CROSS-COUPLED
LMS
FOR QUADRATIC CONSTRAINED FILTER
DESIGN
Miguel
A. Lagunas, A a
I.
Pe ez-Nei a
Modulo
D5,
Campus No d
UPC
G an Capi a s/n
08034
BARCELONA
SPAIN
e-mail miguelgps. sc.upc.es
ABSTRACT
Quad a ic cons ained maximiza ion appea s
as
he cen al p oblem in many il e design me hods
and signal p ocessing applica ions, whene e he
p ocessing o il e ing en ails he maximiza ion o
a
signal o noise a io. Fu he mo e, sel e e ence
sys ems o MSE il e s, beam o me s and equalize s
which manage, ei he ime o equency di e si y o he
desi ed signal, a ise o he same ma hema ical p oblem.
This wo k epo s an adap i e algo i hm based
on wo coupled LMS-like algo i hms. I is shown how
he Lag ange mul iplie , in ol ed in cons ained
minimiza ion p oblems, plays he ole o
a
gain con ol
in o ming he e o signal o he il e upda es. This
gain is powe con olled by he ou pu signals keeping
he
p ope equilib ium be ween he ou pu s o he wo
LMS loops.
1.
INTRODUCTION
The o mula ion o il e ing p oblems use o be
in e ms o con olling simul aneously wo quad a ic
o ms. The i s quad a ic o m is he esponse o
he
il e o be designed o he desi ed signal while he
second one is he co esponding esponse o he noise
plus in e e e s o jamme s. The design c i e ia use o
be o minimize he noise con ibu ion, ye p ese ing
he desi ed signal esponse cons an . The al e na i e o
maximizing he signal con ibu ion is useless since i
does no
allow
dynamic con ol o he ou pu signal,
and i equi es an au oma ic gain con ol sys em in
addi ion o he il e s age.
Quad a ic cons ained maximiza ion is
also
aced in
hose sys ems, like equency hopping [l] [2],
equency di e si y
[3]
[4]
and ime di e si y sp ead
spec um o communica ion sys ems, ha use ime
slo s o equency bands o ansmi eplicas o he
desi ed signal. I is easy o p o e ha il e ing ou
he
This wo k has been suppo ed by CICYT (TIC96-
0500c10-01,TIC98-0412, TIC98-0703) o Spain and
CIRIT (l998SGR-00081) o Ca alunya.
desi ed signal om di e si y eplicas and ejec ing
in e e e s o jamme s, unco ela ed in he di e si y
slo s o bands, a ises o
a
quad a ic cons ained
maximiza ion p oblem. The quad a ic cons ain consis s
in se ing
a
gi en alue o he c oss co ela ion
be ween he ou pu s o he il e ed e sion o
he
di e si y componen s. The same p oblem appea s in he
maximiza ion o he spec al sel cohe ence o
ciclos a iona y signals
[5]
o blind beam o ming and
il e ing. The quad a ic cons ain in his case is also
a
c oss co ela ion unc ion.
The solu ion o he quad a ic cons ained
maximiza ion, in
a
block p ocessing, educes o
a
gene alized eigen alue p oblem, which in ol es he
wo ma ixes de ining he objec i e and he cons ain .
The pu pose o his wo k is o mo e he inne
ad an ages o he adap i e LMS algo i hm o p o ide
sample by sample upda es o he solu ion o hose
signal p ocessing p oblems o mula ed
as
quad a ic
cons ained maximiza ion. The de i a ion is based in
he g adien o he Lag angian, wi h ank one es ima es
o he in ol ed au o and c oss-co ela ion ma ixes.
The adequa e choice o he s ep sizes and he c ucial
ole o he Lag ange mul iplie a e epo ed. The
a chi ec u e ha suppo s he esul ing algo i hm is
a
c oss-coupled LMS loop which esembles he schemes
epo ed some yea s ago o independen sou ce
sepa a ion [7].
2.
QUADRATIC CONSTRAINED
MAX.
