1. SYEN 3330 Digital SystemsJung H. Kim Chapter 2-4 1 SYEN 3330 Digital Systems Chapter 2 – Part 4

2. SYEN 3330 Digital Systems Chapter 2-4 2 Standard Forms

3. SYEN 3330 Digital Systems Chapter 2-4 3 Standard Sum-of-Products (SOP)

4. SYEN 3330 Digital Systems Chapter 2-4 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.

5. SYEN 3330 Digital Systems Chapter 2-4 5 AND/OR Two-level Implementation of SOP Expression

6. SYEN 3330 Digital Systems Chapter 2-4 6 Standard Product-of-Sums (POS)

7. SYEN 3330 Digital Systems Chapter 2-4 7 Standard Product-of-Sums (POS)

8. SYEN 3330 Digital Systems Chapter 2-4 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.

9. SYEN 3330 Digital Systems Chapter 2-4 9 OR/AND Two-level Implementation

10. SYEN 3330 Digital Systems Chapter 2-4 10 SOP and POS Observations

11. SYEN 3330 Digital Systems Chapter 2-4 11 Equivalent Cost Circuits

12. SYEN 3330 Digital Systems Chapter 2-4 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.

13. SYEN 3330 Digital Systems Chapter 2-4 13 Karnaugh Maps (K-map)

14. SYEN 3330 Digital Systems Chapter 2-4 14 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

15. SYEN 3330 Digital Systems Chapter 2-4 15 Two Variable Maps A Two variable Karnaugh Map:

16. SYEN 3330 Digital Systems Chapter 2-4 16 K-Map and Function Tables Function Table K-Map

17. SYEN 3330 Digital Systems Chapter 2-4 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

18. SYEN 3330 Digital Systems Chapter 2-4 18 Three Variable Maps

19. SYEN 3330 Digital Systems Chapter 2-4 19 Example Functions

20. SYEN 3330 Digital Systems Chapter 2-4 20 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:

21. SYEN 3330 Digital Systems Chapter 2-4 21 Combining Squares Example

22. SYEN 3330 Digital Systems Chapter 2-4 22 Alternate K-Map Diagram x y z m0m0 m1m1 m3m3 m2m2 m4m4 m5m5 m7m7 m6m6 x yz m0m0 m1m1 m3m3 m2m2 m4m4 m5m5 m7m7 m6m6 0 1 00011110 x m0m0 m1m1 m3m3 m2m2 m4m4 m5m5 m7m7 m6m6 0 1 00011110 yz z x y