TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.7, July 201
4, pp
. 5491 ~ 54
9
8
DOI: 10.115
9
1
/telkomni
ka.
v
12i7.471
6
5491
Re
cei
v
ed O
c
t
ober 1
1
, 201
3; Revi
se
d Ja
nuary 7, 201
4
;
Accepte
d
Febru
a
ry 1, 20
14
Variable Step Size Blind Equalization Based on Sign
Gradient Algorithm
Ying Xiao*
1
, Fuliang
Yin
2
1,2
F
a
culty
of Electronic Inform
ation a
nd El
ectrical
En
gin
eeri
ng, Dal
i
an U
n
iv
ersit
y
of T
e
chn
o
lo
g
y
2
Colle
ge of Info
rmation a
nd C
o
mmunic
a
tio
n
Engi
neer
in
g, Dalia
n Nati
on
alit
y Un
iversit
y
Lia
oni
ng D
a
li
a
n
, 0864
11
87
65
624
7
*Corres
p
o
n
id
n
g
author, e-ma
i
l
: xia
o
y
i
ng@
12
6.com
1
, fly
i
n@dlut.edu.cn
2
A
b
st
r
a
ct
T
h
is w
o
rk pro
poses
a v
a
ria
b
le st
ep s
i
z
e
sign
gra
d
ie
nt
alg
o
rith
m to
s
o
lve t
he
prob
l
e
m of b
l
i
n
d
equ
ali
z
at
io
n
u
nder i
m
pu
lse n
o
ise envir
on
ment.
T
h
is
a
l
gor
i
t
hm s
uppr
esse
s the i
m
p
u
ls
e
nois
e
int
e
rfere
n
ce
effectively
bec
ause
of th
e si
gn
oper
atio
n
on th
e i
n
st
ant
ane
ous
gra
d
ie
nt bas
ed
on
c
onstant
mod
u
l
u
s
alg
o
rith
m (CM
A
) cost functio
n
. Meanw
hil
e
, the excess
mea
n
squar
e error
can be furth
e
r
reduc
ed by us
e of
a vari
abl
e ste
p
si
z
e
a
l
gor
ith
m
base
d
o
n
the
it
erative ti
mes a
nd th
e rel
i
a
b
il
ity of the
output
sign
al. Si
mu
lati
o
n
results show
th
at, the variab
le
step si
z
e
bl
ind
equa
li
z
a
ti
on
b
a
sed o
n
sig
n
g
r
adi
ent al
gorith
m
h
a
s the fastest
conver
genc
e r
a
te an
d the l
o
w
e
st st
eady state resi
due
err
o
r co
mp
ared w
i
th
the fractio
n
a
l low
ord
e
r C
M
A,
the non
li
near tr
ansfor
m
CMA and stop-
an
d-g
o
CMA.
Key
w
ords
:
blind e
q
u
a
li
z
a
t
i
o
n
, sign gra
d
i
e
n
t
algorith
m
, CM
A, imp
u
lse n
o
is
e, variab
le step
si
z
e
Copy
right
©
2014 In
stitu
t
e o
f
Ad
van
ced
En
g
i
n
eerin
g and
Scien
ce. All
rig
h
t
s reser
ve
d
.
1. Introduc
tion
Blind equalization technology has pote
n
tial
applications both in
cooperative and non-
cooperative communication systems
be
cause
it does not require a
n
y training se
quence
s
[1]. In
various blind equalization algorithms, CMA has be
en widely applied for its simple and robust
convergen
ce
performance
[2]. The theoretical analys
i
s
of CMA do
es not
consid
er the ch
ann
el
noise which i
s
often assu
med as Gau
ssian whit
e n
o
ise, but the
wireless
cha
nnel noise ofte
n
shows prop
erties of impulse noise rathe
r
than
Gaussian white noise [3]. For imp
u
lse noise
with
no more
than
two order m
o
ments, CMA
becom
es
un
stable or even failure is po
ssible. No
nlinear
transform CM
A [4] and fractional lower o
r
der
CMA [5] can suppre
s
s the impulse
noise and
sh
ow
better performance than
the traditional CMA.
