Inter
national
J
our
nal
of
Electrical
and
Computer
Engineering
(IJECE)
V
ol.
10,
No.
3,
June
2020,
pp.
2383
2391
ISSN:
2088-8708,
DOI:
10.11591/ijece.v10i3.pp2383-2391
r
2383
Discr
ete
wa
v
elet
transf
orm-based
RI
adapti
v
e
algorithm
f
or
system
identification
Mohammad
Shukri
Salman
1
,
Alaa
Eleyan
2
,
Bahaa
Al-Sheikh
3
1,3
Colle
ge
of
Engineering
and
T
echnology
,
American
Uni
v
ersity
of
the
Middle
East,
K
uw
ait
2
Electrical
and
Electronic
Engineering,
A
vrasya
Uni
v
ersity
,
T
rabzon,
T
urk
e
y
3
Biomedical
Systems
and
Medical
Informatics
Engineering
Department,
Y
armouk
Uni
v
ersity
,
Jordan
Article
Inf
o
Article
history:
Recei
v
ed
Mar
20,
2019
Re
vised
No
v
5,
2019
Accepted
No
v
25,
2019
K
eyw
ords:
Adapti
v
e
filters
Discrete
w
a
v
elet
transform
LMS
algorithm
Recursi
v
e
in
v
erse
algorithm
System
identification
ABSTRA
CT
In
this
paper
,
we
propose
a
ne
w
adapti
v
e
filtering
algorithm
for
system
identifica-
tion.
The
algorithm
is
based
on
the
recursi
v
e
in
v
erse
(RI)
adapt
i
v
e
algorithm
which
suf
fers
from
lo
w
con
v
er
gence
ra
tes
in
some
applications;
i.e.,
t
he
eigen
v
alue
spread
of
the
autocorrelation
matri
x
is
relati
v
ely
high.
The
proposed
algorithm
applies
discrete-w
a
v
elet
transform
(D
WT)
to
the
input
signal
which,
in
turn,
helps
to
o
v
er
-
come
the
lo
w
con
v
er
gence
rate
of
the
RI
algorithm
with
relati
v
ely
small
step-size(s).
Dif
ferent
scenarios
has
been
in
v
estig
ated
in
dif
ferent
noise
en
vironments
in
system
identification
setting.
Experiments
demonstrate
the
adv
antages
of
the
proposed
D
WT
recursi
v
e
in
v
erse
(D
WT
-RI)
filter
in
terms
of
con
v
er
gence
rate
and
mean-square-error
(MSE)
compared
to
the
RI,
discrete
cosine
transform
LMS
(DCT
-LMS),
discrete-
w
a
v
elet
transform
LMS
(D
WT
-LMS)
and
recursi
v
e-least-squares
(RLS)
algorithms
under
same
conditions.
Copyright
c
2020
Insitute
of
Advanced
Engineeering
and
Science
.
All
rights
r
eserved.
Corresponding
A
uthor:
Mohammad
Shukri
Salman,
Colle
ge
of
Engineering
and
T
echnology
,
American
Uni
v
ersity
of
the
Middle
East,
K
uw
ait.
T
el:
+965
2225
1400,
e
xt:
1765
Email:
mohammad.salman@aum.edu.kw
1.
INTR
ODUCTION
Adapti
v
e
filtering
plays
an
essential
role
in
man
y
signal
processing
appli
cations.
The
least
-mean-
square
(LMS)
algorithm’
s
simplicity
made
it
v
ery
commonly
used
in
adapti
v
e
signal
processing
applications
such
as
system
identification
[1],
ec
ho
cancelation
[2],
channel
equalizat
ion
[3,
4],
and
interference
cancela-
tion
[3,
5].
The
main
dra
wback
of
the
LMS
algorithm
is
that
it
suf
fers
from
lo
w
con
v
er
gence
rate
when
the
input
signal
is
highly
correlated.