In o de o jus i y he in e es o he adap i e
algo i hm, o be epo ed in Sec ion
111,
his sec ion
desc ibes b ie ly he mos in e es ing applica ions
which a e o mula ed in e ms o
a
quad a ic
cons ained maximiza ion. Quad a ic cons ained
maximiza ion shows up in hose p oblems whe e he
objec i e is o maximize
a
signal o noise a io
(SNR),
de ined in e ms o he co ela ion ma ixes o he
desi ed signal
R
and noise
R
The maximiza ion o
he
SNR
de ined in
(l),
o he il e coe icien s
deno ed wi h ec o
4,
=S
=n
0-7803-5682-9/99/$10.0001999
IEEE.
839
can be o mula ed
as
an au oma ic gain con ol
cons ain , which se s
a
cons an alue o he
nume a o , and minimizes he denomina o .
No e ha he same solu ion applies o he case whe e
he denomina o includes he il e ou pu powe .
Fu he mo e, o dynamic con ol easons in successi e
s ages, loca ed a he ou pu o he il e , i is be e o
cons ain he denomina o and maximize he nume a o
o
(3)
han he al e na i e ou lined abo e o he
maximiza ion o
(1).
A
H
.R .A
I,,
=s
-
-
In hese wo al e na i es, he pa ame e
@s
plays he
ole o an au oma ic gain con ol ha de e mines he
powe le el o he desi ed signal a he il e ou pu .
The solu ion
o
he quad a ic cons ained
maximiza ion is gi en by
(4)
A.R .A=& .A
=s
-
-n
-
.,
SNR,,
=am,
The same o mula ion appea s in
communica ion sys ems whe e, ei he ime, code o
equency, di e si y is used o he desi ed signal
[6].
Focussing he case o ime di e si y, le
us
assume ha
he il e ing equa ions a e ea u ed by he inpu
snapsho s
Xsn
and
XYn
and he il e weigh
ec o
A
.
(5)
Indexes and
s
indica e ha he successi e samples
o ming he snapsho s a e aken om wo di e en
di e si y slo s.
As
a
consequence, he desi ed signal is
p esen ,
as a
ime eplica, in bo h snapsho s and he
noise and in e e e s a e unco ela ed om slo
o
s.
Taking in o accoun ha he desi ed signal is
p esen in he wo il e inpu s,
a
sui able c i e ia o
emo e undesi ed signals is o minimize he mean
squa e e o
(MSE)
be ween he wo ou pu s. In
addi ion, in o de o a oid he i ial solu ion he MSE
is cons ained by he c oss co ela ion o he ou pu s.
This is o mula ed in
(6).
No e ha he cons ained minimiza ion shown in
(6)
emo es hose signals which a e p esen ei he in
scena io o
s
and hose which p esen in bo h
scena ios p esen null
o
small co ela ion among he
di e si y slo s. In he o he hand, he desi ed signal is
enhanced due o i s high c oss co ela ion.
Equa ion
(6)
is o mula ed
as
(7),
whe e
R
and
R
a e he au oco ela ion ma ixes o
=S
=
snapsho s &,and &,and
R
is he c oss-
=
s
co ela ion be ween he men ioned snapsho s, i.e.
(7)
The equency dual o he abo e p oblem is
he case o equency di e si y
[4).
In his case and
s
deno e wo di e en equency bands ( he signal
o ming he snapsho s a e he complex en elope o he
co esponding band-pass signals)
.
The single il e design o bo h di e si y
componen s does no gua an ee maximum signal o
noise a io. In ac , u he imp o emen s can be
expec ed, no only in he a e age
SNR,
bu also in he
segmen ed
SNR
when di e en il e s a e se o each
di e si y componen
as
indica ed in
(8).
The use o wo
di e en il e s is manda o y in maximum likelihood
(ML) ecei e s o communica ions and ML p ocessing
applica ions.
The cons ained minimiza ion
is
o mula ed in
(9),
(9.4
-S
AH.R
=sy-
.A
+AH.R
-Y
=ys-S
.A
=2.eS
and he solu ion o
(9),
also ound in he li e a u e
as
he
c oss-Sco e solu ion
[5]
is shown in
(10).