Also
stop-and-go
algorithm [6] depends on
the
reliability o
f
the output signal to decide whether
the equalizer upd
ating or not,
which can avoid
the short-tim
e
high ampl
itude impulse noise
interference
to
obtain robu
st convergen
ce
performance under the impulse noise en
vironment.
However, the
fractional lo
wer o
r
der CMA obtai
ns the robu
st co
nver
gence
at the co
st of
slow converg
ence rate an
d the nonline
a
r transfo
rm CMA can only suppress the distinct imp
u
lse
noise which t
he robu
st con
v
ergence p
e
rformance
can
not be ensu
r
e
d
. The stop-a
nd-go algo
rithm
is often ill-convergence if the impulse
noise in
terference is se
rious. This paper propo
s
e
s
a
variable step
size
sign gra
d
ient algorithm to impr
ove CMA blind e
qualization, in which the
blind
equalizer upd
ating based o
n
sign gradie
n
t and the ste
p
size vary with the iterative times and the
reliability of
the output s
i
gnal. Thus a new
blind equalization algorithm
which has f
a
st
convergen
ce
rate and
robu
st converge
n
c
e pe
rfor
man
c
e und
er the
impulse noise
environment
is
obtained. Simulation results show th
at, compar
ed
with the fr
actional low
order
CMA, th
e
nonlinear tran
sform CMA a
nd stop-an
d-go CMA, t
he variable step
size
sign gra
d
ient CMA blind
equalization has the fastest convergence
rate
and the
lowest steady state residue error.
2. Proposed
Variable Step Siz
e
Si
gn Gradient CMA Blind Equaliz
ation
2.1. The Basi
c Principle of CM
A
The basi
c
pri
n
ciple of the
baseban
d m
odel of
CMA
blind equalization [7] is shown a
s
Figure 1. Th
e
send
sign
al
()
x
n
i
s
tran
smitted on the un
kno
w
n
channel
()
hn
interfered by
no
is
e
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5491 – 54
98
5492
()
nn
, and the received signal
()
yn
is obtained. T
he goal of bli
nd equalization is to imple
m
ent
equalization by
equalizer
()
wn
only rely on the ob
served
signal
()
yn
without the information of
the channel
()
hn
and the se
nd
signal
()
x
n
. The output signal
()
x
n
of the equal
izer
can be
detected by the decision d
e
vice
(.
)
D
to recover the transmitted symbol s
equence a
s
ˆ
()
x
n
.
+
D(
.
)
CM
A
()
x
n
()
nn
()
yn
)
(
~
n
x
)
(
ˆ
n
x
()
wn
()
hn
Figure 1. The
Basic Prin
cip
l
e of CMA Blind Equali
z
ati
o
n
The cost function of CMA is given by [8]:
2
2
2
1
()
(
(
)
)
2
CM
A
Jn
E
x
n
R
(1)
Where
2
R
is the constant modulus which can
be calculated by:
42
2
()
()
RE
x
n
E
x
n
(2)
According to
the stochastic gradient de
scent
algorith
m
, CMA updates the equalizer weights b
y
[9]:
(1
)
(
)
(
)
CM
A
wn
wn
J
n
(
3
)
Where
is the step size. Let
the iterative
error
()
en
is:
2
2
()
()
en
R
x
n
(4)
The equalizer weights updating formula
Equation (3) ca
n be rewritten as:
*
(
1
)
(
)
()()
()
wn
wn
e
n
x
n
y
n
(
5
)
2.2. The Impulse Noise M
odel
Although CM
A blind equal
ization does
not con
s
i
der
the channel
noise interference, it
shows robu
st convergen
ce
performance
under Gau
ssi
an noise con
d
ition, and it c
an obtain better
performance
if use the
fractionally s
paced equal
i
z
er [10]. However, many communication
channel
s are interference
with impulse noisewhi
c
h re
sults in instability of CMA.
