Indonesian
Journal
of
Electrical
Engineering
and
Computer
Science
V
ol.
3,
No
.
2,
A
ugust
2016,
pp
.
410
419
DOI:
10.11591/ijeecs
.v3.i2.pp410-419
410
Using
A
Fuzzy
Number
Err
or
Correction
Appr
oac
h
to
Impr
o
ve
Algorithms
in
Blind
Identification
Elmostafa
Atify
*
,
Cherki
Daoui
,
and
Ahmed
Boumezzough
Labor
ator
y
of
Inf
or
mation
Processing
and
Decision
Suppor
t
F
aculty
of
Sciences
and
T
echniques
P
.B
523
Bni-Mellal
*
e-mail:
at.elmost@gmail.com
Abstract
As
par
t
of
a
detailed
study
on
b
lind
identification
of
Gaussian
channels
,
the
main
pur
pose
w
as
to
propose
an
algor
ithm
based
on
cum
ulants
and
fuzzy
n
umber
approach
in
v
olv
ed
throughout
the
whole
process
of
identification.
Our
objectiv
e
w
as
to
compare
the
ne
w
design
of
the
algor
ithm
to
the
old
one
using
the
higher
order
cum
ulants
,
namely
Alg1,
Algat
and
the
Giannakis
algor
ithm.
W
e
w
ere
ab
le
to
demonstr
ate
that
the
proposed
method
-fuzzy
n
umber
error
correction-
increases
the
perf
or
mance
of
the
algor
ithm
b
y
calculating
the
r
atio
of
squared
errors
of
ALGaT
and
AlgatF
.
The
method
can
be
applied
to
an
y
algor
ithm
f
or
more
impro
v
ement
and
effinciency
.
K
e
yw
or
ds:
Blind
identification,
Fuzzy
n
umber
,
FIR
channel,Gaussien
channel,
cum
ulant.
Cop
yright
c
2016
Institute
of
Ad
v
anced
Engineering
and
Science
1.
Intr
oduction
process
of
identification
has
no
w
become
cr
ucial
and
is
prominent
in
se
v
er
al
fields
,
in-
cluding
astroph
ysics
,
geology
,
data
tr
ansmission,
r
adio
comm
unication,
mobile
r
adio
.
Thanks
to
Y
.Sato
that
the
issue
of
identification
in
its
v
ar
ious
aspects
w
as
r
aised.
It
has
contib
ueted
to
the
resolution
of
man
y
prob
lems
[1].
The
tr
ansmission
of
inf
or
mat
ion
through
a
ph
ysical
medium
ma
y
undergo
se
v
er
al
ph
ysical
alter
ations
or
modifications
,
aff
ecting
the
nature
and
e
v
en
the
direction
of
the
initial
inf
or
mation.
The
y
are
essentially
ph
ysical
phenomena
whose
impact
is
quite
consider
ab
le
on
the
authenticity
of
the
message
induced
b
y
the
inf
or
mation.
Major
e
xamples
include
absor
ption,
refr
actio
n,
re-
flection
or
diffusion.
These
cases
of
impact
gener
ate
a
signal
distor
tion
through
atten
uation
and
interf
erence
betw
een
symbols
(IES).
Moreo
v
er
,
in
digit
al
comm
unication,
such
phenomena
alter-
ing
the
amplitude
and
the
phase
can
be
modeled
b
y
a
m
ulti-path
tr
ansmission
channel
inf
ected
with
a
white
noise
.
Such
a
model
uses
digital
Finite
Impulse
Response
filter
(FIR)
inf
ected
b
y
a
Gaussian
white
noise
[2,
3,
4,
5,
6,
7].
Identification
methods
allo
w
us
to
dete
r
mine
the
channel
impulse
response
of
FIR.
T
ac
k-
ing
into
account
on
related
liter
ature
,
one
can
identify
more
methods
and
identifications
that
can
be
classified
into
three
categor
ies
according
to
resolution
methods
[8]
[9].
This
in
v
olv
es
the
use
of
o
v
ersiz
ed
linear
algebr
aic
systems
,
e
xplicit
solutions
and
solutions
using
cum
ulants
which
are
easy
to
implement
throughly
.
Ho
w
e
v
er
,f
or
the
first
tw
o
methods
of
resolution
there
is
a
clear
ineffi-
ciency
since
minimizing
functions
presents
man
y
neighbor
ing
local
mini
ma
whose
computational
comple
xity
is
big.
The
third
method
of
resolution
using
a
higher
order
cum
ulant
is
also
unreliab
le
and
less
efficient.
It
is
f
or
this
reason
that
w
e
think
action
should
be
tak
en
to
impro
v
e
the
reliability
of
the
identification
through
the
use
of
fuzzy
n
umber
method
in
the
implementation
of
higher
order
cum
ulants
used
in
the
case
of
non-Gaussian
frequency
distr
ib
utions
.
The
fuzzy
n
umber
error
cor-
rection
method
allo
ws
us
to
eliminate
e
xtreme
v
alues
that
ma
y
aff
ect
the
calculation
of
targeted
v
alues
.
