1. Gate-Level Minimization1 DIGITAL LOGIC DESIGN by Dr. Fenghui Yao Tennessee State University Department of Computer Science Nashville, TN

2. Gate-Level Minimization2 What is minimization?  Simplifying boolean expressions  Algebraic manipulations is hard since there is not a uniform way of doing it  Karnaugh map or K-map techniques is very commonly used

3. Gate-Level Minimization3 Two-Variable K-Map

4. Gate-Level Minimization4 Example

5. 5 ExampleExample

6. 6 Three-Variable K-Map

7. Gate-Level Minimization7 Three-Variable K-Map

8. Gate-Level Minimization8 Example

9. 9 Note  In K-maps, you can have groups of 2, 4, 8, or 16  You cannot have groups of other combinations such as a group of 6

10. Gate-Level Minimization10 Exercises

11. Gate-Level Minimization11 Example  Represent F in the minimal format and draw the network diagram

12. Gate-Level Minimization12 Example  Represent F in the minimal format and draw the network diagram

13. Gate-Level Minimization13 Example  Represent F in the minimal format and draw the network diagram

14. Gate-Level Minimization14 Four-Variable K-Map

15. Gate-Level Minimization15 Four-Variable K-Map

16. Gate-Level Minimization16 Example  Represent F in the minimal format and draw the network diagram

17. Gate-Level Minimization17 Example  Represent F in the minimal format and draw the network diagram

18. Gate-Level Minimization18 Example  Represent F in the minimal format and draw the network diagram

19. Gate-Level Minimization19 Prime Implicants  You must cover all of the minterms  You must avoid redundancy  You must follow some rules  Prime Implicant  A product term that is generated by combining the maximum number of adjacent squares in the map  Essential Prime Implicant  A minterm that is covered by only one prime implicant

20. Gate-Level Minimization20 Maxterm Simplification  Remember

21. Gate-Level Minimization21 Example  Simplify F in product of sums

22. Gate-Level Minimization22 Example (cont)  Step – 1  Fill the K-map for F

23. Gate-Level Minimization23 Example (cont)  Step – 1  Fill the K-map for F

24. Gate-Level Minimization24 Example (cont)  Step – 2  Fill zeros in the rest of the squares

25. Gate-Level Minimization25 Example (cont)  Step – 3  Cover zeros. This is your F’

26. Gate-Level Minimization26 Important

27. Gate-Level Minimization27 Don’t Care Conditions  A network is usually composed of sub- networks  Net-1 may not produce all combinations of A,B, and C  In this case, F don’t care about those combinations ABCABC F Net-1Net-2

28. Gate-Level Minimization28 Don’t Care Conditions 0 0 0 1 0 0 1 x 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 1 0 x 1 1 X can be considered as 0 or 1, whichever is more convenient

29. Gate-Level Minimization29 NAND/NOR Implementations  AND, OR, and NOT gates can be used to construct the digital systems  However, it is easier to fabricate NAND and NOR gates  So try to replace AND, OR, and NOT gates with NAND or NOR gates

30. Gate-Level Minimization30 NAND Implementation  First implement with AND-OR  Put bubble at the output of each AND gate  Put bubbles at the inputs of each OR gate  Place necessary inverters

31. Gate-Level Minimization31 Example

32. Gate-Level Minimization32 Example

33. Gate-Level Minimization33 Example

34. Gate-Level Minimization34 NOR Implementation  First implement with AND-OR  Put bubble at the inputs of each AND gate  Put bubbles at the output of each OR gate  Place necessary inverters

35. Gate-Level Minimization35 Example

36. Gate-Level Minimization36 Study Problems  Course Book Chapter – 3 Problems  3– 1  3 – 3  3 – 5  3 – 7  3 – 12  3 – 15  3 – 18

37. Gate-Level Minimization37 Questions