Alpha station
a
ry
distribution is often used to describ
e the impulse
noise model and
it has been
applied to ma
ny
communicatio
n
systems [11]. The alp
ha stati
onary distribution is commonly describe
d
by
chara
c
teristic function [12]
which is given by:
e
x
p
1
sgn(
)
t
an
1
2
()
2
e
x
p
1
sgn(
)
l
g(
)
1
jat
t
j
t
t
jat
t
j
t
t
(
6
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Variabl
e Step Size Blind Equali
z
ation B
a
se
d on Sign
Gradi
ent Algo
rithm
(Ying XIAO)
5493
The cha
r
acte
ristic expone
nt
(0
,
2
]
controls the
degree
of the impulse noi
se, that is
the impulse noise become
s
stronge
r if
become
s
sm
aller. On the contrary, the impulse noise
become
s
sm
aller
if
becom
es lage
r. If
02
, the alpha
stationary distribu
tion is named
as
fractional low order stationary distribut
ion (FLOA). Th
e symmetry coefficient
[1
,
1
]
and the
alpha distribu
tion is named as
symmetrical distributionif
0
. If
the symmetry coefficient
0
and the characteristic exponent
2
, the a
l
pha stationary distribution
is as same as the
Gaussia
n
distribution.
The dispersion
co
efficient
0
is almost as same
as the variance of the
Gaussia
n
dist
ribution which
cont
rols the
energy of the alpha distrib
u
tion. The location parameter
aR
expresse
s th
e mean o
r
median value of the
stationary distrib
u
tion. Alpha stationary
distribution is
called as sym
m
et
ry alpha stationary distribution (
SS
) if the symmetry coefficien
t
0
and the location param
eter
0
a
,
SS
has so
me same p
r
operties a
s
Gaussian
distribution such as slickne
ss, unimodality and bell
typ
e
, etc. The study al
so sho
w
s that the alpha
stationary distribution with
the chara
c
teristic expon
ent
12
can sufficiently describ
es the
impulse noise
in the real
world. Therefo
r
e
the chann
el noise i
s
a
s
sumed to b
e
the FLOA-
SS
with
12
. The important difference
between t
he FLO
A
distribution and the Ga
ussian
distribution is that
the FL
OA
distributio
n has no more than
-order moments which result in
failure of CMA blind equalization based on
stochastic
gradient descent algorithm.
2.3. The Sign Gradien
t
Algorithm
The alpha
st
ationary noise is different
from the
White Gaussia
n
noise, a
n
d
CMA
cannot ensure robust
co
nvergence p
e
rformanc
e.
Recently, so
me improved CMA blind
equalization algorithms have been proposed for
blind equalization under impulse noise
environment, and three typical improved algorit
hms are the fractional low order CMA,
nonlinear transform CMA and stop-an
d-go algorit
hm. The
fractional low order CMA blind
equalization
redefined the
co
st function accordin
g
to the statistic pro
perty o
f
the alpha
distribution which is given
by:
2
(,
)
(
(
)
)
p
q
Jp
q
E
x
n
R
(7)
Where
pq
is a fra
c
tion betwee
n
0 and
, meanwhile,
pq
must
meet the co
ndition
p
q
to
ensure
that the co
st function of
the fracti
onal low
orde
r CMA is lim
ited. By using the fractional l
o
w
order mo
men
t
s of the recei
v
ed signal, the fracti
onal lo
w orde
r
CMA obtains ro
bu
st convergen
ce
performance. However, the fractional low order
CM
A does not use
the high order statistic of the
received sign
al and the co
nvergence rat
e
is slow.
