In Table 1, we show he aul co e age as he numbe o
unscanned lip- lips in ci cui S5378 inc eases.
As
shown in Table
1, he aul co e age does no d op signi ican ly un il as many as
50 lip- lops a e scanned.
Table
2
Faul co e age wi h unscanned lip- lops
To al O iginal Sel Unscanned Unscanned
FFs
co e age loop FFs a io
Ci cui Faul
co e age
I
S27
13
S298 114
I
100
3
2
66.7 100
100 14 2 14.3 93.57
In Table
2,
we show he aul co e age o each ci cui wi h he
gi en numbe o lip- lops unscanned. The hi d column gi es he
ull scan aul co e age using ou implemen a ion o he o iginal
FAN algo i hm. The i h column gi es he numbe o unscanned
lip- lops. The se en h column gi es he aul co e age. We
s opped unscanning when he aul co e age d opped signi ican ly.
When he ci cui con ains enough sel -loop ee lip- lops, he p o-
posed pa ial scan selec ion me hod allows mo e han 20%
lip-
lops o be unscanned wi h li le dec ease in he aul co e age.
Fo he 838 ci cui , wi h only one unscanned lip- lop, he aul
co e age d opped om 100 o 91.74%, bu when we assumed he
p ese signal a he unscanned lip- lop, we we e able o ob ain
100% aul co e age.
S382
s444
Conclusion:
We ha e in oduced an in e es ing pa ial scan es
me hod wi h no da a holding o e head. The expe imen al esul s
show ha he p oposed me hod can achie e almos ull scan aul
co e age while ha ing a signi ican numbe o lip- lops
unscanned. The p oposed es me hod will be mos e ec i ely used
o ci cui s
in
which he a ea occupied by memo y elemen s
domi-
na es. These kinds o ci cui s a e p e alen in mul imedia applica-
ions. We will s udy he pa ial scan selec ion algo i hm mo e and
will imp o e he ATPG algo i hm by adop ing a ious echniques
o mo e ad anced ATPGs [l,
21.
21 100 15 4 19.0 95.32
21 97.34 15
3
14.3 90.11
0
IEE
1997
Elec onics Le e s Online
No:
19971414
Dong
Ho
Lee and Hyun Chul Noh
(School
o ’
Elec onics and
Elec ical Enginee ing, KyungPook Na ional Uni e si y, Tuegu, 701-
702,
Ko ea)
26
Augus 1997
Re e ences
POMERANZ,
I.,
REDDY,
L.N.,
and
REDDY,
s.M.: ‘COMPACTEST:
A
me hod o gene a e compac es se s o combina ional ci cui s’.
P oc. In . Tes Con ., 1991, pp. 194203
SCHULZ,
M., TRISCHLER,
E., and
SARFERT,
T.:
‘SOCRATES: A highly
e icien au oma ic es pa e n gene a ion sys em’. P oc. In . Tes
Con ., 1987, pp. 101&1026
CHENG,
K.T.,
and
AGRAWAL,
.D.:
‘A pa ial scan me hod
o
sequen ial ci cui s wi h eedback‘,
IEEE T ans.,
1990,
C-39,
pp.
546548
LEE,
D.H., and
REDDY,
s.M.:
‘On
de e mining scan lip- lops in
pa ial scan designs’. P oc. In . Con . Compu e -Aided Design,
No embe
1990,
pp.
322-325
PARIKH,
P.s., and
ABRAMOVICI,
M.:
‘Tes abili y-based pa ial scan
analysis’,
J.
Elec on. Tes .,
1995,
7,
pp. 61-70
FUJIWARA,
H., and
SHIMONO,
T.:
‘On
he accele a ion o es
gene a ion algo i hms’. P oc. 13 h In . Symp. Faul Tole an
Compu ing, 1983, pp. 98-105
In eg al equa ion
ME1
apiplied o h ee-
dimensional a bi a y su aces
J.M.
Rius,
J.
Pa on,
E.
Ubeda
and
J.R.
