Indonesian
Journal
of
Electrical
Engineering
and
Computer
Science
V
ol.
2,
No
.
2,
Ma
y
2016,
pp
.
334
343
DOI:
10.11591/ijeecs
.v2.i2.pp334-343
334
Extension
of
Linear
Channels
Identification
Algorithms
to
Non
Linear
Using
Selected
Or
der
Cum
ulants
Mohammed
Zidane
1,*
,
Said
Safi
2
,
and
Mohamed
Sabri
1
1
Depar
tment
of
Ph
ysics
,
F
aculty
of
Sciences
and
T
echnology
,
Sultan
Moula
y
Slima
ne
Univ
ersity
,
Morocco
2
Depar
tment
of
Mathematics
and
Inf
or
matics
,
P
olydisciplinar
y
F
aculty
,
Sultan
Moula
y
Sliman
e
Univ
ersity
,
Morocco
*
corresponding
author
,
e-mail:
zidane
.ilco@gmail.com
Abstract
In
this
paper
,
w
e
present
an
e
xtension
of
linear
comm
unication
channels
identification
algor
ithms
to
non
linear
channels
using
higher
order
cum
ulants
(HOC).
In
the
one
hand,
w
e
de
v
elop
a
theoretical
analysis
of
non
linear
quadr
atic
systems
using
second
and
third
order
cum
ulants
.
In
the
other
hand,
the
relationship
linking
cum
ulants
and
the
coefficients
of
non
linear
channels
presented
in
the
linear
case
is
e
xtended
to
the
gener
al
case
of
the
non
linear
quadr
atic
systems
identification.
This
theoretical
de
v
elopment
is
used
to
de
v
elop
three
non
linear
algor
ithms
based
on
third
and
f
our
th
order
cum
ulants
respectiv
ely
.
Numer
ical
sim-
ulation
results
e
xample
sho
w
that
the
proposed
methods
ab
le
to
estimate
the
impulse
response
par
ameters
with
diff
erent
precision.
K
e
yw
or
ds:
Higher
order
cum
ulants
,
Blind
identification,
Non
linear
quadr
atic
systems
.
Cop
yright
c
2016
Institute
of
Ad
v
anced
Engineering
and
Science
1.
Intr
oduction
Applications
of
higher
order
cum
ulants
theo
r
y
in
the
b
lind
identification
domain
are
widely
used
in
v
ar
ious
w
or
ks
[1]-[3],
[6,
7].
Se
v
er
al
models
are
identified
in
the
liter
ature
such
as
the
lin-
ear
and
non
linear
systems
.
In
the
par
t
of
the
linear
case
,
w
e
ha
v
e
impor
tant
results
estab
lished
that
the
b
lind
identification
is
possib
le
only
from
the
second
order
cum
ulants
(autocorrelation
func-
tion)
of
the
output
stationar
y
signal,
b
ut
t
hese
methods
is
not
ab
le
to
identify
correctly
the
channel
models
e
xcited
b
y
non
Gaussian
signal
and
aff
ected
b
y
Gaussian
noise
,
because
the
additiv
e
Gaussian
noise
will
be
v
anish
in
the
higher
order
cum
ulants
domain.
The
sensitivity
of
the
second
order
cum
ulants
to
the
additiv
e
Gaussian
noise
appealed
to
other
b
lind
identification
methods
e
x-
ploiting
the
cum
ulants
of
order
super
ieur
than
tw
o
[4,
5].
There
are
se
v
er
al
motiv
ations
behind
this
interest,
first
the
methods
based
on
HOC
are
b
lind
to
an
y
kind
of
a
Gaussian
process
,
whereas
autocorrelation
function
(second
order
cum
ulants)
is
not.
Consequently
,
cum
ulants
based
meth-
ods
boost
signal
to
noise
r
atio
(SNR)
when
signals
are
corr
upted
b
y
Gaussian
measurement
noise
.
Second,
the
HOC
methods
are
useful
in
identifying
non
minim
um
phase
systems
and
in
reconstr
ucting
non
minim
um
phase
signals
when
the
signals
are
non
Gaussian.
The
linear
models
are
not
efficient
f
or
representing
and
modeling
all
systems
,
because
the
major
ity
of
systems
are
represented
b
y
non
linear
models
[7,
8].
Ho
w
e
v
er
,
wh
en
linear
modeling
of
the
channe
l
is
not
adequate
,
the
non
linear
modeling
appeared
lik
e
an
alter
nativ
e
efficient
solution
in
most
real
cases
.
Moreo
v
er
,
quadr
atic
non
linear
systems
are
widely
used
in
v
ar
ious
engineer
ing
fields
such
as
signal
processing,
system
filter
ing,
predicting,
identification
and
equalization
[9,
10].
In
this
contr
ib
ution,
firstly
w
e
present
a
theoretical
de
v
elopment
of
non
linear
quadr
atic
systems
using
higher
order
cum
ulants
.
Indeed,
w
e
de
v
elop
the
relationship
linking
third
order
cum
ulants
and
the
coefficients
of
non
linear
channels
,
then
the
method
de
v
eloped
[11]
f
or
linear
channels
,
is
e
xtended
to
the
gener
a
l
case
of
the
non
linear
quadr
atic
systems
identification.
Sec-
ondly
,
these
theoretical
analyses
are
used
to
de
v
elop
an
e
xtension
of
linear
algor
ithms
based
on
third
and
f
our
th
order
cum
ulants
proposed
b
y
safi,
et
al
:
[5]
and
Abderr
ahim,
et
al
:
[12]
f
or
linear
Receiv
ed
J
an
uar
y
21,
2016;
Re
vised
March
27,
2016;
Accepted
Apr
il
11,
2016
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
335
systems
to
non
linear
algor
it
hms
f
or
identification
of
quadr
atic
systems
.
