A NEW NON-LINEAR SYSTEM FOR ESTIMATING AND
SUPPRESSING NARROWBAND INTERFERENCE IN PN
SPREAD SPECTRUM MODULATION
A.
I.
Pk ez-Nei a,
J.
Roca,
Ad
A.
Lagunas'
Uni e si a Poli ecnica de Ca alunya.
Dep .
Signal
Theo y
and
Comm.
C1
Jo di
Gi ona,
1-3.
Campus
No d
UPC.
Edi icio
D-5
08034
Ba celona- SPAIN
ABSTRACT
This wo k de elops
a
no el dynamic uzzy logic sys em ha ,
based on
a
uzzy basis unc ion expansion, success ully sol es
he non-linea p oblem o na owband in e e ence p edic ion
and ejec ion in DS-SS.
A
uzzy basis unc ion ep esen a ion
p o ides a na u al amewo k o combining bo h nume ical and
linguis ic in o ma ion in
a
uni o m ashion. The esul is a low
complexi y non-linea adap i e line enhance , which o e s
a
as e con e gence a e and
an
o e all be e pe o mance o e
o he well-known non-linea line enhance s.
1.
INTRODUCTION
Sp ead Spec um
(SS)
communica ions o e
a
p omising
solu ion o an o e c owded equency spec um amid g owing
demand o mobile and pe sonal communica ion se ices. The
p oposed applica ions o comme cial use o sp ead-spec um
in ol e he o e laying o sp ead spec um signals on exis ing
na owband
(NB)
use s, hus, implying s ong in e e ence
o
he
SS
sys em. While
SS
has inhe en noise supp ession
capabili y, sys em pe o mance can be u he enhanced a he
decision de ice i an in e e ence ejec ion il e o line enhance
is
used be o e desp eading
[
11.
In
he case whe e
a
single an enna is used and he s a is ics o he
in e e en signals a e unknown, he ejec ion il e is usually a
ans e sal adap i e il e (adap i e line enhance
o
ALE)
and
elies on bo h, he pseudo-whi e p ope ies
o
he
SS
signal and
he p edic abili y o he na owband in e e ence, bo h p esen in
he ecei ed signal z( ).
The ecei ed signal z( ) consis s
o
3
addi i e componene s: he
SS
ansmi ed signal s( ), he wide-band noise n( ), and he
na ow-band
(NB)
in e e ence i( )
z( )
=
s( )
+
n( )
+
i( )
(1)
s( )
=
A
c( )
d( )
cos(w, )
(2)
The signal s( ) is
a
modula ed wideband signal gi en by
whe e A is
a
cons an ampli ude,
W,
is he ca ie equency, d( )
is
he in o ma ion,
a
bina y da a sequence aking on he
equip obable alues o
1
each
o
which las s o
T
seconds, and
c( ) is he sp eading sequence, usually
a
pseudo andom noise
(PN)
code
o
chip sequence, which also akes he alues
o
l
bu
which
las s
o
Tc
seconds, whe e
T,.c<T
In
ecep ion,
o
eco e he in o ma ion d( ), z( ) is chip-ma ched
and sampled a he chip a e o he
PN
sequence. We hus ha e
zk
=
sk
+nk
+
ik
(3)
whe e
(s },
(n } and (i }
a e
he disc e e- ime sequences om
(s( )), (n( )) and (i( )] espec i ely. (sk}, (n } and (ik} a e
assumed o be mu ually independen . We ha e assumed ha n( )
is bandlimi ed and becomes whi e a e sampling. Fo he
in e e ence, we ha e conside ed ha
i s
bandwid h
is
small
compa ed wi h Inc. Finally, since he PN sequence is andom,
we can assume
(sk},
o be a sequence o i.i.d. andom a iables
aking alues o
k
1
wi h equal p obabili y.
In
equa ion
(3),
(s ),
and (n }
a e
wideband signals and a e
poo ly co ela ed when sampled a he chipping a e. The e o e
when he
ALE
ies o es ima e he nex sample
o
he signal, i
would succeed only in es ima ing he high y co ela ed
in e e ence and consequen ly manages o supp ess i . No e,
howe e , ha he sequence
(sk}
is highly non-Gaussian. Thus,
he op imum il e o p edic ing a na ow-band p ocess in he
p esence o such a sequence will, in gene al, be nonlinea . Only
i he
SS
signal lies below he noise
loo ,
hen he Gaussian
assump ion o sl;+nk
is
mo e easonable and a linea il e
achie es good esul s
[l].
