Gate-Level Minimization: Karnaugh Map & Adjacent Squares

1. Chapter 3: Gate-Level Minimization

2. Karnaugh Map • Adjacent Squares • Number of squares = number of combinations • Each square represents a minterm • 2 Variables  4 squares • 3 Variables  8 squares • 4 Variables  16 squares • Each two adjacent squares differ in one variable • Two adjacent minterms can be combined together Example: F = x y + xy’ = x ( y + y’ ) = x

3. Two-variable Map Note: adjacent squares horizontally and vertically NOT diagonally

4. Two-variable Map • Example

5. Two-variable Map • Example

6. Three-variable Map

7. Three-variable Map • Example

8. Three-variable Map • Example Extra

9. Three-variable Map • Example

10. Three-variable Map • Example

11. Four-variable Map

12. Four-variable Map • Example

13. Four-variable Map • Example Simplify: F = A’ B’ C’ + B’ C D’ + A’ B C D’ + A B’ C’

14. Four-variable Map • Example Simplify: F = A’ B’ C’ + B’ C D’ + A’ B C D’ + A B’ C’

15. Four-variable Map • Example Simplify: F = A’ B’ C’ + B’ C D’ + A’ B C D’ + A B’ C’

16. Four-variable Map • Example Simplify: F = A’ B’ C’ + B’ C D’ + A’ B C D’ + A B’ C’

17. Four-variable Map • Example Simplify: F = A’ B’ C’ + B’ C D’ + A’ B C D’ + A B’ C’

18. Four-variable Map • Example Simplify: F = A’ B’ C’ + B’ C D’ + A’ B C D’ + A B’ C’

19. Five-variable Map

20. Five-variable Map A = 0 A = 1

21. Implicants Implicant: Gives F = 1

22. Prime Implicants Prime Implicant: Can’t grow beyond this size

23. Essential Prime Implicants Essential Prime Implicant: No other choice Not essential 8 Implicants 5 Prime implicants 4 Essentialprime implicants

24. Product of Sums Simplification

25. Don’t-Care Condition • Example You can only drop one coin at a time. Used as“don’t care”

26. Don’t-Care Condition • Example A LogicCircuit F B C Don’t care what value F may take

27. Don’t-Care Condition • Example A F B C

28. Don’t-Care Condition • Example F (w, x, y, z) = ∑(1, 3, 7, 11, 15) d (w, x, y, z) = ∑(0, 2, 5) x = 0 x = 1 x = 0 x = 1

29. Universal Gates • One Type • Use as many as you need (quantity), but one type only. • Perform Basic Operations • AND, OR, and NOT • NAND Gate • NOT-AND functions • OR function can be obtained from AND by Demorgan’s • NOR Gate • NOT-OR functions (AND by Demorgan’s)

30. Universal Gates • NAND Gate • NOT: • AND: • OR: DeMorgan’s

31. Universal Gates • NOR Gate • NOT: • OR: • AND: DeMorgan’s

32. NAND & NOR Implementation • Two-Level Implementation

33. NAND & NOR Implementation • Two-Level Implementation

34. NAND & NOR Implementation • Multilevel NAND Implementation

35. NAND & NOR Implementation • Multilevel NOR Implementation

36. Gate Shapes • AND • OR • NAND • NOR

37. Other Implementations • AND-OR-Invert • OR-AND-Invert

38. Implementations Summary • Sum Of Products: • AND-OR • AND-OR-Invert = AND-NOR = NAND-AND • Products Of Sums • OR-AND • OR-AND-Invert = OR-NAND = NOR--OR

39. Exclusive-OR • XOR F = xy = x y+ x y • XNOR F = xy =xy = x y+ x y

40. Exclusive-OR • Identities • x 0 =x • x 1 =x • xx= 0 • xx= 1 • xy=xy=xy • Commutative & Associative • xy=yx • ( xy ) z=x ( yz ) =xyz

41. Exclusive-OR Functions • Odd Function F = xyz F = ∑(1, 2, 4, 7) • Even Function F = xyz F = ∑(0, 3, 5, 6)

42. Parity 1 0 1 0 1 0 1 0 1 0 0 0  1 0 1 0 1 1 0 1 0 1 1 0 0 0 1   Parity Generator Parity Checker

43. Parity Generator • Odd Parity • Even Parity 1 0 1 0 1 1 0 1 0 Odd number of ‘1’s 1 0 1 0 0 1 0 1 0 Even number of ‘1’s

44. 1 0 1 0 0 Error Check Parity Checker • Odd Parity • Even Parity 1 0 1 0 1 Error Check

45. Homework • Mano • Chapter 3 • 3-1 • 3-3 • 3-5 • 3-7 • 3-9 • 3-15 • 3-16 • 3-18 • 3-22

46. Homework • Mano

47. Homework

48. Homework

49. Homework

50. Homework