CS 105 Digital Logic Design

1. CS 105Digital Logic Design Chapter 3 Gate-Level Minimization

2. Outline • 3.1 Introduction • 3.2 The Map Method • 3.3 Four-Variable Map • 3.4 Product of sums simplification • 3.6 Don‘t Care Conditions • 3.7 NAND and NOR Implementaion • 3.8 Other Two-Level Implementaion • 3.9 Exclusive-OR function

3. 3.1 Introduction (1-1) • Gate-Level Minimization • Notes about simplification of Boolean expression • Simplification Methods: • Algebraic minimization • Karnaugh map or K-map.

4. 3.2 The Map Method (1-12) • A Karnaugh map is a graphical tool for assisting in the general simplification procedure. Combination of 2, 4, … 2n adjacent squares • Canonical form: ( sum of minterms , product of maxterms. • Standard form: ( simplifier : sum of product , product of sum

5. Y’ Y X’ X 3.2 The Map Method (2-12) Two-variable maps: • Number of squares (minterms) is , where n is the number of variables.

6. 3.2 The Map Method (4-12) Two-Variable maps (cont.) Example 1: F(X,Y) = XY’ + XY From the map, we see that F (X,Y) = X. Note: There are implied 0s in other boxes. This can be justified using algebraic manipulations:F(X,Y) = XY’ + XY = X(Y’ +Y) = X.1 = X 1 1 X

7. 1 1 1 3.2 The Map Method (5-12) Two-Variable maps (cont.) Example 2: G(x,y) = m1 + m2 + m3 Y G(x,y) = m1 + m2 + m3 = X’Y + XY’ + XY From the map, we can see that : G = X + Y X

8. 1 1 x yF 0 0 1 0 1 1 1 0 0 1 1 0 3.2 The Map Method (6-12) Two-Variable maps (cont.) Example 3: F = Σ(0, 1) Using algebraic manipulations: F = Σ(0,1) = x’y + x’y’ = x’ (y+y’) = x’ X’

9. 3.2 The Map Method (7-12) Three-variable maps: • 3 variables  8 squares ( minterms).

10. 3.2 The Map Method (8-12) Three-variable maps (cont.): • using algebraic manipulations: F = X’Y’Z’ + X’YZ’ + XY’Z’ + XYZ’ = Z’ (X’Y’ + X’Y + XY’ + XY) = Z’ (X’ (Y’+Y) + X (Y’+Y)) = Z’ (X’+ X) = Z’ Example 1: F(X,Y) = X’Y’Z’ + X’YZ’ + XY’Z’ + XYZ’ Y Z x 1 1 1 1

11. 3.2 The Map Method (9-12) three-Variable maps (cont.) Example 2: F=AB’C’ +ABC +ABC +ABC + A’B’C + A’BC’ From the map, we see that F=A+BC +BC B C 11 01 00 10 A 1 1 0 1 1 1 1 1

12. 3.2 The Map Method (10-12) three-Variable maps (cont.) Example 4 : F (x, y, z)= Σ (2, 3, 6, 7) Y using algebraic manipulations: F(x , y, z) = x’yz + xyz + x’yz’ + xyz’ = yz (x’ + x) + yz’ (x’ + x) = yz + yz’ = y (z + z’) = y y z 11 01 00 10 x 1 1 0 1 1 1

13. 3.2 The Map Method (11-12) three-Variable maps (cont.) Example (3-1) , (3-2) :

14. 3.2 The Map Method (12-12) three-Variable maps (cont.) Example (3-3) , (3-4) :

15. 3.3 Four-Variables Map (1-9) • 4 variables  16 squares ( minterms).

16. 3.3 Four-Variables Map (2-9) Flat Map Vs. Torus

17. 3.3 Four-Variables Map (3-9) Example 1 (3-5) : F(w,x,y,z) = ∑ ( 0,1,2,4,5,6,8,9,12,13,14) y z 11 01 00 10 w x 1 1 00 1 W’Z’ 1 1 1 01 Y’ XZ’ 11 1 1 1 10 1 1 F = y‘ + w‘z‘ + xz‘

18. 3.3 Four-Variables Map (4-9) Example 2 (3-6) : F = A’B’C’ + B’CD’ + A’BCD’ + AB’C’ C D B’C’ 11 01 00 10 A B 1 00 1 1 A’CD’ 01 1 11 1 1 1 10 B’D’ F = B‘D‘ + B‘C‘ + A‘CD‘

19. 3.3 Four-Variables Map (5-9) Simplification using Prime Implicants • A Prime Implicantis a product term obtained by combining the maximum possible number of adjacent squares in the map . • Essential :covered by only one prime implicant.

