Indonesian
Journal
of
Electrical
Engineering
and
Computer
Science
V
ol.
1,
No
.
1,
J
an
uar
y
2016,
pp
.
138
152
DOI:
10.11591/ijeecs
.v1.i1.pp138-152
138
Bit
Err
or
Rate
Anal
ysis
of
MC-CDMA
Systems
with
Channel
Identification
Using
Higher
Or
der
Cum
ulants
Mohammed
Zidane
1,*
,
Said
Safi
2
,
Mohamed
Sabri
1
,
and
Ahmed
Boumezzough
3
1
Depar
tment
of
Ph
ysics
,
F
aculty
of
Sciences
and
T
echnology
,
Sultan
Moula
y
Slimane
Univ
ersity
,
Moro
cco
2
Depar
tment
of
Mathematics
and
Inf
or
matics
,
P
olydisciplinar
y
F
aculty
,
Sultan
Moula
y
Slima
ne
Univ
ersity
,
Morocco
3
Depar
tment
of
Ph
ysics
,
P
olydisciplinar
y
F
aculty
,
Sultan
Moula
y
Slimane
Univ
ersity
,
Morocco
*Corresponding
author
,
e-mail:
zidane
.ilco@gmail.com
Abstract
The
aim
of
this
paper
is
to
contr
ib
ute
to
study
the
prob
lems
of
the
b
lind
identification
and
equal-
ization
using
Higher
Order
Cum
ulants
(HOC)
in
do
wnlink
Multi-
Carr
ier
Code
Division
Multiple
Access
(MC-
CDMA)
systems
.
F
or
this
prob
lem,
tw
o
b
lind
algor
ithms
based
on
HOC
f
or
Broadband
Radio
Access
Netw
or
k
(BRAN)
channel
are
proposed.
In
the
one
hand,
to
assess
the
perf
or
mance
of
these
approaches
to
identify
the
par
ameters
of
non
minim
um
phase
channels
,
w
e
ha
v
e
considered
tw
o
theoretical
channels
,
and
one
pr
actical
frequency
selectiv
e
f
ading
channel
called
BRAN
C
,
dr
iv
en
b
y
non
gaussian
signal.
In
the
other
hand,
w
e
use
the
Minim
um
Mean
Square
Error
(MMSE)
equaliz
er
tech
nique
after
the
channel
identification
to
co
rrect
the
channel
distor
tion.
Theoretical
analysis
and
n
umer
ical
sim
ulation
results
,
in
noisy
en
vironment
and
f
or
diff
erent
signal
to
noise
r
atio
(SNR),
are
presented
to
illustr
ate
the
perf
or
mance
of
the
proposed
methods
.
K
e
yw
or
ds:
HOC
,
Blind
identificatio
n
and
equalization,
BRAN
channel,
MC-CDMA
systems
,
Bit
error
r
ate
,
MMSE
equaliz
er
Cop
yright
c
2016
Institute
of
Ad
v
anced
Engineering
and
Science
.
All
rights
reser
ved.
1.
Intr
oduction
The
channel
identification
is
of
pr
imar
y
impor
tance
in
digital
comm
unication
systems
,
appears
as
a
major
concer
n.
In
f
act
se
v
er
al
met
hods
ha
v
e
been
e
xplored.
Ne
v
er
theless
,
the
most
commonly
used
are
adaptiv
e
methods
that
send
occasionally
a
tr
aining
sequence
kno
wn
to
the
tr
ansmitter
and
receiv
er
.
The
presence
the
tr
aining
sequence
in
this
approach
reduces
the
spectr
al
efficiency
and
hence
tr
ansmission
r
ate
.
Looking
f
orw
ard
better
tr
adeoffs
betw
een
the
quality
of
inf
or
mation
reco
v
er
y
and
suitab
le
bit
r
ates
,
the
use
of
b
lind
techniques
is
of
g
reat
interest.
In
this
paper
,
w
e
f
ocus
on
the
prob
lem
of
b
lind
identification
and
equalization
of
the
Broad-
band
Radio
Access
Netw
o
r
ks
channels
,
using
Higher
Order
Cum
ulants
(HOC)
in
MC-CDMA
sys-
tems
.
Se
v
er
al
w
or
ks
de
v
eloped
in
the
liter
ature
sho
w
that
the
b
lind
identification
channels
using
HOC
ha
v
e
attr
acted
consider
ab
le
attention
[1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12].
Thus
,
the
first
methods
utilizing
the
output
of
the
second
order
cum
ulants
(autocorrelation
function)
of
the
ob-
ser
v
ed
sequences
f
or
identification
of
minim
um
phase
channel
dr
iv
en
b
y
gaussian
distr
ib
ution
input
[6,
9,
10].
Ho
w
e
v
er
,
in
se
v
er
al
applications
,
the
obser
v
ed
signals
are
non
gaussian
and
the
system
to
be
identified
has
non
minim
um
phase
and
is
contaminated
b
y
a
gaussian
noise
.
Moreo
v
er
,
the
non
minim
um
phase
channels
cannot
be
ident
ified
correctly
using
the
autocorre-
lation
function
sequence
[10],
because
the
second
order
cum
ulants
f
or
a
gaussian
process
are
not
identically
z
ero
.