The nonlinea
r transform
CMA blind equalization carried nonlin
ear transform on th
e
received sign
al to suppre
s
s the impulse noise,
and
the nonlinear transform p
l
ays soft limiting
effect on
the r
e
ceived signa
l. The nonline
a
r transform function is ofte
n given by:
()
2
(
1
e
x
p
(
2
(
)
)
)
1
f
xn
x
n
(
8
)
Then the cost function of th
e nonlinear transform CMA is:
2
2
2
1
()
(
(
)
)
2
NCM
A
f
Jn
E
x
n
R
(9)
The blind equalizer weights updating of t
he nonlinear transform CM
A is given
by:
*
(1
)
(
)
(
)
(
)
(
)
f
wn
w
n
en
x
n
y
n
(
1
0
)
However, the soft limiting e
ffect of the no
nlinear
transform can only suppre
ss the
large amplitude
impulse noise
, so the robust convergence of
the nonlinear transform
cannot be ensure
d
.
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5491 – 54
98
5494
Stop-and-go
CMA updates the blind e
qualizer ac
co
rding to the r
e
liability of th
e output
signal, which
can avoid th
e impulse noise effe
ct on the instantaneous gradient. However, the
reliability of th
e output sign
al is difficult to be
judged, which results in
performan
ce
degrad
a
tion
of
stop-and-go CMA.
Based on the
above discu
ssion, blind e
qualization u
nder the imp
u
lse noise en
vironment
still doesn’t h
a
ve satisfactory algorithms. From
the
signal transm
i
ssion in the
communicatio
n
system, the impulse noise
has
influence on the recei
v
ed signal
()
yn
, th
e output signal
()
x
n
and
the error function
()
en
of the gr
adient based
on CMA for the blind equalizer weights updating.
The gradient of CMA
for blind equaliz
ation weights updating is given
by:
*
()
()()
()
CM
A
J
ne
n
x
n
y
n
(11)
Sign error gradient desce
nt algor
ithm [13] has b
een
proved to be
an effective algorithm
to reduce the
computational complexity
and impr
ove the performance of the stochastic g
r
ad
ient
desce
nt algorithm. Signum operation
can
suppress
the impulse
noise effectively. However th
e
sign error gra
d
ient descent algorithm ca
rries signum o
peration only
on the error
()
en
, t
h
e effec
t
of impulse noise on the received signal
()
yn
and the outp
u
t signal
()
x
n
still
hinder the robust
convergen
ce
performance. Hereby we
ca
rry the signum operation on the error
()
en
, the
received
signal
()
yn
and th
e output sign
al
()
x
n
at the
sa
me time, a n
e
w algo
rithm
called
sign
gradien
t
algorithm is obtained. The
sign gradient is given by:
*
()
(
(
)
)
(()
)
(
()
)
SCM
A
J
n
s
i
g
n
e
ns
i
g
n
x
ns
i
g
n
y
n
(12)
Where
(.)
sig
n
denote
s
signum ope
ration and for
the complex
x
the signumop
eration is give
n by:
()
(
(
)
)
*
(
()
)
si
gn
x
s
ign
r
eal
x
j
s
ign
i
mag
x
(13)
Then the blin
d equalizer weights updating formul
a of
the sign gradient CMA is given by:
(1
)
(
)
(
)
SCM
A
wn
wn
J
n
(
1
4
)
2.4. The Variable Step Size Algo
rithm
Sign gradient
CMA can
obt
ain robu
st co
nverge
nce
pe
rformance
un
der the impul
se noi
se
environment. However, the signum
op
eration cau
s
e
s
the qua
ntization error for cal
c
ulating
the
gradient which results in large
steady state residual
error. Al
though the excess mean squa
re
error of CMA
still has no effectiv
e quantitative analysis method [14].
the step size
can control the
excess mean
square error
after the algo
rithm conv
ergence [15]. As
long as the step size is sm
all
enough, the e
x
cess mean square erro
r can be reduce
d
to a desired degree.
