TELKOM
NIKA
, Vol. 11, No. 9, September 20
13, pp.
5141
~51
4
9
ISSN: 2302-4
046
5141
Re
cei
v
ed Fe
brua
ry 26, 20
13; Re
vised
June 5, 201
3; Acce
pted Jun
e
19, 2013
Resear
ch of Signal De-noising Technique Based on
Wavelet
Shigang Hu
*
1
, Yinglu Hu
2
, Xiaofeng Wu
1
, Jin Li
1
, Z
a
ifang Xi
1
, Jin Zhao
1
1
School of Infor
m
ation a
nd El
e
c
trical Eng
i
ne
e
r
ing,
Hu
nan U
n
iversit
y
of Scie
nce an
d T
e
chnolo
g
y
,
Xi
an
gta
n
411
20
1, Chin
a
2
T
he 41st Institute of Chin
a El
ectronics T
e
chnol
og
y Grou
p Corp
oratio
n, Qingd
ao 2
665
55,
Chin
a
*Corres
p
o
ndi
n
g
author, e-ma
i
l
:
hsg99
528
@1
26.com
A
b
st
r
a
ct
Durin
g
the pr
o
c
ess of signa
l testing, it is ofte
n
expos
ed to in
terference a
nd
influ
ence
of all
kinds of
nois
e
sig
nal, s
u
ch as
data c
o
llecti
on
an
d trans
missi
on
an
d so n
o
ise
ma
y be i
n
troduc
e
d
. So in
practi
ca
l
app
licati
ons, b
e
fore a
nalys
is
of
the data
measur
ed, the
n
eed for
de-n
o
i
s
ing pr
ocess
i
n
g
. T
he sign
al
de-
noisi
ng
is a
me
thod for fi
lterin
g the
hig
h
freq
uency
no
is
e
of the si
gna
l a
n
d
makes th
e si
g
nal
more
preci
s
e
.
T
h
is pa
per
de
als w
i
th the
g
ener
al th
eory
of w
a
velet
tra
n
sform, th
e a
p
p
licati
on
of w
a
velet transfor
m
in
sign
al
de-n
o
isi
ng as
w
e
ll
as the a
n
a
l
ysis
of the ch
aracte
r
i
sti
cs of no
ise-p
o
ll
uted si
gn
al. Ma
tlab is
use
d
to
b
e
carried
out the
simulati
on w
h
e
r
e the differe
nt w
a
velet an
d
dif
f
erent thresh
ol
d of the sa
me
w
a
velet for sig
nal
de-n
o
isi
ng
are
app
lie
d. An i
n
d
i
cator of w
a
vel
e
t de-n
o
isi
ng
i
s
prese
n
ted, it
is the i
ndic
a
to
r of smo
o
thn
e
s
s
.
T
h
roug
h a
naly
s
is of the
exp
e
ri
ment, co
nsi
deri
ng
MSE, S
NR a
nd s
m
o
o
t
hness, it c
an
be a
go
od w
a
y to
eval
uate the e
qua
lity of w
a
velet thre
sh
old
de-n
o
isi
ng. T
he results show
that the w
a
ve
let transform c
a
n
achi
eve exc
e
ll
ent results in si
gna
l de-n
o
is
ing
;
denois
ed
si
gn
al usi
ng soft-thresho
l
d
meth
o
d
is smo
o
ther
a
n
d
soft-threshol
d
meth
od
is
mor
e
suita
b
le
for
mor
e
si
gna
l d
e
t
ail co
mpon
ent
; SNR, MSE a
nd s
m
o
o
thn
e
s
s
are
all
i
m
p
o
rtant
in
dexes
to
eval
uate t
h
e
perfor
m
ance
of n
o
ise
re
ductio
n
; thres
hol
d ru
le, w
a
vele
t
deco
m
positi
on level and
w
a
ve
let
function a
ll i
m
p
a
ct the de-
n
o
isin
g perfor
m
ances.
Ke
y
w
ords
: w
a
velet transfor
m
, de-nois
i
ng, th
resho
l
d, MAT
L
AB
Copy
right
©
2013 Un
ive
r
sita
s Ah
mad
Dah
l
an
. All rig
h
t
s r
ese
rved
.
1. Introduc
tion
Wavelet
anal
ysis, a
s
a
kin
d
of n
o
vel th
eory,
is an
i
m
porta
nt out
come
in th
e
history
of
mathemati
c
s developme
n
t. From the point of ma
thematics, Wavelet analy
s
is i
s
a kin
d
o
f
mathemati
c
al
micro
s
cop
e
[1-2]; from the
view of
appli
c
ation, Wavel
e
t analysi
s
is a tool of time
-
freque
ncy an
alysis, ove
r
co
ming the tradi
tional fourie
r
analysi
s
’
s
sh
ortco
m
ing
s
which i
s
co
mpl
e
te
locali
zation
i
n
freq
uen
cy
domain
but n
onlo
c
ali
z
ation
in time do
m
a
in, espe
ciall
y
suiting to t
h
e
analysi
s
of n
on-steady
sig
nal. Wavel
e
t transfo
rm
i
s
of locali
zatio
n
in both tim
e
and frequ
e
n
cy
domain
s
, an
d
the freq
uen
cy distributio
n
of ce
rtain
tim
e
ca
n be
cal
c
ulated, al
so t
he mixed
sig
nal
whi
c
h i
s
com
posed
of different f
r
eq
uen
cie
s
can
be
d
e
com
p
o
s
ed
i
n
to differe
nt f
r
equ
en
cy ba
n
d
s
with different
frequen
cy range
s [3]. Wavelet anal
ysis is asso
ciat
ed clo
s
ely wi
th many other
subj
ect
s
[4].
