cmpe 413/cmsc 711 - electrical & computer...
TRANSCRIPT
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
1(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dig
ital
Dev
ice
Com
pone
nts
Asi
mpl
epr
oces
sor
illus
trat
esm
any
ofth
eba
sic
com
pone
nts
used
inan
ydi
g-it
al s
yste
m:
• D
atap
ath:
The
cor
e --
all
othe
r co
mpo
nent
s ar
e su
ppor
t uni
ts th
at s
tore
eith
er th
e re
sult
s of
the
data
path
or
dete
rmin
e w
hat h
appe
ns in
the
next
cycl
e.
Con
trol
Mem
ory
Dat
apat
h
Input-Output
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
2(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dig
ital
Dev
ice
Com
pone
nts
• M
emor
y:A
bro
ad r
ange
of c
lass
es e
xist
det
erm
ined
by
the
way
dat
a is
acc
esse
d:R
ead-
Onl
y vs
. Rea
d-W
rite
Sequ
enti
al v
s. R
ando
m a
cces
sSi
ngle
-por
ted
vs. M
ulti
-por
ted
acce
ssO
r by
thei
r da
ta r
eten
tion
cha
ract
eris
tics
:D
ynam
ic v
s. S
tati
cSt
ay tu
ned
for
a m
ore
exte
nsiv
e tr
eatm
ent o
f mem
orie
s.
• C
ontr
ol:
A F
SM (s
eque
ntia
l cir
cuit
) im
plem
ente
d us
ing
rand
om lo
gic,
PLA
s or
mem
orie
s.
• In
terc
onne
ct a
nd In
put-
Out
put:
Para
siti
c re
sist
ance
, cap
acit
ance
and
indu
ctan
ce a
ffec
ts p
erfo
rman
ce o
fw
ires
bot
h on
and
off
the
chip
.G
row
ing
die
size
incr
ease
s th
e le
ngth
of t
he o
n-ch
ip in
terc
onne
ct,
incr
easi
ng th
e va
lue
of th
e pa
rasi
tics
.
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
3(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dig
ital
Dev
ice
Com
pone
nts
Dat
apat
h el
emen
ts in
clud
e ad
ders
, mul
tipl
iers
, shi
fter
s, B
FUs,
etc
.Th
e sp
eed
of th
ese
elem
ents
oft
en d
omin
ates
the
over
all s
yste
m p
erfo
r-m
ance
so
opti
miz
atio
n te
chni
ques
are
impo
rtan
t.
How
ever
, as
we
will
see
, the
task
is n
on-t
rivi
al s
ince
ther
e ar
e m
ulti
ple
equi
vale
nt lo
gic
and
circ
uit t
opol
ogie
s to
cho
ose
from
, eac
h w
ith
adv.
/di
sadv
. in
term
s of
spe
ed, p
ower
and
are
a.
Als
o, o
ptim
izat
ions
focu
sed
at o
ne d
esig
n le
vel,
e.g.
, siz
ing
tran
sist
ors,
lead
s to
infe
rior
des
igns
.
Data-In
Registers
Adder
Shifter
Multiplexer
Data-Out
Con
trol
Bit-
slic
ed o
rgan
izat
ion
is c
omm
on fo
r da
tapa
ths.
Bit
0
Bit
1
Bit
3B
it 4
Bit
2
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
4(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: A
ddit
ion/
Subt
ract
ion
Let’s
sta
rt w
ith
addi
tion
, sin
ce it
is a
ver
y co
mm
on d
atap
ath
elem
ent a
ndof
ten
a sp
eed-
limit
ing
elem
ent.
Opt
imiz
atio
ns c
an b
e ap
plie
d at
the
logi
c or
cir
cuit
leve
l.Lo
gic-
leve
lopt
imiz
atio
ntr
yto
rear
rang
eth
eBo
olea
neq
uati
ons
topr
oduc
ea
fast
er o
r sm
alle
r ci
rcui
t, e.
g. c
arry
look
-ahe
ad a
dder
.C
ircu
it-l
evel
opti
miz
atio
nsm
anip
ulat
etr
ansi
stor
size
san
dci
rcui
ttop
olog
yto
opt
imiz
e sp
eed.