Our
goal
is
to
optimiz
e
efficiency
the
processes
of
b
lind
identification
and
the
sensib
le
use
of
higher
order
cum
ulants
whose
v
alue
f
or
a
Gaussian
distr
ib
ution
is
z
er
o
.
Moreo
v
er
,
their
use
in
the
case
of
identification
of
the
noisy
chann
el
b
y
a
Gaussian
white
noise
is
v
er
y
frequent.
Our
Receiv
ed
Apr
il
9,
2016;
Re
vised
J
uly
12,
2016;
Accepted
J
uly
25,
2016
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
411
study
will
f
ocus
on
the
b
lind
identification
of
linear
non-Gaussian
process
adjusted
a
v
er
age
(MA)
[2,
3,
4,
6,
7].
T
o
achie
v
e
this
goal,w
e
will
present
a
typology
of
order
cum
ulants
,
algor
ithm
using
three
and
f
our
cum
ulants
to
impro
v
e
the
perf
or
mance
of
standard
algor
ithms
.
Ne
xt,
w
e
will
implement
the
assumptions
related
to
the
channel
model
(MA)
to
identify
it
with
its
useful
relationships
.
W
e
will
mo
v
e
on
to
the
presentation
of
tw
o
estimation
methods
based
on
cum
ulants
including
Alg1
algor
ithm
[1
0]
et
al,
the
algor
ithm
C
(q,
k)
Giannakis
[3].
Finally
,
w
e
will
pro
vide
a
detailed
presentation
of
our
algor
ithm,
which
will
be
f
ollo
w
ed
b
y
a
sim
ulation
to
compare
and
assess
the
eff
ectiv
eness
of
diff
erent
algor
ithms
presented
in
this
w
or
k.
2.
Model
and
Fundamental
realtionships
2.1.
Hypothesis
and
model
W
e
alw
a
ys
design
a
model
of
a
Sin
gle
Input
and
Single
Output
(SISO)
channel
with
a
m
ultipath
phenomenon,Fig.1,
b
y
using
a
linear
digital
FIR.
The
equation
of
the
finite
diff
erences
model
f
or
the
FIR
mo
ving
a
v
er
age
channel
(Mobile
A
v
er
age:
x(k)
y(k)
z(k)
+
additive noi
se
Canal(
)
n(k)
B
Figure
1.
FIR
Model
channel
MA),
is
represented
b
y
the
f
ollo
wing
[10],
[11],
with
out
noiseless:
Y
(
k
)
=
q
X
j
=0
b
(
j
)
:X
(
k
j
)
;
b
(0)
=
1
(
outnoisel
ess
;
)
(1)
and
with
noise:
Z
(
k
)
=
Y
(
k
)
+
n
(
k
)
(2)
where
X
(
k
)
is
a
non-Gaussian
e
xcitation
inaccessib
le
to
independent
components
an
d
identically
distr
ib
uted
(iid)
with
z
ero
mean,
of
v
ar
iance
2
x
,with
at
least
one
non-z
ero
m
>
2
order
and
chec
king
E
[
X
2
m
(
k
)]
<
1
.
n
(
k
)
is
a
white
Gaussian
noise
,
independent
of
the
input
X
(
k
)
and
unkno
wn
po
w
er
spectr
al
den-
sity
.
B
=
[
b
(0)
;
b
(1)
;
::;
b
(
q
)]
represent
the
impulse
response
of
FIR
channel.
b
(
i
)
are
constant
f
or
a
stationar
y
time-in
v
ar
iant
channel.
q
is
the
order
of
the
channel,
assumed
to
be
kno
wn
[12].
2.2.
Fundamental
Relationships
Related
liter
ature
sho
ws
that
there
are
man
y
impor
tant
algor
ithms
based
on
higher
order
cum
ulant.
In
this
par
ag
r
aph,
w
e
present
the
fundamental
relationships
using
the
cum
ulants
in
the
case
of
a
stationar
y
time-in
v
ar
iant
Gaussian
noisy
channel.
2.2.1.
Moment
and
Cum
ulant
In
this
section,
w
e
presen
t
some
definitions
of
higher
order
sta
tistics
,
moments
and
cu-
m
ulants
.
Let
x(k),
where
1
k
N,
is
a
real
discrete
stationar
y
process
with
N
length,
so
its
moment
of
order
m
is
giv
en
b
y
[3]
[4]
[10]
[5]
M
m;x
(
t
1
;
t
2
;
:::;
t
m
1
)
=
E
f
x
(
k
)
x
(
k
+
t
1
)
x
(
k
+
t
2
)
:::x
(
k
+
t
m
1
)
g
(3)
Fuzzy
Number
F
or
Blind
cum
ulants
Identification
(Elmostaf
a
Atify)
Evaluation Warning : The document was created with Spire.PDF for Python.
412
ISSN:
2502-4752
Where
E
f
:
g
represents
the
mathematical
e
xpectation.
