SIGNAL SELECTIVE DOA TRACKING FOR MULTIPLE MOVING TARGETS
Jaume Riba Jason Goldbe g
*
G ego i Vdzquez Miguel Angel Lagunas
Depa men o Signal Theo y and Communica ions,
E.T.S.E. Telecomunicaci6, Campus No d UPC.
Edi ici D-5, c/. G an Capi i. s/n,
08034
Ba celona, Spain,
Email:
{
j iba, jason,g ego i ,miguel}@ sc .upc. es
ABSTRACT
A
new algo i hm o signal selec i e acking o he di ec-
ions-o -a i al (DOA’s) o mul iple mo ing a ge s wi h an
a ay o passi e senso s is p esen ed. A new me hod based
on he p inciples o maximum likelihood es ima ion and cy-
clos a iona i y is used o gene a e ini ial angle es ima es
which, in u n, a e e ined by a Kalman il e . Sou ce angle
dynamics a e used o achie e co ec da a associa ion. High
pe o mance is ob ained wi h ela i ely low compu a ional
complexi y.
1.
INTRODUCTION
T acking he
DOA’s
o mo ing sou ces has begun o ecei e
inc eased a en ion
[l],
[2].
The p oblem consis s o es ima -
ing he DOA’s o spa ially non-s a iona y sou ces such ha
he so-called “measu emen / a ge da a associa ion p ob-
lem” is sol ed (i.e., each o he DOA es ima es is co ec ly
associa ed wi h a pa icula sou ce om one inc emen o
ime o he nex ).
In p ac ice, he DOA acking p oblem is complica ed by
he p esence o co-channel in e e ing signals-no -o -in e es
(SNOI’s). Howe e , a ela i ely new class o signal selec i e
DOA algo i hms has been shown o disc imina e agains un-
desi ed in e e ing signals and noise by exploi ing he known
cyclos a iona y p ope ies o he signals-o -in e es (SOI’S)
[3].
In his pape , such p ope ies a e used o de elop an
in e e ence ole an DOA acking algo i hm. I s basis is
simila in spi i o ha p esen ed in in e e ence ee case
o
[l].
Howe e , unlike
[l],
he so-called “condi ional max-
imum likelihood”
(CML)
p inciple
[4]
is employed.
As
will
be shown, his likelihood unc ion is highly amenable o he
inclusion o signal selec i e DOA in o ma ion ia he cyclic
co ela ion ma ix.
2.
PROBLEM FORMULATION
Conside
N
spa ially s a iona yna ow band desi ed sou ces
(SOI’s) impinging as plane wa es on an a ay o
M
passi e
senso s in he p esence o SNOI’s and addi i e whi e Gaus-
sian noise. The
A4
dimensional snapsho ec o a snapsho
ime index
m
is modeled as:
y(m)
=
A(e)s(m)
+
e(m).
(1)
~~~ ~
*This
wo k
hasbeen
suppo ed
in
pa
by
he
Spanish
MEC
G an
SE92
AC
0890196
and
EEC
Con ac HCM/CHRXCT-
930405
0-7803-3
192-3196 $5.0001996
IEEE
2559
=
[e,,
02,.
.
.
,
BN]~
(whe e
(.)T
deno es anspose) is he
N
dimensional ec o con aining he DOA’s o he desi ed
sou ces.
A(B)
is he
MzN
s ee ing ma ix whose columns,
a(B,),
p
E
{l,...
,
N},
consis o he s ee ing ec o s o he
sou ces.
s(m)
is an
N
dimensional signal ec o
a
ime
m.
e(m)
=
i(m)
+n(m)
deno es he sum o he M dimensional
in e e ence and noise ec o s a ime
m.