Hence
,
when
the
processed
signal
is
non
gaussian
and
the
additiv
e
noise
is
gaussian,
the
noise
will
v
anish
in
the
higher
order
cum
ulants
(
3
r
d
and
4
r
d
cum
ulants)
domain.
In
this
contr
ib
ution,
w
e
propose
tw
o
b
lind
algor
ithms
based
on
f
our
th
order
cum
ulants
,
this
approach
allo
ws
the
resolution
of
the
insolub
le
prob
lems
using
the
second
order
statistics
.
In
order
Receiv
ed
A
ugust
14,
2015;
Re
vised
No
v
ember
25,
2015;
Accepted
December
12,
2015
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
139
to
test
the
efficiency
of
the
proposed
algor
ithms
w
e
ha
v
e
considered
tw
o
theoretical
channels
,
and
one
pr
actical
f
requency
selectiv
e
f
ading
channel
such
as
BRAN
C
[14,
15]
nor
ma
liz
ed
f
or
MC-CDMA
systems
,
e
xited
b
y
a
non
gaussian
sequences
,
f
or
diff
erent
signal
to
noise
r
atio
(SNR)
and
impor
tant
data
input.
In
this
w
or
k,
w
e
address
the
application
of
proposed
methods
based
on
HOC
in
the
conte
xt
of
b
lind
equalization
of
MC-CDMA
systems
.
Indeed
the
pr
inciples
o
f
MC-CDMA
is
that
a
single
data
symbol
is
tr
ansmitted
at
m
ultiple
narro
w
band
subcarr
iers
[17].
Ho
w
e
v
er
,
in
MC-CDMA
systems
,
spreading
codes
are
applied
in
the
frequency
domain
and
tr
ansmitted
o
v
er
independent
sub-carr
iers
.
The
prob
lem
encountered
in
digital
comm
unication
is
the
synchronization
betw
een
the
tr
ansmitter
and
the
receiv
er
.
Ne
v
er
theless
in
most
wireless
en
vironments
,
there
are
man
y
dif-
f
erent
propagation
paths
caused
b
y
man
y
obstacles
in
the
channels
,
such
as
b
uildings
,
mountains
and
w
alls
betw
een
the
tr
ansmitter
and
receiv
er
.
Synchronization
errors
cause
loss
of
or
thogonal-
ity
among
sub-carr
iers
and
consid
er
ab
ly
deg
r
ade
the
perf
or
mance
especially
when
large
n
umber
of
subcarr
iers
presents
[18,
19].
In
this
paper
w
e
are
concer
ned
with
the
prob
lem
of
the
b
lind
identification
of
the
broadband
r
adio
access
netw
or
k
channel
such
as
BRAN
C
,
after
that
w
e
use
the
Minim
um
Mean
Square
Err
or
(MMSE)
equaliz
er
technique
to
correct
the
channel
distor
tion.
The
sim
ulation
results
,
in
noisy
en
vironment,
are
presented
to
illustr
ate
the
perf
or
mance
of
the
proposed
algor
ithms
.
2.
Pr
ob
lem
statement
2.1.
Channel
modeling
W
e
consider
the
single-input
single-output
(SISO)
model
of
the
Finite
Impulse
Response
(FIR)
system
descr
ibed
b
y
the
f
ollo
wing
figure
(Fig.
1):
W
e
assumed
that
the
channel
is
time
in
v
ar
iant
and
it’
s
impulse
response
is
char
acter
iz
ed
b
y
P
H
(z
)
)
(
k
x
)
(
k
y
)
(
k
r
)
(
k
n
Figure
1.
SISO
channel
model
paths
of
magnitudes
p
and
phases
p
.
The
impulse
response
is
giv
en
b
y
the
f
ollo
wing
equation:
h
(
)
=
P
1
X
p
=0
p
e
p
(
p
)
(1)
The
relationship
betw
een
the
emitted
signal
x
(
t
)
and
the
receiv
ed
signal
r
(
t
)
is
giv
en
b
y:
r
(
t
)
=
h
(
t
)
x
(
t
)
+
n
(
t
)
(2)
r
(
t
)
=
+
1
Z
1
P
1
X
p
=0
p
e
p
(
p
)
x
(
t
)
d
+
n
(
t
)
=
P
1
X
p
=0
p
e
p
x
(
t
p
)
+
n
(
t
)
;
(3)
where
x
(
t
)
is
the
input
sequence
,
h
(
t
)
is
the
impulse
response
coefficients
,
p
is
the
order
of
FIR
system,
and
n
(
t
)
is
the
additiv
e
noise
sequence
.
In
order
to
simplify
the
constr
uction
of
the
proposed
algor
ithms
w
e
assume
that:
BER
Analysis
of
MC-CDMA
Systems
with
Channel
Identification
Using
HOC
(M.
Zidane)
Evaluation Warning : The document was created with Spire.PDF for Python.
140
ISSN:
2502-4752
The
input
sequence
,
x
(
k
)
,
is
ind
ependent
and
identically
distr
ib
uted
(i.i.d)
z
ero
mean,
and
non
gaussian;
The
system
is
causal
and
tr
uncated,
i.e
.
h
(
k
)
=
0
f
or
k
<
0
and
k
>
q
,
where
h
(0)
=
1
;
The
system
order
q
is
kno
wn;
The
measurement
noise
sequence
n
(
k
)
is
assumed
z
ero
mean,
i.i.d,
gaussian
and
inde-
pendent
of
x
(
k
)
with
unkno
wn
v
ar
iance
.