The step size controls the convergen
ce ra
te and the convergen
c
e precision [
16] an
d
variable step size algo
rithm is in a co
mpromise b
e
t
ween them
[17]. The basic prin
ciple
o
f
variable step size algo
rithm is that faster convergen
ce
rate is obtain
e
d with a big
step size
at the
initial stage,
and the step
size de
crea
ses g
r
adua
lly along with the iteration proce
ss to obt
ain
higher convergence
pre
c
ision. Ther
efore, variable step size
algor
ithm can red
u
ce the exce
ss
mean square
error, as a
result, the va
riable st
ep si
ze sign gra
d
ient CMA can obtain better
performance
under impul
se noise
environment. The blind equalize
r
updating formula of variable
step size sign
gradient CMA is given
by:
(1
)
(
)
(
)
(
)
SCM
A
wn
wn
n
J
n
(15)
Where
()
n
is the
step size
gai
n control f
unction and it meets the condit
ion
0(
)
1
n
,
which co
ntrols the change
scale of the step size in
th
e iterative
process. The ideal step size gain
control function should m
eet the condi
tions that
()
1
n
at the initial and
()
0
n
[18] a
f
t
e
r
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Variabl
e Step Size Blind Equali
z
ation B
a
se
d on Sign
Gradi
ent Algo
rithm
(Ying XIAO)
5495
convergen
ce. In the most
variable step
size algo
ri
thm, the step size gain
control function is
set
acco
rding to
a nonlinear transform of th
e output er
ror of the blind e
qualizer
and
some p
a
rame
ters
need to be set. For there
is limited received data sa
mples at one iterative
time, to obtain
th
e
accu
rate successive estimation error and the other
information o
f
the ou
tput
with the channel
noise interference i
s
very
difficult. So the par
ameters setting of the nonlinear transform a
r
e
not
universal. We
hereby p
r
op
osed a
ne
w variable step
si
ze algorithm
for sign g
r
adi
e
nt CMA. In this
algorithm, the step si
ze gai
n control function
()
n
does not
rely on the in
formation of the output
signal or output error, and also it needs no m
an-m
ade setting parameters. T
he step size
gain
control function is given by:
(
)
1
(
2
(
1
e
xp(
)
)
1
)
n
(16)
Where
is the
order of magnitude of
the
maximum st
e
p
size which
can be calcul
ated accordi
n
g
to the
input signal of the
blind equalizer
as:
ma
x
m
a
x
1
(17)
Where
max
is the
maximum eigenvalue of the autocor
relation matrix of the input sign
al.
is the
ratio of the number of the
sign
consi
s
tency and
the
iterative times, and the
sign consiste
ncy
refers to that
the erro
r sig
n
based
on
CMA and de
cision de
cide
d (DD) alg
o
rithm is same.
The
cost function of DD algorithm is given b
y
:
2
1
()
(
(()
)
(
)
)
2
DD
J
ns
i
g
n
x
n
x
n
(
1
8
)
Let the error function of DD algorithm is:
()
(()
)
(
)
D
en
s
i
g
n
x
n
x
n
(
1
9
)
If
the blind e
qualizer obtai
ns right
conve
r
gence, the si
gn of the error
()
D
en
acco
rding to
DD algo
rithm and the e
rror
()
en
according to
CMA will be t
he same
. Therefore, we
can use the
sign co
nsiste
ncy to judge the con
s
tella
tion open
state. Based on this idea,
can be
calculated
by:
kN
(
2
0
)
Where
N
is the iterative times and
k
is calculat
ed by:
1(
(
)
)
(
(
)
)
(
(
))
(
(
))
DD
DD
k
k
if
sign
e
n
sign
e
n
k
k
if
si
gn
e
n
si
gn
e
n
(
2
1
)
Thus, the new variable step size algorithm is given b
y
:
ma
x
((
)
)
(
(
)
)
=
()
DD
s
ig
n
e
n
s
ig
n
e
n
ne
l
s
e
(
2
2
)
According to Equation (22), the
step size is
diminished when the error sign of the CMA
and DD algorithm is same, otherwise, the blind
equalizer weights i
s
updated with the maximu
m
step size. CMA blind equalization based on sign gradi
ent algorithm with the
variable step siz
e
method can o
b
tain faster convergence
rate at t
he initial stage and f
u
rther lowe
r the excess m
e
a
n
square e
rror.
Meanwhile, the parameters
and
of the step size gain
control function both can
be calculated in the program without ma
n-made setting.