Num
er
ical
sim
ulations
results
are
giv
en
to
illustr
ate
the
accur
acy
of
the
proposed
methods
.
2.
Non
linear
comm
unication
c
hannel
and
h
ypotheses
W
e
consider
a
non
linear
channel
(Fig.
1)
descr
ibed
b
y
the
f
ollo
wing
equation:
y
(
k
)
=
q
X
i
=0
h
(
i;
i
)
x
2
(
k
i
)
+
w
(
k
)
;
(1)
where
y
(
k
)
represent
the
output
model
is
gener
ated
through
a
quadr
atic
non
linear
model
dr
iv
en
b
y
a
stationar
y
input
sequence
x
(
k
)
,
h
(
i;
i
)
and
q
are
the
par
ameters
and
the
order
of
non
linear
channel,
respectiv
ely
,
w
(
k
)
is
additiv
e
Gaussian
noise
.
N
on
l
i
ne
a
r
c
om
m
uni
c
a
t
i
on
c
ha
nne
l
)
(
k
x
)
(
k
z
)
(
k
y
)
(
k
w
Figure
1.
Non
linear
channel
model
F
or
this
system
w
e
assume
that:
H1:
The
order
q
is
kno
wn;
H2:
The
input
sequence
x
(
k
)
is
independent
and
identically
distr
ib
uted
(i.i.d)
z
ero
mean,
station-
ar
y
,
non
Gaussian
and
with:
C
k
;x
(
1
;
2
;
:::;
k
1
)
=
k
;x
;
1
=
2
=
:::
=
k
1
=
0
;
0
otherwise
where
k
;x
denotes
the
k
th
order
cum
ulants
of
the
input
signal
x
(
k
)
at
or
igin
H3:
The
system
is
supposed
causal
and
bounded,
i.e
.
h
(
i;
i
)
=
0
if
i
<
0
and
i
>
q
with
h
(0
;
0)
=
1
;
H4:
The
measurement
noise
sequence
w
(
k
)
is
assumed
to
be
z
ero
mean,
i.i.d,
Gaussian,
inde-
pendent
of
x
(
k
)
with
unkno
wn
v
ar
iance;
H5:
The
system
is
supposed
stab
le
,
i.e
.
j
h
(
i;
i
)
j
<
1
.
3.
Theoretical
de
velopment
f
or
non
linear
comm
unication
c
hannels
using
HOC
In
this
section,
w
e
firstly
f
ocus
on
additional
concepts
and
definitions
used
throughout
this
paper
.
Indeed,
w
e
discuss
the
identifiability
of
the
quadr
atic
non
linear
model
in
the
second
and
third
order
cum
ulants
domain,
then
based
an
the
relationship
proposed
b
y
Stogioglou
and
McLaughlin
[11]
in
the
linear
case
,
w
e
de
v
elop
an
e
xtension
of
this
relation
in
non
linear
case
.
3.1.
Identification
of
quadratic
non
linear
models
in
the
second
and
thir
d
or
der
cum
ulants
In
this
subsection,
w
e
use
the
Leono
v
Shir
y
ae
v
f
or
m
ula
[6]
and
the
definition
of
the
cum
u-
lants
to
demonstr
ate
the
equations
linking
the
thr
id
order
cum
ulants
and
the
diagonal
par
ameters
of
non
linear
systems
.
Thus
,
the
cum
ulants
,
order
r
,
and
the
moments
of
the
stochastic
signal
are
link
ed
b
y
the
f
ollo
wing
relationships
of
the
Leono
v
and
Shir
y
a
y
e
v
f
or
m
ula
[6]:
C
u
m
[
z
1
;
:::;
z
r
]
=
X
(
1)
k
1
(
k
1)!
E
h
Y
i
2
v
1
z
i
i
:E
h
Y
j
2
v
2
z
j
i
:::E
h
Y
k
2
v
p
z
k
i
;
(2)
where
the
addition
oper
ation
is
o
v
er
all
the
set
of
v
i
,
1
i
p
r
and
v
i
compose
a
par
tition
of
1
;
2
;
:::;
r
.
In
Eq.
(2)
k
is
the
n
umber
of
elements
compose
a
par
tition.
F
or
the
second
order
cum
ulants
,
w
e
are
r
=
2
and
1
p
2
.
The
possib
le
par
titions
are:
(1,2)
and
(1)(2)
thus:
C
u
m
[
z
1
;
z
2
]
=
E
[
z
1
z
2
]
E
[
z
1
]
E
[
z
2
]
;
(3)
Extension
of
Linear
Channels
Identification
Algor
ithms
to
Non
Linear
Using
Selected
Order
Cum
ulants
(M.
Zidane)
Evaluation Warning : The document was created with Spire.PDF for Python.
336
ISSN:
2502-4752
C
2
;z
(
)
=
C
um
(
z
1
;
z
2
)
=
C
um
h
z
(
k
)
;
z
(
k
+
)
i
=
E
h
q
X
i
=0
h
(
i;
i
)
x
2
(
k
i
)
q
X
j
=0
h
(
j
;
j
)
x
2
(
k
+
j
)
i
E
h
q
X
i
=0
h
(
i;
i
)
x
2
(
k
i
)
i
E
h
q
X
j
=0
h
(
j
;
j
)
x
2
(
k
+
j
)
i
(4)
C
2
;z
(
)
=
q
X
i
=0
h
(
i;
i
)
h
(
i
+
;
i
+
)
E
h
x
2
(
k
i
)
x
2
(
k
i
)
i
q
X
i
=0
h
(
i;
i
)
h
(
i
+
;
i
+
)
E
h
x
2
(
k
i
)
i
E
h
x
2
(
k
i
)
i
(5)
Under
the
assumption
H2
and
Eq.