In
[2],
Mas eliez de eloped an App oxima e Condi ional Mean
o ACM il e , wi h a s uc u e simila o ha o he Kalman
il e , in o de o es ima e he s a e o a linea sys em wi h non-
Gaussian obse a ion noise. Vijayan and
Poo
[3]
employed his
algo i hm
o
sol e he
NB
in e e ence supp ession p oblem in
he
DS-SS.
When he
AR
pa ame e s a e unknown, Vijayan and
Poo also de eloped an adap i e nonlinea
LMS
algo i hm based
on
he ACM il e ing algo i hm. This algo i hm was la e
modi ied by Rush and Poo
[4]
and Wu
[5].
All hese non-linea
algo i hms depa om he s a e-space ep esen a ion
o
he
sys em
i,
=
(Pi,-,
+
w,
(44
Z,
=
hi,
+ ,
(4.b)
whe e he in e e ence has been modeled
as
a
Gaussian AR
p ocess o o de P. The s a e ec o
is
ik=[ik.
i .,,..
i -p+,]
and
is
gene a ed by he Gaussian p ocess wk=[wk
0
...
o ,
and he
ma ix
@,
o med by he AR pa ame e s. The obse a ion ec o
is
h=[l
0
...
01
and
he measu emen noise
is
he non-Gaussian
sequence k=
sk+nk.
This
wo k has been suppo ed by Na ional Resea ch Plan o Spain CICYT unde g an numbe TIC-96-0500-C10-01
3261
0-7803-4428-6/98
$10.00
0
1998
IEEE
Neu al Ne wo k and Radial Basis Func ion sys ems ha e also
been s udied as adap i e line enhance s
[6]
and
[7].
Howe e , he
gene al p oblem o non-linea p edic o s is ha hey equi e
g ea e complexi y han i s linea coun e pa , which is ypically
encompassed
in
a single
DSP
chip. Addi ionally, o non-linea
il e ing, hough he na u e o he e o su ace is no known, i is
highly likely ha he e a e mul iple local minima. The e o e, a
g adien sea ch echnique canno be gua an eed o con e ge o
he globally op imum pa ame e es ima es.
In his wo k we depa om he Kalman and
ACM
il e s a e
space o mula ion in
(4)
and design an adap i e uzzy p edic o
which ou pe o ms he esul s o he ecen wo ks o
[3]
and [SI
and speeds up he con e gence o he adap a ion p ocess.
Also
he p oposed echnique p esen s an a ac i e pa allel algo i hmic
s uc u e, whe e, a each ins an o ime, jus a ixed numbe o
p oduc s and
sums
and one di ision
a e
needed o p oduce he
ou pu .
2.
ADAPTIVE
FUZZY
LINE
ENHANCER
The uzzy ejec ion il e o design depa s om he s a e space
ep esen a ion o mula ed in
(4).
Taking ad an age o he
ela ionship equa ed
in
(4.b), he uzzy sys em p edic s he
in e e ence sample ik om he obse a ion zk.
In
con as o
o he non-linea in e e ence cancelle s, he p oposed sys em
does no equi e he ma hema ical model o he in e e ence in
(4.a): no
AR
pa ame e s ha e o be nei he known no es ima ed.
The uzzy sys em o design jus elies on he slow a ying na u e
o he NB in e e ence in o de o model i s beha io by means o
linguis ic IF-THEN ules, which eplaces equa ion (4.a). The
in e e ence ange is quan ized in egions
o
izy
se s
and, om
a
e e ence poin , he e olu ion o he in e e ence among his
egions is ollowed by means o linguis ic IF-THEN ules
o
he
ype:
"IF
ik;l
is in he egion o posi i e high alues and ikm2 is in
he egion
o
posi i e high alues and
ik-,
is in he egion
o
posi i e high alues
THEN
ik
is in he egion o posi i e high
alues".
These IF-THEN ules
a e
he co e o he uzzy sys em o
design. Addi ionally, he s a is ical knowledge o he
measu emen noise ( k] and model noise
(Wk}
can be easily
in oduced in he design o he p oposed sys em
uzijica ion
s age.
A
uzzy sys em is a unc ional ne wo k (Fig.
1)
ep esen ed as
se ies expansions o uzzy basis unc ions
g,
(x)
M
Y
=
m)
=
g,
(XI
ei
(6)
1'1
whe e
6
E
R
a e
cons an s. Using he S one-Weie s ass
heo em, linea combina ion
o
uzzy basis unc ions p o e o be
capable o uni o mly app oxima e any eal con inuous unc ion
on
a compac se o a bi a y accu acy [8].
