SyntacticModelingand SignalProcessingof MultifunctionRadars: AStochasticContext-Free GrammarApproach

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I NVI TE
D
P A P
E
R
Syntactic
Modeling
and
Signal
Processing
of
Multifunction
R
adars:
A
S
tochastic
C
ontext-Free
Grammar
Approach
Using
i
mproved
p
attern
recognition
t
echniques,
the
t
arget
o
f
a
radar
s
ystem
c
an
model
and
identify
that
system
and
estimate
how
m
uch
o
f
a
threat
it
poses.
By
Nikita
Visnevski,
Member
IEEE
,
Vikr
am
Kr
ishnamurthy,
Fellow
IEEE
,
A
l
e
x
W
a
n
g
,
and
Simon
H
aykin,
Fellow
IEEE
ABSTRACT
|
Mul
t
ifun
c
t
ion
r
a
dars
(
M
FRs
)
ar
e
s
ophis
t
i
c
a
t
e
d
s
e
nso
r
s
w
ith
c
omplex
dyn
a
mi
c
a
l
m
odes
t
hat
a
re
widel
y
used
i
n
s
u
rveil
l
an
c
e
an
d
t
rac
k
in
g
.
This
paper
d
e
m
ons
trates
th
a
t
sto-
c
h
astic
c
ontext-
free
gr
a
m
m
a
rs
(SCFGs)
a
r
e
ade
q
uat
e
model
s
f
o
r
c
aptu
r
ing
the
e
s
s
ential
features
of
the
M
FR
dynamics
.
Sp
e
c
i
f
ic
a
l
l
y
,
M
F
R
s
a
r
e
m
o
d
e
le
d
a
s
s
y
s
t
e
m
s
t
h
a
t
B
spea
k
[
a
la
n
g
u
a
g
e
t
h
a
t
i
s
c
h
a
r
a
c
t
e
r
i
z
e
d
b
y
a
n
S
C
F
G
.
Th
e
p
a
p
e
r
s
h
ow
s
th
a
t
su
c
h
a
g
rammar
i
s
m
o
dul
a
t
ed
by
a
M
arkov
c
hain
re-
pres
enting
radar

s
p
olic
y
o
f
o
peration.
The
p
aper
als
o
d
e
mon-
s
t
rat
e
s
h
o
w
some
well-kno
w
n
s
tatistical
s
i
gnal
pro
c
essing
te
c
hniq
u
e
s
c
an
be
applied
t
o
M
FR
s
i
gnal
pro
c
ess
ing
usin
g
th
e
s
e
s
tochs
t
ic
syntactic
m
odel
s
.
W
e
d
er
i
v
e
t
wo
statis
tical
e
s
timation
appr
o
a
c
h
es
f
o
r
M
FR
signal
proc
essin
g
V
a m
a
x
i
m
u
m
l
i
kel
i
hoo
d
s
equen
c
e
e
stimator
to
e
s
timate
radar’s
p
ol
i
c
ies
o
f
o
p
eratio
n
,
an
d
a
maximu
m
likeliho
o
d
parameter
e
s
t
imat
o
r
to
in
fer
t
he
radar
p
arameter
values.
T
w
o
lay
e
rs
of
signal
pro
c
ess
ing
are
i
n
t
r
o
duc
ed
in
t
h
i
s
paper
.
The
f
irst
layer
i
s
c
onc
erned
w
ith
t
h
e
estimation
of
MFR’s
p
oli
c
i
es
o
f
operat
i
o
n.
It
i
n
volves
si
g
n
al
proce
s
sin
g
in
the
C
F
G
dom
a
i
n
.
Th
e
s
ec
ond
layer
is
c
once
r
ned
w
ith
identi
f
ication
of
tasks
t
he
radar
i
s
e
ngaged
i
n
.
It
in
v
olves
s
i
gnal
proces
sing
i
n
the
f
in
i
t
e-s
t
ate
dom
a
i
n
.
Both
o
f
these
s
ign
a
l
p
r
o
ces
s
ing
techn
i
ques
a
re
im
p
o
r
t
ant
e
leme
n
t
s
o
f
a
bi
g
ger
radar
s
ign
a
l
p
ro
c
e
ss
ing
problem
t
hat
i
s
o
f
t
en
enc
oun
-
tered
i
n
e
lectro
n
i
c
w
arfare
appl
i
c
ations
V
the
p
r
o
blem
of
the
estimation
of
t
h
e
l
e
v
el
of
threat
that
a
r
adar
po
s
e
s
t
o
e
ac
h
indiv
i
dual
t
arge
t
a
t
a
ny
poin
t
i
n
t
ime.
KEYW
ORDS
|
Electronic
w
a
rfare;
G
a
lton–Watson
branching
pr
o
c
es
s;
in
s
ide-
outsi
d
e
algo
r
i
thm;
maximum
likelih
o
o
d
e
sti
-
mat
i
o
n
;
m
u
lt
if
u
n
c
t
io
n
r
ad
ar
(
M
F
R
)
;
s
t
o
c
has
t
i
c
c
o
nt
ex
t-
f
r
e
e
g
r
ammars
(S
CFGs);
synt
actic
m
odel
i
ng;
s
yntact
ic
patt
ern
recog
n
ition
I.
INT
R
ODUC
TIO
N
Statistical
pattern
recognition
h
as
been
a
m
ajor
tool
used
in
building
electronic
w
a
rfare
(
EW)
s
ystems
to
analy
z
e
radar
s
ignals.
C
o
n
ventional
r
ad
a
r
s
h
ave
b
een
historically
characterized
by
fixed
p
arameters
s
uch
a
s
r
adio
frequency
,
pulse-width,
and
p
eak
a
mplitud
e
[1],
[2
]
.
For
t
h
e
se
radar
characterizat
i
o
ns,
p
arametric
m
od
e
l
s
a
re
sufficient
f
or
solving
s
ignal
p
rocessing
problems
such
as
emitter
i
den-
tification
a
nd
threat
evaluation.
R
eferenc
e
s
[
3]
and
[
4
]
discuss
t
emplate
m
atching
o
f
t
h
e
intercepted
r
adar
signal
agai
nst
a
n
E
W
l
ibrary
for
b
oth
t
h
e
emit
ter
t
yp
e
a
nd
emitt
e
r
m
o
d
e
identificatio
n.
Histogram
t
echniques
a
re
d
e
scribed
in
[
5
]
t
o
s
tu
dy
t
h
e
t
emporal
p
e
r
iodi
citi
es
in
rad
a
r
s
ig
nals
such
as
pu
lse
r
ep
e
t
i
t
ion
i
nte
r
vals
.
Manuscript
r
eceived
October
1,
2006;
revised
J
anuary
24,
2007.
N.
Visnevski
is
with
the
General
Electric
Company,
Global
Research
Center,
N
iskayuna,
NY
12309
USA
(e-mail:
nikita.visnevski@research.ge.com).
V.
Krishnamurthy
and
A.
Wang
are
w
ith
the
Department
of
Electrical
and
Computer
Engineering,
University
of
British
Columbia,
Vancouver,
B
C
V6T1Z4,
Canada
(e-mail:
vikramk@ece.ubc.ca;
alexw@ece.ubc.ca).
S.
Haykin
is
with
the
Adaptive
Systems
Laboratory,
Department
of
Electrical
and
Computer
Engineering,
McMaster
University,
Hamilton,
ON
L8S
4K1,
Canada
(e-mail:
haykin@mcmaster.ca).
D
i
g
i
t
a
l
O
b
j
e
c
t
I
d
e
n
t
if
ie
r:
10
.1
10
9/J
PR
OC
.2
007
.8
93
252
1000
Proc
ee
dings
o
f
t
he
IE
EE
|
V
o
l
.9
5
,
N
o
.5
,M
a
y
2
0
0
7
00
18-
9
21
9/$2
5.
00
!
2007
IE
EE
With
t
h
e
a
dvent
o
f
m
odern
r
adars,
especially
multi-
function
radars
(MFRs),
statistical
pattern
recognition
app
r
oaches
described
a
bov
e
became
i
nadequate.
MFRs
are
radio-frequenc
y
s
ensors
that
are
w
idely
u
sed
i
n
m
odern
surveillance
and
t
racking
s
y
s
tems,
a
nd
they
have
t
h
e
capabili
ty
to
perform
a
multitude
o
f
d
ifferent
tasks
sim
u
lta
n
eo
usly
.
T
he
list
of
these
t
asks
often
i
ncludes
su
ch
ac
tiv
i
tie
s
as
se
ar
ch,
a
cq
uis
i
tion,
m
ultip
l
e
t
a
r
ge
t
tracki
n
g
,
and
w
eapon
g
uid
a
nce
[
6].
M
FRs
u
se
el
ectronic
beam
-steerin
g
a
nten
nas
t
o
p
erform
multip
l
e
tasks
s
imul-
taneously
b
y
m
ultiplexing
t
hem
i
n
t
ime
u
sing
short
t
ime
slices
[7].
At
t
h
e
s
ame
t
ime
t
hey
h
ave
t
o
m
ainta
i
n
l
ow
probability
o
f
b
eing
detected
and
j
ammed.
Indeed,
M
FRs
are
a
n
e
xce
l
l
e
nt
ex
ampl
e
o
f
h
i
g
h
l
y
c
ompl
ex
man
-
made
large-scale
d
ynam
ical
system
s.
MFRs’
ability
t
o
a
dap
t
ively
and
a
c
t
ively
s
wit
c
h
m
odes
and
c
hange
s
ystem
p
arameters
g
r
eatl
y
l
i
m
its
t
h
e
app
l
i
cabil
ity
o
f
t
h
e
p
a
rame
tric
statisti
cal
pat
t
ern
r
ecognition
app
r
oaches.
T
he
dimensionality
of
t
h
e
operational
s
tate
sp
a
c
e
f
o
r
such
radars
is
too
l
arge
for
t
h
e
stat
i
s
tical
a
p
proach
to
be
viable.
T
h
i
s
pa
p
e
r
p
r
o
po
s
e
s
a
d
i
f
f
e
r
e
n
t
a
p
pr
o
a
c
h
t
o
r
a
da
r
modeling
and
r
adar
signal
processing
V
one
b
ased
on
syntactic
patt
e
r
n
r
ecognit
i
on.
T
he
origins
o
f
s
yntactic
modeling
can
be
traced
t
o
t
he
classic
w
orks
of
Noam
Chomsky
o
n
f
ormal
l
anguages
and
t
ransformational
g
ram-
mars
[8]–[11].
T
h
e
cen
t
r
a
l
e
lements
o
f
t
his
w
ork
a
re
t
h
e
c
o
n
c
e
p
ts
o
f
a
fo
r
m
al
lan
g
uage
and
i
ts
g
r
a
mma
r
.
L
anguages
are
t
yp
i
cal
ly
infinit
e
se
ts
of
st
rings
d
rawn
from
a
f
i
n
ite
alphab
e
t
of
symbols.
Grammars,
on
the
o
ther
hand
,
a
re
vi
ew
ed
as
fi
nite
-d
i
m
ensi
onal
mod
e
ls
of
langu
a
ges
that
completely
characterize
t
hem.
Ma
n
y
different
kinds
o
f
g
ramm
ars
a
nd
langua
g
e
s
h
ave
been
identified
and
i
nvestigated
f
or
p
r
actical
applications.
Among
t
hem,
the
finit
e
-state
grammars
(FSGs)
and
t
h
e
conte
x
t
-
free
gr
am
mars
(CFGs),
as
well
as
their
s
tochastic
counterparts,
a
re
currently
the
m
ost
w
idely
u
sed
c
lasses
o
f
gram
mars.
S
t
ochastic
finit
e
-state
g
r
ammars
(SFS
G
s),
also
known
a
s
h
idden
M
arkov
models,
achiev
e
d
a
great
s
uccess
in
the
s
peech
c
ommunit
y
[12],
[
13].
They
were
used
in
modern
t
r
acking
s
y
s
tems
[14
]
and
i
n
m
achine
v
ision
[
15].
On
the
o
ther
hand,
stochastic
context-fr
ee
gr
amma
rs
(SC
F
Gs)
a
re
studied
i
n
[
16]
f
or
gesture
r
ecognition
and
the
i
mple
me
ntation
o
f
a
n
o
nl
in
e
p
arki
ng
l
o
t
m
on
itori
n
g
task.
I
n
[
17
]
a
nd
[18]
they
were
used
in
modeling
t
h
e
dy
n
a
mics
of
a
b
ursty
w
ireless
c
ommunicat
i
o
ns
channel.
References
[19]
and
[
20]
d
e
s
cribe
s
yntactic
modeling
app
l
ied
t
o
b
ioinforma
t
ics,
and
[
21]
a
nd
[22]
apply
t
hese
models
to
the
s
tud
y
of
biological
sequence
analysis
and
RNA.
Finally,
app
l
ication
o
f
s
y
n
tact
i
c
modeling
to
pattern
recognition
i
s
c
overed
in
depth
i
n
[
23].
In
this
paper,
we
const
r
uct
a
M
a
rkov
-
m
odu
l
ated
SCFG
to
m
o
del
a
n
a
nt
i
-
aircra
f
t
defense
M
FR
c
a
lled
Mercury.
T
h
e
more
tra
d
itional
a
pproaches
such
as
hidden
M
arkov
a
nd
stat
e
s
pace
models
are
s
uitable
f
or
target
modeling
[14],
[24]
but
n
ot
rad
a
r
m
odeling
b
ecause
MFRs
are
l
arge-scale
d
y
n
a
m
i
c
a
l
s
y
s
t
e
m
s a
n
d
t
h
e
i
r s
c
h
e
d
u
l
i
n
g i
n
v
o
l
v
e
s p
l
a
n
n
i
n
g
and
p
r
eempting
t
hat
m
akes
state
s
pace
ap
p
r
oach
difficult.
In
add
i
tion
to
rad
a
r
m
odeling,
we
also
consider
statistical
rad
a
r
s
ignal
p
rocessing.
T
he
proposed
linguistic
model
o
f
M
F
R
s
i
s
n
a
tu
r
a
l
l
y
d
i
v
i
d
e
d
i
n
to
tw
o
l
e
v
e
l
s
o
f
a
b
s
tr
a
c
t
i
o
n
:
t
a
sk
scheduling
lev
e
l
a
nd
radar
c
ontrol
level.
We
show
tha
t
t
h
e
M
FRs’
SCFG
representation
at
task
scheduling
level
i
s
self-embedding
and
cannot
b
e
r
educed
to
a
f
inite-state
form.
T
hus,
signal
processing
has
t
o
b
e
p
erformed
in
the
CF
G
d
o
m
a
i
n
.
Th
e
M
F
R
s

S
C
F
G
r
e
p
r
e
s
e
n
t
a
ti
o
n
a
t
t
h
e
r
a
d
a
r
control
l
evel,
o
n
t
h
e
other
h
and,
is
non-self-em
beddin
g
(NSE).
Thus,
a
finite-state
model
f
or
such
a
g
ram
m
ar
is
obtainable.
F
inally,
w
e
i
ntroduce
a
s
yst
e
matic
a
pp
r
o
ach
to
convert
t
he
radar
c
ontrol
lev
e
l
S
C
F
G
r
epresent
a
t
ion
t
o
i
ts
finite-st
a
te
count
e
rpart.
T
h
e
r
eason
f
or
such
a
t
wo-level
ap
p
r
oach
to
rad
a
r
modeling
is
t
h
at
of
computational
c
ost.
Although
SCFGs
p
r
ov
i
d
e
a
compac
t
r
ep
r
e
sentation
f
or
a
c
omp
l
ex
syst
e
m
such
as
MFR
s
,
t
hey
a
re
associat
e
d
with
computationally
intensiv
e
s
ignal
p
rocessing
algorithms
[2
1
]
,
[
23],
[25],
[26].
B
y
c
ont
r
ast,
finite-state
representations
a
re
not
nea
r
ly
as
compact
(the
number
of
st
a
t
es
in
the
f
inite-state
automaton
r
ep
r
e
senting
a
n
M
FR
can
be
v
e
ry
large
[
27]),
but
t
he
associated
signal
processing
techniques
are
w
ell-
known
a
nd
much
less
comp
u
t
ationally
d
e
m
and
i
ng
(see
d
i
scuss
i
on
on
com
p
l
e
xity
of
syntactic
p
a
rsing
algorithms
in
[21]).
It
is
therefore
a
dvantageous
t
o
m
od
e
l
MFRs
as
S
C
FGs,
and
p
erform
signal
p
r
o
c
e
ssing
o
n
their
f
inite-state
equivalent
s
a
s
m
uch
a
s
p
ossible.
T
r
aditionally,
MFRs’
signal
modes
w
ere
r
epresented
by
v
o
lumes
o
f
p
aramet
e
r
ized
d
a
ta
record
s
k
nown
as
Elec-
t
r
o
n
ic
Int
e
lligence
(ELINT)
[
1].
T
h
e
dat
a
reco
rd
s
a
re
annotated
b
y
l
ines
of
text
exp
l
a
ining
w
hen,
why,
and
h
ow
a
signal
may
c
hange
f
rom
o
ne
mode
to
ano
t
h
e
r.
This
m
a
kes
radar
m
ode
e
s
t
imation
a
nd
threat
evaluation
fa
ir
ly
di
f
f
i
c
u
l
t
.
I
n
[
2
8
]
,
SC
F
G
i
s
i
n
tr
o
d
u
c
e
d
a
s
a
f
r
a
m
e
w
o
r
k
t
o
model
M
FRs’
signal
and
i
t
i
s
s
hown
that
MFRs’
dy
n
a
mic
beh
a
v
i
or
can
be
ex
pli
c
itly
descri
bed
u
si
ng
a
f
in
ite
s
e
t
of
rules
c
orres
p
ond
i
ng
to
the
p
roduction
r
ules
of
the
S
C
F
G.
S
C
FG
has
s
e
v
eral
pote
ntial
a
dvantag
e
s.
1)
SCFG
is
a
c
omp
a
ct
formal
represent
a
tion
that
can
form
a
h
omog
eneou
s
basi
s
f
or
mode
li
ng
comp
l
e
x
system
dy
n
a
mics
[23]
,
[
25],
[26]
,
a
nd
wit
h
which
it
al
lows
mod
e
l
d
e
s
igne
rs
to
exp
ress
d
i
ffere
nt
aspects
o
f
M
FR
control
r
ules
in
a
s
ingle
f
rame-
work
[28],
a
nd
automates
t
he
threat
estimat
i
on
process
b
y
e
nc
o
d
i
n
g
h
uman
knowledge
i
n
t
he
grammar
[
29
],
[30
]
.
2
)
T
h
e
r
ecursive
embed
d
ing
s
tructure
o
f
MFRs’
con-
t
r
o
l
r
u
l
e
s i
s m
o
r
e n
a
t
u
r
a
l
l
y
m
o
d
e
l
e
d i
n
S
C
F
G
.
A
s
we
show
later,
the
M
arkov
i
an
t
y
pe
model
h
as
d
e
pendency
that
has
v
ariable
l
ength,
and
t
he
growing
s
tate
space
is
difficult
t
o
handle
since
t
he
maximum
r
ange
dependency
must
be
considered.
3)
SCFGs
a
re
m
o
re
efficient
i
n
m
o
d
e
l
ing
h
idd
e
n
branching
p
r
o
cesses
w
hen
c
ompared
t
o
s
t
o
chastic
regular
g
rammars
or
hidden
M
ark
o
v
m
odels
w
ith
Vi
sn
evski
et
al.
:
S
y
n
t
a
ct
i
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
Mu
lt
if
u
n
c
t
i
o
n
R
ad
ars
V
o
l
.
9
5
,
N
o
.
5
,
May
2007
|
Proceedi
n
gs
of
the
I
EEE
1001
the
s
ame
n
umber
o
f
p
arameters.
T
h
e
p
r
e
dictive
power
o
f
a
n
S
CFG
m
easured
i
n
t
erms
of
entropy
i
s
greater
t
han
t
hat
o
f
t
he
stochastic
regul
a
r
gra
m
mar
[
31].
S
C
FG
is
equivalent
to
a
m
ultitype
Galton–W
atson
b
ranch
i
ng
process
w
ith
f
ini
t
e
number
of
rewrite
r
ules,
a
nd
its
e
ntropy
calcula-
tion
is
d
i
scussed
i
n
[
32].
In
summary
,
the
m
ain
r
esults
of
the
p
aper
are
a
s
follows.
1)
A
careful
detailed
model
o
f
t
he
dy
n
a
mics
of
an
MFR
u
sing
form
al
language
production
rules.
By
mode
li
ng
th
e
M
FR
dynami
cs
usin
g
a
li
ng
ui
stic
formalism
s
uch
a
s
a
n
S
CFG,
an
MFR
can
be
viewed
as
a
d
iscrete
e
v
e
nt
system
that
B
speak
s
[
some
known,
or
partially
k
nown,
f
ormal
l
anguage
[33].
O
bservations
o
f
r
adar
emissions
can
be
viewed
as
st
rings
f
rom
t
h
i
s
l
anguage,
corrup
t
ed
by
the
n
oise
in
the
o
bserv
a
tion
environment.
2)
Formal
procedure
o
f
s
ynthesis
of
stochastic
automaton
m
od
e
l
s
f
rom
t
he
compact
syntactic
rules
o
f
C
FG.
U
n
d
er
the
c
ondition
that
the
C
FG
is
NS
E
,
the
C
FG
representation
can
be
converted
t
o
its
f
inite-state
c
ounte
r
part,
w
here
the
s
ignal
processi
ng
i
s
comp
u
t
ati
o
na
l
l
y
i
ne
xpensi
ve.
3
)
Novel
u
se
of
Mark
ov
mo
d
u
lated
S
CFGs
to
model
radar
e
m
i
ssions
generat
e
d
by
MFR.
The
c
om
p
l
ex
embedding
s
t
r
ucture
of
the
r
adar
signa
l
is
capt
u
re
d
b
y
t
h
e
li
ng
ui
stic
mode
l,
SCFG,
a
nd
th
e
MFR’s
i
nt
e
r
nal
s
tate,
i
ts
policies
of
op
e
r
ation,
is
modeled
b
y
a
Markov
chain.
This
mode
ling
app
r
oach
enables
t
he
combination
o
f
t
he
gram-
mar’s
s
yntactic
modeling
p
o
wer
w
ith
t
he
rich
theory
o
f
Markov
decision
process.
4)
S
t
atist
i
cal
signal
processing
of
SCFGs.
The
t
hreat
evaluation
problem
i
s
r
educed
to
a
s
tate
estima-
t
i
on
problem
o
f
H
MM.
T
h
e
m
a
ximum
l
ikelihood
esti
mator
i
s
d
eri
v
ed
based
o
n
a
hybri
d
of
th
e
forward-backward
and
t
he
inside-out
side
algo-
ri
th
m.
(Inside
-
ou
tsid
e
a
lgori
t
hm
is
an
e
x
te
nsi
o
n
o
f
H
M
M’s
f
orward-backward
algorithm
[
34].)
T
h
e
r
est
o
f
t
he
paper
i
s
o
rganized
as
follows.
S
e
ction
I
I
p
r
ovides
a
s
elf-contained
t
heoret
i
cal
background
of
syn-
t
a
ctic
modeling
methodology.
Sect
i
o
n
I
II
describes
t
he
MFR
i
n
d
e
t
ail
a
nd
its
r
ole
i
n
e
lectronic
w
arfare.
S
ection
IV
a
n
d
S
ection
V
p
resent
t
h
e
t
hreat
e
stimation
a
lgorithms
a
nd
a
d
etailed
d
escription
of
the
s
ynthesis
procedure
o
f
s
to-
ch
a
s
t
i
c
a
u
t
o
m
a
t
o
n
m
o
de
l
s
f
r
o
m
t
h
e
s
y
n
ta
c
t
i
c
r
u
l
e
s
o
f
C
F
G
.
Finally,
Section
V
I
c
onclud
e
s
t
h
e
p
aper.
II.
ELEMENT
S
O
F
SYNTA
CTIC
MODE
LING
T
h
is
section
p
r
e
sents
i
mp
o
r
tant
elements
from
the
t
heory
of
syntactic
modeling,
s
yntactic
p
a
ttern
recognition,
and
sy
n
t
a
x
analysis.
T
h
e
aim
i
s
t
o
p
rovid
e
the
r
eader
w
ith
a
b
r
ief
o
verview
o
f
t
he
concepts
tha
t
will
be
used
in
the
r
est
of
the
p
a
p
er,
a
nd
a
m
ore
c
omprehensive
discussion
can
be
found
i
n
[
23].
We
use
t
he
definitions
a
nd
notations
common
to
the
t
h
e
ory
o
f
f
ormal
l
anguages
and
c
omp
u
ta-
tional
li
ng
ui
stics
[
25]
,
[26
]
.
We
will
start
b
y
i
ntroducing
the
c
oncept
of
formal
languages
.
T
hese
languages
a
re
most
accurat
e
ly
defined
in
th
e
s
et-th
e
ore
t
ic
te
rms
a
s
c
ol
le
ctions
of
stri
ng
s
h
avi
n
g
a
c
ertai
n
pre
d
efi
n
e
d
structu
r
e.
In
prac
t
i
ce,
a
fi
nite
-
dimensional
m
od
e
l
of
the
l
anguage
i
s
r
equired,
and
i
t
sh
ould
he
lp
answeri
n
g
t
h
e
two
f
undame
ntal
questi
ons
o
f
the
t
h
e
ory
o
f
f
ormal
l
angu
ages.

Given
a
language,
h
ow
can
we
derive
any
s
tring
from
th
is
language?
(The
problem
o
f
s
tring
ge
ne
ra
t
i
on
.)