(5),
the
second
order
cum
ulants
becomes:
C
2
;z
(
)
=
(
4
;x
2
2
;x
)
q
X
i
=0
h
(
i;
i
)
h
(
i
+
;
i
+
)
(6)
F
or
the
third
order
cum
ulants
,
w
e
are
r
=
3
and
1
p
3
.
The
possib
le
par
titions
are:
(1,2,3),
(1)(2,3),
and
(1)(2)(3)
so:
C
um
[
z
1
;
z
2
;
z
3
]
=
E
[
z
1
z
2
z
3
]
E
[
z
1
]
E
[
z
2
z
3
]
E
[
z
2
]
E
[
z
1
z
3
]
E
[
z
3
]
E
[
z
1
z
2
]
+
2
E
[
z
1
]
E
[
z
2
]
E
[
z
3
]
(7)
Thus
,
the
third
order
cum
ulants
becomes:
C
3
;z
(
1
;
2
)
=
C
um
h
z
(
k
)
;
z
(
k
+
1
)
;
z
(
k
+
2
)
i
=
E
h
q
X
i
=0
h
(
i;
i
)
x
2
(
k
i
)
q
X
j
=0
h
(
j
;
j
)
x
2
(
k
+
1
j
)
q
X
l
=0
h
(
l
;
l
)
x
2
(
k
+
2
l
)
i
[3]
E
h
q
X
i
=0
h
(
i;
i
)
x
2
(
k
i
)
i
E
h
q
X
j
=0
h
(
j
;
j
)
x
2
(
k
+
1
j
)
q
X
l
=0
h
(
l
;
l
)
x
2
(
k
+
2
l
)
i
+
[2]
E
h
q
X
i
=0
h
(
i;
i
)
x
2
(
k
i
)
i
E
h
q
X
j
=0
h
(
j
;
j
)
x
2
(
k
+
1
j
)
i
E
h
q
X
l
=0
h
(
l
;
l
)
x
2
(
k
+
2
l
)
i
(8)
Under
the
assumption
H2
and
Eq.
(8),
the
third
order
cum
ulants
becomes:
C
3
;z
(
1
;
2
)
=
(
6
;x
3
2
;x
4
;x
+
2
3
2
;x
)
q
X
i
=0
h
(
i;
i
)
h
(
i
+
1
;
i
+
1
)
h
(
i
+
2
;
i
+
2
)
(9)
In
the
gener
al
case
,
the
relationship
betw
een
n
th
order
cum
ulants
and
t
he
coefficients
of
non
linear
impulse
response
channel
can
be
wr
itten
under
f
or
m:
C
n;z
(
1
;
2
;
:::;
n
1
)
=
n;x
q
X
i
=0
h
(
i;
i
)
h
(
i
+
1
;
i
+
1
)
:::h
(
i
+
n
1
;
i
+
n
1
)
;
(10)
where
n;x
=
P
i
p
i
m
i
i;x
,
with
p
i
2
Z
and
m
i
2
N
IJEECS
V
ol.
2,
No
.
2,
Ma
y
2016
:
334
343
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
337
3.2.
Relationship
linking
cum
ulants
and
the
coefficients
of
non
linear
c
hannel
In
this
secti
on,
w
e
de
v
elop
a
non
linear
e
xtension
of
the
Stogioglou
and
McLaughlin’
s
linear
relationship
[11].
Pr
oof:
Let
:
Z
s;n
=
X
i
X
j
h
(
i;
i
)
h
(
j
;
j
)
h
s
Y
k
=1
h
(
i
+
k
;
i
+
k
)
ih
s
Y
k
=1
h
(
j
+
k
;
j
+
k
)
ih
n
1
Y
k
=
s
+1
h
(
i
+
j
+
k
;
i
+
j
+
k
)
i
;
(11)
where
s
is
an
arbitr
ar
y
integer
n
umber
satisfying:
1
s
n
2
.
Changing
the
order
of
summation
in
Eq
(11),
and
w
e
m
ultiply
this
equation
b
y
n;x
,
w
e
will
obtain:
n;x
Z
s;n
=
X
i
h
(
i;
i
)
h
s
Y
k
=1
h
(
i
+
k
;
i
+
k
)
i
n;x
X
j
h
(
j
;
j
)
h
s
Y
k
=1
h
(
j
+
k
;
j
+
k
)
ih
n
1
Y
k
=
s
+1
h
(
i
+
j
+
k
;
i
+
j
+
k
)
i
;
(12)
where
,
n;x
represents
the
n
th
order
cum
ulants
of
the
e
xcitation
signal
at
or
igin
in
non
linear
case
defined
b
y
n;x
=
P
i
p
i
m
i
i;x
.
F
rom
Eqs
(10)
and
(12)
w
e
obtain:
n;x
Z
s;n
=
q
X
i
=0
h
(
i;
i
)
h
s
Y
k
=1
h
(
i
+
k
;
i
+
k
)
i
C
n;y
(
1
;
:::;
s
;
i
+
1
;
:::;
i
+
n
s
1
)
(13)
In
the
same
w
a
y
,
if
w
e
sum
on
i
afterw
ards
on
j
in
(11),
w
e
will
find:
n;x
Z
s;n
=
X
j
h
(
j
;
j
)
h
s
Y
k
=1
h
(
j
+
k
;
j
+
k
)
i
n;x
X
i
h
(
i;
i
)
h
s
Y
k
=1
h
(
i
+
k
;
i
+
k
)
ih
n
1
Y
k
=
s
+1
h
(
i
+
j
+
k
;
i
+
j
+
k
)
i
(14)
The
same
,
from
Eqs
(10)
and
(14)
w
e
obtain:
n;x
Z
s;n
=
q
X
j
=0
h
(
j
;
j
)
h
s
Y
k
=1
h
(
j
+
k
;
j
+
k
)
i
C
n;y
(
1
;
:::;
s
;
j
+
1
;
:::;
j
+
n
s
1
)
;
(15)
F
rom
Eqs
.
