PERFORMANCE OF ADAPTIVE BAYESIAN
EQUALIZERS IN OUTDOOR ENVIRONMENTS
Jos6
Luis VALENZUELA, Fe nando CASADEVALL
Depa amen de Teo ia del Senyal i Comunicacions
E.T.S.E.B Uni e si a Poli kcnica de Ca alunya
E-Mail: [email p o ec ed]
Apdo.
30002,08080
Ba celona, SPAIN Tel. +34
3
401 59 48
FAX.
Abs ac -Ou doo communica ions a e a ec ed by mul ipa h
p opaga ion ha imposes an uppe limi on he sys em da a a e
and es ic s possible applica ions. In o de o o e come he
deg ading e ec in oduced by he channel, con en ional
equalize s implemen ed wi h digi al il e s ha e been
adi ionally used. In his pape a new app oach based on
neu al ne wo ks is conside ed. In pa icula , he beha io o he
adap i e Bayesian equalize implemen ed by means o Radial
Basis unc ions applied o he channel equaliza ion o adio
ou doo en i onmen s has been analyzed. The me hod used o
ain he equalize coe icien s is based on a channel esponse
es ima ion. We compa e he esul s ob ained wi h h ee channel
es ima ion me hods: The leas sum o squa e e o s (LSSE)
channel es ima ion algo i hm, ecu si e leas squa e
(IUS)
algo i hm employed only o ob ain one channel es ima ion and,
inally, he
RLS
algo i hm used o es ima e he channel e e y
decided symbol o he whole ame.
I. INTRODUCTION
Reliable da a ansmission in mobile digi al
communica ion sys ems o en equi es he use o channel
equaliza ion in o de o compensa e o he in e symbol
in e e ence
(ISI)
induced by he mul ipa h p opaga ion.
T adi ionally, decision- eedback equalize s
(DFE)
implemen ed
wi h FIR il e s and maximum likelihood es ima ion (MLSE)
ha e been employed o his pu pose. Equalize design o
hi d gene a ion mobile sys ems and, in pa icula , sys ems ha
use an ad anced TDMA mobile access o Uni e sal Mobile
Telecommunica ion Sys em (UMTS), mus o e come p oblems
ha a e much mo e complex han hose encoun e ed in he
ac ual second gene a ion.
MLSE equaliza ion is op imum in he sense ha i
ob ains he bes pe o mance by de ec ing he en i e ansmi ed
sequence ha minimizes he squa ed e o . Howe e , i s high
complexi y makes i useless in many p ac ical implemen a ions.
Mo eo e , i has been p o ed ha MLSE equaliza ion has
impo an deg ada ion o ime- a ying channels
[
11.
Symbol-decision equalize s ha e been commonly
implemen ed wi h
DFE
s uc u es based on he linea il e
+34
3
401
72
00
app oach. These s uc u es a e simple han MLSE despi e a
ce ain pe o mance loss in a s a ic en i onmen .
The op imal solu ion o he symbol-decision
equalize s equi es nonlinea bounds ha can be de i ed
adop ing a Bayesian app oach, also known
as
he
maximum
a
pos e io i
(MAP) equalize . The Bayesian could be
implemen ed wi h o wi hou a decision- eedback s uc u e. I
has been shown epea edly ha eedback s uc u es pe o m
be e han no- eedback s uc u es. Recen ly, he applica ion o
some new equaliza ion s uc u es such
as
Mul ilaye Pe cep on
(MLP) and Radial Basis Func ion (RBF) ha e been in o de o
implemen he nonlinea bounds. The equalize s implemen ed
wi h MLP s uc u es could achie e nonlinea decision bounds,
al hough he algo i hms, such as he backp opaga ion
algo i hm, used o each he ne coe icien s need a g ea
numbe o samples
[2]
and an immense compu a ional e o
and his makes hem imp ope o equalizing mobile
en i onmen s.
On he o he hand,
RBF
s uc u es ha e been
p esen ed in ecen pape s as e ec i e equalize s in s a ic
channels wi h wo main algo i hms o upda ing he
coe icien s. The clus e ing app oach
[3]
iden i ies hese
coe icien s using a clus e ing algo i hm. This algo i hm is
compu a ionally e y simple and e ec i e o nonlinea
channel dis o ion. Howe e , as he backp opaga ion algo i hm
o MLP, i needs a la ge amoun o aining samples. The
second app oach es ima es he channel model and calcula es
he
RBF
coe icien s. The main ad an ages o his me hod
a e
he sho aining sequences needed and he possibili y o
employing adap i e channel es ima o s ha allow he esul s
ob ained wi h
a
non-adap i e algo i hm o be ou pe o med.
