
o
r
s
m
aller
t
h
an
1
.
A
s
ex
p
lain
ed
i
n
th
e
m
eth
o
d
s
ec
tio
n
,
b
ased
o
n
(
1
3
)
,
th
e
f
ilte
r
will
o
p
er
ate
as
a
co
m
p
r
ess
ed
ex
p
o
n
en
tial f
ilter
if
>
2
,
a
s
tr
etch
ed
-
co
m
p
r
ess
ed
ex
p
o
n
e
n
tial f
ilter
if
1
<
≤
2
,
a
Gau
s
s
ian
f
ilter
if
=1
,
an
d
as
a
s
tr
etch
ed
ex
p
o
n
en
tial
f
ilter
wh
e
n
<
1
.
Fin
ally
,
Fig
u
r
e
s
7
(
a)
an
d
7
(
b
)
s
h
o
ws
th
e
o
b
tain
ed
r
esu
lts
as
h
is
to
g
r
am
s
r
elatin
g
to
SNR
an
d
MSE
v
al
u
es,
as
alr
ea
d
y
r
ep
o
r
ted
in
T
ab
le
4
,
f
o
r
th
e
d
if
f
er
en
t
r
an
g
es
.
I
n
ad
d
itio
n
,
t
o
f
u
r
th
e
r
in
v
esti
g
ate
th
e
ef
f
ec
tiv
en
ess
o
f
d
ev
elo
p
ed
f
ilter
,
a
n
o
r
m
al
wh
ite
n
o
is
e
h
as b
ee
n
ad
d
ed
to
th
e
E
C
G
s
ig
n
als
co
n
s
titu
tin
g
th
e
d
ataset
d
escr
ib
ed
in
s
ec
tio
n
2
in
o
r
d
er
to
v
er
if
y
th
e
SC
E
L
P
f
ilter
’
s
ef
f
icac
y
an
d
co
m
p
ar
e
its
p
er
f
o
r
m
an
ce
with
d
if
f
er
e
n
t
d
en
o
is
in
g
f
ilter
s
'
ty
p
o
lo
g
ies
lik
e
th
e
Gau
s
s
ian
,
Mittag
–
L
ef
f
ler
,
an
d
Sav
itzk
y
-
Go
lay
o
n
es.
I
n
m
o
r
e
d
etail,
n
in
e
s
ec
o
n
d
s
wer
e
ex
tr
ac
ted
f
r
o
m
ea
ch
E
C
G
s
ig
n
al,
an
d
th
en
a
n
o
r
m
al
wh
ite
n
o
is
e
(
(
)
)
was
ad
d
ed
.
T
h
is
p
r
o
ce
s
s
is
u
s
ed
to
test
th
e
ca
p
a
b
ilit
y
o
f
f
ilt
er
s
to
elu
cid
ate
th
e
in
h
er
e
n
t
s
ig
n
al
(
(
)
)
f
r
o
m
th
e
co
n
tam
in
ate
d
,
n
o
is
e
-
r
id
d
e
n
s
i
g
n
al
(
(
)
)
.
T
h
u
s
,
a
wh
ite
n
o
is
e
ar
r
ay
was
g
en
er
ated
th
r
o
u
g
h
MA
T
L
AB
an
d
th
en
ad
d
e
d
to
t
h
e
E
C
G
s
ig
n
al
b
y
p
o
in
t
-
by
-
p
o
i
n
t
a
d
d
itio
n
o
f
th
e
E
C
G
an
d
n
o
is
e
v
al
u
es (
1
6
)
;
f
o
r
t
h
is
p
u
r
p
o
s
e,
a
Ma
tlab
co
d
e
h
as b
e
en
im
p
lem
en
ted
,
r
ep
o
r
ted
b
elo
w.
(
)
=
(
)
+
(
)
(
1
6
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8
7
0
8
I
n
t J E
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&
C
o
m
p
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n
g
,
Vo
l.
1
6
,
No
.
1
,
Feb
r
u
ar
y
20
2
6
:
2
3
0
-
245
238
load('AbnormalECGSignalone.mat')
% load('Normal ECG_Filter.mat')
noiseSignal=randn(size(X2));
newSignal=noiseSignal + X2;
y=newSignal
(
a)
(
b
)
(
c)
(
d
)
Fig
u
r
e
5
.
Fil
ter
ed
a
b
n
o
r
m
al
E
C
G
s
ig
n
als b
y
co
m
p
r
ess
ed
an
d
s
tr
etch
ed
ex
p
o
n
e
n
tial f
ilter
s
f
o
r
d
if
f
er
e
n
t
v
alu
es:
=
1
0
.
0
(
a)
=
6
.
0
,
(
b
)
=
1
.
6
,
(
c)
=
0
.
6
,
an
d
(
d
)
all
test
ed
f
ilter
s
h
av
e
σ
=
10
(
a)
(
b
)
(
c)
(
d
)
Fig
u
r
e
6
.
