TELKOM
NIKA Indonesia
n
Journal of
Electrical En
gineering
Vol.12, No.7, July 201
4, pp
. 5379 ~ 53
8
6
DOI: 10.115
9
1
/telkomni
ka.
v
12i7.552
9
5379
Re
cei
v
ed
No
vem
ber 2
1
, 2013; Re
vi
sed
Febr
uary 18,
2014; Accept
ed March 6, 2
014
Multi-user Detection Based on
Gaussian Sum Particle
Filter in Impulsive Noise
Li Zhihui*
1
, Xian
Jinlong
2
Coll
eg
e of Information Sci
enc
e and En
gi
neer
ing, He
na
n Uni
v
ersit
y
of T
e
chnol
og
y
Lia
nhu
a street hig
h
tech zon
e
, Z
hengzh
ou ci
t
y
, 45
000
1, Hen
an, Chi
na.Ph:+
86-1
862
37
16
9
1
5
*Corres
p
o
ndi
n
g
author, em
ail
:
zhihui
_li
5
1
1
@
sina.c
o
m
1
, you
y
o
u
_
70@
12
6.com
2
A
b
st
r
a
ct
In order to i
m
p
r
ove the p
e
rforma
nce of
multi
-
user detector,
th
is paper
ana
lyses a new
al
gorith
m
,
Gaussia
n
su
m particle filter (
G
SPF
).
T
h
is algorith
m
a
pprox
imates the filte
r
ing a
nd pr
edic
t
ive distrib
u
tion
s
by w
e
ig
hted G
aussi
an
mixt
ur
es an
d is
bas
i
c
ally
ba
nks of
Gaussia
n
p
a
rti
c
le filter
s
(GPF
). T
hen, GSPF
is
used
in dy
na
mic state
spa
c
e (DSS)
mo
dels w
i
th n
o
n
-
Gaussia
n
no
i
s
e. T
he no
n-
Gaussia
n
n
o
is
e i
s
appr
oxi
m
ate
d
by La
pl
ace
noi
se an
d Al
ph
a n
o
ise. As
a re
s
u
lt, GSPF
can ef
fectively re
duc
e the
bit err
o
r r
a
te
of the system
. The sim
u
lation result
s show
that the GSP
F
has the ve
rsatility and super perfor
m
anc
e
i
n
MUD. It proves that the im
pr
ov
ed algorithm
has im
portant value
for the research of MUD system
.
Ke
y
w
ords
:
pa
rticle filter, Gau
ssian su
m p
a
rti
c
le filter
, mu
lti-
user detecti
on,
Bayesia
n
esti
mati
on
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
CDMA
suffe
rs fro
m
multipl
e
-a
ccess i
n
te
rfere
n
ce (MA
I
) with
all cell
ular
system,
and the
fact
of stro
ngl
y
powered users
ma
sking
wea
k
e
r
user
s is kno
w
n
as
the ne
ar-far e
ffect. Since
th
e
multi-u
s
er d
e
t
ection which
is se
rved a
s
one of the
key techn
o
l
ogy of 3G a
nd B3G mob
ile
comm
uni
cati
ons i
s
put fo
rwa
r
d by K.S.Schneid
e
r
in
1979, it qui
ckly received
a great d
eal
of
attention
d
ue to
its potentia
l
for red
u
ci
ng
the
effe
ct
s of MAI,
comb
ating with
the
n
ear-far proble
m
and thereby increa
sing th
e cap
a
ci
ty of CDMA sy
ste
m
. In 1986, optimum mul
t
i-use
r
dete
c
t
i
on
algorith
m
i
s
p
u
t forward by
Verd
u [1]. T
he p
e
rfo
r
man
c
e
of this ne
w al
gorith
m
can a
pproa
ch
the
singl
e-u
s
e
r
re
ceiver.
Becau
s
e th
e
optim
um M
U
D i
s
e
x
ponential
in
compl
e
xity, this
metho
d
can
not be
a
c
hi
eved b
a
si
cal
l
y in en
gine
ering.
The
n
many ap
pro
x
imate dete
c
tors have
b
een
pre
s
ente
d
to
redu
ce the
compl
e
xity, such a
s
de
co
rrelation d
e
te
ctor a
nd MM
SE detector
[2].
H
o
w
e
ve
r
,
th
es
e
de
te
c
t
or
s
u
s
e
in
te
r
i
m
ha
r
d
de
c
i
s
i
o
n
s, there pe
rformance is
su
b
optimal. In these
algorith
m
s, th
e ba
ckground
noise i
s
a
s
sumed to
be
Gau
ssi
an
whi
t
e noise. In a
c
tual life, a l
o
t of
noises
are non-Gaussi
an noise
[3]. So the performances of
t
hese algorithms
will reduce
seri
ou
sly in non-Gau
s
sia
n
noise envi
r
onm
ent.
