Mul iple
e lec ions
and
imp o emen
o
edge
sca e ing
in
GRECO
RCS
p edic ion
code
Juan
M.
Rius,
M.VaII-Ilosse a,
Lluis
Jo e
G up
A.M.R.,
Dp .
Teo ia
del
Senyal
Comunicacions,
UPC
Apdo.
30002,
08080
Ba celona,
TI.
+34-3-4017219,
Fax
+34-3-4017232
Abs ac
GRECO
code
o
monos a ic
RCS
p edic ion
in
eal
ime
has
been
ex ended
by
conside ing
mul iple
e lec ions
be ween
su aces
and
imp o ing
he
edge
di ac ion
coe icien s.
Mul iple
e lec ions
a e
analysed
h ough
a
e y
e icien
ay- acing
algo i hm
based
on
he
g aphical
p ocessing
echnique.
Me hod
o
equi alen
cu en s
o
edge
sca e ing
has
been
imp o ed
by
Mi zne 's
and
Michaeli's
inc emen al
leng h
di ac ion
coe icien s
(ILDC).
This
communica ion
p esen s
he
gene al
ea u es
o
GRECO
code,
in
pa icula
he
ad an ages
o
he
new
g aphical
p ocessing
echnique.
Emphasis
will
be
placed
in
he
new
ea u es
o
GRECO
s ill
unpublished:
he
ay- acing
algo i hm
and
he
implemen a ion
o
inc emen al
leng h
di ac ion
coe icien s.
G aphical
Elec omagne ic
Compu ing
(GRECO)
Du ing
he
las
ou
yea s,
he
de elopmen
o
g aphical
p ocessing
echniques
o
high- equency
monos a ic
RCS
p edic ion
has
gi en
ise
o
GRECO
code
(1),
(2),
(3).
Real- ime
compu a ion
is
achie ed
h ough
g aphical
p ocessing
o
an
image
o
he
a ge
p esen
a
he
sc een
o
a
wo ks a ion,
using
he
ha dwa e
capabili ies
o
a
3-D
g aphics
accele a o .
The
wo
main
ad an ages
o
he
g aphical
p ocessing
app oach
o e
classical
echniques
a e:
a)
Ha dwa e
g aphics
accele a o
emo es
hidden
su aces
and
edges:
hey
do
no
con ibu e
o
he
su ace
o
line
in eg als
when
hey
a e
pe o med
by
he
g aphical
p ocessing
echnique.
b)
Reduced
CPU
ime
and
RAM
equi emen s.
They
a e
independen
o
a ge
elec ical
size
and
complexi y.
The
main
ea u es
o
GRECO
code
a e
he
ollowing:
a)
The
I-DEAS
compu e
aided
design
package
o
geome ic
modelling
o
solids
has
been
used
o
modelling
a ge
geome y.
The
a ge
is
desc ibed
as
a
collec ion
o
pa ame ic
su aces,
de ined
wi h
wo-dimensional
NURBS
(non-uni o m
a ional
B-
splines).
Pa ame ic
su aces
equi e
less
mass
s o age
memo y
ha
he
ace ing
app oach,
and
adjus
mo e
accu a ely
o
he
eal
a ge
su ace.
b) Physical
Op ics
(PO)
app oach
o
pe ec ly
conduc ing
su aces
and
Impedance
Bounda y
Condi ion
(IBC)
+
Physical
Op ics
o
ada
abso ben
coa ings.
c)
Me hod
o
Equi alen
Cu en s
(MEC)
wi h
ei he
Physical
Theo y
o
Di ac ion
(PTD),
Mi zne 's
o
Michaeli's
Inc emen al
Leng h
Di ac ion
Coe icien s
(ILDC)
o
pe ec ly
conduc ing
edges.
A
compa a i e
s udy
o
he
esul s
o
he
h ee
kinds
o
ILDC's
will
be
p esen ed
in his
communica ion.
d)
Double
e lec ion
analysis
by
geome ic
op ics
(GO)
ay-
acing
o
he
i s
e lec ion
and
bis a ic
physical
op ics
o
he
second.