(9.b)
In summa y, quad a ic cons ained
maximiza ion appea s in many signal p ocessing
applica ions in ol ing signal o noise a io
maximiza ion and di e si y echniques, which a e
cen al in communica ions sys ems which su e se e e
ading
[6].
Nex sec ion epo s an adap i e algo i hm
ha upda es, a he snapsho a e, he wo il e s weigh
ec o s using an LMS algo i hm.
3.
ADAPTIVE
ALGORITHM
Since he case o wo il e s is mos gene al
and con enien o op imal pe o mance, he adap i e
algo i hm is desc ibed o sol e his case. The il e ing
equa ions, indica ing he dependence wi h ime o bo h
weigh ec o s, a e:
A e o ming he Lag angian and aking he g adien
wi h espec bo h weigh ec o s,
he adap i e algo i hm is o mula ed
as
(13),
whe e
ps
and a e he co esponding s eep sizes.
To
de i e an s ochas ic me hod he ma ixes
in ol ed in he de e minis ic o mula ion abo e a e
es ima ed by i s ins an aneous alues, i.e. ank one
es ima ion, alues. A e using he ins an aneous
es ima es and using he il e ing equa ions
(I
l),
he
s ochas ic algo i hm is o mula ed
as
(14)
I is in e es ing o no e ha bo h he s ep size,
as
well
as
he Lag ange mul iplie , a e ime a ying. Also
he
upda ing equa ions e eal in which manne he
quad a ic cons ain modi y he e o e m. This
sugges , conside ing only equa ion (14.a), ha y,(n) ac s
as
a
ime e e ence o he il e ou pu , and he
Lag ange mul iplie
is
jus
a
gain con ol o he
e e ence. Fu he mo e, depic ing in
a
scheme he
il e ing and upda ing loops, i can be obse ed in
Figu e
1
ha he a chi ec u e is basically wo LMS
loops which
a e
ela ed by he men ioned gain con ol.
AS
Xsn
I
I
I
L
Fig.
1:
C oss-coupled adap i e loops o enhance
co ela ed signals
To
de i e he adequa e se ing o
he
gain
con ol
A,,
powe s
W,,
and
W,
a e de ined
as
he
ins an aneous powe s o he co esponding snapsho s.
The s ep sizes a e se in acco dance o he ule o he
no malized LMS in o de o ob ain missadjusmen
e o s close o
a.
In o de o ob ain he gain con ol pa ame e ,
he upda ed coe icien s om (14) a e used in he
cons ain equa ion. The p ope choice o he
pa ame e
is
such ha he upda ed coe icien s sa is y
he cons ain . A e assuming ha he o iginal weigh s,
A,,
and
Am
,
al eady sa is y he cons ain , equa ion
(17)
is ound,
4.4,
.a+&.( &
+
P n)(l
-a)+
(hS
.(a-
2)=0
(
17)
whe e
P,,
and
P,
a e he ins an aneous powe o he
il e ou pu s.
2
2
As
i can be concluded, he alue o
he
gain con ol
depend on
he
missadjusmen pa ame e . Ne e heless,
u he insigh
in
he gain pa ame e can be gained
when assuming ha , o easonable le els o
missadjusmen noise, he pa ame e
a
is e y small and
i can be conside ed ze o in
(18).
Wi h his
app oxima ion he gain pa ame e can be compu ed
di ec ly om he au oma ic gain pa ame e
@,
and he
powe s a he il e 's ou pu s.
Psn
=
1~s
(n)l
Pyn
=
IY~
(n)l
(18)
This exp ession e eals ha , a e
con e gence, he gain con ol ends o one, since he
au oma ic gain will be equal o wo imes he powe o
he desi ed signal and he denomina o , a e adequa e
il e ing, will be he same. Be o e con e gence, he
se ing o he designe o he au oma ic gain con ol
dic a es di e en alues o he gain con ol. I should be
eminded ha he alue o he cons ain is ob ained
ei he om he inpu powe o he desi ed signal
o
om he il e weigh s; in o he wo ds, in o de o ha e
adequa e dynamic ange in he coe icien s and/o in he
il e ing ope a ion, when using ini e egis e leng h
bo h o he coe icien s and o he inpu signals, i is
necessa y o selec he co esponding alue o he
au oma ic gain con ol
&.