Mu
lti-use
r
dete
c
tion techn
o
lo
gy still need
s
contin
uou
s i
m
provem
ent.
In 1993,
N.J.Gordo
n
p
r
op
ose
d
pa
rticl
e
filt
er (PF
)
m
e
thod, which
is u
s
e
d
for
tricking
sign
als in the
referen
c
e [4].
Most
re
ce
ntly, particl
e
filter is u
s
ed
for simultaneo
us l
o
cali
zatio
n
an
d
mappin
g
a
n
d
huma
n
tra
cking[5][6]. This tech
nique
can be
ap
plie
d to no
n-lin
e
a
r n
on-Gau
s
sian
system. In other words, th
e algorith
m
has a st
rong
robu
stne
ss. PF algorithm,
a Bayesian
-b
ase
d
solutio
n
, is a
pplied to CDMA MUD in [
7
]. But
particle degradatio
n is a sig
n
ificant dra
w
ba
ck of
the particl
e filter. Gau
ssi
an
sum pa
rticle
filter
(GSPF) algorithm
ca
n solve this p
r
oble
m
. In this
pape
r, we an
alyze the sta
ndard pa
rticl
e
filter and G
aussia
n
sum
particl
e filter. The simul
a
tion
res
u
lt
s
sho
w
t
hat
GS
P
F
ca
n ef
f
e
ct
iv
ely
r
edu
ce the bit error rate of the syste
m
.
This p
ape
r is orga
nize
d a
s
follow: Se
cti
on 2 incl
ude
s the mod
e
l of the CDMA
system
and the p
r
e
s
entation of two ki
nd
s of non-Gau
s
si
a
n
noises; th
e appli
c
ation
of the stan
dard
particl
e
filter algorith
m
for multi-u
s
er de
tection
i
s
pre
s
ente
d
in Se
ction 3; Multi
-
user
dete
c
tion
based o
n
GS
PF is p
r
e
s
en
ted in Se
ctio
n 4; Simulati
on result
s a
r
e de
scribe
d i
n
Section
5
and
con
c
lu
sio
n
is
pre
s
ente
d
in Section 6.
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: 5379 – 53
86
5380
2. Introduc
tion of Sy
stem Model
2.1. CDM
A Sy
stem Model
Con
s
id
erin
g
a syn
c
hrono
u
s
CDMA
syst
em with
K u
s
ers and
sym
bol interval i
s
T. The
received si
gn
al can b
e
given by:
1
()
()
(
)
()
()
K
kk
k
k
rt
A
t
g
t
b
t
n
t
(
1
)
In the above
formula,
k
A
is
amplitude
of the k-th
user’s si
gnal;
{1
,
1
}
k
g
is the
norm
a
lized d
e
termini
s
tic
si
gnature
wave
form a
ssi
gne
d to k-th u
s
e
r
;
{1
,
1
}
k
b
is the
k-th u
s
er
symbol;
()
nt
is backgro
und n
o
i
s
e.
The matched
filter output
k
y
of the k-th user is expre
s
sed
as [8]:
0
T
kk
y
rt
g
t
d
t
,
0
1
1
=(
)
(
)
b
K
T
kk
i
k
i
i
k
i
b
ik
A
bA
b
n
t
g
t
d
t
T
kk
k
k
A
bM
A
I
z
(
2
)
Whe
r
e
kk
A
b
is the
sign
al of the
k-th
user;
k
M
AI
is
multiple access
interfe
r
en
c
e
(MAI);
k
z
is noi
se.
,
ik
is the cross-correlation bet
we
en the i-th
user and the
k-t
h
use
r
. It is defined to be:
,
0
1
()
()
b
T
ik
k
i
b
g
tg
t
d
t
T
(
3
)
Whe
n
ik
,
,
01
ik
; when
ik
,
1
ik
. The la
rg
er th
e
ik
is
, th
e
s
t
r
o
ng
er
th
e
cro
s
s-co
rrelat
ion between
different users is.
In orde
r to si
mplify analysi
s
, the re
ceive
d
vector
can
be expre
s
sed
as matrix.
y
RAb
z
(
4
)
}
,
,
,
{
2
1
K
A
A
A
diag
A
is a di
ago
n
a
l matrix of
the po
wer
of the co
rre
s
po
nding
received sig
nal.
T
K
b
b
b
b
]
,
,
,
[
2
1
is u
s
e
r
s’ data,
R
is
a symmet
r
ic correl
a
tion matrix.
T
K
z
z
z
z
]
,
,
,
[
2
1
, it is a
co
mplex-valu
ed
vector with
inde
pend
en
t real
and
i
m
agina
ry
comp
one
nts
and covaria
n
c
e matrix eq
u
a
l to
2
R
.