Wi h he in ini e ime a e aging ope a ion de ined as
(.)m
=
limIC.+m he spa ial co ela ion
ma ix o he a ay snapsho ec o is gi en by:
I(
R
E
(E
[~(m)~~(m)])~
=
A@) AH(8)
+
Ree
(2)
(E
[s(m)~”(m)])~,
Ree
z
(E
[e(~n)e~(m)])~
whe e
E[.]
and
(.)H
espec i ely deno e he expec a ion op-
e a o and he ec o /ma ix conjuga e anspose ope a-
ion. The “asymme ic” cyclic co ela ion ma ix o he
a ay snapsho ec o o cycle equency
a
and ime lag
T
is de ined as
[3]:
R*(T)
(E
[y(m
+
)y“(m)]
e-Jznam)m
(3)
y )
=
(E
[s(m
+
)sH(m)]
e-J2? am)m
(4)
whe e
I’“( )
is called he “cyclic in e signal cohe ence ma-
ix,” and i is assumed ha he
SOI’s
a e cyclos a iona y
a
CY
and ha he noise and in e e ence a e no .
In he spa ially nons a iona y case, i is assumed ha
he DOA ajec o y as a unc ion o ime o he p h sou ce
is desc ibed by he ollowing cons an eloci y model wi h
upda e in e al
T,:
xp(k+I)
=
[BP(k
+
I),&(k
+
I)]”
=
Fx,(k)+w,(k).
(5)
whe e
i,(.)
deno es angula eloci y. The angle upda e ime
index
k
co esponds o ime
=
kT.,,
and he s a e d i ing
noise
wP(k)
is assumed s a iona y ze o mean Gaussian o
co a iance
Qp.
I DOA es ima es a e equi ed e e y
T,
seconds, he es-
ima ed (cyclic
o
o dina y) co ela ion ma ix mus also
be upda ed e e y
T,
seconds. P o ided he DOA’s change
slowly wi h ime, i is easonable o assume ha hey e-
main app oxima ely cons an o e each angle upda e in e -
al,
T,
[l]:
B(k)
s~
l( ),
E
[kT,,
(k
+
l)Ts).
Thus, i
B
snapsho s a e a ailable e e y
T,
seconds, he co ela ion
ma ix o e he k’ h in e al can be es ima ed as:
g(k)
=
(y(kB
+
m)yH(kB
+
m))B
(6)
=
A(B)V-)AH@)
whe e he ini e ime a e aging ope a ion is de ined
as:
co esponds o ime
=
(kB
+
m)T,/B.
In addi ion, an
es ima e o he cyclic co ela ion ma ix
o
he a ay da a
o lag
T
o e he ime in e al
E
[ICT,,
(IC
+
l)Ts)
can be
o med as:
Ra(q
IC)
=
(y( ~
+
m
+
T)yH( ~
+
m)e--32? am)B(7)
B-1
(.)
- -
I
E,=,(.),
and he snapsho ime index
B
+
m
3.
CML
AND
CYCLOSTATIONARITY
This sec ion explains how cyclos a iona y can be used o
imp o e he CML es ima es in he p esence o in e e ence.
To simpli y no a ion, he dependence on he angle ec o
e
and he angle upda e ime index
k
will be d opped whene e
possible. The CML es ima o , which models he sou ces as
de e minis ic unknowns, is gi en by
[4]:
whe e (.) deno es he ma ix ace ope a ion, and
PA
and
A#
a e, espec i ely, he p ojec o and he pseudoin e se
o he s ee ing ma ix.
The CML unc ion can be used in he con ex
o
DOA
acking ia a i s o de Taylo se ies expansion o i s
de i a i e abou
e,
a
DOA
angle ec o known o be close
o he ML
DOA
es ima e
h h
e,,
M
-
[L”(
E)]
-I
L’(
$)
,
8
F5
BML.
(9)
whe e he
Nz1
g adien and
NxN
(app oxima e) Hessian
o he likelihood unc ion a e gi en, espec i ely as
[5]:
L’(8)
=
2
Re {diag (A’GPiD)}
(10)
L”(E)
=
-2
Re
{
(DHPiD)
0
(A#6A#H)T}(11)
a
a
Pi
=
(I
-
PA),
D
=
[
a l,a(&),
. . . ,
-a(Oi )]
a0,
wi h Re{.}, diag(.), and
(.)