A
new
and
e y
e icien
ay- acing
algo i hm
has
been
de eloped
and
will
be
also
p esen ed
in
his
communica ion.
Ray-T acing
Double
e lec ions
be ween
su aces
a e
analysed
by
a
hyb id
GO-PO
scheme.
The
GO
e lec ion
a
he
i s
su ace
assumes
ha
specula
e lec ion
occu s,
acco ding
o
s a iona y
phase
p inciple.
The
PO
e lec ion
a
he
second
su ace
ensu es
ha
he
co ec
sca e ed
ield
is
ob ained
when
he e
is
no
specula
e lec ion
o
he
obse e .
Fo
each
pixel
on
he
a ge
su ace,
a
e lec ed
ay
is
aced
along
he
GO
specula
di ec ion.
The
impac
o
he
e lec ed
ay
wi h ano he
su ace
is
de ec ed
on
he
sc een
by
ollowing
he
ay-pa h
and
compa ing
he
z
coo dina es
o
he
ay
wi h
ha
o
he
su ace
a
he
same
x,y
loca ion.
In
case
ha
he e
is
a
second
e lec ion,
he
ield
sca e ed
o
he
obse e
is
compu ed
using
bis a ic
PO.
Vec o
o mula ion
o
bis a ic
GO
and
PO
o
espec i ely
he
i s
and
second
e lec ions
allows
o
ob ain he
sca e ing
ma ix
o
he
a ge .
This
scheme
is
alid
o
bo h
plana
and
cu ed
su aces.
In
he
o me
case,
he
e lec ed
ays
a e
pa allel,
which
is
equi alen
o
a
plane
wa e
inciden
o e
he
second
su ace.
In
he
la e
case,
he
di e gence
ac o
due
o
he
cu a u e
o
he
su ace
is
implici
in
he
g aphical
ay- acing
algo i hm:
he
numbe
o
ay-
hi s
on
he
second
su ace
is
in e sely
p opo ional
o
he
di e gence
o
he
e lec ed
ays.
Inc emen al
Leng h
Di ac ion
Coe icien s
I
is
well
known
ha
U lm se 's
PTD
coe icien s
a e
accu a e
o
analysing
high- equency
monos a ic
edge
di ac ion
only
when
he
incidence
di ec ion
is
pe pendicula
o
he
edge,
o
which
monos a ic
obse a ion
is
along
he
Kelle
cone
(5).
This
is
he
case
o
he
main
edge
con ibu ion
o
o al
RCS,
so
ha
oblique
edges
con ibu ion
can be
usually
neglec ed.
Fo
ha
eason,
and
due
also
o
hei
simplici y,
PTD
coe icien s
oge he
wi h
Physical
Op ics
cons i u e
he
mos
usual
app oach
o
monos a ic
RCS
p edic ion.
Excellen
esul s
ha e
been
ob ained
by
GRECO
code
using
a
e y
e icien
linea
app oxima ion
o
PTD
coe icien s
(2),
(3).
An
excep ion
o
he
abo e
ule
is
he
analysis
o
s eal h
ai c a ,
in
which
specula
su aces
and
edges
a e
delibe a ely
a oided.
Fo
igo ous
analysis
o
oblique
edges
i
is
necessa y
o
use
MEC
wi h
some
kind
o
inc emen al
leng h
di ac ion
coe icien s,
which
a e
co ec
o
obse a ion
ou side
he
Kelle
cone
(5).
Usually
Mi zne 's
ILDC
(5)
ha e
been
chosen
o
his
pu pose
in
a
numbe
o
RCS
p edic ion
codes
bu ,
on
he
o he
hand,
hey
a e
singula
o
some
obse a ion
di ec ions
(6).
This
p oblem
has
been
o e come
by
Michaeli's
ILDC
(6),
which
emo e
mos
o
he
singula i ies
o
Mi zne 's
and,
in
pa icula ,
a e
ini e
o
all
monos a ic
obse a ions.