Let’s
sta
rt w
ith
som
e ba
sic
defin
itio
ns b
efor
e co
nsid
erin
g op
tim
izat
ions
:
Ci
BA
G(A
.B)
P(A
+B)
P’(A
+ B
)Su
mC
o
00
00
00
00
00
1
00
01
00
10
01
11
0
01
1
01
10
1
10
0
01
11
01
01
01
10
1
11
0
11
00
11
11
11
01
1
dele
teC
arry
sta
tus
dele
tepr
opag
ate
prop
agat
epr
opag
ate
prop
agat
ege
nera
tege
nera
te
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
5(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: A
ddit
ion/
Subt
ract
ion
G(A
.B):
(gen
erat
e)O
ccur
s w
hen
a C
ois
inte
rnal
ly g
ener
ated
wit
hin
the
adde
r (o
ccur
s in
de-
pend
ent o
f Ci).
P(A
+B):
(pro
paga
te)
Indi
cate
s th
at C
i is
prop
agat
ed(p
asse
d) to
Co.
P’(A
XO
R B
): (p
ropa
gate
)U
sed
inso
me
adde
rsfo
rth
eP
term
sinc
eit
can
bere
used
toge
nera
teth
esu
m te
rm.
D(A
.B):
(del
ete)
Ensu
res
that
a c
arry
bit
will
be
dele
ted
at C
o.
The
Bool
ean
expr
essi
ons
for
S an
d C
o ar
e:
Sum
= A
.B.C
i + A
.B.C
i +A
.B.C
i +A
.B.C
i = A
XO
R B
XO
R C
Car
ry =
A.B
+ A
.Ci +
B.C
i
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
6(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: A
ddit
ion/
Subt
ract
ion
But S
and
Co
can
be w
ritt
en in
term
s of
G a
nd P
’:
Co(
G, P
’) =
G +
P’C
i (o
r P
in th
is c
ase)
.
S(G
, P’)
= P’
XO
R C
i
Not
e th
at G
and
P’ a
re IN
depe
nden
t of C
i.
(Als
o, C
o an
d S
can
be e
xpre
ssed
in te
rms
of d
elet
e (D
)).
Rip
ple-
carr
y ad
der:
The
crit
ical
pat
h (w
orst
cas
e de
lay
over
all
poss
ible
inpu
ts) i
s a
ripp
le fr
omls
b to
msb
.
Ci,0
Co,
0
=Ci,1
A0
B 0
S 0
Co,
1
A1
B 1
S 1
Co,
2
A2
B 2
S 2
Co,
3
A3
B 3
S 3
FAFA
FAFA
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
7(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: A
ddit
ion/
Subt
ract
ion
The
dela
y in
this
cas
e is
pro
port
iona
l to
the
num
ber
of b
its,
N, i
n th
e in
put
wor
ds:
t add
er =
(N -
1)t c
arry
+ t s
um
whe
re t c
arry
and
t sum
are
the
prop
agat
ion
dela
ys fr
om C
i to
Co
& S
.
One
pos
sibl
e w
orst
cas
e bi
t pat
tern
(fro
mls
b to
msb
) is:
A: 0
0000
001;
B: 0
1111
111
Con
vinc
e yo
urse
lf th
at th
is is
true
.
Not
eth
atw
hen
opti
miz
ing
this
stru
ctur
e,it
isfa
rmor
eim
port
antt
oop
tim
ize
t car
ry th
an t s
um.
The
inve
rtin
g pr
oper
ty o
f a fu
ll ad
der
can
be u
sed
to a
chie
ve th
is g
oal:
Ci
Co
AB
FA
S
Ci
Co
AB
FA
S
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
8(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: A
ddit
ion/
Subt
ract
ion
Thus
, S(A
, B, C
i) =
S(A
,B,C
i)
Co(
A, B
, Ci)
= C
o(A
,B,C
i)
One
pos
sibl
e (u
n-op
tim
ized
) im
plem
enta
tion
:
Ci
A B
A BC
o
A B Ci
STr
ansi
stor
leve
l dia
gram
use
s32
tran
sist
ors.
(see
Wes
te a
nd E
shra
ghia
n).
Ci.P
(A +
B)
G(A
.B)
P’ X
OR
Ci
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
9(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: A
ddit
ion/
Subt
ract
ion
Co
is r
euse
d in
the
S te
rm a
s:
Sum
= A
.B.C
i + (A
+ B
+ C
i)Co
Even
wit
h so
me
desi
gn tr
icks
, e.g
., tr
ansi
stor
s on
the
crit
ical
pat
h, C
i pla
ced
clos
est t
o th
e ou
tput
and
sym
met
rica
l des
ign,
this
impl
emen
tati
on is
slo
w.