(13)
and
(15)
w
e
obtain
Stogioglou-McLaughlin
relation
in
non
linear
case:
q
X
j
=0
h
(
j
;
j
)
h
s
Y
k
=1
h
(
j
+
k
;
j
+
k
)
i
C
n;y
(
1
;
:::;
s
;
j
+
1
;
:::;
j
+
n
s
1
)
=
q
X
i
=0
h
(
i;
i
)
h
s
Y
k
=1
h
(
i
+
k
;
i
+
k
)
i
C
n;y
(
1
;
:::;
s
;
i
+
1
;
:::;
i
+
n
s
1
)
;
(16)
where
1
s
n
2
.
4.
Extension
of
linear
algorithms
to
non
linear
algorithms
In
this
section,
w
e
descr
ibe
an
e
xtension
of
linear
comm
unication
channels
identification
algor
ithms
to
non
linear
using
HOC
.
Ho
w
e
v
er
,
the
linear
algor
ithms
based
on
third
and
f
our
th
order
cum
ulants
proposed
in
the
liter
ature
[5,
12]
f
or
linear
case
is
e
xtended
to
the
non
linear
case
f
or
identification
of
quadr
atic
systems
.
Extension
of
Linear
Channels
Identification
Algor
ithms
to
Non
Linear
Using
Selected
Order
Cum
ulants
(M.
Zidane)
Evaluation Warning : The document was created with Spire.PDF for Python.
338
ISSN:
2502-4752
4.1.
Fir
st
algorithm:
Algcum1
The
F
our
ier
tr
ansf
or
m
of
Eqs
.
(6)
and
(9)
giv
es
us
the
bispectr
a
and
the
spectr
um
respec-
tiv
ely:
S
3
;z
(
!
1
;
!
2
)
=
(
6
;x
3
2
;x
4
;x
+
2
3
2
;x
)
H
(
!
1
!
2
;
!
1
!
2
)
H
(
!
1
;
!
1
)
H
(
!
2
;
!
2
)
(17)
S
2
;z
(
!
)
=
(
4
;x
2
2
;x
)
H
(
!
;
!
)
H
(
!
;
!
)
(18)
If
w
e
suppose
that
!
=
!
1
+
!
2
,
Eq.
(18)
becomes:
S
2
;z
(
!
1
+
!
2
)
=
(
4
;x
2
2
;x
)
H
(
!
1
!
2
;
!
1
!
2
)
H
(
!
1
+
!
2
;
!
1
+
!
2
)
(19)
Then,
from
Eqs
.
(17)
and
(19)
w
e
obtain
the
f
ollo
wing
equation:
S
3
;z
(
!
1
;
!
2
)
H
(
!
1
+
!
2
;
!
1
+
!
2
)
=
6
;x
3
2
;x
4
;x
+
2
3
2
;x
4
;x
2
2
;x
H
(
!
1
;
!
1
)
H
(
!
2
;
!
2
)
S
2
;z
(
!
1
+
!
2
;
!
1
+
!
2
)
(20)
The
in
v
erse
F
our
ier
tr
ansf
or
m
of
Eq.
(20)
demonstr
ates
that
the
third
order
cum
ulants
,
the
auto-
correlation
function
and
the
impulse
response
channel
par
ameters
are
combined
b
y
the
f
ollo
wing
equation:
q
X
i
=0
C
3
;z
(
1
i;
2
i
)
h
(
i;
i
)
=
6
;x
3
2
;x
4
;x
+
2
3
2
;x
4
;x
2
2
;x
q
X
i
=0
h
(
i;
i
)
h
(
2
1
+
i;
2
1
+
i
)
C
2
;z
(
1
i
)
(21)
If
w
e
use
the
autocorrelation
function
proper
ty
of
the
stationar
y
process
such
as
C
2
;z
(
)
6
=
0
only
f
or
q
q
and
v
anishes
else
where
if
w
e
tak
e
1
=
q
,
Eq.
(21)
tak
es
the
f
or
me:
q
X
i
=0
C
3
;z
(
q
i;
2
i
)
h
(
i;
i
)
=
6
;x
3
2
;x
4
;x
+
2
3
2
;x
4
;x
2
2
;x
h
(0
;
0)
h
(
2
+
q
;
2
+
q
)
C
2
;z
(
q
)
;
(22)
else
,
if
w
e
suppose
that
2
=
q
,
Eq.
(22)
will
become:
C
3
;z
(
q
;
q
)
h
(
q
;
q
)
=
6
;x
3
2
;x
4
;x
+
2
3
2
;x
4
;x
2
2
;x
h
(0
;
0)
C
2
;z
(
q
)
(23)
Using
Eqs
.
(22)
and
(23)
w
e
obtain
the
f
ollo
wing
relation:
q
X
i
=0
C
3
;z
(
q
i;
2
i
)
h
(
i;
i
)
=
C
3
;z
(
q
;
q
)
h
(
2
+
q
;
2
+
q
)
(24)
The
system
of
Eq.
(24)
can
be
wr
itten
in
matr
ix
f
or
m
as:
0
B
B
B
B
B
B
@
C
3
;z
(
q
1
;
q
1)
:::
C
3
;z
(
2
q
;
2
q
)
C
3
;z
(
q
1
;
q
)
:::
C
3
;z
(
2
q
;
2
q
+
1)
:
:
:
:
:
:
:
:
:
C
3
;z
(
q
1
;
1)
:::
C
3
;z
(
2
q
;
q
)
1
C
C
C
C
C
C
A
0
B
B
B
B
B
B
B
B
B
B
B
B
@
h
(1
;
1)
:
:
:
h
(
i;
i
)
:
:
:
h
(
q
;
q
)
1
C
C
C
C
C
C
C
C
C
C
C
C
A
=
0
B
B
B
B
B
B
B
B
B
B
@
0
C
3
;z
(
q
;
q
+
1)
:
:
:
:
:
C
3
;z
(
q
;
0)
1
C
C
C
C
C
C
C
C
C
C
A
;
(25)
where
=
C
3
;y
(
q
;
q
)
.
