1. Lecture Adders
Half adder

2. Full Adder
si is the modulo-2 sum of ci, xi, yi.

3. An n-bit Ripple Adder FA Xn-1 Yn-1 Sn-1 Cn-1 Cn X0 Y0 ……… C1 FA C0 S0
MSB
LSB

4. Adder/subtractor - = add 2’s complement of the subtrahend
y xor 0 = y; y xor 1 = ~y s 1 n – x c
-bit adder y
Add
Sub
control

5. Overflow v.s. Carry-out n-bit signed number: -2n-1 to 2n-1 -1
Detect overflow for signed number:
Overflow = Cn-1 ⊕ Cn
Overflow = Xn-1 Yn-1 ~Sn-1 (110) + ~Xn-1 ~Yn-1 Sn-1 (001)
where X and Y represent the 2’s complement numbers, S = X+Y. (sign bits 0, 0 ≠ 1 )
0111
1111
Carry-out: for unsigned number

6. Propagate and Generate ripple carry
ci+1 = xiyi + (xi+yi)ci = gi + pici
A ripple-carry adder: critical path = 2n + 1 x 1 y g p s
Stage 1
Stage 0 c 2

7. Problem: ripple carry adder is slow

8. Carry-lookahead adder
Motivation:
If we didn't know the value of carry-in, what could we do?
When would we always generate a carry? gi is true.
gi = ai bi
When would we propagate the carry? pi is true.
pi = ai + bi
ci+1 = (bi.ci)+(ai.ci) +(ai.bi)
= (ai.bi) + (ai+ bi) .ci
c2 = g1 + p1 . c1
c3 = g2 + p2 . c2
c4 is computed once inputs (a0-a3, b0 to b3, and c0) are valid.

9. Propagate and Generate Carry-Lookahead
c1 = g0 + p0c0 c2 = g1 + p1g0 + p1p0c0 3 gate delays x 1 y g p s c 2 D1 D2 D3

10. Hierarchical CLA: Build a 16-bit adder using a block of 4-bit CLA.
Instead of producing the a carry-out from the most significant bit of the block, each block produces the generate and propagate signal for the entire block.
For block 0, looking into c4, if all the 4 propagate functions are 1, then the carry-in c0 is propagated through the entire block, hence P0 = p3.p2.p1.p0, the rest in c4 represents the cases when this block produces a carry-out with generates, thus G0 = g3 + p3.g2 + p3.p2.g1 + p3.p2.p1.g0
So, c4 = G0 + P0.c0
Block x 7 4 – y 3 1
Second-level lookahead c s P G 15 12 8 16

11. Second Level Equation c4 = G0 + (P0.c0)
The same principle as in the first level is used in the second level.
The result of the second level is computed as soon as its inputs are valid.
c4 = G0 + (P0.c0)
c8 = G1 + P1c4 = G1 + P1G0 + P1P0c0
c12 = G2 + (P2.G1) + (P2.P1.G0) +(P2.P1.P0.c0)
c16 = G3 + (P3.G2) +(P3.P2.G1) + (P3.P2.P1.G0)
+(P3.P2.P1.P0.c0)
In block 0

12. A Multiplier Array
and
adder

13. Array of Adders for Unsigned Multiplication
4-bit example