Cross-coupled LMS for quadratic constrained filter design
Abstract
Quadratic constrained maximization appears as the central problem in many filter design methods and signal processing applications, whenever the processing or filtering entails the maximization of a signal to noise ratio. Furthermore, self reference systems for MSE filters, beamformers and equalizers which manage either time or frequency diversity of the desired signal, give rise to the same mathematical problem. This work reports an adaptive algorithm based on two coupled LMS-like algorithms. It is shown how the Lagrange multiplier, involved in constrained minimization problems, plays the role of gain control in forming the error signal for the filter updates. This gain is power controlled by the output signals keeping the proper equilibrium between the outputs of the two LMS loops.
Full text
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.
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