Mosig
Indexing e m: Me hod
o
mom m s, Bounda y elemen me hod,
Nume ical me hods, Elec omagne ic wa e sca e ing
The au ho s p esen a new o mula ion
o
he in eg al equa ion o
he measu ed equa ion
o
in a iance as a con ned ield in eg al
equa ion disc e ised by he me hod o momen , in which he use
o nume ically de i ed es ing unc ions
esul s
in an
app oxima ely spa se linea sys em wi h s o age memo y
equi emen s and a CPU ime o compu ing he ma ix
coe icien s p opo ional o he numbe o unknowns.
In oduc ion:
Bounda y elemen me hods (BEMs) a e widely used
o he nume ical analysis o elec omagne ic adia ion and sca -
e ing. Howe e , hei applica ion is limi ed o elec ically small o
esonan size objec s due o he ac ha he compu a ional
equi emen s inc ease apidly wi h 1.he elec ical size. One ecen
app oach o achie ing mo e e icien BEMs is he in eg al equa-
ion o mula ion o he measu ed equa ion o in a iance (IE-MEI)
[l
-
31.
This
Le e p esen s he o mula ion o he IE-ME1 o 3D a bi-
a y sca e ing su aces as a special case o a combined ield in e-
g al equa ion (CFIE), disc e ised Iby he me hod o momen s
(MOM), in which he choice o di e en es ing unc ions o he
elec ic and magne ic ields esul s i i an app oxima ely spa se lin-
ea sys em o sol e o he induced cu en , whe e mos o he
ma ix elemen s can be neglec ed. These new es ing unc ions a e
nume ically de i ed
by a p ocedu e bo owed om he measu ed
equa ion o in a iance (MEI) me hod [4] o iginally de eloped o
nume ically ind he unca ion bounda y coe icien s o ini e di -
e ence and ini e elemen meshes [5].
A
signi ican ea u e o
nume ically de i ed es ing unc ions is ha hey a e
adap i e,
i.e.
speci ic o he pa icula shape o he sca e e bounda y and o
he loca ion o he unc ion in he bounda y.
Fo mula ion:
The elec ic ield in eg al equa ion (EFIE) and he
magne ic ield in eg al equa ion (MIFIE) disc e ised by he MOM
may each be exp essed in ma ix o m, espec i ely, as
I
di
a e he (subdomain) basis unc ions,
J’,
=
Zi
c,
2,
and
Gm
a e he (subdomain) es ing unc ions, he ma ix elemen s in eqn.
1 a e
em
=
(Gm,Ei)
zEi
=
(Gm,lA~~C )
d,i
=
(Gm,.i;Z)
whe e
LEJ
and
LHJ
a e, espec i ely, he linea ope a o s ha
ob ain he elec ic and magne ic ields due o an elec ic cu en .
We le -mul iply he EFIE and he MFIE in eqn.
1
by wo a bi-
a y ma ices [A] and
[B],
espec i ely, and add he wo equa ions
o o m a CFIE
-[E9
=
[ZE][C]
-
[W]
=
([ZH]
-
[D])
[C]
(I)
+.
+.
hk
=
(Gm, i
x
E L)
z,”;
=
(Gm, i
x
LHJGZ)
(2)
-[AI[@]
-
[Bl[Hil
=
([Al[ZEl
+
[Bl[ZHl
-
[BI[Dl)
[Cl
(3)
Le -mul iplica ion o he EFIE o he MFIE by
a
ma ix is equi -
alen o changing he es ing unc ions. Fo ins ance:
m
m
I
sugges s ha he new es ing unc ions a e
Zm
a,,,
$,
o he EFIE
and
Zm
b,,,
G,
o he MFIE. The ows o [A] and
[B]
ma ices a e
he coe icien s o he new es ing unc ions expanded in he old
ones
w,.
Spa se
[A],
[B]
ma ices ep esen subdomain es ing
unc ions: he ze os co espond o o iginal es ing unc ions (col-
LUIUI
index) ha do no o e lap wi h he new ones ( ow index).
