CDMA BLIND CHANNEL EQUALIZATION:
A
WEIGHTED
SUBSPACE
APPROACH
Ra ael Ruiz, Ma ga i a Cab e a
Depa men o Signal Theo y and Communica ion
Uni e si a Poli ecnica de Ca alunya
C/
So Eulalia de Anzizu,
dn.
08034
Ba celona,
SPAIN
e-mail: [email p o ec ed], [email p o ec ed]
Abs ac
-
This
pape conside s he p oblem o blind
demodula ion o mul iuse in o ma ion symbols in a
di ec -sequence code-di ision mul iple access (DS-
CDMA) en i onmen , Channel es ima ion and symbol
de ec ion in p esence o bo h mul iple access
in e e ence
(MAI)
and in e symbol in e e ence (ISI)
is ca ied ou wi h second o de s a is ics me hods om
he ecei ed da a. This p oblem
is
simila o Di ec ion
O A i al (DOA) es ima ion, whe e many solu ions
like he
MUSIC
algo i hm o “weigh ed” echniques
(as
De e minis ic
Maximum
Likelihood
o
Weigh ed
Subspace
Fi ing
me hod) ha e been de eloped.
In
his
p oposal hese echniques a e ex ended o blind DS-
CDMA channel iden i ica ion p oblem in an uni ied
amewo k
known
as
Subspace
Fi ing.
In
his
amewo k he es ima ed and he ecei ed
da a
a e
“ i ing” h ough he subspaces in a leas
squa e
sense.
Then,
in
o de o achie e
a
be e es ima ion o he
channel, a modi ied Gauss-New on ype algo i hm is
sugges ed. Simula ions a e ca ied ou compa ing he
p oposed solu ions wi h a classical signal subspace-
based blind channel iden i ica ion scheme.
I.
INTRODUCTION
The e is ac ually a g owing in e es
in
he design
o
high- a e DS-CDMA ne wo ks. This kind o digi al
communica ions a e subjec ed o in e symbol
in e e ence (ISI) (due o channels wi h mul ipa h
phenomena) and mul iple-access in e e ence
(MAI)
(which is inhe en o any nono hogonal CDMA
sys em). This p oblems can be
so
se e e ha co ec
ecep ion o he ansmi ed symbols is no easible
anymo e. I is necessa y he e o e o equalize he
channel, supp essing join ly bo h
MAI
and ISI.
This wo k
is
pa ially suppo ed by he Na ional
Resea ch Plan o Spain, CICYT, TIC96-0500-C10-01,
TIC98-0412, TIC98-0703 and by he Gene ali a
o
Ca alonia, CIRIT, 1998SGR-0008 1.
Since Tong,
Xu
and Kaila h showed in [l] ha i is
possible o ob ain
an
es ima ion o he channel om a
second o de s a is ic o he ecei ed signal, second
o de s a is ics ha e showed i s use ul in channel
es ima ion. Subspace-based me hods a e based on he
singula alue decomposi ion
(SVD)
o a
ma ix
cons uc ed om he obse ed signal, which p o ides a
obus disc imina ion be ween desi ed and dis u bing
signals
in e ms
o signal and noise subspaces. In
pa icula , se e al wo ks (see
[2]
and e e ences
he ein) ha e add essed he use o MUSIC- ype
me hods o pa ame e es ima ion
in
CDMA
sys ems.
The gene al objec i e is o
ind
a low- ank subspace
wi h a
shi
s uc u e ha
has
minimal dis ance o he
ue signal space,
o
equi alen ly, ha
is
as o hogonal
o he noise subspace
as
possible. Vibe g and O e s en
o mula e in
[4]
di e en me hods
in
a common
subspace i ing based hmewo k, p o iding an
o e iew o he DOA es ima ion p oblem and
cla i ying he algeb aic ela ions be ween he
algo i hms.
In
his
con ibu ion we ocus on
MUSIC
[3],
WSF
[4]
and
MDL
[6]
algo i hms. These me hods a e
summa ized
and adap ed o blind channel iden i ica ion
in a subspace i ing app oach. The pape
is
o ganized
as ollows.