Or
in
more
compact
f
or
m,
the
Eq.
(25)
can
be
wr
itten
as
f
ollo
ws:
M
h
s
=
d;
(26)
IJEECS
V
ol.
2,
No
.
2,
Ma
y
2016
:
334
343
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
339
where
M
is
the
matr
ix
of
siz
e
(
q
+
1)
(
q
)
elements
,
h
s
is
a
column
v
ector
constituted
b
y
the
un-
kno
wn
impulse
response
par
ameters
h
(
i;
i
)
f
or
i
=
1
;
:::;
q
and
d
is
a
column
v
ector
of
siz
e
(
q
+
1)
as
indicated
in
the
Eq.
(26).
The
least
squares
solution
of
the
system
of
Eq.
(26),
per
mits
b
lindly
identification
of
the
par
am-
eters
h
(
i;
i
)
and
without
an
y
inf
or
mation
of
the
input
selectiv
e
channel.
Thus
,
the
solution
will
be
wr
itten
under
the
f
ollo
wing
f
or
m:
b
h
s
=
(
M
T
M
)
1
M
T
d
(27)
4.2.
Second
algorithm:
Algcum2
Safi,
et
al
:
[5]
use
the
Stogioglou-McLaughlin
relation
f
or
de
v
eloping
an
algor
ithm
based
only
on
f
our
th
order
cum
ulants
in
the
linear
case
,
this
algor
ithm
is
e
xtended
to
the
non
linear
case
.
Thus
,
if
w
e
tak
e
n
=
4
into
(16)
w
e
obtain
the
f
ollo
wing
equation:
q
P
j
=0
h
(
j
;
j
)
h
(
j
+
1
;
j
+
1
)
h
(
j
+
2
;
j
+
2
)
C
4
;y
(
1
;
2
;
j
+
1
)
=
q
P
i
=0
h
(
i;
i
)
h
(
i
+
1
;
i
+
1
)
h
(
i
+
2
;
i
+
2
)
C
4
;y
(
1
;
2
;
i
+
1
)
(28)
If
1
=
2
=
q
et
1
=
2
=
0
,
(28)
tak
e
the
f
or
m:
h
(0
;
0)
h
2
(
q
;
q
)
C
4
;y
(0
;
0
;
1
)
=
q
X
i
=0
h
3
(
i;
i
)
C
4
;y
(
q
;
q
;
i
+
1
)
;
(29)
with
q
1
q
(30)
Then,
from
(29)
and
(30)
w
e
obtain
the
f
ollo
wing
system
of
equations:
0
B
B
B
B
B
B
B
B
B
B
B
B
@
C
4
;y
(
q
;
q
;
q
)
:::
C
4
;y
(
q
;
q
;
0)
:
:
:
:
:
:
:
:
:
C
4
;y
(
q
;
q
;
0)
:::
C
4
;y
(
q
;
q
;
q
)
:
:
:
:
:
:
:
:
:
C
4
;y
(
q
;
q
;
q
)
:::
C
4
;y
(
q
;
q
;
2
q
)
1
C
C
C
C
C
C
C
C
C
C
C
C
A
0
B
B
B
B
B
B
B
B
B
B
B
B
B
@
1
h
2
(
q
;q
)
:
:
:
h
3
(
i;i
)
h
2
(
q
;q
)
:
:
:
h
3
(
q
;q
)
h
2
(
q
;q
)
1
C
C
C
C
C
C
C
C
C
C
C
C
C
A
=
0
B
B
B
B
B
B
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
C
C
C
C
C
C
A
(31)
In
more
compact
f
or
m,
the
system
of
(31)
can
be
wr
itten
in
the
f
ollo
wing
f
or
m:
M
b
q
=
d;
(32)
where
M
,
b
q
and
d
are
defined
in
the
system
of
(32).
The
least
squares
solution
of
the
system
of
(32)
is
giv
en
b
y:
b
b
q
=
(
M
T
M
)
1
M
T
d
(33)
This
solution
giv
e
us
an
estimation
of
the
quotient
of
the
par
ameters
h
3
(
i;
i
)
and
h
2
(
q
;
q
)
,
i.e
.,
b
q
(
i;
i
)
=
\
h
3
(
i;i
)
h
2
(
q
;q
)
,
i
=
1
;
:::;
q
.
Thus
,
in
order
to
ob
tain
an
estimati
on
of
the
par
ameters
b
h
(
i;
i
)
,
i
=
1
;
:::;
q
w
e
proceed
as
f
ollo
ws:
The
p
ar
ameters
h
(
i;
i
)
f
or
i
=
1
;
:::;
q
1
are
estimated
from
the
estimated
v
alues
b
b
q
(
i;
i
)
using
the
f
ollo
wing
equation:
b
h
(
i;
i
)
=
sig
n
h
b
b
q
(
i;
i
)
(
b
b
q
(
q
;
q
))
2
in
abs
(
b
b
q
(
i;
i
))
(
b
b
q
(
q
;
q
))
2
o
1
=
3
(34)
Extension
of
Linear
Channels
Identification
Algor
ithms
to
Non
Linear
Using
Selected
Order
Cum
ulants
(M.
Zidane)
Evaluation Warning : The document was created with Spire.PDF for Python.