In his pape we p esen he esul s ob ained wi h he
RBF s uc u es in a ime- a ying channel, using di e en
channel es ima o s, and compa e hem wi h he esul s ob ained
using a con en ional decision- eedback equalize . The sys em
pe o mance is e alua ed by means
o
he bi e o a e (BER)
as a unc ion
o
he signal- o-noise a io
(SNR)
and he
no malized delay sp ead
[4]
o se e al mobile speeds.
0-7803-3659-3/97
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0.00
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997
IEEE
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11. TRANSMISSION MODEL
Fig.
1
shows he low-pass equi alen model o he
ansmission sys em.
DECEM
Fig.
1
Low-pass ansmission model.
means
o
he Powe Delay P o ile, he numbe o disc e e
coe icien s, he delay coe icien and he Rayleigh dis ibu ed
ampli ude o each ap, a ying acco ding o a Dopple
spec um. In his channel model h ee Dopple spec um
classes a e conside ed.
The ecei ed and sampled signal a he inpu
o
he
equalize can be o mula ed as:
whe e
nknT,,,)
is a complex Gaussian andom unco ela ed
p ocess wi h a iance equal o
0:
and
h[(n-k)T]
is he sampled
esponse o he channel model gi en by:
In gene al he ansmi ed baseband
QAM
signal can
be o mula ed as:
whe e
{ad,
{bJ
a e independen da a sequences o he in-
phase and quad a u e channel. These da a sequences ake hei
alues om he se
kl,
k3,..,k(M1”-1) wi h
M=4
o
4-QAM.
The o e all il e ing ans e unc ion
Hd’=H#)HR( 3
is a
aised-cosine ype wi h a oll-o ac o ,
p
equal o
0.5,
which
is equally spli be ween he ansmi e and he ecei e . The
unc ion
HJ )
models he channel beha io ha in oduces
selec i e ading in he adio link. The ecei ed signal can be
exp essed as:
ki-m
whe e nk ) is he Gaussian il e ed noise added by he channel
and h( ) is he o e all impulse esponse gi en by:
whe e
being
l
he in e se Fou ie ans o m ope a o ,
p
he oll-o
ac o o he aised-cosine unc ion and
Tis
he symbol pe iod.
In his pape , we ha e conside ed
an
ou doo non-
s a ic channel model based on COST-207 documen
[5,6].
In
pa icula we ha e used he model denomina ed Typical U ban
en i onmen . This documen cha ac e izes he channel by
whe e
is
he delay o he channel ap i de ined in COST
207
documen ,
(h,+jh,,)
is
he ins an aneous complex ampli ude o
he ap
i
whose mean powe and ime a ia ion a e also
de ined in
COST
207 documen and being
wi h
y=(l-k)
(7)
IJI.
BAYESIAN DECISION-FEEDBACK EQUALIZER
The
MAP
equalize eaches he op imum pe o mance
in a symbol-decision sys em based on a p obabilis ic decision
algo i hm. Fo a ini e sampled linea impulse esponse h(kT),
he ecei ed symbols wi hou noise a e:
1=p
Fo an inpu sequence wi h leng h equal o
N,
he
algo i hm decides he symbol wi h highe p obabili y:
I
is
clea ha o a modula ion wi h
M
symbols he e
a e
ML
possible s a es, wi h
L=p+q+l.
Then, when he decided
symbol co esponds o he cen al impulse esponse sample, he
possible s a es a ins an
k
a e:
21
44
The Bayes heo em could be applied o equa ion
(1
1)
gi ing he ollowing exp ession:
whe e
SZ,
is he se ha encompasses all
ML
s a es and
R(dJ
is he se o s a es when he symbol
k
is
dk.
Then, each e m o
equa ion (13) could be ac o ized as:
wi h
and
Fig.
2
Radial Basis Func ion ne .
Compa ing his se o equa ions wi h he RBF
s uc u e shown in Fig.
2,
i is clea ha by calcula ing he
cen e s
(ci)
o
each node as he noiseless es ima ed ecei ed
signal, he ne could pe o m as a MAP algo i hm.
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IV. CHANNEL ESTIMATORS
In his pape we ha e e alua ed h ee kinds
o
channel
es ima o s. Fi s , we ha e used he
LSSE
algo i hm. This
algo i hm is simila o he es ima o used in he GSM sys em
based on a co ela ion p ocedu e wi h a aining sequence and
has been chosen o he Uni e sal Mobile Telecommunica ion
Sys em (UMTS) [7,8]. The main cha ac e is ic o his
algo i hm is he low compu a ional cos equi ed o es ima e
he impulse esponse. Howe e , his algo i hm canno be
implemen ed in an adap i e o m.