No
wad
a
ys m
a
thematical a
nd engi
nee
ri
ng fields a
r
e
paying much intere
st to the
developm
ent of new theo
ry and method
con
c
e
r
nin
g
wavelet with its application.
Signal d
e
-n
oi
sing
is one
o
f
the impo
rta
n
t re
se
a
r
ch t
opics i
n
the fi
eld of
sign
al
analysi
s
[5-7]. At present, there a
r
e two d
e
-n
o
i
sing met
hod
s, the traditi
onal filtering
method a
nd
the
wavelet de
-n
oisin
g
metho
d
, when in th
e actual te
st, different noi
se and si
gnal
with the ch
oice of
different d
e
-n
oisin
g
meth
o
d
s. T
r
aditio
n
a
l de
-noi
sin
g
method
is b
a
se
d on
fou
r
i
e
r a
nalysi
s
,
ca
n
only be u
s
e
d
in the
circumstan
ce
s t
hat sig
nal
a
nd noi
se i
s
very small
b
and ove
r
lap
or
compl
e
tely separate from
, and sep
a
ra
ted the si
gn
al and noi
se
by the method of filtering.
Ho
wever, in
pra
c
tice, the
sign
al sp
ectrum and
n
o
ise spe
c
trum a
r
e overl
app
e
d
, the traditio
nal
filtering meth
od can
not a
c
hieve
an
effective remo
v
a
l of noi
se, a
nd the
purpo
se of
extra
c
ting
useful
sig
nal.
Wavelet a
n
a
l
ysis i
s
a n
e
w
mathe
m
ati
c
al the
o
rie
s
a
nd metho
d
s
develop
ed in
the
mid-1
980
s, a
nd it kno
w
n
as the "mi
c
ro
scope" of
ma
thematical
an
alysis.
Wavel
e
t analysi
s
is a
time-freq
uen
cy analysi
s
m
e
thod
of si
gn
al, with th
e
chara
c
te
risti
c
s of multi-re
so
lution a
nalysi
s
,
can b
e
focu
sed on any of the details of
sign
al
to multi-re
sol
u
tion time-fre
que
ncy
analysi
s
. And it
is su
peri
o
r to
Fouri
e
r an
alysis al
gorith
m
.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 9, September 201
3: 514
1 – 5149
5142
The theo
ry of wavelet a
nd pri
n
ci
ple
of signal
d
e
-noi
sin
g
ba
sed
on wavelet are
introdu
ce
d in
this pape
r. Smoothne
ss is used a
n
i
ndi
cator. Th
e qu
ality of wavelet de-n
o
isin
g
is
evaluated u
s
i
ng the indi
cat
o
rs
RMSE , SNR an
d sm
oothne
ss. Th
e MATLAB si
mulation resu
lts
confirm that
wavelet th
re
shold m
e
thod
is effe
ctive a
nd e
n
joys so
me a
d
vantag
es i
n
removi
ng
noises.
2. Wav
e
let Theor
y
and the Principle of De
-noising
Wavelet tran
sform i
s
a
n
a
nalytical met
hod
whi
c
h u
n
i
ts the time d
o
main a
nd freque
ncy
domain
[8-1
0
]. It has the
cap
a
city of th
e multi-re
solu
tion analy
s
is
whi
c
h i
s
lo
cal
i
zed i
n
time
or
spa
c
e. And it
is a lo
cal tim
e
-fre
que
ncy
analysi
s
met
hod that the
wind
ow
size fixed but its shape
can
be
ch
ang
ed, the time
a
nd fre
que
ncy
wind
ow
all
ca
n be
ch
ang
ed
. Here, its
scaling a
nalyzi
n
g
function
ca
n
vary its wi
dth
depe
ndin
g
o
n
the fr
e
que
n
c
y inform
atio
n to be
analy
z
ed. T
he
scal
in
g
analyzi
ng fun
c
tion ha
s a la
rge
width in the tempo
r
al
domain fo
r lo
w freq
uen
cy comp
one
nts
and
a sm
all
width
for the
high
frequ
en
cy co
mpone
nts.
It i
s
very
normal
for d
e
tectin
g
tran
sient
sig
nal
entrain
ment
anomaly
and
demo
n
st
rate
s the
si
gnal'
s
comp
one
nts. Th
erefo
r
e,
it is
call
ed
a
mathemati
c
al
microsco
pe
for an
alyzin
g sig
nal
s
[1
1-
1
3
]. Ju
s
t
bec
a
u
s
e
o
f
th
is c
h
ar
ac
te
r
i
s
t
ic
,
wavelet tran
sform ha
s the adapta
b
ility of the signal proce
s
sing.