340
ISSN:
2502-4752
The
b
h
(
q
;
q
)
par
ameters
is
estimated
as
f
ollo
ws:
b
h
(
q
;
q
)
=
1
2
sig
n
h
b
b
q
(
q
;
q
)
in
abs
(
b
b
q
(
q
;
q
))
+
1
b
b
q
(1
;
1)
1
=
2
o
(35)
4.3.
Thir
d
algorithm:
Algcum3
Abderr
ahim,
et
al
:
[12]
use
also
the
relationship
(16),
based
on
f
our
th
order
cum
ulants
,
f
or
de
v
eloping
an
algor
ithm
based
an
f
our
th
order
cum
ulants
in
linear
case
.
This
algor
ithm
is
e
xtended
to
the
gener
al
case
of
the
non
linear
quadr
atic
systems
,
with
1
=
2
=
0
,
1
=
q
and
2
=
0
:
q
X
i
=1
h
3
(
i;
i
)
C
4
;y
(
q
;
0
;
i
+
1
)
h
(
q
;
q
)
C
4
;y
(0
;
0
;
1
)
=
C
4
;y
(
q
;
0
;
1
)
;
(36)
where
1
=
q
;
:::;
q
:
In
more
compact
f
or
m,
the
system
of
Eq.
(36)
can
be
wr
itten
in
the
f
ollo
wing
f
or
m:
M
=
A
(37)
=
[
h
(
q
;
q
)
h
3
(1
;
1)
h
3
(2
;
2)
::::h
3
(
q
;
q
)]
T
is
a
column
v
ector
of
siz
e
(
q
+
1)
;
A
=
[0
:::
0
C
4
y
(
q
;
0
;
0)
C
4
y
(
q
;
0
;
1)
::::
C
4
y
(
q
;
0
;
q
)]
T
is
a
v
ector
of
siz
e
(2
q
+
1)
;
The
least
squares
solution
of
the
system
of
Eq.
(37),
will
be
wr
itten
under
the
f
ollo
wing
f
or
m:
b
=
(
M
T
M
)
1
M
T
d
(38)
The
par
ameters
h
(
i;
i
)
f
or
i
=
1
;
:::;
q
are
estimated
from
the
estimated
v
alues
b
(
i;
i
)
using
the
f
ollo
wing
equation:
b
h
(
i;
i
)
=
3
q
b
(
i
+
1
;
i
+
1)
(39)
5.
Sim
ulation
results
and
Anal
ysis
In
this
section,
w
e
testy
the
p
roposed
approach
to
identify
the
non
linear
impulse
re-
sponse
par
ameters
of
the
non
linear
channels
.
F
or
this
reason,
w
e
select
the
model
(Eq.
40)
char
acter
izing
a
non
linear
quadr
atic
system,
with
kno
wn
par
ameters
and
then
w
e
tr
y
to
reco
v
er
these
par
ameters
using
proposed
algor
ithms
.
y
(
k
)
=
x
2
(
k
)
+
0
:
15
x
2
(
k
1)
0
:
35
x
2
(
k
2)
+
0
:
90
x
2
(
k
3)
z
eros:
z
1
=
1
:
1439
;
z
2
=
0
:
4969
+
0
:
7348
i
;
z
3
=
0
:
4969
0
:
7348
i:
(40)
The
model
of
the
selected
channel
is
a
non
minim
um
phase
because
one
of
its
z
eros
are
outside
of
the
unit
circle
(Fig.
2).
The
sim
ulation
is
perf
or
med
with
MA
TLAB
softw
are
an
d
f
or
diff
erent
signal
to
noise
r
atio
(SNR)
defined
b
y
the
f
ollo
wing
relationship:
S
N
R
=
10
l
og
10
h
2
z
(
k
)
2
n
(
k
)
i
(41)
T
o
measure
the
accur
acy
of
the
diagonal
par
ameter
estimation
with
respect
to
the
real
v
alues
,
w
e
define
the
Nor
maliz
ed
Mean
Square
Error
(NMSE)
f
or
each
r
un
as:
N
M
S
E
=
q
X
i
=0
h
h
(
i;
i
)
b
h
(
i;
i
)
h
(
i;
i
)
i
2
(42)
IJEECS
V
ol.
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.
2,
Ma
y
2016
:
334
343
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
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341
−1
−0.5
0
0.5
1
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
3
Real Part
Imaginary Part
Figure
2.
The
z
eros
of
Model
In
order
to
test
the
rob
ustness
of
the
proposed
algor
ithms
w
e
conseder
a
non
minim
um
phase
channel
descr
ibe
b
y
(Eq.
40).
The
sim
ulation
results
,
using
selected
algor
ithms
,
of
impulse
response
par
ameters
estimation
are
sho
wn
in
the
T
ab
le
1
f
or
diff
erent
v
alues
of
SNR
a
nd
f
or
N=2400.
F
rom
T
ab
le
1,
w
e
can
conclude
that
f
or
all
v
alues
of
SNR
considered,
the
NMSE
v
alues
of
the
T
ab
le
1.