In
ou case y=Tk and
he inpu
x
is
he measu emen zk and wo ee- o wa ded
in e e ence es ima ed alues:
x
=
[z,
jk-3
ik-2].
The mos
impo an ad an age o using uzzy basis unc ions, a he han
polynomials, adial basis unc ions, neu al ne wo ks, e c.,
is
ha
a
linguis ic IF-THEN ule is na u ally ela ed o
a
uzzy basis
unc ion (FBF). In o he wo ds, he FBF p o ide a gene al
amewo k
o
ansla e abs ac concep s in o compu able en i ies.
B
(.)
=
m:
I
Rule
base:
ma ix
R
11
P
e
Y
I
I
Figu e
1.
Adap i e uzzy line enhance .
In ig. 1 we dis inguish
4
main pa s: he
izzpe
maps he c isp
inpu s
x
o uzzy se s de ined
on
he inpu space; he se
o
s a emen s comp ise he
uzzy ule
base, which is a i al pa o
a
Fuzzy Logic Sys em, he
iiy
in e ence
engine combines he
s a emen s in he ule base acco ding o app oxima e easoning
heo y o p oduce a mapping om uzzy se s in he inpu space
X
(i.e.
Ai(.)
in ig. 1) o uzzy se s in he ou pu space
Y
(i.e. Bi(.)
in ig.2). Finally, he
de izi e
maps he agg ega ed ou pu
uzzy se s o he single c isp poin in he ou pu space, which in
ou sys em is he in e e ence es ima e o ik o be used by he
communica ion ecei e . Nex , he design o he p oposed uzzy
logic sys em
(FLS)
is desc ibed.
2.1
The
ma hema ical amewo k o heo y
o
uzzy se s p o ides
a
na u al basis o uzzy logic, which is
a
gene aliza ion o bina y
logic.
In
o he wo ds, he logical in e encing using uzzy se s is
known as uzzy logic [8-111.
In
uzzy se heo y he e is
no
sha p
bounda y be ween hose objec s ha belong o he class and
hose do no .
In
addi ion, an elemen may also be a membe
o
mo e han one se . Membe ship unc ion in a uzzy se is a ma e
o
deg ee.
A
uzzy se
F
in
a
uni e se o discou se,
U,
is
cha ac e ized by a membe ship unc ion
pF
,
which akes alues
in he in e al
[O,l];
ha is,
pF
:
U
3
[0,1].
Thus,
a
uzzy se
F
consis s o
a
gene ic elemen
U
and i s g ade
o
membe ship
unc ion; ha is,
F
=((u,p,(u))IuE
U>.
A
uzzy a iable
is
cha ac e ized by
a
e m se o se o uzzy se s (i.e.
o
linguis ic
o uzzy alues)
o
U.
In his wo k
A(.)
and
x
will be used
o
he
inpu e m se and he inpu a iable, espec i ely.
Also
B(.) and
y
will be used o he ou pu e m se and he ou pu a iable,
espec i ely. Due o noise, he measu ed inpu s a e ague and,
he e o e, he sys em classi ies
o
quan ize hem in o e lapping
egions o uzzy se s Ai(.), o whom he inpu s belong wi h some
membe ship deg ee (e.g. A3(.) s ands o he uzzy alue:
"
posi i e high alue").
Thus,
hese se s con o m in
a
na u al
si ua ion when desc ibing he possible alues o he
3
measu emen s in
x.
The
Fou
S ages
o
he
FLS
3262
Figu e
2.
Fuzzy e m se o he a iable “ il e inpu ”
Figu e 2 plo s Gaussian membe ship unc ions, howe e , he e
a e di e en me hods o de e mine
a
uzzy membe ship unc ion
[IO]. I is wo h no ing ha a membe ship unc ion may be
subjec i e, bu no a bi a y. Since ou p oblem employs
s a is ical inpu s, he design based on hei p obabili y densi y
unc ions shall be app op ia e. In his way, we ela e he uzzy
membe ship unc ions o physical p ope ies o he sys em. F om
(4.b) we know he condi ional p obabili y densi y ( .d.p) unc ion
o ik-1.
p(ik-l
/z~-~).
The e o e, i we ela e he uzzy se s
AI
wi h his .d.p., we can say ha whene e he inpu alls inside
hese uzzy se s, his inpu will be ela ed o some deg ee wi h ik-
I.
This dynamic designed uzzy se s (dynamic because hei
posi ion depend on he alue o &.I) ac as a e e ence o
loca ing alues ik+ ik.3 in
a
slow
a ying na owband.