Fil
ter
ed
n
o
r
m
al
E
C
G
s
ig
n
als b
y
co
m
p
r
ess
ed
an
d
s
tr
etch
ed
ex
p
o
n
en
tial f
ilter
s
f
o
r
d
if
f
er
en
t
v
alu
es:
=
1
0
(
a
)
=
6
,
(
b
)
=
1
.
6
,
(
c)
=
0
.
6
,
an
d
(
d
)
all
test
ed
f
ilter
s
h
av
e
σ
=
10
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J E
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&
C
o
m
p
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n
g
I
SS
N:
2088
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8
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8
A
n
o
ve
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r
ess
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ex
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o
n
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tia
l lo
w
-
p
a
s
s
filt
er a
n
d
its
a
p
p
lica
tio
n
…
(
R
o
b
erto
d
e
F
a
z
io
)
239
T
ab
le
2
.
Per
f
o
r
m
an
ce
c
o
m
p
ar
i
s
o
n
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f
p
r
o
p
o
s
ed
co
m
p
r
ess
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etch
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p
o
n
en
tial f
ilter
f
o
r
d
if
f
er
en
t
β
v
alu
es
ap
p
lied
to
n
o
r
m
al
E
C
G
s
ig
n
al
s
(
to
tally
5
0
)
f
r
o
m
t
h
e
Ph
y
s
io
n
et
d
ataset
F
i
l
t
e
r
t
y
p
e
F
i
l
t
e
r
p
a
r
a
m
e
t
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r
M
e
a
n
S
N
R
(
d
B
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e
a
n
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o
m
p
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1
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7
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C
o
m
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2
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0
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4
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S
t
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e
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h
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d
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1
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7
S
t
r
e
t
c
h
e
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m
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e
d
e
x
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t
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f
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6
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S
t
r
e
t
c
h
e
d
-
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p
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e
x
p
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n
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t
i
a
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f
i
l
t
e
r
β
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2
1
4
.
6
7
2
9
0
.
0
1
1
4
0
G
a
u
ss
i
a
n
f
i
l
t
e
r
β
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.
0
1
1
.
2
5
8
0
0
.
0
1
3
2
8
S
t
r
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t
c
h
e
d
e
x
p
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t
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a
l
f
i
l
t
e
r
β
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.
6
9
.
6
6
7
8
0
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0
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3
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5
S
t
r
e
t
c
h
e
d
Ex
p
o
n
e
n
t
i
a
l
f
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l
t
e
r
β
=
0
.
2
7
.
2
5
9
5
0
.
0
1
5
9
8
T
ab
le
3
.
Per
f
o
r
m
an
ce
c
o
m
p
ar
i
s
o
n
o
f
p
r
o
p
o
s
ed
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m
p
r
ess
ed
-
s
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etch
ed
ex
p
o
n
en
tial f
ilter
f
o
r
d
if
f
er
en
t
β
v
alu
es
ap
p
lied
to
a
b
n
o
r
m
al
E
C
G
s
ig
n
als (
to
tally
5
0
)
f
r
o
m
th
e
Ph
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s
io
n
et
d
ataset
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l
t
e
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t
y
p
e
F
i
l
t
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r
p
a
r
a
m
e
t
e
r
M
e
a
n
S
N
R
(
d
B
)
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e
a
n
M
S
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o
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d
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l
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o
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l
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8
0
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3
7
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6
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t
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c
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d
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t
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6
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4
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3
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5
S
t
r
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t
c
h
e
d
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c
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d
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t
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r
β
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.
6
1
3
.
7
5
7
4
0
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0
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1
7
8
S
t
r
e
t
c
h
e
d
-
c
o
m
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d
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t
i
a
l
f
i
l
t
e
r
β
=
1
.
2
1
1
.
5
3
5
4
0
.
0
1
3
4
8
G
a
u
ss
i
a
n
f
i
l
t
e
r
β
=
1
.
0
8
.
6
9
3
9
0
.
0
1
6
8
5
S
t
r
e
t
c
h
e
d
e
x
p
o
n
e
n
t
i
a
l
f
i
l
t
e
r
β
=
0
.
6
6
.
2
0
4
8
0
.
0
1
8
9
3
S
t
r
e
t
c
h
e
d
e
x
p
o
n
e
n
t
i
a
l
f
i
l
t
e
r
β
=
0
.
2
5
.
7
2
4
0
0
.
0
2
4
8
6
T
ab
le
4
.