No e ha any ealis ic implemen a ion o he
algo i hm equi es smoo hed e sion o he
ins an aneous powe s de ined be o e. Values abo e
0.99 o he smoo hing pa ame e p o e o be adequa e
o p ese e he pe o mance expec ed om he ini ial
se ing o missadjusmen noise. Finally, i is impo an
o ema k ha he c oss-coupled a chi ec u e o Figu e
1
is close o hose schemes p oposed some yea s ago
o independen sou ce sepa a ion.
4.
SIMULATIONS
Using wo di e en scena ios
o
and
s
ime
slo s, he desi ed signal was p esen as a eplica wi h
SNR
equal o
10
dB in independen whi e gaussian
noise. The signal consis s in a digi al phase modula ed
signal con aining he same symbols in bo h scena ios.
In addi ion, wo unco ela ed in e e e s we e added o
he desi ed wi h
SNR
equal o
10
dB abo e he desi ed.
Figu e 2 depic s he lea ning cu e o he c oss-coupled
algo i hm (smoo hed squa e e o be ween he wo il e
ou pu s) o a missadjusmen o
10%.
Figu e
3
shows
he e olu ion o he gain con ol pa ame e .
0
*‘‘,3 I’m
0
5M)
1000
1500
2000
2500
3000
3500
4000
4500
5000
Fig.
2:
Lea ning cu e o he algo i hm
Qd
’
amIm,€m nlam
’ ’
’
’
’
’
’
’
’
Fig.
3:
E olu ion o he gain con ol.
5.
CONCLUSSIONS
An adap i e algo i hm, based in he
ins an aneous g adien , has been epo ed o quad a ic
cons ained maximiza ion. The co esponding
a chi ec u e shows a coupled pai o LMS loops
con olled by a gain con ol o sel e e ence. The gain
con ol mix up he wo il e s ou pu s in o de o
achie e he p ope con ol o bo h adap i e loops. The
gain con ol is powe con olled by he loga i hmic
di e ence be ween he au oma ic con ol se ing and he
co esponding il e ou pu s. In his way, he
a chi ec u e exhibi s dynamic con ol o weigh and
inpu ini e leng h ep esen a ions. The algo i hm and
he a chi ec u e a e o in e es in hose p oblems
in ol ing
SNR
maximiza ion and hose p oblems whe e
he desi ed signal p esen s ime
o
equency di e si y.
6.
REFERENCES
[
11
M.Naja , M.A.Lagunas. “Adap i e a ay
beam o ming o equency hopping modula ion”.
Signal P ocessing VIII, Vol. 11, pp. 939-942. Sep
1996. T ies e, Eusipco 96. I aly.
[2]
D.
To ie i,
K.
Bakh u. “F equency compensa ion in
an adap i e an enna sys em
o
equency-hopping
communica ions”. IEEE T ans. on Ae ospace
Elec onics Sys ems. Vol. AES-23, no. 4, pp. 448-466,
July 1987.
[3] G.
K.
Kaleh.
‘‘
F equency di e si y sp ead spec um
communica ions o coun e band-limi ed gaussian
in e e ence”. IEEE T ans. on Communica ions, Vol.
[4] M.
A.
Laguna, A.I. Pe ez, M. G. Amin,
J.
Vidal.
“
Spa ial p ocessing o equency di e si y schemes”.
Submi ed o IEEE- SP.
[5]B.G. Agee, S.V. Schell,
W.A.
Ga dne . “Spec al
sel cohe ence es o al:
A
new app oach o blind
adap i e signal ex ac ion using an enna a ays”. P oc.
IEEE, Vol. 78, pp. 753-767,
Ap il
1990.
[6] J. P oakis. “Digi al communica ions”, Thi d
Edi ion, Mac G aw-Hill, Chap e s 12-15, 1995.
[7]
Y.
Ba -ness, J. Rokach. “C oss-coupled
boo s apped in e e ence cancelle ”. P oc. In e n.
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pp. 886-893, July 1996.