The
prob
lem
statement
is
to
identify
the
par
ameters
of
the
system
h
(
k
)
(
k
=1
;::;q
)
using
the
HOC
of
the
measured
output
process
y
(
k
)
.
2.2.
Basic
Relationships
In
this
section,
w
e
descr
ibe
the
main
gener
al
relationships
betw
een
cum
ulants
and
im-
pulse
response
coefficients
.
Br
illinger
and
Rosenb
latt
sho
w
ed
that
the
m
th
order
cum
ulants
of
y
(
k
)
can
be
e
xpressed
as
a
function
of
impulse
response
coefficients
h
(
i
)
as
f
ollo
ws
[20]:
C
m;y
(
t
1
;
t
2
;
:::;
t
m
1
)
=
m;x
q
X
i
=0
h
(
i
)
h
(
i
+
t
1
)
:::h
(
i
+
t
m
1
)
;
(4)
where
m;x
represents
the
m
th
order
cum
ulants
of
the
e
xcitation
signal
x
(
i
)
at
or
igin.
If
m
=
2
into
Eq.
(4)
w
e
obtain
the
second
order
cum
ulant
(A
utocorrelation
function):
C
2
;y
(
t
1
)
=
2
;x
q
X
i
=0
h
(
i
)
h
(
i
+
t
1
)
(5)
The
same
,
if
m
=
4
,
Eq.
(4)
yield
to:
C
4
;y
(
t
1
;
t
2
;
t
3
)
=
4
;x
q
X
i
=0
h
(
i
)
h
(
i
+
t
1
)
h
(
i
+
t
2
)
h
(
i
+
t
3
)
(6)
P
e
yre
,
et
al
:;
presents
the
relationship
betw
een
diff
erent
m
and
n
cum
ulants
of
the
output
signal,
y
(
k
)
,
and
the
coeff
ecients
h
(
i
)
,
where
n
>
m
and
(
n;
m
)
2
N
f
1
g
,
are
link
ed
b
y
the
f
ollo
wing
relationship
[13]:
q
X
j
=0
h
(
j
)
C
n;y
(
j
+
1
;
j
+
2
;
:::;
j
+
m
1
;
m
;
:::;
n
1
)
=
q
X
i
=0
h
(
i
)
h
n
1
Y
k
=
m
h
(
i
+
k
)
i
C
m;y
(
i
+
1
;
i
+
2
;
:::;
i
+
m
1
)
;
(7)
where
=
n;x
m;x
Pr
oof:
Let:
P
mn
=
X
ij
h
(
i
)
h
(
j
)
h
m
1
Y
k
=1
h
(
i
+
j
+
k
)
ih
n
1
Y
k
=
m
h
(
i
+
k
)
i
(8)
IJEECS
V
ol.
1,
No
.
1,
J
an
uar
y
2016
:
138
152
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
141
Firstly
,
if
w
e
sum
on
i
afterw
ards
on
j
in
(8),
w
e
will
find:
P
mn
=
X
j
h
(
j
)
X
i
h
(
i
)
h
m
1
Y
k
=1
h
(
i
+
j
+
k
)
ih
n
1
Y
k
=
m
h
(
i
+
k
)
i
(9)
If
w
e
m
ultiply
(9)
b
y
n;x
and
tak
e
th
e
Br
illinger
and
Rosenb
latt
f
or
m
ula
(4)
into
account,
w
e
will
obtain:
n;x
P
mn
=
X
j
h
(
j
)
C
n;y
(
j
+
1
;
j
+
2
;
:::;
j
+
m
1
;
:::;
m
;
:::;
n
1
)
(10)
Changing
the
order
of
summation
in
(8)
yields:
P
mn
=
X
i
h
(
i
)
h
n
1
Y
k
=
m
h
(
i
+
k
)
i
X
j
h
(
j
)
h
m
1
Y
k
=1
h
(
i
+
j
+
k
)
i
(11)
If
w
e
m
ultiply
the
r
ight
and
left
sides
of
(11)
b
y
m;x
and
tak
e
t
he
relation
(4)
into
account,
w
e
will
obtain:
m;x
P
mn
=
X
i
h
(
i
)
h
n
1
Y
k
=
m
h
(
i
+
k
)
i
C
m;y
(
i
+
1
;
i
+
2
;
:::;
i
+
m
1
)
(12)
F
rom
(10)
and
(12),
w
e
obtain
the
relation
of
the
P
e
yre
,
et
al
:
3.
Pr
oposed
algorithms
3.1.
Fir
st
Algorithm:
Algo1
If
w
e
tak
e
n
=
4
and
m
=
2
into
Eq.