0
(.)
deno ing espec i ely “ eal
pa o ,” “column ec o o med by diagonal elemen s o ,”
and he Hadama d (i.e., elemen -by-elemen ) p oduc op-
e a ion.
3.1.
E ec
o
SNOI’s on CML Es ima es
To unde s and how he p esence o SNOI’s a ec
lem begin by ew i ing
(8)
as
ollows:
L(B)
=
(PAG)
=
(AFA~)
h
=
A#EA#~.
The likelihood unc ion is e-exp essed in e ms
oLA,
he
angle dependen s ee ing ma ix, and he ma ix
I’
which
is an angle and da a dependen es ima e
o
he in e sig-
nal cohe ence m2 ix
I’.
When he e is no addi i e noise
o in e e ence,
I’
is asymp o ically unbiased when
(13)
is
IAs
seen in Sec ion
4.
a
can be ob ained eadily om he
DOA
es ima e
o
he
p e ious
i e a ion
o
some
unc ion he eo .
e alua ed a he ue
DOA’s.
No e ha he p esencz o ad-
di i e whi e noise p oduces an asymp o ic bias in
l?
(e en
when
(13)
is e alua ed a he ue
DOA’s).
Ne e heless,
he CML
DOA
es ima o emains asymp o ically unbiased
since (PAu21)
=
u2N
does no a y wi h angle.
Nex , when in e e ing SNOI’s a e p esen , good pe o -
mance can be expec ed only i
all
o he ollowing condi-
ions a e sa is ied: (i) The o al numbe o signals is less
han he numbe o senso s.
(ii)
The
DOA’s
o all he sig-
nals (SNOI’s included) a e es ima ed- esul ing in
a
co e-
sponding inc ease o he compu a ional complexi y o he
CML es ima o . (iii) Pos -p ocessing is pe o med o de-
e mine which
DOA
es ima es co espond o he SOI’s. I
he SNOI’s a e no aken in o accoun , he pe o mance o
he es ima o can de e io a e d ama ically esul ing highly
biased
DOA
es ima es. In e ms o
(12)
and
(13),
when
in e e ing SNOI’s a e p esen ,
^
is biased by he addi i e
noise
as
well
as
he p esence o he in e e e s. I is his la e
sou ce
o
bias which, in u n, esul s in an asymp o ically
biased es ima o o
e.
Thus, he in luence o he SNOI’s
should be able o be educed i hei in luence on
;
can be
educed. The goal is o modi y he CML es ima o using
he cyclos a iona y p ope ies o he
SOI’s
so
as
o yield
signal selec i e
SO1
DOA
es ima es while a oiding he need
o sa is y he condi ions lis ed abo e.
3.2.
An Imp o ed In e signal Cohe ence Ma ix
Es ima e based
on
he Cyclic Co ela ion Ma-
ix
I will now be explained ha o a cycle equency
u
o he
SOI’s bu no o he SNOI’s, he des uc i e in luence o he
SNOI’s on
;
can be educed by exploi ing in o ma ion om
he es ima ed cyclic co ela ion ma ix. Begin by deno ing
he SO1 signals a hei espec i e ansmi e s
as
sp(m)
=
s
(m
-
P)
whe e he ime shi s,
{ p} =l,
a e due o he
combined e ec o ansmission delay and di e ences in he
phases o he ansmi e and ecei e clocks. The cyclic
c oss co ela ion be ween sou ces
p
and
p
a hei espec i e
ansmi e s
is
w i en as:
-P
I a en ion is es ic ed o he cases o unco ela ed sou ces
and/o equency non-selec i e mul ipa h, hen
I’
can be
app oxima ed
as
[6]:
M
--- a( )TH
1
M
-TH a( )
1
(15)
Pa
(7)
Pa
(TI
whe e
pa(.)
=
ya
(T)/Y
and
-PP
-PPI
e--g2? a l
T
=
[
...
O
]
.
