Binary Addition

Section 4.5

Binary Addition Example

Binary Addition (2)

Derivation of ∑

Question: What primitive best implements ∑? • Inputs: A, B• Outputs: ∑=

B A ∑

0 0 0

1 0 1

0 1 1

1 1 0

Derivation of Carry Out

Question: What primitive best implements Co? • Inputs: A, B• Outputs: Co =A∙B

B A Co

0 0 0

1 0 0

0 1 0

1 1 1

Implementation of a Half-Adder

Limitation of a Half Adder

A half-adder does not account for carry-in.

Truth Table for a Full Adder

carry-in

Karnaugh Map For the Sum Bit

= = =

Karnaugh Map For the Carry-Out Bit

C=𝑥 ′ 𝑦𝑧+𝑥 𝑦 ′ 𝑧+𝑥𝑦=𝑧 (𝑥⊕𝑦 )+𝑥𝑦

Implementation of a Full Adder

𝑆=𝑧 (𝑥⊕ 𝑦 )+𝑥𝑦C=𝑧⊕(𝑥⊕𝑦 )

(carry-in)

Four-Bit Adder

C4 is calculated last because it takes C0 8 gates to reach C4.

Alternative Naming Convention for the Full Adder

𝐺𝑖=𝐴𝑖𝐵 𝑖P i=(𝐴𝑖⊕𝐵𝑖)

C 𝑖+1=𝐺𝑖+𝑃𝑖𝐶𝑖S i=(𝑃𝑖⊕𝐶𝑖)

Hardware Simplificationinput carry

)

2 gate delays for C3!

Four-bit adder with Carry Lookahead