AB
Ci
ABA BC
i
AB
Ci
AB
A B Ci
Ci
BA
AB
Ci
Co
S
Are
the
n an
d p
tree
s du
als
of e
ach
othe
r?28
tran
sist
ors
Co
Sym
met
rica
lde
sign
elim
inat
esdi
ffus
ion
caps
and
redu
ces
seri
es R
.
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
10(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: A
ddit
ion/
Subt
ract
ion
The
load
cap
acit
ance
in p
revi
ous
vers
ion
on C
o co
nsis
ts o
f 2 d
iffus
ion
capa
ci-
tanc
es (i
nver
ter)
and
6 (n
ext b
it) g
ate
capa
cita
nces
:
Thi
sve
rsio
nin
crea
ses
Co’
slo
adto
4di
ffus
ion
caps
,2in
tern
al(s
um)g
ate
caps
plus
the
6 (n
ext b
it) g
ate
caps
.
CinC
<n>
C<3
>S<
3>
S<2>
S<1>
S<0>
S<n>
B<n>
A<n
>
B<3>
A<3
>
A<2
>
A<1
>
A<0
>
B<2>
B<1>
B<0>
Sign
of
C<n
+1>
Ove
rflow
C<3
>S<
3>
S<2>
S<1>
S<0>
B<3>
A<3
>
A<2
>
A<1
>
A<0
>
B<2>
B<1>
B<0>
the
resu
lt
Subt
ract
Elim
inat
es th
e in
vert
er d
elay
per
bit
for
carr
y!
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
11(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: A
ddit
ion/
Subt
ract
ion
Seri
al a
dditi
on c
an b
e us
ed if
are
a is
a c
once
rn:
Inth
isca
se,y
ouw
ante
qual
Sum
and
Car
ryde
lays
inor
dert
om
inim
ize
cloc
kcy
cle
tim
e.
Bit-
leve
l pip
elin
ing
can
be u
sed
to b
reak
the
depe
nden
cy b
etw
een
addi
tion
tim
e an
d th
e nu
mbe
r of
bit
s by
inse
rtin
g FA
s be
twee
n ea
ch r
egis
ter
bit.
n bi
t shi
ft r
egis
ter
Clk
n bi
t shi
ft r
egis
ter
adde
nd
auga
ndC
in
Cou
t
Clk
Set
Clr
Reg
1-bi
t
resu
lt
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
12(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: A
ddit
ion/
Subt
ract
ion
Tran
smis
sion
-gat
eA
dder
:
Not
e: S
and
Co
dela
y ti
mes
are
app
roxi
mat
ely
equa
l -- g
ood
for
mul
tipl
iers
.
See
Wes
te a
nd E
shra
ghia
n fo
r an
18
tran
sist
or im
plem
enta
tion
.
XO
R
XN
OR
AB Ci
S Co
Tota
l tra
nsis
tors
is 2
6
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
13(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: A
ddit
ion/
Subt
ract
ion
Dyn
amic
Add
er D
esig
n:np
-CM
OS
adde
r
A0
B 0
Ci0C
i
A1
B 1
S 0φ
φA0B 0
φφφ
φφA1
B 1
Ci1Ci2
φφφ
Ci0
A0
B 0
Ci0
A0
B 0φφ
A1
φ B 1C
i1
φ
Ci1 A
1
B 1
S 1
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
14(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: A
ddit
ion/
Subt
ract
ion
Dyn
amic
Add
er D
esig
n:M
anch
este
r C
arry
-Cha
in a
dder
.A
cha
in o
f pas
s-tr
ansi
stor
s ar
e us
ed to
impl
emen
t the
car
ry c
hain
.
Prec
harg
e: A
ll in
term
edia
te n
odes
, e.g
. Co,
0, c
harg
ed to
VD
D.
Eval
uate
: Nod
eC
o,k
is d
isch
arge
d, fo
r ex
ampl
e, if
ther
e is
an
inco
min
g
carr
y, C
i,0 a
nd th
e pr
evio
us p
ropa
gate
sig
nals
are
hig
h, P
0 to
Pk-
1.
Onl
y 4
diff
usio
n ca
paci
tanc
es a
re p
rese
nt p
er n
ode
but t
he d
istr
ibut
ed R
C-
natu
re o
f the
cha
in r
esul
ts in
del
ay th
at is
qua
drat
ic w
ith
num
ber
of b
its.
Buff
ers
and/
or tr
ansi
stor
siz
ing
can
be u
sed
to im
prov
e pe
rfor
man
ce.