In
Sec ion
II
signal model o a DS/SS
CDMA sys em and i s subspace app oach a e
o mula ed. Blind channel iden i ica ion me hod o
DS-CDMA p oposed by
[2]
is
also
ou lined. In Sec ion
UI
we ex end
he
subspace- i ing
iamewo k p oposed
by Vibe g
[4]
o blind DS-CDMA channel
iden i ica ion. The cos hc ions whose miniiza ionl
maximiza ion will allow es ima ing he channel a e hen
o mula ed. A e
ha ,
he Gauss-New on algo i hm o
blind channel iden i ica ion [8]
is
ou lined.
In
sec ion
IV
some simula ion esul s a e p esen ed and discussed.
Finally, we p esen some conclusions and ou line
u u e
wo k in sec ion
V.
0-7803-5565-2/99/$10.00
0
1999
IEEE
2373
11.
PROBLEM
FORMULATION
Hw=
Signal
Model
H(L)
-.e
H(0)
0
i
0
...
Conside a K-use bina y communica ion sys em
h ough a Gaussian channel. Le
sk(n)
deno e he
k h
use ansmi ed symbol a ime
nT
(whe e
T
is he
symbol du a ion) and
{Cko)}
be he signa u e sequence
o
M
l
's
assigned o he use . The ansmi ed signal
due o he
k h
use is:
Ns-l
M
Xk
(0
=
4
*
CSk
(n)CCk
(A.
P(
-
nT
-jTc
-
5k)
n=O
j=1
Ol lT
(1)
Whe e
Ns
is he numbe o ansmi ed symbols,
M
is
he p ocessing gain,
p( )
is a no malized chip wa e o m
o du a ion
T,
=
TIM
and
Ak
and
Tk
deno e, espec i ely,
he ampli ude and delay
o
k h
use . The esul ing signal
ho ough
a
ime-in a ian channel
hk(0
is:
Ns-1
M
~k( )
=
A,
*
CS~
(~)CC,
(j)-Zk(
-
nT
-
jT,)
n=O
j4
Ol lT
(2)
Wi h
&( )=p( -7k)*hk( );
zL( )sO
/ EIO,Lk
.TI.
The signal is il e ed wi h he chip-ma ched il e and
sampled a he chip a e
T,.
The esul ing disc e e- ime
signal componen due o he
k h
use a he i h chip
pe iod
o
he n h symbol in e al will be:
Lh
~,+(n)=Cs,(n-~).h,,(~)
i
=
z..M;~=
Z..K
(3)
I=O
Wi h
L
(m)
=
s,"
E++
(
+
m~,
1.
p( )d
Olml(L,
+l).M-l
(4)
M
(j)
=
h,
(IM
+
i>
;
hk(m)
=
'cck(j)* k
(m-j)
J=1
Olml(L,+l).M
(5)
The ecei ed disc e e- ime signal a he i h chip pe iod
o
he n h symbol in e al
is
hen:
,
(n)
=
Y,
(n)
+
,
(n)
K
Y,
(n)
=
C
Y,k
(n)
(6)
Wi h
i;(n)
he il e ed AWGN noise. De ining he
ollowing ma ix and ec o s
[2]:
k=l
(n)
=
;
y(n)
=
;
(n)
=
H(n)
=
;
s(n)=
(7)
By s acking
W
successi e samples
o
he ecei ed da a
we ha e he
MWxZ
ec o s:
Subspace App oach
The addi i e noise
c(n)
is modeled
as
a Gaussian,
s a iona y,
whi e
and
ze o
mean
andom
p ocess.
The
co a iance
ma ix
o he ecei ed signal is:
(13)
C,
=Ebw
.RF]=Hw .Hi
+021M,
2374
Whe e he unc ion
E[*L
deno es ma hema ical
expec a ion and supe sc ip is he he mi ic ope a o .
We assume ha each use 's in o ma ion
symbols
a e
independen ly iden ically dis ibu ed and he symbol
s eams o di e en use s a e independen :
.Ew
.s;]=
I,
wi h
=
K(w+L).
The signal subspace
is
de ined as he space spanned by
columns o
Hw,
and he noise subspace is i s o hogonal
complemen . Bo h subspaces can be ob ained om he
SVD
o co a iance ma ix:
CRR
=
UAU"
=
[U, U,].
p
*)Js
uo
(14)
Wi h
A,
=
diag(il1
...
&),
Aid
o
i=l,
...,
(signal
eigen alues) and
A,
=diag(&+I
...