We
de me
he ‘combined impedance
ma ix’
as
[Zq
=
[A][Zq
+
[B][Zq.
Now, eqn. 3 becomes
--
-[AlP 1
-
PI[W
=
ELECTRONICS LETTERS
20 h
No embe
1997
Vol.
33
No.
24
2029
whe e ma ix
[Zc]
is ull,
[D]
is spa se (because
2
and
2,
a e sub-
domain unc ions) and
[A]
and
[B]
a e a bi a ily chosen as
spa se
and such ha
[Zq[Cl
=
0.
The e o e, eqn. 5 can be app oxima ed
by he spa se sys em o equa ions
whe e
[Cj
is
he unknown ec o .
The abo e is he 3D o mula ion o he IE-MEI, equi alen o a
CFIE- ype equa ion wi h di e en es ing o he EFIE and he
MFIE, such ha he elec ic and magne ic impedance ma ices
cancel each o he . The 2D o mula ion o he IE-ME1 [I, 21,
[q
=
[B]-'
[A][EI
+
[w,
is a special case o eqn.
6
wi h
[D]
=
[I]
(iden i y
ma ix), due o he use o pulse basis and del a es ing.
[AI[E"l
+
PI[ 7
=
[BI[DI[Cl
(6)
Coe Jicien compu a ion (MEIme hod):
In eqn.
5,
we need
[Zq[q
0.
Since
[E]
=
[Zq[q
and
[PI
=
[Zq[q,
whe e
[E]
and
[PI
a e, espec i ely, he disc e isa ion o elec ic and magne ic sca -
e ed ields, i ollows ha
[Zq[q
=
[A][E]
+
[BI[Hsl
=
0.
To
nume ically ind he coe icien s o ma ices
[A]
and
[B],
he
ME1
me hod en o ces
[A][E;]
+
[B][H;]
=
0,
whe e
[E;]
=
[Z7[opl,,
[Hi]
=
[Zq[o]
a e he disc e isa ion o he ields due o
P
a bi a y
cu en s
%p
called
me ons.
[CY
J
is a ma ix whose columns a e he
disc e isa ion o he me ons
2
o
p
=
1,
...,
P.
The esul is
a
se
o linea sys ems, one o each ow
o
he ma ix equa ion
[Al[ZE1[%l
+
[Bl[aHI[%l
=
[ZC1[.Pl
=
0
(7)
whe e he unknowns a e he elemen s o
[A]
and
[B]
ma ices.
Ob iously, eqn. 7 does no gua an ee ha
[Zc}
[q
0,
especially
I
he sys ems (eqn. 7) a e o e de e mined and canno be exac ly
sa is ied. Howe e , i has been obse ed ha in he 2D case, some
me on se s p oduce
a
negligible esidual
[Zq[q
in
he IE-ME1
eqn. 6 [1, 2, 6, 71 o in he equi alen unca ion bounda y condi-
ion
[4,
81.
Me ons:
Fo
3D
a bi a y sca e e su aces, he easies me ons
o implemen a e
del a
me ons. Each del a me on is equal o
a
basis unc ion,
.',
=
zp,
and hus
[CYJ
=
[I].
Wi h his se o me -
ons, eqn. 7 becomes
[Zq
=
0.
Row
n
o eqn. 7 co esponds o he
linea sys em
CL,,^:,
+
bnmzii)
=
z,",
=
0
i
=
1,
...,
N
(8)
m
whe e he non-ze os o
a,,
and
b,,
a e he unknowns. To a oid
he i ial solu ion, one o he coe icien s mus be a bi a ily se ,
o ins ance
a,,
=
1. Since i is desi able ha ma ices
[A]
and
[B]
a e spa se, he numbe
o
unknowns in eqn.
8
is much smalle
han
N.
Con e sely, he e a e
N
del a me ons, and hus
N
equa-
ions in he sys em eqn.
8.
