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7/27/2019 Integer Add and Mult
http://slidepdf.com/reader/full/integer-add-and-mult 1/42
1
E CE 3 6 6
C om p u t e r Ar c h i t
e c t ur e
I n s t r u c t or : S h a n t a n uD u t t
D e p a r t m e n t o
f E l e c t r i c a l a n d C om p u t e r E n gi n e e r i n g
Uni v e r s i t y of I l l i n oi s a t C h i c a g o
L e c t ur e N o t e s # 1 0
C O
MP UT E RARI T H
ME T I C :
I n t e g e r A d d i t i on a n d M ul t i pl i c a t i on
c
S h a n t a n uD u t t , UI C
1
7/27/2019 Integer Add and Mult
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2
ADDE R S
¡
F ul l A d d e r F A :
Y i
X i
C i +1
C i
S i
F A i
I n p u t s
¢
£
¤
¥
£
¤ ¦
£
O u t p u t s a r e
¨
£
¤¦
£
©
, w
h e r e ¨
£
¢
£
¥
£
¦
£
¦
£
©
¢
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¥
£
c
S h a n t a n uD u t t
, UI C
2
7/27/2019 Integer Add and Mult
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3
Ri p pl e - C ar r yA d d e
r ( R CA )
F A 0 y
0
x 0
c 0
y1
x1
c 1
F A1
x2
y2
F A2
c 2
c 3
y 3
x 3 F
A 3
c 4
y4
x4 F
A4
c 5
y 5
x 5
F A 5
c 6
y 6
x 6 F
A 6
F A 6
c 7
y 7
x 7
c o u t
S 7
S 6
S 5
S 4
S 3
S 2
S 1
S 0
P r o b l e m : D e l a yi s
g a t e d e l a y s or
— e a c h F Ah a s a 2 - g a t e d e l a y .T h u s
i s e a c h
g a t e h a s a d e l a y of 2 n s , d e l a yf or a 3
2 - b i t R C Ai s 6 4 n s .
c
S h a n t a n uD u t t
, UI C
3
7/27/2019 Integer Add and Mult
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4
O v e r fl o wi nA d d
i t i on
¡
O v er fl o w o c c ur s wh e n
t h e r e s ul t of t h e o p e r a t i on d o e s n o t fi
t i n t h e r e pr e -
s e n t a t i on b e i n g u s e d
¡
F or
e x a m pl e ,i f 4 - b i t
un s i gn e d n um b e r s 6 =
!
!
"
a n d 1 2 =!
!
"
a r e
a d d
e d t h e s um ( 1 8 ) o v
e r fl o w s s i n c e i t s b i n a r y e q ui v a l e n t
!
!
"
d o e s n o t
fi t i n4 b i t s
¡
O v e r fl o wi s d e t e c t e d f or un s i gn e d a d d i t i on wh e n t h e c a r r y
o u t ¦
#
of t h e
fi n a
l F ul l A d d e r i s 1
¡
F or
2 ’ s c om pl e m e n t r e pr e s e n t a t i on of s
i gn e d n um b e r s o v
e r fl o w o c c ur s
wh e n t h e c a r r yi n t o t h e M S B ( m o s t s i gni fi c a n t b i t ) , wh i c h i s a l s o t h e s i gn
b i t ,
i s d i f f e r e n t f r om t h e c a r r y o u t of t h a
t b i t .T h e c a r r y o u t of t h e M S B
a l w a y s r e pr e s e n t s t h e s i gn b i t of t h e s i gn- e x t e n d e d
!
b i t r e pr e s e n t a t i on
of t h e s um
c
S h a n t a n uD u t t
, UI C
4
7/27/2019 Integer Add and Mult
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5
ADDE R S ( C on
t d . )
S p e e d i n g u p a d d
i t i on
¡
Af a s t e r a d d e r : T h e t r a d i t i on a l c a r r y-l o ok a h e a d a d d e r ( C L A
) :
D e fi n e t w o e x t r a f un c t i on s :
$
£
¢
£
¥
£
%
£
¢
£
¥
£
Y i
X i
S i F
A i i
P
i G
i C
T o c ar r y − g en.
l o gi c
F r om c ar r y −
g en.
l o gi c
¡
$
£
i s t h e g e n er a t e b i t , wh i c h i s 1 onl yi f a c a r r y o u t i s t o
b e g e n e r a t e d
i r r e s p e c t i v e of t h e i n p u t c a r r y .T h i s o b vi o u s l yi s t h e c a s e onl y wh e n
¢
£
¥
£
!
¡
%
£ i s t h e pr o p a g a t e b i t , wh i c h i s 1 onl yi f t h e o u t p u t c a r r yi s t o b e t h e s a m e
a s t h e i n p u t c a r r y ,i . e . ,
¦
£
©
¦
£ .T h i s wi l l b e t h e c a s e onl y wh e n e i t h e r
¢
£
or ¥
( b u t n o t b o t h ) i s
1
¡
T h u
s t h e c a r r y o u t
¦
£
©
of t h e
' t h s t a g e
c a n b e e x pr e s s e d
a s : ¦
£
©
$
£
%
£
¦
£
c
S h a n t a n uD u t t
, UI C
5
7/27/2019 Integer Add and Mult
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6
T r a d i t i on a l C L AA d d e r ( C on t d . )
¡
C on s i d e r a 4 - b i t a d d e r :
N o t e t h a t
$
£ a n d %
£ c a n b e g e n e r a t e d i n c on s t a n t t i m e ( s p e c i fi c a l l y ,1 g a t e
d e l a y ) b y e a c h m o d i fi e d f ul l a d d e r ( MF A
) .