Given
a
certain
s
t
r
ing
a
nd
a
l
anguage,
how
can
we
tel
l
if
th
is
st
ring
is
p
a
rt
of
t
h
i
s
l
a
nguag
e
?
(
Th
e
problem
o
f
s
tring
pa
r
s
in
g
or
syntax
ana
l
ys
is
.)
The
f
ini
t
e-di
me
nsi
o
na
l
m
odel
s
o
f
l
angu
ages
t
h
at
hel
p
answering
t
hese
fundamental
q
uestions
are
called
gr
a
m
-
ma
rs
.
I
f
w
e
f
ocus
on
the
p
roblem
of
string
generation,
s
uch
grammars
are
t
ypically
called
ge
ne
ra
t
i
v
e
gr
a
m
m
a
r
s
.I
f
,
o
n
the
o
ther
hand,
w
e
a
re
interested
in
string
parsing,
i
t
i
s
customary
t
o
r
efer
to
the
l
anguage
g
r
ammars
a
s
t
r
a
n
s
f
or
-
m
a
tional
gr
am
mar
s
.
1)
Formal
Languages:
Let
A
be
an
arbitr
ary
set
of
symb
ols
that
we
will
call
an
al
phabet
.
I
n
genera
l,
an
al
phabet
does
no
t
have
to
be
finit
e,
but
fro
m
the
prac
tical
standpoin
t
w
e
will
assume
that
A
is
a
finite
set
of
sym
bols.
Using
s
ymbols
from
A
,
o
ne
can
construct
a
n
i
nfinit
e
number
of
strings
b
y
conc
atenati
n
g
them
together.
We
call
an
"
-strin
g
a
n
emp
t
y
s
t
r
in
g
V
a
s
tring
c
onsisting
o
f
n
o
symbols.
Let
u
s
d
e
n
ote
b
y
A
!
an
infinite
set
o
f
all
finit
e
s
t
r
i
n
g
s
f
or
m
e
d
b
y
c
on
cate
n
at
io
n
o
f
s
ym
b
o
ls
f
r
om
A
,a
n
d
l
e
t
us
denote
by
A
"
#A
!
[
"
.
F
or
ex
amp
l
e,
i
f
A#f
a
;
b
;
c
g
,
then
A
!
#f
a
;
b
;
c
;
aa
;
ab
;
ac
;
ba
;
bb
;
bc
;
ca
;
cb
;
cc
;
aa
a
;
..
.
g
(1)
A
"
#f
";
a
;
b
;
c
;
aa
;
ab
;
ac
;
ba
;
bb
;
bc
;
ca
;
cb
;
cc
;
aa
a
;
..
.
g
:
(2)
The
A
!
operation
i
s
called
positive
(
t
ransitiv
e)
closure
of
A
,
and
t
he
A
"
operation
i
s
c
a
l
led
K
l
eene
(
re
f
l
e
x
iv
e
a
nd
transit
i
v
e
)
c
losure
.
Definition
2.1:
Th
e
l
a
ng
u
a
g
e
L
d
e
f
i
n
e
d
o
v
e
r
a
n
a
l
p
h
a
be
t
A
i
s
a
s
et
of
some
fi
nite
-l
engt
h
s
trings
formed
by
concate-
nat
i
ng
sy
m
b
ols
f
rom
A
.
Evid
e
n
tly
,
L
$ A
"
,a
n
d
i
n
p
a
r
t
i
c
u
l
a
r

,
A
,a
n
d
A
"
are
also
languages.
2)
Grammars:
Th
e
d
e
f
ini
t
ion
o
f
t
h
e
formal
l
a
nguag
e
(Def.
2.1)
i
s
e
x
t
remel
y
broad
a
nd
t
h
ere
f
ore,
has
v
e
r
y
limited
practical
a
pplication.
A
m
ore
u
se
ful
w
ay
of
defining
formal
languages
is
through
t
he
use
o
f
gramm
a
rs
[8]–[
11].
Vi
snev
ski
et
al.
:
S
y
n
ta
cti
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
M
u
l
tifu
ncti
on
Rad
a
rs
1002
Proc
ee
dings
o
f
t
he
IEEE
|
V
o
l
.9
5
,
N
o
.5
,M
a
y
2
0
0
7
Definition
2.2:
A
de
terministic
g
ramm
ar
G
i
s
a
f
o
u
r
-
t
u
pl
e
G
#%
A
;
E
;
!
;
S
0
&
(3)
where:
A
is
the
alphabet
(the
set
of
terminal
symbols
of
the
grammar);
E
i
s
th
e
s
e
t
o
f
n
o
n
t
e
r
mi
n
a
l
s
y
m
bo
l
s
o
f
t
h
e
grammar;
!
is
the
finite
set
of
grammatical
production
rules
(syntactic
rules);
S
0
is
the
starting
nonterminal.
In
gene
r
al,
!
is
a
p
art
i
ally
d
e
fined
f
unction
o
f
t
ype
!:
%A
[
E
&
"
!%
A
[
E
&
"
:
(4
)
However,
as
we
will
see
l
ater,
c
ertain
restrictions
a
p
plied
to
the
p
roduction
r
ules
!
allow
u
s
t
o
d
ef
ine
s
ome
v
ery
useful
typ
e
s
o
f
g
ramm
ars.
I
n
t
h
e r
e
s
t o
f t
h
i
s p
a
p
e
r
,u
n
l
e
s
s s
p
e
c
i
f
i
e
d
o
t
h
e
r
w
i
s
e
,
we
will
write
n
onterminal
symbols
a
s
cap
i
t
al
letters,
and
symbols
o
f
t
he
alphabet
using
l
ower
case
l
etters.
This
follows
t
he
d
e
fault
c
o
n
vent
i
o
n
o
f
t
he
theory
of
formal
lan
g
uage
s.
Def.
2.1
p
r
o
vides
a
set-theoret
i
c
d
efinition
o
f
a
formal
language.
N
ow,
u
sing
Def.
2.2
we
can
redefine
the
lan
g
uage
in
te
rms
o
f
i
ts
gramm
a
r
L
#
"
L
%
G
&
.
To
illustrat
e
the
u
se
of
grammars,
consid
e
r
a
s
im
p
l
e
lan
g
uage
L
#
L
%
G
&
whose
g
rammar
G
#%
A
;
E
;
!
;
S
0
&
is
defined
a
s
f
ollows:
A#
f
a
;
b
g
S
0
!
aS
1
j
b
E#
f
S
0
;
S
1
g
S
1
!
bS
0
j
a
:
(5
)
These
a
re
some
of
the
v
alid
strings
i
n
t
his
l
anguage,
and
examples
o
f
h
ow
they
can
b
e
d
eri
v
ed
by
repeated
app
l
ication
o
f
t
he
product
i
on
rules
o
f
(
5):
1)
S
0
)
b
;
2)
S
0
)
aS
1
)
aa
;
3)
S
0
)
aS
1
)
ab
S
0
)
ab
b
;
4)
S
0
)
aS
1
)
ab
S
0
)
ab
aS
1
)
ab
aa
;
5)
S
0
)
aS
1
)
ab
S
0
)
ab
aS
1
)
ab
ab
S
0
)
ab
ab
b
;
6)
S
0
)
aS
1
)
ab
S
0
)
ab
aS
1
)
aba
bS
0
)
..
.
)
ab
ab
ab
..
.
ab
b
;
7)
S
0
)
aS
1
)
ab
S
0
)
ab
aS
1
)
ab
ab
S
0
)
..
.
)
ab
ab
ab
..
.
ab
aa
.
This
language
contains
an
infinite
number
of
strings
that
can
be
of
arbit
r
ary
l
engt
h
.
The
s
trings
st
a
r
t
w
ith
e
it
h
e
r
a
or
b
.
I
f
a
string
st
a
r
ts
with
b
,t
h
e
n i
t o
n
l
y c
o
n
t
a
i
n
s o
n
e
symbol
.
S
trings
termi
n
ate
w
ith
e
ith
e
r
aa
or
bb
,
a
nd
consist
o
f
a
d
i
s
ti
n
c
t
r
e
p
e
a
t
i
n
g
p
a
tt
e
r
n
ab
.
T
h
is
simple
example
i
llustrat
e
s
the
p
ower
of
the
gramm
a
t
i
cal
representation
of
languages.
Very
simple
grammars
can
define
rather
sop
h
isticat
e
d
languages.
3)
Chomsky
Hierarchy
of
Grammars:
In
Def.
2.2,
the
p
r
o
d
u
c
tion
rules
o
f
t
he
grammar
a
re
given
i
n
a
very
general
form.
R
e
f
erence
[10]
used
t
h
e
p
roperti
e
s
o
f
t
he
product
i
on
rules
o
f
g
rammars
to
develop
a
very
useful
hierarchy
t
hat
i
s
known
i
n
t
h
e
literature
as
the
C
homsky
hierarchy
o
f
gramm
a
rs.

R
e
g
u
l
a
r G
r
a
m
m
a
r
s (
R
G
)
:
Only
production
rules
of
the
f
orm
S
!
aS
or
S
!
a
are
a
llowed
.
T
h
i
s
means
t
hat
t
he
l
e
ft-h
and
s
i
d
e
o
f
t
he
product
i
on
must
co
ntain
o
ne
nontermin
a
l
o
n
l
y,
and
t
he
right-
hand
side
could
b
e
e
it
h
e
r
o
ne
terminal
or
on
e
terminal
followed
by
one
n
onterminal.
T
h
e
gram-
mar
o
f
t
h
e
lan
g
uage
in
the
l
ast
e
x
a
mple
of
th
is
section
i
s
a
regular
g
rammar.
R
e
gular
g
rammars
are
s
ometimes
referred
t
o
a
s
finite-stat
e
g
r
ammars
.

CFGs:
Any
p
r
o
duction
r
ule
o
f
t
he
form
S
!
!
is
al
lowe
d.
Th
is
me
ans
t
h
a
t
t
h
e
le
ft-hand
s
ide
o
f
t
he
prod
u
c
tio
n
rule
m
u
st
contain
o
n
e
nonterminal
only
,
w
hereas
t
h
e
r
ight-hand
s
ide
can
be
any
string.

Context-Sensitive
Grammars
(CSG
):
Product
i
on
rules
o
f
t
he
form
"
1
S
"
2
!
"
1
!"
2
are
a
llowed.
He
r
e
"
1
;"
2
2 %
A
[
E
&
"
,a
n
d
!
6
#
"
.I
n o
t
h
e
r
wo
rd
s
,
the
a
llowed
t
r
ansformations
o
f
n
onterminal
S
are
d
ependent
on
its
c
ontext
"
1
and
"
2
.

Unrestricted
Grammars
(UG)
:
Any
p
roduct
i
o
n
rules
o
f
t
he
form
"
1
S
"
2
!
#
are
a
ll
owed.
H
ere
"
1
,
"
2
,
#
2 %
A
[
E
&
"
.
T
he
unrestricted
gra
m
mars
are
also
often
r
eferred
t
o
a
s
ty
pe-0
g
r
am
mars
due
t
o
Ch
o
m
s
k
y
[
1
0
]
.
C
h
omsky
a
lso
c
lassified
languages
i
n
t
e
r
ms
of
the
grammars
t
h
at
can
be
used
to
d
e
fine
t
h
em.
F
ig.
1
illustrates
t
h
is
hierarchy
o
f
l
a
n
guages.
E
ach
inner
c
ircle
o
f
t
his
d
i
agra
m
i
s
a
subset
of
the
o
uter
circle.
T
hus,
context-
sensitive
l
anguage
(
CS
L
)
is
a
s
pecial
(more
r
est
r
icted)
form
of
unrestricted
language
(UL),
c
o
n
text-free
l
anguage
(
C
F
L)
is
a
s
pecial
case
o
f
C
S
L
,
a
nd
regular
l
anguage
(RL)
i
s
a
Fig.
1.
T
h
e
C
ho
m
s
k
y
hi
e
r
a
r
c
h
y
o
f
f
o
r
ma
l
l
a
n
g
u
a
g
e
s
.
Vi
sn
evski
et
al.
:
S
y
n
t
a
ct
i
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
Mu
lt
if
u
n
c
t
i
o
n
R
ad
ars
V
o
l
.
9
5
,
N
o
.
5
,
May
2007
|
Pr
ocee
dings
o
f
t
he
IE
EE
1003
sp
e
c
ial
case
o
f
C
F
L
.
T
able
1
p
rovides
a
condensed
s
ummary
of
the
c
lasses
o
f
g
rammars,
their
p
r
o
d
u
ction
r
ule
s
tructures,
an
d
c
l
a
s
s
e
s
o
f
l
an
g
u
a
g
e
s
t
h
a
t
th
e
y
d
e
f
i
n
e
.
M
o
r
e
d
e
t
a
i
l
e
d
t
r
eatment
o
f
t
he
Chom
sky
h
ierarchy
is
given
b
y
[
21].
Sy
n
t
a
c
t
i
c
m
o
d
e
l
i
n
g
o
f
D
E
S
(
di
s
c
r
e
te
e
v
e
n
t
s
ys
t
e
m
)
d
e
ve
lope
d
i
n
t
h
i
s
p
ape
r
wil
l
make
ex
tensive
u
se
of
F
S
G
a
nd
CF
G
.
C
S
G
a
n
d
UG
w
i
l
l
n
o
t
b
e
u
s
e
d
i
n
o
u
r
m
o
d
e
l
i
n
g
a
p
proach.
4)
Relationship
Between
Regular
Languages
and
Finite-
State
Automata:
Definition
2.3:
A
finite
-state
automaton
(FSA
)
#
is
a
five-t
u
p
le
#
#%
Q
;
$
;$
;
q
0
;
F
&
(6)
wh
ere
:
Q
is
the
set
of
states
of
the
FSA;
$
is
the
set
of
input
symbols
of
the
FSA;
$
is
the
transition
function
of
the
FSA;
q
0
is
the
initial
state
of
the
FSA;
F
is
the
set
of
final
(accepting)
states
of
the
FSA
%
F
'
Q
&
.
FSAs
were
shown
t
o
b
e
e
quivalent
t
o
R
G
a
nd
RL
(see
[25],
[
26],
and
[
35]–[38]).
In
fact,
u
sing
Def.
2.2
and
Def.
2.3
w
e
can
observ
e
t
hat
i
f
Q
#E
,
$
#A
,a
n
d
q
0
#
S
0
,
we
can
relate
$
and
!
in
such
a
w
ay
that
L
%
#
&#
L
%
G
&
.
L
%
#
&
is
also
called
t
he
language
accept
e
d
by
the
F
S
A
#
.T
h
e
s
e
t o
f
final
(
accep
t
ing)
states
F
is
the
s
et
of
states
such
that
any
input
s
tri
n
g
f
rom
L
%
#
&
ca
u
s
es
#
to
tra
n
sition
int
o
one
o
f
t
h
ese
s
tates.
An
FSA
e
quivalent
o
f
t
he
grammar
(
5
)
is
s
h
o
w
n i
n F
i
g
.2
.
5)
Context-Free
Languages
and
CFGs:
The
n
ext,
less
restricted
member
o
f
the
C
h
o
msky
hierarchy
o
f
g
rammars
is
th
e
CFG
.
L
anguages
that
can
be
accep
t
ed
by
FS
A
s
are
limited
i
n
t
e
r
ms
of
st
rings
t
hat
t
hey
can
conta
i
n.
T
h
e
m
ost
fa
m
o
us
example
o
f
a
language
that
cannot
b
e
accepted
by
FSAs
is
the
l
anguage
o
f
palindr
o
m
e
s
.
1
It
wa
s
s
hown
to
be
a
CFL
[
26].
A
s
imple
l
anguage
o
f
p
alindromes
ca
n
,
for
exampl
e
,
b
e
d
e
f
ined
by
th
e
f
ol
lowi
ng
set
o
f
p
rodu
ction
rules:
P
!
bPb
j
aPa
j
b
j
a
j
"
(7)
and
a
n
e
xampl e
stri ng
from
thi s
l anguage
i s
ba
ba
ba
aa
ba
b
a
b
.
A
ccording
t
o
T
ab
l
e
1,
the
g
rammar
in
(7)
is
a
C
F
G
.
CFGs
are
o
ft
e
n
associated
wit
h
tree-like
g
rap
h
s
i
nstead
of
FSAs
since
t
he
d
e
pe
nd
e
n
cy
bet
w
ee
n
t
h
e
ele
m
ents
of
the
strings
o
f
t
he
CFL
a
re
nest
e
d
[2
5
]
,
[
26],
[35],
[
39],
[4
0
]
.
Due
t
o
t
his
f
act,
t
h
e
task
of
processing
t
h
e
s
trings
fro
m
a
C
F
L
i
s
a
m
o
r
e
c
o
m
p
u
t
a
t
i
o
n
a
l
l
y
c
o
m
pl
e
x
pr
o
c
e
d
u
r
e
t
h
a
n
that
of
an
RL.
O
n
t
he
other
h
and,
[41]
have
shown
t
hat
CFG
c
ould
be
more
compact
descriptions
of
the
R
L
t
han
RG.
I
t
i
s
o
ften
co
nvenient
to
d
e
scrib
e
complex
f
inite-stat
e
systems
i
n
t
he
context-free
form
,
b
ut
it
is
less
com
p
u
t
a-
tionally
int
e
nsive
t
o
p
erform
analy
s
is
of
these
s
ystems
using
F
SA
.
T
his
f
act
is
at
the
c
enter
o
f
t
he
large
s
ca
l
e
DES
m
o
d
e
ling
methodology
t
h
a
t
w
e
a
re
going
t
o
d
e
v
elop
in
the
rest
o
f
this
sec
t
ion.
As
Fig.
1
c
learly
demonstrates,
R
L
a
re
a
p
rop
e
r
s
ubset
o
f
th
e
c
l
a
s
s
o
f
CF
L
.
H
o
w
e
ve
r
,
g
i
v
e
n
a
g
e
n
e
r
a
l
C
F
G
,
o
n
e
cannot
t
e
l
l
i
f
t
his
g
ram
m
ar
d
e
scribes
a
n
R
L
o
r
a
CFL
(
this
task
was
s
how
n
to
be
undecidable
[
40]).
W
e
w
ill
n
ow
look
a
t
t
h
e
p
r
o
pe
r
t
y
o
f
s
e
lf-e
mbe
dding
of
the
C
FG
and
s
ee
how
this
property
helps
i
n
d
eter
mining
the
c
la
ss
of
the
languages
described
b
y
s
uch
C
F
G
.
6)
Non-Self-Embedding
CFGs:
Definition
2.4:
A C
F
G
G
#%
A
;
E
;
!
;
S
0
&
is
self-emb
edd
i
ng
if
there
e
xists
a
nonterminal
s
y
m
bol
A
2 E
such
t
h
at
a
string
"
A
!
can
be
derived
f
rom
i
t
i
n
a
finit
e
num
b
er
of
der
i
va
tio
n
steps,
with
"
,
!
6
#
"
bei
n
g
a
ny
st
ring
o
f
terminal
and
n
onterminal
symbols.
For
e
xample,
t
he
nonterminal
s
ymbol
P
in
the
palindrom
e
gramm
a
r
(
7)
is
such
a
s
elf-em
beddin
g
nonter-
m
i
nal,
an
d
t
he
CFG
o
f
p
a
l
indromes
is
self-embedd
i
ng.
Table
1
Deterministic
Grammars,
Production
Rules,
and
Languages
Fi
g
.
2.
FS
A
e
qu
i
v
a
l
e
n
t
t
o
t
h
e
g
r
a
m
ma
r
e
x
a
mp
le
(
5
)
.
S
t
a
t
e
S
0
is
th
e
st
ar
ti
ng
s
t
a
t
e
,
an
d
T
is
an
ac
ce
pt
in
g
s
ta
te
,
a
s
i
n
d
i
c
at
ed
by
th
e
d
o
u
b
l
e
c
i
r
c
l
e.
1
A
palindrome
is
a
string
that
reads
the
same
way
both
left-to-right
and
right-to-left.
Vi
snev
ski
et
al.
:
S
y
n
ta
cti
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
M
u
l
tifu
ncti
on
Rad
a
rs
1004
Proc
ee
dings
o
f
t
he
IE
EE
|
V
o
l
.9
5
,
N
o
.5
,M
a
y
2
0
0
7
Definition
2.5:
A C
F
G
G
#%
A
;
E
;
!
;
S
0
&
is
no
n
-
self-
em
bed
d
ing
if
there
exists
n
o
n
onterminal
sy
m
b
ols
f
or
whic
h
t
h
e
cond
i
t
ion
o
f
t
he
Def.
2.4
can
be
satisfied.
[11]
has
d
emonstrated
t
hat
i
f
a
CFG
i
s
N
S
E
,
i
t
g
enerates
a
f
inite-state
l
anguage
(
FSL).
I
n
S
ection
V-A,
we
will
describe
an
algorithm
t
o
v
erify
t
h
e
NS
E
p
roperty
o
f
C
F
G
s,
and
s
how
h
ow
t
o
obtain
FSAs
for
t
hese
gram
mars.
7)
Stochastic
Languages
and
Stochastic
Grammars:
A
number
of
pract
i
cal
ap
p
l
ications
contain
c
ertain
amount
s
of
uncert
a
i
nty
t
hat
a
re
often
r
epresented
by
probabilistic
distributions.
These
f
actors
r
equire
extension
o
f
t
he
concept
s
descr
i
bed
a
bove
into
the
d
omain
o
f
s
tochastic
lan
g
uage
s.
Definition
2.6:
A
weigh
t
ed
gr
am
mar
G
w
is
a
f
iv
e-tup
l
e
G
w
#%
A
;
E
;
!
;
P
w
;
S
0
&
(8)
where:
A
is
the
alphabet
(the
set
of
terminal
symbols
of
the
grammar);
E
i
s
th
e
s
e
t
o
f
n
o
n
t
e
r
mi
n
a
l
s
y
m
bo
l
s
o
f
t
h
e
grammar;
!
is
the
finite
set
of
grammatical
production
rules
(syntactic
rules);
P
w
is
the
set
of
weighting
coefficients
defined
over
the
production
rules
!
;
S
0
is
the
staring
nonterminal.
Here
is
a
s
impl
e
e
xampl
e
of
a
w
e
i
gh
t
e
d
g
ramm
ar:
S
0
!
9
aS
1
S
0
!
1
b
S
1
!
1
bS
0
S
1
!
9
a
:
(9)
This
g
r
ammar
h
as
be
en
ob
tained
from
grammar
(
5
)
by
a
sso
ciating
w
ith
i
ts
produc
tions
t
he
set
o
f
w
eights
P
w
#f
%
9
;
1
&
;
%
1
;
9
&g
.
N
ot
e
t
hat
t
h
e
set
o
f
w
ei
ghts
P
w
does
n
o
t h
a
v
e t
o b
e n
o
r
m
a
l
i
z
e
d
.
Definition
2.7:
A
s
t
ocha
st
ic
g
r
a
mma
r
G
s
i
s
a
f
i
v
e
-
tu
p
l
e
G
s
#%
A
;
E
;
!
;
P
s
;
S
0
&
(10)
where
A
,
E
,
!
,a
n
d
S
0
are
t
he
same
as
in
Def.
2.6,
a
nd
P
s
is
th
e
s
et
of
p
r
obabil
ity
d
ist
r
ibu
t
ions
over
the
s
et
of
produ
c
-
tion
rules
!
.
Clea
rly,
stochastic
gramma
rs
a
r
e
s
imply
a
more
restrict
e
d
case
o
f
t
he
weighted
grammars.
Here
is
a
s
imple
example
o
f
a
stochast
i
c
gram
mar:
S
0
!
0
:
9
aS
1
S
0
!
0
:
1
b
S
1
!
0
:
1
bS
0
S
1
!
0
:
9
a
:
(11)
T
h
i
s
g
r
ammar
h
as
been
obtained
from
grammar
(
5)
b
y
a
p
p
l
yi
n
g
t
o
i
t
s
p
r
o
d
u
c
t
i
o
n
s
t
h
e
p
r
o
b
a
b
i
l
i
ty
d
i
s
t
r
i
b
u
ti
o
n
s
P
s
#f
%
0
:
9
;
0
:
1
&
;
%
0
:
1
;
0
:
9
&g
.
S
t
ochastic
and
w
eighted
g
ramm
ars
a
re
classified
and
analyzed
on
the
b
asis
of
their
u
nderlying
cha
r
ac
ter
i
s
t
ic
gr
amm
a
r
s
[23
]
,
[
42].
A
ch
ar
a
c
t
e
r
i
s
t
ic
gr
am
ma
r
G
c
is
ob
ta
ined
from
t
h
e
s
toc
h
astic
g
ra
mma
r
G
s
(weighted
gramm
a
r
G
w
)
b
y
r
emovi
n
g
t
he
probabil
ity
d
istri
b
uti
o
n
P
s
(set
o
f
w
eight
s
P
w
)
f
rom
t
he
grammar
d
efinition.
If
the
r
esult
i
ng
charac
t
e
ristic
grammar
i
s
a
n
F
SG,
t
he
st
o
c
hastic
grammar
i
s
called
s
tochastic
f
inite-state
g
ram-
m
a
r (
S
F
S
G
)
.
I
f
t
h
e c
h
a
r
a
c
t
e
r
i
s
t
i
c g
r
a
m
m
a
r i
s a C
F
G
,
t
h
e
stochastic
gramm
a
r
i
s
r
eferred
t
o
a
s
a
n
S
CFG.
For
example,
grammar
(
11)
is
an
SFS
G
,
a
nd
grammar
(
5)
is
its
c
haracteristic
g
rammar.
Characteristic
g
rammars
play
important
r
oles
in
d
e
riv
i
ng
sy
n
t
a
c
tic
m
odels
o
f
r
eal-life
systems.
The
t
yp
i
cal
procedure
i
s
illustrated
i
n
F
ig.
3
.
T
he
characteristic
grammar
i
s
a
brid
g
e
between
t
he
internal
d
e
terministic
rules
o
f
t
he
system,
a
nd
the
s
t
o
chastic
e
nvironment
in
which
t
his
s
ystem
i
s
o
p
e
rating
or
observed.
8)
Stochastic
Finite-State
Languages,
Markov
Chains,
and
Hidden
Markov
Models:
Just
as
FS
A
s
constitute
one
o
f
t
he
Fig.
3.
De
ri
va
ti
o
n
pr
oc
ed
ur
e
f
or
th
e
s
to
ch
as
ti
c
g
r
a
mm
ar
s.
Fi
r
s
t
,
a
d
e
t
e
r
m
i
n
i
s
t
i
c
g
r
a
m
m
a
r
f
or
th
e
s
ys
te
m
i
s
c
o
n
s
t
r
u
c
t
e
d
.
T
he
n,
a
f
t
e
r
c
on
si
de
ra
ti
on
s
o
f
p
o
s
s
i
bl
e
s
o
u
r
c
es
of
un
ce
rt
ai
nt
ie
s
,
th
e
d
e
t
e
r
m
i
n
i
st
i
c
g
r
a
m
m
a
r
i
s
m
od
if
i
e
d
i
n
t
o
a
c
h
a
r
a
c
t
e
r
i
st
ic
g
r
a
m
m
a
r
th
at
ac
c
o
m
m
o
d
a
t
e
s
fo
r
t
h
e
s
e
un
ce
rt
ai
nt
ie
s
.
Fi
na
ll
y,
a
p
r
o
b
a
b
i
l
i
ty
di
st
r
i
b
u
t
i
o
n
is
as
s
i
g
n
e
d
to
th
e
c
ha
ra
c
t
e
r
i
s
t
i
c
g
ra
m
m
a
r
,
yi
el
di
ng
a
s
to
ch
as
ti
c
g
r
a
m
m
ar
of
th
e
s
ys
te
m.
Vi
sn
evski
et
al.
:
S
y
n
t
a
ct
i
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
Mu
lt
if
u
n
c
t
i
o
n
R
ad
ars
V
o
l
.
9
5
,
N
o
.
5
,
May
2007
|
Proceedi
n
gs
of
the
I
EEE
1005
representation
fo
rms
o
f
F
SLs,
discrete-stat
e
discrete-tim
e
Markov
chains
are
n
aturally
consid
e
r
ed
the
e
quivalent
r
e
p
r
e
s
e
n
t
a
t
i
o
n
s o
f t
h
e
S
F
S
L
s [
3
3
]
.T
h
i
s r
e
p
r
e
s
e
n
t
a
t
i
o
n h
a
s
b
e
en
succe
ssful
ly
u
t
il
ize
d
in
bioi
nformati
cs
and
c
ompu
ta-
t
i
onal
genomics
[19],
[
2
1
]
a
s
w
ell
a
s
i
n
n
atural
language
a
n
d
s
peech
p
rocessing
[12].
A
d
iscrete-state
d
iscrete-t
i
me
Markov
chain
can
be
d
e
fined
a
s
a
stochast
i
c
timed
a
utomaton
[33]:
Definition
2.8:
A
discret
e
-time
Mar
k
ov
cha
i
n
de
f
i
n
e
d
o
v
e
r
a
d
isc
r
ete
s
tate
space
is
a
t
uple
#
#%
A
;%
&
(12)
wh
ere
:
A
is
the
N
(
N
state
transition
probability
matrix,
%
is
the
N
(
1
vector
of
initial
state
probability
distribution,
and
N
is
the
number
of
states
in
the
Markov
chain.
We
wi
ll
il
lustrate
the
r
e
l
atio
nshi
p
b
etwee
n
SFSGs
a
nd
Markov
chains
b
y
lo
oking
at
the
t
r
a
nsition
s
t
r
ucture
of
the
grammar
(
11).
W
e
can
construct
a
Markov
chain
t
h
a
t
w
ill
reflect
t
he
transitio
n
s
w
ithin
!
of
(11)
as
#
#
A
#
0 0
:
9 0
:
1
0
:
1 0 0
:
9
0 0
0
2
4
3
5
;%
#
1
0
0
2
4
3
5
0
@
1
A
(13)
wh
ere
A
and
%
are
d
efi
n
ed
wit
h
respec
t
t
o
t
he
state
ord
e
ring
f
S
0
;
S
1
;
T
g
a
s
s
h
o
w
n i
n F
i
g
.4
.
T
h
e
exa
m
p
l
e
abov
e
i
ll
ustrate
s
a
s
trong
p
arall
e
l
b
etwe
en
FSAs
in
the
case
o
f
d
eterministic
grammars,
and
M
arkov
c
h
ains
in
the
case
o
f
s
tochastic
o
nes.
However,
Markov
c
h
ains
defined
b
y
D
ef.
2
.
8
do
not
accommodate
for
t
he
al
p
h
a
b
e
t
A
of
t
h
e
g
rammar.
Therefore,
Markov
chains
can
only
cap
t
u
re
transition
dynamics
of
the
g
rammar,
but
d
o
not
a
ddress
g
eneration
a
nd
t
r
ansformation
aspec
t
s
o
f
t
he
SFS
G
s
d
iscussed
e
arlier.
H
idden
M
arkov
models
(HM
M
s)
addres
s
t
his
i
ssue.
HMMs
[12
]
,
[
13],
[19]
are
p
articularly
s
uit
a
ble
f
or
represen
ting
stochastic
languages
o
f
t
he
finite-state
discrete-ev
e
nt
sy
s
t
ems
o
bserved
i
n
n
oisy
env
i
ronm
ents.
They
separate
the
u
ncerta
i
n
ty
in
the
m
odel
attributed
to
the
o
bservation
process
f
rom
t
he
uncertainties
a
ssociated
with
the
s
ystem’s
f
unctionality.
Ge
nerall
y
s
peaking,
HMMs
are
M
arkov
chains
indirectly
observed
through
a
noisy
p
rocess
[1
3],
[
33],
[
43]–[4
5
]
.
Definition
2.9:
A
HMM
&
is
a
t
hre
e
-
t
u
p
le
&
#%
A
;
B
;%
&
(14)
where:
A
is
the
N
(
N
state
transition
probability
matrix
of
the
underlying
Markov
chain,
B
is
the
N
(
M
observation
probability
matrix
t
h
a
t
e
s
t
a
b
l
i
s
h
e
s
p
r
o
b
a
bi
l
i
ty
d
i
s
t
r
i
b
u
ti
o
n
s
o
f
observing
certain
discrete
symbols
associated
with
a
certain
state
of
the
chain,
%
is
the
N
(
1
vector
of
initial
state
probability
distribution
of
the
underlying
Markov
chain,
N
is
the
number
of
states
of
the
underlying
Markov
chain,
and
M
is
the
number
of
possible
discrete
observation
symbols.
To
illustrate
how
H
MMs
r
ela
t
e
t
o
S
FSGs,
w
e
w
ould
like
t
o
revisit
t
he
grammar
(
11).
T
he
Marko
v
chain
f
or
this
grammar
i
s
d
efined
by
(13).
N
ow
we
can
extend
this
chain
bringi
ng
in
th
e
a
l
p
habe
t
A
of
t
h
e
g
ramm
ar
(11)
through
the
s
tructure
of
the
o
bservation
p
r
obab
i
l
ity
m
at
rix
B
.
However,
t
h
is
ext
e
n
sion
requires
a
t
ransformation
o
f
the
s
tructure
of
the
M
arkov
chain
i
n
F
ig.
4
.
D
ef.
2.9
associates
the
o
bservation
p
r
obab
i
l
ity
m
at
rix
B
with
the
states
of
the
c
hain,
w
hereas
S
F
SGs
a
ssociate
generation
o
f
nonterminals
with
transitions
o
f
t
he
state
m
achine.
The
f
o
r
m
e
r
c
a
s
e i
s k
n
o
w
n
i
n
t
h
e l
i
t
e
r
a
t
u
r
e a
s t
h
e
Mo
o
r
e
m
achine
,
a
n
d t
h
e
l
a
t
t
e
r i
s r
e
f
e
r
r
e
d t
o
a
s t
h
e
M
e
aly
m
ac
hine
[2
6
]
.
Therefore,
to
accommodat
e
f
or
the
s
tructural
c
onstraint
s
of
the
H
MM,
t
h
e
Markov
chain
i
n
F
ig.
4
has
t
o
b
e
c
onv
e
rted
to
the
M
oore
machine
f
orm
a
s
d
escribed
in
detail
in
[26].
The
r
esulting
HMM
h
as
the
f
ollowing
st
ructure:
&
#
A
#
0 0
:
1 0
:
9 0
0
:
9 0
0
0
:
1
0 0
0 0
0 0
0 0
2
6
6
4
3
7
7
5
;
0
B
B
@
B
#
1 0
0 1
1 0
0 1
2
6
6
4
3
7
7
5
;%
#
0
:
9
0
0
0
:
1
2
6
6
6
4
3
7
7
7
5
1
C
C
C
A
(15)
Fig.
4
.
E
x
a
m
pl
e
o
f
a
Ma
r
k
o
v
c
h
a
i
n
f
o
r
th
e
S
F
S
G
(
11
).
No
te
th
at
Ma
rk
o
v
c
h
a
i
n
s
on
ly
c
a
p
t
u
r
e
t
h
e
tr
a
n
s
i
t
i
o
n
dy
na
mi
c
s
of
th
e
g
r
a
m
m
a
r
s
in
ce
th
e
t
e
r
m
i
n
a
l
s
y
m
bo
ls
of
th
e
g
ra
m
m
a
r
do
no
t
f
e
a
tu
re
i
n
th
e
M
a
r
ko
v
c
h
a
i
n
s
t
r
u
c
t
u
r
e
.
Vi
snev
ski
et
al.
:
S
y
n
ta
cti
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
M
u
l
tifu
ncti
on
Rad
a
rs
1006
Pr
oc
ee
di
ng
s
o
f
t
he
IE
EE
|
V
o
l
.9
5
,
N
o
.
5
,M
a
y
2
0
0
7
where
A
as
well
as
rows
of
%
and
B
are
d
e
f
ined
with
respect
t
o
t
he
stat
e
o
rd
e
r
ing
f
S
1
;
S
2
;
T
1
;
T
2
g
as
shown
i
n
Fig.
5,
and
c
olumns
of
B
are
d
efined
with
resp
e
c
t
t
o
t
h
e
ordering
f
a
;
b
g
.
II
I.
E
LEC
TRONIC
WARFA
R
E
APPLICA
T
IO
N
V
E
LECT
R
ONIC
SUPPO
RT
AND
M
FR
With
t
h
e
a
bove
background
in
syntactic
modeling,
w
e
a
re
now
r
eady
to
st
u
d
y
M
FRs,
and
d
ev
i
s
e
e
lectronic
s
upp
o
rt
algorit
h
ms
that
deal
with
their
e
ver
i
ncr
e
asing
s
ophistica-
ti
on
of
their
r
emote
s
ensing
c
a
pabil
i
ti
es.
Electronic
warfare
(
EW)
can
be
broad
l
y
d
efined
as
any
military
a
ction
w
ith
t
he
objective
o
f
c
ontrolling
t
he
elect
r
o
m
agnetic
s
pectrum
[
46].
An
im
p
o
rt
a
n
t
a
spect
o
f
EW
is
the
r
a
d
ar-targ
e
t
in
teracti
o
n.
I
n
ge
ne
ral,
t
h
is
i
n
ter-
action
can
be
examined
from
two
e
ntirely
d
ifferent
points
of
view
V
the
v
iewp
o
i
nt
of
the
r
adar
and
t
h
e
viewpoint
o
f
the
t
arget.
From
the
r
a
d
ar’s
point
o
f
v
iew,
its
p
rimary
goal
is
to
detect
t
a
rget
s
a
nd
t
o
id
e
n
tify
their
c
ritical
parameters.
From
the
t
arget’s
p
oint
of
view,
t
he
goal
is
t
o
protec
t
i
tself
from
a
r
adar-equipped
t
hreat
b
y
c
ollecting
r
adar
emissions
and
e
val
u
ati
n
g
t
h
r
eat
i
n
r
eal
t
i
m
e
(
el
ectron
ic
su
p
p
ort).
I
n
t
h
i
s
pa
p
e
r
,
th
e
t
a
r
g
e
t