This
makes he sys em o e de e mined
and, in gene al, i canno be sol ed exac ly,
[Z,q
#
0.
The mini-
mum
~~[Zn~~~*
is ob ained by he leas squa es p ocedu e.
I mus be no ed ha del a me ons a e equi alen o he
G*
me ons o Jb ic and Lee
[SI,
aking in o accoun he ac ha he
combined impedance ma ix
Zc
plays he ole o he null- ield
En",,
in
[8].
2.5
al
m
0.
-
2
1.5
c
._
I
.
7x
1
.o
'
c
0
0.5
.
0'
I
-200
-1
00
0
100
200
e
187411
Fig.
1
Induced cu en in a sphe e
o
diame e
2h
o
inciden plane
wa e
IE-ME1 esul s compa ed wi h analy ical solu ion
analy ical
IE-ME1
____
Wi h del a me ons, he ope a ion coun o ob ain
[A]
,
[B]
ma ices is p opo ional o
Nz:
Fo each ow o
[A]
and [B], a sys-
em o
N
simul aneous equa ions (eqn.
8)
mus be sol ed by leas
squa es wi h an ope a ion coun o he o de o
aN
x
1Mz
+
/3M,
whe e
M
is he numbe o
a,,,
On,
coe icien s o compu e.
Howe e , we ha e ound ha o cing
z,",
=
0
only
o he
P
-
2M
basis unc ions
i
closes o es ing unc ion
n
leads o he same
alues o coe icien s
a,,,
ba n.
Tha is, o cing
z,"I
=
0
only o
nea ield mu ual impedances makes i also ue o a ield ones.
Now, he numbe o simul aneous equa ions
P
does no inc ease
wi h he numbe o
unknowns
N:
we need
aP
x
Mz
+
pM3
=
AF
ope a ions o each ow o [A],
[B]
and he o al ope a ion coun is
hus p opo ional o
N.
Resul s:
The p oposed IE-ME1 echnique has been applied o se -
e al sca e ing
3D
p oblems. I has been ound ha he accu acy
ob ained wi h ou cu en o mula ion is p oblem-dependen ,
which con iis ha he selec ion o he me on se is he c i ical
s ep.
As
a ypical-case example, Fig.
1
shows he cu en induced
in a sphe e o diame e 2
h
by a plane wa e inciden along he
z
axis, wi h he elec ic ield pola ised along he
2
di ec ion. The IF-
ME1 esul s a e compa ed wi h he analy ical solu ion. The plo
co esponds o
a
cu o
J,
in he
yz
plane, wi h he o igin a he
cen e o he sphe e. The su ace has been disc e ised in
2048
i-
angles wi h sides anging om 0.1
h
o 0.15
h.
The numbe o
unknowns
o Rao, Wil on
&
Glisson basis unc ions
[9]
is
N
=
3072.
The spa si y o
[A], [B]
ma ices is 1.2%. The o al amoun
o memo y used is 6MB.
The accu acy o he IE-ME1 esul is easonably good, bu
poin wise accu acy could no be enough o some angles. Fo
hese cases, a subs an ial imp o emen could ce ainly be ob ained
by using
a
di e en se o me ons ha p oduces
a
smalle esidual
[ZTCl.
Acknowledgmen s:
The au ho is indeb ed o
K.K.
Mei
o
Ci y
Uni e si y
o
Hong Kong o he aluable discussions on he ME1
me hod.
This
wo k was pa ially suppo ed by he Spanish
'Comisibn In e minis e ial de Ciencia
y
Tecnologia' (CICYT)
unde p ojec TIC 95-0983. Pa o he wo k was done a Ci y
Uni e si y o Hong Kong, whe e J.M. Rius was a isi ing ellow
om Janua y
4
o Feb ua y 3, 1997. G adua e s uden s J. Pa bn
and
E.
Ubeda a e suppo ed by he Gene ali a de Ca alunya,
Comissiona pe a Uni e si a s i Rece ca, unde g an s 1997
FI
00679 and 1997 FI 00747, espec i ely.