(16)
e-3Z a ~
0
Since,
as
desc ibed ea lie , he e ec o addi i e noise and
in e e ence is nulled ou in he cyclic co ela ion ma ix
(and, by implica ion, in he cyclic in e signal cohe ence ma-
ix),
(15)
p o ides he basis o ob aining an es ima e o
I’
which is less sensi i e o (i.e., less biased by) he SNOI’s. In
p ac ice, howe e , any es ima e o
I’
will equi e es ima es
o bo h
T
and
P(T).
Fo he case o he SOI’s being digi al communica ions
signals o he same, linea modula ion ype,
i
is u he
25
60
shown in
[SI
ha a mul iple lag es ima e o he in e signal
cohe ence ma ix can be implemen ed e icien ly as:
2
Pa
=A#(g)6."A#H(g)(17)
h
Ra=(x(m)xH(m)e-'2nnm)B, x(m)
=
y(m)
*g(m)(18)
whe e
i(m)
is
a
sampled e sion o he ansmission pulse,
and
(.)
*
(.)
deno es con olu ion. is a good ini ial es i-
ma e o
gML
(which will be seen o exis he case o he
DOA acking p oblem).
R"
is a ype o mul iple lag es i-
ma e o he cyclic co ela ion ma ix o med by empo ally
p e- il e edsnapsho da a. Thus, he mul iple lag cyclic co -
ela ion ma ix es ima e o
(18)
is o med a e each senso
ou pu has unde gone a simple ime-domain ma ched il e -
ing
ype ope a ion.
Las ly, pa icula ly simple es ima es o he ime shi ma-
ix o (16) can be ob ained by gene alizing he echnique
p esen ed in
[7]
o he mul isenso , mul isou ce case by ex-
ploi ing he ac ha he diagonal elemen s o
FnTH
o
(17)
should be eal. Speci ically, he diagonal elemen s o
he ime shi ma ix can be es ima ed as
[6]:
h
h
[*Ipp=
['~]pp/~["]pp~,
PE
{1,2,.*.,W. (19)
3.3.
A
Signal Selec i e
CML
Func ion
An ob ious ob ious signal selec i e modi ica ion
o
(8)
would be o use
F,
o
(17)
in
(8)
ins ead o
,
he in e -
e ence sensi i e es ima e o
(13)
no mally used. Inse ing
(17)
in o
(12)
yields:
h
Di ec maximiza ion o his cos unc ion would yield in e -
e ence esis an DOA es ima es. Un o una ely, ob aining
exp essions o he g adien and Hessian o
(20)
appea s
-
o be di icul . Howe e , a good al e na i e is o use
I?,
di ec ly in he exp essions o he g adien and Hessian o
he o iginal likelihood unc ion o
(8)
(which assumes ab-
sence o in e e e s). In pa icula , he expec ed alue o
he g adien may be w i en as:
E
[L'(g)]
=
2
Re{diag (A#(g)E[6.]PA(g)D(g))))
=
2
Re{ diag
(A"
(E)
[A(e)I'AH(&)
+
Re,]
WE)
D(u>}
M
2
Re{diag(I'AH(e)PA(g)
.(E))}
(21)
+2
Re(diag(A'(8) R,;PA(E)D(g))}
whe e he ac s ha i
E
M
e
hen
A#( i)
A@)
z
I
and
ha he addi i e noise is spa ially whi e ha e been used.
The i s and second e ms in
(21)
speci y he espec i e
con ibu ions due o he
SOI's
and he
SNOI's.
App oxi-
ma ing he desi ed i s e m by using
(17),
he es ima e o
he cyclic in e signal cohe ence ma ix p esen ed ea lie in
his sec ion, yields:
L'(
E)
M
2
Re{ diag
(F(
E)
AH(&)Pi(
E)
D(
g)
)
)22)
M
Re{?diag(A#(g)
&aHPi(g)
D(&))
+
?Hdiag(A#(&)6.aPi(@D($))}
whe e he app oxima ions
A#($
A(&)
M
I
and
A(&)A#(g)
M
PA@)
bo h o
M
&
in addi ion o he
second app oxima ion in (15) ha e been used.
