C-H1

Lecture Adders

• Half adder

C-H2

Full Adder

0 0 0 1 0 1 1 1

c i 1 +

0 0 0 0 1 1 1 1

0 0 1 1 0 0 1 1

0 1 0 1 0 1 0 1

c i x i y i

00 01 11 10

0

1

x i y i c i

1

1

1

1

s i x i y i c i =

00 01 11 10

0

1

x i y i c i

1

1 1 1

c i 1 + x i y i x i c i y i c i + + =

c i

x i

y i s i

c i 1 +

(a) Truth table

(b) Karnaugh maps

(c) Circuit

0 1 1 0 1 0 0 1

s i si is the modulo-2 sum of ci, xi, yi.

C-H3

An n-bit Ripple Adder

FA

Xn-1 Yn-1

Sn-1

Cn-1Cn FA

X0 Y0

S0

C0C1

………

MSB LSB

Adder/subtractor

• - = add 2’s complement of the subtrahend

• y xor 0 = y; y xor 1 = ~y

C-H4

s 0 s 1 s n 1 –

x 0 x 1 x n 1 –

c n n -bit adder

y 0 y 1 y n 1 –

c 0

Add Sub control

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 )

C-H5

Carry-out: for unsigned number

011101111111

Propagate and Generateripple carry

• ci+1 = xiyi + (xi+yi)ci = gi + pici

• A ripple-carry adder: critical path = 2n + 1

C-H6

x 1 y 1

g 1 p 1

s 1

Stage 1

x 0 y 0

g 0 p 0

s 0

Stage 0

c 0 c 1 c 2

Propagate and GenerateCarry-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

C-H7

Propagate and GenerateCarry-Lookahead

c1 = g0 + p0c0

c2 = g1 + p1g0 + p1p0c0

3 gate delays

C-H8

x 1 y 1

g 1 p 1

s 1

x 0 y 0

s 0

c 2

x 0 y 0

c 0

c 1

g 0 p 0 D1

D2

D3

A Multiplier Array

C-H9

and

adder

Array of Adders for Unsigned Multiplication

C-H10

4-bit example