0
IEE 1997
Elec onics Le e s Online
Nol:
19971419
J.M.
Rius,
J.
Pa on and E. Ubeda
(Depa men
o
TSC, Ed$ci
0-3,
Uni e si a Poli &cnica de Ca alunya, Jo di Gi ona
1-3,
08034
Ba celona, Spain)
5
Augus
1997
J.R.
Mosig
(LEMA,
ELB, Ecole Poly echnique
FkdP ale
de Lausanne
(EPFL),
CH-IO15
Lausanne, Swi ze land)
Re e ences
1
RIUS, J.M.:
'In eg al o mula ion o he measu ed equa ion
o
in a iance',
Elec on. Le .,
1996,
32,
(l),
pp. 23-25
2
RIUS,
J.M.,
POUS,
R.,
and
CARDAMA,
A.:
'In eg al o mula ion o he
measu ed equa ion o in a iance: a no el spa se-ma ix bounda y
elemen me hod',
ZEEE T ans.
Magn.,
1996,
32,
(3),
pp. 962-967
3
RIUS,
J.M.,
CARPINTERO, c.P., CARDAMA,
A.,
and
MOSIG, J.R.:
'F quency ex apola ion in
he
in eg al equa ion MEI',
Elec on.
Le .,
1996,
32,
(25),
pp.
2324-2326
4 MEI,
KK.,
POUS,
R.,
CHEN,
z.,
LIU,
Y.W,
and
PROUTY,
M.D.:
'The
measu ed equa ion o in a iance:
a
new concep in ield
compu a ions',
IEEE T ans. An ennas P opag.,
1994,
42,
(3),
pp.
320-328
abso bing bounda y condi ions',
IEEE T ans. An ennas P opag.,
1995,
43,
(5), pp. 478437
6
RIUS,
J.M.,
CARPINTERO, c.P., CARDAMA, A.,
and
MOSIG, J.R.:
'The
heo e ical e o in he in eg al equa ion MEI',
Elec on. Le .,
1996,
32,
(23),
pp. 2131-2132
7
RIUS,
J
M.,
CARPINTERO,
c
P
,
CARDAMA,
A.,
and
MICHALSKI, K.A.:
'Analysis o elec ically la ge conca e sca e e s wi h he in eg al
equa ion MEI',
Mic ow. Op . Technol. Le .,
1997,
14,
(5),
pp. 287-
289
5
STUPFEL,
B.,
and
MITTRA, R.:
'A heo e ical s udy
O
nume ical
2030
ELECTRONICS LETTERS
20 h
No embe
7997
Vol.
33
No.
24
8
JEVTIC,
J.O.,
and
LEE,
R.:
'A
heo e ical and nume ical analysis
o
he
measu ed equa ion o in a iance',
IEEE T ans. An ennas P opag.,
9
RAO,
s.M.,
WILTON,
D.R.,
and
GLISSON,
A.w.:
'Elec omagne ic
sca e ing by su aces o a bi a y shape',
IEEE T ans. An ennas
P opag.,
1982,
30,
(3),
pp. 409418
1994,
42,
(8),
pp. 1097-1105
Measu ing image edge de ec o accu acy
using ealis ically simula ed edges
R.C.
S aun on
Indexing e ms: Edge de ec ion, Image p ocessing
Edge de ec o es s using edges de i ed om he acquisi ion
sys em's edge sp ead unc ion a e desc ibed. The es edges used
a e he mos di icul ha could exis in a p ac ical sys em. The
esul s we e signi ican ly di e en o hose ob ained by adi ional
me hods, indica ing ha simple de ec o s may
su ice
o some
applica ions.
In oduc ion:
The analogue on end componen s o a digi al
imaging sys em lowpass il e he signals hey p ocess. The cam-
e a's lens and i s cha ge coupled de ice (CCD) a ay a e wo-
dimensional (2D) il e s, whe eas he came a elec onics and he
digi ise 's an ialiasing d e a e one-dimensional (ID). Thei com-
bined e ec , he on end modula ion ans e unc ion (MTF),
can be measu ed in 2D [I] and has been used he e o enable he
simula ion o ealis ic edges on which o es and compa e de ec-
o s.