The second simula ed sys em uses he classical
RLS
in
o de o calcula e he impulse esponse. When he es ima ed
esponse is ob ained a he end o he aining sequence, he
algo i hm is s opped. This algo i hm needs sho e aining
sequences han he
LSSE
algo i hm. I only needs
2N-I
whe e
N
is he leng h o he es ima ed impulse esponse
[9],
bu i
has a highe compu a ional cos . This algo i hm is mo e
app op ia e han he leas mean squa e (LMS) algo i hm
because o he low- ime con e gence achie ed wi h LMS
which p oduces a
loss
o acking in as a ying channels [4].
We ha e also e alua ed he sys em pe o mance
u ilizing he same
IUS
algo i hm bu using he decided symbol
when he aining sequence is inished. This echnique
compensa es o he ime a ia ion e ec s o he mobile
channel.
In he i s and second sys ems, he equalize
coe icien s a e calcula ed when he aining sequence is
inished and he equalize keeps hese coe icien s
o
he
whole in o ma ion ame. Howe e , he hi d sys em needs o
calcula e he coe icien s e e y i e a ion.
Finally we ha e conside ed a ame wi h
20
aining
symbols and
IO0
in o ma ion symbols when he
LSSE
es ima o is employed, and
IO
aining symbols and
110
in o ma ion symbols when he
IUS
es ima o
is
used.
V. RESULTS
We ha e examined he e ec i eness o he
RBF
used
as an equalize in igh ing he mul ipa h and ading in oduced
by he adio channel. The c i e ion used o e alua e he sys em
quali y is he bi e o a e,
Pe,
e sus he signal- o-noise a io
and he no malized delay sp ead. The equalize s ha e he
ollowing cha ac e is ics: he RBF ne has been implemen ed
wi h 3 inpu signal nodes and 3 eedback-decided symbol
nodes. The con en ional DFE has he same s uc u e in o de
o ha e he same in o ma ion in each equalize . The equalize
coe icien s a e calcula ed om an es ima ed impulse esponse
o
5
aps. The en i onmen modeled is a ypical u ban channel
wi h 12 aps and
1
mic osecond o delay sp ead [5,6].
10-2
cc
LL
m
10-3
0.01 0.1
1
TRb
Fig.
3.
B.E.R. e sus no malized delay sp ead in a s a ic
en i onmen .
In Fig.
3,
we show he e olu ion o he mean alue o
he bi -e o p obabili y agains he no malized delay sp ead
alue,
Tb,
Tb
being he bi pe iod,
20
dB o signal- o-noise
a io, and a s a ic en i onmen . In he igu e, he esul s
ob ained o bo h equalize s, RBF and con en ional DFE, a e
compa ed. Thei coe icien s ha e been calcula ed aking he
LSSE
algo i hm and
IUS
algo i hm as channel es ima o s. I
could be no iced ha equalize s ained wi h he
LSSE
algo i hm ha e a pe o mance loss o all he alues o he
no malized delay sp ead. I is also shown ha i he delay
sp ead is highe han
0.2,
he RBF pe o mance is be e han
he con en ional DFE. These esul s seem o be easonable,
since a
RBF
s uc u e is able o c ea e mo e complica ed
decision egions in his case as hey a e needed.
10-2
w
m
10-3
10-1
5 10
15
20 25 30 35
SNR
(dB1
Fig.
4.
B.E.R. e sus
SNR
in a low dispe si e and s a ic
en i onmen .
In Figs.
4
and
5
we show he e olu ion o he BER
e sus he signal- o-noise a io conside ing
a
null equency
Dopple . In hese g aphics, he e ec i eness o he
RLS
algo i hm compa ed o he
LSSE
algo i hm is shown. I could
be no iced ha a gain o
2
dB up o a
SNR
o
25
dB o a
no malized delay sp ead equal o
0.02
is ob ained. The sys em
wi h he
RLS
algo i hm
also
has a gain o abou
1
o
2
dB o
a no malized delay sp ead equal o
1.0.
On he o he hand, i
is shown ha he RBF equalize is be e han a con en ional
DFE
o
a
no malized delay sp ead equal o
1.0,
ob aining a
gain app oxima ely o
1
dB.
iO-1
.....
.....
2-.
...........
.-.;
............
-L..
.
i1
' &.
.
10-2
1
...........................
.....
m
i 0-3
m4
5
IQ
15
20 25 30
SNR
(dB)
Fig.
5.
B.E.R. e sus
SNR
in a high dispe si e and s a ic
en i onmen .
When we conside he Dopple e ec (Figu es
6
o
9),
i could be no iced ha he sys em has a g ea
loss
o
pe o mance, especially o low alues o he no malized delay
sp ead. I is shown in Figu e
6
ha only when acking
RLS
is
used, (in Figu es k. ls), does he bi e o descend o
app oxima ely In hese igu es i could be no iced ha
when he Dopple e ec is conside ed, he
LSSE
es ima o
pe o ms be e han he no- acking
RLS
es ima o , (in Figu e
no k. ls), o
low
alues o he no malized delay sp ead.