R
is a
symmetri
c
matrix, Coli
cky fa
ctori
z
ati
on
can b
e
em
ployed. The
r
e
is a u
n
iqu
e
l
o
we
r
triangul
ar mat
r
ix
F
s
u
ch that
T
R
FF
. We apply
T
F
to Equation (4
), we ca
n obtai
n [9]:
T
yF
yF
A
b
z
(
5
)
It can b
e
pro
v
ed that the
covarian
ce
ma
trix of
z
is
2
I
, wh
ere
I
is the identity matri
x
.
Becau
s
e
the
noise be
co
m
e
s in
dep
end
e
n
t and
i
denti
c
ally di
strib
u
ted, white
noi
se,
y
is
called
the whitene
d
matche
d filter output. S
c
al
ar
expre
s
sion
of the
re
ceiv
ed
sign
al
can
be
expresse
d
as:
,
1
K
kk
l
l
l
k
l
yF
a
b
z
(
6
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Multi-u
s
er
De
tection Based
on Gau
ssi
an
Sum
Particle Filter in Im
pulsive Noi
s
e (L
i Zhihui)
5381
The pu
rpo
s
e
of the detecti
on is
to detect signal
s of the use
r
s
}
,
,
,
{
2
1
:
1
K
K
b
b
b
b
from
the matche
d filter output si
gnal
s
}
,
,
,
{
2
1
:
1
K
K
y
y
y
y
.
2.2. Non-Ga
ussian Nois
e Simulation
In order to
simplify math
ematical
an
a
l
ysis, b
a
ckg
r
ound
noi
se
i
s
often
a
s
su
med
as
Gau
ssi
an n
o
i
s
e. But this a
s
sumption
is
not very a
c
cu
rate. In p
r
a
c
tice, ma
ny kin
d
s of
noi
se
s
do
not have
G
a
u
ssi
an
natu
r
e,
su
ch
as thu
n
der an
d lig
htning, i
c
e
aval
anche
s, all
ki
nds of m
a
chi
n
e
motors, neo
n
sign
s, etc.
These ki
nd
s
of noise
sh
o
w
si
gnifica
nt pea
k amplit
ude in the ti
me
domain. In order to preve
n
t the detection perfo
rm
a
n
c
e de
cline u
nder the “noi
se spi
k
e
s
”. It is
very necessa
ry to establi
s
h a mo
re a
ccurate
m
odel.
Two m
odel
s
of the non
-G
aussia
n
noi
se will
be discu
s
sed
briefly in the followin
g
s.
Firstly, intro
d
u
ce th
e L
apl
ace
noi
se
wh
ich i
s
on
e
kin
d
of no
n-G
a
u
ssi
an n
o
ise. Lapla
c
e
probability density function
(PDF
) can be expressed as [3]:
2
2
12
()
e
x
p
(
)
2
px
x
(
7
)
In the ab
ove
formula,
2
is varian
ce
or power
of the noi
se. La
pl
ace P
D
F
ha
s an
obviou
s
sm
e
a
ring. T
h
is i
s
the main dif
f
eren
ce b
e
tween the L
apl
ace
noise an
d the Gau
s
si
an
noise.
Secon
d
ly, introdu
ce
an
oth
e
r
kind
of n
on-G
a
u
ssi
an
noi
se, Alph
a sta
b
le n
o
ise. If the
rand
om varia
b
le X wa
s subje
c
t to the Alpha stabl
e
distrib
u
tion
, its characte
ristic fu
nction
is
expre
s
sed a
s
[10, 11]:
(
)
e
xp{
[
1
sgn(
)
(
,
)
]
}
uj
a
u
u
j
u
u
(
8
)
1
|
|
log
)
/
2
(
1
)
2
/
tan(
)
,
(
u
u
(
9
)
1,
0
sgn(
)
0
,
0
1,
0
u
uu
u
(
1
0
)
In these form
ulas,
0,
2
is
calle
d characte
ri
stic in
dex, it de
termine
s
th
e
degree
of
the di
strib
u
tion p
u
lse
cha
r
acte
ri
stics. The small
e
r of
the
, the
more
obvio
u
s
of
the
pul
se
cha
r
a
c
t
e
ri
st
ic
s.
W
hen
02
, this di
strib
u
tion
is n
a
med
fra
c
tional
lo
wer
orde
r Alp
ha
stable
distrib
u
tion. Whe
n
2
, it is
Gau
ssi
an
dist
ribution
which its m
ean
is
a
an
d vari
an
ce is
2
2
.
In other
wo
rds, the G
a
u
ssi
an di
stri
b
u
tion is
a speci
a
l case of the
-sta
bl
e distri
bution
.