φ Ci,0
φ
P 0
G0
P 1
G1
P 2
G2
P 3
G3
P 4
G4
Co,
4C
o,0
Co,
1C
o,2
Co,
33
2.5
21.
51
3.5 4
3.5
32.
52
1.5
32.
52
1.5
1
Co,
4
Tran
sist
or s
izes
larg
est h
ere
sinc
e w
orst
cas
e is
to d
isch
arge
all
node
sC
o,k.
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
15(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: A
ddit
ion/
Subt
ract
ion
Con
side
r th
e w
orst
cas
e de
lay
of th
e ca
rry
chai
n:
Elm
ore
dela
y is
giv
en b
y:
The
dela
y of
the
RC
net
wor
k is
then
:t p
= 0
.69(
C1R
1 +
C2(
R1
+ R
2) +
C3(
R1
+ R
2 +
R3)
+ C
4(R
1 +
R2
+ R
3 +
R4)
+
C5(
R1
+ R
2 +
R3
+ R
4 +
R5)
+ C
6(R
1 +
R2
+ R
3 +
R4
+ R
5 +
R6)
Sinc
e R
1 ap
pear
s6
tim
es in
the
expr
essi
on, i
t mak
es s
ense
to m
inim
ize
its
cont
ribu
tion
.
Not
eth
atre
duci
ngR
bya
fact
or,e
.g.k
,ate
ach
stag
ein
crea
ses
the
capa
cita
nce
by a
fact
ork
and
incr
ease
s ar
ea.
Ak-
fact
or o
f 1.5
, red
uces
del
ay b
y 40
% a
nd in
crea
ses
area
by
3.5X
.
R1
R2
R3
R4
R5
R6
C1
C2
C3
C4
C6
C5
Ou
t
t p0.
69C
ii
1=N ∑
Rj
j1
=i ∑
=
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
16(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: A
ddit
ion/
Subt
ract
ion
Car
ry-B
ypas
s ad
der:
Ass
ume
Ak
and
B k (f
or k
= 1
...3)
are
set
suc
h th
at a
ll P k
(pro
paga
te) a
re
high
.In
this
cas
e, a
n in
com
ing
carr
y C
i,0 =
1, p
ropa
gate
s al
ong
the
com
-
plet
e ch
ain
and
Co,
3=
1.
In o
ther
wor
ds:
if(P
0P1P
2P3
==1)
then
Co,
3=
Ci,0
else
eith
erD
ELET
Eor
GEN
ERA
TE
occu
rred
.
Ci,0
Co,
0
P 0G
0
Co,
1
P 1G
1
Co,
2
P 2G
2
Co,
3
P 3G
3
FAFA
FAFA
Mux
BP =
P0P
1P2P
3
Co,
3
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
17(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: A
ddit
ion/
Subt
ract
ion
Line
ar C
arry
-Sel
ect a
dder
:O
new
ayar
ound
wai
ting
for
the
inco
min
gca
rry
isto
com
pute
the
resu
ltof
both
poss
ible
valu
esin
adva
nce
and
lett
hein
com
ing
carr
yse
lect
the
corr
ect r
esul
t.
A S
quar
e-R
oot C
arry
-Sel
ectA
dder
(del
ay =
O(N
1/2 ))
is c
onst
ruct
ed b
yin
crea
sing
the
num
ber
of in
put b
its
in e
ach
bloc
k fr
omls
bto
msb
.
Setu
p
0-ca
rry
prop
agat
ion
1-ca
rry
prop
agat
ion
Mux
Sum
Gen
erat
ion
Car
ry v
ecto
r
Co,
k-1
Co,
k+3
0 1
P,G
This
blo
ck a
dds
bits
k to
k+3
.
Sele
ct o
pera
tion
is m
uch
fast
er th
anti
me
to c
ompu
te e
ithe
r of
the
two
poss
ible
car
ry v
ecto
rs.
For
Squa
re-R
oot C
arry
-Sel
ect,
high
er o
rder
blo
cks
take
mor
eop
eran
d bi
ts th
an lo
wer
ord
erbl
ocks
.
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
18(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: A
ddit
ion/
Subt
ract
ion
Car
ry lo
ok-a
head
add
er (a
void
ing
the
ripp
le a
ltog
ethe
r):
Com
pute
the
carr
ies
to e
ach
stag
e in
par
alle
l.