Estimated
par
ameters
of
the
first
channel
f
or
diff
erent
S
N
R
and
e
xcited
b
y
sample
siz
es
N
=
2400
S
N
R
b
h
(
i;
i
)
std
ALGcum
1
ALGcum
2
ALGcum
3
b
h
(1
;
1)
std
0
:
1391
0
:
0960
0
:
0770
0
:
4375
0
:
1310
0
:
3395
0
dB
b
h
(2
;
2)
std
0
:
4225
0
:
0940
0
:
5359
0
:
1832
0
:
5057
0
:
1666
b
h
(3
;
3)
std
0
:
8326
0
:
1120
0
:
8250
0
:
1420
0
:
5371
0
:
2283
N
M
S
E
0
:
0538
0
:
5261
0
:
3766
b
h
(1
;
1)
std
0
:
1539
0
:
0622
0
:
1016
0
:
3243
0
:
2059
0
:
2406
8
dB
b
h
(2
;
2)
std
0
:
3747
0
:
0527
0
:
3376
0
:
2095
0
:
4090
0
:
0708
b
h
(3
;
3)
std
0
:
7879
0
:
0757
0
:
8153
0
:
0984
0
:
6404
0
:
1436
N
M
S
E
0
:
0212
0
:
1141
0
:
2507
b
h
(1
;
1)
std
0
:
1523
0
:
0455
0
:
1366
0
:
2959
0
:
1742
0
:
2425
16
dB
b
h
(2
;
2)
std
0
:
3680
0
:
0500
0
:
2640
0
:
2366
0
:
3718
0
:
0724
b
h
(3
;
3)
std
0
:
8155
0
:
0589
0
:
8395
0
:
0826
0
:
6524
0
:
1158
N
M
S
E
0
:
0117
0
:
0729
0
:
1056
b
h
(1
;
1)
std
0
:
1583
0
:
0475
0
:
1282
0
:
2880
0
:
1522
0
:
2609
24
dB
b
h
(2
;
2)
std
0
:
3611
0
:
0547
0
:
2983
0
:
1876
0
:
3662
0
:
1072
b
h
(3
;
3)
std
0
:
8350
0
:
0609
0
:
8234
0
:
0812
0
:
6565
0
:
1048
N
M
S
E
0
:
0093
0
:
0501
0
:
0755
b
h
(1
;
1)
std
0
:
1479
0
:
0492
0
:
1664
0
:
2536
0
:
1438
0
:
2580
32
dB
b
h
(2
;
2)
std
0
:
3643
0
:
0451
0
:
2939
0
:
2011
0
:
3722
0
:
1007
b
h
(3
;
3)
std
0
:
8419
0
:
0643
0
:
8298
0
:
0930
0
:
6723
0
:
0955
N
M
S
E
0
:
0060
0
:
0437
0
:
0697
T
r
ue
par
ameters
h
(
i;
i
)
h
(1
;
1)
=
0
:
150
h
(2
;
2)
=
0
:
350
h
(3
;
3)
=
0
:
900
first
proposed
method
such
as
(Al
gcum1)
are
lo
w
er
than
the
other
methods
(Algcum2,
Algcum3),
this
is
due
to
the
comple
xity
of
the
systems
of
equations
f
or
each
algor
ithm,
non
linea
r
of
the
par
ameters
in
the
(Algcum2,
Algcum3)
algor
ithms
.
The
perf
or
mance
of
the
(Algcum2)
method
deg
r
ade
than
the
(Algcum3)
in
v
er
y
noise
en
vironment
(SNR=0
dB),
b
ut
it
becomes
more
eff
ectiv
e
than
(Algcum3)
when
the
noise
v
ar
iance
is
relativ
ely
small,
this
is
due
the
f
act
that
the
higher
order
cum
ulants
f
or
a
Gaussian
noise
are
not
identically
z
ero
,
b
ut
the
y
ha
v
e
v
alues
close
to
z
ero
f
or
higher
data
length.
This
is
v
er
y
clear
in
the
(Fig.
3).
In
the
par
t,
of
comple
xity
of
these
algor
ithms
the
first
proposed
algor
ithm
e
xploiting
(
q
+
1)
Extension
of
Linear
Channels
Identification
Algor
ithms
to
Non
Linear
Using
Selected
Order
Cum
ulants
(M.
Zidane)
Evaluation Warning : The document was created with Spire.PDF for Python.
342
ISSN:
2502-4752
0
5
10
15
20
25
30
0
0.1
0.2
0.3
0.4
0.5
Signal to Noise Ratio SNR(dB)
Normalized Mean Square Error (NMSE)
NMSE using Algcum1
NMSE using Algcum2
NMSE using Algcum3
Figure
3.
NMSE
f
or
each
algor
ithm
and
f
or
diff
erent
SNR
and
f
or
a
data
length
N
=
2400
equations
,
compar
ing
de
second
and
third
proposed
methods
e
xploiting
(2
q
+
1)
f
or
identify
the
impulse
response
par
ameters
channel.
In
the
Fig.
4
w
e
ha
v
e
presented
the
estimation
of
the
magnitude
and
the
phase
of
the
impulse
response
using
the
proposed
algor
ithms
,
f
or
data
length
N
=
2400
and
v
er
y
noise
en
vi-
ronment
SNR
=0
dB
.
F
rom
the
Fig.
4
w
e
remar
k
that
the
magnitude
estimation
ha
v
e
the
same
appear
ance
using
tw
o
first
proposed
methods
b
ut
using
(Algcum3)
algor
ithm
w
e
ha
v
e
a
minor
dif-
f
erence
betw
een
the
estimated
and
tr
ue
ones
.
Concer
ning
the
phase
estimation,
w
e
ha
v
e
same
allure
compar
ativ
ely
to
the
real
model
using
all
proposed
algor
ithms
.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
−600
−400
−200
0
200
Normalized Frequency (
×
π
rad/sample)
Phase (degrees)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
−20
−10
0
10
Normalized Frequency (
×
π
rad/sample)
Magnitude (dB)
True channel
Estimated using Algcum1
Estimated using Algcum2
Estimated using Algcum3
(True, Algcum1, Algcum2)
Algcum3
Figure
4.
Estimated
magnitude
and
phase
of
the
non
linear
model
channel
impulse
response
when
the
data
input
is
N=2400
and
an
SNR=0
dB
IJEECS
V
ol.
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.
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y
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:
334
343
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
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343
6.
Conc
lusion
In
this
contr
ib
ution,
w
e
ha
v
e
considered
the
prob
lem
of
b
lind
identification
of
non
linear
channel
using
selected
order
cum
ulants
.