Man
y
LMS
v
ariants
[6-10]
had
been
introduced
to
o
v
ercome
the
limitations
of
the
original
LMS
algorithm
and
to
achie
v
e
better
performance
in
terms
of
mean-square-error
(MSE)
and
con
v
er
gence
rate.
T
ransform
domain
v
ariable
step-size
algorithms
are
ef
fecti
v
e
and
performing
well
in
highly
correlated
noise
en
vironment
[7,
11].
It
has
been
sho
wn
in
[12]
that
transforming
the
input
signal
into
another
domain
can
reduce
the
eigen
v
alue
spread
of
input
signal
autocorrelation
matrix
which,
in
turn,
accelerates
the
con
v
er
gence
rate
of
the
adapti
v
e
filters.
The
discrete-w
a
v
elet-transform
LMS
(D
WT
-LMS)
[10]
and
the
discrete-cosine
transform
LMS
(DCT
-LMS)
[13]
algorithms
were
proposed
to
decrease
the
eigen
v
alue
spread
of
the
autocorrelation
matrix
of
t
h
e
input
signal.
Ho
we
v
er
,
due
to
the
fix
ed
step-size
of
the
LMS
algorithm,
these
algorithms
pro
vide
poor
performance
in
terms
of
the
con
v
er
gence
rate.
The
recursi
v
e-least-square
(RLS)
algorithm
w
as
proposed
to
of
fer
f
aster
con
v
er
gence
rate,
especi
ally
,
for
highly
correlated
input
signals
[6,
14,
15].
Ho
we
v
er
,
it
has
a
dis
adv
antage
of
being
highly
computation-
J
ournal
homepage:
http://ijece
.iaescor
e
.com/inde
x.php/IJECE
Evaluation Warning : The document was created with Spire.PDF for Python.
2384
r
ISSN:
2088-8708
ally
comple
x.
The
perform
ance
of
RLS
algorithm
and
its
v
ariants
depends
on
the
for
getting
f
actor
(
)
in
terms
of
con
v
er
gence
rate,
misadjustment,
tracking
capability
and
stability
[16].
The
for
getting
f
actor
can
tak
e
v
alues
between
zero
and
unity
and
needs
to
compromise
between
the
abo
v
e
mentioned
performance
criteria.
When
the
for
getting
f
actor
is
close
to
unity
,
the
algorithm
achie
v
es
lo
w
misadjustment
and
high
stability
,
b
ut
it
compromises
its
tracking
capabilities.
A
smaller
v
alue
of
(
)
w
ould
impro
v
e
the
tracking
capability
of
the
RLS
algorithm
[16],
b
ut
it
w
ould
increase
the
misadjustment
and
might
af
fect
the
stability
of
the
algorithm.
In
[17]
the
authors
in
v
estig
ated
the
influence
of
the
for
getting
f
actor
of
the
RLS
adapti
v
e
filter
in
system
identification.
A
possible
solution
to
o
v
ercome
this
problem
is
to
use
a
v
ariable
for
getting
f
actor
(VFF-RLS)
algorithm
[18,
19].
Recursi
v
e
in
v
erse
(RI)
algorithm
[20,
21]
had
been
proposed
to
o
v
ercome
the
dra
wbacks
and
li
mita-
tions
of
the
abo
v
e
mentioned
adapti
v
e
filters.
It
had
been
sho
wn
that
the
RI
algorithm
performs
considerably
better
than
the
LMS
algorithm
and
its
v
ariants.
It
w
as
also
sho
wn
t
hat
its
performance,
in
terms
of
con
v
er
-
gence
rate
and
e
xcess
MSE,
is
comparable
to
that
of
the
RLS
under
dif
ferent
settings,
with
less
computational
comple
xity
.
Ho
we
v
er
,
still
the
RI
algorithm
requires
a
v
ery
small
initial
v
alue
for
its
step-size
if
the
eigen
v
alue
spread
of
input
signal
autocorrelation
matrix
is
relati
v
ely
high.