¦
$(
%
(
¦
(
¦
"
$
%
¦
$
%
$ (
%
(
¦
(
$
%
$(
%
%
(
¦
(
¦
)
$
"
%
"
¦
"
$
"
%
"
$
%
"
%
$ (
%
"
%
%
(
¦
(
¦
0
$
)
%
)
¦
)
$
)
%
)
$
"
%
)
%
"
$
%
)
%
"
%
$ (
%
)
%
"
%
%
(
¦
(
T h e
¦
£ s a r e g e n e r a t e d b y a c ar r y- g e n er a t i o n u n i t a s s h o wn b e l o w .
Y
X S
F
A
P
G C
Y
X S
F A
P
G C
Y
X S
F A
P
G
Y
X S
F A
P
G C 0 0
0
0
0
0
1
1
1
1
1
2
2
2
1
2
0
2
2
3 3
3
3
3 3
1
2
C 3
C 4
C A R R Y GE N E R A T I O
N L O GI C
¡
I n t h i s C L A t h e
¦
£ s u s e
2 -l e v e l l o gi c a n d t h u s h a v e a d e l a y of 3 g a t e d e l a y s
( 1 f
or e a c h $
£
¤
%
£ , a n d 2 f or t h e
¦
£ ’ s ) , a s o
p p o s e d t o
1
2
3
g a t e d e l a y s
f or a 4 - b i t R C A .
c
S h a n t a n uD u t t
, UI C
6
7/27/2019 Integer Add and Mult
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7
T r a d i t i on a l C L AA d d e r ( C on t d . )
¡
Di s a d v a n t a g e : F or a 1 6 - b i t a d d e r ,f or e x a m pl e , w e c a nn o t g o on g e n e r -
a t i n
g t h e ¦
£ s i n t h i s m
a nn e r , s i n c e t h e h a r d w a r e b e c om e s s
e x e c s s i v e a n d
m e s s y
¡
A1
6 - b i t a d d e r c a n b e
p a r t i t i on e d i n t o gr o u p s of 4 4 - b i t C L A a d d e r s , wi t h
t h e
i n t e r - gr o u p c a r r i e s
r i p pl i n g t h r o u gh t h e f o ur gr o u p s :
C 0
4 − b i t C L A
4
4
4
S 3 − 0
X 3 − 0
Y 3 − 0
C 4
4 − b i t C L A
4
4
4
S
X
Y
C
4 − b i t C L A
4
4
4
S
X
Y
C
4 − b i t C L A
4
4
4
X
Y
C
7 −4
7 −4
7 −4
1 1 − 8 1
1 − 8
1 1 − 8
1 5 −1 2
1 5 −1 2
S 1 5 −1 2
8
1 2
1 6 1
6
S =
¡
S u c
h a 1 6 - b i t a d d e r h a s a d e l a y of 1 2 g a t e
d e l a y s ( =2 4 n s f or
a g a t e d e l a y of
2 n s ) a s o p p o s e d t o a d
e l a y of
1
!
4
= 3 2 g a t e d e l a y s ( =
1
5
= 6 4 n s ) i n
a 1 6 - b i t R C A
¡
I n g
e n e r a l f or a n
- b i t gr o u p- C L A a d d e r wi t h 4 - b i t C L A c e l l s , t h e d e l a y
i s 5
6
2
g a t e d e l a y s a s o p p o s e d t o
g a t e
d e l a y s f or t h e r i p p
l e - c a r r y a d d e r
c
S h a n t a n uD u t t
, UI C
7
7/27/2019 Integer Add and Mult
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8
T r a d i t i on a l C L AA d d e r ( C on t d . )
¡
An
o t e of c a u t i on :
7
-i n p u t g a t e s h a v e gr e a t e r d e l a y s t h a n
2 -i n p u t g a t e s
f or 7
8
. W e h a d a s s um e d t h a t f or a
4 - b i t C L A t h e 5 -
i n p u t OR a n d
A N
D g a t e s h a v e t h e s a m e d e l a y a s a 2 -i n
p u t g a t e .
¡
T h e
d e l a y of a 5 -i n p u t g a t e wi l l b e gr e a t e r b e c a u s e :
V d d
Gn d
V d d
A B
A
B
A B
A B
A B
A B C D
E
V d d
Gn d
V d d
A B C D E
A
B
C
D
E
A B C D E
A B C D E
A B
C
D
E
A +B + C +D +
E
s
( c ) 5 −
i / p OR
u s i n g2 −i / p g a t e s : d el a y =4 t +7 R C
s
( a ) 2
−i / p OR
g a t e: d el a y =2 t + 3 R C
s
( b ) 5 −i / p OR
g a t e: d el a y =2 t + 6 R C
c
S h a n t a n uD u t t
, UI C
8
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9
T r a d i t i on a l C L AA d d e r ( C on t d . )
¡
L e t
9
@
b e t h e m a x . of t h e s wi t c h i n g on
a n d t h e s wi t c h i n g
of f t i m e of a
M O
S t r a n s i s t or ,i . e . , t h e m a x . of t h e t i m e f or c h a r g e t o
c ol l e c t a t t h e
c h a nn e l or t h e t i m e f or t h e c h a r g e t o b e r e m o v e d f r om t h e c h a nn e l
¡
L e t
A
b e t h e r e s i s t a n c
e of a c h a nn e l , a n d
B
6
t h e i n p u t g a t e c a p a c i t a n c e
of a
t r a n s i s t or
¡
T h e
n , a 2 -i / pA ND / OR g a t e h a s a d e l a y of
9
@
5
A
B
a 5 -i / p g a t e h a s a d e l a y of
9
@
4
A
B
wh i l e a 5 -i / pA ND / OR c k t .f or m e d of c a s c a d e d 2 -i / p g a t e s wi l l h a v e a
d e l a y of
2
9
@
C
A
B
¡
T h u
s , wh i l e a 5 -i / pA
ND / OR g a t e wi l l b e s l o w e r t h a n a 2
-i / pA ND / OR
g a t e ,i t wi l l b e f a s t e r t h a n a c a s c a d e d i m pl e m e n t a t i on of a 5
-i / pA ND / OR
c k t .T h e l a t t e r i s pr i m a r i l y b e c a u s e m or e
t r a n s i s t or s s wi t c h
oni n p a r a l l e l
i n a
5 -i / pA ND / OR g a
t e t h a ni n a 5 -i / pA
ND / OR c k t .