Not
e th
at th
e lo
w-o
rder
term
s, e
.g.,
P 0 a
nd G
0, a
ppea
r in
the
expr
essi
on fo
r
ever
y bi
t, m
akin
g th
e fa
nout
load
larg
e.
Co,
k =
Gk
+ P k
. C
o,k-
1w
here
Gk
= A
k . B
kP k
= A
k +
B k
The
carr
y ou
t of t
he k
th s
tage
is c
ompu
ted
as:
For
exam
ple,
for
4 st
ages
of l
ook-
ahea
d:C
0 =
G0
+ P 0
Ci
C1
= G
1 +
P 1G
0 +
P 1P 0
Ci
C2
= G
2 +
P 2G
1 +
P 2P 1
G0
+ P 2
P 1P 0
Ci
C3
= G
3 +
P 3G
2 +
P 3P 2
G1
+ P 3
P 2P 1
G0
+ P 3
P 2P 1
P 0C
i
The
depe
nden
cy b
etw
een
Co,
k an
d C
o,k-
1 ca
n be
elim
inat
ed b
yex
pand
ing
Co,
k-1.
Co,
k =
Gk
+ P k
. (G
k-1
+ P k
-1.C
o,k-
2)
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
19(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: A
ddit
ion/
Subt
ract
ion
Car
ry lo
ok-a
head
add
er:
One
pos
sibl
e im
plem
enta
tion
wit
hout
usi
ng s
impl
e lo
gic
gate
s.
Size
and
fan-
in o
f the
gat
es li
mit
the
size
to a
bout
four
.
P 0 P 1 P 2 P 3Ci,0
G3
G2
G1
G0
C0,
3
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
20(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: A
ddit
ion/
Subt
ract
ion
Car
ry lo
ok-a
head
add
er:
C3
= G
3 +
P 3(G
2 +
P 2(G
1 +
P 1(G
0 +
P 0C
i,0))
)
Fact
orin
g te
rm C
3 yi
elds
:
Dom
ino
CM
OS
impl
emen
tati
on:
Clk
P<0>
Ci,0
G<0
>
P<1>
G<1
>
P<2>
G<2
>
P<3>Clk
G<3
>
C<3
>
Wor
st c
ase
is p
ull-
dow
nth
roug
h 6
seri
es n
-cha
nnel
Oth
er h
igh
spee
d ve
rsio
nsgi
ven
in W
este
and
Esh
ragh
ian.
tran
sist
ors.
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
21(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: A
ddit
ion/
Subt
ract
ion
The
Loga
rith
mic
look
-ahe
ad a
dder
: O(l
og2N
) del
ay:
The
num
ber
of lo
gic
leve
ls is
pro
port
iona
l to
log 2
N, f
an-i
n is
lim
ited
and
the
layo
ut is
com
pact
(jig
saw
puz
zle)
(see
Rab
aey
for
deta
ils).
(G0,
P0)
(G1,
P1)
(G2,
P2)
(G3,
P3)
(G4,
P4)
(G5,
P5)
(G6,
P6)
(G7,
P7)
Co,
0
Co,
1C
o,2
Co,
3
(C4-
7,P 4
-7)
Co,
4
Co,
5
Co,
6
Co,
7
Forw
ard
bina
ry tr
ee
Inve
rse
bina
ry tr
ee
The
dot o
pera
tor
( )
is d
efine
d as
: (g,
p) .
(g’,
p’) =
(g +
pg’
, pp’
)
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
22(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: C
ompa
riso
nM
agni
tude
Com
para
tors
:M
ay b
e bu
ilt fr
om a
n ad
der,
com
plem
ente
r (X
OR
gat
es) a
nd a
zer
ode
tect
uni
t.
Thi
nkab
outt
hem
odifi
cati
ons
nece
ssar
yto
mak
eit
asi
gned
com
para
tor
(Hin
t: A
cou
ple
of X
OR
gat
es).
B<3>
A<3
>
A<2
>
A<1
>
A<0
>
B<2>
B<1>
B<0>
B =
A
Zer
o de
tect
NO
R g
ate.
B >=
A
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
23(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: B
inar
y C
ount
ers
Asy
nchr
onou
s: B
ased
on
the
Togg
le r
egis
ter.
Not
a g
ood
choi
ce fo
r pe
rfor
man
ce a
nd te
stab
ility
(wit
h no
res
et).
CQ Q
TQ
TQ
TQ
TQ
TQ
TQ
TQ
TQ
Q<0
>Q
<1>
Q<2
>Q
<3>
Clk ClkTT
TT
TT
TT
Q<3
>
"Rip
ple
Car
ry"
Bin
ary
coun
ter
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
24(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: B
inar
y C
ount
ers
Sync
hron
ous
coun
ter.