Addi ionally, o ob ain he uzzy se o zk we no e ha
P(Z,
/ik-l)
can be assumed o be Gaussian by a easoning
equi alen o he one used by Mas eliez in [2] o de elop he
ACM il e . Thus, he uzzi ica ion o zk is done by means o he
same uzzy se e m A(.)
as
he one depic ed in ig.2. Finally, he
ou pu uzzy se s
B(.),
which quan ized in a uzzy way he
possible alues o he es ima ed in e e ence alues
k
,
ha e
been designed as M Gaussian unc ions no malized o
1
and o
equal a iance.
M
is he numbe
o
IF-THEN ules. Thei means
a e ini ially he same
as
hose in ig.2, howe e , hey can be
modi ied by
a
LMS
ype algo i hm
as
we commen la e in his
sec ion 2. We no e wo gene al design conside a ions:
1)
because
o
he ela ionship be ween .d.p and membe ship unc ions, he
mo e noise p esen , he wide he uzzy se s ha e o be;
2)
o sa e
compu a ion he inpu uzzy se s
o
ig. 2 and he ou pu uzzy
se s can be designed as iangles wi h he same wid h
as
he
Gaussian noise a iance.
Once he inpu uzzy se s a e designed, he
uui ie
maps a c isp
measu emen o alue in o a uzzy se . The mos widely used
uzzi ie is he single on uzzi ie : he c isp poin xi is mapped
in o
a
uzzy se
F
wi h suppo x whe e
pF(x)=6(x-x,~.
We
no e howe e ha in cases when he signal- o-noise a io
(SNR)
is
low o he e
is
high inpu unce ain y, non-single on uzzy se s
[8]
a e mo e use ul
as
he simula ions in his pape show.
The
uzzy
ule
base
consis s o
a
se o linguis ic ules in he
o m o
“IF
a
se
o
condi ions a e sa is ied,
THEN
a
se
o
conseqiiences
a e
in e ed”.
Suppose we ha e a ule base
consis ing o M uzzy i - hen ules R, (m=1
...
M)
R,
:
IF
iL-,
is
A,,
and
lk-?
is
A,,
and
whe e
{0,+1,+2,+3}.
The p edic o
o
in e e ence
ik
cons uc ed based on he
M
ules. Each ule R, can be iewed
as
a
uzzy implica ion which is a uzzy se R,(.) in XxY wi h
is
A,,
THEN
k
is
B,
is
PRm(X?y)=PAm,(X)*PAw (‘)*PAd
(’)*PB,(Y)’
whe e
he
mos commonly used ope a ions o
“*”
a e “p oduc ” and ‘“in’’
[8].
In
his wo k we ha e used he “p oduc ” ope a ion.
The uzzy ules can be sis ema ically de i ed
by
conside ing
all
he possible combina ions among he 7 membe ship unc ions
(73=343). Howe e , in his wo k, his ule explo ion is
d ama ically educed o 72 ules by a oiding hose i ele an
ules o
slow
a ying in e e ences such
as:
R,
:
IF
-,
is
A-,
and
-,
is
A,
and
z,
is
A-,
THEN
is
B.!
The
uw
in e ence engine
o uzzy associa i e memo ies is
decision making logic which employs uzzy ules om he uzzy
ule base o de e mine a mapping om he uzzy se s in he inpu
space
X
o he uzzy se s in he ou pu s space Y. Le
F
be
a
uzzy
se in
X;
hen each
R,
de e mines a uzzy se
FOR,
in Y based on
he sup-s a com osi ion
[81:
single on uzzi ica ion
pF
(x)
=
6(x
-
x,
)
and esul s in
pFoRm
(y)
=
SUPE,y
bF
(XI
*pRm
(x,
YIP
1’
he case o
p,G.Rm
(y)
=
PA,
(k-3)*/’!AM
(L-2)*pAk
(2k)*/’!Bm(y)
=
wm(x)*p8m
( )
whe e w, is he i ing s eng h o weigh o he m h ule. In
summa y,
all
he
M
ules
o
he FAM’s a e ac i a ed pa allely
and imply a ixed numbe o sums and mul iplica ions. A ins an
o ime
“k”,
he esul o he in e ence o each ule can be
exp essed as
a
ma ix mul iplica ion
[
111 ( ig. 1). Finally, he
indi idual s a emen solu ions a e agg ega ed o p o ide he
o e all solu ion
A e he uzzy in e ence, he
de uui ie
pe o ms a mapping
om he uzzy se s in
Y
o c isp poin s in
Y.