s
v
i
e
w
po
i
n
t
i
s
t
h
e
f
o
c
u
s
,
a
n
d
M
F
R
s
are
t
he
specific
threat
considered.
The
f
ram
e
work
of
EW
con
s
ide
r
ed
in
th
is
paper
consists
of
three
l
ayers:
receiv
e
r
/d
e
i
nterleaver,
p
ulse
train
analy
z
er,
a
nd
syntactic
processor
[
47].
The
l
ay
e
r
s
a
re
depict
e
d
in
Fig.
6
a
nd
a
b
rief
d
e
scrip
t
ion
i
s
g
iven
here:
The
r
eceiver
p
rocesses
t
he
rada
r
p
ulses
i
ntercept
e
d
by
t
h
e
antenna,
a
nd
outputs
a
sequence
of
pulse
d
escrip
t
o
r
word
s
,
which
i
s
a
d
a
ta
structure
c
ont
a
ining
p
arameters
such
as
carrier
frequency,
p
u
lse
a
mplitud
e
,
o
r
p
ulse
widt
h
.
The
d
ei
nte
r
le
aver
p
r
oce
sse
s
t
he
pulse
d
e
s
criptor
w
ords,
groups
t
h
em
according
t
o
t
heir
possible
o
riginating
rada
r
em
itters
and
s
tores
t
hem
i
n
t
heir
corresponding
t
r
ack
files.
The
p
ulse
t
r
ain
a
naly
z
e
r
p
r
o
cesses
t
he
track
file,
a
nd
f
u
r
t
h
e
r g
r
o
u
p
s t
h
e
p
u
l
s
e d
e
s
c
r
i
p
t
o
r w
o
r
d
s
i
n
t
o
r
a
d
a
r
words.
(See
S
e
ction
I
II-A
for
d
efinit
i
o
ns.)
Finally,
the
sy
n
t
ac
t
i
c
p
rocessor
a
nalyzes
t
he
syntactic
structure
o
f
t
he
r
a
da
r
w
o
r
ds
,
e
s
t
i
m
a
t
e
s
th
e
s
t
a
t
e
o
f
t
h
e
r
a
d
a
r
s
y
s
t
e
m
an
d
i
t
s
th
r
e
a
t
l
e
ve
l
,
a
n
d
o
u
t
p
u
ts
t
h
e
r
e
s
u
l
t
s
o
n
a
p
i
l
o
t
i
n
s
t
r
u
m
e
n
-
t
a
tion
panel.
Because
the
r
eceiver,
deinterleaver,
and
p
u
lse
t
rain
analyzer
have
been
well
studied,
the
s
yntactic
p
r
o
c
e
s
s
o
r i
s t
h
e
f
o
c
u
s o
f t
h
i
s p
a
p
e
r
.
T
h
e
s
yntactic
processor
captures
t
he
knowledge
o
f
t
he
B
langu
a
ge
[
that
MFRs
speak.
It
is
a
c
om
p
l
ex
system
of
rules
a
nd
constrain
t
s
t
hat
a
llow
radar
a
nalysts
t
o
d
istin-
gu
ish
B
grammatical
[
rad
a
r
s
ignal
f
rom
B
ungrammatical
[
one.
In
other
w
ords,
a
n
a
nalogy
is
drawn
b
etween
the
st
r
u
ctural
descript
i
o
n
o
f
t
he
radar
s
ignal
a
nd
the
s
yntax
o
f
a
l
anguage,
and
t
h
e
structural
description
c
ould,
t
herefore,
be
speci
f
ied
b
y
t
he
est
a
bli
s
hment
o
f
a
g
r
ammar
[
4
8
].
As
far
as
EW
is
concerned,
the
o
p
t
imal
app
r
oa
c
h
is
to
collect
a
corp
u
s
of
radar
s
amp
l
es,
a
nd
induce
the
g
rammar
directly
without
h
um
an
interv
e
n
tion.
H
owever,
b
ecause
of
the
d
e
gree
of
complexity
and
p
otential
lack
of
data
on
the
M
FR
signal,
g
rammatical
induct
i
o
n
a
pproach
is
impract
i
cal.
A
s
a
result,
i
n
t
his
p
a
p
er,
t
he
grammar
i
s
c
onstrained
to
be
S
C
FG,
a
nd
its
c
ontext-free
b
ack
b
o
ne
is
specified
b
y
r
ad
a
r
anal
ysts
from
studyi
ng
M
F
Rs’
si
gnal
generat
i
o
n
m
echa-
nism
.
S
ection
III-A
d
esc
r
ibes
MFRs’
system
architecture
and
t
he
building
blocks
making
up
the
r
adar
signal,
a
nd
S
e
ction
I
II-B
discusses
t
he
shortcomings
of
HMM
a
nd
explains
why
S
CFG
i
s
p
referred.
A.
MFR
Signal
Model
and
Its
System
Architecture
In
ord
e
r
t
o
d
escribe
M
FRs’
system
archit
e
c
ture,
w
e
begin
w
ith
t
h
e
building
block
s
making
up
MFRs’
signal
generation
process,
and
t
hey
a
re
d
e
fined
a
s
f
ollows.

Radar
w
ord
:
A
fi
xe
d
a
r
r
an
ge
me
nt
of
fin
i
te
nu
mber
of
pulses
that
is
optimized
f
or
extracting
a
pa
r
t
icular
target
information;
for
e
xample,
p
ulses
wi
th
a
f
ix
ed
pul
s
e
r
e
p
eti
t
ion
f
re
que
n
cy.
Fig.
6.
T
h
e
e
l
e
c
t
r
o
n
i
c
w
ar
fa
re
(
E
W
)
fr
am
ew
or
k
c
on
si
de
r
e
d
i
n
t
h
i
s
pa
pe
r.
Th
e
r
a
d
a
r
si
g
n
a
l
em
it
te
d
b
y
t
h
e
MF
R
i
s
c
ap
tu
r
e
d
a
t
t
h
e
E
W
s
y
s
t
e
m
on
bo
ar
d
t
h
e
ta
rg
et
a
f
t
e
r
b
e
i
n
g
co
r
r
u
p
t
e
d
b
y
t
h
e
s
t
o
c
h
a
s
t
i
c
e
n
v
i
ro
nm
en
t.
T
h
e
E
W
s
y
s
t
e
m
c
o
n
s
i
s
t
s
o
f
a
n
a
n
t
e
n
n
a
,
a
re
c
e
i
v
e
r
/
de
i
n
t
e
rl
ea
ve
r,
a
p
u
l
s
e
tr
a
i
n
a
n
a
l
y
z
e
r
,
a
n
d
a
sy
nt
ac
ti
c
p
r
o
c
e
s
s
o
r
.
F
i
g.
5.
E
x
a
m
pl
e
o
f
a
n
H
M
M
fo
r
t
h
e
SF
S
G
(1
1)
.
E
ac
h
s
t
a
t
e
is
la
be
le
d
by
tw
o
s
y
m
bo
l
s
se
pa
r
a
t
e
d
b
y
a
sl
as
h.
Th
e
f
i
r
s
t
sy
m
b
o
l
id
en
ti
fi
es
th
e
st
at
e
o
f
t
h
e
sy
st
em
,
a
n
d
th
e
s
ec
on
d
d
e
t
e
r
m
i
n
e
s
t
h
e
ou
tp
ut
pr
od
uc
e
d
by
th
e
s
ys
te
m
i
n
t
h
i
s
s
t
a
t
e
.
T
o
a
cc
om
mo
da
te
fo
r
t
h
e
te
r
m
i
n
a
l
sy
m
b
o
l
s
o
f
t
h
e
g
r
a
m
m
a
r
(
11
)
t
h
r
o
u
g
h
th
e
u
s
e
of
th
e
o
b
s
e
r
v
a
t
i
o
n
pr
o
b
a
b
i
l
i
t
y
m
a
t
r
i
x
B
,
t
h
e
st
ru
ct
ur
e
o
f
t
h
e
Ma
r
k
o
v
ch
a
i
n
i
n
F
ig
.
4
ha
d
t
o
b
e
t
r
a
n
s
f
o
r
m
e
d
to
th
e
M
o
o
r
e
m
a
c
h
i
n
e
.
Co
ns
e
q
u
e
nt
ly
,
t
h
e
un
de
rl
yi
ng
Ma
r
k
o
v
ch
ai
n
o
f
t
h
i
s
H
M
M
ha
s
d
i
f
fe
r
e
n
t
se
t
o
f
d
i
s
c
r
et
e
s
t
a
t
e
s
f
S
1
;
S
2
;
T
1
;
T
2
g
.
Vi
sn
evski
et
al.
:
S
y
n
t
a
ct
i
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
Mu
lt
if
u
n
c
t
i
o
n
R
ad
ars
V
o
l
.
9
5
,
N
o
.
5
,
May
2007
|
Proce
e
dings
o
f
t
he
IE
EE
1007

R
a
da
r
p
hra
s
e
(
ra
dar
t
as
k)
:
C
oncat
enation
o
f
f
inite
number
of
radar
w
ords.
E
a
c
h
p
h
r
ase
m
ay
be
imple-
mented
by
more
t
h
an
one
c
oncat
e
n
ation
o
f
r
adar
words.
Exam
p
l
es
are
s
earch
a
nd
target
acquisition.

Ra
d
a
r
p
o
l
i
c
y
:
P
reoptimized
s
chemes
that
allocate
resources
t
o
r
ad
a
r
phrases.
An
exa
m
p
l
e
is
rules
o
f
engagem
e
nt
or
policies
of
operatio
n.
Fig.
7
i
l
l
ustrat
e
s
how
a
rad
a
r
p
hrase
a
nd
rad
a
r
w
ords
are
r
e
lat
e
d.
Fig.
7(a)
shows
t
wo
radar
w
ords
that
are
represented
b
y
s
y
m
bols
B
a
[
and
B
b
,
[
where
v
ertical
bars
represent
r
adar
pulses.
F
ig.
7
(b)
i
llustrates
a
sequence
of
ra
d
a
r
w
ords
for
a
radar
p
h
r
ase,
and
w
hich
is
constructed
from
concatenat
i
o
n
o
f
a
and
b
into
a
s
eque
nce
B
ab
aa
.
[
T
h
e
g
eneration
p
rocess
of
radar
w
ords
is
governed
a
c
co
rd
i
n
g
t
o
M
FRs’
system
archit
e
c
ture
2
as
il
lu
strated
i
n
Fig.
8.
An
MFR
c
onsists
o
f
t
hree
main
co
mponents:
s
it
u
a
-
t
i
on
assessm
ent,
system
m
a
nager,
and
p
hrase
s
cheduler/
ra
d
a
r
c
ontroller.
T
h
e
s
ituation
assessment
m
od
u
l
e
p
ro-
vi
d
e
s
f
e
e
d
b
a
c
k
o
f
t
h
e
t
a
c
t
i
c
e
n
vi
r
o
n
m
e
n
t,
a
n
d
t
h
e
s
y
s
t
e
m
manager,
based
o
n
t
he
feedback,
s
elects
a
r
adar
policy
.
E
a
ch
rad
a
r
p
olicy
i
s
a
resource
allocation
s
cheme
t
hat
represents
trad
e
o
ffs
between
d
ifferent
performance
m
ea-
sures,
and
i
t
d
ic
t
a
tes
h
ow
the
p
hrase
s
cheduler/radar
c
o
ntroller
will
op
e
r
ate.
Examples
of
the
r
adar
policies
are
long
range
t
rack
acquisition
and
s
hort
range
s
elf
p
rotect
t
a
rget
acquisition
policies:
t
he
major
p
erformance
mea-
sures
i
n
t
hese
two
p
olicies
a
re
false
a
larm
rate
and
t
rack
latency;
track
latency
(
false
a
larm
rate)
i
s
t
olerable
in
long
ra
n
g
e
t
rack
acquisit
i
on
policy
(short
r
ange
self
protect
ta
r
g
e
t
a
c
q
u
i
s
i
ti
o
n
p
o
l
i
c
y
)
a
n
d
m
a
y
b
e
s
ac
r
i
f
i
ce
d
f
o
r
l
o
w
e
r
fa
l
s
e
a
larm
rate
(track
latency).
Th
e
s
c
h
e
d
u
l
i
n
g
a
n
d
g
e
n
e
r
a
t
i
o
n
o
f
r
a
da
r
w
o
r
ds
,
o
n
t
h
e
other
h
and,
is
dictated
by
two
c
ontrollers,
p
hrase
sc
h
e
d
u
ler
a
nd
radar
c
ontroller,
and
t
heir
corresponding
q
u
eu
es,
p
l
a
nni
n
g
q
u
e
u
e
and
c
ommand
qu
eue
,
respe
c
tive
ly.
Th
e
r
e
a
s
o
n
f
o
r
h
a
v
i
n
g
t
h
e
q
u
e
u
e
s
i
s
d
r
i
ve
n
b
y
t
h
e
n
e
e
d
f
o
r
MFR
t
o
b
e
b
oth
a
daptive
a
nd
fast
[4
9
]
.
T
he
planning
queue
stores
scheduled
r
adar
phrases
t
hat
a
re
o
r
d
e
red
b
y
t
ime
a
n
d
p
r
i
o
r
i
t
y,
a
n
d
i
t
a
l
l
o
w
s
t
h
e
s
c
h
e
d
u
l
i
n
g
to
b
e
m
o
di
f
i
e
d
b
y
phrase
scheduler.
Due
t
o
t
he
system’s
finite
response
tim
e
,
radar
p
hra
s
es
in
the
p
l
a
nning
queue
a
re
retrieved
s
equen-
tially
and
e
ntered
to
the
c
ommand
queue
w
here
no
further
planning
or
a
d
aptation
is
allowed.
Radar
c
ontroller
m
aps
the
r
adar
phrases
i
n
t
h
e
command
queue
t
o
r
adar
words
and
w
hich
are
f
ixed
for
execution.
More
specif
ically
,
the
phrase
schedu
ler
models
MFRs’
abil
ity
to
pl
an
ahead
its
course
of
action
and
to
pro-ac
tively
mo
nitor
the
feasibili
ty
of
its
schedu
led
tasks
[5
0].
Such
an
abil
ity
is
essentia
l
because
MFR
switch
es
betwee
n
radar
phr
ases,
and
confli
cts
such
as
execut
ion
ord
er
and
syst
em
loadin
g
must
be
resolved
ah
ead
of
time
based
on
the
predi
cted
system
perfo
rmance
and
th
e
tac
tic
enviro
nment.
In
additi
on,
planni
ng
is
al
so
neces
sary
if
MFR
is
interfa
ced
wit
h
a
n
externa
l
devic
e,
w
here
the
exec
ution
of
certa
in
phr
ases
needs
to
meet
a
fix
ed
time
line
.
Radar
contr
oller,
on
the
other
hand,
mod
els
M
FR’s
abilit
y
t
o
convert
radar
phr
ases
to
a
multi
tude
of
differen
t
radar
words
de
pending
on
the
tact
ic
enviro
nment.
Such
an
arr
angeme
nt
follo
ws
the
mac
ro/mic
ro
architec
ture
as
descr
ibed
in
Blackm
an
and
Po
poli
[14];
th
e
phr
ase
schedu
ler
dete
rmines
which
phr
ase
the
MFR
is
to
pe
rform
that
best
u
tilize
the
system
resour
ces
to
achi
e
v
e
the
missio
n
goal,
and
th
e
rada
r
contr
oll
er
de
termin
es
how
the
particula
r
phr
ase
is
to
be
perfo
rmed.
The
M
FR’s
op
e
r
ational
d
etails
that
are
t
o
b
e
m
od
e
l
ed
are
d
escribed
here.
P
hrase
s
cheduler
p
r
oc
e
sses
t
h
e
radar
phrases
i
n
t
h
e
planning
q
u
eue
s
equentially
f
rom
l
eft
t
o
right.
(If
t
h
e
queue
i
s
e
mpty
,
a
n
a
pprop
r
i
ate
r
adar
phrase
is
inserted.)
To
process
d
ifferent
types
o
f
r
adar
phrases,
phrase
scheduler
calls
their
c
orresponding
cont
r
o
l
r
ules;
Fig.
7.
Ra
da
r
w
o
r
d
s
c
a
n
b
e
v
i
e
w
e
d
a
s
f
un
da
m
e
n
t
a
l
bu
il
di
ng
bl
oc
k
s
o
f
th
e
M
F
R
s
i
g
n
a
l
.
(
a
)
Sh
ow
s
t
w
o
di
s
t
i
n
c
t
ra
da
r
w
or
ds
la
be
l
e
d
a
s
a
an
d
b
.
(
b)
Il
lu
st
ra
te
s
h
o
w
a
r
ad
ar
ph
ra
s
e
as
re
pr
es
en
te
d
b
y
a
p
u
l
s
e
se
qu
en
ce
c
a
n
b
e
d
e
c
o
m
po
se
d
i
n
t
o
a
se
r
i
e
s
of
ra
da
r
w
o
r
d
s
a
s
d
e
f
in
ed
in
(a
)
.
2
The
system
architecture
does
not
include
multiple
target
tracking
functionalities
such
as
data
association.
The
paper
focuses
on
a
single
target’s
self
protection
and
threat
estimation,
and
thus
models
only
the
radar
signal
that
a
single
target
can
observe.
Fi
g
.
8.
Th
e
f
ig
ur
e
i
ll
us
tr
a
t
e
s
MF
Rs