The
cum
ulant
of
order
n
of
a
non-Gaussian
stationar
y
process
is
giv
en
b
y:
C
m;x
(
t
1
;
t
2
;
:::;
t
m
1
)
=
M
m;x
(
t
1
;
t
2
;
:::;
t
m
1
)
M
m;G
(
t
1
;
t
2
;
:::;
t
m
1
)
(4)
This
relationship
sho
ws
the
impor
tance
of
cum
ulants
estimators
relativ
ely
to
the
time
when
it
comes
noise
Gaussian
in
nature
.
2.2.2.
Higher
or
der
cum
ulants
The
most
used
moments
in
pr
actice
are
moment
s
of
order
m
lo
w
er
or
equal
to
5.
In
this
section
w
e
giv
e
the
e
xpression
cum
ulant
based
moments
.
The
giv
en
e
xpressions
are
simplified
in
the
case
of
the
samples
adjusted
to
a
z
ero
mean
(centered
System
C.S
).
The
cum
ulant
of
order
m
=
1
is
giv
en
b
y:
C
1
;x
=
M
1
;x
=
E
f
x
(
k
)
g
:
(5)
is
equal
to
0
f
or
a
z
ero-mean
sample:
centered
sample
.
The
e
xpression
(5)
is
equal
to
0
f
or
a
z
ero-mean
sample:
centered
sample
(
C.S
).
The
cum
ulant
of
order
m
=
2
is
giv
en
b
y:
C
2
;x
(
t
1
)
=
M
2
;x
(
t
1
)
(
M
1
;x
)
2
(6)
In
the
case
of
a
system
(
C.S
)
e
xpression
(6)
becomes:
C
2
;x
(
t
1
)
=
M
2
;x
(
t
1
)
(7)
The
cum
ulant
of
order
m
=
3
is
wr
itten
as:
C
3
;x
(
t
1
;
t
2
)
=
M
3
;x
(
t
1
;
t
2
)
M
1
;x
(
M
2
;x
(
t
1
)+
M
2
;x
(
t
1
t
2
))
+
2(
M
1
;x
)
2
(8)
F
or
a
system
(
C.S
)
e
xpression
(8)
becomes:
C
3
;x
(
t
1
;
t
2
)
=
M
3
;x
(
t
1
;
t
2
)
(9)
F
or
a
system
(
C.S
),
the
cum
ulant
of
order
m
=
3
is
wr
itten
as:
C
4
;x
(
t
1
;
t
2
;
t
3
)
=
M
4
;x
(
t
1
;
t
2
;
t
3
)
M
2
;x
(
t
1
)
M
2
;x
(
t
3
t
2
)
M
2
;x
(
t
2
)
M
2
;x
(
t
3
t
1
)
M
2
;x
(
t
3
)
M
2
;x
(
t
2
t
1
)
(10)
2.2.3.
Brilling
er
and
Rosenb
latt
Equation
The
common
point
of
all
con
v
entional
methods
of
identifying
adjusted
a
v
er
age
(MA)
mod-
els
is
the
use
of
Br
illinger
and
Rosenb
latt
f
or
m
ula
[2]
which,
under
the
abo
v
e
assumptions
is:
C
m;Z
(
1
;
:::;
m
1
)
=
C
m;Y
(
1
;
:::
m
1
)
=
m;x
P
q
i
=0
b
(
i
)
b
(
i
+
1
)
:::b
(
i
+
m
1
)
(11)
F
or
m
=
2
,
the
autocorrelation
is:
C
2
;Z
(
)
=
C
2
;Y
(
)
+
C
2
;N
(
)
(12)
where
C
2
;N
(
)
is
the
autocorrelation
of
the
noise
sk
e
wing
results
and
C
2
;Y
(
)
is
the
autocorrela-
tion
of
the
non-noisy
signal
e
xpressed
b
y:
C
2
;Y
(
)
=
2
;x
q
X
i
=0
b
(
i
)
b
(
i
+
)
;
(
2
;x
=
2
x
)
(13)
IJEECS
V
ol.
3,
No
.
2,
A
ugust
2016
:
410
419
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
413
According
to
(11),
one
can
easily
demonstr
ate
that
the
order
cumm
ulants
m
et
n
,
with
(
m
>
n
),
meet
the
f
ollo
wing
relationship:
P
q
i
=0
b
(
i
)
C
m;Y
(
i
+
1
;
::i
+
n
1
;
n
;
:::;
m
1
)
=
"
m;n
P
q
i
=0
b
(
i
)
h
Q
m
1
j
=
n
b
(
i
+
j
)
i
C
n;Y
(
i
+
1
;
::i
+
n
1
)
(14)
Where
"
m;n
=
m;x
n;x
.
This
gener
al
equation
estab
lishes
se
v
er
al
basic
algor
ithms
and
will
also
be
the
basis
of
our
proposed
algor
ithm.
2.3.
Based
unique
or
der
cumm
ulants
algorithms
The
algor
ithms
based
only
on
higher
order
cumm
ulants
are
interesting
when
the
pro-
cessed
signal
is
contaminated
b
y
an
additiv
e
Gaussian
noise
.
Indeed
cumm
ulants
of
higher
or
equal
to
three
orders
of
a
Gaussian
distr
ib
ution
is
z
ero
.