W
e
ha
v
e
de
v
eloped
a
theoretical
analysis
f
or
non
linear
quadr
atic
systems
,
these
tools
will
ser
v
e
us
later
f
or
proposing
an
e
xtension
of
linear
algor
ithms
to
non
linear
.
Ho
w
e
v
er
,
w
e
ha
v
e
de
v
eloped
three
approaches
based
on
the
three
and
f
our
th
order
cum
ulants
,
respectiv
ely
,
f
or
b
lind
identification
of
diagonal
par
ameters
of
quadr
atic
systems
.
F
rom
sim
ulation
results
and
compar
ison
betw
een
these
methods
one
can
see
that
the
first
proposed
algor
ithm
(Algcum1)
can
alw
a
ys
achie
v
e
better
perf
or
mance
than
other
(Algcum1,
Algcum2),
and
is
adequate
f
or
estimating
diagonal
quadr
atic
systems
.
The
future
w
or
k
of
this
paper
is
the
non
linear
Broadband
Radio
Access
Netw
or
k
(BRAN)
channels
identification
and
equalization
especially
MC-CDMA
systems
using
the
presented
meth-
ods
.
Ref
erences
[1]
M.
Zidane
,
S
.
Safi,
M.
Sabr
i
and
A.
Boumezz
ough,
“Higher
Order
Statistics
f
or
Identification
of
Minim
um
Phase
Channels
,
”
W
or
ld
Academ
y
of
Science
Engineer
ing
and
T
echnology
,
In-
ter
national
Jour
nal
of
Mathematical,
Computational,
Ph
ysical
and
Quantum
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neer
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8,
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.
831-836,
(2014).
[2]
M.
Zidane
,
S
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Safi,
M.
Sabr
i
and
A.
Boumezz
ough,
“Blind
Identification
Channel
Using
Higher
Order
Cum
ulants
with
Application
to
Equalization
f
or
MC-CDMA
System,
”
W
or
ld
Academ
y
of
Science
Engineer
ing
and
T
e
c
h
nology
,
Inter
national
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of
Electr
ical,
Robotics
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M.
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Safi,
M.
Sabr
i,
A.
Boumezz
ough
and
M.
F
r
ik
el,
“Broadband
Radio
Access
Netw
or
k
Channel
Identification
and
Do
wnlink
MC-
CDMA
Equalization,
”
Inter
national
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nal
of
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or
mation
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Zeroual,
“Blind
non
minim
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phase
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3
r
d
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4
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order
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ik
el,
A.
Zeroual,
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M.
M’Saad,
“Higher
Order
Cum
ulants
f
or
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and
Equalization
of
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ier
Spreading
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um
Systems
,
”
Jour
nal
of
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elecomm
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or
mation
T
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.
74-84,
1/2011.
[6]
V
.
P
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Leono
v
and
A.
N.
Shir
y
ae
v
,
“On
a
method
of
calculation
of
semi-in
v
ar
iants
,
”
Theor
y
of
probability
and
its
applications
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ol.
4,
No
3,
pp
.
319-329,
(1959).
[7]
J
.
Antar
i,
A.
Elkhadimi,
D
.
Mammas
,
and
A.
Zeroual,
“De
v
eloped
Algor
ithm
f
or
Super
vising
Identification
of
Non
Linear
Systems
using
Higher
Order
Statistics
:
Modeling
Inter
net
T
r
affic
,
”
Inter
national
Jour
nal
of
Future
Gener
ation
Comm
unication
and
Netw
or
king
,
v
ol.
5,
No
4,
pp
.
17-28,
(2012).
[8]
J
.
Antar
i,
S
.
Cha
baab
,
R.
Iqdour
,
A.
Zeroual,
S
.
Safi,
“Identificat
ion
of
quadr
atic
systems
using
higher
order
cum
ulants
and
neur
al
netw
or
ks:
Application
to
model
the
dela
y
of
video-
pac
k
ets
tr
ansmission,
”
Jour
nal
of
Applied
Soft
Computing
(ASOC),
Else
vier
,
v
ol.
11,
No
1,
pp
.
1-10,
(2011).
[9]
H.
Z.
T
an,
T
.
W
.
S
.
Cho
w
,
“Blind
identification
of
quadr
atic
non
linear
models
using
neur
al
netw
or
ks
with
higher
order
cum
ulants
,
”
IEEE
T
r
ansactions
on
Industr
ial
Electronics
,
v
ol.
47,
No
3,
pp
.
687-696,
(2000).
[10]
H.
Z.
T
an,
Z.
Y
.
Mao
,
“Blind
identifiability
of
quadr
atic
non
linear
systems
in
higher
order
statistics
domain,
”
Inter
national
Jour
nal
of
Adaptiv
e
Control
and
Signal
Process
,
v
ol.
12,
No
7,
pp
.
567-577,
(1998).
[11]
A.
G.
Stogioglou
and
S
.
McLaughlin,
“MA
par
ameter
estimation
and
cum
ulant
enhancement,
”
IEEE
T
r
ansactions
on
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,
v
ol.
44,
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.
1704-1718,
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[12]
K.
Abderr
ahim,
R.
B
.
Abdennour
,
F
.
Msahli,
M.
Ksour
i,
and
G.
F
a
vier
,
“Identification
of
non
minim
um
phase
finite
impulse
response
systems
using
the
f
our
th
order
cum
ulants
,
”
Prog
ress
in
system
and
robot
analysis
and
control
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Spr
inger
,
v
ol.
243,
pp
.
41-50,
(1999).
Extension
of
Linear
Channels
Identification
Algor
ithms
to
Non
Linear
Using
Selected
Order
Cum
ulants
(M.
Zidane)
Evaluation Warning : The document was created with Spire.PDF for Python.