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1 0
¡
I n a 4 - b i t C L A a d d e r , w e a r e e s s e n t a l l y
r e pl a c i n g a s e r i e s of 2 -i / p c a s -
c a d e d ( m ul t i -l e v e l ) l o gi c b y a m ul t i pl e -i n p u t 2 -l e v e l l o gi c
¡
N o t e t h a t t h e d e l a y of
a n
- b i t gr o u p- C L
A a d d e r wi t h 4 - b i t c e l l s wi l l b e
s om
e wh a t m or e t h a n
5
6
2
2 -i / p g a t e d e l a y s .
c
S h a n t a n uD u t t
, UI C
1 0
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1 1
F a s t e r a d d e r s
¡
C a n
w e d o b e t t e r t h a n
a d e l a y t h a t i s l i n e
a r i n t h e n um b e r o
f b i t s
?
¡
Y e s
, b y u s i n g a c ar r y- s
e l e c t a d d e r (
D
t i m e ) , OR b y u s i n g a p ar a l l e l
pr e fi
x c i r c ui t t o g e n e r a t e a l l c a r r i e s i n t i m
e
E
G
H
¡
C a r
r y- S e l e c t a d d e r :
c i n
0 1
A d d
M ux
A d d
c i n
c i n
c o u t
0 1
A d d
M ux
A d d
c i n
c o u t
n−1
s n
c i n
c o u t
0 1
A d d
M ux
A d d
c i n
c o u t
c i n
c o u t
A d d
s 0
0
0
, y
n−1
x
x
, y
¡
I f g
r o u p s i z e i s
¢ , t i m e t a k e ni s
¢
6
¢ I
( 2 -i / p ) g a t e d e l a y s
¡
F or
¢
Q
, t h e t i m
e mi ni mi z e s t o
2
D
I
2
2
D
( 2 -i / p ) g a t e
d e l a y s
c
S h a n t a n uD u t t
, UI C
1 1
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1 2
T h
e p a r a l l e l - pr e fi x C
L A a d d e r
¡
F or
t h e ' t h f ul l a d d e r , d e fi n e t h e s ym b ol s
R : k
i l l i n c omi n g c a r r y
( wh e n¢ S
£
¥
S
£
!
)
% : p
r o p a g a t e i n c omi n g c a r r y ( wh e n
¢
£
¥
£
!
)
$ : g
e n e r a t e a c a r r y ( wh e n
¢
£
¥
£
!
)
W e
c a n e n c o d e t h e s e
s ym b ol s a s
R
¤
%
!
¤
$
!
a n d c a l l t h e s e
p a i r of b i t s
U
£ .E a c h F A c a n pr o d u c e
U
£ i n
c on s t a n t t i m e
¡
A s
a m a t t e r of f a c t ,
U
£
$
£
¤
%
£ , wh e r e
$
£
a n d %
£
a r e t h e
g e n e r a t e a n d
pr o p a g a t e b i t s of a F A
d i s c u s s e d e a r l i e r
¡
D e fi n e o p e r a t or
V a s :
U
£
!
R
%
$
R
R
R
R
U
£
©
%
R
%
$
$
$
$
$
¡
N o t e t h a t
Vi s a n a s s o c i a t i v e o p e r a t or ,i . e . , W
V
Y
V
¦
W
V
Y
V
¦
¡
Al s o ,Vi s n o t a c o m m u t a t i v e o p e r a t or ,i . e
. ,WV
Y
`
Y
V
W
c
S h a n t a n uD u t t
, UI C
1 2
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1 3
T h e p a
r a l l e l - pr e fi x C L A a d d e r ( c on t d . )
¡
D e fi n e
a
£ f or t h e
' t h s t a g e a s
a
£
U
£
V
U
£ b
V d
d
d
V
U
(
a
£ i s t h e
' t h pr e fi x of t h e a s s o c i a t i v e c om p u t a t i on :
U
#
b
V
U
#
b
"
V d
d
d
V
U
(
¡
I f w
e c a n c om p u t e e a c
h a
£ q ui c k l y , t h e n w e c a n o b t a i n t h e c
a r r y-i n¦
£
©
of
t h e
'
!
t h F A a s f ol
l o w s :
I f a
£
R
t h e n¦
£
©
e l s e
i f a
£
%
t h e n¦
£
©
¦
(
e l s e
i f a
£
$
t h e n¦
£
©
!