s
y
s
t
e
m
a
r
c
h
i
t
e
c
t
u
r
e.
T
h
e
s
i
t
u
a
t
i
o
n
as
se
ss
m
e
n
t
m
o
d
u
l
e
ev
a
l
u
a
t
e
s
t
h
e
ta
ct
ic
e
n
v
i
ro
nm
en
t
a
nd
pr
ov
i
d
e
s
fe
ed
ba
c
k
to
th
e
s
ys
te
m
m
an
ag
er
.
T
he
sy
st
em
m
a
n
a
g
e
r
,
ba
se
d
o
n
t
h
e
fe
ed
ba
c
k
,
s
e
l
e
c
ts
th
e
r
a
d
a
r
po
li
c
y
i
n
w
h
i
c
h
t
h
e
ph
ra
se
sc
he
du
l
e
r
/
r
a
d
a
r
co
nt
ro
l
l
e
r
wi
ll
op
er
at
e.
Th
e
p
h
r
a
s
e
s
c
h
e
d
u
l
e
r
in
it
ia
te
s
a
nd
sc
he
du
le
s
ra
da
r
p
h
r
as
es
in
th
e
p
l
a
n
n
i
n
g
q
u
e
u
e
an
d
t
h
e
ph
ra
se
s
f
i
x
e
d
fo
r
ex
ec
ut
i
o
n
a
r
e
m
o
v
e
d
t
o
t
h
e
c
o
m
m
an
d
q
u
e
ue
.
T
h
e
ph
ra
se
s
i
n
t
h
e
co
m
m
a
n
d
qu
eu
e
a
re
m
a
p
p
e
d
to
a
p
p
r
op
ri
a
t
e
r
ad
ar
w
o
r
d
s
b
y
t
h
e
r
a
d
a
r
co
nt
ro
l
l
e
r
a
n
d
a
r
e
se
nt
to
MF
R
f
or
ex
e
c
u
t
i
o
n
.
Vi
snev
ski
et
al.
:
S
y
n
ta
cti
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
M
u
l
tifu
ncti
on
Rad
a
rs
1008
Pr
oc
ee
di
ng
s
o
f
t
he
IE
EE
|
V
o
l
.9
5
,
N
o
.
5
,M
a
y
2
0
0
7
the
r
ule
t
a
k
es
the
r
adar
phrase
being
p
rocessed
a
s
i
np
u
t
,
and
r
esp
o
nds
b
y
a
p
p
ending
appropriate
r
adar
phrases
i
nt
o
the
c
ommand
q
u
eu
e
a
nd/or
t
he
pl
anni
ng
queu
e.
T
h
e
select
i
o
n
o
f
t
he
control
r
ules
is
a
f
unc
t
ion
o
f
r
adar
policies,
and
w
hich
are
e
xpressed
b
y
h
ow
probable
each
rule
would
be
selected.
S
imilar
to
phrase
scheduler,
the
r
ad
a
r
con-
troller
p
r
o
cesses
t
he
rada
r
p
h
r
ases
in
the
c
om
mand
queue
sequentially
and
m
aps
t
he
radar
p
hrases
t
o
radar
w
ord
s
according
t
o
a
set
o
f
c
ont
r
ol
rules.
B.
Inadequacy
of
HMM
for
Modeling
MFR
Th
e
d
ist
i
ngu
i
sh
ing
f
e
a
ture
s
o
f
M
FRs
c
ompa
red
t
o
conventional
radars
are
t
heir
ability
t
o
s
wit
c
h
b
etween
radar
t
asks,
a
nd
their
u
se
of
sched
u
lers
to
p
l
an
ahead
t
h
e
course
of
action
[
49].
In
ord
e
r
t
o
m
odel
such
features,
a
s
will
be
shown
l
at
e
r
in
the
n
ext
s
ection,
p
a
r
tial
production
rules
o
f
t
he
form
1)
B
!
bB
and
2)
B
!
AB
j
BC
j
bB
are
d
ev
i
s
ed
(See
Section
I
II-D
for
d
e
t
ails);
Fig.
9
i
llustrates
the
p
roduction
r
ules
and
t
heir
derivatio
n
s.
The
s
ignifi-
cance
of
the
r
ules
is
that
since
H
M
M
is
equivalent
t
o
stochastic
regular
g
rammar
[
51]
[
rules
o
f
t
he
fo
rm
1)],
and
MFRs
follo
w
rules
t
h
a
t
s
trictly
c
ontain
regular
g
rammar
[
r
u
l
e
s
o
f
t
h
e f
o
r
m2
) c
a
n
n
o
t b
e r
e
d
u
c
e
d
t
o
1
)
]
,
H
M
Mi
s n
o
t
sufficient
t
o
m
od
e
l
MFRs’
signal.
F
urthermo
re,
f
or
sources
with
hidd
e
n
branching
p
rocesses
(
MFRs),
SCFG
is
shown
t
o
b
e
m
o
r
e
e
f
f
i
c
i
e
n
t
t
h
a
n
H
M
M
;
t
h
e
e
s
t
i
m
a
te
d
S
CF
G
h
as
lower
e
ntropies
than
that
of
HMM
[
31].
Remark:
The
s
et
of
prod
u
c
tio
n
rules
p
resented
above
i
s
a
self-embed
d
i
ng
C
F
G
a
nd
thus
its
l
an
guage
i
s
n
ot
regular,
and
cannot
b
e
r
epresented
by
a
M
arkov
c
hain
[10].
F
or
t
h
e
rules
p
resented
,
s
elf-em
b
e
dding
property
can
be
shown
b
y
a
s
imple
d
e
r
ivat
i
o
n
B
!
AB
!
AB
C
:
In
addition
to
t
h
e
s
elf-embedding
p
r
op
e
r
ty
derived
from
the
s
cheduling
p
rocess,
the
generat
i
o
n
o
f
w
ords
by
the
r
ad
a
r
cont
roller
poses
a
nother
p
r
oblem.
For
e
a
c
h
r
adar
phrase
scheduled,
a
v
a
r
iable
n
um
ber
o
f
r
ad
a
r
words
m
ay
be
generated.
If
HMM
i
s
a
pplied
t
o
s
t
u
dy
the
s
equence
o
f
rad
a
r
w
ords,
t
he
Markovian
d
ependency
m
ay
be
of
variable
l
e
ngth.
I
n
t
hi
s
case,
maxi
mu
m
l
ength
d
e
p
e
n
de
ncy
n
ee
ds
to
be
used
to
define
the
s
tate
space,
a
nd
the
e
xponential
growing
s
tate
space
m
i
ght
b
e
a
n
i
ssue.
C.
A
Syntactic
Approach
to
MFR
In
term
s
o
f
n
atural
language
processing,
w
e
m
odel
the
M
F
R a
s a s
y
s
t
e
m
t
h
a
t
B
speaks
[
according
t
o
a
n
S
C
F
G.
Based
o
n
t
he
discussion
in
Section
I
II-A,
t
h
e
syntactic
ap
p
r
oach
is
suitable
in
modeling
the
p
hrase
s
cheduler
because
the
s
cheduler
operates
according
t
o
a
set
o
f
f
ormal
rules.
On
t
h
e
o
ther
hand,
t
he
radar
c
ontroller
i
s
s
uitably
modeled
b
ecause
of
two
r
easons:
1
)
e
ach
sequence
of
rad
a
r
words
i
s
s
emantically
ambiguous,
i.e.,
m
any
s
tatistic
a
l
ly
d
i
stinct
p
a
tterns
(radar
w
ord
s
equence)
possess
the
s
am
e
semantic
meanings
(radar
p
hrase),
a
nd
2)
each
rad
a
r
p
h
rase
consists
o
f
finite
num
b
er
of
rad
a
r
w
ords,
a
nd
rad
a
r
words
a
re
relatively
easy
to
discriminate.
A
Simple
Example
of
MFR:
As
an
il
lust
rative
ex
ample
showing
t
h
e
correspond
e
nce
b
et
w
e
en
the
g
rammar
a
nd
t
h
e
M
FR,
c
onsider
p
roduction
r
ules
of
the
f
orm
1
)
A
!
aA
and
2
)
A
!
BA
,w
h
e
r
e
A
and
B
are
c
onsidered
a
s
r
ad
a
r
p
h
rases
i
n
t
he
planning
queue
a
nd
a
as
a
r
adar
phrase
in
th
e
c
o
m
m
a
n
d
q
u
e
u
e
.
Th
e
r
u
l
e
A
!
aA
is
interpreted
a
s
d
i
rect
i
n
g
t
he
phrase
scheduler
t
o
a
ppend
a
to
the
q
ueue
l
i
s
t
i
n
th
e
c
o
m
m
a
n
d
qu
e
u
e
,
a
n
d
A
in
t
h
e
p
l
a
nn
ing
q
ueue
.
Si
m
i
l
a
r
l
y,
A
!
BA
is
interp
rete
d
a
s
d
el
ayi
n
g
t
he
ex
ecut
i
o
n
of
A
in
the
s
che
d
ul
in
g
q
ue
ue
and
i
n
s
erti
ng
B
in
fron
t
o
f
i
t
.
S
u
p
p
o
s
e
t
h
e p
l
a
n
n
i
n
g q
u
e
u
e
c
o
n
t
a
i
n
s
t
h
e r
a
d
a
r
phra
se
A
,
a
possible
realization
o
f
t
he
radar
w
ords’
g
e
nerati
on
p
r
ocess
i
s
i
ll
ustr
ated
i
n
Fi
g.
10.
(
The
f
i
g
ure
F
i
g.
9.
T
h
e
f
i
g
u
r
e
i
l
l
us
tr
at
es
th
e
d
e
r
i
v
a
t
i
o
n
p
r
o
c
e
s
s
wi
th
tw
o
d
i
f
f
e
r
e
n
t
ty
pe
s
o
f
p
r
o
du
c
t
i
o
n
r
ul
es
.
(
a)
Th
e
d
e
r
i
v
a
t
i
o
n
o
f
t
h
e
r
u
l
e
of
re
gu
la
r
g
r
a
m
m
a
r
t
y
p
e
.
(b
)
T
h
e
de
ri
va
ti
on
of
th
e
r
ul
e
o
f
C
FG
ty
pe
.
Fig.
10
.
A
p
o
s
s
i
b
l
e
r
e
a
l
i
z
a
t
i
o
n
of
th
e
s
c
h
e
d
u
l
in
g
p
r
o
c
e
s
s
r
e
p
r
e
s
en
te
d
by
a
g
ra
m
m
a
t
i
c
a
l
de
ri
va
ti
on
pr
o
c
e
s
s.
A
an
d
B
ar
e
r
a
d
a
r
p
h
ra
se
s
i
n
t
h
e
p
la
nn
in
g
q
ue
ue
,
a
an
d
b
ar
e
r
a
d
a
r
ph
ra
se
s
i
n
t
h
e
co
m
m
a
n
d
qu
eu
e
an
d
w
an
d
y
ar
e
r
ad
ar
wo
r
d
s
.
Vi
sn
evski
et
al.
:
S
y
n
t
a
ct
i
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
Mu
lt
if
u
n
c
t
i
o
n
R
ad
ars
V
o
l
.
9
5
,
N
o
.
5
,
May
2007
|
Pr
oceed
ings
of
the
I
EE
E
1009
a
l
so
illustrates
t
he
operat
i
o
n
o
f
t
he
radar
c
ontroller;
the
t
r
iangle
represents
t
h
e
m
apping
of
the
r
adar
p
h
rase
to
the
ra
d
a
r
w
ords,
w
hich
are
d
enoted
by
y
and
w
.)
It
can
be
seen
that
as
long
as
the
c
omma
n
d
queue
p
hrases
appear
only
to
t
h
e
l
eft
o
f
p
lann
ing
q
ue
ue
phrase
s
i
n
t
he
rul
e
,
th
e
c
o
m
m
a
n
d
qu
e
u
e
a
n
d
th
e
p
l
a
n
n
i
n
g
q
u
e
u
e
a
r
e
w
e
l
l
represented.
D.
A
Syntactic
Model
for
an
MFR
Called
Mercury
T
h
e
s
yntact
i
c
mod
e
ling
technique
i
s
d
i
s
cussed
i
n
t
his
subsection,
a
nd
the
d
iscussion
is
based
o
n
a
n
a
nti-aircraft
d
e
fense
r
adar
called
M
ercury.
T
he
rad
a
r
i
s
c
lassified
and
its
i
ntelli
gence
report
i
s
sanitize
d
a
nd
provide
d
in
A
p
pe
ndix
A.
Table
2
provide
s
an
e
x
h
austive
l
i
s
t
o
f
a
l
l
p
o
ssible
M
ercury
p
h
rases,
and
a
ssociates
t
h
em
wit
h
the
functional
stat
e
s
of
the
r
adar.
T
his
t
able
was
o
btained
f
rom
sp
e
c
ifi
cation
s
i
n
the
s
an
iti
z
e
d
i
n
te
ll
ig
en
ce
report
an
d
i
s
c
e
ntral
t
o
g
rammatical
d
e
riv
a
tions
t
hat
f
ollow.
T
h
e S
C
F
G
m
o
d
e
l
i
n
g t
h
e
M
F
R i
s
G
#f
N
p
[
N
c
;
T
c
;
P
p
[
P
c
;
S
g
,w
h
e
r
e
N
p
and
N
c
are
n
onterminals
r
epresent-
ing
t
he
sets
of
radar
p
h
r
ases
in
the
p
lanning
queue
a
nd
c
o
mmand
qu
eue
r
esp
e
cti
v
el
y,
T
c
is
terminals
r
ep
r
e
senting
th
e
s
e
t
o
f
r
a
d
a
r
w
o
r
ds
,
P
p
is
product
i
on
ru
les
m
apping
N
p
to
%
N
c
[
N
p
&
!
,
P
c
is
the
s
et
of
produ
c
tion
ru
les
m
apping
N
c
to
T
!
c
,a
n
d
S
is
t
h
e
s
tarting
s
ymbol.
It
sh
oul
d
be
note
d
t
h
at
the
s
election
of
p
r
od
u
c
tion
r
u
les
i
s
p
robabilistic
b
e
cause
MFR’s
i
nner
workings
cannot
b
e
k
nown
com-
p
l
ete
l
y.
I
n
thi
s
su
bsecti
on,
e
ach
compon
ents
o
f
MF
R
a
s
illustra
t
e
d
i
n
8
w
ill
b
e
d
iscussed
i
n
t
urn.
1)
Phrase
Scheduler:
T
h
e p
h
r
a
s
e s
c
h
e
d
u
l
e
r
m
o
d
e
l
s
t
h
e
MFR’s
a
bility
to
p
l
an
and
t
o
p
reempt
radar
p
hra
s
es
based
on
the
r
adar
command
and
t
h
e
d
y
namic
t
actic
environ-
ment.
I
ts
operational
r
ules
for
t
he
schedulin
g
a
nd
rescheduling
o
f
p
h
rases
a
re
modeled
b
y
t
he
production
rule
P
p
,
a
nd
it
is
found
t
hat
P
p
could
b
e
c
onstructed
from
a
sm
all
s
et
of
basic
r
ules.
S
uppose
N
p
#f
A
;
B
;
C
g
and
N
c
#f
a
;
b
;
c
g
,
T
he
basic
c
ontrol
rules
t
hat
a
re
available
t
o
the
p
hrase
s
c
h
ed
u
l
er
are
l
i
s
ted
b
el
ow
Markov
B
!
bB
j
bC
Adaptive
B
!
AB
j
BC
Terminat
i
n
g
B
!
b
:
The
i
nterpre
t
ation
o
f
t
h
e
rule
s
f
ol
lows
th
e
e
xampl
e
gi
ven
a
t
t
h
e
e
n
d
o
f
t
he
previ
o
us
subse
c
tion.
A
rule
is
Markov
if
it
sent
a
r
adar
p
h
rase
t
o
the
c
om
mand
queue,
and
rescheduled
e
ither
t
he
sam
e
o
r
different
radar
p
hrase
i
n
th
e
p
lann
ing
q
ue
ue
.
A
rul
e
i
s
ad
a
p
tive
if
it
ei
the
r
pre
-
empted
a
r
ad
a
r
p
h
rase
for
a
nother
ra
d
a
r
p
hrase
o
r
i
f
i
t
scheduled
a
radar
p
hrase
a
head
of
time
in
the
r
adar’s
time
line
after
t
he
current
p
h
rase.
A
rule
is
terminating
if
it
sent
a
r
adar
phrase
t
o
the
c
ommand
queue
w
itho
ut
sched
u
ling
any
n
ew
phrases.
The
s
ignificance
of
the
M
arkov
rule
is
obvious,
as
it
represents
the
d
ynamics
o
f
F
SA.
A
simple
examp
l
e
o
f
t
he
Markov
rule
is
illustrated
i
n
F
ig.
1
1
b
ased
on
Mercury

s
funct
i
onal
sta
t
es.
A
cc
ording
to
t
h
e
s
pecific
a
tions
i
n
Appen
d
ix
A,
re
lati
v
e
to
each
in
d
i
vidual
targe
t
,
t
h
e
Me
rcury
emitt
e
r
can
be
in
one
o
f
t
he
seven
functional
states
V
search,
acquisition,
n
onadap
t
i
ve
track
(NAT),
three
s
tages
o
f
range
r
esolution,
and
t
rack
m
a
intenance
(
TM).
The
t
ransi-
tions
b
etw
een
these
f
unct
i
o
nal
s
tates
can
be
captured
b
y
the
s
tate
machine
i
llustrated
in
Fig.
11.
The
M
ercury’s
st
a
t
e
m
achine
i
s
g
eneralized
by
in-
cluding
t
h
e
adaptive
and
t
erminating
rules.
The
i
nclusion
Table
2
List
of
All
Mercury
Emitter
Phrase
Combinations
According
t
o
t
he
Functional
State
o
f
t
he
Radar
Fi
g
.
11.
Me
r
c
u
r
y
e
m
i
t
t
e
r
fu
nc
ti
on
al
it
y
a
t
t
h
e
hi
g
h
le
ve
l
.
Th
er
e
a
re
se
ve
n
f
u
n
c
t
i
o
na
l
s
t
a
t
e
s
o
f
t
h
e
em
it
te
r
(
s
e
a
r
ch
,
a
c
q
u
i
s
i
t
i
o
n
,
N
A
T
,
th
r
e
e
s
t
a
g
e
s
o
f
r
a
n
g
e
re
so
lu
ti
on
,
a
n
d
TM
)
.
Th
e
t
r
a
n
s
i
t
i
o
n
s
be
tw
ee
n
t
h
e
s
t
a
t
e
s
ar
e
d
e
f
in
ed
ac
c
o
r
d
i
n
g
t
o
t
h
e
sp
ec
if
i
c
a
t
i
o
n
o
f
Ap
pe
nd
ix
A.
T
h
e
s
t
a
te
ma
c
h
i
n
e
i
s
s
h
o
w
n
as
th
e
M
o
o
r
a
ut
om
at
on
wi
th
ou
tp
ut
s
d
e
f
i
n
e
d
by
th
e
s
ta
te
s
.
A
c
or
r
e
s
p
o
n
d
i
n
g
ph
ra
s
e
fr
om
Ta
bl
e
2
is
g
e
n
e
r
a
te
d
i
n
e
v
e
ry
st
at
e
o
f
t
h
e
a
u
t
o
m
a
t
o
n
.
Vi
snev
ski
et
al.
:
S
y
n
ta
cti
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
M
u
l
tifu
ncti
on
Rad
a
rs
1010
Pr
oce
e
di
ngs
o
f
t
he
IEEE
|
V
o
l
.9
5
,
N
o
.
5
,M
a
y
2
0
0
7
of
t
h
e
a
daptive
r
ule
m
odel
s
M
F
Rs’
abil
ity
t
o
r
e
s
che
d
ul
e
radar
p
h
r
ases
w
h
en
the
s
ystem
l
oading
or
the
t
act
i
c
envi-
r
o
n
m
e
n
t
c
h
a
n
g
e
s
.
T
h
e
t
w
o
a
da
p
t
i
v
e
r
u
l
e
s
m
o
de
l
a
f
t
e
r
th
e
MFRs’
ability
t
o
1
)
p
reem
p
t
and
2
)
P
lan
t
h
e
radar
p
hrases.
The
p
reem
p
t
ability
i
s
d
em
onstrated
b
y
t
he
rule
B
!
AB
where
t
he
radar
p
hrase
B
is
pree
mpted
w
h
e
n
a
hi
gh
er
priority
task
A
enters
the
q
ueue.
O
n
t
he
other
h
and,
t
h
e
abil
ity
t
o
p
lan
i
s
captured
i
n
t
he
rul
e
B
!
BC
where
t
h
e
phrase
C
i
s
sch
e
du
led
a
he
ad
of
time
if
its
p
redi
cted
perform
a
nce
e
xceed
s
a
threshold.
On
the
o
ther
hand
,
t
h
e
terminating
r
ule
r
eflects
t
he
fact
that
t
h
e
q
ueues
h
ave
finite
lengt
h
,
a
nd
the
g
rammatical
derivation
process
m
ust
terminate
a
nd
yield
a
terminal
st
ring
of
finite
length.
Al
l
t
h
e
control
r
ul
es,
exce
p
t
t
he
ad
apti
ve
ru
le
,
c
oul
d
be
app
l
ied
t
o
a
ny
radar
p
hrases
available.
The
a
daptive
r
ule
schedul
e
s
p
hrases
ahead
o
f
t
i
m
e
a
nd
t
h
us
requi
r
es
a
reasonable
prediction
on
the
t
arget’s
k
inematics;
it
would
not
b
e
a
p
p
licable
to
phra
s
e
s
w
here
t
h
e
p
rediction
i
s
lacking.
Applying
the
r
ules
to
Mercury’s
r
adar
phrases,
t
h
e
produ
c
tion
ru
le
P
p
co
u
l
d
b
e
c
o
n
s
t
r
u
c
t
e
d
a
n
d
i
t
i
s
l
i
s
te
d
i
n
Fi
g.
12.
2)
Radar
Controller
and
the
Stochastic
Channel:
In
this
sect
ion,
we
develop
d
eterminist
ic
,
c
hara
cteris
tic,
and
stochastic
phrase
structure
g
rammars
of
t
h
e
M
ercury’s
radar
c
ontroller
a
s
d
escribed
in
Append
i
x
A.
The
g
rammar
is
derived
a
s
a
word-lev
e
l
syntactic
m
o
d
e
l
o
f
t
he
emitter.
We
consider
how
t
he
d
y
namics
of
radar
w
ords
that
make
up
one
o
f
t
he
ind
i
vidual
v
e
rtical
slots
i
n
F
ig.
2
3
captures
internal
processes
o
ccurring
within
the
r
adar
emit
t
e
r.
Deterministic
G
rammar:
According
t
o
D
ef.
2.2,
a
d
e
te
rmin
istic
g
rammar
i
s
d
e
f
ine
d
through
i
t
s
al
phabe
t
,
t
h
e
s
et
of
non
t
erminals,
a
nd
the
s
et
of
grammatical
p
r
oduction
rules.
Using
t
he
Mercury
s
pecifica
t
i
on
of
Ap
p
e
ndix
A,
we
can
define
t
h
e
a
lphabet
a
s
A#
f
w
1
;
w
2
;
w
3
;
w
4
;
w
5
;
w
6
;
w
7
;
w
8
;
w
9
g
(16)
wh
ere
w
1
;
...
;
w
9
are
t
he
wo
rd
s
o
f
t
he
Mercury
e
mitter.
At
ev
e
r
y
f
unctional
s
tate,
t
he
radar
e
mit
s
a
p
hrase
consist
i
ng
of
four
words
d
rawn
from
t
h
e
T
able
2.
Som
e
p
h
rases
a
re
unique
and
d
i
r
ect
l
y
i
dent
i
f
y
t
he
functio
n
al
st
a
t
e
o
f
t
he
radar
(
i.e.
)
w
1
w
2
w
4
w
5
*
can
only
be
encountered
d
u
ring
search
opera
t
ions).
Ot
h
e
r
p
hrases
are
c
haracteristic
t
o
sev
e
ral
r
adar
states
(i.e.
)
w
6
w
6
w
6
w
6
*
can
be
u
t
il
ize
d
in
acquisition,
N
AT
,
a
nd
T
M
).
T
h
ese
p
hrases
fo
r
m
strings
i
n
t
h
e
r
ad
a
r
language
that
we
are
i
nterested
i
n
m
odeling
sy
n
t
ac
t
i
cally.
T
o
c
o
mple
te
th
e
d
e
r
ivation
o
f
t
he
gramm
a
r,
we
must
de
f
i
n
e
t
h
e
r
u
l
e
s
f
o
r
t
h
e
h
XPhr
as
e
i
nonterminals
where
X
st
a
n
d
s
for
t
h
e
corresponding
name
o
f
the
e
mitter
s
t
a
te
in
w
h
i
c
h t
h
i
s p
h
r
a
s
e i
s e
m
i
t
t
e
d
.
Using
d
a
t
a
f
rom
T
able
2
,
we
defi
ne
th
e
t
ripl
ets
T
6
!
w
6
w
6
w
6
T
8
!
w
8
w
8
w
8
T
7
!
w
7
w
7
w
7
T
9
!
w
9
w
9
w
9
an
d
t
he
b
l
ock
s
of
f
o
ur
w
o
r
d
s
Q
1
!
w
1
w
1
w
1
w
1
Q
4
!
w
4
w
4
w
4
w
4
Q
7
!
w
7
w
7
w
7
w
7
Q
2
!
w
2
w
2
w
2
w
2
Q
5
!
w
5
w
5
w
5
w
5
Q
8
!
w
8
w
8
w
8
w
8
Q
3
!
w
3
w
3
w
3
w
3
Q
6
!
w
6
w
6
w
6
w
6
Q
9
!
w
9
w
9
w
9
w
9
:
Th
e
h
Thr
eeWSe
archPhrase
i
and
h
FourWSearchPhrase
i
rules
a
re:
h
Fo
u
r
WSearchPhrase
i
!
w
1
w
2
w
4
w
5
j
w
2
w
4
w
5
w
1
j
w
4
w
5
w
1
w
2
j
w
5
w
1
w
2
w
4
h
Thr
eeWSearchPhrase
i
!
w
1
w
3
w
5
w
1
j
w
3
w
5
w
1
w
3
j
w
5
w
1
w
3
w
5
:
Th
e
h
A
c
qP
h
r
a
s
e
i
rules
a
re
h
AcqPhrase
i!
Q
1
j
Q
2
j
Q
3
j
Q
4
j
Q
5
j
Q
6
:
F
i
g.
1
2
.
P
r
o
d
u
c
ti
on
r
u
l
e
s
o
f
M
e
r
c
u
r
y