(7)
w
e
obtain
the
f
ollo
wing
equation:
q
X
j
=0
h
(
j
)
C
4
;y
(
j
+
1
;
2
;
3
)
=
q
X
i
=0
h
(
i
)
h
3
Y
k
=2
h
(
i
+
k
)
i
C
2
;y
(
i
+
1
)
(13)
q
X
j
=0
h
(
j
)
C
4
;y
(
j
+
1
;
2
;
3
)
=
q
X
i
=0
h
(
i
)
h
(
i
+
2
)
h
(
i
+
3
)
C
2
;y
(
i
+
1
)
;
(14)
the
autocorrelation
function
of
the
FIR
systems
v
anishes
f
or
all
v
alues
of
j
t
j
>
q
,
equiv
alently:
C
2
;y
(
t
)
=
6
=
0
;
j
t
j
q
;
0
otherwise
.
If
w
e
suppose
that
1
=
q
the
Eq.
(14)
becomes:
q
X
j
=0
h
(
j
)
C
4
;y
(
j
+
q
;
2
;
3
)
=
h
(0)
h
(
2
)
h
(
3
)
C
2
;y
(
q
)
;
(15)
if
w
e
fix
e
that
3
=
0
the
Eq.
(15)
becomes:
q
X
j
=0
h
(
j
)
C
4
;y
(
j
+
q
;
2
;
0)
=
h
2
(0)
h
(
2
)
C
2
;y
(
q
)
(16)
The
considered
system
is
causal.
Thus
,
the
inter
v
al
of
the
2
is
2
=
0
;
:::;
q
.
Else
if
w
e
suppose
that
2
=
0
,
and
using
the
cum
ulants
proper
ties
C
m;y
(
1
;
2
;
:::;
m
1
)
=
0
,
if
one
of
the
v
ar
iab
les
k
>
q
,
where
k
=
1
;
:::;
m
1
;
the
Eq.
(16)
becomes:
C
4
;y
(
q
;
0
;
0)
=
h
3
(0)
C
2
;y
(
q
)
(17)
Thus
,
w
e
based
on
Eq.
(17)
f
or
eliminating
C
2
;y
(
q
)
in
Eq.
(16),
w
e
obtain
the
equation
constituted
of
only
the
f
our
th
order
cum
ulants:
q
X
j
=0
h
(
j
)
C
4
;y
(
j
+
q
;
2
;
0)
=
h
(
2
)
C
4
;y
(
q
;
0
;
0)
(18)
BER
Analysis
of
MC-CDMA
Systems
with
Channel
Identification
Using
HOC
(M.
Zidane)
Evaluation Warning : The document was created with Spire.PDF for Python.
142
ISSN:
2502-4752
The
system
of
Eq.
(18)
can
be
wr
itten
in
matr
ix
f
or
m
as:
0
B
B
B
B
B
B
@
C
4
;y
(
q
+
1
;
0
;
0)
:::
C
4
;y
(2
q
;
0
;
0)
C
4
;y
(
q
+
1
;
1
;
)
:::
C
4
;y
(2
q
;
1
;
0)
:
:
:
:
:
:
:
:
:
C
4
;y
(
q
+
1
;
q
;
0)
:::
C
4
;y
(2
q
;
q
;
0)
1
C
C
C
C
C
C
A
0
B
B
B
B
B
B
B
B
@
h
(1)
:
:
h
(
i
)
:
:
h
(
q
)
1
C
C
C
C
C
C
C
C
A
=
0
B
B
B
B
B
B
@
0
C
4
;y
(
q
;
1
;
0)
:
:
:
C
4
;y
(
q
;
q
;
0)
1
C
C
C
C
C
C
A
;
(19)
where
=
C
4
;y
(
q
;
0
;
0)
.
Or
in
more
compact
f
or
m,
the
Eq.
(19)
can
be
wr
itten
as
f
ollo
ws:
M
h
e
1
=
d;
(20)
where
M
is
the
matr
ix
of
siz
e
(
q
+
1)
(
q
)
elements
,
h
e
1
is
a
column
v
ector
constituted
b
y
the
unkno
wn
impulse
response
par
ameters
h
(
i
)
i
=1
;:::;q
and
d
is
a
column
v
ector
of
siz
e
(
q
+
1)
as
indicated
in
the
Eq.
(19).
The
least
squares
solution
of
the
system
of
Eq.
(20),
per
mits
b
lindly
identification
of
the
par
ameters
h
(
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
e
1
=
(
M
T
M
)
1
M
T
d
(21)
3.2.
Second
algorithm:
Algo2
The
Z-tr
ansf
or
m
of
the
second
order
cum
ulant
is
str
aightf
orw
ard
and
giv
es
the
f
ollo
wing
equation:
S
2
;y
(
z
)
=
2
;x
H
(
z
)
H
(
z
1
)
(22)
The
Z-tr
ansf
or
m
of
equation
(6)
is
equation
(23):
S
4
;y
(
z
1
;
z
2
;
z
3
)
=
4
;x
H
(
z
1
)
H
(
z
2
)
H
(
z
3
)
H
(
z
1
1
z
1
2
z
1
3
)
(23)
If
w
e
suppose
that
z
=
z
1
z
2
z
3
Eq.
(22)
becomes:
S
2
;y
(
z
1
z
2
z
3
)
=
2
;x
H
(
z
1
z
2
z
3
)
H
(
z
1
1
z
1
2
z
1
3
)
(24)
Then,
from
Eqs
.