The co
ntinual
wavelet tran
sform
a
tion is
defined a
s
:
WT
x
(a,b)=
a
1
dt
t
x
a
b
t
)
(
)
(
(1)
In pra
c
tice, di
screte
wavel
e
t transfo
rm i
s
use
d
fre
que
ntly, and it is
al
so
req
u
ire
d
t
hat the
sign
al can
be recon
s
tructed. M
a
ke
a=a
0
m
an
d b
=
na
0
m
b
0
, then the
di
screte
wave
let
transfo
rmatio
n is:
)
(
0
0
2
0
,
nb
t
a
a
m
m
n
m
(2)
Take a
0
>
1
and
b
0
0
, Then the discrete
wa
velet transfo
rm is:
W T
x
(m, n)=
dt
t
x
t
n
m
)
(
)
(
,
(3)
Take a
0
=
2
and
b
0
=1, Equat
ion (2
) ca
n be
written a
s
a functio
n
syste
m
s.
k
x
t
j
j
k
j
2
2
)
(
2
,
j, k
z
(
4
)
The ab
ove eq
uation i
s
sta
n
dard
orth
ogo
nal ba
si
s a
c
cordin
g to the
binary exp
a
n
s
ion
and
transl
a
tion on
L2 (R)
.
In the actual appli
c
ation, b
e
sid
e
s havin
g the
flexibility to choose
wavelet ba
sis function,
the wavelet al
so ha
s the fol
l
owin
g signifi
cant adva
n
ta
ges:
Orthog
onality
:
reduce the signal re
dun
da
ncy.
Short su
ppo
rt: not only show the lo
cality of ti
me
domain, but
also redu
ce
the amount
of
comp
utation.
Approximatio
n and Regul
a
r
ity: analyze
the sin
gula
r
ity of the function.
Symmetry: make
ea
sily to deal
with the
bord
e
r i
n
the
data comp
re
ssion, avoi
d b
o
rde
r
di
storti
on,
and re
du
ce q
uantization error.
The wavel
e
t transfo
rm is
a sign
al pro
c
essi
ng te
chni
que that rep
r
ese
n
ts a tran
sient or
non-station
a
ry signal in term
s of time and scale
d
i
stributio
n. Due to its ligh
t
computatio
nal
compl
e
xity, the wavelet transfo
rm i
s
a
n
excell
ent
t
ool for
on
-lin
e data
com
p
ressio
n, an
al
ysis,
and de
-noi
sin
g
.
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Re
sea
r
ch of Signal De
-noi
sing Te
ch
niq
ue ba
sed o
n
Wa
velet (Shi
gang
Hu)
5143
Suppo
se the
r
e is an ob
se
rved sign
al.
f
(
t
)=
s
(
t
)+
n
(
t
)
(5)
Whe
r
e
s
(
t
) i
s
the origin
al si
gnal,
n
(
t
) i
s
G
aussia
n
white
noise
with mean 0 an
d variance
2
.
The flow
cha
r
t of signal de-noisi
ng is a
s
sho
w
n in Fig
u
re 1.
1) Sel
e
ct t
he a
pprop
ri
ate wavelet
and
wavel
e
t de
comp
o
s
ition l
e
vel, ma
ke
wav
e
le
t
decompo
sitio
n
for the noise sign
al
f
(
t
), get corre
s
p
o
n
d
ing coefficie
n
t of wavelet decompo
sitio
n
;
2)
Deal
with t
he threshold f
o
r
coeffici
ent
s
w
j
,
k
whi
c
h
come
from th
e wavel
e
t de
comp
ositio
n,
and
obtain the e
s
timation value
,
j
k
w
of the wavelet coeffici
en
ts of origina
l signal
s
(
t
) ;
3) M
a
ke wavelet inverse
transfo
rm fo
r t
hese e
s
timat
ed value
,
j
k
w
, and get th
e
si
gnal afte
r d
e
-
noisi
ng.
Figure 1. Flow Ch
art of Signal De-n
oisi
ng
There a
r
e h
a
rd thre
sh
old m
e
thod a
nd
sof
t
threshold m
e
thod fo
r the
wavelet
coeffi
cient
s
estimation
which
put forward
by Do
noh
o. The b
a
si
c i
dea i
s
to
rem
o
ve the
small
coeffici
ent a
n
d
shri
nk or
ke
e
p
the la
rge
coefficient. By hard
th
re
sh
ol
d functio
n
, we co
mpa
r
e
wavelet co
effici
ent
absolute valu
e with the th
resh
old valu
e, turn the
ab
solute value
which i
s
le
ss than o
r
e
qual
to
the threshold
into
ze
ro, a
n
d
keep
the
o
ne
whi
c
h i
s
g
r
eate
r
tha
n
th
e threshold;
by soft th
re
sh
old
function,
we t
u
rn th
e ab
sol
u
te value
whi
c
h i
s
le
ss tha
n
the threshol
d into
zero, a
nd turn the
o
ne
whi
c
h is g
r
e
a
ter than the
threshold int
o
the D-va
l
u
e of coefficie
n
t and the th
reshold. O
n
this
basi
s
, the
r
e
are
many im
proved
threshold
algo
rith
m, the
key o
f
the de
-n
oising i
s
to fin
d
a
suitabl
e num
ber of
λ
a
s
a
threshold, th
en if the wav
e
let co
efficie
n
ts are lo
wer than
λ
, s
e
t
the
coeffici
ent 0, and if the wa
velet coefficie
n
ts are
highe
r than
λ
, maint
a
in or contract.