The
main
d awback
o
Michaeli's
ILDC
is
a
a he
complex
o mula ion,
e en
o
he
monos a ic
case,
which
makes
eal- ime
RCS
p edic ion
di icul
o
GRECO
code.
Resul s
An
in e es ing
RCS
p oblem
which
includes
con ibu ions
om
bo h
double
e lec ions
and
edge
di ac ion
is
he
900
dihed al
p oposed
a
Ma seille'92
RCS
wo kshop
(4).
The
geome y
o
he
objec
is
shown
in
ig.
1.
No ice
he
monos a ic
sweep
along
he
bisec o
plane,
q
=
450,
ins ead
o
he
plane
pe pendicula
o
he
dihed al
aces,
0
=
90°.
This
unusual
sweep
makes
he
con ibu ion
o
oblique
edges
mo e
impo an
han
ha
o
su ace
e lec ion,
becoming
an
excellen
es
o
he
accu acy
o
he
di e en
kind
o
edge
di ac ion
coe icien s.
Figu e
2
shows
he
RCS
om
single
and
double
su ace
e lec ions
compu ed
by
a
double
PO+PO
su ace
in eg al
compa ed
o
he
p edic ions
o
he
GO+PO
ay- acing
echnique
implemen ed
in
GRECO
code.
No ice
ha
o
g azing
incidence
he
la e
is
a
ew
dB
unde
he
PO+PO
esul ,
due
o
he
ac
ha
mos
GO
ays
specula y
e lec ed
on
a
ace
o
he
dihed al
do
no
hi
he
o he
one,
while
o
hese
ays
PO
p edic s
non-null
sca e ed
ields
ha
each
he
o he
ace
o
he
dihed al.
The
esul s
o
GRECO
code
using
he
h ee
kinds
o
edge
di ac ion
coe icien s
oge he
wi h
he
ay- acing
algo i hm
a e
shown
in
ig.
2
( hick
line
plo s).
These
esul s
ha e
been
386
iso
II..
-e~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~4
"
-:I
Side
iow
Fig.
1:
Dihed al,
1=
1.5
X,
h=
2.25
x.
a
=
9
RCS
sweep
along
bisec o
plane.
RCS
Es
0
10
20
30 40 50
60
e
Fig.
2:
Con ibu ion
o
single
and
double
su ace
e lec ions
o
RCS
o
he
dihed al.
Resul s
o
PO+PO
double
su ace
in eg al
(dashed
line)
compu ed by
gaussian
cuad a u e,
and
GO+PO
ay- acing
(solid
line)
compu ed
by
he
g aphical
p ocessing
echnique
(dashed
line).
100
20
140
16"
3
a
ILDC
MiChoe i
compa ed
wi h
he
compac
ange
measu emen s
o
S.
Mish a,
pe o med
a
he
Canadian
Space
Agency
( hin
line
plo ).
I
should
be
no ed
ha
he e
a e
wo
main
sou ces
o
e o
in
he
GRECO
code
analysis
o
his
objec :
he
elec ical
size
is
no
la ge
enough
o
he
high
equency
me hods
o
be
accu a e
and
he e
exis s
an
impo an
con ibu ion
o
su ace-edge
in e ac ions
ha
a e
no
aken
in o
accoun
by
GRECO.
The
bes
esul s
o
00
pola isa ion
a e
ob ained
wi h
U im se
s
PTD
o
Michaeli
s
ILDC,
excep
o
some
di ec ions
o
obse a ion
whe e
Mi zne
s
ILDC
a e
mo e
accu a e.
In
he
case
o
ZZ
pola isa ion,
he
beha iou
o
Mi zne
s
ILDC
is
wo se
han
ha
o
PTD
o
Michaeli
s
ILDC.
Conclusions
GRECO
code
has
been
imp o ed
by
he
implemen a ion
o
a
GO-
PO
g aphical
ay- acing
algo i hm
o
analysis
o
mul iple
su ace
e lec ions
and
Michaeli
s
ILDC
o
oblique
edge
di ac ion.
The
esul s
o
a
90°
dihed al
wi h
oblique
monos a ic
incidence
ha e
been
compa ed
o
RCS
measu emen s.