Rep
lace
AN
D g
ate
wit
h an
add
er fo
r up
/dow
n co
unti
ng c
apab
ility
.W
este
and
Esh
ragh
ian
also
sho
w a
ver
sion
that
can
be
init
ializ
ed.
DQ
1-bi
tR
eg
Clk
Cle
ar
DQ
1-bi
tR
eg10
Clk
Cle
ar
DQ
1-bi
tR
eg10
Clk
Cle
ar
DQ
1-bi
tR
eg10
Clk
Cle
ar
Clk
Q<0
>Q
<1>
Q<2
>Q
<3>
Cle
ar
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
25(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: M
ulti
plic
atio
nM
ulti
plic
atio
n ca
n be
bro
ken
dow
n in
to tw
o st
eps:
Com
puta
tion
of p
arti
al p
rodu
cts.
Acc
umul
atio
n of
the
shif
ted
part
ial p
rodu
cts.
Mul
tipl
iers
may
becl
assi
fied
byth
efo
rmat
inw
hich
data
wor
dsar
eac
cess
ed:
Ser
ial
Ser
ial/
para
llel
Par
alle
l
The
para
llel f
orm
com
pute
s th
e pa
rtia
l pro
duct
s in
par
alle
l.
1100
0101
1100
0000
1100
0000
X 0111
100
Bina
ry m
ulti
plic
atio
n eq
uiva
lent
toA
ND
ope
rati
on
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
26(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: M
ulti
plic
atio
nPa
ralle
l Uns
igne
d M
ulti
plic
atio
n:
XX
i2i
i0
=
m1
– ∑=
YY
j2j
j0
=
n1
– ∑=
Mul
tipl
ying
2 u
nsig
ned
bina
ry in
tege
rs r
esul
ts in
:
PX
Y×
Xi2i
Yj2
j
j0
=
n1
– ∑i
0=
m1
– ∑P
k2k
k0
=
mn
1–
+ ∑=
==
X3
X
2
X1
X
0Y
3
Y2
Y
1
Y0
X3Y
0 X
2Y0
X1Y
0 X
0Y0
X3Y
1 X
2Y1
X1Y
1 X
0Y1
X3Y
2 X
2Y2
X1Y
2 X
0Y2
X3Y
3 X
2Y3
X1Y
3 X
0Y3
P 7
P 6
P 5
P 4
P 3
P
2
P 1
P 0
Ther
e ar
e m
*n s
umm
ands
prod
uced
by
a se
t of m
*nA
ND
gat
es in
par
alle
l.
Mul
tipl
ican
dM
ulti
plie
r
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
27(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: M
ulti
plic
atio
nPa
ralle
l Mul
tipl
icat
ion:
Mul
tipl
icat
ion
is c
arri
ed o
ut u
sing
a b
itw
ise
AN
D o
f the
ope
rand
s, X
i
and
Yi.
Mos
t of t
he w
ork
(and
del
ay) i
s in
sum
min
g th
e pa
rtia
l pro
duct
s.
Sum
the
Co
X A N
xN m
ulti
plie
r re
quir
es:
N(N
-2) f
ull a
dder
sN
hal
f add
ers
N2 A
ND
gat
es
Y
A
BC
i
Mul
tipl
icat
ion
Part
ial p
rodu
cts
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
28(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: M
ulti
plic
atio
nA
rray
mul
tipl
ier:
Y0
X0
X1
X2
X3
P 0Y
1X
0X
1X
2X
3
P 1
HA
FAFA
HA
Y2
X0
X1
X2
X3
P 2
HA
FAFA
FA
Y3
X0
X1
X2
X3
P 3
HA
FAFA
FA
P 4P 5
P 6P 7
Ther
e ar
e a
larg
enu
mbe
r of
nea
rly
iden
tica
l cri
tica
lpa
ths
in th
isci
rcui
t.
MN t m
ult =
(M-1
)+(N
-2)t
carr
y +
(N-1
)tsu
m +
t and
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
29(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: M
ulti
plic
atio
nFr
om th
e de
lay
expr
essi
on a
nd th
e fa
ct th
at a
ll cr
itic
al p
aths
hav
e th
e sa
me
leng
th, m
inim
izin
g t m
ult r
equi
res
min
imiz
ing
both
t car
ry a
nd t s
um.