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ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5491 – 54
98
5496
3. Simulations and Dis
c
u
ssion
In the simulations, equivalent probability bi
nary sequence is adopt
ed to act as sending
signal and QPSK modulation is utilize
d
. The typi
cal shallow sea and deep sea underwater
acoustic
cha
nnel with impulse noise i
n
terfer
ence a
r
e used in o
u
r simulations. The chan
nel
models have verified by th
e sea experiment. For
the
shallow sea
channel mod
e
l, the parameters
set as follow: the carrier fr
equency is 10kHz, the channel bandwi
dth is 2kHz, the transmit baud
rate is 1000
sym
bol/
s
, the wind
spee
d is 20
kn, the sende
r an
d the receive
r
are lo
cated
in
underwater 1
0
meters and
the distance
is 5000 meters. For the
deep sea
ch
annel model, th
e
parameters
set as follow: the depth of sea is
5000 meters, the sound source is located in 1000
meters underwater, the receiver is loca
ted in
the 90
0 meters und
e
rwater, the distance between
the sound so
urce and the
receiver is 56
kilomete
rs, the carrie
r
fre
quency is 1kHz, the transmit
baud rate is 100
sym
bol/
s
. The parameters of eight rays of
the cha
n
nel models [19] can be sh
own
as Table 1.
Table 1. The
Paramete
rs o
f
Eight Rays of the Chan
n
e
l Model
s
Ra
y
number
Shallow
sea cha
nnel
Deep sea chann
el
Time dela
y
t/
m
s
Amplitude Time
dela
y
t/
m
s
Amplitude
1 0.000
1.0000
0.0000000
0.4954
2 0.026
-1.0000
0.0265385
-0.1464
3 0.026
-0.3286
0.0319367
0.5079
4 0.100
0.3286
0.0647739
-0.1555
5 0.100
0.3286
0.2056037
0.8399
6 0.240
-0.3286
0.2320864
1.0000
7 0.420
-0.1080
0.2359591
0.6914
8 0.420
0.1080
0.3671784
0.2187
The length of the blind equalizer
N
=
21
for the shallow sea
chan
nel and
N
=
3
4
for
the
deep sea cha
nnel.In order to verify
the
performanc
e of the
variable step size blind equalization
based on sign gradient al
gorithm (VSSG-CMA) prop
osed in this
paper, the fractional low order
CMA (FL-CM
A), the nonlinear transform
CMA
(NT
-
CMA) and stop
-and-go
CMA
(SAG-CMA)
are
done in the si
mulations for
compari
s
on.
The compa
r
ison is in term
s of the residu
al inter-symb
o
l
interference (ISI) [20
]
which is given b
y
:
22
ma
x
2
ma
x
ii
i
i
CC
IS
I
C
(23)
Where
C
is the combined impulse resp
o
n
se of the channel and the equalizer.
Because the
-stationary distribution with the cha
r
a
c
teristic exp
onent
has
no
statistical moments above
order, the signal-to-noise
(
SNR
) define
d
based on the two order
statistics can
not describe
the degre
e
of the im
pulse noi
se i
n
terference.
Therefore, the
generalized
SN
R
(
GSNR
)
which can me
asure the impulse noise
in the signal [21] is defined as:
2
10
l
g
(
)
GS
NR
x
n
(24)
Where
is the dispersion co
efficient of
th
e
-stationary distribution impulse noise
an
d
2
()
x
n
is the signal energy. Figure 2 and Figure 3 sho
w
the
ISI
comparison re
sults u
nder the
shallow sea
channel and
the deep
se
a
channel
with
GSNR
=1
5dB
respe
c
tively.
The step
size
is
s
e
t to
0.0
015
in the shallow se
a channel and
0.0024
in the d
eep sea ch
annel
simulation. From Figure
2
and Figure 3 can
see
that VSSG-CMA has the fastest convergence
rate and the
lowest steady
state
residual error,
which proved
the effectiveness of VSSG-CMA
blind equalization under the
impulse noise environment.