¡
T h e
¦
£ ’ s c a n b e c om p u
t e d i n c on s t a n t t i m
e a f t e r t h e
a
£ ’ s a r e
a v a i l a b l e
c
S h a n t a n uD u t t
, UI C
1 3
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1 4
T h e p a
r a l l e l - pr e fi x C L A a d d e r ( c on t d . )
C om p u
t i n g t h e
a
£ ’ s q ui c k l y :
¡
D e fi n e
a
£
f
g
a s
a
£
f
g
U
£
V
U
£ b
Vd
d
d
V
U
g
T h u
s ,a
£
a
£
f
(
¡
S i n c e
Vi s a s s o c i a t i v e , w e h a v e
a
£
f
g
a
£
f h
V
a
h
b
f
g .
i
k −1
k
j
!
r i ,k
r k −1 , j
r i , j
c
S h a n t a n uD u t t
, UI C
1 4
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1 5
T h e p a
r a l l e l - pr e fi x C L A a d d e r ( c on t d . )
¡
W e
c a n u s e t h e a b o v e pr o p e r t y t of or m1 -l e v e l
a
£
f
g ’ s b y c om
b i ni n g2 a d j a -
c e n t U
£ ’ s , t h e n2 -l e v e l
a
£
f
g ’ s b y c om b i ni n g t w o a d j a c e n t 1 -l e v e l
a
£
f
g ’ s , e t c .
T h i s yi e l d s a t r e e - s t r u c t ur e d c i r c ui t wi t h
a Vl o gi c a t e v e r yn o d e ; t h i s c k t .
gi v e s u s onl y t h o s e
a
£ ’ s f or wh i c h
'
p
I
!
f or s om e
r
!
!
!
!
q 0
q
q
q
1
2
3
r
r
!
!
!
q
q
q
q
r
r
4
5
6
7
r
r
r
7 ,4
7 , 6
5 ,4
3 ,2
7 , 0
3 , 0
1 , 0
c
S h a n t a n uD u t t
, UI C
1 5
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1 6
T h e p a
r a l l e l - pr e fi x C L A a d d e r ( c on t d . )
¡
T o o b t a i n a l l
a
£ ’ s , t h e t r e e h a s t o b e a u gm
e n t e d a s s h o wn b e
l o w .
q 0
q
q
1
q2
3
r
q
q
q
r
r
4
5
q 6
7
r
r
7 ,4
7 , 6
5 ,4
r 3
,2
7 , 0
r 3 , 0
1 , 0
! !
! !
! !
! !
! !
p
!
r 3 , 0
r 1 , 0
r 5 , 0
r
r
r 6 , 0
4 , 0
2 , 0
! !
! !
a
b
c
x y
!
!
a
b
c
x
y
L e g en d :
¡
T h i s c i r c ui t i s c a l l e d a
p a r a l l e l pr e fi x c i r c
ui t , a n d c a n b e u s e d t o o b t a i n t h e
pr e fi x e s of a n y a s s o c i a t i v e o p e r a t i on ( l i k e A ND , OR , a d d i t i on ,m ul t i pl i -
c a t i on , e t c . )
¡
T h e
d e l a yi s
E
G
H
I
!
V-l o gi c s t e p s ,
E
G
H
s t e p s t o g o u p t h e t r e e a n d
E
G
H
I
!
s t e p s t o c om
e d o wn
¡
E x t r a h a r d w a r e u s e d :
s
u
v
#
I
!
I
!
I
!
I
!
Vl o g
i c uni t s
¡
VL S I a r e a r e q d . :
E
G
H
–h e i gh t of t r e e i s
E
G
H
, wi d t h of t r e e i s
c
S h a n t a n uD u t t
, UI C
1 6
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1 7
T h e p a
r a l l e l - pr e fi x C L A a d d e r ( c on t d . )
An o t h e r VL S I i m pl e m e n t a t i on of a p a r a l l e l
pr e fi x t r e e :
q 0
2
q
i q
i +1
r i +
1 ,i
qn−1
n−1
r
r 0
c
S h a n t a n uD u t t
, UI C
1 7
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1 8
T h e p a
r a l l e l - pr e fi x C L A a d d e r ( c on t d . )
T h e fi n
a l h a r d w a r e :
F A
F A
F A
F
A
F A
S
S
S
S
S
C
C
C
i n
C 0
C
o u t
C
X
Y 0
0
X
Y 1
1
X
Y 2
2
X
Y 1 4
1 4
X
Y 1 5
1 5
P ar al l el P r ef i x M o d ul e
1
2
q
1
2
q
q
1 4
1 4
q
1 5
1 5
q2
2
2
2
2
c
S h a n t a n uD u t t
, UI C
1 8
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1 9
C O
MP UT E RARI T HME T I C
S UBT RA C T I O N
¡
S u b
t r a c t i onw
I
x
c a n b e d on e u s i n g a n a d d e r , s i n c e
w
I
x
w
I
x
¡
T h i s m e a n s t h a t
x
, wh i c h w e a s s um e i s
i n2 ’ s c om pl e m e n t n o t a t i on ,h a s
t o b
e n e g a t e d .A2 ’ s c om pl e m e n t n um b e r i s n e g a t e d b y c om
pl e m e n t i n gi t
a n d
a d d i n g a 1 ,i . e . ,I
x
y
x
!