(23)
and
(24)
w
e
obtain
the
f
ollo
wing
equation:
H
(
z
1
z
2
z
3
)
S
4
;y
(
z
1
;
z
2
;
z
3
)
=
H
(
z
1
)
H
(
z
2
)
H
(
z
3
)
S
2
;y
(
z
1
z
2
z
3
)
;
(25)
with
=
4
;x
2
;x
The
in
v
erse
Z-tr
ansf
or
m
of
Eq.
(25)
demonstr
ates
that
the
4
th
order
cum
ulants
,
the
autocorrelation
function
and
the
impulse
response
channel
par
ameters
are
combined
b
y
the
f
ollo
wing
equation:
q
X
i
=0
C
4
;y
(
t
1
i;
t
2
i;
t
3
i
)
h
(
i
)
=
q
X
i
=0
h
(
i
)
h
(
t
2
t
1
+
i
)
h
(
t
3
t
1
+
i
)
C
2
;y
(
t
1
i
)
(26)
If
w
e
use
the
autocorrelation
function
proper
t
y
of
the
stationar
y
process
such
as
C
2
;y
(
t
)
6
=
0
only
f
or
q
t
q
and
v
anishes
else
where
.
If
w
e
suppose
that
t
1
=
2
q
the
Eq.
(26)
becomes:
q
X
i
=0
C
4
;y
(2
q
i;
t
2
i;
t
3
i
)
h
(
i
)
=
h
(
q
)
h
(
t
2
q
)
h
(
t
3
q
)
C
2
;y
(
q
)
;
(27)
else
if
w
e
suppose
that
t
2
=
q
the
Eq.
(27)
becomes:
q
X
i
=0
C
4
;y
(2
q
i;
q
i;
t
3
i
)
h
(
i
)
=
h
(
q
)
h
(0)
h
(
t
3
q
)
C
2
;y
(
q
)
(28)
IJEECS
V
ol.
1,
No
.
1,
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an
uar
y
2016
:
138
152
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
143
F
or
eliminating
h
(
q
)
in
(28),
w
e
consider
the
relation
of
Br
illinger
and
Rosenb
latt
descr
ibe
with
f
ollo
wing
equation
f
or
m
=
4
:
C
4
;y
(
t
1
;
t
2
;
t
3
)
=
4
;x
q
X
i
=0
h
(
i
)
h
(
i
+
t
1
)
h
(
i
+
t
2
)
h
(
i
+
t
3
)
(29)
If
t
1
=
t
2
=
t
3
=
q
Eq.
(29)
becomes:
C
4
;y
(
q
;
q
;
q
)
=
4
;x
h
3
(
q
)
;
(30)
else
if
t
1
=
t
2
=
q
and
t
3
=
0
(29)
reduces:
C
4
;y
(
q
;
q
;
0)
=
4
;x
h
2
(
q
)
(31)
F
rom
(30),
(31)
w
e
obtain:
h
(
q
)
=
C
4
;y
(
q
;
q
;
q
)
C
4
;y
(
q
;
q
;
0)
(32)
Thus
,
w
e
based
on
(32)
f
or
eliminating
h
(
q
)
in
(28),
w
e
obtain
the
f
ollo
wing
equation:
q
X
i
=0
C
4
;y
(2
q
i;
q
i;
t
3
i
)
h
(
i
)
=
C
4
;y
(
q
;
q
;
q
)
C
4
;y
(
q
;
q
;
0)
h
(
t
3
q
)
C
2
;y
(
q
)
(33)
The
considered
system
is
causal.
Thus
,
the
inter
v
al
of
the
t
3
is
t
3
=
q
;
:::;
2
q
The
system
of
Eq.
(33)
can
be
wr
itten
in
matr
ix
f
or
m
as:
0
B
B
B
B
B
B
@
C
4
;y
(2
q
1
;
q
1
;
q
1)
:::
C
4
;y
(
q
;
0
;
0)
C
4
;y
(2
q
1
;
q
1
;
q
)
:::
C
4
;y
(
q
;
0
;
1)
:
:
:
:
:
:
:
:
:
C
4
;y
(2
q
1
;
q
1
;
2
q
1)
:::
C
4
;y
(
q
;
0
;
q
)
1
C
C
C
C
C
C
A
0
B
B
B
B
B
B
B
B
@
h
(1)
:
:
h
(
i
)
:
:
h
(
q
)
1
C
C
C
C
C
C
C
C
A
=
0
B
B
B
B
B
B
@
C
4
;y
(2
q
;
q
;
q
)
C
4
;y
(2
q
;
q
;
q
+
1)
:
:
:
C
4
;y
(2
q
;
q
;
2
q
)
1
C
C
C
C
C
C
A
;
(34)
where
=
C
4
;y
(
q
;q
;q
)
C
4
;y
(
q
;q
;
0)
C
2
;y
(
q
)
.
The
least
squares
solution
of
the
system
of
Eq.
(34),
per
mits
b
lindly
identification
of
the
par
am-
eters
h
(
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
e
2
=
(
M
T
M
)
1
M
T
d
(35)
4.
Application
of
MC-CDMA
systems
The
m
ulticarr
ier
code
division
m
ultiple
access
(MC-CDMA)
systems
is
based
on
the
com-
bination
of
code
division
m
ultiple
access
(CDMA)
and
or
thogonal
frequency
division
m
ultiple
xing
(OFDM)
which
is
potentially
rob
ust
to
channel
frequency
selectivity
.
4.1.