In order to further prove th
e effectiveness of VSSG-CMA b
lind equalization, the steady
state residual
errors
of VSSG-CMA, FL-CMA
, NT-CMA and SAG-CMA are compared by 500
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TELKOM
NIKA
ISSN:
2302-4
046
Variabl
e Step Size Blind Equali
z
ation B
a
se
d on Sign
Gradi
ent Algo
rithm
(Ying XIAO)
5497
times Monte Carlo simulat
i
on under different
GSNR
conditions an
d the results are sho
w
n i
n
Table 2. From Table 2 can see that, VSSG-CMA
blind equalization has th
e lowest
steady state
residual error in the
four blind equalization algorithms.
Figure 2
.
IS
I
in Shallow Se
a Channel
F
i
g
u
r
e
3
.
ISI
in Deep Sea Channel
T
a
b
l
e
2. The Compari
s
ion of St
eady Sta
t
e Residual Errors (dB)
Channel
GS
NR
(dB
)
Algorithm
0 5
10
15
20
25
The
shallow
sea
channel
FL-CMA
-8.5
-9.7
-10.3
-13.2
-15.6
-18.3
SAG-CMA
-8.8
-12.2
-14.5
-16.1
-18.4
-19.6
NT-CMA
-9.4
-13.5
-15.2
-17.3
-18.9
-21.2
VSSG
-CMA
-10.6
-16.8
-17.4
-18.0
-22.6
-28.6
The deep
sea
channel
FL-CMA
-9.0
-10.4
-12.2
-14.2
-16.4
-18.8
SAG-CMA
-9.8
-12.6
-16.5
-21.2
-24.5
-24.5
NT-CMA
-11.2
-12.8
-17.2
-21.3
-24.6
-24.8
VSSG
-CMA
-14.8
-18.5
-24.9
-28.4
-32.8
-36.5
4. Conclusio
n
In this work, we propo
sed
a sign gradie
n
t algorithm for CMA to solve the problem of blin
d
equalization
under impulse noise environment. The
signum ope
r
ation on the iterative gradien
t
can sup
p
re
ss the impulse noise effectively, wh
ich ensure
s
the blind equalization algorithm to
obtain
robust convergen
ce performance.
Furthermore
,
a variable st
ep size
algori
t
hm is design
ed
acco
rding to
the iterative
times and the reli
ability of the output signal without man-m
ade
parameters
setting
to impr
ove the perfo
rmance of
sig
n
gradient algorithm. The simulation results
show the effectiveness of the variable step size
sign
gradient CMA blind equalization under the
impulse noise
environment.
Ackn
o
w
l
e
dg
ements
This
wo
rk
wa
s supp
orted
i
n
pa
rt by The
Nation
al Natural S
c
ien
c
e
Found
ation o
f
Chin
a
(612
014
18),
Funda
mental
Re
se
arch F
und
s
for
the
Ce
ntral Uni
v
ersitie
s
(DC1201
0218
) a
n
d
Liaoni
ng Prov
ince
High School Tale
nt Suppo
rt Prog
ra
m (LJQ20
131
26).
0
10
00
20
00
30
00
40
00
50
00
-2
0
-1
5
-1
0
-5
0
5
I
t
er
ati
v
e
ti
m
e
s
n
IS
I
/d
B
SA
G
-
C
M
A
VS
S
G
-
C
M
A
NT
-
C
M
A
FL
-
C
M
A
0
1000
2000
3000
4000
50
00
-3
0
-2
5
-2
0
-1
5
-1
0
-5
0
5
I
t
er
at
i
v
e t
i
m
e
s
n
IS
I
/d
B
FL
-
C
M
A
SA
G
-
C
M
A
NT
-
C
M
A
V
SSG
-
C
M
A
Evaluation Warning : The document was created with Spire.PDF for Python.
ISSN: 23
02-4
046
TELKOM
NI
KA
Vol. 12, No. 7, July 201
4: 5491 – 54
98
5498
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