Hard-th
r
e
s
hol
d method is e
x
presse
d as f
o
llows:
,,
,
,
,
0,
jk
j
k
jk
ww
jk
w
w
(6)
Soft-thre
shol
d method is d
e
fined a
s
:
,,
,
,
()
(
)
,
,
0,
jk
jk
jk
jk
si
g
n
w
w
w
jk
w
w
(7)
Whe
r
e
2
l
og(
)
N
In the case of white
noise, it
s sta
ndard deviati
on ca
n be e
s
timated
from the medi
an of its detai
l coefficie
n
ts (
d
j
), with
j
1···
L
, and is
computed a
s
follows:
()
0.
6
745
j
M
AD
d
(8)
Whe
r
e
MAD
i
s
the media
n
absolute devi
a
tion
of the correspon
ding
seq
uen
ce.
Comm
only u
s
ed
thre
sh
ol
d sele
ction
rule
s a
r
e:
ri
grsure, sqtwolog, h
eursu
re a
nd
minimaxi rul
e
s
. Stein
u
nbia
s
ed
ri
sk
threshold
(rig
rsu
r
e
)
i
s
an
adaptive th
re
shol
d sele
ction
prin
ciple b
a
sed on no
n-partial likelih
ood e
s
tima
t
o
r. For a gi
ven thre
shol
d, its likelih
ood
estimating
va
lue i
s
first co
mputed. T
h
e
n
minimi
ze
n
on-li
kelih
ood
of
λ
, the
thresh
old
co
uld
be
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 9, September 201
3: 514
1 – 5149
5144
determi
ned.
The d
e
finition
of the
universal th
re
shol
d
is
sho
w
n
a
s
2l
o
g
L
, Where
L
is
the sampl
e
number,
is standa
rd dev
iation of the noisy si
g
nal. Heuri
s
tic Th
re
shold combin
e
s
the two th
re
shol
ds
co
ncerne
d ab
ove
.
The choi
ce
of the vari
able th
re
shol
d ha
s the
b
e
st
predi
cto
r
. Wh
en the sig
nal
to noise ratio is ve
ry small,
the fixed thresh
old is b
e
tter than
ri
gr
su
re
rule. Minimal
Great Va
rian
ce Th
re
shol
d (Minimaxi Rul
e
) is
sho
w
n a
s
:
2
(0.
393
6
0
.
1829
l
o
g
)
,
3
2
0,
3
2
nn
n
.
In orde
r to
compa
r
e the
e
ffect of red
u
cing noi
se i
n
the u
s
e of diff
erent
wavelet
bases
and diffe
rent
combi
nation
of thre
shold
rules, th
ree
p
e
rform
a
n
c
e i
ndexe
s
are u
s
ed to
evalu
a
te
the noise red
u
ction effe
ct, whi
c
h are sig
nal to noise
ratio (
SNR
), mean squa
re
erro
r (
MSE
) and
smoot
h
n
e
ss .
Signal to noise ratio refers to the sign
al powe
r
to noise po
we
r ratio. It
is often used a
s
the de-noi
sin
g
effect eval
uation ind
e
x. S
NR is m
e
a
s
ured in d
e
ci
bels. Th
e larger
SNR
is
, the
better the de
-noisi
ng effect
. The definit
ion of SNR is
shown in Equa
tion 9.
22
11
10
l
g
(
(
)
[
(
)
(
)
]
)
NN
d
nn
SNR
s
n
f
n
s
n
(9)
Whe
r
e,
N
i
s
the numb
e
r of
sample p
o
int
s
,
s
(
n
) is o
r
igi
nal sig
nal wit
hout noi
se,
f
d
(
n
) i
s
de-noi
sed
sign
al.
Mean sq
ua
re
er
ro
r
me
asure
s
th
e d
e
g
r
ee
of
simila
rity of the
d
e
-noi
se
d
sig
nal a
nd
origin
al sig
n
a
l
without noise. Error i
s
sm
aller, wh
i
c
h ill
ustrate
s
the d
e
-noi
se
d sig
n
a
l more faithf
ul
to the o
r
igin
a
l
sig
nal,
whi
c
h is mea
n
s,
error
i
s
bette
r noi
se
redu
ction effect. T
he d
e
finition
of
MSE is sho
w
n in Equation
10.