s
ph
ra
s
e
sc
he
du
le
r.
Vi
sn
evski
et
al.
:
S
y
n
t
a
ct
i
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
Mu
lt
if
u
n
c
t
i
o
n
R
ad
ars
V
o
l
.
9
5
,
N
o
.
5
,
May
2007
|
Pr
ocee
dings
o
f
t
he
IEE
E
1011
Th
e
h
NATPhrase
i
rules
a
re
h
NATPhrase
i!
S
1
T
6
j
Q
6
S
1
!
w
1
j
w
2
j
w
3
j
w
4
j
w
5
:
T
h
e
r
ange
resolution
rules
a
re
h
RR
1
Phrase
i!
w
7
T
6
h
RR
2
Phrase
i!
w
8
T
6
h
RR
3
Phrase
i!
w
9
T
6
:
Finally,
the
t
rack
maintenance
r
ules
are
h
TM
Phra
se
i!
h
FourWTrack
ijh
ThreeWTrack
i
h
FourW
T
r
a
ck
i!
Q
6
j
Q
7
j
Q
8
j
Q
9
h
ThreeWTrack
i!
S
1
T
6
j
S
2
T
7
j
S
2
T
8
j
S
2
T
9
S
2
!
S
1
j
w
6
S
1
!
w
1
j
w
2
j
w
3
j
w
4
j
w
5
:
A
c
co
rd
i
n
g
t
o
t
he
Cho
m
sky
h
ierarchy
d
i
scussed
i
n
S
e
ction
I
I,
this
grammar
i
s
a
CFG.
Characteristic
Grammar:
The
cha
racte
risti
c
gramm
ar
of
the
Mercu
ry
emit
ter
must
exten
d
the
det
ermi
nisti
c
gramm
ar
to
acco
mmod
ate
for
the
possi
ble
unce
rtai
nties
in
the
real
life
env
ironm
ent.
These
unce
rtai
nties
are
due
to
the
errors
in
rec
eptio
n
and
ident
ific
ation
of
the
rada
r
words.
Conc
eptu-
ally
,
this
proce
ss
can
be
des
cribe
d
b
y
the
mode
l
o
f
the
stoc
hasti
c
erasu
re
cha
nnel
with
propag
ation
errors
.
T
o
accommodate
f
or
the
c
hannel
i
mpairment
m
od
e
l
,
w
e
hav
e
to
make
tw
o
m
odifications
to
the
d
eterministic
gramm
a
r
o
f
t
h
e
Mercury
e
mitter.
First
of
all,
the
a
lp
h
a
bet
(16)
has
t
o
b
e
e
xpanded
t
o
i
nclude
the
c
haracter
V
the
c
h
aracter
indicating
t
hat
n
o
r
eliable
r
adar
signal
d
e
tection
was
p
ossible
A
c
#f
Ø
g [
A
#f
Ø
;
w
1
;
w
2
;
w
3
;
w
4
;
w
5
;
w
6
;
w
7
;
w
8
;
w
9
g
:
(17)
Fin
a
ll
y,
we
introduce
a
n
a
d
d
i
t
ion
a
l
l
e
v
e
l
of
in
d
i
recti
o
n
into
t
h
e
g
rammar
by
adding
nine
new
n
onterminals
W
1
;
..
.
;
W
9
and
n
ine
n
ew
p
r
oduction
rules
W
i
!
Ø
j
w
1
j
w
2
j
w
3
j
w
4
j
w
5
j
w
6
j
w
7
j
w
8
j
w
9
(18)
wh
ere
i
#
1
;
..
.
;
9
.
Th
e
r
eason
f
or
th
is
modifi
cation
w
il
l
b
e
come
ap
p
a
rent
when
we
associate
p
robabilities
wit
h
the
p
r
oduction
rules
o
f
t
h
e
grammar.
Stochastic
Grammar:
The
c
hann
el
impairment
m
o
del
h
as
the
f
ollowing
transition
probabilit
y
m
atrix:
P
o
#)
p
1
p
2
p
3
p
4
p
5
p
6
p
7
p
8
p
9
*
T
(19)
where
p
i
#)
p
i
;
j
*
0
+
j
+
9
is
the
p
r
o
bability
that
the
t
ransmit-
ted
r
adar
word
i
b
e
i
n
g i
n
f
e
r
r
e
d
b
y t
h
e
p
u
l
s
e t
r
a
i
n
a
n
a
l
y
s
i
s
layer
o
f
t
he
EW
receiver
as
rad
a
r
w
ord
j
via
t
he
noisy
a
nd
corrupted
observ
a
t
ions.
The
s
tochastic
r
ad
a
r
grammar
can
be
obtained
from
the
characterist
i
c
g
rammar
b
y
a
ssociating
p
r
o
b
a
bility
v
e
ctors
of
(19)
with
the
c
orresponding
product
i
ons
o
f
(
18)
W
i
!
p
i
Ø
j
w
1
j
w
2
j
w
3
j
w
4
j
w
5
j
w
6
j
w
7
j
w
8
j
w
9
(20)
where
i
#
1
;
..
.
;
9.
Th
u
s
,
t
h
e
c
o
m
p
l
e
te
s
t
o
c
h
a
s
t
i
c
g
r
a
m
-
m
a
r
o
f
t
he
Mercury
e
mit
t
er
is
shown
i
n
F
ig.
1
3.
Strictly
speaking,
t
his
g
rammar
s
hould
b
e
c
alled
weighted
grammar
r
ather
t
han
s
tochastic.
As
shown
b
y
[23],
a
stoch
a
stic
CF
G
m
ust
s
ati
s
fy
the
l
imit
i
n
g
s
tochasti
c
consistency
c
riterion.
H
owever,
[
23]
d
em
onstrates
t
hat
us
eful
synta
c
tic
p
attern
r
e
co
gnition
t
echniq
ues
a
pply
equal
l
y
w
el
l
t
o
b
oth
t
he
stochast
i
c
,
a
nd
the
w
eig
h
te
d
CFGs.
T
herefo
re,
w
e
a
re
not
c
o
n
cerned
with
satisfying
the
stochastic
consist
e
ncy
c
riterion
at
this
p
o
int.
E.
MFR
and
System
Manager
V
Markov
Modulated
SCFG
From
the
p
revious
s
ect
i
on,
t
he
phrase
scheduler
a
nd
the
r
adar
controller
are
m
odeled
by
constructing
the
context-free
backbone
of
the
M
FR
grammar.
The
t
hird
Fi
g
.
13
.
We
ig
ht
e
d
g
r
a
m
m
a
r
o
f
t
h
e
Me
rc
ur
y
e
m
i
t
t
e
r
.
T
hi
s
g
r
a
mm
ar
,
li
k
e
it
s
d
e
t
e
r
m
i
n
i
s
t
i
c
co
un
te
rp
ar
t,
i
s
a
C
F
G
.
Vi
snev
ski
et
al.
:
S
y
n
ta
cti
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
M
u
l
tifu
ncti
on
Rad
a
rs
1012
P
r
oc
ee
di
n
g
s
o
f
t
he
IE
EE
|
V
o
l
.9
5
,
N
o
.5
,M
a
y
2
0
0
7
component
o
f
t
he
MFR
a
s
d
escri
b
ed
in
S
e
cti
o
n
I
II-A
is
t
h
e
syst
e
m
manager.
The
o
peratio
n
of
the
s
ystem
m
anager
is
deciding
on
the
o
ptimal
policy
of
operation
f
or
each
time
period,
a
nd
which
i
s
m
odeled
by
assigning
p
rod
u
ctio
n
r
ule
probabilities
t
o
t
he
context-free
backbone.
T
he
ev
o
l
ution
of
t
h
e
p
ol
ici
e
s
o
f
o
perati
on
is
drive
n
by
the
i
nte
r
action
between
t
he
targets
a
nd
the
r
adar,
a
nd
the
t
he
production
rules’
probabilities
c
onvenient
l
y
r
epresent
the
r
esource
allocation
s
cheme
d
eployed.
The
s
tate
sp
a
c
e
o
f
M
FR
is
constructed
b
ased
on
different
tactic
scenarios.
Let
k
#
0
;
1
;
...
denote
discrete
ti
m
e
.
T
he
po
li
cie
s
of
op
er
ati
o
n
x
k
is
model
e
d
a
s
a
M-state
Mark
ov
chain.
Define
the
t
ransit
i
o
n
p
robability
m
a
t
r
ix
a
s
A
#)
a
ji
*
M
(
M
;
w h e r
e
a
ji
#
P
%
x
k
#
e
i
j
x
k
,
1
#
e
j
&
f o r
i
;
j
2 f
1
;
2
;
..
.
;
M
g
.
F
or
example,
d
epending
on
the
loading
c
ondition
of
the
M
FR,
t
wo
states
may
b
e
d
efined
according
t
o
t
he
amount
of
resources
a
llocated
to
MFR’s
target
searching
a
nd
target
acquisition
funct
i
ons.
More
specifically,
o
ne
state
m
ay
consider
the
s
cenario
w
here
t
h
e
number
of
targets
d
etecte
d
i
s
l
ow,
a
nd
thus
hi
gher
proport
i
on
of
rada
r
r
esources
are
a
llocated
to
search
functions.
The
o
t
h
er
state
m
ay
consider
the
s
cenario
w
here
the
n
umber
o
f
t
argets
d
e
tect
e
d
is
high,
a
nd
resources
a
re
allocated
to
t
a
rget
acquisition
and
t
racking
f
unctions.
In
each
state,
the
M
FR
will
B
spea
k
[
a d
i
f
f
e
r
e
n
t
B
l
a
nguage,
[
which
i
s
d
efined
by
its
s
tate
-dependent
grammar.
Denote
the
g
r
a
mmar
a
t
s
tate
e
i
as
G
i
#
f
N
p
[
N
c
;
T
c
;
P
i
p
[
P
i
c
;
S
g
,
a
nd
it
is
no
t
e
d
t
hat
t
he
grammars’
context-free
ba
c
k
bone
is
id
e
n
tical
for
a
ll
i
e
x
c
e
p
t
th
e
probabi
l
i
t
y
d
ist
r
ibu
t
ions
d
e
fine
d
o
ve
r
t
h
e
i
r
producti
on
rules.
Each
state
o
f
M
FR
is
characterized
by
its
p
olicy
o
f
operations,
a
nd
which
d
etermines
t
he
resource
allocation
to
the
t
argets.
O
bviously,
t
h
e
more
resource
allocated
t
o
the
t
a
r
get
,
t
he
hi
gh
er
th
e
t
hreat
M
F
R
p
o
ses
o
n
t
he
target.
From
t
h
is
perspectiv
e
,
t
h
r
eat
e
stim
at
i
o
n
o
f
M
FR
is
red
u
ced
to
a
s
tate
estimation
problem.
One
pra
cti
cal
is
sue
is
tha
t
the
si
gnal
gen
era
ted
by
rad
ar
sys
tems
has
fi
nite
leng
th,
and
thi
s
fini
ten
ess
con
stra
int
must
be
sat
isf
ied
by
the
SCFG
is
the
mod
el
is
to
be
sta
ble.
We
dis
cu
ss
thi
s
poi
nt
by
fir
st
def
ini
ng
the
sto
cha
sti
c
mea
n
matr
ix.
Definition:
Let
A
;
B
2
N
,
t
he
stochastic
mean
matrix
M
N
is
a
j
N
j (j
N
j
square
matrix
wit
h
its
%
A
;
B
&
th
e
n
t
r
y
b
e
i
n
g
the
expected
n
umber
o
f
v
ariabl
es
B
re
sul
t
i
n
g
f
r
o
m
rewriting
A
M
N
%
A
;
B
&#
X
'
2%
N
[
T
&
"
s
:
t
:
%
A
!
'
&2
P
P
%
A
!
'
&
n
%
B
;
'
&
where
P
%
A
!
'
&
is
the
p
robability
o
f
a
pplying
t
he
produ
c
tion
rule
A
!
'
,a
n
d
n
%
B
;
'
&
i
s
th
e
n
u
m
b
e
r
o
f
in
st
a
n
ces
o
f
B
in
'
[5
2].
The
f
in
ite
n
e
s
s
c
onstrai
n
t
i
s
s
atisfi
ed
i
f
th
e
g
rammar
i
n
each
st
a
t
e
s
at
i
s
fies
the
f
ollowing
t
h
eorem.
Theorem:
If
the
s
pectral
r
adius
o
f
M
N
is
l
e
ss
than
one
,
t
h
e
g
en
erati
o
n
p
roce
ss
o
f
the
S
C
F
G
w
il
l
t
ermi
nate
,
a
nd
th
e
de
r
i
ve
d
s
e
n
t
e
n
c
e
i
s
f
i
n
i
t
e
.
Proof:
The
p
roof
c
a
n
b
e
f
ound
in
[52].
IV.
SIGNAL
PROCESSING
IN
CFG
D
OMAIN
A.
Overview
of
MFR
Signal
Processing
at
Two
Layers
of
Abstractions
In
the
p
revious
s
ection,
w
e
h
ave
c
onsidered
t
he
represent
a
tion
problem
w
here
the
M
FR
is
specified
a
s
a
M
a
r
k
o
v
m
o
d
u
l
a
t
e
d S
C
F
G
,
a
n
d i
t
s
p
r
o
d
u
c
t
i
o
n
r
u
l
e
s d
e
r
i
v
e
d
from
the
r
adar
words’
synt
a
c
tic
s
tructure.
I
n
t
his
a
nd
the
next
sectio
n,
we
will
deal
with
the
s
ignal
p
rocessing
of
the
MFR
s
ignal
a
nd
pre
s
ent
a
lgori
t
hms
f
or
state
a
nd
p
a
ram
e
ter
e
stim
at
i
o
n.
The
s
ignal
p
rocessing
problem
i
s
il
lustrat
e
d
i
n
F
ig
.
1
4,
whe
r
e
i
t
i
s
d
e
c
o
m
posed
i
nto
t
w
o
layers
of
abstrac
t
ions;
t
he
radar
c
ontroller
l
ayer
and
t
he
p
h
rase
scheduler.
The
high
er
layer
of
proces
sing
is
dis
cusse
d
i
n
Sect
ion
IV,
w
h
e
r
e
t
h
e
s
t
a
t
e
a
n
d
pa
r
a
m
e
te
r
e
st
i
m
at
i
o
n
a
r
e
b
o
t
h
pr
o-
ce
ss
e
d
in
th
e
C
F
G
fr
am
e
w
or
k
.
B
a
s
e
d
o
n
t
h
e
e
s
t
i
ma
te
d
ra
da
r
p
h
r
as
e
s
pa
ss
e
d
f
r
o
m
th
e
r
a
d
a
r
co
nt
ro
l
l
a
y
e
r
(h
ar
d
de
ci
si
on
i
s
a
p
p
l
i
e
d
i
n
t
h
e
pa
ss
in
g
o
f
r
ad
ar
ph
ra
s
e
s)
,
t
h
e
M
F
R

s
p
o
l
i
c
i
e
s
o
f
o
p
e
ra
t
i
o
n
i
s
e
s
ti
m
a
t
e
d
b
y
a
h
y
b
r
i
d
o
f
th
e
Vi
te
rb
i
a
n
d
th
e
i
n
s
i
d
e
a
l
g
o
r
i
t
h
m
.
I
n
a
d
d
i
t
i
o
n
,
m
a
x
i
mu
m
li
k
e
l
i
h
o
o
d
pa
ra
m
e
te
r
e
st
i
m
a
t
o
r
of
th
e
u
n
k
n
o
w
n
sy
st
e
m
pa
ra
m
e
te
rs
w
i
l
l
b
e
de
r
i
v
e
d
b
a
s
e
d
on
th
e
E
x
p
e
c
t
a
t
i
on
M
a
x
i
m
i
za
ti
on
al
g
o
r
i
th
m.
In
Sect
i
o
n
V
,
a
p
r
incip
l
ed
approach
is
discussed
t
o
d
eal
w
i
th
the
s
ignal
p
r
o
cessing
of
the
N
SE
CFG.
A
s
ynthesis
p
r
oc
e
d
ure
t
hat
c
onvert
s
a
n
N
S
E
CFG
t
o
i
ts
finite-state
count
e
rpart
i
n
p
olynomial
t
ime
i
s
i
ntroduced.
The
r
ad
a
r
control
l
er
wi
ll
be
shown
t
o
b
e
N
SE,
a
nd
its
f
ini
t
e-state
represent
a
tion
will
be
derived
.
Once
the
r
adar
controller
is
Fig.
14
.
S
i
g
n
a
l
pr
oc
e
s
s
i
n
g
in
tw
o
l
ev
el
s
o
f
a
b
s
t
r
a
c
ti
on
s,
th
e
w
o
r
d
l
ev
el
an
d
t
h
e
ph
ra
se
le
ve
l.
Vi
sn
evski
et
al.
:
S
y
n
t
a
ct
i
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
Mu
lt
if
u
n
c
t
i
o
n
R
ad
ars
V
o
l
.
9
5
,
N
o
.
5
,
May
2007
|
Proceedings
of
the
I
EEE
10
13
in
i
t
s
f
in
ite
-
stat
e
f
orm,
i
t
s
s
ig
nal
p
r
o
ce
ssi
ng
c
o
ul
d
b
e
pe
r
f
o
r
m
e
d
w
i
t
h
a
n
F
S
A
.
Following
t
he
state
s
p
a
ce
notation
introd
u
c
ed
in
the
p
r
e
v
ious
sectio
n,
l
e
t
x
0
:
n
#%
x
0
;
x
1
;
...
;
x
n
&
b
e
th
e
(
u
n
-
known)
state
s
equence,
and
#
1
:
n
#%
#
1
;#
2
;
..
.
;#
n
&
be
the
c
o
rresponding
intercept
e
d
r
adar
signal
stored
in
the
t
rack
file.
E
ach
#
k
#%
w
1
;
w
2
;
..
.
;
w
m
k
&
is
a
s
trin
g
o
f
c
on
cat
e
-
nated
t
erminal
s
ymbols
(radar
w
ords),
and
m
k
is
th
e
le
ng
t
h
o
f
#
k
.
In
ord
e
r
t
o
f
a
c
ilitate
t
h
e
d
i
scussion
of
the
e
stimation
a
l
gori
thms,
i
t
i
s
c
onve
nient
t
o
i
ntrodu
ce
the
f
ol
lowi
ng
v
a
riables:

Forward
v
ariable:
f
i
%
k
&#
P
%
#
1
;#
2
;
..
.
;#
k
;
x
k
#
e
i
&

Backward
variable:
b
i
%
k
&#
P
%
#
k
!
1
;#
k
!
2
;
...
;#
n
j
x
k
#
e
i
&

Inside
variable:
!
i
j
%
k
;
p
;
q
&#
P w
pq
j
A
j
pq
;
x
k
#
e
i
!"

Outside
v
ariable:
"
i
j
%
k
;
p
;
q
&#
P w
1
%
p
,
1
&
;
A
j
pq
;
w
%
q
!
1
&
m
j
x
k
#
e
i
!"
wh
ere
w
pq
is
t
h
e
s
ubsequence
of
t
e
rminals
f
rom
p
th
p
o
sition
of
#
i
to
q
th
position,
a
nd
A
j
pq
is
the
n
onterminal
A
j
2
N
p
wh
ich
d
eri
v
es
w
pq
,o
r
A
j
)
"
w
pq
.
F
ig.
1
5
i
ll
ustrate
s
the
p
roba
bilities
asso
ciated
with
inside
and
o
ut
side
v
a
riables.
B.
Bayesian
Estimation
of
MFR’s
State
via
Viterbi
and
Inside
Algorithms
The
e
st
i
m
ator
of
MFR’s
s
tate
at
t
i
me
k
is
^
x
k
#
arg
m
ax
i
P
%
x
k
#
e
i
j
#
1
:
n
&
and
w
hich
could
b
e
c
omputed
u
sing
the
V
iterbi
algorithm.
Define
$
i
%
k
&#
max
x
0
;
x
1
;
...
;
x
k
,
1
P
%
x
0
;
x
1
;
...
;
x
k
#
i
;#
1
;#
2
;
..
.
;#
k
&
the
V
iterbi
algorithm
c
omput
e
s
t
he
best
state
s
equence
induct
i
v
e
l
y
a
s
f
oll
o
ws:
1)
I
n
iti
a
li
zation:
$
i
%
1
&#
%
i
o
i
%
#
1
&
,f
o
r
1
+
i
+
M
.
2)
Induct
i
o
n:
$
i
%
k
!
1
&#
max
1
+
j
+
M
$
i
%
k
&
a
ji
%

&
#$
o
i
%
#
k
!
1
&
;
where
1
+
k
+
n
,
1 a
n
d
1
+
i
+
M

i
%
k
!
1
&#
arg
m
ax
1
+
j
+
M
$
i
%
k
&
a
ji
;
where
1
+
k
+
n
,
1 a
n
d
1
+
i
+
M
:
3)
T
e
rminatio
n:
^
x
n
#
arg
m
ax
1
+
j
+
M
$
j
%
n
&
4)
Pat
h
backtrac
k
i
ng:
^
x
k
#