11
, it is symmet
r
y paramete
r
whi
c
h
can
contro
l the
gra
d
ient of the d
i
stributio
n. Whe
n
0
, this distri
bu
tion is a sy
mmetric
-sta
ble dist
ributio
n and d
enot
ed as
SS
. When
1
and
0
, this dist
ribution i
s
call
ed Cau
c
hy di
stributio
n.
is
s
c
attering coeffic
i
ent. It is
simila
r to the varian
ce
of the Gau
s
sian di
stri
but
ion. Alpha stable noi
se’
s
powe
r
can
be
expre
s
sed a
pproxim
ately as
2
, but
2
is n
o
t equal
com
p
letely to the
true po
we
r.
Signal to
Noi
s
e Ratio (SNR) ca
n be
expre
s
ses a
s
/2
SNR
S
(
S
is the si
g
nal’s p
o
wer).
a
is a re
al
numbe
r. Wh
e
n
0
a
,
1
, this distri
bution is n
a
m
ed stan
dard
-stable di
strib
u
tion.
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: 5379 – 53
86
5382
3. Multi-user
Dete
ction
Based on Sta
ndard Par
t
icle Filter Algo
rithm
Particle filter (PF) algo
rithm is a Mo
nt
e Carl
o m
e
thod which is ba
sed o
n
Bayesian
theory. In this algo
rithm, poste
rio
r
dist
ributi
on of th
e states i
s
repre
s
e
n
ted
by importan
c
e
sampli
ng a
n
d
re
sampli
ng
. Its core i
d
ea is to ex
p
r
ess the
po
sterior
distri
bu
tion by a set
o
f
sampl
e
s with
associ
ated weig
hts,
and usin
g
them
to com
pute e
s
timate
s of the sign
als. All
the
receiving
dat
a an
d the
p
r
io
r info
rmation
are
co
mbin
ed
in the
po
ste
r
i
o
r
distri
bution
1:
1:
(|
)
k
k
p
by
.
If
the sam
p
les
were from t
h
e accurate
posteri
o
r
probability distribution
1:
1:
(|
)
k
k
p
by
, each sa
mple
has the
sam
e
weight. Ho
wever, i
n
p
r
a
c
tice,
1:
1:
(|
)
k
k
p
by
ha
s n
o
t the typical sol
u
tion. So
the
sampli
ng p
r
o
c
e
ss i
s
very
difficult to achieve. In
ord
e
r to solve this p
r
obl
em, the parti
cle
s
are
often obtaine
d from an i
m
portan
c
e
den
sity function
1:
1:
(|
)
k
k
qb
y
. The weight
s of particl
es
are
defined a
s
[1
2, 13]:
1:
1:
1:
1:
(|
)
(|
)
i
i
k
k
k
i
k
k
px
y
qx
y
s
N
i
,
,
2
,
1
(
1
1
)
Whe
r
e
i
is the traje
c
tory i
ndex. The im
porta
n
c
e
den
sity function
can b
e
de
co
mposed
into:
1:
1
:
1
1
:
1
1:
1
:
1
:
1
(|
)
(
|
,
)
(
|
)
kk
k
k
kk
k
qx
y
q
x
x
y
q
x
y
(
1
2
)
Assu
me that
the parti
cles
1:
1
i
k
x
from
1:
1
1:
1
(|
)
i
k
k
px
y
have been gen
erate
d
with weig
hts
1:
1
i
k
. If partic
l
es
i
k
x
are
sam
p
led
from the
pro
posal di
stribu
tion and
app
ende
d to
1:
1
i
k
x
,
1:
i
k
x
can b
e
obtain
ed.
The po
steri
o
ri
proba
bility density functio
n
can b
e
expressed a
s
:
1:
1
1
:
1
1:
1
:
1
(|
)
(
|
)
(
|
)
(
|
)
kk
k
k
k
kk
k
px
y
p
y
x
px
x
p
x
y
(
1
3
)
By sub
s
titutin
g
(14) with
(12) an
d (11
)
,
we
can
get t
he u
pdate
d
f
o
rmul
a of i
m
portan
c
e
weig
hts.
11
:
1
1
1:
1
1
1:
1
1
:
1
1
:
1
1:
1:
1
1
:
(|
)
(
|
)
(
|
)
(
|
)
(
|
)
(|
,
)
(
|
)
(
|
,
)
ii
i
i
ii
i
i
i
kk
k
k
kk
k
kk
k
k
k
ii
i
i
i
kk
k
k
k
kk
k
p
y
x
p
x
x
px
y
p
y
x
px
x
qx
x
y
q
x
y
q
x
x
y
(14
)
In the standa
rd pa
rticle filter algo
rithm, we
choo
se th
e prio
ri pro
b
a
b
ility density functio
n
as the impo
rt
ance den
sity function.