10-1
'
:
1
.....................
*.
...........-.....I-
............
) u
10-7
I
5
IQ
15
20
25
30
SNR
(dB)
Fig.
6.
B.E.R. e sus
SNR
in a low dispe si e en i onmen
wi h 50Hz o Dopple .
2146
E
:
Fig. 7. B
Dopple .
10"
10-2
10-3
.-
0.01 0.1
1
T
Rb
..ER. e sus no malized delay sp ead wi h
10-1
10-2
;
aj
w
10-3
1
n-4
50
Hz o
.-
5
10 15 20
25
30
SNR
(dB)
Fig.
8.
B.E.R. e sus
SNR
in a high dispe si e en i onmen
wi h
IOOHz
o Dopple .
Fig.
10-1
n
.;!
10-2
m
10-3
0.01 0.1 1
T
Rb
9. B.E.R. e sus delay sp ead wi h 100 Hz
o
'
Dopple .
VI. CONCLUSIONS
In his pape he pe o mance o he MAP algo i hm
implemen ed wi h
RBF
using h ee es ima o s in a ypical
u ban adio en i onmen ha e been analyzed and compa ed
wi h a con en ional DFE. I has been shown ha he
MAP
algo i hm o e s e ec i eness o adap i e equaliza ion o he
h ee analyzed channel es ima o s wi h espec o he
con en ional DFE and ha i p o ides a iable solu ion o he
p oblem
o
channel dis o ion in digi al communica ion
sys ems. I has been shown ha his new echnique allows he
signal- o-noise a io in high dispe si e channels o be educed
mo e han 1 dB. On he o he hand, i has been p o ed ha he
LSSE es ima o p oduces losses in pe o mance o low
dispe si e s a ic channels. Howe e , i is mo e obus espec
o he no- acking RLS algo i hm when he dopple e ec is
aken in o accoun . This e ec is p oduced because he
simula ed sys em wi h he LSSE es ima o has a longe aining
sequence. When he no- acking
RLS
algo i hm has he same
leng h o aining sequence, we ha e p o ed ha i has he
same beha io as he LSSE algo i hm. Howe e , his does no
p oduce he same e ec o acking
RLS
algo i hms. We hink
ha in his pape he necessi y o using
RLS
acking
echniques o compensa e he e ec s p oduced by ime- a ying
dispe si e channels has been demons a ed.
VII. ACKNOWLEDGMENTS
The wo k desc ibed in his pape has been ca ied ou
wi hin he CICYT TIC94-0870-C02-01 p ojec , in he
amewo k o Na ional Plan o Spain.
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[1]S.
Chen,
S.
McLaughlin, B. Mulg ew, P. M. G an ,
"Adap i e Bayesian Decision Feedback Equalize o
Dispe si e Mobile Radio Channels",
IEEE T ans. on Comm.,
ol. 43, no. 5, pp. 1937-1945, May 1995.
[2]M. Peng, C. L. Nikias y J.
G.
P oakis,
"Adap i e
Equaliza ion wi h Neu al Ne wo ks: New Mul ilaye
Pe cep on S uc u es and hei E alua ion",
IEEE-ICASSP-92,
ol. 2, San F ancisco, Ma ch 1992.
[3]S. Chen,
B.
Mulg ew, P. M. G an ,
"A
Clus e ing Technique
o Digi al Communica ions Channel Equaliza ion
using
Radial
Basis Func ion Ne wo ks",
IEEE T ans. on Neu al Ne ., ol. 4,
[4] J.G. P oakis,
"Digi al Communica ions",
2nd ed. New
Yo k, McG aw-Hill, 1982.
[5] COST 207,
"Digi al Land Mobile Radio Communica ions",
Final Repo . O ice o O icial Pub. o he Eu opean
Commun., Luxembou g, 1989.
[6] R. W. Lo en z,
"Co ec ions o COST
207
P oposals o
Simula ion
o
Radio Channels",
COST 231
TD
(91)
26,
Da ms ad , Feb ua y 1991.
[7] S.N. C ozie , D.D. Falcone , S.A. Mahmoud,
"Leas Sum
o
Squa ed E o s (LSSE) Channel Es ima ion",
IEE P oc.-F
o1.138,no. 4, pp. 371-378, Augus 1991.
[SI
A. U ie, M. S ee on, C. Mou o ,
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Mobile Access Sys em o UMTS'
PIMRC 94, pp. 685-690,
Sep . 94. The Hague.
[9]
J.G.
P oakis,
"Adap i e Equaliza ion o TDMA Digi al
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IEEE T ans. on Vehicula Tech., ol. 40, no.
2, pp. 333-341, May 1991.
pp. 570-579.
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