1. Lecture 2 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. Propagate and Generate Carry-Lookahead
ci+1 = xiyi + (xi+yi)ci = gi + pici
= gi + pi (gi-1 + pi-1ci-1)
= gi + pigi-1 + pipi-1gi-2 + …+ pipi-1 …p2p1g0 + pipi-1…p1p0c0
For instance,
c1 = g0 + p0c0
c2 = g1 + p1g0 + p1p0c0

8. 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

9. A Multiplier Array
and
adder

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