1. Kogge Stone adder
Peeyush Pashine(2011H140033H)

2. Prefix adder
• What is a prefix circuit ?

3. Prefix Adder
• Given: • Associativity
– n inputs (gi, pi) – (A o B) o C = A o ( B o C)
– An operation o
• Compute:
– yi= (gi, pi) o … o (g1, p1) ( 1 <= i <= n)
a, i=1
 (g’’, p’’) o (g’, p’) = (g, p) gi=
aibi , otherwise
 g=g’’ + p’’g’
1, i=1
 p=p’’p’ pi=
ai xor bi , otherwise
CSE 246 3

4. Group PG logic

5. Prefix Adder: Graph Representation
ai bi • Example:
Ripple Carry Adder
(gi , pi)
x y
xoy xoy
CSE 246 5

6. Prefix adder(continued…)
Prefix circuit theory provides a solid
theoretical basis for wide range of design
trade-offs between
• Delay
• Area
• Wire complexity

7. Basic type of prefix circuits

8. Prefix Adders: Brent – Kung Adder
15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0
sc(16) = 26
dc(16) = 6
 total = 32
CSE 246 8

9. (A0, B0) S0
(A1, B1) S1
(A2, B2) S2
(A3, B3) S3
(A4, B4) S4
(A5, B5) S5
(A6, B6) S6
(A7, B7) S7
(A8, B8) S8
(A9, B9) S9
(A10, B10) S10
(A11, B11) S11
(A12, B12) S12
Kogge stone adder radix 2
(A13, B13) S13
(A14, B14) S14
(A15, B15) S15

10. Kogge
Stone
adder
Brent –kung
adder

11. Kogge stone Prefix Adder
8 7 6 5 4 3 2 1
15 13 11 9 7 5 3
26 22 18 14 10 6 3
36 28 21 15
CSE 246 11

12. Prefix Adders: Conditional Sum Adder
8 7 6 5 4 3 2 1
 alphabetical
tree:
 Binary tree
 Edges do not
cross
• For output yi, there is an alphabetical tree
covering inputs (xi, xi-1, …, x1)
CSE 246 12

13. Prefix Adders: Conditional Sum Adder
8 7 6 5 4 3 2 1
 The nodes in this
tree can be
reduced to
(g, p) o c = g+pc
• From input x1, there is a tree covering all
outputs (yi, yi-1, …, y1)
CSE 246 13

14. (a 0 , b 0 ) S0
(a 1 , b 1 ) S1
(a 2 , b 2 ) S2
(a 3 , b 3 ) S3
(a 4 , b 4 ) S4
(a 5 , b 5 ) S5
(a 6 , b 6 ) S6
(a 7 , b 7 ) S7
(a 8 , b 8 ) S8
(a 9 , b 9 ) S9
(a 10 , b 10 ) S 10
(a 11 , b 11 ) S 11
(a 12 , b 12 ) S 12
(a 13 , b 13 ) S 13
Kogge stone radix 4 adder
(a 14 , b 14 ) S 14
(a 15 , b 15 ) S 15

15. Comparison between radix 2 and radix
4 koggstone