2.3.1.
Algorithm
Based
on
4th
Or
der
Cum
ulant
using
equations
2q
+1:
Alg1
F
rom
equation
(11)
The
matr
ix
f
or
m
of
the
algor
ithm
is
giv
en
b
y
Alg1
[11]
0
B
B
B
B
B
B
B
B
B
B
B
@
0
0
C
4
;y
(
q
;
q
;
0)
.
.
.
.
.
.
.
.
.
0
C
4
;y
(
q
;
q
;
0)
C
4
;y
(
q
;
q
;
q
)
.
.
.
.
.
.
0
.
.
.
C
4
;y
(
q
;
q
;
q
)
0
0
1
C
C
C
C
C
C
C
C
C
C
C
A
0
B
B
B
B
B
B
B
B
B
B
@
1
b
2
(
q
)
.
.
.
b
3
(
i
)
b
2
(
q
)
.
.
.
b
3
(
q
)
b
2
(
q
)
1
C
C
C
C
C
C
C
C
C
C
A
=
0
B
B
B
B
B
B
@
C
4
;y
(0
;
0
;
q
)
.
.
.
C
4
;y
(0
;
0
;
0)
.
.
.
C
4
;y
(0
;
0
;
q
)
1
C
C
C
C
C
C
A
(15)
in
a
more
compact
f
or
m,
the
system
of
equations
(15)
can
be
wr
itten
as
f
ollo
ws:
M
b
q
=
d
(16)
with
M
,
and
h
q
are
defined
in
the
equation
system
(15).
The
solution
in
the
sense
of
least
squares
,
LS
,
of
the
system
of
equation
(16)
is
giv
en
b
y:
b
h
(
q
)
=
(
M
T
M
)
1
M
T
d
(17)
this
solution
giv
es
us
an
estimate
of
the
quotient
of
par
ameters
b
3
(
i
)
and
b
3
(
q
)
,
b
y:
h
q
(
i
)
=
\
b
3
(
i
)
b
3
(
q
)
;
i
=
1
;
:::;
q
:
(18)
So
,
to
estimate
the
par
ameters
b
b
(
i
)
,
i
=
1
;
:::;
q
w
e
proceed
as
f
ollo
ws:
The
par
ameters
b
(
i
)
f
or
i
=
1
;
:::;
q
1
are
estimated
from
estimates
of
b
h
q
(
i
)
v
alues
using
the
f
ollo
wing
equation:
b
b
(
i
)
=
sig
n
h
b
h
q
(
i
)(
b
h
q
(
q
))
2
i
f
abs
(
b
h
q
(
i
))(
b
h
q
(
q
))
2
g
1
=
3
(19)
a
v
ec
sig
n
(
x
)
=
8
<
:
1
;
if
x
>
0;
0
;
if
x
=
0;
1
;
if
x
<
0
:
and
abs
(
x
)
=
j
x
j
indicates
the
absolute
v
alue
of
x.
Fuzzy
Number
F
or
Blind
cum
ulants
Identification
(Elmostaf
a
Atify)
Evaluation Warning : The document was created with Spire.PDF for Python.
414
ISSN:
2502-4752
The
par
ameter
b
b
(
q
)
is
estimated
as
f
ollo
ws:
b
b
(
q
)
=
1
2
sig
n
h
b
h
q
(
q
)
i
8
<
:
abs
(
b
h
q
(
q
))
+
1
b
h
q
(1)
!
1
=
2
9
=
;
(20)
2.3.2.
Algorithme
’C(q,k)’
of
Giannakis
F
rom
(11),
Giannakis
sho
w
ed
that
the
coefficients
(FIR)
can
be
e
xpressed
b
y
the
f
ollo
wing
f
or
m
ula:
b
(
)
=
C
m;Y
(
q
;
;
0
;
:::;
0)
C
m;Y
(
q
;
0
;
:::;
0)
(21)
with
=
0,...,q
and
the
cum
ulant
of
order
m
of
e
xcitation
is:
m;x
=
C
2
m;Y
(
q
;
0
;
:::;
0)
C
m;Y
(
q
;
q
;
:::;
0)
(22)
F
or
m
=
3,
w
e
ha
v
e:
b
(
)
=
C
3
;Y
(
q
;
)
C
3
;Y
(
q
;
0)
et
3
;x
=
C
2
3
;Y
(
q
;
0)
C
3
;Y
(
q
;q
)
2.4.
Pr
oposed
Algorithm
In
this
section
the
impulse
response
B
=
[
b
(0)
;
b
(1)
;
:::;
b
(
q
)]
is
proposed
to
estimate
a
q
order
RIF
channel
using
an
algor
ithm
that
combines
cum
ulants
of
order
3
and
4,
as
a
pre
viously
proposed
h
ypothesis
.
It
also
e
xplains
the
method
that
impro
v
es
the
proposed
algor
ithm.
2.4.1.