2
1
((
)
(
)
)
d
n
M
SE
f
n
s
n
N
(10)
MSE and SNR d
o
not f
u
lly reflect th
e de-noi
sing
effect. Here
is anoth
e
r
evaluation
indicator, sm
oothne
ss ind
e
x. The def
inition is sh
own in Equation 1
1
.
22
11
[
(
1)
(
)
]
/
[
(
1)
(
)
]
NN
dd
nn
r
f
nf
n
f
nf
n
(11)
Whe
r
e,
f
(
n
) is the original n
o
ise
d
sign
al. The indi
ca
to
r can reflect th
e degree of smoothing of the
de-n
o
ised si
g
nal,
3. Simulation results a
n
d analy
s
is
Take
he
avy sine
si
gnal
with Gau
s
sian
white
noi
se a
s
a
n
exam
pl
e, wh
ere
the
SNR is
12.516
6dB, MSE is 0.994
7, the smooth
ness is 1
(as
sho
w
n in Fig
u
re 2
)
.
Firstly, FFT
method
wa
s
applie
d on
h
eavy sin
e
si
g
nal de
-n
oisi
n
g
an
d si
mula
ted on
MATLAB.
The basi
c
idea of FFT de-noi
si
ng
metho
d
is
suppressin
g
the high
freque
ncy po
rtion of the sig
nal
an
d
retaining
the l
o
w
frequ
ency
signal.
F
FT
de-noi
si
ng
pro
c
e
ss
can
be divid
ed i
n
to
the follo
win
g
step
s. (1)
Make
FF
T o
p
e
ration
of
sig
nal; (2) a
c
cording
to the freq
uen
cy spe
c
t
r
um of si
g
nal, noise
portion i
s
suppresse
d; (3)
Ma
ke
inve
rse
fourie
r
tran
sform fo
r the
transfo
rme
d
spectrum
and
obtain
the d
e
-
noi
sing
sign
al. Figure 3
i
s
the
freque
ncy
sp
ectru
m
of
h
eavy sin
e
si
gnal
with
noise u
nde
r FFT.
It ca
n be
seen
t
hat
the sign
al en
ergy
i
s
con
c
entrated
in th
e low fr
eq
uen
cy
and
the si
gnal en
ergy
quickly de
cay
s
to
zero
after
the
frequen
cy of 5Hz an
d the
r
e is almo
st no ene
rgy at 20Hz. As sh
o
w
n in Figu
re
4,
the si
gnal
is
filtered
by
l
o
w-
pa
ss filters
with diffe
re
nt width
s
re
spectively. FF
T noi
se
re
du
ction
doe
s not
wel
l
pre
s
e
r
ve th
e pe
ak
and
mutation p
o
rt
ion of
usefu
l
sign
al. It can’t di
stingui
sh
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Re
sea
r
ch of Signal De
-noi
sing Te
ch
niq
ue ba
sed o
n
Wa
velet (Shi
gang
Hu)
5145
betwe
en the
high fre
que
n
c
y portio
n
of
the sig
nal
an
d high frequ
e
n
cy interfe
r
e
n
ce i
ndu
ced
by
noise effectively. If the low-pa
ss f
ilter i
s
too narro
w, there a
r
e still
a
lot of noise in
the signal aft
e
r
filtering; if the low-pa
ss filt
er
is to
o wi
de
, a part of th
e useful sig
n
a
l wo
uld be fi
ltered o
u
t as
the
noise.
Figure 2. Orig
inal and
Noisy Heavy Sine Signal
Figure 3. Fre
quen
cy Spect
r
um of He
avy Sine
Signal with Noise u
nde
r F
FT
Figure 4. sig
nal de-noi
sin
g
by FFT method
Comp
ari
ng with FFT fouri
e
r tra
n
sfo
r
m
de-n
o
isi
ng, wavelet de-noi
s
ing i
s
mo
re
suitabl
e
for no
n-statio
nary
signal, a
nd it ca
n rem
o
ve the noi
se
in the si
gnal
effectively, while retai
n
ing
the
original signal
for more
deta
ils.
The
wavelet t
r
ansform has
concentrating ability. Generally,
durin
g the wavelet transf
o
rmatio
n, the wavelet co
e
fficient of useful signal
s
i
s
larg
e, and
the
energy is
con
c
entrated; th
e wavel
e
t co
efficient of
de
comp
osed n
o
i
se i
s
sm
all. Acco
rdi
n
g to
the
sign
al an
d n
o
ise i
n
wavel
e
t decomp
o
si
tion co
efficie
n
ts, noi
se
ca
n be
sup
p
ressed th
ro
ugh t
h
e
different de
compo
s
ition le
vel of wavele
t coeffici
ent t
h
re
shol
d pro
c
e
ssi
ng. The
coeffici
ent which
is larg
er than
the thresh
ol
d is saved, a
nd the c
oeffi
cient whi
c
h is
smalle
r than
the threshold
is
turn to z
e
ro.
The noisy
h
eavy
sin
e
si
gnal wa
s de
comp
osed o
n
5 scale
s
u
s
ing
db
4
(da
ube
chie
s
wavelet
s
) wa
velet.