A
v
er
ag
e
SNR
an
d
MSE
v
alu
es r
elate
d
to
n
o
r
m
al
an
d
ab
n
o
r
m
al
E
C
Gs (
f
r
o
m
t
h
e
Ph
y
s
io
n
et
d
ataset)
d
en
o
is
ed
b
y
th
e
d
e
v
elo
p
e
d
co
m
p
r
ess
ed
/s
tr
etch
ed
ex
p
o
n
en
ti
al
f
ilter
f
o
r
d
if
f
er
en
t
r
an
g
es
N
o
r
mal
EC
G
s
(
5
0
s
i
g
n
a
l
s)
A
b
n
o
r
ma
l
E
C
G
s (
5
0
si
g
n
a
l
s)
Ex
p
o
n
e
n
t
i
a
l
p
a
r
a
met
e
r
(
)
A
v
e
r
a
g
e
S
N
R
(
d
B
)
A
v
e
r
ag
e
M
S
E
A
v
e
r
a
g
e
S
N
R
(
d
B
)
A
v
e
r
a
g
e
M
S
E
>
2
4
,
0
9
3
5
0,
0299
7
2
.
0
9
7
1
0
,
0
4
8
6
8
1
.
2
≤
≤
2
1
5
,
3
5
1
9
0,
0110
7
1
1
,
6
5
0
7
0
,
0
1
2
9
7
≤
1
9
,
3
9
5
1
0,
0144
0
6
,
8
7
4
2
0
,
0
2
0
2
1
(
a)
(
b
)
Fig
u
r
e
7
.
Hi
s
to
g
r
am
s
with
th
e
o
b
tain
ed
a
v
er
ag
e
SNR
(
a)
an
d
MSE
(
b
)
v
alu
es f
r
o
m
n
o
r
m
al
an
d
ab
n
o
r
m
al
E
C
Gs p
r
o
ce
s
s
ed
b
y
th
e
d
ev
el
o
p
ed
co
m
p
r
ess
ed
/s
tr
etch
ed
ex
p
o
n
en
tial f
ilter
f
o
r
d
if
f
er
e
n
t
r
an
g
es
Fin
ally
,
th
e
d
en
o
is
in
g
p
er
f
o
r
m
an
ce
s
o
f
th
e
d
if
f
er
e
n
t
f
ilter
ty
p
es
h
av
e
b
ee
n
ex
am
i
n
ed
an
d
r
ep
o
r
ted
in
T
ab
le
5
.
Fo
r
a
Gau
s
s
ian
f
ilter
,
th
e
m
o
s
t
c
r
itical
p
ar
a
m
eter
is
t
h
e
s
tan
d
ar
d
d
ev
iatio
n
(
σ
)
o
f
th
e
Gau
s
s
ian
k
er
n
el,
as
it
d
eter
m
in
es
th
e
s
m
o
o
th
in
g
ex
ten
t;
a
lar
g
er
σ
r
esu
lts
in
g
r
ea
ter
s
m
o
o
th
in
g
b
u
t
m
a
y
b
l
u
r
th
e
f
in
e
d
etails,
wh
ile
a
s
m
aller
σ
p
r
eser
v
es
f
in
er
d
etails
b
u
t
m
ig
h
t
n
o
t
r
em
o
v
e
th
e
n
o
is
e
ef
f
ec
tiv
el
y
.
T
h
e
k
e
r
n
el
s
ize
is
ty
p
ically
ch
o
s
en
as
a
f
u
n
ctio
n
o
f
σ
,
o
f
ten
u
s
in
g
a
s
ize
o
f
(
6
σ
+1
)
to
en
s
u
r
e
th
e
f
ilter
en
co
m
p
ass
es
m
o
s
t
o
f
th
e
Gau
s
s
ian
d
is
tr
ib
u
tio
n
[
3
9
]
.
A
d
d
itio
n
ally
,
th
e
b
o
u
n
d
ar
y
co
n
d
itio
n
s
s
u
ch
as
“r
ef
lect
,
”
“c
o
n
s
tan
t
,
”
o
r
“wr
ap
”
m
u
s
t
b
e
s
elec
ted
b
ased
o
n
th
e
in
p
u
t
d
ata’
s
n
atu
r
e
to
a
v
o
id
ed
g
e
ar
tifa
cts.
T
h
e
m
ain
p
a
r
am
eter
s
f
o
r
Mittag
-
L
ef
f
ler
f
ilter
ar
e
th
e
s
ca
lin
g
f
ac
to
r
α
an
d
f
r
ac
tio
n
al
o
r
d
e
r
β,
wh
ich
g
o
v
er
n
th
e
weig
h
t
o
f
th
e
Mittag
-
L
ef
f
ler
f
u
n
ctio
n
in
m
o
d
elin
g
m
em
o
r
y
o
r
s
m
o
o
th
in
g
ef
f
ec
ts
[
4
2
]
;
a
s
m
aller
β
em
p
h
asizes
lo
n
g
-
ter
m
m
em
o
r
y
ef
f
ec
ts
,
wh
ile
a
lar
g
er
β
p
r
o
v
id
es
m
o
r
e
lo
ca
lized
s
m
o
o
th
in
g
.
T
h
e
p
ar
am
eter
s
s
h
o
u
l
d
b
e
tu
n
ed
b
ased
o
n
th
e
d
esire
d
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