Simila ly, he Hessian can be app oxima ed
as:
L"
(
8)
M
-2
Re
{
(
DH
(
2)
Pi
(
8)
D
(
g)
)
0
FT
(
E)
}
.
The use o
(22)
and
(23)
in
(9)
will be seen yield in e e ence
ole an app oxima e
CML
DOA es ima es.
4.
TRACKING ALGORITHM
The de ails o he Kalman il e based acking me hod a e
now desc ibed. Recall he cons an eloci y s a e upda e
equa ion o Eq. 5 used o model he DOA ajec o y. The
co esponding measu emen equa ion is gi en
as:
(23)
~p(k)
=
hTXp(k)
-I-
~p(k)
(24)
whe e
h
=
[l
0IT,
and he measu emen noise p(k) is ze o
mean Gaussian o a iance
p(k).
zp(k), hTxp(k), and
+(k)
co espond espec i ely o
B^ML,
(k),
OP(k),
and he es ima-
ion
e o associa ed wi h he es ima e
(9)
o he DOA o
he p h sou ce a ime
k.
I he measu emen noises asso-
cia ed wi h di e en sou ces a e assumed unco ela ed, he
Kalman il e can be decomposed in o
a
bank o compu-
a ionally e icien scala Kalman il e s
[I].
The speci ic
equa ions a e:
kp(klk
-
1)
=
Fkp(k
-
1(k
-
l),
(25-a)
kp(klk)
=
k,(klk-
1)
(25-b)
+
kp(k)
(zp(k)
-
hTj ,(klk
-
1))
whe e
kp(klk
-
l),
kp(klk),
and
kpl(k)
deno e, espec i ely,
he one-s ep p edic ed s a e ec o , he il e ed s a e ec o
es ima e, and he Kalman gain
all
o he p h sou ce a ime
k.
The inal angle es ima e is gi en by he i s elemen o
A i e a ion
k,
he es ima o
o
(9)
is e alua ed
a
z(klk-
l),
he p edic ed DOA sou ce ec o such ha (25-b), can
be ew i en as:
jip(kIk)
=
gp(klk
-
l)+kp(k)
[d~l,,(k)
-
dp(klk
-
I)]
(26)
%
(k
I
=
qklk
-
1)
-
kp(k)
[
(L"(i(k1k
-
1)))-'
L'(i(klk
-
I))]
P
whe e
[.Ip
deno es he p' h elemenl. o a ec o . No e ha
calcula ion o
EM,
(k)
comp ises he main compu a ional
bu den o he algo i hm (e.g., a
QR
decomposi ion o he
h
2561
A
MxN
ma ix
A(
e(klk
-
1))
in o de o compu e i s pseu-
doin e se and calcula ion o he in e se o he (symme ic)
The Kalman gain is upda ed by he ollowing se o s an-
NXN
ma ix,
L”(~?((LI~
-
1))).
da d equa ions:
C,(klk)
=
Cp(klk
-
1)
-
kp(k)hTCp(klk
-
1)
qk
+
ilk)
=
FC,(~I~)F~
+
Q~.
I has been ound ha p ope con ol o
p(k)
can be
achie ed by
[6]:
whe e
c
is
a
cons an di ec ly ela ed o he maximum an-
icipa ed o al SNOI powe .
5.
RESULTS
Compu e simula ion esul s now illus a e he imp o ed
pe o mance o he new echnique. Conside
N
=
3
mo ing
SOI’s
impinging on a uni o m linea a ay o
M
=
8
senso s
wi h hal -wa eleng h in e -elemen spacing. They ha e ini-
ial DOA’s o
&(O)
=
-70°,
&(O)
=
20°,
and
?,(O)
=
70’
and a e all o uni powe . Sou ce h ee is
a
mul ipa h e-
lec ion o sou ce wo. The
SOPS
consis o ou sample pe
symbol QPSK modula ed digi al communica ions signals
wi h uni oll-o squa e- oo Nyquis pulses. Such signals
exhibi cyclos a iona i y
a
cycle equency
cy
=
0.25.