The
d
a
ube
chie
s wa
velets
a
r
e co
mpactly sup
p
o
rted wavel
e
ts with
extrem
al
pha
se and
highe
st num
ber of vanishing mom
e
n
t
s for a given sup
p
o
r
t width. Wavel
e
t
decompo
sitio
n
diag
ram i
s
sho
w
n i
n
Fig
u
re 5. T
he
a
p
p
roximate sig
n
als are as shown
in
Fig
u
re
6,
the detail
sig
nals
are
sho
w
n a
s
in
Fig
u
re
7.
It can
be seen th
at the detail
si
gnal
s have
more
clo
s
e correlat
ion with the n
o
ise.
0
200
40
0
60
0
800
1
000
1200
140
0
160
0
1
800
2
000
-10
-5
0
5
10
A
m
pl
i
t
ud
e
o
r
i
g
i
nal
s
i
g
nal
0
200
40
0
60
0
800
1
000
1200
140
0
160
0
1
800
2
000
-15
-10
-5
0
5
10
S
a
m
p
l
e
num
b
e
r
A
m
p
l
i
t
ude
no
i
s
e
d
s
i
g
nal
0
50
100
150
200
250
300
350
40
0
450
500
0
5000
10000
15000
F
r
equ
en
c
y
/
H
z
A
m
pl
i
t
ud
e
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 9, September 201
3: 514
1 – 5149
5146
Figure 5. Wa
velet Decomp
osition
Figure 6.
Approximate Sign
als of ea
ch Scale
after Wav
e
let
Dec
o
mpo
s
iti
o
n
Figure 7. Detail Signals of
each Scale af
ter
Wavelet Decomposition
Figure 8. De-noised Signal
using Soft-
threshold, So
ft-threshold a
nd Com
p
romi
sing of
the Hard- and Soft- thres
hold Methods
The
hard- an
d soft-thre
sh
old m
e
thod
s
are
wi
dely u
s
ed in
ap
plications, th
ey h
a
v
e so
me
advantag
es
a
nd di
sadva
n
tage
s. The
db
4 wa
s
cho
s
e
n
here in h
e
a
vy sine si
gnal
de-n
o
isi
ng. T
he
decompo
sitio
n
scale was f
i
ve. The hard
-
, soft-thresh
old and
stand
ard comp
rom
i
sing meth
od
of
the ha
rd
- an
d
soft-th
re
shol
d metho
d
s were
used
re
s
pectively. It can b
e
seen
from Fig
u
re
8 t
hat
the de
-noi
se
d sig
nal
usi
n
g ha
rd-th
r
e
s
h
o
ld meth
od i
s
n
o
t sm
ooth
.
The d
e
-n
oi
sed
sig
nal
u
s
ing
soft-threshold
method is
smooth, but m
a
y lose so
me
signal
cha
r
a
c
teri
stics. Ta
ble 1 is the d
e
-
noisi
ng pe
rfo
r
man
c
e
s
of the thre
e met
hod
s. T
able
1 sho
w
s that
the stand
ard com
p
ro
misi
ng
method
of th
e ha
rd
- a
nd
soft-threshold
overcom
e
s
the short
c
om
ings of oth
e
r two
kin
d
s o
f
method
s to a certai
n extent.
Table 1.
De-n
oisin
g
Performances of th
e Three T
h
re
shol
d Method
s
Methods SNR
MSE
Smoothness
hard-th
reshold
17.1509
0.5834
0.3092
soft-threshold
19.7671
0.4317
0.0178
Compromising of
Soft
and Hard
thresh
old
19.7175
0.4342
0.1125
Comm
only, thre
shol
d
can
be
determin
ed by
ri
g
r
sure, heu
rsure,
sqtwolog
an
d
minimaxi
rule
s
re
spe
c
tively. Db4
wa
velet wa
s u
s
ed
with
soft-thre
sh
old m
e
thod. the
n
o
isy
signal
wa
s
0
200
400
600
800
1000
1200
1400
1600
1800
2000
-1
0
0
10
a5
0
200
400
600
800
1000
1200
1400
1600
1800
2000
-1
0
0
10
a4
0
200
400
600
800
1000
1200
1400
1600
1800
2000
-1
0
0
10
a3
0
200
400
600
800
1000
1200
1400
1600
1800
2000
-1
0
0
10
a2
0
200
400
600
800
1000
1200
1400
1600
1800
2000
-1
0
0
10
a1
S
a
m
p
l
e
num
ber
0
200
400
600
800
1000
1200
1400
1600
1800
2
000
-1
0
1
d5
0
200
400
600
800
1000
1200
1400
1600
1800
2
000
-1
0
1
d4
0
200
400
600
800
1000
1200
1400
1600
1800
2
000
-2
0
2
d3
0
200
400
600
800
1000
1200
1400
1600
1800
2
000
-2
0
2
d2
0
200
400
600
800
1000
1200
1400
1600
1800
2
000
-5
0
5
d1
Sa
m
p
l
e
n
u
mb
e
r
0
200
400
60
0
80
0
10
00
12
00
14
00
16
00
18
00
20
00
-1
0
-5
0
5
10
D
e
no
i
s
e
d
s
i
g
nal
us
i
n
g
har
d
-
t
h
r
e
s
h
o
l
d
m
e
t
h
o
d
A
m
pl
i
t
ude
0
200
400
60
0
80
0
10
00
12
00
14
00
16
00
18
00
20
00
-1
0
-5
0
5
10
D
e
no
i
s
e
d
s
i
g
nal
us
i
n
g
s
o
f
t
-
t
h
r
e
s
ho
l
d
m
e
t
h
o
d
A
m
pl
i
t
ude
0
200
400
60
0
80
0
10
00
12
00
14
00
16
00
18
00
20
00
-1
0
-5
0
5
10
D
e
no
i
s
e
d
s
i
g
n
al
us
i
n
g
s
t
and
ar
d
c
o
m
p
r
o
m
i
s
i
ng
m
e
t
h
o
d
o
f
t
h
e
har
d
-
and
s
o
f
t
-
t
hr
e
s
ho
l
d
A
m
pl
i
t
ude
S
a
m
p
l
e
num
b
e
r
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Re
sea
r
ch of Signal De
-noi
sing Te
ch
niq
ue ba
sed o
n
Wa
velet (Shi
gang
Hu)
5147
decompo
se
d
on 5
scale
s
.