1:
1
1
1:
(|
,
)
(|
)
ii
ii
kk
k
k
k
qx
x
y
px
x
(
1
5
)
By substitutin
g
Equation (1
5) with Equ
a
tion (16
)
, the formul
a is sim
p
lified as:
1
(|
)
ii
i
kk
k
k
p
yx
(
1
6
)
Normali
z
e th
e weig
hts:
1
/
s
N
ii
i
kk
k
i
(
1
7
)
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Multi-u
s
er
De
tection Based
on Gau
ssi
an
Sum
Particle Filter in Im
pulsive Noi
s
e (L
i Zhihui)
5383
Whe
n
the
course of al
g
o
rithm i
s
finishe
d
with th
e last u
s
e
r
K, we ca
n o
b
tain the
gene
rated p
a
r
ticle
s
and th
ere weight
s that can
a
pproximate the poste
rio
r
pro
bability den
sity
function.
For
example, th
e
marginali
z
e
d
po
steri
o
r probability d
e
n
s
ity functio
n
1:
(|
)
k
k
p
xy
ca
n
be expre
s
sed
as:
1:
1
(|
)
(
)
s
N
ii
kK
k
k
K
i
p
xy
x
x
(
1
8
)
In the abov
e formul
a,
()
is
D
i
r
a
c
de
lta
fu
n
c
tion. W
e
de
fine
tw
o
vec
t
o
r
s
,
T
N
k
k
k
k
s
x
x
x
x
]
,
,
,
[
2
1
,
T
N
k
k
k
k
s
]
,
,
,
[
2
1
. Ac
c
o
rding t
o
the Maximum A Pos
t
erior (MAP)
rule, we have:
()
T
k
kk
bs
i
g
n
x
(
1
9
)
A major p
r
obl
em with the
PF algorith
m
is that
mo
st o
f
the particl
es’ weig
hts except for a
very few
are
negligibl
e
. Th
is p
r
obl
em i
s
calle
d p
a
rticl
e
s
deg
rad
a
tion. The
de
ge
nera
c
y p
r
obl
em
implies that a
larg
e compu
t
ation is
wa
st
ed on
upd
ating the
parti
cl
es
who
s
e
co
ntribution
to the
approximatio
n to
1:
(|
)
k
k
p
xy
is almo
st
zero. Re
sam
p
ling can solve this proble
m
. The ba
sic
idea
of resam
p
ling is that sample
s
N
times from the
posterior
probability density function
1:
1
(|
)
(
)
s
N
ii
kK
k
k
K
i
px
y
x
x
, then ne
w
sample
set
*
*
1
{}
s
N
i
k
i
x
can be
obtain
ed. The
wei
g
ht of
each parti
cle
is
1/
j
ks
N
.
A suitable me
asu
r
e of deg
e
nera
c
y of the algorith
m
is d
e
fined a
s
:
s
N
i
i
k
eff
N
1
2
)
(
/
1
(
2
0
)
We
s
e
t a thres
hold
threshol
d
N
. If
threshold
eff
N
N
, resample. As
a result, we hav
e no n
eed
to re
sampli
ng
at every mo
ment. The
co
mplexity
of the alg
o
rithm
can
be redu
ced to a
ce
rta
i
n
extent.
4. Multi-user
Dete
ction
Based on G
a
u
ssian Sum Particle Filter
In orde
r to i
m
prove
syste
m
perfo
rman
ce, th
is p
ape
r discu
s
se
s a
new filter, G
aussia
n
sum p
a
rt
icl
e
f
ilt
er (GS
P
F
).
GS
P
F
com
b
i
nes G
a
u
ssi
an
sum filter and Particle filter. This al
gorit
hm
can in
crea
se
the diversity of the particl
e
s
then imp
r
ov
e the system
perfo
rman
ce.
Assu
me that we have the
predi
ctive dist
ribution:
G
j
kj
kj
k
kj
k
k
u
x
N
y
x
p
1
1
:
1
)
,
;
(
)
|
(
(
2
1
)
Whe
r
e
u
is me
an of the part
i
cle
s
,
is cova
rian
ce of the
particl
es. Th
e
y
are ba
sed o
n
a
priori information.
G
is the
number of parallel GPF.