General
equation
Equation
(14)
is
tr
ansf
or
med
into
an
equation
which
links
m
and
n
such
that
m
=
n
+
1
as
f
ollo
wing:
P
q
i
=0
b
(
i
)
C
m;Y
(
i
+
1
;
::i
+
n
1
;
n
)
=
"
m;n
P
q
i
=0
b
(
i
)
b
(
i
+
n
)
C
n;Y
(
i
+
1
;
::i
+
n
1
)
(23)
2.4.2.
Appr
oac
h
combining
3
and
4
cum
ulants
or
der
Especially
m
=
4
et
n
=
3
,
Equation
(23)
becomes:
P
q
i
=0
b
(
i
)
C
4
;Y
(
i
+
1
;
i
+
2
;
3
)
=
"
4
;
3
P
q
i
=0
b
(
i
)
b
(
i
+
3
)
C
3
;Y
(
i
+
1
;
i
+
2
)
(24)
W
e
tak
e
1
=
2
=
q
et
3
=
,
the
equation
(24)
becomes:
q
X
i
=0
b
(
i
)
C
4
;Y
(
i
+
q
;
i
+
q
;
)
=
"
4
;
3
q
X
i
=0
b
(
i
)
b
(
i
+
)
C
3
;Y
(
i
+
q
;
i
+
q
)
(25)
giv
en
that
C
4
;Y
(
1
;
2
;
3
)
=
C
3
;Y
(
1
;
2
)
=
0,
si
i
>
q
;
the
equation
(25)
becomes:
b
(0)
C
4
;Y
(
q
;
q
;
)
=
"
4
;
3
b
(0)
b
(
)
C
3
;Y
(
q
;
q
)
(26)
W
e
deduce:
b
(
)
=
C
4
;Y
(
q
;
q
;
)
"
4
;
3
C
3
;Y
(
q
;
q
)
(27)
with
"
4
;
3
=
4
;x
3
;x
(28)
IJEECS
V
ol.
3,
No
.
2,
A
ugust
2016
:
410
419
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
415
According
to
equation
(22),
w
e
deduce:
"
4
;
3
=
C
2
4
;Y
(
q
;
0
;
0)
C
4
;Y
(
q
;
q
;
0)
C
3
;Y
(
q
;
q
)
C
2
3
;Y
(
q
;
0)
(29)
then
b
(
)
=
C
4
;Y
(
q
;
q
;
0)
C
2
4
;Y
(
q
;
0
;
0)
C
2
3
;Y
(
q
;
0)
C
3
;Y
(
q
;
q
)
C
4
;Y
(
q
;
q
;
)
C
3
;Y
(
q
;
q
)
(30)
2.4.3.
AlgatF
The
reduction
of
n
umer
ical
calculations
and
the
perf
or
mance
of
the
used
statistical
es-
timator
can
be
a
source
of
some
div
ergence
of
v
alues
compared
to
the
tr
ue
v
alue
.
T
o
minimiz
e
these
error
diff
erences
sign
w
e
will
also
proposes
a
selectiv
e
choice
of
estimated
v
alues
of
im-
pulse
responses
from
the
pre
vious
algor
ithms
in
the
f
ollo
wing
f
or
mat:
Since
each
calculated
v
alue
is
accompanied
b
y
an
error
,
it
is
theref
ore
considered
as
a
fuzzy
n
umber
[13]
defined
b
y
an
inter
v
al
in
the
set
R
b
y
the
f
ollo
wing
figure
,Fig.2:
x
x
x
x
x
Figure
2.
Fuzzy
n
umber
representation
Fig.3,
represents
fuzzy
v
alues
obtained
b
y
iter
ativ
e
sim
ulation.
Fuzzy
v
alues
ma
y
be
intersecting
or
not.
W
e
remo
v
ed
fuzzy
e
xtreme
v
alues
ha
ving
a
z
ero
intersection
with
the
other
fuzzy
v
alues
.
Indeed,
these
fuzzy
v
alues
are
f
ar
from
the
tr
ue
v
alue
.
Note
that
the
n
umber
of
fuzzy
x
x
x
x
x
Figure
3.
Representation
of
a
fuzzy
n
umber
of
estimated
ser
ies
v
alues
,
remaining
after
remo
v
al
of
the
end
m
ust
be
g
reater
than
at
least
half
of
the
iter
ations
.
AlgatF
is
the
method
of
selection
applied
on
ALGaT
giv
en
that
the
fuzzy
v
ar
iab
le
is
se-
lected
b
y:
B
=
q
X
i
=0
b
(
i
)
(31)
where
2x
B
is
the
siz
e
fuzzy
inter
v
al.
The
sum
is
f
ed
to
remo
v
e
the
div
ergence
due
to
the
undesired
occurrence
of
the
min
us
sign
in
one
of
the
component
of
the
impulse
response
.
3.
Sim
ulation
In
this
sim
ulation,
w
e
tak
e
100
iter
ations
and
each
time
a
ne
w
sample
is
tak
en
b
y
a
noisy
Gaussian
noise
with
z
ero
mean.
The
diff
erent
algor
ithms
pro
vide
estimates
f
or
the
same
samples
in
siz
es
400,
800
and
1200
respectiv
ely
.
