Shoaib Parallel Adders (2)

8/17/2019 Shoaib Parallel Adders (2)

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Parallel Prefx Adders

A Case Study

Muhammad Shoaib Bin Alta CS/ECE 755


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Outline

• Motivation

• ntrodu!tion

• "arious #ree adders• Com$arison

• %ayout o &o''e(Stone

• Con!lusion


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Motivation

• Addition) a undamental o$eration

– Basi! blo!* o most arithmeti! o$erations

– Address !al!ulation

• +aster, aster and aster

• -o.

– 0i$$le Carry Adder %oo* Ahead

– Carry Sele!t, !arry S*i$

– 1ood or small number o bits but2

– 3eed some !han'e or .ider adders

Square root select

Linear select

Ripple adder

20 40N

t p

( i n

u n i t d e l a y s )

600

10

0

20

30

40

50


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Pro$a'ate and 1enerate%o'i!

• +or a ull adder, defne .hat ha$$ensto !arries

– 1enerate) Cout 4 inde$endent o C

• 1 4 A 6 B

– Pro$a'ate) Cout 4 C

• P 4 A ⊕ B


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Prefx Adder Euations

• Euations oten a!tored into 1and P

• 1enerate and $ro$a'ate or 'rou$ss$annin' i)8

• Base !ase

• Sum)

: : : 1:

: : 1:

i j i k i k k j

i j i k k j

G G P G

P P P

−

−

= +

=

''

:

:

i i i i i

i i i i i

G G A B

P P A B

≡ =

≡ = ⊕

'0:0 0

0:0 0 0

inG G C

P P

≡ =

≡ =

1:0i i iS P G −= ⊕3:0 3:0 3:3 3:3 2:0G C G P G= = +

3:0 3:0 3:2 3:2 1:0G C G P G= = +


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3otations


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0i$$le Carry Adder

S1

B1

1

!1

"1

"0#0

S2

B2

!2

"2

"1#0

2

S3

B3

3

!3

"3

"2#0

S4

B4

!4

"4

"3#0

4

$in

"0

!0

1# Bit%ise !" lo&ic

2# "roup !" lo&ic

3# Su' lo&ic

$0

$1

$2

$3

$out

$4


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0i$$le Carry Adder

15:0 14:0 13:0 12:0 11:0 10:0 9:0 8:0 7:0 6:0 5:0 4:0 3:0 2:0 1:0 0:0C C C C C C C C C C C C C C C C

15:15 14:14 13:13 12:12 11:11 10:10 9:9 8:8 7:7 6:6 5:5 4:4 3:3 2:2 1:1 0:0G G G G G G G G G G G G G G G G

7:0 7:7 7:7 6:0C G P G= +

3:0 3:3 3:3 2:0C G P G= +

11:0 11:11 11:11 10:0C G P G= +

15:0 15:15 15:15 14:0C G P G= +


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%oo* Ahead Basi! idea

AN-1( BN-1 1( B1

P 1

S 1

• • •

• • • S N-1

P N-1C i ( N-1

S 0

P 0C i (0 C i (1

A

:0 1:0 1:0( , , )

k k k k k k k C f A B C G P C

− −= = +


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%oo*ahead) #o$olo'y

)*pandin& Loo+a,ead equations#

ll t,e %ay#

C o(3

C i (0

V DD

P 0

P 1

P 2

P 3

G0

G1

G2:0 1 1 2:0

( )k k k k k k

C G P G P C − − −

= + +

:0 1 1 1 0 0 0( (.... ( )))k k k k k C G P G P P G P C − −= + + + +


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%o'arithmi! %oo*aheadAdder

A7

F

A6A5A4A3A2A1

A0

A0

A1

A2

A3

A4

A5

A6

A7

F

tp∼ log2(N)

tp∼ N


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Carry loo*ahead #rees

1:0 1 1 0 1 0 0

2:0 2 2 1:0

2:0 2 2 1 2 1 0 2 1 0 0

2 2 1 2 1 0 0 0

2:1 2:1 0

( ) ( )( )

C G PG P P C

C G P G

C G P G P PG P P P C

G P G P P G P C

G P C

= + +

= +

= + + +

= + + +

= +

• #his idea !an be extended to buildhierar!hal trees


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Prefx Adder Stru!ture

• m$lement the ideao Carry %oo*aheadtree


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Brent(&un' Adder

• Sta'es

– 9:lo'3(;

• +an out

– 9• Avoids Ex$losion o

.ires

• Odd Com$utationthen even

• n any ro. at themost one $air


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Brent(&un' Adder3:2 3:3 3:3 2:2C G P G= +15:14 15:15 15:15 14:14C G P G= +