In
this
paper
,
we
introduce
a
solution
for
the
aforementioned
problems
by
proposing
a
ne
w
D
WT
-
based
RI
(D
WT
-RI)
algorithm.
Applying
D
WT
on
the
input
signal
[22,
23]
will
reduce
the
eigen
v
alue
spread
of
the
autocorrelation
matrix
[24,
25]
and
gi
v
es
the
user
the
freedom
to
initiate
the
RI
algorithm
with
relati
v
ely
high
initial
v
alues
for
the
st
ep-size.
This
process
guarantees
a
v
ery
high
performance,
of
the
algorithm,
in
terms
of
both
MSE
and
con
v
er
gence
rate.
The
paper
is
or
g
anized
as
follo
ws:
In
Se
ction
2.,
a
brief
e
xplanation
of
discrete
w
a
v
elet
transform
is
co
v
ered.
Section
3.
presents
the
deri
v
ation
of
proposed
D
WT
-RI
algorithm.
In
Section
4.,
simulation
results
that
compare
the
performance
of
the
proposed
algorithm
to
those
of
the
RI,
DCT
-LMS,
D
WT
-LMS
and
RLS
algorithms,
in
dif
ferent
noise
en
vironments
for
system
identification
setting,
are
gi
v
en.
Finally
,
the
conclusions
are
dra
wn
in
the
last
section.
2.
DISCRETE
W
A
VELET
TRANSFORM
The
theory
of
multiresolution
analysis
w
as
firstly
de
v
eloped
by
Mallat
[26],
to
represent
functions
defined
o
v
er
N
dimensional
space.
W
a
v
elet
transform
is
a
multiresolution
technique
for
analyzing
signals.
It
w
as
de
v
eloped,
as
an
alternati
v
e
to
short
time
F
ourier
transform
(STFT),
to
o
v
ercome
the
time-frequenc
y
resolution
problems.
The
D
WT
uses
filter
banks
to
perform
the
w
a
v
elet
analysis
by
the
construction
of
the
multiresolution
time-frequenc
y
plane.
D
WT
decomposes
the
signal
into
w
a
v
elet
coef
ficients
from
which
the
original
signal
can
be
reconstructed
ag
ain.
The
w
a
v
elet
coef
ficients
represent
the
signal
in
v
arious
frequenc
y
bands
[27,
28].
Figure
1
sho
ws
the
structure
of
discrete
w
a
v
elet
transform
adapti
v
e
filter
(D
WT
AF).
Figure
1.
Structure
of
discrete
w
a
v
elet
transform
transv
ersal
adapti
v
e
filter
.
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
3,
June
2020
:
2383
–
2391
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
2385
According
to
D
WT
theory
,
reconstruction
of
the
original
signal
x
(
k
)
can
be
performed
using
the
follo
wing
finite
sum:
x
(
k
)
=
J
1
X
j
=0
X
n
2
Z
j
;n
j
;n
(
k
)
;
(1)
where
j
;n
are
the
w
a
v
elet
coef
ficients
and
j
;n
(
k
)
are
the
w
a
v
elet
functions
that
form
an
orthogonal
basis.
The
purpose
of
D
WT
adapti
v
e
filter
is
to
generate
the
discrete
reconstruction
of
x
j
(
k
)
which
is
the
projecti
v
e
discrete
form
of
x
(
k
)
in
w
a
v
elet
subspace.
x
j
(
k
)
is
gi
v
en
by:
x
j
(
k
)
=
X
n
2
Z
j
;n
j
;n
(
k
)
:
(2)
If
v
j
(
k
)
is
the
approximation
of
projecti
v
e
x
j
(
k
)
,
then
v
j
(
k
)
=
X
n
2
Z
^
j
;n
j
;n
(
k
)
;
(3)
where
^
j
;n
is
the
discrete
approximation
of
the
w
a
v
elet
coef
ficients
j
;n
,
^
j
;n
=
X
l
x
(
l
)
j
;n
(
l
)
:
(4)
Gi
v
en
that,
h
j
(
l
;
k
)
=
X
n
2
Z
j
;n
(
l
)
j
;n
(
k
)
:
(5)
No
w
,
substituting
(4)
and
(5)
in
(3)
results
in
v
j
(
k
)
=
X
l
x
(
l
)
h
j
(
l
;
k
)
:
(6)
Equation
(6)
is
simply
the
discre
te
con
v
olution
of
the
input
signal
x
(
k
)
and
the
filter
coef
ficients
h
j
(
l
;
k
)
.