T
o
compare
the
samples
using
the
mean
square
error
defined
as
f
ollo
ws:
E
QM
=
1
q
+
1
q
X
i
=0
(
b
(
i
)
h
(
i
))
2
(32)
Fuzzy
Number
F
or
Blind
cum
ulants
Identification
(Elmostaf
a
Atify)
Evaluation Warning : The document was created with Spire.PDF for Python.
416
ISSN:
2502-4752
Consider
the
channel,
non-minim
um
phase
(there
is
a
z
ero
of
the
tr
ansf
er
function
outside
the
unit
circle),
figure
(4)
belo
w
,
ha
ving
the
impulse
response
H
=
[1
1
;
083
0
;
95
0
;
95]
.
The
0
2
−1
1
−1.5
−0.5
0.5
1.5
0
−1
1
−1.2
−0.8
−0.6
−0.4
−0.2
0.2
0.4
0.6
0.8
1.2
Axe réel
Axe des imaginaires
Pôles
Zéros
Zéros et pôles de transmission
Figure
4.
The
z
eros
and
poles
f
or
channel
par
ag
r
aphs
belo
w
summar
iz
e
sim
ulation
on
channel
1
f
or
the
v
ar
ious
algor
ithms
presented
abo
v
e
in
the
case
where
the
noise
signal
to
noise
r
atio
SNR
equal
to
10
dB
and
in
case
SNR
equal
to
20
dB
.
With
S
N
R
=
10
Log
10
2
y
2
br
uit
!
(33)
where
2
i
is
the
standard
de
viation
of
the
statistical
distr
ib
ution
(
i
).
Giv
en
that
h
(1)
=
1.
The
least
precise
v
alue
of
h
(
i
)
comes
three
significant
digits
.
Our
choice
of
the
error
on
B
is
also
to
3
significant
fig
ures
in
the
f
ollo
wing
is
tak
en
into
sim
ulation
B
=
0
;
03
.
3.1.
Case
SNR
is
10
dB
The
f
ollo
wing
tab
le
summar
iz
es
the
results
obtained
f
or
the
proposed
channel,
the
f
our
algor
ithms
namely
Alg1,
Alg
of
Gianakis
,
AlgaT
and
AlgatF
.
The
AlgaT
corrected
b
y
the
proposed
selection
method
in
case
S
N
R
=
10
dB
.
The
descr
iptiv
e
data
tab
le
(1),
allo
ws
us
to
see
a
clear
impro
v
ement
of
EQM.
Indeed,
f
or
a
sample
siz
e
of
400,
w
e
note
that
the
proposed
method
ensures
amelior
ation,
EQM
b
y
a
f
actor
of
2.
In
addition,
the
800
sample
reaches
a
f
act
or
of
about
5.1,
more
than
doub
le
.
This
f
actor
will
increase
and
reach
about
20
in
the
case
of
the
siz
e
of
the
sample
1200
f
or
the
same
method.
This
increase
ensures
thereb
y
a
minimizing
of
the
EQM
v
ersus
other
algor
ithm.
The
impro
v
ement
associated
with,
according
to
the
v
ar
iab
le
fuzzifier
,
the
method
of
the
prob
lem
of
the
sign
is
remar
kab
le
and
is
also
the
e
xample
of
the
estimated
h
4
AlgatF
b
y
the
sample
siz
e
to
800.
This
error
sign
is
corrected
b
y
AlgatF
.
Thus
the
cr
iter
ion
of
choice
is
cr
ucial
and
e
v
en
decisiv
e
in
impro
ving
the
div
ergence
of
the
calculation.
Note
at
this
le
v
el
the
e
xample
of
the
estimated
h
4
b
y
Alga
T
at
AlgatF
f
or
1200.
According
to
the
sample
siz
e
Figure
5,
w
e
note
that
the
cur
v
e
coincides
perf
ectly
with
the
AlgatF
ideal
channel
cur
v
e
.
IJEECS
V
ol.
3,
No
.
2,
A
ugust
2016
:
410
419
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
417
10
−1
10
0
2.0x10
−1
4.0x10
−1
6.0x10
−1
8.0x10
−1
0
−20
20
−30
−10
10
Normalized Frequency (x %pi rad/sample)
Magnitude (dB)
Magnitude
10
−1
10
0
2.0x10
−1
4.0x10
−1
6.0x10
−1
8.0x10
−1
0
−200
200
−100
100
−150
−50
50
150
Normalized Frequency (x %pi rad/sample)
Phase (Degrees)
Ideal chanel
ALG 1 Chanel
Gianakis Chanel
ALGaT Chanel
ALGaTF Chanel
Phase
Figure
5.
N
=
1200
and
SNR
=
10
dB
Magnitude
and
phase
representation
10
−1
10
0
2.0x10
−1
4.0x10
−1
6.0x10
−1
8.0x10
−1
0
−20
−30
−10
10
−25
−15
−5
5
Normalized Frequency (x %pi rad/sample)
Magnitude (dB)
Magnitude
10
−1
10
0
2.0x10
−1
4.0x10
−1
6.0x10
−1
8.0x10
−1
0
−200
200
−100
100
−150
−50
50
150
Normalized Frequency (x %pi rad/sample)
Phase (Degrees)
Ideal chanel
ALG 1 Chanel
Gianakis Chanel
ALGaT Chanel
ALGaTF Chanel
Phase
Figure
6.