¡
T h e
a u gm e n t a t i on t o a
n a d d e r t o p e r f or m
s u b t r a c t i oni s s h o
wn b e l o w :
X
Y
C i n
C o u t
S u b t r a c t / A d d
=1 / 0
F r om C on t r ol U ni t
n
n
1 n
n
n− b i t A d d er
c
S h a n t a n uD u t t
, UI C
1 9
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2 0
C O
MP UT E RARI T HME T I C
M UL T I P L I E R
S
S e r i a l M ul t i pl i c a t i on
A d d - a n d - s h i f t ( A & S ) m ul t i pl i c a t i on :
M a n u a
l E x a m pl e :
I f t h e a
d d i t i on s a r e d on e on e a t a t i m e , w e o
b t a i n a s e q u e n c e o
f p a r t i a l pr o d -
u c t s
(
¤
¤
¤
#
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2 1
E a c h p
a r t i a l pr o d u c t
£
©
i s o b t a i n e d a s
(
£
©
£
£
¢
£
d
x
T h u s
#
#
b
£
(
£
¢
£
d
x
x
d
#
b
£
(
£
¢
£
x
d
w
wh e r e w
i s t h e m ul t i pl i e r
a n d x
t h e m ul t i pl i c a n d
c
S h a n t a n uD u t t
, UI C
2 1
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2 2
A
& S m ul t i pl i c a t i on
( c on t d . ) :
T h e s a m e e f f e c t a s s h i f t i n g t h e m ul t i pl i c a n d l e f t (
£
¢
£
x
) c a
n b e a c h i e v e d
b yk e e pi n g t h e m ul t i pl i c a n d fi x e d a t t h e l e f t -m o s t p o s i t i on a n
d s h i f t i n g t h e
p a r t i a l
pr o d u c t
£ r i gh t .
E x a m p
l e :
I n t h i s
c a s e , t h e p a r t i a l pr o d u c t s o b t a i n e d a r e :
(
¤
£
©
b
£
#
¢
£
x
H o w e v
e r ,
#
i s t h e s a m e :
#
#
b
£
(
£
¢
£
x
c
S h a n t a n uD u t t
, UI C
2 2
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2 3
A
& S m ul t i pl i c a t i on
( c on t d . ) :
H a r d w a r e :
1 6 − b i t A d d er
A c c um ul a t or
Q
M
A n d
C o u t
1 6
1 1 6
1 6
1
1 6
M ul t i pl i er X
M ul t i pl i c an d Y
1
A d d −
an d − sh i f t m ul t i pl i c a t i onf or un
si gn e d n um b er s
C o u t
R e g.
Al g or i t h m :
I ni t i a l i z e A C = 0 ; Q=M ul t i pl i e r ; M=M ul t i pl i c a n d .
D o t h e
f ol l o wi n g s t e p s
t i m e s
I f L S B
of Qi s 1 t h e nA C
=A C +M e l s e A C
=A C ;
S h i f t
B
-A C - Qr e gi s t e r c om b i n a t i onr i gh t b y1 b i t
F i n a l p
r o d u c t i s i nA C - Q
r e gi s t e r
N OT E : O v e r fl o w s a r e t ol e r a t e d i n t h e a d d i t i on s —
B
i s f e d t o t h e M S B of
A C wh
e nr i gh t s h i f t i n g
c
S h a n t a n uD u t t
, UI C
2 3
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2 4
A & S
2 ’ s c om pl e m e n t m
ul t i pl i c a t i on
A s s um
p t i on : B o t h
w
a n d x
a r e i n t h e i r 2 ’ s c om pl e m e n t r e pr e s e n t a t i on
M e t h o d 1 :
¡
I f m
ul t i pl i e r w
i s - v e
g e t i t s 2 ’ s c om pl e m e n t s o t h a t i t b e c om e s p o s i t i v e ,
i . e .w
I
w
¡
I f m
ul t i pl i c a n d x
i s - v e g e t i t s 2 ’ s c om pl e
m e n t s o t h a t i t b e c om e s p o s i t i v e ,
i . e .x
I
x
¡
M ul t i pl y
w
x
¡
I f e x a c t l y on e of
w
a n d x
w a s n e g a t i v e , g e t
’ s 2 ’ s c om pl e m e n t s o t h a t
i t b e c om e s n e g a t i v e ,i . e .
I
¡
Di s a d v a n t a g e : P r e pr o
c e s s i n g a n d p o s t pr o c e s s i n g c a n t a k e
u p t o4 c l o c k
c y c l e s ( c c ’ s )
c
S h a n t a n uD u t t
, UI C
2 4
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2 5
A & S 2 ’ s c
om pl e m e n t m ul t i p
l i c a t i on ( c on t d . ) :
M e t h o d 2 :
¡
Wh
e n t h e m ul t i pl i e r
w
i s + v e p e r f or m
w
x
t a k i n g c a r e t o d o t h e
f ol l o wi n g wh e n e a c h
£
i s s h i f t e d r i gh t :
1 . Wh e n t h e r e i s n o o v
e r fl o wi n t h e a d d i t i on ( r e c a l l t h e c on d
i t i onf or o v e r -
fl o wf or 2 ’ s c om pl e
m e n t a d d i t i on ) , a n
ar i t h m e t i cr i g h t s h
i f t of r e gi s t e r
A C . Qi s p e r f or m e d
wi t h o u t s h i f t i n gi n
B
i n t o t h e M S B of A C
2 .Ar i t h m e t i c r i gh t s h
i f t : M S Bi s s i gn- e x t e n d e d ,i . e . ,i f t h e M S B ( s i gn
b i t ) i s 1 a 1 i s s h i f t e d i n t o t h e M S B of A C , o t h e r wi s e a 0
i s s h i f t e d i n
3 .I f t h e r e i s a n o v e r fl o w , t h e n a s i n t h
e un s i gn e d c a s e , s h i f t
B
i n t o
M S B of A C wh e n s h i f t i n gA C . Qr i gh t .
