1. ECE/CS 352 Digital Systems Fundamentals
Spring 2001
Chapter 2 – Part 4
Tom Kaminski & Charles R. Kime
ECE/CS 352 Digital System Fundamentals

2. ECE/CS 352 Digital System Fundamentals
Standard Forms
ECE/CS 352 Digital System Fundamentals

3. Standard Sum-of-Products (SOP)
ECE/CS 352 Digital System Fundamentals

4. Standard Sum-of-Products (SOP)
The Canonical Sum-of-Minterms form has (5 * 3) = 15 literals and 5 terms. The reduced SOP form has 3 literals and 2 terms.
ECE/CS 352 Digital System Fundamentals

5. AND/OR Two-level Implementation of SOP Expression
ECE/CS 352 Digital System Fundamentals

6. Standard Product-of-Sums (POS)
ECE/CS 352 Digital System Fundamentals

7. Standard Product-of-Sums (POS)
ECE/CS 352 Digital System Fundamentals

8. Standard Product-of-Sums (POS)
The Canonical Product-of-Maxterms form had (3 * 3) = 9 literals and 3 terms. The reduced POS form had 4 literals and 2 terms.
ECE/CS 352 Digital System Fundamentals

9. OR/AND Two-level Implementation
ECE/CS 352 Digital System Fundamentals

10. SOP and POS Observations
ECE/CS 352 Digital System Fundamentals

11. Equivalent Cost Circuits
ECE/CS 352 Digital System Fundamentals

12. Boolean Function Simplification
Reducing the literal cost of a Boolean Expression leads to simpler networks.
Simpler networks are less expensive to implement.
Boolean Algebra can help us minimize literal cost.
When do we stop trying to reduce the cost?
Do we know when we have a minimum?
We will introduce a systematic way to arrive a a minimum cost, two-level POS or SOP network.
ECE/CS 352 Digital System Fundamentals

13. ECE/CS 352 Digital System Fundamentals
Karnaugh Maps (K-map)
ECE/CS 352 Digital System Fundamentals

14. ECE/CS 352 Digital System Fundamentals
Uses of Karnaugh Maps
Provide a means for finding optimum:
Simple SOP and POS standard forms, and
Small two-level AND/OR and OR/AND circuits
Visualize concepts related to manipulating Boolean expressions
Demonstrate concepts used by computer-aided design programs to simplify large circuits
ECE/CS 352 Digital System Fundamentals

15. ECE/CS 352 Digital System Fundamentals
Two Variable Maps
A Two variable Karnaugh Map:
ECE/CS 352 Digital System Fundamentals

16. K-Map and Function Tables
ECE/CS 352 Digital System Fundamentals

17. K-Map Function Representations
For function F(x,y), the two adjacent cells containing 1’s can be combined using the Minimization Theorem:
For G(x,y), two pairs of adjacent cells containing 1’s can be combined using the Minimization Theorem:
Duplicate x y
ECE/CS 352 Digital System Fundamentals

18. ECE/CS 352 Digital System Fundamentals
Three Variable Maps
ECE/CS 352 Digital System Fundamentals

19. ECE/CS 352 Digital System Fundamentals
Example Functions
ECE/CS 352 Digital System Fundamentals

20. ECE/CS 352 Digital System Fundamentals
Combining Squares
By combining squares, we reduce the representation for a term, reducing the number of literals in the Boolean equation.
On a three-variable K-Map:
ECE/CS 352 Digital System Fundamentals

21. Combining Squares Example
ECE/CS 352 Digital System Fundamentals

22. Alternate K-Map Diagram
ECE/CS 352 Digital System Fundamentals