Af
ter re
ceiv
in
g
the ob
se
rv
a
t
ion
k
y
, th
e
filtering dist
ri
bution can be
approxim
ate
d
as [14]:
G
j
kj
kj
k
k
k
kj
k
k
k
u
x
N
x
y
p
C
y
x
p
1
:
1
)
,
;
(
)
|
(
)
|
(
(
2
2
)
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: 5379 – 53
86
5384
Usually, the mean a
nd co
varian
ce of
1:
(|
)
k
k
p
xy
cannot be
obt
ained a
c
curat
e
ly. GSPF
sampl
e
s from
the impo
rtan
ce fun
c
tion
1:
(|
)
k
k
qx
y
,
and the
n
cal
c
ulate
s
the
weights
i
kj
of the
particl
es
i
kj
x
, wh
ere
i
is the tra
j
ectory in
dex
of the parti
cl
e. Then the
Monte Carl
o
estimate
s of
the mean an
d
covaria
n
ce o
f
the state ca
n be expre
ssed as:
ss
N
i
N
i
T
i
kj
kj
i
kj
kj
i
kj
kj
i
kj
i
kj
kj
x
u
x
u
x
u
11
)
)(
(
,
(
2
3
)
In this formula,
s
N
i
s
the
total
num
ber of th
e pa
rticle
s. T
he u
pdate
d
fil
t
ering
dist
ribu
tion
can b
e
expre
s
sed a
s
:
G
j
kj
kj
k
kj
k
k
u
x
N
y
x
p
1
:
1
)
,
;
(
)
|
(
(
2
4
)
Predi
ction
probability di
st
ribution
can
be
obtain
e
d
thro
ugh
the u
pdate
d
filtering
distrib
u
tion.
11
;
1:
1
(|
)
(
|
)
(
,
)
G
k
k
j
k
k
k
kj
kj
k
k
j
px
y
p
x
x
N
x
u
d
x
)
,
;
(
)
1
(
)
1
(
1
)
1
(
j
k
j
k
k
j
k
u
x
N
(
2
5
)
In this
formula,
j
k
u
)
1
(
and
j
k
)
1
(
are
obtaine
d fro
m
GSPF. In this algorit
hm, the
importa
nce function i
s
expressed a
s
:
)
,
;
(
)
|
(
)
(
:
1
k
k
k
k
k
u
x
N
y
x
p
q
(
2
6
)
In summa
ry, the step
s of multi-u
s
er d
e
t
e
cti
on b
a
sed
on Gau
s
sian
sum p
a
rticl
e
filter are
as
follows
[15]:
(1)
Fo
r
1,
2
,
j
G
, Sample
particles
1
{}
s
N
i
kj
i
x
from
the imp
o
rta
n
ce
fun
c
tion
1:
(|
)
k
k
p
xy
.
(2) F
o
r
1,
2
,
s
iN
,
1,
2
,
j
G
, calculate the
wei
ght of the every particl
e.
)
|
(
)
,
;
(
)
|
(
:
1
k
i
kj
kj
kj
i
kj
k
i
kj
k
i
kj
y
x
q
u
x
x
N
x
y
p
(
2
7
)
(3) F
o
r
1,
2
,
j
G
, calculate the mea
n
and varia
n
ce.
11
11
()
(
)
,
ss
ss
NN
ii
i
i
i
T
k
j
kj
kj
k
j
kj
kj
k
j
ii
kj
k
j
NN
ii
kj
kj
ii
xx
u
x
u
u
(
2
8
)
(4)
Update the weights
as
:
Evaluation Warning : The document was created with Spire.PDF for Python.
TELKOM
NIKA
ISSN:
2302-4
046
Multi-u
s
er
De
tection Based
on Gau
ssi
an
Sum
Particle Filter in Im
pulsive Noi
s
e (L
i Zhihui)
5385
G
j
G
j
N
i
i
kj
N
i
i
kj
j
k
kj
s
s
,
2
,
1
,
11
1
)
1
(
(
2
9
)
(5)
Normali
z
e
the weight
s:
G
j
kj
kj
kj
1
(
3
0
)
(6) For
1,
2
,
j
G
, Sample p
a
rt
icle
s from t
he filter p
r
obability dist
ribution
(;
,
)
kk
j
k
j
Nx
u
, denote them
as
1
{}
s
N
i
kj
i
x
.
(7)
Fo
r
1,
2
,
s
iN
,
1,
2
,
j
G
, dra
w
p
a
rticl
e
s from the
state
transitio
n di
st
ribution
(1
)
(|
)
i
k
j
kj
kj
p
xx
x
, then denote
them as
(1
)
1
{}
s
N
i
kj
i
x
.
