Kogge Stone adder

Peeyush Pashine(2011H140033H)

Prefix adder

• What is a prefix circuit ?

CSE 246 3

Prefix Adder

• Given:– n inputs (gi, pi)– An operation o

• Compute:– yi= (gi, pi) o … o (g1, p1) ( 1 <= i <= n)

• Associativity– (A o B) o C = A o ( B o C)

(g’’, p’’) o (g’, p’) = (g, p) g=g’’ + p’’g’ p=p’’p’

gi=

pi=

a, i=1

aibi , otherwise

1, i=1

ai xor bi , otherwise

Group PG logic

CSE 246 5

Prefix Adder: Graph Representation

• Example: Ripple Carry Adder

ai bi

(gi , pi)

x y

xoy xoy

Prefix adder(continued…)

Prefix circuit theory provides a solid theoretical basis for wide range of design trade-offs between

• Delay• Area• Wire complexity

Basic type of prefix circuits

8CSE 246

Prefix Adders: Brent – Kung Adder15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

sc(16) = 26 dc(16) = 6 total = 32

Kogge stone adder radix 2(A

0, B

0)

(A1, B

1)

(A2, B

2)

(A3, B

3)

(A4, B

4)

(A5, B

5)

(A6, B

6)

(A7, B

7)

(A8, B

8)

(A9, B

9)

(A10

, B10

)

(A11

, B11

)

(A12

, B12

)

(A13

, B13

)

(A14

, B14

)

(A15

, B15

)

S0

S1

S2

S3

S4

S5

S6

S7

S8

S9

S10

S11

S12

S13

S14

S15

Kogge Stone adder

Brent –kung adder

11CSE 246

Kogge stone Prefix Adder

8 7 6 5 4 3 2 1

35

36

7

10

9

14

11

15

13

22 18

15

21

26

2836

12CSE 246

Prefix Adders: Conditional Sum Adder

• For output yi, there is an alphabetical tree covering inputs (xi, xi-1, …, x1)

8 7 6 5 4 3 2 1 alphabetical

tree: Binary tree Edges do not

cross

13CSE 246

Prefix Adders: Conditional Sum Adder

• From input x1, there is a tree covering all outputs (yi, yi-1, …, y1)

8 7 6 5 4 3 2 1

The nodes in this tree can be reduced to

(g, p) o c = g+pc

Kogge stone radix 4 adder(a

0, b

0)

(a1, b

1)

(a2, b

2)

(a3, b

3)

(a4, b

4)

(a5, b

5)

(a6, b

6)

(a7, b

7)

(a8, b

8)

(a9, b

9)

(a1

0, b

10)

(a1

1, b

11)

(a1

2, b

12)

(a1

3, b

13)

(a1

4, b

14)

(a1

5, b

15)

S0

S1

S2

S3

S4

S5

S6

S7

S8

S9

S1

0

S1

1

S1

2

S1

3

S1

4

S1

5

Comparison between radix 2 and radix 4 koggstone