k
!
1
%
^
x
k
!
1
&
;
k
#
n
,
1
;
n
,
2
;
..
.
;
1
where
o
i
%
#
k
&
is
the
o
utp
u
t
p
robabilit
y
of
the
s
tring
#
k
gener
ated
by
the
g
rammar
G
i
.
A
n
e
fficient
w
ay
to
calculat
e
the
p
robability
is
by
the
i
nsid
e
a
lgorithm,
a
dynamic
programming
algorithm
t
hat
i
nductively
c
a
lculates
the
probabil
ity
.
T
h
e i
n
s
i
d
e a
l
g
o
r
i
t
h
m
c
o
m
p
u
t
e
s
t
h
e p
r
o
b
a
b
i
l
i
t
y
o
i
%
#
k
&
induct
i
v
e
l
y
a
s
f
oll
o
ws:
1)
I
n
iti
a
li
zation:
!
j
%
k
;
k
&#
P
%
A
j
!
w
k
j
G
i
&
.
2)
Induct
i
o
n:
!
j
%
p
;
q
&#
X
r
;
s
X
q
,
1
d
#
p
P
%
A
j
!
A
r
A
s
&
!
r
%
p
;
d
&
!
s
%
d
!
1
;
q
&
for
8
j
,1
+
p
G
q
+
m
k
.
3)
T
e
rminatio
n:
o
i
%
#
k
&#
!
1
%
1
;
m
k
&
.
Fig.
15
.
In
si
de
a
n
d
o
u
t
s
i
d
e
pr
ob
a
b
i
l
i
t
i
e
s
i
n
S
CF
G.
Vi
snev
ski
et
al.
:
S
y
n
ta
cti
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
M
u
l
tifu
ncti
on
Rad
a
rs
1014
Pr
oce
e
di
ngs
o
f
t
he
IEEE
|
V
o
l
.9
5
,
N
o
.
5
,M
a
y
2
0
0
7
Running
both
t
h
e
V
iterbi
and
t
he
insid
e
algorit
h
ms,
t
h
e
posteriori
dist
ribut
i
on
of
the
s
tates
g
iv
e
n
the
o
bservation
could
b
e
c
omput
e
d.
C.
MFR
Parameter
Estimation
Using
EM
Algorithm
The
E
xpe
c
tat
i
on
Max
i
mi
zation
(
EM)
a
l
g
orith
m
is
a
widely
used
it
e
r
ative
n
umerical
algorit
h
m
f
or
computing
max
i
mum
l
ikeli
h
ood
p
arameter
esti
mates
o
f
p
artially
ob
served
m
o
dels.
S
uppo
se
we
have
obs
e
r
v
ations
!
n
#f
#
1
;
..
.
;#
n
g
available,
where
n
is
a
f
ix
e
d
posi
tive
integer.
L
et
f
P
(
;(
2
%
g
be
a
f
amily
of
probability
measures
o
n
%
&
;
F&
all
a
bsolutely
c
o
n
tinuous
w
ith
r
espect
to
a
f
ix
ed
p
r
oba
b
il
ity
m
e
a
sure
P
o
.
T
he
likelihood
function
for
c
omputing
an
estimate
of
the
p
arameter
(
based
o
n
t
h
e
i
n
formati
o
n
a
vail
able
in
!
n
is
L
n
%
(
&#
E
o
dP
(
dP
o
j
!
n
% &
and
t
h
e
maxi
mu
m
l
i
k
el
ihood
e
sti
m
ate
(
MLE)
is
defi
ne
d
b
y
^
(
2
arg
m
ax
(
2
%
L
n
%
(
&
:
The
E
M
a
lg
ori
t
hm
i
s
an
ite
r
ative
n
ume
r
ical
me
thod
for
comput
i
n
g
t
he
MLE.
Let
^
(
o
be
the
i
nitial
parameter
estimate.
T
he
EM
algorithm
g
enerates
a
s
equence
o
f
parameter
e
stimates
f
(
j
g
j
2
Z
!
as
follows:
Each
iteration
o
f
t
he
EM
algorit
h
m
c
onsists
o
f
t
wo
steps:
St
e
p
1)
(E
x
p
ect
a
tion-step)
Set
~
(
#
^
(
j
and
c
omp
u
te
Q
%-
;
~
(
&
,w
h
e
r
e
Q
%
(;
~
(
&#
E
~
(
log
dP
(
dP
~
(
j
!
n
"#
:
St
e
p
2)
(Maximization-st
e
p
)
F
ind
^
(
j
!
1
2
arg
m
ax
(
2
%
Q
%
(;
(
j
&
:
Using
J
ensen’s
i
nequality,
it
can
be
shown
(see
T
h
e
o
r
em
1
in
[53])
t
hat
s
equence
o
f
m
odel
est
i
mates
f
^
(
j
g
,
j
2
Z
!
from
the
E
M
a
lgorithm
are
s
uch
t
h
a
t
t
he
sequence
of
likelihoods
fL
n
%
^
(
j
&g
,
j
2
Z
!
is
m
o
notonically
increasing
with
equality
if
and
o
nly
i
f
^
(
j
!
1
#
^
(
j
.
In
Section
I
V-B,
MFR’s
s
tat
e
estimation
p
r
oblem
w
a
s
discussed.
However,
the
a
lgorit
h
m
introduced
assumes
complete
knowledge
o
f
s
ystem
p
arameters,
i.e.,
t
he
Marko
v
chain’s
t
ransition
m
atrix
a
nd
the
S
C
F
G’s
p
roduc-
t
i
on
rules.
In
reality,
such
paramet
e
rs
are
o
ft
e
n
unkno
w
n.
In
th
is
subsectio
n
,
E
M
a
lg
ori
t
hm
i
s
appl
ie
d
t
o
a
batch
o
f
noisy
r
adar
signal
in
the
t
rack
file,
a
nd
system
parameters
are
e
st
i
m
ated
it
e
r
atively.
In
EM’s
terminology,
the
i
ntercepted
radar
s
ignal
(rad
a
r
w
o
r
d
s
),
#
1
:
n
,
i
s
t
he
incomplete
observ
a
t
ion
s
e-
quence,
a
nd
we
hav
e
comp
l
e
t
e
data
if
it
is
augmented
wi
t
h
f
x
0
:
n
;
C
1
:
n
g
.
x
0
:
n
is
the
s
tate
seq
u
ence
of
the
M
FR
sy
s
t
em.
C
1
:
n
#%
C
1
;
..
.
;
C
n
&
i
s
t
h
e n
u
m
b
e
r o
f c
o
u
n
t
s e
a
c
h
p
r
od
u
c
tion
rule
is
used
to
deriv
e
#
1
:
n
,a
n
d
i
n
p
a
r
t
i
c
u
-
lar,
C
k
#%
C
1
%
A
!
'
;
#
k
&
;
C
2
%
A
!
'
;
#
k
&
;
..
.
;
C
M
%
A
!
'
;
#
k
&&
for
k
#
1
;
2
;
..
.
;
n
and
C
i
%
A
!
'
;
#
k
&
for
i
#
1
;
2
;
...
;
M
i
s
t
h
e n
u
m
b
e
r o
f
c
o
u
n
t
s g
r
a
m
m
a
r
G
i
applies
t
he
pr
o
d
u
c
t
i
o
n
r
u
l
e
A
!
'
in
deri
v
i
ng
#
k
.
D
e
note
the
p
a
r
ameter
of
int
e
rest
as
%
#f
a
ji
;
P
1
%
A
!
'
&
;
P
2
%
A
!
'
&
;
..
.
;
P
M
%
A
!
'
&g
,w
h
e
r
e
P
i
%
A
!
'
&
is
se
t
o
f
probabilities
o
f
a
ll
th
e
p
roduction
r
ules
defined
i
n
gramm
a
r
i
,
t
he
com
p
l
e
t
e
-data
l
i
k
el
ih
ood
is
written
a
s
L
n
%
(
&#
Y
n
k
#
1
P
%
#
k
;
C
k
j
x
k
;(
&
P
%
x
k
j
x
k
,
1
;(
&
P
%
x
o
j
(
&
:
In
order
t
o
f
acilitat
e
the
d
i
s
cussion
of
the
E
M
a
lgo
r
ithm
,
t
h
e
f
ollowing
two
v
ariables
are
i
ntroduced:
)
i
%
k
&#
P
%
x
k
#
e
i
j
#
1
:
n
&
#
f
i
%
k
&
b
i
%
k
&
P
3
i
#
1
f
i
%
k
&
b
i
%
k
&
*
ji
%
k
&#
P
%
x
k
#
e
j
;
x
k
!
1
#
e
i
j
#
1
:
n
&
#
f
j
%
k
&
a
ji
o
i
%
#
k
!
1
&
b
i
%
k
!
1
&
P
3
j
#
1
P
3
i
#
1
f
j
%
k
&
a
ji
o
i
%
#
k
!
1
&
b
i
%
k
!
1
&
:
T
h
e
E
xp
e
c
tation
step
of
the
E
M
a
lgorit
h
m
y
i
elds
the
foll
owing
e
quation
:
E
~
(
log
L
n
%
(
&
% &
#
X
n
k
#
1
X
x
k
X
A
x
k
X
T
x
k
E
~
(
C
x
k
%
A
!
'
;
#
k
&
% &
(
log
P
x
k
%
A
!
'
&
)
x
k
%
k
&
!
X
n
k
#
1
X
x
k
X
x
k
,
1
log
a
x
k
j
x
k
,
1
'(
*
x
k
,
1
x
k
%
k
,
1
&
!
X
n
k
#
1
X
x
0
lo
g
P
%
x
0
&
)
x
0
%
k
&
Vi
sn
evski
et
al.
:
S
y
n
t
a
ct
i
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
Mu
lt
if
u
n
c
t
i
o
n
R
ad
ars
V
o
l
.
9
5
,
N
o
.
5
,
May
2007
|
Proce
e
dings
o
f
t
he
IE
EE
1015
wh
ere
E
~
(
%
C
x
k
%
A
!
'
;
#
k
&&
can
be
comp
u
t
ed
using
i
nside
an
d
o
u
t
s
i
d
e
va
r
i
a
b
l
e
s
[
5
1
]
.
T
h
e
m
aximization
s
tep
o
f
t
h
e
EM
algorithm
c
ould
be
c
o
mpu
t
ed
by
ap
p
l
y
i
ng
Lag
r
ange
mul
t
ip
l
i
er.
S
i
n
ce
the
parameters
we
wish
to
optimize
are
i
ndependently
separated
i
nto
t
hree
t
e
rms
i
n
t
h
e
sum,
we
can
op
t
i
mize
the
p
arameter
term
by
te
rm.
T
he
estimate
s
o
f
t
he
p
r
obabilities
o
f
t
he
production
rules
can
be
derived
u
sing
t
h
e
f
irst
term
of
the
e
quation,
and
t
he
updating
equat
i
on
is
P
x
k
%
A
!
'
&#
P
n
k
#
1
E
~
(
C
x
k
%
A
!
'
;
#
k
&
% &
)
x
k
%
k
&
P
'
P
n
k
#
1
E
~
(
C
x
k
%
A
!
'
;
#
k
&
% &
)
x
k
%
k
&
:
S
i
milarly
,
the
u
pda
t
i
ng
equation
of
the
t
ransition
m
atrix
a
ji
is
a
ji
#
P
n
,
1
k
#
1
*
ji
%
k
&
P
n
,
1
k
#
1
)
j
%
k
&
U
n
d
e
r
t
h
e
condit
i
o
ns
in
[54],
i
terative
computat
i
o
ns
of
the
expectation
a
nd
maximization
s
teps
above
w
ill
p
roduc
e
a
seq
u
ence
of
parameter
e
s
t
imates
with
monotonically
nondecre
asing
l
i
k
el
ih
ood
.
F
o
r
d
e
t
ai
ls
of
th
e
n
u
m
e
r
ical
examples,
t
he
parameterization
o
f
t
h
e
Markov
chain’s
t
r
an
sitio
n
matrix
b
y
l
o
gi
stic
mode
l,
and
t
h
e
study
o
f
t
h
e
pr
e
d
i
c
t
i
ve
p
o
w
e
r
(
e
n
tr
o
p
y
)
o
f
S
C
F
G
s
,
p
l
e
a
s
e
s
e
e
[
5
5
]
.
V.
SIGNAL
PROCESSING
IN
F
I
N
I
TE- STAT
E
D
O
MAIN
I
n
th
i
s
s
e
c
t
i
o
n
,
w
e
de
a
l
w
i
t
h
t
h
e
l
o
w
e
r
l
a
y
e
r
o
f
s
i
g
n
a
l
p
r
o
-
cessing
as
de
scribed
in
Se
ction
IV-A.
Before
w
e
discuss
finite-s
tate
mo
deling
and
signa
l
proc
essing
of
syntactic
MFR
models,
we
need
to
prov
ide
some
additi
on
al
de
finitions
.
1)
CFGs
and
Production
Graphs:
The
p
roperty
o
f
s
elf-
embedding
of
CFGs
introduced
in
Section
I
I
i
s
n
ot
very
e
a
sy
to
dete
rmine
t
h
r
ough
a
s
impl
e
v
isu
a
l
i
nsp
e
cti
o
n
o
f
gramm
a
t
i
cal
product
i
on
rules.
More
precise
a
nd
form
al
t
e
chni
que
s
of
t
h
e
s
el
f-embedd
i
n
g
a
nal
y
si
s
r
el
y
o
n
t
he
co
n
c
e
p
t
o
f
C
F
G
production
graphs
.
Definition
5.1:
A
productio
n
g
r
ap
h
P
%
G
&
for
a
CFG
G
#%
A
;
E
;
!
;
S
0
&
is
a
d
irected
g
raph
whose
v
ertices
c
o
rrespond
t
o
t
he
nonterminal
s
y
m
bols
from
E
,a
n
d
t
h
e
r
e
ex
ists
an
edg
e
fro
m
verte
x
A
to
vert
e
x
B
if
and
o
nl
y
i
f
t
h
e
re
is
a
p
roduction
i
n
!
such
that
A
!
"
B
!
.
Definition
5.2:
A
labe
led
p
rod
u
c
t
ion
g
r
a
ph
P
l
%
G
&
for
a
CFG
G
#%
A
;
E
;
!
;
S
0
&
is
a
p
rodu
ction
g
raph
P
%
G
&
w
i
th
the
s
et
of
labe
ls
lab
%
!
&
defined
o
ver
t
he
set
o
f
e
dges
of
P
%
G
&
in
th
e
foll
ow
ing
w
ay:
lab
%
A
!
B
&
#
l
if
for
e
v
e
ry
A
!
"
B
!
2
!
;"
6
#
";
!
#
"
,
r
if
for
e
v
e
ry
A
!
"
B
!
2
!
;"
#
";
!
6
#
"
,
b
if
for
e
v
e
ry
A
!
"
B
!
2
!
;"
6
#
";
!
6
#
"
,
u
if
for
e
v
e
ry
A
!
"
B
!
2
!
;"
#
";
!
#
"
.
8
>
>
>
<
>
>
>
:
(21)
Note
that
the
p
rod
u
ction
g
raphs
o
f
D
ef.
5
.1
and
Def.
5.2
a
re
not
r
elat
e
d
to
FS
A
s
o
r
FSLs
d
e
scrib
e
d
e
arlier.
They
are
s
imply
u
seful
g
raphical
representations
o
f
C
FGs.
Let
u
s
c
onsider
a
n
e
xample
grammar
(reproduced
from
[56]
with
modifications)
A#
f
a
;
b
;
c
;
d
g
;
E#
f
S
;
A
;
B
;
C
;
D
;
E
;
F
g
;
!
#
S
!
DA
A
!
bE
aB
B
!
aE
j
S
C
!
bD
D
!
daC
j
a
E
!
D
j
Cc
j
aF
j
Fc
F
!
bd
8
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
:
9
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
;
:
(22)
The
l
abeled
production
graph
f
or
this
grammar
i
s
il
lu
strated
i
n
F
ig
.
1
6
.
Definition
5.3:
A
transi
tion
matr
ix
M
%
G
&
for
the
labele
d
produ
ction
grap
h
P
l
%
G
&
of
a
CFG
G
#%
A
;
E
;
!
;
S
0
&
is
an
N
(
N
matr
ix
whose
dimens
ions
are
equa
l
t
o
the
number
of
Fi
g
.
16.
P
r
o
d
u
c
t
i
o
n
g
r
a
p
h
f
o
r
th
e
g
ra
m
m
a
r
in
(2
2)
.
Vi
snev
ski
et
al.
:
S
y
n
ta
cti
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
M
u
l
tifu
ncti
on
Rad
a
rs
1016
Proc
ee
dings
o
f
t
he
IE
EE
|
V
o
l
.9
5
,
N
o
.5
,M
a
y
2
0
0
7
nont
ermina
l
symbo
ls
in
E
(num
ber
of
vert
ices
in
the
pro-
duc
tion
gra
ph),
and
who
se
eleme
nts
are
defin
ed
as
foll
ows:
m
i
;
j
%
G
&#
0 i
f
A
i
!
"
A
j
!
6
2
!
,
lab
%
A
i
!
A
j
&
if
A
i
!
"
A
j
!
2
!
.
)
(23)
The
tran
sit
ion
matr
ix
of
the
lab
eled
produ
ction
gra
ph
in
Fig.
1
6
with
respect
t
o
t
he
ver
t
ex
o
r
dering
f
S
;
A
;
B
;
C
;
D
;
E
;
F
g
has
the
foll
owing
stru
ctur
e:
M
%
G
&#
0
l
0 0
r
0 0
0 0
l
0 0
b
0
u
0 0
0 0
l
0
0 0
0 0
l
0 0
0 0
0
l
0 0 0
0 0
0
r u
0
b
0 0
0 0
0 0 0
2
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
5
:
(24
)
In
Sec
tion
V-A,
we
will
use
produc
tion
gra
phs
and
tran
sitio
n
matr
ices
in
anal
ysis
of
self
-embe
dding
prop
erty
of
CFG
.
A.
CFG-Based
Finite-State
Model
Synthesis
This
subsect
i
on
is
devoted
t
o
d
e
v
elopment
of
a
p
ro-
cedure
that
a
l
lows
to
automatically
synthesize
a
f
init
e
-
state
model
o
f
a
DES
u
sing
its
C
FG
model
a
s
a
n
i
nput.
W
e
introduce
a
theoretic
a
l
f
ramework
for
d
eterm
i
ning
whet
h
e
r
a
specified
C
F
G
of
the
s
ystem
actually
r
epresent
s
an
FSL
a
nd
provide
a
n
a
utomated
poly
nomial-time
algorit
h
m
f
or
generating
the
c
o
rresponding
FSA
.
This
synthesis
p
r
o
cedure
consists
of
four
ba
s
i
c
s
teps.
1)
Test
of
self-embedding.
A
C
F
G
that
is
deter-
mined
t
o
b
e
N
S
E
d
e
scribes
a
n
F
SL
(see
S
e
c
tion
II).
Therefore,
an
FS
A
can
b
e
sy
n
t
h
e
sized
f
rom
t
his
grammar.
2)
Grammatical
decompos
ition.
First,
t
he
NSE
CFG
i
s
b
r
o
ken
d
own
i
nto
a
set
o
f
s
impler
FSGs.
3)
Componen
t
s
yn
t
h
esis.
Once
the
g
rammar
h
a
s
be
en
decom
p
o
s
ed
i
n
to
a
s
e
t
of
simpl
e
r
g
ram-
mars,
an
FSA
can
be
sy
n
t
h
e
sized
f
or
ev
e
r
y
o
ne
of
t
h
ese
F
SGs.
4)
Co
m
p
os
iti
o
n.
Fi
nal
l
y,
th
e
c
om
pone
n
t
s
f
rom
t
he
previ
o
us
step
are
c
ombi
ne
d
t
og
eth
e
r
t
o
f
orm
a
si
ng
le
FS
A
t
h
a
t
i
s
e
qu
ival
ent
t
o
t
he
orig
inal
NSE
CFG
o
f
t
he
MFR.
Th
is
procedu
r
e
i
s
b
ase
d
on
combined
resul
t
s
p
u
b
li
she
d
i
n
[41]
an
d
[
57].
1)
Verification
of
Non-Self-Embedding:
As
was
m
entioned
earl
ie
r,
if
a
C
FG
of
a
s
yst
e
m
i
s
i
n
t
he
NSE
f
orm,
t
h
is
CFG
has
a
n
e
quiv
a
l
ent
f
inite-state
r
epresentation.
However,
given
a
n
a
rbitrarily
com
p
l
e
x
C
FG,
i
t
i
s
n
o
t
possible
t
o
verify
the
N
SE
property
of
this
grammar
b
y
s
imple
v
isual
in
sp
e
c
tion
.
In
t
h
is
section,
we
present
a
formal
verification
p
r
o
c
e
d
ure
o
f
t
he
NSE
p
ropert
y
o
f
a
n
a
rbitrary
CFG.
T
h
is
p
r
oc
edur
e
i
s
b
as
e
d
on
the
o
n
e
des
c
r
i
b
e
d
i
n
[
4
1
]
,
b
u
t
i
t
h
as
been
modified
to
suite
t
he
needs
o
f
t
he
DES
g
ramm
ars.
Let
u
s
s
tart
by
defining
the
c
on
c
e
pt
of
a
semi
-ri
n
g
[58
]
:
Definition
5.4:
A
sem
i
-ri
n
g
is
a
s
e
t
S
together
w
i
th
ad
d
i
tion
B
!
[
and
m
ult
i
plication
B
(
[
operat
i
o
ns
d
e
fined
over
the
e
lements
o
f
t
his
s
et
in
such
a
w
ay
that
they
satisfy
t
h
e
f
ollo
wing
properties:
1)
additive
associativ
i
t
y:
%8
e
;
g
;
f
2
S
&%
e
!
g
&!
f
#
e
!%
g
!
f
&
;
2
)
ad
d
i
tive
commutativ
i
t
y:
%8
e
;
g
2
S
&
e
!
g
#
g
!
e
;
3)
multiplicati
v
e
a
ssoci
ativity:
%8
e
;
g
;
f
2
S
&%
e
(
g
&
(
f
#
e
(%
g
(
f
&
;
4)
left
and
r
ight
distributivity:
%8
e
;
g
;
f
2
S
&
e
(
%
g
!
f
&#%
e
(
g
&!%
e
(
f
&
a nd
%
e
!
g
& (
f
#
%
e
(
f
&!%
g
(
f
&
.
Let
u
s
n
o
w
defi
ne
a
s
emi
r
i
n
g
o
ver
t
he
set
o
f
l
ab
e
l
s
o
f
p
r
o
d
u
c
tion
graphs
of
CFGs.
T
h
i
s
s
et
of
labels
is
int
r
o
d
u
c
ed
by
Def.
5.2
a
nd
Def.
5
.
3.
The
sum
and
p
r
od
u
c
t
operatio
ns
of
t
h
is
semiring
are
l
isted
i
n
T
able
3.
If
M
%
G
&
is
a
t
ransit
i
o
n
m
atri
x
o
f
t
h
e
C
F
G
G
(see
Def.
5.3
)
,
t
hen,
using
t
he
sem
i
ring
operations
of
Table
3
,
we
define
the
s
teady-state
m
atrix
o
f
t
h
e
production
graph
of
this
grammar
a
s
M
+
N
%
G
&#
X
N
i
#
1
M
%
G
&
) *
i
(2
5
)
wh
ere
N
is
th
e
d
ime
n
si
on
of
the
t
ransi
t
ion
m
atrix
M
%
G
&
.
[41]
h
a
ve
proven
that
if
di
ag
M
+
N
%
G
&
#$
d
o
e
s
not
c
ontai
n
labe
ls
B
b
,
[
the
c
orrespond
i
ng
grammar
G
is
NS
E
.
This
demonstrates
that
the
N
SE
propert
y
of
a
C
FG
can
be
verified
in
polynomial
t
i
me.
T
o
i
l
l
u
strate
t
h
e
a
pp
l
i
cation
o
f
t
h
i
s
a
lg
orith
m
,
l
et
u
s
revisit
t
he
example
CFG
(
22).
T
he
labeled
p
roduct
i
o
n
graph
f
or
thi
s
gram
mar
i
s
s
hown
i
n
F
i
g
.
16,
a
nd
th
e
t
r
ansition
matrix
of
this
grap
h
i
s
g
iven
by
(24
)
.
Table
3
Semiring
Operations
of
Sum
and
Product
Vi
sn
evski
et
al.
:
S
y
n
t
a
ct
i
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
Mu
lt
if
u
n
c
t
i
o
n
R
ad
ars
V
o
l
.
9
5
,
N
o
.
5
,
May
2007
|
Proceedi
n
gs
of
the
I
EEE
1017
dia
g
)
M
+
N
%
G
&*
#
)
l
;
l
;
l
;
l
;
l
;
0
;
0
*
,t
h
e
r
e
f
o
r
e
,C
F
G
(
2
2
)
is
NSE.
In
th
e
r
e
m
ai
nder
of
th
is
se
ction,
we
wi
ll
descri
be
a
t
h
ree-step
procedure
t
hat
accepts
a
n
N
SE
CFG
a
nd
a
u
tomatically
synthesizes
a
n
F
SA
that
is
eq
u
i
v
a
lent
to
t
h
is
grammar.
2)
Grammatical
Decomposition:
We
start
b
y
i
ntroducing
t
h
e
c
oncept
of
the
g
rammatical
.
-
co
m
p
osition
.
Definition
5.5:
If
G
1
#%
A
1
;
E
1
;
!
1
;
S
1
&
and
G
2
#%
A
2
;
E
2
;
!
2
;
S
2
&
are
t
wo
C
F
Gs
with
E
1
\E
2
#;
an
d
E
1
\A
2
#;
,
th
e
n
.
-
co
m
p
osition
o
f
t
h
e
s
e
g
r
a
m
m
a
r
s i
s d
e
f
i
n
e
d
a
s
G
#
G
1
.
G
2
#%
A
;
E
;
!
;
S
&
wh
ere
A#A
1
n E
2
[ A
2
,
E#
E
1
[ E
2
,
!
#
!
1
[
!
2
,a
n
d
S
#
S
1
.
[
4
1
]
h
a
v
e
de
m
o
n
s
tr
a
t
e
d
t
h
a
t
f
o
r
a
n
y
N
S
E
C
F
G
G
there
ex
i
s
t
n
FSGs
G
1
;
G
2
;
..
.
;
G
n
s uc
h
t hat
G
#
G
1
.
G
2
.
..
.
.
G
n
.
T
hey
h
ave
a
lso
s
hown
that
every
F
SG
G
i
of
this
decomposit
i
o
n
i
s
e
quiv
a
l
ent
t
o
s
ome
s
trongly
c
o
nnected
component
o
f
t
h
e
product
i
on
grap
h
P
%
G
&
.
T
h
e
g
rammatical
decomposition
p
rocedure
consists
of
t
h
e
f
ollowing
steps.
1)
Let
P
1
%
G
&
;
P
2
%
G
&
;
..
.
;
P
n
%
G
&
be
n
s t r o
ngl y
connected
c
omponent
s
o
f
t
he
p
r
oduction
graph
P
%
G
&
.T
h
e
n
E
i
of
the
F
S
G
G
i
i
s
t
h
e s
a
m
e a
s t
h
e
s
e
t
of
vertices
in
P
i
%
G
&
.
2)
T
h
e
s
et
of
terminal
symbols
o
f
t
he
FSG
G
i
is
fou
n
d
t
h
rough
t
he
following
r
ecursive
relationship:
A
n
#A
A
n
,
1
#A
[
E
n
.
.
.
A
2
#A
[
E
3
[
..
.
[ E
n
A
1
#A
[
E
2
[
..
.
[ E
n
3
)
The
s
et
of
gram
matical
production
rules
!
i
$
!
is
defined
a
s
!
i
#f
A
!
"
j
A
2 E
i
g
.
4)
Finally,
t
h
e
s
tart
sy
m
b
ol
S
1
for
t
he
FSG
G
1
is
chosen
as
S
0
o
f
th
e
o
r
i
g
i
n
a
l
N
S
E
C
F
G
G
,a
n
d
S
i
for
i
#
2
;
...
;
n
is
chosen
to
be
an
arbitrary
n
ont
e
r-
minal
f
rom
t
he
co
rresponding
set
E
i
.
One
o
f
t
he
most
efficient
p
rocedures
t
o
d
ecom
p
o
se
a
d
i
re
cted
grap
h
i
nto
a
set
o
f
s
t
r
ongl
y
c
onne
cted
components
invol
v
es
Dulmag
e