(8) F
o
r
1,
2
,
j
G
, update the weight
s.
kj
j
k
)
1
(
(
3
1
)
(9) Cal
c
ulate the
predi
cted mean
j
k
u
)
1
(
and co
varian
ce
j
k
)
1
(
.
s
N
i
i
j
k
j
k
s
j
k
x
N
u
1
)
1
(
)
1
(
)
1
(
1
(
3
2
)
s
N
i
T
i
j
k
j
k
i
j
k
j
k
s
j
k
x
u
x
u
N
1
)
1
(
)
1
(
)
1
(
)
1
(
)
1
(
)
)(
(
1
(
3
3
)
(10
)
By sub
s
tituting
Eq
u
a
tion (24, 25
) wi
th Eq
uati
on
(32,
33
), Poste
r
io
r p
r
obability
den
sity function ca
n be obt
ained. Tu
rn to (1),
an
d esti
mate the sign
als of the nex
t user.
5.
Simulation Resul
t
s
In this
simul
a
tion, PF an
d GSPF a
r
e
applie
d to sy
nch
r
on
ou
s DS_CDMA sy
stem. We
sele
ct 8 u
s
e
r
s, 31-bit gold
spread
-spe
ct
rum
cod
e
. Ch
annel n
o
ises
are a
dditive
Gau
ssi
an n
o
ise,
Lapla
c
e n
o
ise and Alph
a stable n
o
ise.
Alpha sta
b
le
noise parame
t
ers:
1.8
,
0
,
1
,
0
a
. The rang
e of signal to
noise ratio (SNR)
for all use
r
s i
s
-4
~10 dB. The numbe
r of
particl
es i
s
50
; the number
of parallel GP
F is 5.
From th
e
si
mulation, it i
s
clea
r that t
he p
e
rfo
r
ma
nce
of GSP
F
is
better than PF
in
Gau
ssi
an
noi
se
enviro
n
m
ent. GSPF al
gorithm
c
an
obviou
s
ly im
prove th
e p
e
r
forma
n
ce of
the
sy
st
em.
Figure 2 anal
yzes the e
rro
r cod
e
perfo
rmanc
e of GSPF detection
aiming to Ga
ussian
noise, Lapl
ace noi
se a
nd A
l
pha
stable
n
o
ise.
We
ca
n
find that the e
rro
r
cod
e
pe
rf
orma
nce of th
e
Gau
ssi
an noi
se
s
a
n
d
the Lapla
c
e
noi
se
a
r
e almo
st the
same. Th
e
e
rro
r co
de
perfo
rman
ce
of
the Alpha sta
b
le noises
sli
ghtly wea
k
en
ed, as t
he true power of the Alpha sta
b
le noise is
not
2
. But, in the
s
i
mulation,
we as
sume its pow
e
r
is
2
.
The re
sult al
so proves
th
at
GSPF
algorith
m
ha
s strong
rob
u
st
ness.
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: 5379 – 53
86
5386
Figure 1. The
BER of PF Detection a
nd
GSPF
Dete
ction
Figure 2. The
BER of GSPF Dete
ction u
nder
Thre
e Kinds
of Noise
6. Conclusio
n
This
pap
er a
nalyze
s
th
e e
rro
r
cod
e
p
e
rforman
c
e
of
PF dete
c
tion
and
GSPF d
e
tection
aiming to Ga
ussian n
o
ise, Lapla
c
e noi
se, and Alpha
stable n
o
ise. It can be se
e
n
that the error
perfo
rman
ce
of GSPF i
s
better th
an P
F
. It also
ca
n be
seen
from Fig
u
re
2
that the
GS
PF
provide n
ear-optimum pe
rforma
nce in non-Gau
s
sia
n
noise e
n
vironment, whi
c
h is con
s
i
s
te
nt
with m
u
lti-u
s
er
dete
c
tion
result i
n
the
real
con
d
ition.
The
r
efo
r
e,
GSPF alg
o
rit
h
m
can
obvi
ously
improve th
e p
e
rform
a
n
c
e
o
f
the syste
m
.
Mean
while
, t
h
is n
e
w alg
o
ri
thm ha
s a
strong a
dapta
b
il
ity
in non
-Ga
u
ssian noi
se
en
vironme
n
t. All in all,
the rese
arch
re
su
lts have im
p
o
rtant refere
nce
value for the
resea
r
ch of MUD
syste
m
. The next wo
rd to be
done
is to modify algorith
m
so t
hat
detectio
n
perf
o
rma
n
ce achi
eves better
re
sults in n
on-Gau
ssi
an.
Referen
ces
[1]
R Lup
as, S
Verdu. Li
near
multistage d
e
tector
s for sync
hro
nous c
ode d
i
visi
on
multipl
e
acces
s
chan
nels.
IEEE Trans. On Comm
.
19
91; 3
5
:
123-13
6.
[2]
Dou z
h
o
ngz
ha
o, Le
i
xi
an
g.
CDMA W
i
re
les
s
Co
mmu
nic
a
tions
T
heor
y, T
s
inghua Univ
ersit
y
Press
,
Beiji
ng. 20
04.
[3]
Steven M Ka
y.