7:0 7:4 7:4 3:0C G P G= +15:8 15:12 15:12 11:8C G P G= +

4:0 4:4 4:4 3:0

C G P G= +14:0 14:14 14:14 13:0

C G P G= +


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S*lans*y Adder

• Sta'es

– %o' 3

• +an out

–


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S*lans*y Adder

3:2 3:3 3:3 2:2C G P G= +15:14 15:15 15:15 14:14C G P G= +

3:0 3:2 3:2 1:0C G P G= +

15:12 15:14 15:14 13:12C G P G= +

7:0 7:4 7:4 3:0C G P G= +15:0 15:8 15:8 7:0C G P G= +


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&o''e(Stone Adder

• Sta'es

– %o' 3

• +an out

– 9 at ea!h sta'e

• %on' .ires

• More P1 !ells

Po.er• =idely >sed


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&o''e(Stone Adder3:2 3:3 3:3 2:2C G P G= +

15:14 15:15 15:15 14:14C G P G= +

3:0 3:2 3:2 1:0C G P G= +15:12 15:14 15:14 13:12C G P G= +

7:0 7:4 7:4 3:0C G P G= +15:0 15:8 15:8 7:0C G P G= +


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-an(Carlson Adder

• Mix o &o''e(Stoneand Brent(&un'

• Sta'es

– %o' 3 ?• +an out

– 9

• #rades lo'i!al levelor .ire len'th

• n any ro. at themost one $air


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-an(Carlson Adder3:2 3:3 3:3 2:2C G P G= +15:14 15:15 15:15 14:14C G P G= +

5:0 5:2 5:2 1:0C G P G= +15:8 15:12 15:12 11:8C G P G= +

7:0 7:4 7:4 3:0C G P G= +15:0 15:8 15:8 7:0C G P G= +


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&no.les Adder

• >sin' &o''e(stoneand S*lans*y

• Sta'es

– %o' 3

• +an out

– @

• =ires


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&no.les Adder3:2 3:3 3:3 2:2C G P G= +15:14 15:15 15:15 14:14C G P G= +

5:0 5:2 5:2 1:0C G P G= +

15:8 15:12 15:12 11:8C G P G= +

7:0 7:4 7:4 3:0

C G P G= +15:0 15:8 15:8 7:0C G P G= +

4:3 4:4 4:4 3:3C G P G= +

8:0 8:1 8:1 0:0

C G P G= +


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%adner(+is!her Adder

• By Combinin'Brent(&un' andS*lans*y

• Sta'es – %o' 3 ?

• +an out

– 3/ ?

• =ires


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%adner(+is!her Adder3:2 3:3 3:3 2:2C G P G= +15:14 15:15 15:15 14:14C G P G= +

5:0 5:4 5:4 3:0C G P G= +13:8 13:12 13:12 11:8C G P G= +

7:0 7:4 7:4 3:0

C G P G= +15:0 15:8 15:8 7:0C G P G= +

8:0 8:8 8:8 7:0

C G P G= +


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Com$arison Amon' Adders

N=16 N=32 N=64 N=128

Brent(&un'

@7 D 9

S*lans*y @ 9F @D9 7D&o''e(Stone

9 7 9D

-an(Carlson

9 5 7

&no.les 7 97 7@ 95%adner(+is!her

5 D

Carryn!re

57 975 FD D@

Lo&ical e--ort o- carry propa&ate adders( .a/id arris( 2003

n ter' o- delays

- %ire capacitance ne&lected o&&eStone is est


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"alen!y o a #ree

• "alen!y

– 3umber o 'rou$s!ombine to'ether toma*e lar'er 'rou$s

– Earlier exam$les.ere o valen!y 9

– -i'h "alen!y• %ess lo'i! levels

• Ea!h sta'e has 'raterdelay

–


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S$arseness o #ree

• Com$ute Carriesor blo!*s only

• 0edu!e

– =ire !ount – 1ate !ount

– Po.er


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m$lementation o &S Adder

•


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m$lementation o &S Adder

Pro$a'ate 1enerate: : : 1:

: : 1:

i j i k i k k j

i j i k k j

G G P G

P P P

−

−

= +

=

'

'


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%ayout o &S Adder

64 bit Adder


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%ayout o &S Adder

• Area !om$letely dominated by .ires

–

•


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+e. Observations

• =ire delay ex!eeds lo'i! delay inmany !ases

• #he .ire delay in!reases .ith .idtho adder

• EHe!t o eature siIe

•@< sta!*in' !an hel$ in de!reasin'area, $o.er and delay


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Con!lusion

• +ast Adders reuired or 3J@9

• rre'ular hybrid s!hemes are $ossible

• &o''e(Stone, &no.els reuire lar'e number

o $arallel .irin' tra!*s• %ar'e .ires .ill in!rease .irin' !a$a!itan!es

• Choi!e is yours2

• #rade oH bet.een delays and Area• @< inte'ration !an hel$ in redu!in' the

delays urther


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Shoaib Parallel Adders (2)