MC-CDMA
T
ransmitter
Fig.
2
e
xplains
the
pr
inciple
of
the
tr
ansmitter
f
or
do
wnlink
MC-CDMA
systems
.
The
MC-
CDMA
signal
is
giv
en
b
y
x
(
t
)
=
a
i
p
N
p
N
p
1
X
k
=0
c
i;k
e
2
j
f
k
t
;
(36)
where
f
k
=
f
0
+
k
T
c
,
N
u
is
the
user
n
umber
and
N
p
is
the
n
umber
of
subcarr
iers
,
and
w
e
consider
L
c
=
N
p
.
BER
Analysis
of
MC-CDMA
Systems
with
Channel
Identification
Using
HOC
(M.
Zidane)
Evaluation Warning : The document was created with Spire.PDF for Python.
144
ISSN:
2502-4752
S
p
r
e
a
d
i
n
g
S
/
P
I
F
F
T
D
a
t
a
s
y
m
b
o
l
a
n
a
1
c
1
c
n
T
r
a
n
s
m
i
t
t
e
d
s
i
g
n
a
l
Figure
2.
The
tr
ansmitter
f
or
do
wnlink
MC-CDMA
systems
4.2.
MC-CDMA
Receiver
The
do
wnlink
receiv
ed
MC-CDMA
signal
at
t
he
input
receiv
er
is
giv
en
b
y
the
f
ollo
wing
equation:
r
(
t
)
=
1
p
N
p
P
1
P
p
=0
N
p
1
P
k
=0
N
u
1
P
i
=0
<f
p
e
j
p
a
i
c
i;k
e
2
j
(
f
0
+
k
=T
c
)(
t
p
)
g
+
n
(
t
)
(37)
In
Fig.
3
w
e
represent
the
receiv
er
f
or
do
wnlink
MC-CDMA
systems
.
At
the
reception,
w
e
demodulat
e
the
signal
according
the
N
p
subcarr
iers
,
and
then
w
e
m
ul
tiply
D
e
s
pr
e
a
di
ng
P
/
S
F
F
T
a
n
a
1
c
1
c
n
Re
c
e
i
v
e
d
s
i
gn
a
l
E
q
u
a
l
i
z
a
t
i
on
Ch
a
n
n
e
l
i
d
e
nt
i
f
i
c
a
t
i
o
n
a
i
ˆ
Figure
3.
The
receiv
er
f
or
do
wnlink
MC-CDMA
systems
the
receiv
ed
sequence
b
y
the
code
of
the
user
.
After
the
equalization
and
the
despreading
oper
a-
tion,
the
estimation
b
a
i
of
t
he
emitted
user
symbol
a
i
,
of
the
i
th
user
can
be
wr
itten
b
y
the
f
ollo
wing
equation:
b
a
i
=
N
u
1
X
q
=0
N
p
1
X
k
=0
c
i;k
g
k
h
k
c
q
;k
a
q
+
g
k
n
k
=
N
p
1
X
k
=0
c
2
i;k
g
k
h
k
a
i
|
{z
}
I
(
i
=
q
)
+
N
u
1
X
q
=0
N
p
1
X
k
=0
c
i;k
c
q
;k
g
k
h
k
a
q
|
{z
}
I
I
(
i
6
=
q
)
+
N
p
1
X
k
=0
c
i;k
g
k
n
k
|
{z
}
I
I
I
;
(38)
where
the
ter
m
I,
II
and
III
of
Eq.
(38)
are
,
respectiv
ely
,
the
signal
of
the
considered
user
,
a
signals
of
the
others
users
(m
ultiple
access
interf
erences)
and
the
noise
pondered
b
y
the
equalization
coefficient
and
b
y
spreading
code
of
the
chip
.
4.3.
Minim
um
Mean
Square
Err
or
(MMSE)
equaliz
er
f
or
MC-CDMA
The
MMSE
technique
minimiz
e
the
mean
square
error
f
or
each
subcarr
ier
k
betw
een
the
tr
ansmitted
signal
x
k
and
the
output
detection:
E
[
j
"
j
2
]
=
E
[
j
x
k
g
k
r
k
j
2
]
(39)
IJEECS
V
ol.
1,
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.
1,
J
an
uar
y
2016
:
138
152
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IJEECS
ISSN:
2502-4752
145
The
minimization
of
the
fun
ction
E
[
j
"
j
2
]
,
giv
es
us
the
optimal
equaliz
er
coefficient,
under
the
min-
imization
of
the
mean
square
error
cr
iter
ion,
of
each
subcarr
ier
as:
g
k
=
h
k
j
h
k
j
2
+
1
k
;
(40)
where
k
=
E
[
j
x
k
h
k
j
2
]
E
[
j
n
k
j
2
]
.