This
is in
con
tras
t wit
h th
e ad
der
whe
re m
inim
izin
g t c
arry
was
key
.
The
tran
smis
sion
gat
e ad
der
is a
goo
d ch
oice
her
e.
Para
llel S
igne
d M
ulti
plic
atio
n:
Pa–
m1
–2m
1–
a i2i
i0
=
m2
– ∑+
b n1
––
2n1
–b i2i
i0
=
n2
– ∑+
=
a m1
–b n
1–
2mn
2–
+a ib
j2ij
+
j0
=
n2
– ∑i
0=
m2
– ∑a ib n
1–
2n1
–i
+
i0
=
m2
– ∑–
a m1
–b i2m
1–
i+
i0
=
n2
– ∑–
+=Ba
ugh-
Woo
ley
algo
rith
m:
Aa m
1–
2m1
–a i2i
i0
=
m2
– ∑+
–=
Bb m
1–
2m1
–b i2i
i0
=
m2
– ∑+
–=
Expa
ndin
g sh
ows
that
the
last
two
row
s of
sim
ply
adds
in th
eir
nega
tion
s.
Onl
y 3
addi
tion
al a
dder
s re
quir
ed o
ver
the
unsi
gned
ver
sion
.
sum
man
ds a
re a
ll ne
gati
ve s
o th
e al
gori
thm
Let A
and
B r
epre
sent
sig
ned
inte
gers
.
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
30(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: M
ulti
plic
atio
nPa
ralle
l Sig
ned
Mul
tipl
icat
ion:
a7b7 ()ADD
a7b7 ()ADD
a7b1 ()AND
a7b7 ()AND
a7b6 ()AND
a7b5 ()AND
a7b4 ()AND
a7b3 ()AND
a7b2 ()AND
a3b4 ()
ANDADD
a3b0 ()AND
a7b0 ()AND
a5b0 ()AND
a4b0 ()AND
a2b0 ()AND
a1b0 ()AND
a0b0 ()AND
a6b0 ()AND
P0
P1
P2
P3
P4
P5
P6
a6b1 ()
ANDADD
a5b1 ()
ANDADD
a4b1 ()
ANDADD
a3b1 ()
ANDADD
a2b1 ()
ANDADD
a1b1 ()
ANDADD
a0b1 ()
ANDADD
a6b2 ()
ANDADD
a5b2 ()
ANDADD
a4b2 ()
ANDADD
a3b2 ()
ANDADD
a2b2 ()
ANDADD
a1b2 ()
ANDADD
a0b2 ()
ANDADD
a6b3 ()
ANDADD
a5b3 ()
ANDADD
a4b3 ()
ANDADD
a3b3 ()
ANDADD
a2b3 ()
ANDADD
a1b3 ()
ANDADD
a0b3 ()
ANDADD
a6b4 ()
ANDADD
a5b4 ()
ANDADD
a4b4 ()
ANDADD
a2b4 ()
ANDADD
a1b4 ()
ANDADD
a0b4 ()
ANDADD
a6b5 ()
ANDADD
a5b5 ()
ANDADD
a4b5 ()
ANDADD
a3b5 ()
ANDADD
a2b5 ()
ANDADD
a1b5 ()
ANDADD
a0b5 ()
ANDADD
a6b6 ()
ANDADD
a5b6 ()
ANDADD
a4b6 ()
ANDADD
a3b6 ()
ANDADD
a2b6 ()
ANDADD
a1b6 ()
ANDADD
a0b6 ()
ANDADD
a6b7 ()
ANDADD
a5b7 ()
ANDADD
a4b7 ()
ANDADD
a3b7 ()
ANDADD
a2b7 ()
ANDADD
a1b7 ()
ANDADD
a0b7 ()
ANDADD
ADD
ADD
ADDADDADDADDADDADD
P8
P7 P9 P10 P11 P12 P13 P14 P15
a7a7a6a5a4a3a2a1a0 a6a5a4a3a2a1a0
b0b0
b1b1
b2b2
b3b3
b4b4
b5b5
b6b6
b7b7
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
31(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: M
ulti
plic
atio
nC
arry
-Sav
e M
ulti
plie
r:C
arry
bit
s ca
n be
pas
sed
diag
onal
ly d
ownw
ards
inst
ead
of to
the
left
.
Her
eth
eca
rry
bits
are
noti
mm
edia
tely
adde
dbu
trat
her
“sav
ed”
for
the
next
add
er s
tage
.