Using
Orthogonality
and
time-steadiness,
filter
indices
can
be
re
written
as:
h
j
(
l
;
k
)
=
h
j
(
l
k
)
:
(7)
Therefore,
v
j
(
k
)
=
X
l
x
(
l
)
h
j
(
l
k
)
:
(8)
3.
DISCRETE
W
A
VELET
TRANSFORM
RECURSIVE
INVERSE
ALGORITHM
3.1.
Recursi
v
e
in
v
erse
(RI)
algorithm
The
optimum
solution
for
the
FIR
filter
coef
ficients
can
be
obtained
using
the
W
iener
-Hopf
equation
[6]:
C
(
k
)
=
R
1
(
k
)
p
(
k
)
;
(9)
where
k
is
the
time
parameter
(
k
=
1
;
2
;
:
:
:
),
C
(
k
)
is
the
filter
weight
v
ector
calculated
at
time
k
,
R
(
k
)
is
the
estimate
of
the
tap-input
v
ector’
s
autocorrelation
matrix,
and
p
(
k
)
is
the
estimate
of
the
cross-correlation
v
ector
between
the
desired
output
signal
and
the
tap-input
v
ector
.
Recursi
v
e
estimation
of
R
(
k
)
and
p
(
k
)
in
(9)
can
be
obtained
as
follo
ws;
R
(
k
)
=
R
(
k
1)
+
x
(
k
)
x
T
(
k
)
;
(10)
p
(
k
)
=
p
(
k
1)
+
d
(
k
)
x
(
k
)
;
(11)
where
is
the
for
getting
f
actor
which
is
usually
v
ery
close
to
unity
and
x
(
k
)
is
the
tap-input
v
ector
.
Discr
ete
wavelet
tr
ansform-based
RI
adaptive
...
(Mohammad
Shukri
Salman)
Evaluation Warning : The document was created with Spire.PDF for Python.
2386
r
ISSN:
2088-8708
Using
the
recursi
v
e
solution
of
the
W
iener
-Hopf
equation
for
one
iteration
with
v
ariable
step
size
gi
v
es
the
weight
update
equation
of
the
RI
algorithm
[20]:
C
(
k
)
=
[
I
(
k
)
R
(
k
)]
C
(
k
1)
+
(
k
)
p
(
k
)
:
(12)
Where
(
k
)
is
the
v
ariable
step-size
[20]
which
satisfies
the
con
v
er
gence
criterion
[6]:
(
k
)
<
2
max
(
R
(
k
))
=
1
1
k
2
max
(
R
xx
)
=
max
1
k
;
(13)
where
max
is
the
maximum
eigen
v
alue
of
R
(
k
)
and
R
xx
=
E
x
(
k
)
x
T
(
k
)
.
F
or
more
details
about
deri
v
ation
and
con
v
er
gence
analysis
of
the
RI
algorithm,
the
reader
may
refer
to
[20].
The
RI
algorithm
has
a
major
adv
antage
compared
to
the
RLS
algorithm
in
that
it
does
not
require
the
update
of
the
in
v
erse
autocorrelation
matrix.
Also,
its
computational
comple
xity
is
much
less
than
that
of
the
RLS
algorithm.
Due
to
the
update
of
in
v
erse
autocorrelation
matrix,
RLS
type
algorithms
may
f
ace
numerical
stability
problems
[29],
which
is
not
the
case
for
the
RI
algorithm.
