\(3.2.\)Introduction to Digital System

1.Logic gate representation

2.Using boolean algebra to simplify circuits

3.Build an Adder

  \(a.\)Half adder

  A half adder's output consists of:

• The sum. It can be computed by xor method.
• The carry. It can be computed by and method.

  \(b.\)Full adder

  To implement full adder, we just need to add A, B and the carry at the position \(C_i\).

  The output of a full adder is the same as the half adder. Then how can we calculate each of them?

• For \(C_0\), it's decided by the sum of A and B & the current carry in the position. There are two cases that make the carry 1:

  1. A + B is 1, and current carry is 1.
  2. A + B is 2, and current carry can be 0 or 1.

  So we can compute \(C_0\) by this expression:

\[ C_0 = C_i(A+B) + AB \]

• The sum is easy, it becomes 1 when 1 or 3 of A, B and current carry is 1:

\[ S = A \; xor B \; xor C_i \]

  {{c.}}4-bit Adder

  To make a multi-bit Adder, we simply connect full adder together: