MACHINE ARCHITECTURE AND THEIR HISTORICAL CONTEXT

13. Boolean Algebra

2.2 Boolean Algebra

Boolean algebra is the mathematical foundation for digital logic circuit design. It deals with binary variables and logical operations. The basic operations in Boolean algebra are:

• AND (·)
• OR (+)
• NOT (')

Boolean expressions are simplified using laws like De Morgan’s laws, distributive, associative, and commutative properties.

2.3 Designing Combinational Circuits

Combinational circuits are circuits whose outputs depend solely on the current input. Examples include adders, multiplexers, and decoders. The design process involves:

1. Expression of the problem: Convert the given logic into a Boolean expression.
2. Simplification: Use Boolean algebra to simplify the expression.
3. Implementation: Create the circuit based on the simplified Boolean expression.

2.4 Designing Sequential Circuits

Sequential circuit design includes two main parts:

1. State Diagram: The first step in designing sequential circuits is to create a state diagram. It shows the states of the system, inputs, outputs, and the transitions between the states.
2. State Table: Based on the state diagram, construct a state table that represents the possible transitions and outputs.
3. Flip-Flop Assignment: Choose the type of flip-flop (e.g., D, T, JK) to implement the state table.
4. Simplify Logic: Use Boolean algebra to simplify the logic for the flip-flop inputs and outputs.
5. Circuit Implementation: Design the final logic circuit by interconnecting flip-flops and logic gates.