The
estimated
receiv
ed
symbol,
b
a
i
of
symbol
a
i
of
the
user
i
is
descr
ibed
b
y:
b
a
i
=
N
p
1
X
k
=0
c
2
i;k
j
h
k
j
2
j
h
k
j
2
+
1
k
a
i
|
{z
}
I
(
i
=
q
)
+
N
u
1
X
q
=0
N
p
1
X
k
=0
c
i;k
c
q
;k
j
h
k
j
2
j
h
k
j
2
+
1
k
a
q
|
{z
}
I
I
(
i
6
=
q
)
+
N
p
1
X
k
=0
c
i;k
h
k
j
h
k
j
2
+
1
k
n
k
|
{z
}
I
I
I
(41)
If
w
e
suppose
that
the
spreading
code
are
or
thogonal,
i.e
.,
N
p
1
X
k
=0
c
i;k
c
q
;k
=
0
8
i
6
=
q
(42)
Eq.(41)
will
become:
b
a
i
=
N
p
1
X
k
=0
c
2
i;k
j
h
k
j
2
j
h
k
j
2
+
1
k
a
i
+
N
p
1
X
k
=0
c
i;k
h
k
j
h
k
j
2
+
1
k
n
k
(43)
5.
Sim
ulation
results
5.1.
Algorithms
test
In
this
subsection
w
e
test
the
perf
or
mance
of
the
proposed
algor
ithms
,
f
or
that
w
e
ha
v
e
considered
tw
o
theoretical
channels
.
The
channel
output
w
as
corr
upted
b
y
an
additiv
e
gaussian
noise
f
or
diff
erent
signal
to
noise
r
atio
and
f
or
50
Monte
Car
lo
r
uns
.
Where
the
signal
to
noise
r
atio
(SNR)
is
defined
b
y:
S
N
R
=
10
l
og
h
2
y
(
k
)
2
n
(
k
)
i
(44)
T
o
measure
the
accur
acy
of
par
ameter
estimation
with
respect
to
th
e
real
v
alues
,
w
e
define
the
Mean
Square
Error
(MSE)
f
or
each
r
un
as:
M
S
E
=
1
q
q
X
i
=0
h
h
(
i
)
b
h
(
i
)
h
(
i
)
i
2
;
(45)
where
b
h
(
i
)
,
i
=
1
;
:::;
q
are
the
estimated
par
ameters
in
each
r
un,
and
h
(
i
)
are
the
real
par
ameters
in
the
model.
5.1.1.
Fir
st
c
hannel
In
this
e
xample
,
w
e
consider
a
non
minim
um
phase
impulse
response
channel,
giv
en
b
y
the
f
ollo
wing
equation:
y
(
k
)
=
x
(
k
)
+
0
:
327
x
(
k
1)
0
:
815
x
(
k
2)
+
0
:
470
x
(
k
3)
z
eros:
z
1
=
1
:
2650
;
z
2
=
0
:
4690
+
0
:
3893
j
;
z
3
=
0
:
4690
0
:
3893
j
:
(46)
In
the
T
ab
le
1
w
e
represent
the
estimated
impulse
response
par
ameters
using
proposed
algo-
r
ithms
.
BER
Analysis
of
MC-CDMA
Systems
with
Channel
Identification
Using
HOC
(M.
Zidane)
Evaluation Warning : The document was created with Spire.PDF for Python.
146
ISSN:
2502-4752
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
=
2048
.
S
N
R
b
h
(
i
)
Al
g
o
1
Al
g
o
2
b
h
(1)
0
:
4867
0
:
2360
0
:
3144
0
:
1033
0
dB
b
h
(2)
0
:
8575
0
:
1798
0
:
7385
0
:
1227
b
h
(3)
0
:
3081
0
:
1821
0
:
0980
0
:
0583
M
S
E
0
:
0900
0
:
1592
b
h
(1)
0
:
4735
0
:
1621
0
:
4344
0
:
1223
4
dB
b
h
(2)
0
:
9292
0
:
1757
0
:
8076
0
:
0848
b
h
(3)
0
:
4937
0
:
2102
0
:
1731
0
:
0658
M
S
E
0
:
0557
0
:
1267
b
h
(1)
0
:
3648
0
:
0907
0
:
4681
0
:
1104
12
dB
b
h
(2)
0
:
8036
0
:
0947
0
:
8522
0
:
1205
b
h
(3)
0
:
3357
0
:
1479
0
:
2839
0
:
0993
M
S
E
0
:
0238
0
:
0863
F
rom
the
T
ab
le
1
w
e
can
conclude
that:
The
MSE
v
alues
,
obtained
using
the
first
algor
ithm
(
Al
g
o
1)
are
small
f
or
all
S
N
R
than
those
giv
en
b
y
t
he
second
algor
ithm
(
Al
g
o
2)
,
t
his
imply
,
that
the
estimated
par
ameters
are
v
er
y
close
to
the
or
iginal
ones
if
w
e
use
the
first
algor
ithm
(
Al
g
o
1)
.
Using
the
tw
o
methods
the
v
ar
iances
of
the
estimated
par
ameters
are
acceptab
le
.
In
the
f
ollo
wing,
Fig.
4,
w
e
represent
the
estimation
of
the
magnitude
and
phase
of
the
channel
impulse
response
f
or
a
data
input
N
=
2048
and
f
or
S
N
R
=
0
dB
.
The
Fig.
4
proof
that
the
proposed
algor
ithms
giv
es
a
v
er
y
good
estimation
f
or
phase
response
,
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
−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)
Measured Channel
Estimated using Algo1
Estimated using Algo2
(Measured; Estimated using Algo1)
(Estimated using Algo2)
(Estimated using Algo2)
(Measured; Estimated using Algo1)
Figure
4.