3.2.
The
pr
oposed
algorithm
The
performance
of
the
RI
algorithm
can
be
impro
v
ed
further
by
applying
the
D
WT
to
the
input
s
ignal.
In
this
case,
the
weights
update
equation
in
(12)
will
become
as:
C
(
k
+
1)
=
[
I
(
k
)
R
v
v
(
k
)]
C
(
k
)
+
(
k
)
p
v
d
(
k
)
:
(14)
where
R
v
v
(
k
)
=
R
v
v
(
k
1)
+
v
(
k
)
v
T
(
k
)
;
(15)
and
p
v
d
(
k
)
=
p
v
d
(
k
1)
+
d
(
k
)
v
(
k
)
:
(16)
Where
v
(
k
)
=
Wx
(
k
)
is
the
transformed
input
signal
and
W
is
the
w
a
v
elet
transform
matrix
of
size
J
N
.
(
k
)
=
0
1
k
;
(17)
where
0
is
a
constant
selected
as:
0
<
max
=
2(1
)
max
(
R
v
v
)
and
R
v
v
=
E
v
(
k
)
v
T
(
k
)
:
The
adapti
v
e
estimation
error
is
gi
v
en
as:
e
(
k
)
=
d
(
k
)
y
(
k
)
;
(18)
where
y
(
k
)
=
v
T
(
k
)
C
(
k
)
=
J
1
X
j
=0
v
j
(
k
)
c
j
(
k
)
=
J
1
X
j
=0
X
l
c
j
(
k
)
h
j
(
l
k
)
x
(
l
)
;
(19)
where
y
(
k
)
is
the
transformed
v
ersion
of
y
(
k
)
sho
wn
in
Figure
2
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
3,
June
2020
:
2383
–
2391
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
2387
Figure
2.
Block
diagram
of
an
adapti
v
e
system
identification
setting
0
0.5
1
1.5
2
2.5
3
3.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Frequency
Magnitude
Figure
3.
Unkno
wn
system
transfer
function
4.
SIMULA
TION
RESUL
TS
In
the
simulations,
the
performance
of
the
proposed
D
WT
-RI
algorithm
is
compared
t
o
those
of
the
RI
[20],
DCT
-LMS
[9],
D
WT
-LMS
[10]
and
RLS
[6]
algorithms
in
the
system
identification
setting
described
in
Figure
2.
The
filter
length
for
all
algorithms
is
J
=
16
taps
and
the
signal-to-noise
rat
io
(SNR)
w
as
selected
to
be
30
dB
for
all
the
e
xperiments.
The
recei
v
ed
signal
x
(
k
)
w
as
generated
us
ing
the
D
WT
on
the
signal
x
i
(
k
)
gi
v
en
by
[7]:
x
i
(
k
)
=
1
:
79
x
i
(
k
1)
1
:
85
x
i
(
k
2)
+
1
:
27
x
i
(
k
3)
0
:
41
x
i
(
k
4)
+
v
0
(
k
)
:
(20)
where
v
0
(
k
)
is
a
Ga
ussian
process
with
zero
mean
and
v
ariance
2
=
0
:
3849
.
The
unkno
wn
system
is
the
bandpass
filter
sho
wn
in
Figure
3.
The
simulation
results
for
Gaussian
and
impulsi
v
e
noise
were
obtained
by
a
v
eraging
100
and
300
Monte-Carlo
independent
runs,
respecti
v
ely
.
4.1.
Additi
v
e
white
gaussian
noise
In
order
to
test
the
performance
of
the
proposed
algorithm,
the
signal
i
s
assumed
to
be
corrupted
with
an
additi
v
e
white
Gaussian
noise
(A
WGN)
process.
Simulations
were
carried
out
with
the
follo
wing
parame-
ters:
F
or
the
D
WT
-RI
and
RI
algorithms:
=
0
:
99
and
0
=
0
:
0013
.