The d
e
-n
oise
d re
sult
s u
s
in
g
four th
re
sh
old rules is
shown in Fi
gure 9.
The
summ
ary of the resu
lts is sho
w
n
in Tabl
e
2.
De-noi
sing
perfo
rman
ce
s vary
with t
he
threshold
rule
s, the
r
e i
s
no
sig
n
ifica
n
t di
ffe
rence
between S
N
R a
n
d
MSE, but
the diffe
ren
c
e
of
smooth
n
e
ss i
s
relatively ap
pare
n
t.
Table 2. De
-n
oisin
g
Performances of th
e Four
Thre
sh
old Ru
les
Rules SNR
MSE
Smoothness
rigrsure
24.3310
0.2553
0.0100
heursure
24.7757
0.2425
4.8270e-0
4
sqtw
olog
24.6553
0.2459
4.7410e-0
4
minimaxi 24.7530
0.2432
0.0037
Figure 9. De-noised si
gnal
usin
g soft-
threshold, soft-thre
shol
d an
d comp
romi
si
ng of
the hard
-
an
d
soft- thre
shol
d method
s
Wavelet de
co
mpositio
n level is one of the im
porta
nt factors whi
c
h
effect the de-noi
sing
perfo
rman
ce
s. Here
wavel
e
t ba
se
s fun
c
tions
of db
4
were u
s
e
d
wi
th soft-th
re
sh
old meth
od
a
n
d
rigrsu
re rul
e
. The influence of wavelet decomp
o
sit
i
on level on noise red
u
cti
on perfo
rma
n
c
e
wa
s studi
ed. It can be see
n
from the Fi
gure
10,
with
the increa
si
ng of wavelet
deco
m
po
sition
level, signal d
e
-noi
sin
g
perf
o
rma
n
ce is i
m
prove
d
obvi
ously.
1
234
56
0
2
4
6
8
10
12
14
16
18
20
22
24
26
Den
ois
i
ng Per
f
o
m
a
n
c
e
w
a
ve
le
t
d
e
c
o
m
p
o
s
it
io
n
le
ve
l
SNR
MSE
Sm
oo
t
h
n
e
s
s
db4 w
a
v
e
l
e
t
,
r
i
g
r
u
r
e
r
u
l
e
,
s
o
f
t
-
t
hr
es
hol
d m
e
t
hod
Figure 10. Influen
ce of Wav
e
let De
comp
osition L
e
vel on De
-noi
sin
g
Perform
a
n
c
e
The
sele
ction
of wavelet i
s
also very
im
por
tant
for
si
gnal
de-noi
si
ng. Wavelets of db
H
were u
s
ed
with cla
s
sic
hard
-
threshol
d and
soft-t
h
re
shol
d met
hod
s and
ri
grsure rul
e
. The
decompo
sitio
n
level wa
s 6
.
Here, the d
e
-noi
se
d re
sult using
different da
ube
ch
ies wavelets
is
sho
w
n in Ta
ble 3. The de-noi
sin
g
effect is ev
alua
ted by three evaluation fu
nction
s. The
de-
noisi
ng effect
using soft-th
reshold i
s
be
tter than
that using ha
rd
-thre
s
hol
d. Different wavelet
s
can
ge
ne
rate
differe
nt de
-noisi
ng
effect
. In the
wavel
e
ts of
db
H, d
b4, db
5,and
db8
are
sup
e
r
ior
to the other wavelets for th
e de-n
o
isi
ng
of heavy sine
signal
with G
aussia
n
white
noise.