The ollowing
cen oid o cen e o mass de uzzi ie
[8]
is
he mos commonly
used me hod. I uses
all
and
only
he in o ma ion in he ou pu
se B in i s domain “y” in a Bayesian sense (see eq. (8)).
m=l
whe e
7,
=
cen oid
{B,
(y)}
and
8,
=
F,,,
No e ha we ha e inally come o he unc ional exp ession
(6).
which depends linea ly on he ou pu pa ame e
8,.
The e o e,
we p opose o use an LMS (leas mean squa e) ype algo i hm in
o de o adjus
0,
and e ine he uzzy sys em esul . This LMS is
modi ied o inco po a e he app oxima e condi ional mean non
linea i y exac ly in he same way as done in [3]. Tha is he
adap i e algo i hm is applied o each uzzy sys em ou pu
ik
in
o de o minimize
(
zk
-
i”,
-
sign
(
zk
-
i;
)II*
.
II
3.
SIMULATIONS
In his sec ion, we epo on simula ions ca ied ou o e alua e
he pe o mance o he p oposed algo i hms. We ollow he
commonly used SNR imp o emen , de ined in [3-51. The SNR a
he inpu was a ied by changing he powe o he in e e ing
examples
s udied
in
[3]
and
[SI.
Ou
pe o mance measu e
is
he
3263
signal. The a iance o he backg ound he mal noise was kep
cons an a
a
*
=
0.01.
The
SS
p ocessing gain is
10.
AH
esul s
we e ob ained based on
10
ials and, o each ial,
3000
da a
poin s we e compu ed. Table
I
summa izes he esul s o he 3
se s o simula ions. I can be seen ha adap i e non linea
il e ing uzzy echniques o e conside able imp o emen o e
con en ional linea il e s (i.e. TS-LMS: Two Sided Leas Mean
Squa e il e ) and he non linea algo i hm designed in [5].
We no e ha , i o simpli y compu a ion iangula membe ship
unc ions a e used ins ead o Gaussian ones, he esul s jus
deg ade in 1 dB. Also, in he case o
AR
in e e ence no
di e ence exis s i he
72
ules a e educed o 32. In he case o
single one sinusoidal in e e ence, o high
SNR
a ios he
pe o mance is no
so
good as in [5].
This
ac is due o he
quickly speed o change o he alue o he in e e ing signal. I
we wish be e esul s, we ha e o assign mo e membe ship
unc ions o he inpu s o co e he a ia ions o he in e e ing
signal. O he poin o ema k is ha he LMS adap a ion is e en
no needed in he case
o
he
AR
in e e ence. In any case, he
LMS con e ges in ew samples ( ig. 3) due o he good ule
ini ializa ion and he good p ope ies o he
FBF.
Finally, we ha e also ca ied ou a s udy o noisy scena ios. As
no hing is said in [3-5]
o
his case, we compa e ou algo i hm
wi h he linea
ALE
o 4 aps epo ed in [l]. Figu e 4 shows he
pe o mance o he p oposed algo i hm when no LMS adap a ion
is ca ied ou . The bes esul s a e o he non-single on uzzy
il e , which is mo e sui able o noisy scena ios 181. Logically,
he pe o mance o he linea il e is imp o ed o low noise. Fo
high noise, he designed sys em imp o es he linea p edic o LP
and ob ains BER o he same o de o magni ude han he LP
wi h ma ched il e . We no e ha in he linea simula ions
AR
pa ame e s a e conside ed known, while in he uzzy sys em no
in e e ence knowledge is assumed.
4.
CONCLUSIONS
In his wo k we ha e add essed he p oblem o in e e ence
ejec ion in
SS
sys ems. We p esen a low compu a ional
algo i hm ha imp o es he pe o mance ob ained wi h ecen
non-linea algo i hms. Jus he slow a ying na u e o he
NB
in e e ence is assumed and used o he ule ini ializa ion, which
helps o a oid local minima and o speed up con e gence,
0.1.
0.s.J
0.3
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0.16
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5.
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-A-
Single on
Fil e
Figu e
4.BER a e desp eade o
100
ones. The signal-
o-in e e ence a io pe chip is -20 dB.
AR
in e e ence
I
TS-LMS/DR2D
[SI
I
26.9/37
I
22.3132.6
I
17.6/28.1
1
13/23.4
I
I
Fuzzy
I
35.7
I
31.6
I
26.8
I
21.9
I
Sinusoidal in e e ence (1 one), no =O.
15
-1
Sinusoidal in e e ence
(100
ones in
Table
1.
SNR
imp o emen (dB) o single on uzzi ica ion.
3264