F
or
DCT
-LMS
algorithm:
=
0
:
9985
,
=
8
10
4
,
=
0
:
02
,
=
2
10
3
and
M
=
10
.
F
or
D
WT
-LMS
algorithm:
=
0
:
02
.
F
or
the
RLS
algorithm:
=
0
:
99
.
Figure
4
sho
ws
that
the
RI,
RLS
and
D
WT
-RI
algorithms
con
v
er
ge
to
the
same
Discr
ete
wavelet
tr
ansform-based
RI
adaptive
...
(Mohammad
Shukri
Salman)
Evaluation Warning : The document was created with Spire.PDF for Python.
2388
r
ISSN:
2088-8708
MSE.
Ho
we
v
er
,
the
D
WT
-RI
and
RLS
con
v
er
ge
much
f
aster
than
the
RI
algorithm
(950
iterations
f
aster).
On
the
other
hand,
the
DCT
-LMS
and
D
WT
-LMS
algorithms
con
v
er
ge
to
a
higher
MSE
(mse=5dB
w
orse)
than
the
other
algorithms
with
lo
w
rate
of
con
v
er
gence.
200
400
600
800
1000
1200
1400
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Iteration
MSE
DWT−RI
RLS
RI
DCT−LMS
DWT−LMS
Figure
4.
The
ensemble
MSE
for
RI,
RLS,
DCT
-LMS,
D
WT
-LMS
and
D
WT
-RI
in
A
WGN
200
400
600
800
1000
1200
1400
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
10
0
Iteration
MSE
RI
DWT−LMS
DWT−RI
RLS
DCT−LMS
Figure
5.
The
ensemble
MSE
for
RI,
RLS,
DCT
-LMS,
D
WT
-LMS
and
D
WT
-RI
in
A
WIN
(
=
0
:
2
,
=
100
)
4.2.
Additi
v
e
white
impulsi
v
e
noise
Due
to
man-made
noise,
underw
ater
acoustic
noise,
atmospheric
noise,
etc,
the
noise
added
to
the
recei
v
ed
signal
may
not
be
modeled
as
Gaussian.
This
type
of
noise
which
has
a
hea
vy-tailed
distrib
ution
is
characterized
by
outliers
and
may
be
better
modeled
using
a
Gaussian
mixture
model.
In
order
to
test
the
rob
ustness
of
the
proposed
algorithm,
and
to
study
the
ef
fects
of
the
impulsi
v
e
components
(outliers)
o
f
the
noise
process
in
the
system
identification
setting,
an
impulsi
v
e
noise
process
is
generated
by
the
probability
density
function
[30]:
f
=
(1
)
G
0
;
2
n
+
G
0
;
2
n
with
v
ariance
2
f
gi
v
en
as:
2
f
=
(1
)
2
n
+
2
n
,
where
G
0
;
2
n
is
a
Gaussian
probability
density
function
with
zero
mean
and
v
ariance
2
n
that
represents
the
nominal
background
noise.
G
0
;
2
n
represents
the
impulsi
v
e
component
of
the
noise
model,
where
is
the
probability
and
1
is
the
strength
of
the
impulsi
v
e
components.
(a)
Firstly
,
the
signal
is
assumed
to
be
corrupted
with
an
additi
v
e
white
impulsi
v
e
noise
(A
WIN)
process.
The
white
impulsi
v
e
noise
process
is
generated
with
the
parameters:
=
0
:
2
and
=
100
.
Simulations
were
carried
out
with
the
follo
wing
parameters:
F
or
the
D
WT
-RI
and
RI
algorithms:
=
0
:
99
and
0
=
0
:
001
.
F
or
DCT
-LMS
algorithm:
=
0
:
9985
,
=
8
10
4
,
=
0
:
02
,
=
2
10
3
and
M
=
10
.
F
or
D
WT
-LMS
algorithm:
=
0
:
018
.
F
or
the
RLS
algorithm:
=
0
:
99
.