N
=
800
and
SNR
=
20
dB
Fuzzy
Number
F
or
Blind
cum
ulants
Identification
(Elmostaf
a
Atify)
Evaluation Warning : The document was created with Spire.PDF for Python.
418
ISSN:
2502-4752
T
ab
le
1.
Estimed
v
alues
of
h
i
f
or
SNR
=
10
dB
and
N
=
400,800,1200
Algor
ithms/Sample
siz
e
h
1
h
2
h
3
h
4
EQM
Ideal
channel
1
-1,083
-0,95
0,95
0
Alg1/400
1
-0,346
-0,253
0,553
0.488
AlgGianakis/400
1
2,119
-1,724
1,984
1,544
AlgaT/400
1
-0,240
-0,494
0,220
0,539
AlgatF/400
1
-0,672
-0,592
0,699
0,268
Alg1/800
1
-0,610
-0,434
0,734
0,328
AlgGianakis/800
1
-0,555
0,155
-0,672
0,909
AlgaT/800
1
-0,227
-0,342
-0,226
0,706
AlgatF/800
1
-1,317
-0,833
1,116
0,139
Alg1/1200
1
-0,904
-0,458
0,917
0,235
AlgGianakis/1200
1
1,150
-2,407
2,539
1,388
AlgaT/1200
1
-0,615
-1,566
3,554
1,215
AlgatF/1200
1
-1.121
-0.874
0,847
0,060
3.2.
Case
SNR
is
20
dB
The
obtained
results
are
summar
iz
ed
in
T
ab
le
2,
f
or
channel
1,
f
or
the
f
our
ALG1
algo-
r
ithms
Alg
of
Gianakis
,
ALGaT
and
ALGatF
the
ALGa
T
corrected
b
y
the
selection
method
pro-
posed
in
case
S
N
R
=
20
dB
.
T
ab
le
2.
Estimed
v
alues
of
h
i
f
or
SNR
=
20
dB
and
N
=
400,800,1200
Algor
ithms/Sample
siz
e
h
1
h
2
h
3
h
4
EQM
Ideal
channel
1
-1,083
-0,95
0,95
0
Alg1/400
1
-0,417
-0,324
0,562
0,444
AlgGianakis/400
1
1,0773
-1,846
0,467
1,068
AlgaT/400
1
-1,340
-0,970
1,734
0,369
AlgatF/400
1
-0,986
-0,966
1,005
0,051
Alg1/800
1
-0,724
-0,562
0,845
0,242
AlgGianakis/800
1
-0,206
-0.080
0,045
0,685
AlgaT/800
1
-0,751
-0,720
1,063
0,188
AlgatF/800
1
-1,085
-1,040
0,871
0,053
Alg1/1200
1
-0,874
-0,487
0,961
0,227
AlgGianakis/1200
1
-0,055
-0,233
-0,253
0,777
AlgaT/1200
1
-2,572
-1,058
2,962
1,121
AlgatF/1200
1
-1,284
-0,935
1,177
0,136
According
to
the
descr
iptiv
e
data
T
ab
le
2,
w
e
can
also
see
a
big
impro
v
ement
in
the
EQM
f
or
SNR
=
20
dB
.
Indeed,
f
or
400
the
siz
e
sample
,
one
notes
that
the
proposed
method
ensures
the
impro
v
ement
of
the
EQM
of
a
f
actor
7
:
3
.
Fur
ther
more
,
the
sample
reaches
800
orders
of
a
f
actor
3
:
5
,
more
than
doub
le
.
This
f
actor
will
increase
and
reach
the
8
:
2
in
the
case
of
the
sample
siz
e
1200.
This
increase
also
ensures
minimization
of
the
EQM
v
ersus
other
algor
ithm.
The
impro
v
ement
asso-
ciated
with
the
so
called
prob
lem
of
the
method
ensures
good
con
v
ergence
to
the
tr
ue
v
alues
of
the
impulse
response
.
Figure
6
sho
w
that
the
cur
v
es
of
AlgatF
coincides
perf
ectly
with
the
cur
v
e
of
the
ideal
channel.
IJEECS
V
ol.
3,
No
.
2,
A
ugust
2016
:
410
419
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
419
4.
Conc
lusion
Se
v
er
al
b
lind
identification
algor
ithms
based
on
higher
order
cum
ulants
are
usually
used.
Among
them
three
e
xamples
Alg1,
Alg
of
Gi
annakis
and
AlgaT
,
w
ere
selected
.
W
e
applied
the
method
based
on
the
concept
of
fuzzy
n
umber
on
the
latter
one
to
obtain
the
corrected
algor
ithm
AlgatF
.
In
sim
ulation
,
w
e
considered
a
non-minim
um
phase
channel
and
the
estimated
impulse
response
of
100
iter
ations
f
or
SNR
of
about
10
dB
and
20
dB
f
or
the
v
ar
ious
algor
ithms
.