T h i s w or k s b e c a u s e i n t h i s c a s e t h e
!
b i t o u t p u t of t h e a d d e r , wh e r e
B
i s t h e M S B ,i s t h e e x a c t 2 ’ s c om pl e m e n t r e pr e s e n t a t i on of t h e s um .
C h e c k t h i s b y s i gn
e x t e n d i n g t h e i n p u
t s t o
!
b i t s a n d c om p u t e t h e
s um— t h e o u t p u t w
i l l b e t h e s a m e a s
f or t h e
- b i t i n p u t s , b u t wi t h o u t
o v e r fl o w o u t of t h e
!
t h b i t
c
S h a n t a n uD u t t
, UI C
2 5
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2 6
A & S 2 ’ s c
om pl e m e n t m ul t i p
l i c a t i on ( c on t d . ) :
M e t h o d 2 ( c on t d . ) : E x a m
pl e :
c
S h a n t a n uD u t t
, UI C
2 6
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2 7
A & S 2 ’ s c
om pl e m e n t m ul t i p
l i c a t i on ( c on t d . ) :
M e t h o d 2 ( c on t d . ) :
¡
I f t h e m ul t i pl i e r
w
i s
n e g a t i v e , p e r f or m
t h e fi r s t
I
!
a d
d i t i on s a s e x-
pl a i n e d a b o v e , a n d t h e
n s u b t r a c t x
a s t h e fi n a l s t e p
¡
T h i s w or k s b e c a u s e
t h e v a l u e of a 2
’ s c om pl e m e n t n
um b e r w
¢
#
b
¢
(
i s gi v e n b y
w
I
#
b
¢
#
b
(
£
#
b
"
£
¢
£
T h u
s
w
x
I
#
b
¢
#
b
(
£
#
b
"
£
¢
£
x
. Wh e nw
i s n e g a t i v e ,
¢
#
b
i s 1
a n d t h u s
(
£
#
b
"
£
¢
£
x
I
x
d
#
b
wh e r e t h e fi r s t t e r mr
e pr e s e n t s n or m a l
m ul t i pl i c a t i o
nf or t h e fi r s t
I
!
b i t s of
w
c
S h a n t a n uD u t t
, UI C
2 7
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2 8
A & S 2 ’ s c
om pl e m e n t m ul t i p
l i c a t i on ( c on t d . ) :
M e t h o d 2 —E x a m pl e :
¡
N o t e t h a t m ul t i pl i c
a t i on
i s p e r f or m
e d on
t h e b a s
i s :
w
w
x
¡
T h e
m a gni t u d e of a n e g a t i v e
w
i s gi v e n b y
w
#
I
w
#
b
I
(
£
#
b
"
£
¢
£
T h u
s
w
x
I
#
b
I
(
£
#
b
"
£
¢
£
x
I
#
b
¢
#
b
(
£
#
b
"
£
¢
£
x
,
wh i c h i s wh a t w e a r e d oi n g
c
S h a n t a n uD u t t
, UI C
2 8
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2 9
A & S 2 ’ s c
om pl e m e n t m ul t i p
l i c a t i on ( c on t d . ) :
M e t h o d 2 ( c on t d . ) : H a r d w
a r e :
1 6 − b i t A
d d er
A c c um ul a t or
Q
M
A n d
C o u t
1 6
1 1 6
1 6
1
1 6
M ul t i pl i er X
M ul t i pl i c an d Y
1
Ov f l .
d e t .
A C [ 0 ]
L o gi c
1 1
C 1 5
A d d − an d − sh i f t m ul t i pl i c a t i onf o
r 2 ’ s c om pl em en t n um b er s
1
A C [ 0 ]
L o gi c :
C o u t
I f ov f l t h en
o u t p u t =
el s e
o u t p u t =A C [ 0 ]
c
S h a n t a n uD u t t
, UI C
2 9
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3 0
S p e e d i n g U p S e r i a l M ul t i pl i c a t i on
B o o t h ’ s Al g or i t h m
¡
I d e a : C on s i d e r t h e f ol l o wi n g s u b s t r i n g of
w
:
j
lm
n
o
m
n
o
j
m
n
o
¡
T h u
s i n s t e a d of a d d i n g
a n d s h i f t i n g
x
4 t i m e s ( c or r e s p on d i n
g t o t h e s t r i n g
of 4
1 ’ s i n 0 1 1 1 1 0 ) , w e c a n s u b t r a c t
x
( I
x
!
) wh e n
w e s e e t h e 1 s t
1 c omi n g a f t e r a 0 i n
w
,i . e . , w e d e t e c t
t h e 2 - b i t s u b s t r i n g
1 0 i n t h e l a s t
2 b i t s of t h e c ur r e n t
w
, j u s t s h i f t 4 t i m e
s a n d t h e n a d d
x
(
x
!
)
wh e n w e s e e t h a t t h e c
ur r e n t s t r i n g of 1 ’ s i n
w
h a v e e n d e d ,
i . e . , w e d e t e c t
t h e
2 - b i t s u b s t r i n g 0 1 i n t h e l a s t 2 b i t s of
t h e c ur r e n t w
.T h i s s a v e s u s t w o
a d d
s
¡
T h u
s wh e n t h e m ul t i p
l i e r c on t a i n s l on g ( gr e a t e r t h a nl e n g t h 2 ) s t r i n g s of
1 ’ s ,B o o t h ’ s m ul t i pl i c a t i oni s f a s t e r
c
S h a n t a n uD u t t
, UI C
3 0
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3 1
B
o o t h ’ s Al g or i t h m
( c on t d . )
¡
B o o t h ’ s m ul t i pl i c a t i on a l s o t a k e s c a r e of a n e g a t i v e m ul t i p
l i e r a u t om a t i -
c a l l y . C on s i d e r t h e f ol l o wi n gw
:
j
j
S i n
c e t h e m ul t i pl i c a t i on a l g or i t h m c on t a i n s
s t e p s (
4
i n t h e a b o v e
e x a m pl e ) , t h e 1 0 0 0 0 0 0 i s i gn or e d a n d w e
e n d u p s u b t r a c t i n g
!