,IMw),
A+=d
o
i= +l,
...,
MW
(noise eigen alues). The
MW
x
Y
ma ix
U,
and he
MW
x
MW-
ma ix
U,
a e:
U,
=[ul
e..
U,];
U,
=b,+,
u n]
(15)
Since columns o
U,
span he signal subspace and
columns
o
U,
he noise subspace, o hogonali y
be ween subspaces p o ides:
U:
.Hw
=O
o
i=Z,..,MW-(W+L)
(16)
Blind channel es ima ion in DS-CDMA
The ollowing subspace algo i hm is an ex ension o he
me hod de eloped in
[3]
o mul iple inpu signals. Le
U,"
...
U;],
whe e
U:
is
he i h
(MW- )xA4
pa i ion o
U,"
,
we de ine he
M(L+Z)xK
ma ix
H
and
he
(MW- )(W+L) xM(L+l)
ma ix
G
as:
Then
(1
6)
can be ewi en as:
I
G
has mo e ows han columns, hen
i s
igh null
subspace speci ies
H
up o a nonsingula ambigui y
XXK
ma ix ac o , Le:
H
=
H.D
.
To
u he sol e
his
ambigui y, he ini e alphabe p ope y o he inpu da a
has o be exploi ed, which
is
p ohibi i ely expensi e. In
o de
o
o e come
his
p oblem, Wang inco po a ed
in
[2]
he sp eading signa u es.
Le deno e he
M(L+Z)
x
m
ma ix
o
he composi e
signa u e
o
he k h use
Ck
as:
U,H*HIY=O
e
G*H=O
(18)
(19)
..
Wi h
m
he channel esponse leng h
o
&(m)
in
(4);
ha ing ha
H=[hl
h
...
h,],
we cm ew i e
(5)
as:
h,
=A,C, ,
Wi h
,,
=
[
k?
]
(20)
k
("J
-
1)
m,x,
Then, om
(1
8)
we ha e ha :
And he e o e, he igh null ec o o
G
.
C,
is
k
up o
an ambigui y scala ac o .
G.H=O*G*hk
=Ak*G*ck* k
=o
(21)
111. WEIGTED SUBSPACE FITTING
Cos unc ions
Since co a iance
ma ix
Cm
is
in
p ac ice es ima ed
om a limi ed amoun
o
ecei ed da a, only an
app oxima ion
o
Cm
is a ailable. Then,
(21)
has o be
sol ed
in
a leas
squa e
sense. Adap ing o blind
channel iden i ica ion he basic subspace i ing
p oblem in
DOA
es ima ion p oposed by Vibe g
&
O e sen
[4],
we ha e ha , gi en some ep esen a ion
o
he da a
M,
we should o ind an es ima ion
o
H
and
T
such ha
H,T
=
a g
2~11~
-H,
(H).TII:
(22)
Whe e
[diis
he quad a ic F oebius no m.
This
is
a
sepa able p oblem, and subs i u ing he solu ion o he
i s pa ,
T
=
Hw+.M
in o
(22)
( he supe sc ip
deno es he pseudoin e se ope a o ), his gi es he
gene ic subspace i ing cos unc ion:
H
=
a g
m$V(H)
wi h
V(H)=/(I-P,,).MII:
=T&.M.M")
(23)
Whe e
P"=Hw-Hw+
is he
LS
p ojec o on o he column
space o
Hw,
P;
=
I
-
P,,
is he o hogonal
LS
p ojec o
and T (*) deno es he ace ope a o .
As
M
is a
ep esen a ion o he da a, di e en choices o
M
will
p o ide di e en cos unc ions (c i e ions).
MUSIC
c i e ion.
The
MUSIC
c i e ion is based on o hogonali y be ween
signal and noise subspaces. The cos unc ion de i ed
om
(
16)
can be w i en
as:
2375
This is he MUSIC cos unc ion. Since in
DOA
p oblem
(24)
does no gi e accu a e esul s when he
signals a e highly co ela ed, Schmid
E o !