Estimated
magnitud
e
and
phase
of
the
first
channel
impulse
response
,
using
the
pro-
posed
algor
ithms
,
when
the
data
input
is
N
=
2048
and
an
S
N
R
=
0
dB
the
estimated
phase
are
closed
to
the
tr
ue
ones
,
and
an
impor
tant
estimation
on
the
magnitude
using
first
algor
ithm
(
Al
g
o
1)
,
b
ut
using
the
second
algor
ithm
(
Al
g
o
2)
w
e
ha
v
e
more
diff
erence
betw
een
measured
and
estimated
magnitude
.
IJEECS
V
ol.
1,
No
.
1,
J
an
uar
y
2016
:
138
152
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEECS
ISSN:
2502-4752
147
5.1.2.
Second
c
hannel:
Macc
hi
c
hannel
The
Macchi
channel
is
defined
b
y
the
f
ollo
wing
equation:
y
(
k
)
=
0
:
8264
x
(
k
)
0
:
1653
x
(
k
1)
+
0
:
8512
x
(
k
2)
+
0
:
1636
x
(
k
3)
+
0
:
8100
x
(
k
4)
;
z
eros:
z
1
=
0
:
5500
+
0
:
9526
j
;
z
2
=
0
:
5500
0
:
9526
j
;
z
3
=
0
:
4500
+
0
:
7794
j
;
z
4
=
0
:
4500
0
:
7794
j
:
(47)
The
Macchi
channel
is
a
non
minim
um
phase
because
tw
o
of
its
z
eros
are
outside
of
the
unit
circle
.
In
the
T
ab
le
2
w
e
ha
v
e
summar
iz
ed
the
sim
ulation
results
,
using
proposed
algor
ithms
,
when
the
length
data
input
is
N
=
2048
.
F
rom
the
T
a
b
le
2
w
e
obser
v
e
that
the
par
ameters
estimation
of
the
Macchi
channel
impulse
T
ab
le
2.
T
r
ue
and
estimated
par
ameters
of
macchi
channel
e
xcited
b
y
input
sequence
of
N
=
2048
samples
and
f
or
diff
erent
S
N
R
S
N
R
b
h
(
i
)
Al
g
o
1
Al
g
o
2
b
h
(1)
0
:
8080
0
:
2683
0
:
6118
0
:
1812
b
h
(2)
0
:
1128
0
:
2446
0
:
0502
0
:
2504
0
dB
b
h
(3)
0
:
6271
0
:
1363
0
:
5062
0
:
1376
b
h
(4)
0
:
2035
0
:
1468
0
:
2007
0
:
1320
b
h
(5)
0
:
5041
0
:
1776
0
:
0864
0
:
0397
M
S
E
0
:
0621
0
:
2610
b
h
(1)
0
:
9593
0
:
6516
0
:
7412
0
:
3183
b
h
(2)
0
:
2370
0
:
5398
0
:
0908
0
:
2261
4
dB
b
h
(3)
0
:
8363
0
:
3581
0
:
7028
0
:
2154
b
h
(4)
0
:
1924
0
:
1009
0
:
1853
0
:
0860
b
h
(5)
0
:
8470
0
:
3873
0
:
3811
0
:
3359
M
S
E
0
:
0412
0
:
0904
b
h
(1)
0
:
8336
0
:
2350
0
:
7935
0
:
2648
b
h
(2)
0
:
1777
0
:
2687
0
:
2475
0
:
2782
12
dB
b
h
(3)
0
:
8711
0
:
2001
0
:
8297
0
:
2536
b
h
(4)
0
:
1416
0
:
1433
0
:
1110
0
:
1329
b
h
(5)
0
:
8762
0
:
1695
0
:
6047
0
:
2900
M
S
E
0
:
0052
0
:
0695
response
,
using
the
first
algor
ithm
(
Al
g
o
1)
,
are
not
diff
erent
to
the
tr
ue
ones
compared
with
the
results
obtained
with
second
algor
ithm
(
Al
g
o
2)
.
The
MSE
giv
e
us
a
good
idea
about
the
precision
of
these
algor
ithms
.
In
t
he
Fig.
5
w
e
ha
v
e
plotted
the
estimation
of
the
magnitude
and
phase
of
Macchi
chan
nel,
the
case
of
the
S
N
R
=
4
dB
and
f
or
data
length
of
N
=
2048
.
In
the
Fig.
5
w
e
remar
k
that
the
estimated
magnitude
and
phase
response
using
the
first
algor
ithm
(
Al
g
o
1)
ha
v
e
the
same
allure
in
compar
ison
with
the
tr
ue
ones
.
Concer
ning
the
second
algor
ithm
(
Al
g
o
2)
w
e
ha
v
e
more
diff
erence
betw
een
the
estimated,
magnitude
and
phase
,
and
the
tr
ue
ones
.
T
o
conclude
,
the
first
algor
ithm
(
Al
g
o
1)
is
ab
le
to
estimate
the
phase
and
magnitude
of
the
non
minim
um
phase
channel
impulse
response
in
v
er
y
noisy
en
vironments
with
v
er
y
good
precision.
BER
Analysis
of
MC-CDMA
Systems
with
Channel
Identification
Using
HOC
(M.
Zidane)
Evaluation Warning : The document was created with Spire.PDF for Python.