Figure
5
sho
ws
that
the
D
WT
-RI
and
RLS
algorithms
con
v
er
ge
to
the
same
MSE
at
the
same
time,
The
RI
and
DCT
-LMS
algorithms
still
con
v
er
ge
to
the
same
MSE
b
ut
with
much
lo
w
rate
of
con
v
er
gence
than
those
of
the
D
WT
-RI
and
RLS
algorithms.
The
D
WT
-LMS
algorithm
con
v
er
ges
to
much
higher
MSE
with
almost
the
same
rate
of
con
v
er
gence
of
those
of
RLS
and
D
WT
-RI
algorithms.
(b)
Secondly
,
in
order
to
emphasiz
e
on
the
capability
of
the
proposed
D
WT
-RI
algorithm
in
suppressing
impulsi
v
e
noise,
e
v
en,
with
high
impulsi
vity
strength,
an
A
WIN
process
is
generated
with
the
parame-
ters:
=
0
:
2
and
=
100000
.
Simulations
were
carried
out
with
the
follo
wing
parameters:
F
or
the
D
WT
-RI
and
RI
algorithms:
=
0
:
99
and
0
=
0
:
0001
.
F
or
DCT
-LMS
algorithm:
=
0
:
9985
,
=
8
10
4
,
=
0
:
02
,
=
2
10
3
and
M
=
10
.
F
or
D
WT
-LMS
algorithm:
=
0
:
018
.
F
or
the
RLS
algorithm:
=
0
:
99
.
The
RI
algorithm
f
ails
to
con
v
er
ge
under
these
settings.
Figure
6
sho
ws
that
the
RLS
algorithm
con
v
er
ges
to
the
steady-state
MSE
(MSE
=
10
dB)
after
400
time
itera-
tions,
while
the
D
WT
-RI
algorithm
con
v
er
ges
to
a
lo
wer
MSE
(MSE
=
14
dB)
than
the
RLS
at
the
same
time.
The
DCT
-LMS
algorithm
con
v
er
ges
to
a
lo
wer
MSE
t
han
that
of
the
D
WT
-RI
algorithm
b
ut
with
v
ery
lo
w
con
v
er
gence
rate,
whereas
the
D
WT
-LMS
al
go
r
ithm
con
v
er
ges
to
a
v
ery
high
MSE.
This
sho
ws
the
adv
antage
of
domain
transform
in
reducing
the
self-correlation
of
the
input
signal
and
con
v
er
ging
to
a
lo
wer
MSE
v
alues
with
higher
con
v
er
gence
rates.
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
3,
June
2020
:
2383
–
2391
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
2389
500
1000
1500
2000
2500
3000
10
−3
10
−2
10
−1
10
0
10
1
Iteration
MSE
DWT−RI
RLS
DCT−LMS
DWT−LMS
Figure
6.
The
ensemble
MSE
for
RI,
RLS,
DCT
-LMS,
D
WT
-LMS
and
D
WT
-RI
in
A
WIN
(
=
0
:
2
,
=
100000
)
5.
CONCLUSION
A
ne
w
discrete
w
a
v
elet
transform
based
RI
adapti
v
e
filtering
algorithm
is
proposed.
The
performa
n
c
es
of
the
D
WT
-RI,
DCT
-LMS,
D
WT
,LMS,
RI
and
RLS
algorithms
are
compared
in
A
WGN
and
A
WIN
en
viron-
ments
in
system
identification
setting.
Under
the
same
conditions,
the
D
WT
-RI
algori
thm
outperforms
the
aforementioned
algorithms
in
terms
of
MSE
and/or
con
v
er
gence
rate.
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ariable
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ariable
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r
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2088-8708
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ariable
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ariable
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(ICCA),
pp.
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2017.
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Mannan
and
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Habib,
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Adapti
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BIOGRAPHIES
OF
A
UTHORS
Mohammad
Shukri
Salman
recei
v
ed
the
B.Sc.,
M.Sc.
and
Ph.D.
