ENEE244-02xx Digital Logic Design

1. ENEE244-02xxDigital Logic Design Lecture 12

2. Announcements • HW4 due today • HW5 is up on course webpage. Due on 10/16. • Recitation quiz on Monday, 10/13 • Will cover the material from lectures 10,11,12.

3. Agenda • Examples of finding minimal sums for 3-variable and 4-variable Boolean functions.

4. Using Karnaugh Maps to Find Minimal Sums 00 01 11 10 0 1

5. Using Karnaugh Maps to Find Minimal Sums 00 01 11 10 0 1

6. Using Karnaugh Maps to Find Minimal Sums • Step 1: Find all prime implicants • Verify these are the only prime implicants. 00 01 11 10 0 1

7. Using Karnaugh Maps to Find Minimal Sums • Step 2: Find all essential prime implicants • Essential 1-cells are in green. • Both prime implicantsubcubes contain essential 1-cells. • So minimal sum is: 00 01 11 10 0 1

8. Using Karnaugh Maps to Find Minimal Sums 00 01 11 10 0 1

9. Using Karnaugh Maps to Find Minimal Sums 00 01 11 10 0 1

10. Using Karnaugh Maps to Find Minimal Sums • Step 1: Find all prime implicants • Verify these are the only prime implicants. 00 01 11 10 0 1

11. Using Karnaugh Maps to Find Minimal Sums • Step 2: Find all essential prime implicants • Essential 1-cells are in green. • All three prime implicantsubcubes contain essential 1-cells. • So minimal sum is: 00 01 11 10 0 1

12. Using Karnaugh Maps to Find Minimal Sums 00 01 11 10 0 1

13. Using Karnaugh Maps to Find Minimal Sums 00 01 11 10 0 1

14. Using Karnaugh Maps to Find Minimal Sums • Step 1: Find all prime implicants • Verify these are the only prime implicants. 00 01 11 10 0 1

15. Using Karnaugh Maps to Find Minimal Sums • Step 2: Find all essential prime implicants • Essential 1-cells are in green. • Two of the prime implicantsubcubes contain essential 1-cells. • So minimal sum will contain the terms: 00 01 11 10 0 1

16. Using Karnaugh Maps to Find Minimal Sums • Step 3: Check if all 1-cells are covered • Yes, so Final sum: 00 01 11 10 0 1

17. Using Karnaugh Maps to Find Minimal Sums 00 01 11 10 00 01 11 10

18. Using Karnaugh Maps to Find Minimal Sums 00 01 11 10 00 01 11 10

19. Using Karnaugh Maps to Find Minimal Sums • Step 1: Find all prime implicants • Verify these are the only prime implicants. 00 01 11 10 00 01 11 10

20. Using Karnaugh Maps to Find Minimal Sums • Step 2: Find all essential prime implicants • Essential 1-cells are in green. • All three prime implicantsubcubes contain essential 1-cells. • So minimal sum is: 00 01 11 10 00 01 11 10

21. Using Karnaugh Maps to Find Minimal Sums 00 01 11 10 00 01 11 10

22. Using Karnaugh Maps to Find Minimal Sums 00 01 11 10 00 01 11 10

23. Using Karnaugh Maps to Find Minimal Sums • Step 1: Find all prime implicants • Verify these are the only prime implicants. 00 01 11 10 00 01 11 10

24. Using Karnaugh Maps to Find Minimal Sums • Step 2: Find all essential prime implicants • Essential 1-cells are in green. • Only two prime implicantsubcubes contain essential 1-cells. • So minimal sum must contain terms: 00 01 11 10 00 01 11 10

25. Using Karnaugh Maps to Find Minimal Sums • Step 3: Determine optimal choice of remaining prime implicants • Can cover remaining 1-cells with a single prime implicant: • Final sum: 00 01 11 10 00 01 11 10