T
r
anslated
b
y
Pe
ng-fe
i Lu
o. Stat
istics based
on si
gn
a
l
process
i
n
g
: estimatio
n
an
d
detectio
n
theor
y, Electro
n
ic In
dustr
y
Pr
ess, Beiji
ng. 20
06.
[4]
NJ Gorda
n
, DJ
Salmo
nd, AF
M Smith. Nov
e
l
ap
pr
oac
h to
n
on-li
ne
ar a
nd
n
on-Gauss
i
an
B
a
yesi
an stat
e
estimation
. IEEE Proc.F
. 1993; 140(2): 10
7-1
13.
[5]
Mei Wu, Fujun Pei.
Improve
d
Distrib
uted P
a
rticle Fi
lter fo
r
Simulta
n
e
ous
Loca
liz
ation
a
nd Ma
pp
ing
.
T
E
LKOMNIKA Indon
esi
an Jou
r
nal of Electric
al Eng
i
ne
eri
n
g
.
2013; 1
1
(12): 761
7-76
26.
[6]
Indah
Ag
ustie
n
Sir
adj
ud
din,
M Ra
hmat
W
i
d
y
anto,
T
Basaru
ddi
n. P
a
rticle F
i
lter
w
i
t
h
Ga
ussia
n
W
e
ighti
ng for
Huma
n T
r
acking.
T
E
LKOMN
I
KA Indon
esia
n Jour
nal
of E
l
ectrical
Eng
i
n
eeri
ng.
2
012
;
10(6): 14
53-
14
57.
[7]
E Punsk
a
ya,
C Andr
ie
u, A
Douc
et, W
J
F
i
tzgeral
d.
Partic
le filter
ing
for
mu
ltius
e
r d
e
te
ction i
n
fad
i
n
g
CDMA cha
nne
l
s
.
Proc. 11th IEEE Workshop
SSP. 2001; 38
-41.
[8]
Xi
an J
i
nl
on
g,
Geng Y
u
lo
ng.
Mult
i-user
Det
e
ction B
a
se
d o
n
Particl
e
F
ilte
r
in Impu
lsiv
e
Noise
. IJACT
.
201
3; 5(4): 250
-258.
[9]
Hou R
u
i, Z
han
g Santo
ng, Z
hu Gang.
CDM
A Multiuser D
e
tection Usi
ng
Unsce
nted Par
t
icle Filteri
n
g
,
Internatio
na
l C
onfere
n
ce o
n
W
i
reless Com
m
unic
a
tions, N
e
t
w
o
r
ki
ng an
d Mobil
e
Com
p
u
t
ing, W
iCOM.
200
7; 902-
905.
[10]
Xi
an
Jin
l
o
ng, L
i
Jia
n
w
u
.
Multi
-
user R
e
ce
iver
Base
d
on M
M
SE Criteri
a
i
n
L
apl
ace
No
i
s
e a
nd A
l
p
ha-
.stable No
ise
. WASE International Conference on
Information Engineering, ICIE.
2010; 4:
38-41.
[11]
T
s
akalides, P
a
nag
iotis, N
i
kias
, Chr
y
sostom
o
s
L.
W
i
d
e
b
and
array
sig
n
a
l
p
r
ocessi
ng w
i
th
alp
ha-stab
l
e
distrib
u
tions.
IEEE Militar
y
C
o
mmunic
a
tio
n
s
Confer
e
n
ce M
I
LCOM. 1995; 1: 135-1
39.
[12]
Z
hu Z
h
i
y
u. T
he particle filt
er a
l
gorithm a
nd its
appl
icatio
n,
Scienc
e Press, Beiji
ng. 20
10.
[13]
Xi
an Ji
nlo
ng,
Geng Yu
lon
g
. Multi-
user
Det
e
ction B
a
sed
o
n
Au
xil
i
ar
y Pa
rt
icle F
ilter i
n
Impulsiv
e
No
ise,
IJACT
.
2013; 5(4): 259-2
67.
[14]
Xu
e F
e
n
g
, Li
u
Z
hong, Z
h
a
ng
Xi
aor
ui.
Gauss
i
an S
u
m Partic
le F
ilter fo
r P
a
ss.ive T
r
ackin
g
,
Journa
l of
System Si
mu
la
tion
. 200
6; 28(
supp
l.2): 900-
9
02.
[15]
Lin Qi
ng, Yin J
i
anj
un, Z
h
a
ng
Jian
qiu, H
u
Bo
. G
aussian s
u
m particle fi
lter
methods for
n
on-li
ne
ar no
n-
Gaussian methods.
Systems
Engi
neer
in
g an
d Electron
ics.
201
0; 32(1
2
); 2493-
249
9.
Evaluation Warning : The document was created with Spire.PDF for Python.