W
e
w
ere
ab
le
to
demonstr
ate
that
the
proposed
method
increases
the
perf
or
mance
of
the
algor
ithm
b
y
calculating
the
r
atio
of
squared
errors
of
ALGaT
and
AlgatF
.
The
method
can
be
applied
to
an
y
algor
ithm
f
or
more
impro
v
ement
and
efficiency
.
F
or
future
research,
w
e
intend
to
test
the
eff
ect
of
the
method
on
a
small
n
umber
of
iter
ations
so
as
to
minimiz
e
the
e
x
ecution
time
of
the
algor
ithms
.
Ref
erences
[1]
Y
.Sato
,
“A
method
of
self-reco
v
er
ing
equalizatio
n
f
or
m
ultile
v
el
amplitude-modulation
sys-
tems
,
”
IEEE
tr
ansaction,
comm
,
v
ol.
23,
no
.
6,
pp
.
pp
679
–
682,
J
une
1975.
[2]
D
.
Br
illinger
and
L.
Rosenb
latt,
Computation
and
inter
pretation
of
kth
order
spectr
a
.
Spectr
al
Analysis
of
Times
Signals
,
Ne
w
Y
or
k
:
Wile
y
,
1967.
[3]
G.
Giannakis
and
A.
Sw
ami,
“Higher
order
statistics
,
”
Else
vier
Science
Pub
l
,
1997.
[4]
M.
I.BADI,
E.A
TIFY
and
S
.SAFI,
“Blind
identification
of
tr
ansmission
channel
with
the
method
of
higher-order
cumm
ulants
,
”
Inter
national
Jour
nal
of
Adv
ances
in
Science
and
T
echnology
,
v
ol.
6,
no
.
3,
2013.
[5]
E.
K.Abid-Mer
iam
and
F
.Loubaton,
“Predection
erreur
method
f
or
second-ordre
b
lind
identifi-
cation,
”
IEEE
T
r
ansaction,Signal,
Processing
,
v
ol.
45,
no
.
3,
pp
.
pp
694
–
705,
1997
-
March.
[6]
S
.
Safi
and
A.
Zeroual,
“Blind
par
ametr
ic
identification
of
linear
stochastic
non
gaussian
fir
systems
using
higher
order
cum
ulants
,
”
Inter
national
Jour
nal
of
Systems
Sciences
T
a
ylor
F
r
ancis
,
Signal
Processing
,
v
ol.
44,
no
.
15,
pp
.
pp:855–867,
2004.
[7]
X.
D
.
Zhang
and
Y
.
S
.
Zhang,
“Fir
system
identification
using
higher
order
statistics
alone
,
”
IEEE
T
r
ansactions
,
Signal
Processing
,
v
ol.
42,
no
.
12,
pp
.
pp:2854
–
2858,
1994.
[8]
D
.
Dembl,
“Identification
du
modle
ar
ma
lineaires
l’aide
de
statistiques
d’ordres
ele
vs
.
appli-
cation
l’egalisation
a
v
eugle
,
”
Ph.D
.
disser
tation,
J
uillet-1995.
[9]
G.F
a
vier
,
“Identification
de
modles
par
amtr
iques
ar
,ma
et
ar
ma
a
v
ec
des
statistiques
d’ordres
supr
ieur
et
analyse
des
prf
or
mances
,
”
GRETSI
,
pp
.
pp:137
–
140,
Septembre-1993.
[10]
I.
Badi,
M.
Boutalline
,
S
.
Safi,
and
B
.
Bouikhalene
,
“Blind
identification
and
equalization
of
channel
based
on
higher-order
cumm
ulants:
Application
of
mc-cdma
systems
,
”
in
Multimedia
Computing
and
Systems
(ICMCS),
2014
Inter
national
Conf
erence
on
,
Apr
il
2014,
pp
.
800–
807.
[11]
A.
S
.Safi,
“Ma
system
identification
using
higher
ordre
cum
ulants
applications
to
modelling
solar
r
adiation,
”
Jour
nal
of
Statistical
Computation
and
Sim
ulation
,
v
ol.
72,
no
.
7,
pp
.
pp
533
–
548,
2002.
[12]
A.
S
.Alsh
ebeili
and
F
.
Cetin,
“Cum
ulant
based
identification
approaches
f
or
minim
um
phase
fir
system,
”
IEEE
T
r
ansaction,
Signal
Processing
,
v
ol.
41,
no
.
4,
pp
.
pp
1576
–
1588,
1993
-
Apr
.
[13]
A.
N.
Gani
and
S
.
N.
M.
Assar
udeen,
“An
algor
ithmic
approach
of
solving
fuzzy
linear
system
using
f
our
ier
motzkin
elimination
method,
”
Adv
ances
in
Fuzzy
Sets
and
Systems
,
v
ol.
10,
no
.
2,
pp
.
pp:95
–
109,
2011.
Fuzzy
Number
F
or
Blind
cum
ulants
Identification
(Elmostaf
a
Atify)
Evaluation Warning : The document was created with Spire.PDF for Python.