Me
nde
l
soh
n
de
compositi
o
n
[
59]
o
f
t
h
e
p
r
oduction
grap
h

s
a
djac
e
n
cy
matrix.
T
his
d
ecomposition
finds
a
perm
utation
o
f
t
he
v
e
rtex
ordering
that
renders
t
he
a
d
jacency
m
atrix
i
nto
u
p
p
er
block
t
riangular
f
orm.
Each
bl
o
c
k
t
r
i
a
n
g
u
l
a
r
c
o
m
po
n
e
n
t
o
f
t
h
e
t
r
a
n
s
f
o
r
m
e
d
a
d
j
a
ce
n
c
y
mat
r
ix
correspond
s
to
a
s
trongly
c
onnected
comp
o
n
ent
o
f
the
p
roduction
g
raph.
No
w
c
on
sider
a
n
example
of
the
d
ecom
po
s
i
tion
procedu
r
e
a
pplied
t
o
g
rammar
(
22).
I
ts
p
r
oduction
graph
includes
four
strongly
connected
comp
o
n
ents
shown
i
n
Fi
g.
17.
T
h
e
four
FS
G
c
omp
o
ne
nts
o
f
t
h
i
s
C
F
G
are
G
1
#
A
1
#f
a
;
b
;
c
;
d
;
C
;
D
;
E
g
;
E
1
#f
S
;
A
;
B
g
;
!
1
#
S
!
DA
;
A
!
bEa
B
;
B
!
aE
j
S
8
>
<
>
:
9
>
=
>
;
;
S
1
#
S
0
B
B
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
C
C
A
;
(26a)
G
2
#
A
2
#f
a
;
b
;
c
;
d
;
C
;
D
;
F
g
;
E
2
#f
E
g
;
!
2
#f
E
!
D
j
Cc
j
aF
j
Fc
g
;
S
2
#
E
0
B
B
B
@
1
C
C
C
A
;
(26b)
G
3
#
A
3
#f
a
;
b
;
c
;
d
g
;
E
3
#f
C
;
D
g
;
!
3
#
C
!
bD
;
D
!
da
C
j
a
( )
;
S
3
#f
X
j
X
2 E
3
g
0
B
B
B
B
B
B
B
@
1
C
C
C
C
C
C
C
A
;
(26c)
G
4
#
A
4
#f
a
;
b
;
c
;
d
g
;
E
4
#f
F
g
;
!
4
#f
F
!
bd
g
;
qS
4
#
F
0
B
B
B
@
1
C
C
C
A
(26d)
where,
in
(26c),
X
may
r
epre
sent
ei
ther
of
the
n
on-
terminal
s
c
ontained
in
E
3
.
Fi
g
.
17
.
St
ro
ng
ly
co
nn
ec
te
d
c
om
po
ne
nt
s
o
f
t
h
e
pr
od
uc
ti
on
g
r
a
p
h
sh
ow
n
i
n
F
i
g
.
1
6
.
Vi
snev
ski
et
al.
:
S
y
n
ta
cti
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
M
u
l
tifu
ncti
on
Rad
a
rs
1018
Proc
ee
dings
o
f
t
he
IE
EE
|
V
o
l
.9
5
,
N
o
.5
,M
a
y
2
0
0
7
3)
Synthesis
of
Finite-State
Components:
The
n
ext
s
tep
in
v
o
lves
synthe
sis
o
f
i
n
d
i
v
idual
F
SAs
f
or
e
ach
of
the
F
S
G
s
G
1
;
G
2
;
...
;
G
n
ob
ta
ined
at
t
h
e
s
tep
o
f
g
r
a
m
m
at
ic
al
dec
o
mpos
it
ion.
This
is
a
s
traightfor
war
d
mechanical
procedu
r
e
w
e
l
l
d
escri
b
ed
i
n
the
l
i
t
eratu
r
e
[
23],
[
2
5
]
,
[26
]
,
[
35]
,
[40
]
,
[
42].
The
F
SA
for
t
he
exam
p
l
e
g
rammars
(26)
are
s
hown
in
F
i
g
.
18.
T
h
e
y
h
ave
t
h
e
foll
ow
ing
s
tructu
re:
#
1
#
$
1
#A
1
;
Q
1
#E
1
[
Q
0
1
;
$
1
#
$
%
!
1
&
;
q
0
#f
S
g
;
F
1
#
H
2
Q
0
1
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
;
(27a
)
#
2
#
$
2
#A
2
;
Q
2
#E
2
[
Q
0
2
;
$
2
#
$
%
!
2
&
;
q
0
#f
E
g
;
F
2
#
H
2
Q
0
2
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
;
(27b
)
#
3
#
$
3
#A
3
;
Q
3
#E
3
[
Q
0
3
;
$
3
#
$
%
!
3
&
;
q
0
#f
X
gj
X
2 E
3
f g
;
F
3
#
H
2
Q
0
3
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
;
(27c
)
#
4
#
$
4
#A
4
;
Q
4
#E
4
[
Q
0
4
;
$
4
#
$
%
!
4
&
;
q
0
#f
F
g
;
F
4
#
H
2
Q
0
4
0
B
B
B
B
B
B
@
1
C
C
C
C
C
C
A
(2
7
d
)
wh
ere
Q
0
i
are
t
he
set
s
of
int
e
rmediate
states
required
for
construction
of
the
a
ut
omaton
i
that
are
n
ot
present
i
n
t
he
set
o
f
n
onterminals
o
f
t
he
c
o
rresponding
grammar
G
i
.
N
o
t
e
t
h
a
t
w
e
pr
e
s
e
n
t
h
e
r
e
s
i
m
p
l
i
f
i
e
d
v
e
r
s
i
o
n
s
o
f
F
SA
s
highlighting
only
the
i
mportant
structural
aspects.
Specif-
ically,
i
n
F
ig.
1
8(c)
and
(
d)
transitions
l
abeled
with
nonterminals
B
da
[
and
B
bd
,
[
when
rigo
rously
treated,
require
t
h
e
insertion
o
f
a
n
i
nterm
e
diat
e
s
t
a
te.
A
lso,
before
constructing
an
FSA,
the
g
rammar
should
have
been
conve
r
ted
t
o
t
he
unit
product
i
on
free
form
so
that
eve
r
y
edge
in
the
g
raphs
i
n
F
ig.
1
8
c
orresp
o
n
ds
t
o
t
h
e
generat
i
o
n
of
a
s
in
gl
e
n
onte
rmin
al.
W
e
w
il
l
s
uppress
t
h
e
se
in
terme
-
d
i
ate
s
teps
in
the
r
est
o
f
t
his
p
aper,
a
nd
,
w
ithout
loss
of
g
e
n
e
r
a
l
i
t
y
,
w
i
l
l
a
do
pt
a
s
l
i
g
h
t
l
y
s
i
m
pl
i
f
i
e
d
f
o
r
m
o
f
F
SA
r
e
p
r
e
s
e
n
ta
t
i
o
n
.
4)
Composition
of
the
FSA:
T
h
e
f
i
n
a
l
s
t
e
p
o
f
th
e
F
SA
sy
n
t
hesis
p
rocedure
for
a
given
N
SE
CFG
i
nv
o
l
ves
c
om-
p
o
sition
of
the
F
S
A
components
obt
a
ined
at
the
p
revious
st
e
p
.
A
recursiv
e
B
depth-first
[
algorithm
t
hat
p
e
r
forms
t
h
i
s
ope
r
ati
o
n
w
as
d
e
ve
lope
d
b
y
[
57]
a
nd
it
s
m
odi
f
ie
d
v
e
rsion
was
p
resented
by
[56].
H
ere
w
e
p
resent
an
alternative,
B
breadth-first
[
algorithm.
T
h
e
F
SA
comp
o
s
ition
p
r
o
cedure
is
formalized
in
t
e
rms
of
two
a
lgorithms
p
resented
below.
T
h
e
m
ain
p
rocedure
B
createFSA
[
i
n
iti
a
li
zes
t
h
e
compositi
o
n
o
pe
ration
and
calls
the
f
unct
i
o
n
B
ex
pandFS
A
[
that
p
e
rforms
the
actual
F
S
A
c
ompositi
on.
W
e
w
il
l
i
llu
strate
th
is
proce
d
ure
b
y
a
n
example
c
omposing
FS
A
c
omponents
s
how
n
in
Fig.
18.
Algori
thm
1
Proced
ure
B
c
r
e
a
te
F
S
A
[
creates
a
n
F
SA
1)
procedure
cr
eate
FSA
%
S
0
;
#
1
;
..
.
;
#
n
&
.
Create
s
an
FSA
f
rom
c
omponents
#
1
;
...
;
#
n
2)
$
f
S
0
g
.
Init
i
a
li
zing
set
o
f
t
ransit
i
o
ns
3)
Q
f
q
0
;
H
g
.
Addi
ng
in
t
e
rme
d
iate
states
4)
$
f
$
%
q
0
;
S
0
&#
f
H
gg
.
Adding
in
itial
tr
a
n
s
i
t
i
o
n
5)
q
0

q
0
.
Set
t
ing
i
nitial
state
6)
F
f
H
g
.
Setting
terminal
states
7)
#
%
$
;
Q
;$
;
q
0
;
F
&
.
Initi
a
li
zing
th
e
F
S
A
8)
#

ex
pandFSA
%
#
;
#
1
;
...
;
#
n
&
.
Call
in
g
t
h
e
expansion
p
roced
u
re
9)
end
p
rocedu
re
Algori
thm
2
Fu
nct
i
on
B
exp
a
n
dFSA
[
inserts
F
SA
compo-
nents
i
nto
t
he
FSA
#
1)
func
t
i
o
n
ex
pand
FSA
%
#
;
#
1
;
..
.
;
#
n
&
.
Inserts
FSA
c
om
pone
nt
s
i
n
t
o
#
2)
for
a
l
l
#
i
and
"
2
$
do
3)
if
"
2
Q
i
the
n
.
If
transition
matches
a
state
4)
q
fr
om

arg
q
%
$
%
q
"
;"
&&
.
Saving
from
stat
e
F
i
g.
1
8
.
Co
m
p
o
n
e
n
t
s
of
th
e
F
S
A
fo
r
t
h
e
gr
am
ma
r
(
2
2
)
.
St
at
es
th
at
ar
e
n
o
t
l
a
b
e
l
e
d
i
n
t
h
e
s
e
g
r
a
p
h
s
ar
e
c
on
si
de
re
d
i
n
t
e
r
m
e
d
i
a
t
e
an
d
h
a
v
e
n
o
d
i
r
e
c
t
r
ep
re
s
e
n
t
a
t
i
o
n
i
n
t
h
e
se
t
o
f
n
o
n
t
e
r
m
in
al
s
o
f
t
h
e
g
r
a
m
m
a
r
s
.
(
a)
Co
r
r
e
s
po
nd
s
t
o
t
h
e
g
r
a
m
m
a
r
(
26
a)
.
(
b
)
Co
r
r
e
s
p
o
n
d
s
to
th
e
g
ra
mm
ar
(
2
6
b
)
.
(
c
)
C
o
r
r
e
sp
on
ds
to
th
e
g
r
a
m
m
ar
(2
6c
).
(
d
)
C
or
re
sp
on
ds
to
th
e
g
r
a
m
m
a
r
(2
6d
)
.
Vi
sn
evski
et
al.
:
S
y
n
t
a
ct
i
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
Mu
lt
if
u
n
c
t
i
o
n
R
ad
ars
V
o
l
.
9
5
,
N
o
.
5
,
May
2007
|
Proc
eedings
of
the
I
EEE
10
19
5)
q
to

$
%
q
from
;"
&
.
Sav
i
ng
to
state
6)
$

$
n
"
[
$
i
.
Expanding
s
et
of
transitions
7)
Q

Q
[
Q
i
.
Expand
i
n
g
s
et
of
states
8)
$

$
[
$
i
.
App
e
nding
t
ransition
structure
9)
fo
r
a
ll
$
%
q
j
;!
&#
#
q
fr
om
do
10
)
$
%
q
j
;!
&
".
Rerouting
q
fr
om
inpu
ts
11
)
end
f
or
12
)
for
a
l
l
$
%
q
j
;!
& 2
F
i
do
13
)
$
%
q
j
;!
&
q
to
.
Rerouting
q
to
input
s
14
)
en
d
f
or
15)
Q

Q
n
F
i
.
Removing
term
.
states
of
#
i
16
)
Q

Q
n
q
fr
om
.
Removing
q
fr
om
state
17
)
en
d
i
f
18
)
end
f
or
19
)
re
tur
n
#
.
Returning
t
he
com
p
l
e
t
e
FSA
#
20)
end
f
u
n
c
t
ion
T
h
e
i
nitialization
s
tep
i
nvolves
c
rea
t
ion
o
f
a
B
du
mmy
[
FSA
t
hat
c
ontai
n
s
t
wo
states
V
an
intermed
i
a
te
st
a
t
e
a
nd
a
t
e
rminal
state.
It
also
contains
one
t
ransition
f
rom
t
he
intermed
i
a
te
state
t
o
t
h
e
terminal
state
o
n
s
ymbol
S
0
#
S
from
the
o
riginal
N
SE
CFG
(
22
).
T
h
is
FSA
i
s
s
hown
in
Fig.
19(a).
Th
e
f
u
n
c
t
i
o
n
B
ex
pandFS
A
[
accept
s
t
h
e
B
dummy
[
FS
A
a
s
wel
l
as
four
FSA
c
ompon
e
nts
s
hown
i
n
F
i
g.
18.
I
t
t
hen
tr
a
n
s
f
o
r
m
s
t
h
e
B
dummy
[
FSA
i
nto
a
real
aut
o
maton
b
y
c
o
nsec
u
t
ively
i
nserting
FS
A
c
om
p
o
nents
a
nd
rewirin
g
t
r
ansitio
n
s.
T
h
e
s
t
e
p-by-step
c
omposition
procedu
r
e
i
s
i
llustrated
in
Fig.
19(b)–(e).
First,
t
he
FS
A
s
hown
in
Fig.
18(a)
i
s
inserted
int
o
the
F
SA
in
Fig.
19(a)
i
nstea
d
of
the
t
ransition
labe
le
d
S
.
T
he
resulting
i
ntermed
i
ate
F
SA
is
shown
i
n
Fig.
19(b).
Next,
a
ll
the
E
-tran
s
itio
ns
are
r
e
p
l
aced
w
ith
t
h
e
F
S
A
i
n
F
ig
.
1
8(b).
T
h
e
re
sult
i
n
g
i
nte
r
medi
ate
F
SA
is
sh
own
in
Fig.
19(c).
Then,
a
ll
the
F
-transitions
are
r
eplaced
w
ith
th
e
F
S
A
i
n
F
i
g
.
1
8
(
d
)
.
T
h
e
r
e
s
u
l
ti
n
g
i
n
t
e
r
m
e
d
i
a
te
F
S
A
i
s
sh
own
i
n
F
ig.
1
9
(
d).
F
inal
ly,
a
ll
the
C
- a
n
d
D
-transitions
are
r
ep
l
aced
w
ith
t
he
FS
A
i
n
F
ig.
1
8(c).
T
he
final
a
uto-
mato
n
e
qui
v
al
ent
t
o
t
he
g
r
am
mar
(
22)
is
s
h
own
i
n
Fig.
19(e).
Note
t
h
at
the
l
ab
e
l
s
o
f
t
he
states
in
Fig.
19
are
n
ot
t
h
e
u
nique
s
t
a
te
identifiers.
T
h
ese
l
abels
a
re
shown
to
illustrate
the
c
omposition
procedure
a
nd
t
o
provid
e
linkage
w
ith
t
he
states
o
f
the
o
riginal
F
S
A
shown
i
n
F
ig.
1
8.
B.
State
Machine
Synthesis
of
the
Mercury
Radar
Controller
As
s
t
a
t
ed
in
[23
]
,
t
he
an
alysis
of
st
ocha
stic
and
weighted
grammar
s
must
be
performed
u
s
i
n
g
their
characterist
i
c
c
ounterp
a
rts.
However,
since
t
he
character-
isti
c
g
ra
m
m
ar
i
s
so
cl
ose
t
o
t
he
orig
inal
det
e
rminist
i
c
grammar,
we
can
perform
t
h
e
NSE
p
r
o
perty
t
e
s
t
d
irectly
on
the
d
et
e
r
minist
i
c
C
F
G.
Following
t
he
verificat
i
o
n
p
rocedures
a
s
d
escribed
in
Section
V
-A1,
the
d
iagonal
e
lements
o
f
t
he
steady-stat
e
analysis
matrix
can
be
shown
t
o
b
e
d
i
ag
M
+
N
%
G
&
'(
#)
o
o
l l
l l l
l l
o
..
.
o
*
T
and
w
hich
confirms
that
the
M
ercury’s
radar
c
ontroller
grammar
i
s
a
n
N
SE
CFG.
The
r
efore,
an
FS
A
o
f
t
he
radar
controller
can
be
synthesized
f
rom
t
h
e
grammar
o
f
F
ig.
1
3.
Using
t
he
D
u
l
m
ag
e–M
e
ndel
sohn
d
e
compositi
o
n,
29
strongly
connect
e
d
c
omponents
o
f
t
h
e
production
graph
o
f
the
C
F
G
of
the
M
ercury
emitter
w
ere
o
btained
,
and
t
he
product
i
on
g
r
aph
i
s
i
ll
ustrat
ed
in
Fi
g.
20
.
A
s
s
hown
in
Section
I
I,
each
strongly
connected
component
o
f
t
he
product
i
on
graph
c
orresp
o
n
ds
to
an
FSG.
FSAs
for
e
ach
of
these
F
SGs
a
re
show
n
i
n
F
igs.
21
and
22.
The
c
omp
l
et
e
state
m
achine
o
f
t
he
Mercury
e
mitter
can
be
obtained
by
applying
the
F
SA
composit
i
o
n
o
perat
i
on.
3
The
f
inal
step
of
the
s
ynthesis
p
r
ocedure
i
nvolv
e
s
transformation
of
the
d
eterministic
state
m
achine
o
f
t
he
Mercury
e
mitter
i
nto
a
stochastic
mod
e
l,
taking
into
account
t
he
probability
d
i
s
tributions
d
e
termined
by
the
structure
o
f
t
he
st
o
c
h
a
stic
gramm
a
r
s
hown
in
Fig.
13.
A
t
this
st
a
ge,
the
p
robabilistic
elem
ent
s
of
t
h
e
p
roblem
that
led
t
o
t
he
development
o
f
t
he
stochastic
radar
g
rammar
[
e
.g
.,
t
h
e
c
han
n
e
l
impai
r
men
t
probabil
ity
d
i
s
tribution
(
19
)]
are
b
rought
into
the
s
tructure
of
the
r
ad
a
r
model.
This
conversion
procedure
i
s
i
llustrated
in
Section
I
I
i
n
(12)–(15).
T
h
e
p
rocedure
is
essentially
a
simple
mechan-
ical
op
e
r
ation
o
f
c
onverting
t
he
Mealy
s
tate
machine
t
o
t
he
Moor
automaton
a
nd
assigning
t
he
probabilit
i
e
s
o
f
t
ran-
sitions
a
s
s
hown
in
(1
3),
a
nd
the
p
robabilities
of
ob-
servations
as
demonstrated
in
(15).
C.
Finite-State
Signal
Processing
The
f
inite-state
m
odel
of
the
r
adar
controller
grammar
is
b
a
sed
o
n
t
h
e
HMM.
Once
th
is
mode
l
i
s
o
bt
ained,
th
e
Fig.
19
.
Sy
nt
he
si
s
p
r
o
ce
du
re
o
f
th
e
F
S
A
f
o
r
t
h
e
ex
am
pl
e
C
FG
(
2
2
)
.
3
Due
to
the
very
large
size
of
the
final
Mercury
state
machine,
we
do
not
include
it
in
this
paper.
Vi
snev
ski
et
al.
:
S
y
n
ta
cti
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
M
u
l
tifu
ncti
on
Rad
a
rs
1020
Proc
ee
dings
o
f
t
he
IEEE
|
V
o
l
.9
5
,
N
o
.5
,M
a
y
2
0
0
7
fo
r
w
ard
-
ba
ck
w
a
rd
algorithm
a
lso
k
nown
as
HMM-filter
can
be
ap
p
l
ied
t
o
s
tatistic
a
l
signal
processing
based
o
n
t
h
i
s
model
.
This
al
gori
thm
i
s
w
ell
-
known
a
nd
stud
i
e
d
i
n
d
et
ail
in
[27]
and
[
44].
VI.
CON
C
L
U
S
I
ON
The
m
ain
i
dea
o
f
t
his
p
a
p
er
is
to
model
a
nd
c
h
aracterize
M
F
R
s
a
s
a s
t
r
i
n
g g
e
n
e
r
a
t
i
n
g d
e
v
i
c
e
,
w
h
e
r
e t
h
e
c
o
n
t
r
o
l r
u
l
e
s
are
s
pe
cifi
ed
in
te
rms
o
f
a
n
S
CFG.
Th
is
is
unli
ke
model
i
ng
of
targets,
where
h
idd
e
n
M
arkov
and
s
tate
space
models
are
a
d
e
q
u
a
t
e
[
1
4
]
,[
2
4
]
.
T
h
e t
h
r
e
a
t o
f
t
h
e
M
F
R i
s r
e
c
o
g
n
i
z
e
d a
s
i
t
s
p
ol
icy
o
f
o
pe
ration,
a
nd
it
is
mode
le
d
a
s
a
Markov
chai
n
modulating
the
p
robabilit
i
es
of
the
M
FR’s
production
rules.
The
p
aper
shows
h
ow
a
l
arge
scale
d
ynamical
system
such
as
MFRs
is
expressed
b
y
a
com
p
a
c
t
r
epresentation,
and
d
e
m
on
stra
t
e
s
t
he
fl
e
x
i
b
il
ity
o
f
t
r
a
n
s
lati
ng
from
o
n
e
r
e
p
r
e
s
e
n
t
a
ti
o
n
to
a
n
o
t
h
e
r
i
f
t
h
e
s
e
l
f
-
e
m
b
e
d
d
i
n
g
pr
o
p
e
r
t
y
i
s
satisfied.
Based
o
n
t
h
e
SCFG
representat
i
on,
a
maxim
u
m
l
i
kel
i
hood
se
quence
esti
mator
i
s
d
eri
v
ed
to
e
v
alu
a
te
t
h
e
threat
poses
b
y
t
he
MFR
,
and
a
m
a
ximum
l
ikeliho
o
d
parameter
e
stimator
is
d
e
riv
e
d
t
o
i
nfer
the
u
nknown
syst
e
m
parameters
with
the
E
M
a
lgorithm.
S
ince
SCFGs
are
m
ultitype
Galton–Watson
b
ranching
p
r
ocesses,
t
h
e
algorit
h
ms
p
r
op
o
s
ed
in
this
paper
can
be
viewed
as
filt
e
r
ing
and
e
sti
m
atio
n
o
f
a
partial
l
y
o
bse
r
v
e
d
m
ulti
t
y
pe
Galton