Unknown
swi ch a gumen .
in oduced a
no maliza ion ma ix
(HF
.Hw)-'
in o
(24),
esul ing
he
Mul i-Dimensional (MO) MUSIC
algo i hm:
H
=
a g
minV,
(H)
,(H)
=
T b;
-H,)I.Hg
.UiT,-U
.H,}=
=
T kH
.U,. U:)=
T (p,
-b
-
Us
-U:)}
(25)
As
he ace o a p ojec ion ope a o is equal o he
dimension
o
he subspace on o which i p ojec s,
minimizing
VI@)
gi es he same esul ha maximizing
V
(H)
=
T $
-6,
.U:)=
T [I
-
Pi)-Us
.U:},
and he
cos unc ion o
MD-MUSIC
algo i hm will be:
H
=
a g
minVM,_,,,
(H)
m
Thus, he
MD-MUSIC
algo i hm is a subspace i ing
me hod whe e he ep esen a ion o da a
is
di ec ly
gi en by he signal subspace,
M
=Us
.
De e minis ic
ML
c i e ion.
Desc ibed by Biihme
[6],
his me hod
y
o maximize
he log likelihood o he ecei ed da a
Y
wi h espec o
Hw
and
S
( he columns o he
MW
x
Ne
ma ix
Y
a e
he
Ne
snapsho s
y(n)
wi h
n=I..Ne,
and he columns o
he
(W+L)
x
Ne
ma ix
S
a e he
Ne
ansmi ed symbol
ec o s
s(n)).
This
is
equi alen o minimize he cos
unc ion
VMDL
(H)
=
11
Y
-H,
.S
11:
.
The solu ion
o
he
i s pa
is
S
=
Hk
-Y
and he cos unc ion becomes:
i
=
WgmZ MDL(H)
wi h
MDL(H)=
T (&
-c,}
(27)
Connec ion wi h subspace i ing can be made using
asymp o ic a gumen s
(see
[4]).
Fo la ge
Ne,
we ha e
A,
+021,
C,
-+
U,
.i*U
+02
.I
and
A
=
A,
-0'
.I.
As
he ace
o
ozPi
is a cons an , he cos unc ion is
asymp o ically (la ge
Ne)
equi alen o:
The e o e, he de e minis ic ML me hod is a subspace
i ing
echnique whe e he da a a e ep esen ed
by
i s
weigh ed signal subspace
M
=
U,
W,&
;
wMDL
=
A.
WSF
c i e ion.
The
de e minis ic
ML
me hod allows
o
us
o in oduce
he weigh ed subspace- i ing concep . This is:
H
=
a g
m, n .ssF
(H)
wi h
V,,(H)=
T {PkUsWwsFU:}
(29)
Whe e
Wmp
is
a posi i e de ini e Weigh ing ma ix.
The ques ion
is
o ind a weigh ing ma ix
Wmp
ha
makes he es ima ion s a is ically ( o la ge
Ne)
e icien ,
i.
e., ha makes he
WSF
es ima es
asymp o ically achie e he C ame -Rao lowe bound on
he a iance
o
he es ima o e o . Vibe g and O e s en
[4]
ha e shown ha he op imal choice o
WmF
is
W,,,
=x2A;'.
Thus,
he
WSF
c i e ion can be
exp essed in
(23)
when he ep esen a ion o he
da a
is
M
=
U,
WzF.
Modi ied
Va iable
P ojec ion
Algo i hm
The p oposed echnique in [8] is he
Modi ied Va iable
P ojec ion
(MVP)
me hod (see
[7]
and e e ences
he ein) applied o (one use ) blind channel
iden i ica ion. The c i e ion unc ion in
(23)
mus be
minimized o e he ec o
h
( o one emi e , ma ix
H
is a column ec o ). Conside he nonlinea leas squa e
p oblem gi en by
(23);
in he damped New on scheme,
h
is i e a i ely es ima ed wi h:
hk+'
=hk-,Uk.G-'.V'
(30)
Wi h
,uk
he s ep leng h,
G
he Hessian ma ix
o
he
cos unc ion and
'
he g adien . E e y i e a ion he
Hessian and he g adien a e e alua ed in
hk.
I
h
is well
ini ialized, he New on me hod gua an ees
an
ul ima e
quad a ic con e gence o
h
.