0
200
400
60
0
800
1000
1200
140
0
1600
1800
2000
-1
0
0
10
D
e
no
i
s
e
d
s
i
g
n
al
us
i
n
g
r
i
g
s
ur
e
r
u
l
e
A
m
p
lit
u
d
e
0
200
400
60
0
800
1000
1200
140
0
1600
1800
2000
-1
0
0
10
D
e
no
i
s
e
d
s
i
g
n
al
us
i
n
g
h
e
ur
s
u
r
e
r
u
l
e
A
m
p
lit
u
d
e
0
200
400
60
0
800
1000
1200
140
0
1600
1800
2000
-1
0
0
10
Den
o
i
s
ed s
i
g
n
a
l
u
s
i
n
g s
q
t
w
ol
og
r
u
l
e
A
m
p
lit
u
d
e
0
200
400
60
0
800
1000
1200
140
0
1600
1800
2000
-1
0
0
10
D
e
no
i
s
e
d
s
i
g
n
al
us
i
n
g
m
i
ni
m
a
xi
r
u
l
e
Am
p
l
i
t
u
d
e
S
a
mp
l
e
n
u
mb
e
r
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NIKA
Vol. 11, No
. 9, September 201
3: 514
1 – 5149
5148
Table 3. Evaluation Results by usin
g Dif
f
erent Da
ube
chie
s Wavele
ts
Wavelet Method
SNR
MSE
Smoothness
db1
hard-th
reshold
20.5932
0.3881
0.0418
db2
hard-th
reshold
17.8403
0.5329
0.1626
db3
hard-th
reshold
20.5096
0.3919
0.1050
db4
hard-th
reshold
20.2284
0.4048
0.0936
db5
hard-th
reshold
20.3036
0.4013
0.0232
db6
hard-th
reshold
20.2395
0.4043
0.0220
db7
hard-th
reshold
20.1787
0.4071
0.0456
db8
hard-th
reshold
21.6280
0.3445
0.0311
db9
hard-th
reshold
18.9472
0.4691
0.0565
db10
hard-th
reshold
19.4107
0.4447
0.0916
db1 soft-threshold
23.0930
0.2911
0.0107
db2 soft-threshold
24.2863
0.2537
0.0096
db3 soft-threshold
25.0027
0.2336
0.0051
db4 soft-threshold
25.6989
0.2156
0.0038
db5 soft-threshold
25.1663
0.2293
0.0021
db6 soft-threshold
24.5422
0.2463
0.0030
db7 soft-threshold
25.0242
0.2330
0.0027
db8 soft-threshold
25.5575
0.2192
0.0017
db9 soft-threshold
23.7345
0.2703
0.0045
db10
soft-threshold
24.6691
0.2428
0.0054
4. Conclusio
n
Wavelet m
e
thod
usi
ng fo
r de-noi
sing
is an im
po
rtant
aspe
ct of
wa
velet analy
s
is appli
ed
to the a
c
tual.
This a
r
ticle
descri
bed
se
veral
co
mm
o
n
ly used p
r
in
ciple
s
of wav
e
let de
-noi
sin
g
method, an
d achi
eved wavelet de-noi
sin
g
method b
a
sed on thresh
old in the mat
l
ab. The resul
t
s
are as
follows:
1)
Comp
ari
n
g with
FFT f
ourie
r tran
sfo
r
m de
-n
oisi
n
g
, wavelet
d
e
-noi
sin
g
ca
n retai
n
more d
e
tails
of the original
signal a
nd is
more effe
ctive.
2)
Hard
-threshold meth
od
is more ro
ug
h than
othe
r ways,
but sm
oother sig
nal
can
b
e
obtaine
d fro
m
the soft-thresh
old meth
o
d
, and m
any
more detail
s
of
the
sig
nal comp
one
nts can
be re
se
rved, so soft-thre
sh
old method
which i
s
more suitabl
e for m
o
re si
gnal d
e
tail comp
one
n
t
.
3) In
ad
ditio
n
to S
N
R an
d MSE, smo
o
thne
ss is a
n
imp
o
rtant
index to
eval
uate the
perfo
rman
ce
of noise
redu
ction. In som
e
ca
se
s, it
is very difficult to evaluate the effect of de-
noisi
ng corre
c
tly usin
g o
n
ly SNR a
n
d
MSE. The
adopting
of smooth
n
e
s
s will ma
ke t
h
e
evaluation m
o
re comp
reh
ensive.
4) Th
re
shold
rule, wavele
t decom
positi
on le
vel and
wavelet fun
c
tion are all im
portan
t
factors which
impact the de
-noi
sing p
e
rfo
r
man
c
e
s
.
Ackn
o
w
l
e
dg
ements
This wo
rk
wa
s sup
porte
d by
the
Nation
al
Natural
Science Fo
und
ation of
Chin
a (G
ra
nt
No
s 61
2740
77 an
d 612
7402
6), the
Scien
c
e a
n
d
Tech
nolo
g
y Plan Fou
n
d
a
tion of Hun
a
n
Province
(Grant Nos 20
1
2
GK310
3
an
d 20
12GK
3
1
02) and
the
Scientific Re
sea
r
ch F
und
of
Hun
an Provin
cial Edu
c
atio
n Dep
a
rtme
nt (12C010
8).
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