HA
HA
HA
HA
FAFA
FAH
A
FAFA
FAH
A
HA
FAFA
HA
Vect
or-m
ergi
ng a
dder
t mul
t = (N
-1)t
carr
y +
t and
+ t m
erge
Cri
tica
l pat
h is
uni
quel
y de
fined
:
Cos
t: A
litt
le e
xtra
area
:
Adv
anta
ge:
(Ass
umin
g t a
dd =
t car
ry).
Min
imiz
ing
t mer
ge is
use
ful,
e.g.
use
car
ry-s
elec
t or
look
ahea
d.
4x4
vers
ion
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
32(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: M
ulti
plic
atio
nSe
rial
Uns
igne
d M
ulti
plic
atio
n:
Seri
al/P
aral
lel U
nsig
ned
Mul
tipl
ier
show
n in
Wes
te a
nd E
shra
ghia
n.
Clk
seri
al r
egis
ter
CinC
lk
Reg
1-bi
t
G2
G1
X Y
rese
t
Xi a
nd Y
i del
iver
ed s
eria
llyto
the
inpu
ts o
f G1
at d
iffer
ent r
ates
.
P7
P0
Com
pute
s th
e su
mm
ands
row
-wis
e fr
om r
ight
to le
ft.
Dis
adv:
Qua
drat
ic d
elay
: tm
ult =
M x
N x
t car
ry
If a
rea
is a
conc
ern.
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
33(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: M
ulti
plic
atio
nBo
oth
Enco
ding
:A
spe
cial
enc
odin
g of
the
mul
tipl
ier
wor
d re
duce
s th
e nu
mbe
r of
requ
ired
add
itio
n st
ages
and
spe
eds
up m
ulti
plic
atio
n su
bsta
ntia
lly.
Rad
ix-4
sch
eme:
The
num
ber
of p
arti
al p
rodu
cts
(and
add
itio
ns) i
s ha
lved
, res
ulti
ng in
area
and
spe
ed a
dvan
tage
.
The
disa
dvan
tage
is a
som
ewha
t mor
e in
volv
ed m
ulti
plie
r ce
ll.A
ND
ope
rati
on r
epla
ced
wit
h in
vers
ion
and
shif
t log
ic.
Vir
tual
ly e
very
mul
tipl
ier
in u
se e
mpl
oys
the
Boot
h sc
hem
e.
YY
j4j
with
Yj
21
01
2,
,,
–,–{
}∈
()
j0
=
N1
–(
)2⁄
∑=
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
34(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: M
ulti
plic
atio
nW
alla
ce M
ulti
plie
r:Tr
ees
can
be u
sed
to r
epla
ce th
e lin
ear
part
ial-
sum
add
ers:
Y0
Y1
Y2
FA FA FA FA
Y3
Y4
Y5
Ci-
1
Ci-
1
Ci-
1
Ci
Ci
Ci
Ci
Sum
Y0
Y1
Y2
Y3
Y4
Y5
# of
rip
ple
stag
es is
N-2
Slic
e of
a 6
-bit
carr
y-sa
ve m
ult.
Ci-
1C
i
Ci
Ci
FAFA
FA FA
Ci-
1
CSu
m
Adv
: O(l
og2N
) mul
t tim
e.D
isad
v: V
ery
irre
gula
r --
diffi
cult
to la
yout
.
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
35(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: Sh
ifte
rsR
ight
/Lef
t 1-b
it s
hift
er:
Rig
ht/L
eft
SS
SS
Mux
Mux
Mux
Mux H
0H
1H
2H
3
01
01
01
01
A3
I R
A2
A1
A0
I L
Prin
cipl
es o
f V
LSI
Des
ign
Subs
yste
m D
esig
nC
MPE
413
/CM
SC 7
11
36(D
ecem
ber
11, 2
000
3:44
pm
)U
MB
CU
M B
C
UNIVERSITY OF M
AR
YL
AN
D B
ALTIM
ORE COUNTY
1 9
6 6
Dat
apat
h O
pera
tors
: Sh
ifte
rsBa
rrel
shi
fter
:s<
3>s<
2>s<
1>s<
0>
l<6:
0>
r<3>
r<2>
r<1>
r<0>
shif
tre
sult
1 2 4 8
l<3:
0>l<
4:1>
l<5:
2>l<
6:3>
Ari
thm
etic
and
logi
cal s
hift
s an
d ro
tate
s po
ssib
leby
mux
ing
l<6:
0> to
the
appr
opri
ate
valu
es.