Conside as he g adien o he cos unc ion. The i h
elemen o he g adien yields:
Conside now he Hessian ma ix, he 0 h componen o
he
app oxima e
(a
Kau manns's
modi ica ion is
applied o he algo i hm, see
[7])
Hessian
ma ix
is:
G,
=2T {[(HLT
.H
.Pi
.HY .H&].M-M"}
(32)
Then,
(30), (31)
and
(32)
gi e he
MVP
algo i hm
(de i a ion
o
(31)
and
(32)
can
be
ound
in
[7]
and
[SI).
In
o de
o
ob ain a good ini ializa ion o he
es ima es, we can use he MUSIC algo i hm, which will
allow o achie e a global minimum. The s ep leng h
2376
ac o should be chosen in o de o gua an ee global
con e gence. I is
known
ha quad a ic con e gence
o
New on- ype algo i hms is only achie ed i he s ep
leng h ac o con e ges
o
uni y.
IV.
SIMULATION
RESULTS
A
50
Mon e Ca lo simula ion is ca ied ou in o de
o
e alua e he pe o mance o he weigh ed subspace
algo i hms
in
a digi al DS-CDMA communica ion
sys em. The
K=5
use s emi a bu s
o
Ns=128
BPSK
symbols. The Gold signa u e sequence has
M=25
l
‘s
(i.e.
Tc=T/15).
The highes o de o he channels (ISI) is
se o
L=2
and he wid h o he empo al window
is
W=4.
Each use has
A =l(O
dB MAI) and we conside
he asynch onous CDMA channel. The pe o mance
measu e plo ed he e is he mean
o
he p obabili y o
bi e o compu ed o each use in a gi en signal- o-
noise a io
(SNR)
de ined
as
A:
/oz.
P elimina y esul s
can
be
seen
in
Fig.
1.
WSF and
MDL me hods gi e i ually he same esul s, sligh ly
wo s
han
he
MUSIC
me hod. Since
50
Mon e Ca lo
simula ions do no p o ide a su icien s a is ics, we can
expec he same esul han in case o one emi e ,
whe e he h ee me hods p oposed he e gi e he same
pe o mance.
Bi
Em
Ra a
J
1
2
3
4
5
6
SNR
Figu e
1.
Bi e o a e.
V.
CONCLUSIONS
A Mon e Ca lo simula ion is ca ied ou in o de
o
e alua e he di e en weigh ed subspace algo i hms.
As
i is shown in he igu es, weigh ed subspace me hods
p o ide he same accu a e esul s han he
MUSIC
me hod de eloped by Moulines
e al.
in
[3]
and
ex ended
o
DS-CDMA by Wang
in
[2].
As
was shown
in
[8],
he
Mod$ed Va iable P ojec ion
algo i hm
de eloped and applied in a blind channel iden i ica ion
con ex can imp o e he MUSIC esul s in a single use
con ex when weigh ed subspace me hods a e in ol ed.
I can be expec ed ha
his
algo i hm can be applied o
DS-CDMA
sys ems wi h
same
esul s. On he o he
hand, he weigh ing ma ices in
MDL
and
WSF
me hods
we e de ined o DOA p oblem; as channel es ima ion
is
no he same p oblem ano he weigh ing ma ix can
be de i ed o
his
speci ic si ua ion. Ano he
simula ions ha e been ca ied ou in o de o es he
beha io
o
weigh ed algo i hms wi h. la ge bu s s.
Resul s ha e shown ha a la ge sample size, less
di e ence in Mean Squa e E o (MSE) exis s be ween
MUSIC
and he p oposed algo i hms. The e o e we can
conclude ha he weigh ing ma ices de ined in
MDL
and
WSF
c i e ions a e use bl in low sample sizes.
REFERENCES
[
13
Tong,
L.,
Xu
G.
and Kaila h
T.
“A
new app oach o
blind iden i ica ion and equaliza ion o mul ipa h
channels”.
P oc. 2S h Asiloma Con$ On Signals,
Sys ems and Compu e s,
pp.
856-860,
Paci ic G o e,
CA., No .
1991.
[2]
X.
Wang and
H.
Vicen Poo , “Blind Equaliza ion
and Mul iuse De ec ion in Dispe si e CDMA
Channels”,
IEEE T ans. On Commun.,
Vol.
46,
pp.
91-
103.
Jan.
1998.
[3]
E.
Moulines, P. Dohamel,
J.
Ca doso
&
S.
May a gue, ”Subspace Me hods o he blind
iden i ica ion o mul ichannel
FIR
il e s”,
IEEE
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