Basic Adders and Counters: Lecture 3 - Half-adder and Full-adder Implementations, Study notes of Electrical and Electronics Engineering

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Basic Adders and Counters

Lecture 3

Half-adder

x y

c s

HA

x + y = ( c s ) 2

2 1

x y c s 0 0 1 1

0 1 0 1

0 0 0 1

0 1 1 0

Half-adder

Alternative implementations (1)

s = xy + xy

b)

a)

s = x ⊕ y

c = xy

c = x + y

c) c = xy

s = xc + yc = xc ⋅ yc

Half-adder

Alternative implementations (2)

Full-adder

x y

cout s

FA

x + y + cin = ( cout s ) 2

2 1

x y cout s 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 0 0 1 0 1 1 1 0 1 1 0 1 0 0 1

cin

0 1 0 1 0 1 0 1

cin

Full-adder

Alternative implementations (1)

a) (^) s = (x ⊕ y) ⊕ cin

cout = xy + cin (x ⊕ y)

s

c c

Latency of a k -bit ripple-carry adder

Tripple-add = TFA(x,y→cout) +

+ ( k -2) ⋅ TFA(cin→cout) +

+ TFA(cin →s)

Latency ≈ k ⋅ TFA

Latency ∝ k

Addition of a constant (1)

xk-1 xk-2... x 1 x 0 yk-1 yk-2... y 1 y 0

variable

• constant

xk-1 xk-2... xh+1 xh xh-1... x 0 yk-1 yk-2... yh+1 1 0... 0

variable

• constant

xh xh-1... x 0

sk-1 sk-2... s 1 s 0

sk-1 sk-2... sh+

Addition of a constant (2)

HA/^... MHA

HA/ MHA

HA/ MHA

HA/ MHA

xk-1 xk-2... xh+2 xh+1 xh^ xh-1... x 0

..

sk-1 sk-2... (^) sh+2 sh+1 xh xh-1... x 0

If yi = 0 Half-adder (HA) yi = 1 Modified half-adder (MHA)

ck

Modified half-adder

x y

c s

MHA

x + y + 1 = ( c s ) 2

2 1

x y c s 0 0 1 1

0 1 0 1

0 1 1 1

1 0 0 1

HA HA HA HA

xk-1 xk-2... x 2 x 1

..

sk-1 sk-2... s 2 s 1

x 0

x 0

ck

Incrementer

MHA MHA MHA MHA

xk-1 xk-2... x 2 x 1

..

sk-1 sk-2... s 2 s 1

x 0

x 0

ck

Decrementer

Expected length of the carry chain

that starts at position i (1)

Expected length( i , k ) =

k i

k i

k j i

j i

j i

−^ −

Length of the carry chain

Probability of the given length

Probability of propagation till the end of adder

Distance till the end of adder

Expected length of the carry chain

that starts at position i (2)

Expected length( i , k ) =

k i

For i << k

Expected length of the carry propagation is ≈≈≈≈ 2

Carry completion detection

Two-rail code

bi ci

0 0

0 1

1 0

carry not yet known

carry known to be 1

carry known to be 0

Meaning

Manchester Carry Chains and Adders

Generate signal: gi = xiyi Propagate signal: pi = xi ⊕ yi Anihilate (absorb) signal: ai = xi yi = x + y

Transfer signal: ti = gi + pi = ai = xi + yi cout =1 given cin = 1

ci+1 = gi + cipi = = (gi + cigi) + cipi = = gi + ci (gi + pi) = = gi + ci ti

X + XY = X Absorption law