2.1 Digital Logic

1. Number Systems, Logic Levels, Logic Gates, Boolean Algebra

Number Systems

• Binary System: A number system that uses only two digits, 0 and 1. It is the foundation of digital electronics.
• Decimal System: The standard number system used in everyday life, consisting of digits 0-9.
• Octal System: A base-8 number system using digits 0-7. It is often used in computing as a shorthand for binary numbers.
• Hexadecimal System: A base-16 number system using digits 0-9 and letters A-F (representing values 10-15). It is commonly used in programming to represent binary data more compactly.

Logic Levels

• High (1): Represents a logic high or “true” state, typically corresponding to a voltage near the supply voltage (e.g., 5V).
• Low (0): Represents a logic low or “false” state, typically corresponding to a ground or low voltage (e.g., 0V).

Logic Gates

Logic gates are the basic building blocks of digital circuits. They perform logical operations on one or more binary inputs to produce a binary output.

• AND Gate: Output is 1 only if both inputs are 1.
• OR Gate: Output is 1 if at least one input is 1.
• NOT Gate (Inverter): Output is the inverse of the input. If input is 1, output is 0, and vice versa.
• NAND Gate: Output is the inverse of the AND gate. Output is 1 except when both inputs are 1.
• NOR Gate: Output is the inverse of the OR gate. Output is 1 only when both inputs are 0.
• XOR Gate: Output is 1 if the inputs are different.
• XNOR Gate: Output is 1 if the inputs are the same.

Boolean Algebra

• Boolean Variables: These variables represent two possible states, 0 and 1.
• Basic Operations:
  • AND: A * B or A AND B
  • OR: A + B or A OR B
  • NOT: ¬A or NOT A
• Boolean Laws:
  • Commutative: A + B = B + A, A * B = B * A
  • Associative: (A + B) + C = A + (B + C), (A * B) * C = A * (B * C)
  • Distributive: A * (B + C) = (A * B) + (A * C)
  • Identity: A + 0 = A, A * 1 = A
  • Null: A + 1 = 1, A * 0 = 0
  • Complement: A + ¬A = 1, A * ¬A = 0

2. Sum-of-Products and Product-of-Sums Methods

Sum-of-Products (SOP)

• Definition: SOP is a Boolean expression where several product terms (AND operations) are summed (OR operations).
• Example: The Boolean expression A * B + C is in SOP form. The terms A * B and C are the product terms, and they are summed with the OR operator.
• Application: SOP is often used in designing digital circuits with AND and OR gates.

Product-of-Sums (POS)

• Definition: POS is a Boolean expression where several sum terms (OR operations) are multiplied (AND operations).
• Example: The Boolean expression (A + B) * (C + D) is in POS form. The terms (A + B) and (C + D) are sum terms, and they are multiplied with the AND operator.
• Application: POS is used in digital circuit design when the expression needs to be implemented with NAND gates.

3. Truth Tables and Karnaugh Maps

Truth Tables

• A truth table is a tabular representation of all possible input combinations and their corresponding outputs for a Boolean function or logic circuit.
• Steps to Create a Truth Table:
  1. List all possible input combinations.
  2. Determine the output for each combination based on the Boolean expression or circuit.
  3. Present the results in a table format.
• Example for a 2-input AND gate:

Karnaugh Maps (K-map)

• A Karnaugh map is a graphical representation used to simplify Boolean expressions. It helps identify patterns in the truth table to minimize the Boolean expression.
• Steps to Use K-map:
  1. Construct a K-map grid with cells representing all possible input combinations.
  2. Place the output values from the truth table into the corresponding cells.
  3. Group adjacent cells with 1s in powers of two (1, 2, 4, 8, etc.).
  4. Write the simplified Boolean expression based on the grouped cells.
• Example for a 2-variable K-map:

The simplified Boolean expression for this K-map is: A + B.

Conclusion

Understanding number systems, logic levels, gates, and Boolean algebra is fundamental to digital electronics. Sum-of-Products (SOP) and Product-of-Sums (POS) methods simplify logic expressions for circuit design. Truth tables outline all input-output possibilities, while Karnaugh maps minimize Boolean expressions, optimizing circuit efficiency. These concepts enable the design of reliable and efficient digital systems.