Edge
model: S ep edges a e conside ed o be he mos di icul
edges o de ec accu a ely [2] and, a e modi ica ion o allow o a
pixel shaped windowing o he da a while sampling, ha e been
used o es de ec o s
[3,
41.
He e, he windowing is eplaced by a
ealis ic physical model o modi ying he s ep edge based
on
he
sys em's edge sp ead unc ion (ESF). The ESF can be measu ed
uniquely o each sys em and is one s ep in he calcula ion o he
MTF. Resul s om he wo edge models ha e been compa ed
he e,
as
ha e he g adien and angula accu acy o a numbe o
commonly used i s -o de de ec o s. The high esolu ion ESF was
scanned e ically h ough
a
2D space o p oduce an e ec i ely
con inuous image o an edge.
A
sampling g id was hen applied a
en housand andom posi ions and o ien a ions wi h espec o
~
A
60
50
40
30
20
signal- o-noise a io,
dB
Fig.
I
E ec o noise
on
SD
o g adien es ima ion (each mean alue
scaled
o
100)
(i)
s anda d cubic ace
5
x
5
(ii) Sobel
3
X
3
(iii) P ewi 5
X
5
(i ) Sobel 5
X
5
( ) In eg a ed di ec ional de i a i e (IDD) 5
X
5
( i) IDD 7
X
7
Edge con as
=
255 uni s
ELECTRONICS LETTERS
20 h
No embe
1997
Vol.
his edge. The esampled images we e cons ained
so
ha he edge
always passed h ough he cen al pixel, and we e used o p o ide
accu a e es da a.
The new edge model was ob ained om he
ID
ESF
by com-
bining he esul s om many CCD elemen s, and he es da a
we e easonably noise ee. Gaussian dis ibu ed andom noise
was added o he sampled in ensi y alues. The signal- o-noise
a io was de ined as
SNR
=
2010g,,
(5)
[dB]
CT
whe e
c
=
255 is he edge con as and
CJ
he s anda d de ia ion
(SD)
o
he noise.
L
ioo
.-
I
.-
a
l
I
( i)
_,.,
,"
I
,"
60
50
40
30
20
signal- o-mise m io
1764/2/
Fig. 2
E ec o noise
on
SD
o angle
ewe
T aces labelled as in Fig. 1. Edge con as
=
255 uni s
Ope a o pe o mance:
Fo i s de i a i e de ec o s, e o s occu
in g adien and angle measu emen s. Pai s o de ec o s we e used
o e u n ho izon al and e ical g adlien ec o s. These we e ana-
lysed in he usual way o p o ide he magni ude
o
he edge g adi-
en and he angle in o ma ion. The 2D MTF o he acquisi ion
sys em
[I]
was consul ed in choosing he mos sui able ESF, he
sha pes and hence mos di icul o de ec , on which o es he
ope a o s. The ope a o empla es aned
in
size om
3
x
3
o
7
x
7
pixels and a e desc ibed
in
[4]. A
1,a ge empla e should ha e a
highe noise immuni y and p oduce mo e accu a e esul s, bu
may ake longe o compu e.
10'
a
?i
C
.-
I
'5
U
E
U
m
c
L
OI
-
ioo
60
50
4CI
30
20
signal- o-noise a io,
dB
/7Mi3]
Pig.
3
Compa ison o e ec o noise
on
SD
o g adien es ima ion o
wo edge models (each mean alue scaled o
100)
Pixel a e age model:
(i) Sobel
3
x
3
(ii) IDD
5
x
5
(iii) IDD 7
x
7
ESF model:
(i ) Sobel
3
X
3
( ) IDD 5
X
5
( i) IDD 7
X
7
33
No.
24
203
1