The
e o s
due
o
he
no
la ge
elec ical
size
o
he
objec
and
he
su ace-edge
in e ac ions
deg ade
signi ican ly
he
accu acy
o
he
esul s.
Howe e ,
i
can
be
no iced
ha
e en
in
hese
ad e se
condi ions,
GRECO
is
able
o
p edic
he
app oxima e
le el
o
he
RCS,
o
he
deg ee
o
accu acy
usually
equi ed
when
complex
ada
a ge s
a e
analysed.
The
conclusion
o
he
compa ison
be ween
he
h ee
kinds
o
di ac ion
coe icien s
is
he
ollowing:
al hough
Mi zne
s
o
Michaeli
s
ILDC
a e
in
heo y
mo e
accu a e
han
U im se
s
PTD
o
analysing
oblique
edges,
he
imp o emen
o e
PTD
is
no
signi ican
when
o he
unp edic ed
con ibu ions
o
RCS
a e
p esen ,
which
is
always
he
case
o
complex
ai c a .
Mo eo e ,
he
con ibu ion
o
oblique
edges
is
e y
a ely
signi ican
agains
ha
o
pe pendicula
su aces
o
edges,
so
ha
he
linea
app oxima ion
o
PTD
coe icien s
p esen ed
in
(2)
and
(3)
is
p obably
he
mos
e icien
app oach
o
eal- ime
monos a ic
RCS
p edic ion.
Acknowledgmen s
This
wo k
has
been
suppo ed
by
he
Spanish
"Comisi6n
In e minis e ial
de
Ciencia
y
Tecnologia"
(CICYT)
unde
he
p ojec :
TIC
88-288E.
Re e ences
(1)
Juan
M.
Rius,
M.
Fe ando,
L.
Jo e,
"GRECO:
G aphical
Elec omagne ic
Compu ing
o
RCS
P edic ion
in
Real
Time",
IEEE
An ennas
and
P opag.
Magazine,
Ap il
1993
(2)
Juan
M.
Rius,
M.
Fe ando,
L.
Jo e,
"RCS
o
Complex
Rada
Ta ge s
in
Real
Time",
accep ed
o
publica ion
in
IEEE
T ans.
on
An ennas
and
P opag.
(3)
Juan
M.
Rius,
"Seccion
ec a
de
blancos
ada
complejos
en
iempo
eal",
Doc o a e
Thesis
Disse a ion,
Uni e si a
Poli ecnica
de
Ca alunya,
July
1991.
(4)
"SER
Ma seille-92,
d udes
compa a i es
de
Su aces
Equi alen es
Rada
de
co ps
in ini emen
conduc eu s
ou
A
impddance
de
su ace
e
de
cibles
inhomogenes",
wo kshop
on
RCS
o ganized
by
Dassaul -A ia ion,
Socid d
Mo hesim
and
he
Uni e si e
de
P o ence.
(5)
E.F.
Kno ,
J.F.
Shae e ,
M.T.
Tuley,
"Rada
C oss
sec ion:
I s
P edic ion,
Measu emen
and
Reduc ion",
A ech
House,
Inc.,
Dedham,
MA,
1985.
(6)
A.
Michaeli,
"Elimina ion
o
In ini ies
in
Equi alen
Edge
Cu en s,
Pa
I:
F inge
Cu en
Componen s",
IEEE
T ans.
on
An ennas
and
P opag.,
Vol.
AP-34,
No.7,
July
1986
Meoszmemen
by
5.
Mb o.
Can adin
Space
Agency
Figu e
3
Resul s
o
GRECO
code
(solid,
dashed
o
dosed
hick
line)
o
he
h ee
kinds
o
edge
di ac ion
coe icien s
compa ed
wi h
compac - ange
measu emen s
(solid
hin
line).
387
-0
I
eX
2
ACS
zz
25
x2
25
20
15
10
-10k
-15
-20_
0
0
20
40
60
go
100
170
140
160
80
PO
+
R
+
PTD
USun se
IOC
M zne