De
grees
in
Electrical
and
Electronics
Engineering
from
Eastern
Mediterranean
Uni
v
ersity
(EMU),
in
2006,
2007
and
2011,
respecti
v
ely
.
From
2006
to
2010,
he
w
as
a
teaching
assistant
of
Electrical
and
Ele
ctronics
Engi-
neering
department
at
EMU.
In
2010,
he
has
joined
the
Department
of
Electrical
and
Electronic
Engineering
at
European
Uni
v
ersity
of
Lefk
e
(EUL)
as
a
senior
lecturer
.
F
or
the
period
2011-2015,
he
has
w
ork
ed
as
an
Assist.
Prof.
in
the
Department
of
Electrical
and
Electronics
Engineering,
Me
vlana
(Rumi)
Uni
v
ersity
,
T
urk
e
y
.
Currently
,
he
is
with
Electrical
Engineering
Department
at
the
American
Uni
v
ersity
of
Middle
East
in
K
uw
ait.
He
has
serv
ed
as
a
general
chair
,
program
chair
and
a
TPC
me
mber
for
man
y
international
conferences.
His
re
search
interests
include
signal
processing,
adapti
v
e
filters,
image
processing,
sparse
representation
of
signals,
control
systems
and
communications
systems.
Int
J
Elec
&
Comp
Eng,
V
ol.
10,
No.
3,
June
2020
:
2383
–
2391
Evaluation Warning : The document was created with Spire.PDF for Python.
Int
J
Elec
&
Comp
Eng
ISSN:
2088-8708
r
2391
Alaa
Eleyan
recei
v
ed
the
B.Sc.
and
M.Sc.
de
grees
in
Electrical
Electronics
Engineering
from
Near
East
Uni
v
ersity
,
Northern
Cyprus,
in
2002
and
2004,
respecti
v
ely
.
In
2009,
He
finished
his
PhD
de
gree
in
Electrical
and
Electronics
Engineering
from
Eastern
Mediterranean
Uni
v
ersity
,
Northern
Cyprus.
Currently
,
he
is
w
orking
as
an
associate
professor
at
A
vrasya
Uni
v
ersity
,
T
urk
e
y
.
His
current
research
interests
are
computer
vision,
signal
image
processing,
machine
learning
and
robotics.
He
has
more
than
60
published
journal
articles
and
conference
papers
in
these
research
fields.
Bahaa
Al-Sheikh
recei
v
ed
the
B.Sc.
de
gree
in
Electronics
Engineering
from
Y
armouk
Uni
v
ersity
,
Jordan,
MSc
in
Electrical
Engineering
from
Colorado
State
Uni
v
ersity
,
Colorado,
USA,
and
PhD
in
Biomedical
Engineering
de
gree
from
the
Uni
v
ersity
of
Den
v
er
,
Colorado,
USA,
in
2000,
2005
and
2009,
respecti
v
ely
.
Between
2009
and
2015,
he
w
ork
ed
for
Y
armouk
Uni
v
ersity
as
an
assis-
tant
professor
in
the
department
of
Biom
edical
Systems
and
Medical
Informatics
Engineeri
ng
and
serv
ed
as
the
department
chairman
between
2010
and
2012.
He
serv
ed
as
a
part-tim
e
consultant
for
Sandhill
Scientific
Inc.,
Highlands
Ranch,
Col
orado,
USA
in
Biomedical
Signal
Processing
field
between
2009
and
2014.
Currently
,
he
is
an
Assistant
Professor
at
the
Electrical
Engineering
Department
at
the
American
Uni
v
ersity
of
the
Middle
East
in
K
uw
ait.
His
research
interests
include
digital
signal
and
image
processing,
biomedical
systems
modeling,
medical
instrumentation
and
sound
source
localization
systems.
Discr
ete
wavelet
tr
ansform-based
RI
adaptive
...
(Mohammad
Shukri
Salman)
Evaluation Warning : The document was created with Spire.PDF for Python.