3
(
t i m e s t h e m ul t i pl i c a n d
— e x a c t l y t h e r i gh t a n s w e r !
c
S h a n t a n uD u t t
, UI C
3 1
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3 2
B
o o t h ’ s Al g or i t h m
( c on t d . )
B o o t h ’ s a l g or i t h mi s d e c r
i b e d b y t h e f ol l o wi n g t a b l e f or i t e r a t i on
' :
B i t
'
B i t
'
I
!
E x pl an a t i on
A c t i on
¢
£
( c ur r e n t ) ( pr e v . )
1
0
B e gi nni n g of
a r un of
1 s
I
x
y
!
1
1
Mi d d l e of a r un of 1 s Onl y s h i f t 0
0
1
E n d of a r un of 1 s
x
1
0
0
Mi d d l e of a r un of 0 s Onl y s h i f t 0
N o t e : ( 1 ) F or un s i gn e d m
ul t i pl i c a t i on , w e n e e d t o p a d t h e m ul
t i pl i e r w
wi t h
m y t h i c
a l 0 s on b o t h s i d e s
( r i gh t of L S B p a d d i n gr e q ui r e d t o s t a r t of f t h e pr o-
c e s s )
( 2 ) F or
2 ’ s c om pl e m e n t m
ul t i pl i c a t i on , w e n
e e d t o p a d t h e m ul t i pl i e r
w
wi t h
a m y t h
i c a l 0 onl y t o t h e r i gh t of i t s L S B .T
h i s w or k s b e c a u s e
( f or 2 ’ s c om-
pl e m e n
t , t h e l a s t r un of 1 ’ s i s 1 1 . . .1 , wh e r e
t h e l e f t m o s t 1 i s t h e s i gn b i t ( b i t
I
!
)
a n d s u p p o s e t h e r i gh t m o s t 1 i s t h e
'
’ t h b i t f r oml e f t , t h e n t h e v a l u e
of t h i s
s e q u e n c e i s I
#
I
£
g
#
b
g
I
£
, wh i c h i s e x a c t l y
t h e v a l u e w e
a l l o t e d
t o t h i s s e q u e n c e w
h e n w e s u b t r a c t e d
x
a t t h e
£’ t h b i t p o s i t i on a t t h e
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3 3
b e gi nni n g of t h i s l a s t r un
of 1 ’ s .F ur t h e r ,i f
i s t h e v a l u e w e
a l l o t e d t o t h e
r e s t of
t h e m ul t i pl i e r b e f or e t h e l a s t r un of 1 ’ s , t h e n t h e fi n a l v a l u e w e gi v e
t o t h e m ul t i pl i e r i s I
£
, wh i c h i s i t s c
or r e c t v a l u e i n2 ’ s
c om pl e m e n t :
I
#
I
£
g
#
b
g
£
.
( 3 ) ¢
£
i s t h e
' t h b i t of t h e
B o o t h R e c o d i n g of
w
:
( 4 ) ¢
£
m e a n s n o a r i t h m e t i c o p e r a t i on ,
!
m e a n s a d d
x
,
y
!
m e a n s
s u b t r a c
t x
H a r d w a r e : E x c er c i s e
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3 3
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3 4
B
o o t h ’ s Al g or i t h m
( c on t d . )
E x a m p
l e s :
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S h a n t a n uD u t t
, UI C
3 4
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3 5
B
o o t h ’ s Al g or i t h m
( c on t d . )
¡
P r o b l e m : Wh e n t h e m
ul t i pl i e r c on t a i n s l on g s t r i n g s ( s a y , of l e n g t h
7
) of
a l t e
r n a t i n g1 s a n d 0 s
( !
!
!
!
) , t h e n
w e p e r f or m
7
6
a d d i t i on s a n d
7
6
s u b t r a c t i on s u s i n
gB o o t h ’ s a l g or i t h
m c om p a r e d t o onl y
7
6
a d d i -
t i on
s u s i n gr e g ul a r a d d - a n d - s h i f t
¡
S ol u t i on : L o ok a t 3 c on s e c u t i v e b i t s of t h e m ul t i pl i e r i n s t e
a d of 2 t o d e -
c i d e wh a t t o d o .T h i s i s c a l l e d t h e M o d i fi
e d B o o t h ’ s Al g or i t h m ( MBA )
¡
T h i s wi l l e n a b l e u s t o
t r e a t i s ol a t e d 1 s a n d 0 s d i f f e r e n t l yf r
omr un s of 1 s
a n d
0 s .
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S h a n t a n uD u t t
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3 5
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3 6
M
o d i fi e d B o o t h ’ s A
l g or i t h m
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s wh e n w e s e e a
w e
a d d x
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w o ul d h a v e s u b t r a c t e d
a x
on d e t e c t i n g1 0 a n d a d d e d
x
on
s u b s e q u e n t l y
d e t e c t i n g 0 1 .T h u s a s s umi n g t h e 1 i s t h
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x
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£
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£
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w e a r e d oi n g t h e R
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e n w e s e e a
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i s ol a t e d 1 , t h e n w
e s u b t r a c t x
c or r e s p on d i n g t o t h e i s ol a t e d 0 .T h i s i s
c or r e c t , s i n c e i nBA w e w o ul d
h a v e a d d e d
x
on d e t e c t i n g 0 1 a n d t h e n s u b t r a c t e d
x
on
d e t e c t i n g1 0 .