Watson
branching
p
rocesses.
For
d
e
t
ails
of
numerical
ex
ampl
es
o
f
the
c
onstru
cted
mo
d
e
l
,
an
d
t
he
study
o
f
t
h
e
p
r
e
d
i
c
ti
v
e
po
w
e
r
(
e
n
t
r
o
p
y)
s
t
u
d
y
,
pl
e
a
s
e
s
e
e
[
5
5
]
.
S
e
v
e
r
a
l e
x
t
e
n
s
i
o
n
s o
f t
h
e
i
d
e
a
s i
n t
h
i
s p
a
p
e
r
a
r
e w
o
r
t
h
consid
e
r
ing.
1
)
The
algorithm
s
studied
in
this
paper
are
inhere
ntly
off
-line.
It
is
of
interest
to
study
stocha
stic
ap-
proxim
ation
al
gorithms
for
adaptive
learnin
g
o
f
the
MFR
grammar
and
real-
time
evaluati
on
of
the
thre
at.
For
HMMs,
such
real-t
ime
stat
e
and
para
m-
eter
e
stimatio
n
a
l
gorithms
are
well
known
[60]
.
2)
I
n
t
h
i
s
p
a
pe
r
w
e
h
a
v
e
c
o
n
s
t
r
u
c
t
e
d
SC
F
G
m
o
d
e
l
s
for
t
he
MFR
r
adar
as
it
responds
to
the
d
ynamics
o
f
a
t
a
r
get.
R
ecall
f
rom
F
ig.
6
that
in
this
paper
w
e
a
re
interest
e
d
in
electronic
warfare
f
rom
t
he
target’s
p
o
int
o
f
v
ie
w.
An
i
n
tere
stin
g
e
xte
n
si
on
of
thi
s
pa-
p
e
r
t
hat
w
e
a
re
cu
rrently
conside
r
ing
i
s
o
pti
m
izi
n
g
t
h
e
t
r
a
jectory
o
f
t
he
target
t
o
maximize
the
a
mo
unt
of
information
o
btained
f
rom
t
he
CFG.
Such
t
r
ajectory
opt
i
mizat
i
o
n
can
be
formulated
as
a
st
o
c
hastic
con
t
r
o
l
p
roble
m
in
volvin
g
a
n
S
CFG
(
or
equivalent
l
y
a
G
alton–Watson
branching
p
rocess).
3)
The
S
CFG
s
ignal
p
rocessing
algo
rithm
s
presented
in
this
paper
c
onsider
a
n
i
id
channel
i
mpairment
mode
l.
It
i
s
important
t
o
e
x
t
e
n
d
t
hi
s
t
o
R
ayle
ig
h
fading
channels.
S
equential
M
ont
e
C
a
rlo
m
ethod,
such
as
p
a
rticle
filtering,
can
be
applied
t
o
c
ope
with
fading
channel.
4
)
In
th
is
pape
r
w
e
d
eal
e
xclu
sive
ly
wi
th
opti
mizi
ng
t
h
e
r
adar
signal
at
the
w
ord
l
evel.
A
n
a
logous
to
cross
l
ayer
optimization
o
n
c
ommunication
s
yst
e
m
[61],
c
ross
layer
o
ptimization
can
be
applied
t
o
rad
a
r
s
ignal
p
rocessing
at
the
p
u
l
se
and
w
ord
l
ev
e
l
.
5)
Th
is
paper
d
eal
s
ex
clu
s
ive
l
y
w
ith
m
odel
ing
a
nd
i
d
e
n
t
i
f
y
i
n
g M
F
R
s i
n o
p
e
n l
o
o
p
.T
h
a
t i
s
,
w
e
d
o
n
o
t
m
o
d
e
l t
h
e
M
F
R a
s a f
e
e
d
b
a
c
k
s
y
s
t
e
m
w
h
i
c
h
optimizes
i
t
s
task
allocation
i
n
r
esponse
t
o
t
he
targ
et.
S
ee
[62]
for
a
Lag
r
ang
i
an
M
D
P
f
ormulat
i
on
of
radar
r
esource
a
llocatio
n.
Mo
d
e
ling,
i
dentify-
ing,
and
e
stimating
a
n
M
FR
in
closed
loop
is
a
chal
le
ng
in
g
t
ask
a
nd
wi
ll
require
s
ophi
sticated
real
t
i
me
p
r
ocessing
(See
point
1
above).
h
F
i
g.
20
.
P
r
o
d
u
c
t
i
o
n
g
r
a
p
h
o
f
t
h
e
Me
rc
ur
y
g
r
a
mm
ar
.
Vi
sn
evski
et
al.
:
S
y
n
t
a
ct
i
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
Mu
lt
if
u
n
c
t
i
o
n
R
ad
ars
V
o
l
.
9
5
,
N
o
.
5
,
May
2007
|
Proceed
ings
of
the
I
EEE
1021
APPENDIX
A
.
Functional
Specification
of
the
B
Mercury
[
Emitter
T
h
is
appendix
contains
a
s
anitiz
e
d
version
of
a
t
extual
inte
ll
ig
e
n
ce
re
port
de
scribi
ng
t
h
e
f
unctio
nali
ty
of
th
e
emitter
called
B
Mercury
[
.
4
1)
General
Remarks:
T
h
e t
i
m
i
n
g o
f t
h
i
s e
m
i
t
t
e
r i
s b
a
s
e
d
on
a
c
rystal-controlled
c
lock.
E
ach
cycle
o
f
t
h
e
clock
i
s
known
a
s
a
crystal
c
ount
(X
c
)
and
t
he
assoc
i
ated
time
inte
rval
is
t
h
e
c
l
o
ck
pe
riod.
A
l
l
le
adin
g
e
dg
e
e
missi
on
t
i
mes
a
nd
dead-times
can
be
measured
in
crystal
c
ounts
(i
nteg
er
mul
t
i
p
le
s
o
f
t
he
cl
ock
p
e
r
iod).
Most
of
the
i
nformation
below
r
elat
e
s
to
search,
a
c
quisi
t
i
o
n,
and
t
racking
f
u
n
c
t
ions
onl
y
.
M
issi
le
engag
e
-
ment
mod
e
s
(
launching,
guidance,
a
nd
fusing)
can
also
be
fit
i
nto
t
h
e
st
ructure
b
elow,
b
ut
with
some
modifications.
2)
Radar
Words:
The
t
iming
o
f
t
his
e
mitter
i
s
d
ictated
b
y
a
s
ubst
ructure
called
a
word
.
W
or
d
s
occur
s
equentially
i
n
t
h
e
p
u
l
s
e
t
r
a
i
n
s
o
t
h
a
t
o
n
e
w
o
r
d
be
g
i
n
s
a
s
th
e
p
r
e
v
i
o
u
s
word
is
ending.
T
here
are
n
ine
d
istinct
w
ords,
d
enoted
by
w
1
;
..
.
;
w
9
.
E
ach
has
t
he
same
l
e
ngth
(on
t
h
e
orde
r
o
f
several
m
illiseconds),
and
i
s
a
ssociated
with
a
f
ixed
integer
number
of
crystal
c
ount
s
.
For
t
he
purpose
o
f
t
his
p
aper
we
will
consider
all
w
ords
of
the
r
adar
d
i
stinguishable
f
rom
each
other.
3)
Time-Division
Multiplexing
V
Phrases
and
Clauses:
This
system
is
an
MFR
capab
l
e
of
engaging
five
targets
i
n
a
time-multiplexed
fashion
u
sing
structures
called
c
lauses
and
p
hrases.
A
phrase
is
a
s
equence
o
f
f
our
c
onsecutiv
e
words.
A
c
lause
i
s
a
sequence
of
five
consecutive
p
hrases
(see
F
ig.
2
3).
Each
phrase
within
a
c
lause
i
s
a
llocated
to
one
t
ask,
and
t
hese
tasks
a
re
independent
o
f
e
ach
other.
Fo
r
i
n-
stance,
t
he
radar
m
ay
search
for
t
argets
using
p
hrases
1,
3,
and
4
,
w
hile
trac
k
i
ng
two
d
ifferent
target
s
u
sing
phrases
2
and
5
.
4)
Search-While-Track
Scan:
One
o
f
t
he
gen
e
ric
f
unc-
tional
states
of
the
r
adar
is
a
s
earch
s
can
denoted
b
y
G
FourWSearch
>.
In
the
G
FourWSearch
>
s
can,
t
he
words
4
The
specification
of
this
emitter
was
provided
by
Dr.
Fred
A.
Dilkes
of
Defence
R&D
Canada.
It
is
based
on
specifications
of
some
real-life
anti-aircraft
defense
radars,
but
has
been
altered
and
declassified
before
the
release.
Fig.
21
.
Me
rc
ur
y
s
t
a
t
e
ma
ch
in
e
c
om
po
ne
nt
s.
Fi
g
.
22
.
Me
rc
ur
y
s
t
a
t
e
ma
c
h
i
n
e
c
o
m
p
o
n
e
n
t
s
.
Vi
snev
ski
et
al.
:
S
y
n
ta
cti
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
M
u
l
tifu
ncti
on
Rad
a
rs
1022
Pr
oce
e
di
ngs
o
f
t
he
IEEE
|
V
o
l
.9
5
,
N
o
.
5
,M
a
y
2
0
0
7
a
r
e c
y
c
l
e
d t
h
r
o
u
g
h
t
h
e q
u
a
d
r
u
p
l
e
t o
f w
o
r
d
s
w
1
,
w
2
,
w
4
,
w
5
.
T
h
e
radar
w
ill
c
omplete
o
ne
cy
cle
(four
w
ords)
f
or
each
beam
position
as
it
scans
i
n
s
pace.
Th
is
is
done
se
quenti
all
y
u
s
ing
a
ll
unoccu
pie
d
word
positions
a
nd
is
not
d
ictated
b
y
t
he
clause
or
p
h
rase
structure.
(Note
t
h
a
t
t
h
e
radar
d
oes
n
ot
have
to
start
t
h
e
cycle
w
ith
W
1
a
t
e
ach
beam
position;
i
t
c
ould,
f
or
instance,
radiate
w
4
,
w
5
,
w
1
,
w
2
o
r
a
n
y
o
th
e
r
c
y
cl
i
c
pe
r
m
u
t
a
t
i
o
n
at
each
beam
p
o
sition.)
It
is
possib
l
e
f
or
the
e
ntire
s
ystem
t
o
o
perate
in
a
search-only
s
tat
e
in
which
n
o
t
arget
t
racks
a
re
maintained
during
the
s
earch.
Howev
e
r,
G
FourWSearch
>
can
also
be
multip
l
e
xed
w
i
th
target
t
r
acking
f
unctions.
I
n
t
h
e
latter
case,
some
of
the
w
ords
within
each
c
l
ause
are
o
ccupied
by
t
a
rget
t
r
acking
a
nd
will
not
e
ngage
i
n
s
earch
f
unctions.
Only
t
h
e
a
vailable
phrases
(
those
t
hat
a
re
no
t
o
ccupied
)
a
r
e
c
y
c
l
e
d
t
h
r
o
u
g
h
t
h
e
q
u
a
dr
u
p
l
e
t
o
f
w
o
r
d
s
.
S
i
n
c
e
th
e
number
of
beam
positions
i
n
t
he
scan
is
fixed,
the
r
ate
a
t
w
h
i
c
h
t
h
e
r
a
d
a
r i
s a
b
l
e t
o s
e
a
r
c
h a g
i
v
e
n v
o
l
u
m
e o
f s
p
a
c
e
is
proportional
to
the
n
umber
o
f
a
vailable
words;
as
a
result
,
s
im
ultaneous
t
racking
i
ncreases
the
o
verall
scan
period.
T
h
e
r
a
d
ar
ha
s
a
not
h
er
sc
an
st a
t
e
cal l ed
G
Thre
eWSearch
>
.
Th
is
is
si
mi
lar
t
o
G
FourWSearch
> e
x
c
e
p
t
that
it
uses
only
a
t
riplet
of
words
w
1
,
w
3
,
w
5
(and
dwells
on
each
beam
position
with
only
three
w
o
r
d
s
).
It
can
also
be
multiplexed
w
ith
a
utomatic
tracking.
5)
Acquisition
Scan:
When
the
r
adar
search
scan
detects
a
target
of
interest,
i
t
m
ay
att
e
mpt
t
o
i
nitiate
a
track.
T
h
i
s
requires
the
r
adar
scan
t
o
switch
from
one
o
f
t
he
search
behav
i
ors
t
o
o
ne
of
the
acquisition
p
a
tterns.
All
o
f
t
he
acquisition
scans
f
ollow
t
hese
st
e
p
s
s
equentially.
a)
Switch
from
Search
to
Acquisitio
n
:T
h
e
s
w
i
t
c
h
f
r
o
m
search
to
acquisit
i
on
begins
with
all
a
vailable
word
s
b
e
i
ng
converted
t
o
t
he
same
v
a
riety
o
f
word
:
o
ne
of
w
1
;
..
.
;
w
6
,c
h
o
s
e
n s
o a
s
t
o
o
p
t
i
m
i
z
e
t
o
the
t
arget
D
oppler
shif
t.
Wor
d
s
t
ha
t
a
re
occup
i
ed
with
other
t
racks
c
ontinue
t
o
p
e
r
form
their
t
racking
f
unction
a
nd
are
n
ot
affected
by
the
change
from
search
to
acquisit
i
on.
T
he
available
word
s
p
erform
one
o
f
s
everal
scan
p
a
tterns
i
n
which
e
ach
beam
position
dwells
only
for
t
he
period
o
f
one
w
ord.
b)
Nonadapt
iv
e
t
rack
:
T
hen,
one
o
f
t
h
e
available
phrases
b
ecomes
desi
gnated
to
t
r
ack
the
t
a
r
get
of
int
e
rest.
T
h
i
s
d
e
s
ignation
will
perp
e
t
ua
t
e
until
t
he
track
is
dropped.
Correspon
d
i
ng
ly,
e
ithe
r
t
he
last
three
o
r
a
ll
f
o
ur
of
the
w
ord
s
w
i
thin
t
h
at
designat
e
d
phrase
become
associated
with
the
t
r
a
c
k
an
d
s
w
i
tc
h
t
o
w
6
(a
nonadaptive
t
rack
without
r
ange
resolution).
The
r
emaining
avail-
able
words
c
ontinue
t
o
r
adiate
in
the
v
ar
i
e
t
y
app
r
opriate
t
o
t
h
e
target
Dopp
l
e
r.
c)
Rang
e
resol
ution
:
A
t
this
point
the
rad
ar
has
angu
lar
track
resol
utio
n
but
still
suffe
rs
from
range
ambi-
guit
ies.
Afte
r
some
vari
able
amo
unt
of
tim
e,
the
firs
t
word
in
the
des
ignat
ed
phra
se
will
hop
betw
een
word
s
w
7
,
w
8
,
and
w
9
,
i
n
n
o
pred
ictab
le
order.
It
will
dwel
l
o
n
each
of
thos
e
vari
eties
of
word
s
only
once
in
order
to
resol
ve
the
range
ambig
uity,
but
dwell
-
time
for
each
vari
ety
is
unpre
dicta
ble.
d)
Re
t
u
r
n
fr
o
m
A
c
q
u
i
s
i
t
i
o
n
t
o
S
e
a
r
c
h
:
F
inal
ly,
o
nce
t
he
rad
a
r
h
as
established
t
rack,
i
t
i
s
r
eady
to
terminate
t
h
e
acquisition
scan.
T
hereafter,
until
t
he
track
is
d
r
o
p
p
e
d,
either
the
l
ast
t
hree
or
all
f
our
w
ord
s
of
t
h
e
d
e
s
ig
nated
p
h
r
ase
w
il
l
b
e
o
c
c
upi
e
d
w
i
t
h
t
he
t
r
ack
and
w
ill
n
ot
be
available
f
or
search
funct
i
ons
or
further
acquisitions.
The
r
adar
then
returns
t
o
one
o
f
t
he
search-while-track
functions.
All
occupied
words
m
aintain
t
heir
tracks
a
nd
all
av
a
i
lable
w
ords
(p
o
ssibly
i
ncluding
the
f
irst
wo
rd
of
the
d
es
ignated
t
rac
k
phra
se)
e
xec
u
te
the
ap
p
r
opriat
e
s
can
pa
t
t
ern.
Only
one
acquisition
can
b
e
performed
a
t
a
ny
giv
e
n
t
ime.
6)
Track
Maintenance:
Each
track
is
maintained
by
either
t
h
e
l
ast
t
h
r
ee
or
all
f
o
u
r
w
ords
of
one
o
f
t
he
phrases.
Those
words
a
re
considered
occupied
and
cannot
p
a
r
ticipate
in
search
or
acquisit
i
on
functions
u
ntil
t
h
e
t
arget
i
s
d
rop
p
ed.
T
h
e
r
adar
performs
range
t
racking
b
y
a
daptively
c
hanging
amongst
a
ny
of
the
h
igh
p
u
l
se
repetition
frequency
w
ord
s
%
w
6
;
..
.
;
w
9
&
i
n
order
t
o
a
void
e
c
li
psin
g
a
nd
mai
n
tain
the
i
r
range
g
ates.
Occasionally,
the
s
ystem
m
ay
perform
a
range
v
e
r
ifi-
cat
i
o
n
f
unct
i
o
n
o
n
t
he
track
by
repeating
t
he
range
resol
u
t
i
on
steps
d
escri
b
ed
above.
F
i
g.
23
.
T
h
e
o
ut
pu
t
s
eq
ue
nc
e
o
f
t
h
i
s
r
ad
ar
is
f
o
r
m
ed
so
th
a
t
th
e
c
l
au
se
s
f
ol
lo
w
e
ac
h
o
th
er
s
e
qu
en
ti
a
l
l
y
.
A
s
s
oo
n
a
s
t
he
l
a
s
t
w
o
rd
of
th
e
l
a
s
t
p
h
r
a
s
e
o
f
a
c
l
a
u
s
e
is
e
m
i
t
t
e
d
,
th
e
f
i
r
st
w
o
r
d
of
th
e
f
i
r
st
ph
ra
s
e
of
th
e
n
e
w
cl
a
u
s
e
fo
ll
ow
s.
Al
th
ou
gh
th
e
p
r
o
c
e
ss
is
li
ne
ar
in
ti
m
e
,
i
t
i
s
v
e
r
y
c
on
ve
ni
en
t
t
o
a
na
ly
z
e
th
e
r
ad
ar
ou
tp
ut
se
qu
e
n
c
e
as
a
t
w
o
-
d
i
m
en
si
on
al
ta
bl
e
w
h
e
n
c
l
a
u
s
e
s
ar
e
s
t
a
c
k
e
d
to
g
e
t
h
e
r
no
t
h
o
r
iz
on
ta
l
l
y
,
bu
t
v
e
r
ti
ca
l
l
y
.
In
th
at
c
a
s
e
,
b
o
u
n
d
a
r
i
e
s
of
ph
ra
se
s
a
s
s
o
c
i
a
t
e
d
w
i
t
h
m
ul
ti
pl
e
x
e
d
ta
sk
s
a
li
g
n
,
a
nd
on
e
c
an
ex
am
in
e
e
a
c
h
m
ul
ti
pl
e
x
e
d
ac
ti
vi
ty
in
de
pe
nd
en
tl
y
b
y
r
e
a
d
i
ng
ra
da
r
ou
tp
ut
wi
th
in
on
e
p
h
r
a
s
e
f
r
o
m
t
o
p
to
bo
tt
o
m
.
Vi
sn
evski
et
al.
:
S
y
n
t
a
ct
i
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
Mu
lt
if
u
n
c
t
i
o
n
R
ad
ars
V
o
l
.
9
5
,
N
o
.
5
,
May
2007
|
Proc
eedings
of
the
I
EEE
10
23
Acknowl edgment
T
h
e a
u
t
h
o
r
s
w
o
u
l
d l
i
k
e t
o t
h
a
n
k
D
r
.F
r
e
d D
i
l
k
e
s a
n
d
Dr.
P
ierre
Lavoie
of
the
D
efense
Research
and
D
evelop-
ment
Cana
d
a
for
p
rovid
i
ng
useful
feed
b
ack
on
the
m
a-
terial
p
r
esented
i
n
t
his
p
aper.
W
e
w
ould
like
t
o
thank
Dr.
D
il
kes
f
or
providin
g
t
h
e
sp
e
c
i
f
ication
o
f
t
h
e
Me
rcury
emitt
e
r
t
hat
w
as
used
as
an
example
o
f
m
odeling
o
f
a
realistic
M
FR.
REF
E
R
E
N
C
E
S
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G.
Wil
e
y
,
Elec
tr
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In
tell
igen
ce
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The
Anal
ysi
s
o
f
R
a
dar
Sig
nals
.
N
or
woo
d
,
MA:
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ch
Hou
se,
1993
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N.
J.
Whi
tta
ll,
B
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gnal
sor
tin
g
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n
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S
M
syst
ems
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IEE
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c.
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28,
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85
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kin
son
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on,
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f
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tech
ni
ques
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M
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essi
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c.
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dne
r,
B
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im
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tio
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tter
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tio
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ESM
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gnal
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ssin
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i
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19
94
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mod
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N
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[24
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ar-S
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.
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,
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,
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Ah
o
a
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J.
D.
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sl
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g
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ee
gra
mma
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roni
c
s
u
r
v
eill
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tifu
ncti
on
ra
dars
:
A
stoch
ast
ic
co
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t
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[
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ul
tif
uncti
on
radar
s:
Mode
li
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n
d
sta
tis
tica
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s
i
gna
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pro
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The
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Lan
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es,
and
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19
94
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i
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e
t
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erg
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[55
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A
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W
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g
a
nd
V.
Kri
s
h
namur
thy
,
B
Signa
l
in
terp
retat
ion
of
mul
tifu
ncti
on
radar
s:
Mode
li
ng
a
n
d
sta
tis
tica
l
s
igna
l
p
ro
ces
sing
with
stoch
ast
ic
co
ntex
t
fre
e
gra
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[
IEE
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Tra
ns.
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ss.,
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[56
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F
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i
lke
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a
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N.
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B
Non
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s
elf-
embed
di
ng
conte
xt
-fre
e
gram
mars
for
el
ectro
ni
c
w
ar
fare
,
[
De
fen
ce
Resea
rch
&
Deve
lop
men
t
Ca
nada
,
O
t
taw
a,
ON,
Ca
nada
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Tech
.
R
e
p
.
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E
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T
M
2
0
0
4
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O
ct.
2004
.
Vi
snev
ski
et
al.
:
S
y
n
ta
cti
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
M
u
l
tifu
ncti
on
Rad
a
rs
1024
Pro
cee
di
ngs
o
f
t
he
IEEE
|
V
o
l
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5
,
N
o
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,M
a
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0
0
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[57]
M.
Ned
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im
atio
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r
a
mma
tic
al
Vi
ew
ser.
Adva
nces
in
Pro
babi
listi
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and
Othe
r
Pars
in
g
Tech
nol
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W.
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uag
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ser.
EAT
C
S
M
o
nogr
aph
s
on
T
heor
eti
cal
C
o
mp
uter
Sc
ien
ce,
W.
Brau
er
,
G.
Roz
enb
erg
,
and
A.
Sa
loma
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E
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A.
Po
the
n
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B
Comp
ut
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f
a
spa
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[
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aut
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iv
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on
c
ontr
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in
in
tegr
ate
d
WLAN
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E
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A
pr
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2
0
0
6.
ABOUT
T
HE
AUTHO
R
S
Nikita
Visn
evski
(
M
e
m
b
e
r
,
I
E
EE
)
w
as
b
o
r
n
i
n
Krasnod
a
r
,
R
ussi
a
,
i
n
1974.
H
e
d
i
d
his
unde
rgr
a
d-
uate
studies
i
n
c
o
mp
uter
a
n
d
s
of
tware
e
ngi-
neering
a
t
t
he
S
t
.
P
ete
r
sbur
g
A
cademy
of
Aerospace
E
ng
i
n
eering,
St.
P
et
er
s
b
urg,
Russi
a
(19
9
1–19
95
).
He
re
ce
ived
t
h
e
M
.
S
.
d
e
g
re
e
i
n
elect
r
ical
eng
i
n
eering
f
ro
m
T
ex
a
s
Tec
h
Universi-
ty
,
L
ubb
o
ck,
i
n
1
997
and
t
he
Ph.D.
d
egree
i
n
elect
r
ical
engineer
ing
f
r
o
m
M
cMaster
U
niver
s
i
t
y,
Hamilton,
ON
,
C
anada
i
n
2
005
.
I
n
1996–
1997
he
wor
k
ed
at
the
D
e
f
ense
and
S
p
a
ce
depa
r
t
ment
of
the
Boein
g
Company
i
n
S
eattle,
WA,
a
n
d
the
S
cientific
R
esear
c
h
L
aborato
r
y
of
For
d
Motor
C
o
m
pany
in
Dearb
o
r
n
,
M
I.
Dur
i
ng
this
per
i
od
he
fo
cused
his
r
esearch
o
n
t
he
ap
plication
o
f
a
rtif
icia
l
n
e
u
ral
n
etwo
rk
b
a
sed
intelligen
t
sy
stems
t
o
d
ia
g
nostics
and
c
o
n
trol
o
f
ai
r
-
and
g
r
ound
-based
v
e
h
i
c
l
e
s
.
F
r
o
m
1
9
9
7
u
n
t
il
2
0
0
0
h
e
w
o
r
k
e
d
as
an
a
p
p
l
ic
a
t
i
o
n
s
p
e
c
i
a
l
i
s
t
a
t
the
C
ontr
o
ls
Design
A
utom
a
t
ion
d
ep
artmen
t
o
f
T
he
MathWor
k
s
,
Inc.
(Natick,
MA)
w
her
e
h
e
fo
cused
o
n
s
oftwar
e
d
esign
a
nd
dev
e
lop
m
ent
f
or
Simulink
and
R
eal
T
ime
W
o
r
kshop
pr
oduc
ts.
F
rom
2
000
un
ti
l
2
001
he
worked
as
an
ap
plication
d
ev
e
l
op
me
nt
team
leader
for
T
he
Solect
Te
chno
logy
Gr
oup
(Tor
onto,
ON
,
C
anad
a
)
wher
e
h
e
s
up
erv
i
s
e
d
a
mob
i
l
e
IP
softwar
e
de
velo
pment
t
eam
a
n
d
d
e
velo
ped
m
ob
ile
I
nter
net
a
n
d
v
o
ice
over
IP
ap
plications.
S
ince
A
u
gust
2
005
he
has
b
een
a
s
taf
f
scientist
and
system
arc
h
itect
a
t
t
he
Global
Researc
h
Cen
t
er
of
the
G
ener
al
Elec
tric
Comp
a
n
y,
Nisk
a
y
una,
N
Y
.
He
is
a
m
em
ber
o
f
t
he
Intellige
nt
Emb
e
dded
Sy
stems
L
abor
atory
o
f
t
he
E
l
ectr
onics
and
E
nerg
y
C
o
n
ver
sion
O
r
g
ani-
zation.
Hi
s
p
rimary
research
f
o
cus
a
r
e
as
include
intel
l
i
g
ent
a
nd
co
gnitiv
e
embedded
s
y
s
tems
as
well
as
large-sca
l
e
s
y
s
tems
archi
t
e
c
tures.
Vikram
Krish
n
amur
t
h
y
(Fellow,
IEEE)
was
b
o
r
n
in
1966.
He
rec
e
ived
the
B
.S.
d
e
g
r
e
e
f
r
o
m
t
he
Unive
r
sity
o
f
A
uckland
,
Ne
w
Z
ealand,
i
n
1988
,
a
nd
the
P
h.D.
degr
ee
fr
om
the
A
ustralian
N
ational
Univ
e
r
sity
,
Canberra,
i
n
1
9
9
2.
S
i
n
c
e
2
00
2,
he
has
b
ee
n
a
Profe
s
sor
a
nd
C
a
n
a
da
Research
Chair
a
t
t
he
Departme
nt
of
E
l
ectrica
l
Engine
ering,
Uni
v
e
r
s
i
t
y
of
Briti
s
h
Columbia,
V
ancouv
e
r
,
C
anada
.
Prior
t
o
2
002
,
h
e
was
a
chaired
P
r
o
fes
s
or
in
the
D
e
p
artme
n
t
o
f
Electr
ical
a
n
d
E
lectr
onic
E
ng
i
n
eering
,
Univer
si
t
y
of
Melb
our
ne,
A
ustr
ali
a
,
wher
e
h
e
a
l
s
o
s
erved
a
s
D
eputy
H
ead
o
f
t
he
de
partme
nt.
H
i
s
cur
r
ent
resear
ch
i
n
terests
i
nc
lude
stoch
a
s
t
ic
modelin
g
of
b
i
ologic
a
l
ion
c
han-
ne
ls,
s
tochastic
o
ptimization
a
nd
sche
duling,
a
nd
st
atist
ical
signal
pr
oce
s
s
i
ng
.
H
e
i
s
C
o
e
ditor
(
wi
t
h
S.
H.
Chu
n
g
a
n
d
O.
Ander
s
en)
o
f
t
he
boo
k
Biolo
g
ic
al
M
e
mbrane
Ion
C
hannel
s
V
Dynamics
St
ruct
ure
a
nd
App
l
ications
(Spri
n
ger-Verlag,
2
006).
D
r
.
K
rishn
a
m
u
rthy
has
s
er
ved
a
s
A
s
s
o
c
iate
Editor
for
s
ev
eral
jour
nals
including
I
EEE
T
RA
NSAC
TIO
N
S
O
N
S
IG
NA
L
P
RO
CESSI
NG
,
I
EEE
T
RAN
S
ACTI
ONS
A
ERO
S
PACE
A
N
D
E
LE
CTRON
I
C
S
YSTEMS
,
I
EEE
T
RA
NSA
C
TIO
N
S
O
N
C
IRCU
ITS
A
ND
S
YST
E
M
S
B,
IEEE
T
RAN
S
ACTI
ON
S
O
N
N
A
NOB
IO
SCIEN
C
E
,a
n
d
S
YST
E
M
S
AND
C
ON
TROL
L
ETTER
S
.
Ale
x
Wa
ng
w
a
s b
o
r
n
i
n 1
9
7
9 i
n T
a
i
w
a
n
.
H
e
rec
e
i
v
ed
th
e
B
.S
.
d
eg
ree
(
w
i
t
h
hono
urs)
in
en
gi-
neerin
g
p
hy
si
c
s
with
a
c
o
mmerc
e
m
i
n
or
and
t
he
M.S.
de
gre
e
in
ele
c
trica
l
and
c
omputer
e
ngineer-
ing
f
rom
t
he
U
n
iversi
ty
of
Brit
ish
Co
lu
mbia
,
Va
n
c
ouver,
C
a
n
a
d
a
,
in
2003
an
d
2
005,
r
espec-
ti
v
e
ly
.
H
e
i
s
c
urr
e
ntly
wo
rking
t
oward
t
he
Ph.D.
degree
in
s
t
atistical
signa
l
p
roce
s
s
ing,
un
der
t
he
s
u
p
e
rv
ision
o
f
D
r.
Vikram
K
r
ishnamurthy
,
a
t
t
he
Univ
ersi
t
y
of
Bri
t
ish
Co
lumb
ia.
His
r
ese
a
rch
i
nte
r
est
s
i
n
clu
d
e
r
adar
signal
p
rocessing,
syntactic
p
a
ttern
recogn
i
t
ion
,
and
uncertain
rea
s
oning.
Si
m
o
n
H
ayk
i
n
(Fellow,
IEEE)
received
t
h
e
B
.Sc.
(F
irst-class
Honours
)
,
P
h.D.,
a
n
d
D.Sc.
d
egrees
in
elec
tric
al
eng
i
neer
ing
f
r
o
m
t
h
e
University
of
Birming
h
am,
B
ir
mi
n
g
ham,
U.
K.
Cu
rre
ntly,
h
e
h
olds
the
t
i
t
le
of
Di
s
t
ing
u
is
h
e
d
Univ
ersi
t
y
Prof
essor
i
n
t
he
ECE
D
e
partme
nt
at
McMaster
Univ
e
r
sity
,
Hamilton
,
O
N,
Canada.
H
e
i
s
the
a
uth
o
r
o
f
n
umero
u
s
b
o
o
ks,
i
nc
luding
the
most
wide
ly
used
boo
k
s
:
Co
mmunication
S
y
stems
(4
t
h
ed.,
Wiley),
Adap
ti
v
e
Filter
The
o
ry
(4th
ed
.,
P
r
entice
-Hall),
Ne
ural
Netwo
r
ks:
A
Comp
re
hensive
F
oun
dati
o
n
(2nd
ed.,
P
r
entice
-Hall),
a
nd
th
e
n
ewly
publis
h
e
d
Adapti
v
e
Ra
d
a
r
S
igna
l
P
rocess-
in
g
(Wiley
),
as
well
as
numer
ous
re
fer
eed
jour
nal
p
ap
ers.
Dr
.
H
aykin
i
s
a
Fellow
o
f
t
he
Royal
S
ociety
of
Canada,
r
ecipient
o
f
the
Hono
urar
y
D
egre
e
o
f
D
o
c
tor
o
f
T
echn
i
c
al
S
c
ienc
es
f
r
om
ETH,
Zur
i
c
h
,
Switzer
l
and,
and
t
he
Henr
y
B
ooker
Go
ld
Medal
f
r
o
m
U
RSI,
as
we
ll
a
s
other
p
r
i
z
e
s
a
n
d
awa
r
ds.
Vi
sn
evski
et
al.
:
S
y
n
t
a
ct
i
c
Mo
de
li
ng
an
d
S
i
g
nal
P
r
o
ce
ssi
n
g
o
f
Mu
lt
if
u
n
c
t
i
o
n
R
ad
ars
V
o
l
.
9
5
,
N
o
.
5
,
May
2007
|
Pr
oceed
ings
of
the
I
EE
E
1025