The
senso noise powe is
o
n2
=
1
uni a iance, and
B
=
100
snapsho s pe acke i e a ion a e used.
Fig
1
shows he pe o mance o he o dina y CML acke
(using
(9),
(lo),
and
(11))
wi h he p esence o a uni powe
BPSK SNOI wi h symbol a e wo and
a
hal imes smalle
han ha o he SOI’s and ini ial DOA
d(0)
=
-30’.
CML
es ima o bias due o he in e e ence esul s in se e e pe -
o mance deg ada ion wi h
a
loss
o
da a associa ion and
he acking o he undesi ed SNOI ins ead o SO1 h ee
( om
k
=
200
onwa d). Nex , as shown in Fig.
2,
he
newly p oposed signal selec i e algo i hm yields g ea ly im-
p o ed acking pe o mance. Las ly,
Fig.
3
shows he pe -
o mance o he algo i hm when he e a e mul iple ( i e)
uni powe SNOI’s each
o
he same o m
as
ha in Fig.
1.
He e o dina y CML based acking would ail e en i he
in en ion we e o ack
all
he sou ces since he o al num-
be o sou ces equals he numbe o senso s. Al hough he
SINR o each sou ce is nea ly -8dB, he new echnique is
seen o pe o m well.
6.
CONCLUSION
A new DOA acking algo i hm o mul iple mo ing a -
ge s in he p esence o in e e e s has been p esen ed. The
echnique applies he ou pu o a modi ied signal selec i e
CML es ima o o
a
bank o pos -p ocessing scala Kalman
il e s. High quali y pe o mance is ob ained o c ossing
sou ces and/o ully cohe en sou ces, in low SINR scena -
ios
[I1
o da a’ eco ds o easonable leng h.
REFERENCES
C.R. Rao, C.R. Sas y, and
B.
Zhou. “T acking he
di ec ion o a i al o mul iple mo ing a ge s”.
IEEE
T ansac ions on Signal P ocessing,
ol.
42:
pp.
1133-
1144,
May
1994.
C.K. Swo d, M. Simaan, and
E.W.
Kamen. “Mul i-
ple a ge angle acking using senso ou pu s”.
IEEE
T ansac zons on Ae ospace and Elec onzc Sys ems,
ol.
26:
pp.
367-373,
Ma ch
1990.
S.V. Schell and
W.A.
Ga dne . High- esolu ion di ec ion
inding. In N.K. Bose and C.R. Rao, edi o s,
Handbook
o
S a as zcs
10:
Szgnal P ocessang and
i s
Applzca ons.
No h Holland,
1993.
P. S oica and
A.
Neho ai. “Pe o mance s udy o con-
di ional and uncondi ional di ec ion-o -a i al es ima-
ion”.
IEEE T ansac zons on Acous zcs, Speech and S g-
nal P ocesszng,
ol.
38:
pp.
1783-1795,
Oc obe
1990.
M. Vibe g,
B.
O e s en, and
T.
Kaila h. “De ec ion
and es ima ion in senso a ays using weigh ed subspace
i ing”.
IEEE T ansac zons on Szgnal P ocesszng,
ol.
39:
pp.
2436-2449,
No embe
1991.
J.Riba,
J.
Goldbe g, and
G.
Vkzquez. “Signal selec-
i e di ec ion-o -a i al acking o mul iple mo ing
a ge s”. submi ed
IEEE T ansac ions on Szgnal P o-
cessing,
June
1995.
M. Oe de and
H.
Mey . “Digi al il e and squa e im-
ing eco e y”.
IEEE T ansac aons on Communzca ions,
ol.
36:
pp.
605-612,
May
1988.
Figu e
1.
CML acke wi h SNOI (do ed ajec-
o y).
0
SO
100
150
ZM)
2.50
300
350
400
450
500
k
Figu e
2.
Signal selec i e
DOA
acke .
...
0
50
100
150
200
2.50
300
350
400
450
500
k
Figu e
3.
Signal selec i e acke wi h mul iple
SNOI’s.
2562