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3 7
M o d i fi e d B o o t h ’ s Al g or i t h m ( c on t d . )
¡
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ul d b e n o t e d a s s u c h s o t h a t a f t e r
s h i f t i n gr i gh t , w e d on’ t mi s i n t e r pr e t t h e b i t p a t t e r n a s e n d i n g a r un of 1 s .
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e x a m pl e , c on s i d e r t w o b i t p a t t e r n s s h o wi n g t h e 4 c on s e c u t i v e b i t s of
w
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{
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e 1 s t h a s a ni s ol a t e d 1 a n d t h e 2 n d a r un of 1 s .F or t h
e 1 s t c a s e w e
h a v e a d d e d
x
c or r e s p
on d i n g t o t h e 1 , a n d i n t h e s e c on d c a s e , w e d on o t
d o a n y t h i n g a s w e a r e
i n t h e mi d d l e of a
r un of 1 ’ s ( a s i nBA )
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h a v e t h e p a t t e r n s
!
z
{
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wh
i c h a r e i d e n t i c a l .
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e n e e d t oh a v e n o
t e d t h a t t h e 1
c or r e s p on d s t o a ni s ol a t e d 1 , s o t h a t w e d
on o t d o a n y t h i n g .
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c a s e , t h i s m e a n s e n d o
f a r un of 1 s , a n d s o w e n e e d t o a d d
x
( a s i nBA ) .
¡
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s e c a s e s a r e d i s t i n g ui s h e d b y s e t t i n g a l a t c h
|
t o b e 0 wh
e n a ni s ol a t e d
1 i s
s p o t t e d , a n d t o1 i f a r un of 1 s i s s p o t t e d
¡
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s a c t u a l l y t h e l e a s t s i gni fi c a n t of t h e
3 b i t s t h a t w e o b s e r v e s h o ul d b e
|
a n d n o t t h e pr e vi o u s b i t of
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3 8
i s ol a t e d 1 ( 0 ) a n d t h e
e n d of a r un of 1 ’ s ( 0 ’ s )
|
wi l l b e t h
e s a m e a s t h e
pr e vi o u s l y o b s e r v e d b i t of
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s a f t e r a r i gh t s h i f t , t h e a b o v e 3 - b i t p a t t e r n s wi l l b e
|
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3 9
M o d i fi e d B o o t h ’ s Al g or i t h m ( c on t d . )
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i l a r l y , w e n e e d t o
d i s t i n g ui s h b e t w e e n a ni s ol a t e d 0 a n d t h e e n d of a
r un
of 0 ’ s .I n t h e f or m
e r c a s e ,|
i s s e t t o
1 a n d t o 0 i n t h e l a
t t e r c a s e
¡
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i t p a t t e r n s s h o wi n g
4 c on s e c u t i v e b i t s of
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1 s t h a s a ni s ol a t e d 0 i ni t s 2 n d b i t f or wh i c h w e s u b t r a c
t e d x
a n d t h e
2 n d
p a t t e r nh a s a 0 i n
i t s 2 n d b i t t h a t e n d
s a r un of 0 ’ s .
¡
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s a f t e r a r i gh t s h i f t , t h e a b o v e 3 - b i t p
a t t e r n s wi l l n o t b e
i d e n t i c a l , b u t
wi l l b e
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c or -
r e s p on d i n g t o t h e i s ol a t e d 0 ) c or r e s p on d
i n g t o t h e mi d d l e
b i t a n d i n t h e
s e c on d w e s u b t r a c t
x
, s i n c e t h e mi d d l e b i t b e gi n s a r un of
1 ’ s wh i c h h a s
n o t
y e t b e e n a c c o un t e d f or .
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S h a n t a n uD u t t
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3 9
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4 0
M o d i fi e d B o o t h ’ s Al g or i t h m ( c on t d . )
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r i gh t m o s t b i t i n t h e 3 b i t s t h a t w e a r e l o ok i n g a t i s a c t u a l l y
|
wh i c h
i s i ni t i a l i z e d t o 0
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M o d i fi e d B o o t h ’ s Al g or i t h m ( c on t d . )
W e t h u
s h a v e t h e f ol l o wi n gM o d i fi e d B o o t h
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N OT E : ( 1 ) T h e m ul t i pl i e
r n e e d s t o b e p a d d
e d b ym y t h i c a l 0 s
on b o t h s i d e s
f or un s
i gn e d a n d 2 ’ s c om pl e m e n t m ul t i pl i c a
t i on .
( 2 ) ¢
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i s t h e
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w
( 3 ) T h i s s i g n e d - d i g i t e n c o
d i n gh a s
6
5
0 s o
n t h e a v e r a g e , a s o p p o s e d t o
6
i n t h e r
e g ul a r b i n a r y c o d e .T h u s
6
4
f e w e r a
r i t h m e t i c o p e r a t i on
s a r e r e q ui r e d
on t h e a v e r a g e u s i n gMBAf or m ul t i pl i c a t i on .
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M o d i fi e d B o o t h ’ s Al g or i t h m ( c on t d . )
E x a m p
l e s :
c
S h a n t a n uD u t t
, UI C