20. CD C B B D D B C 1 1 1 1 1 1 1 1 BD BD 1 1 B B 1 1 1 1 A A 1 1 1 1 1 1 1 1 A B D D AD Minterms covered by single prime implicant 3.3 Four-Variables Map (6-9) Simplification using Prime Implicants Example 1: F(A,B,C,D) = ∑ (0,2,3,5,7,8,9,10,11,13,15) ESSENTIAL Prime Implicants C

21. 3.3 Four-Variables Map (7-9) Simplification using Prime Implicants Example 1: F(A,B,C,D) = ∑ (0,2,3,5,7,8,9,10,11,13,15) • Essential prime implicants:BD , B’D’ • Prime implicant: CD , B’C, AD , AB’. • The minterms that not cover by essential implicants are: m3, m9, m11. • The simplified expression is optained from the sum of the essential implicants and other prime implicants that may be needed to cover any remaining minterms. • So this function can be written with these ways: • F = BD + B’D’ + CD + AD • F = BD + B’D’ + CD + AB’ • F = BD + B’D’ + B’C + AD • F = BD + B’D’ + B’C + AB’

22. Y X W Z 3.3 Four-Variables Map (8-9) Simplification using Prime Implicants Example 2: F(W,X,Y,Z) = ∑ (0,2,3,8,9,10,11,12,13,14,15) X’Y X’Z’ 1 1 1 Note: that all of these prime implicants are essential. 1 1 1 1 1 1 1 1 W

23. Y X W Z 3.3 Four-Variables Map (9-9) Simplification using Prime Implicants Example 3: F(W,X,Y,Z) = ∑ (0,2,3,4,7,12,13,14,15) W’YZ W’X’Z’ W’X’Y • Essential: WX • Prime: XYZ , XY’Z’ , W’Y’Z’, W’YZ, W’X’Y , W’X’Z’ 1 1 1 W’Y’Z’ 1 1 XYZ 1 1 1 1 XY’Z’ WX

24. 3.5 Product-of-Sum simplification (1-3) • Mark with 1’s the minterms of F. • Mark with 0’sthe minterms of F’. • Circle 0’s to express F’. • Complement the result in step 3 to obtain a simplified F in product-of-sums form.

25. C B A D 3.5 Product-of-Sum simplification (2-3) Example 1: Simplify :F= ∑(0,1,2,5,8,9,10)in Product-of-Sums Form • F’ = AB + CD + BD’ • F = (F’)’ = (A’+B’) + (C’+D’) + (B’+D) CD 0 1 1 1 0 1 0 0 BD’ AB 0 0 0 0 1 1 0 1

26. 3.5 Product-of-Sum simplification (3-3) Example 2: Simplify: F(x, y, z) =(0, 2, 5,7)in Product-of-Sums Form X’Z’ y z 11 • F’ = XZ + X’Z’ • F = (F’)’ = (X’+Z’) + (X+Z) 01 00 10 x 0 0 0 0 0 1 XZ

27. 3.6 Don't Cares Condition (2-4) • Example : • A logic function having the binary codes for the BCD digits as its inputs. Only the codes for 0 through 9 are used. • The six codes, 1010 through 1111 never occur, so the output values for these codes are “x” to represent “don’t cares.”

28. Y Y X X W W Z Z 3.6 Don't Cares Condition (3-4) Example (3.9) : F(W,X,Y,Z) = ∑ (1,3,7,11,15) d(W,X,Y,Z) = ∑ (0,2,5) x 1 x x 1 x 1 1 x x 0 0 1 0 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 0 1 1 F = YZ + W’X’ F = YZ + W’Z

29. Y X W Z 3.6 Don't Cares Condition (4-4) Example (3.9) : F(W,X,Y,Z) = ∑ (1,3,7,11,15) d(W,X,Y,Z) = ∑ (0,2,5) x 1 x 1 x 0 0 1 0 0 1 0 0 0 0 1 F’ = Z’ + WY’ F = Z ( W’ + Y)

30. 3.7 NAND and NOR Implementation (1-15) • Digital circuits are frequently constructed with NAND or NOR gates rather than with AND and OR gates.

31. 3.7 NAND and NOR Implementation (2-15) NAND Implementation • NAND gate: a universal gate. • Any digital system can be implemented with it.

32. 3.7 NAND and NOR Implementation (3-15) NAND Implementation

33. 3.7 NAND and NOR Implementation (5-15) NAND Implementation Two-Level Implementation F = AB + CD = [(AB + CD)’]’ = [(AB)’*(CD)’]’

34. 3.7 NAND and NOR Implementation (6-15) NAND Implementation Two-Level Implementation Example (3.10): F(X,Y,Z) = ∑ (1,2,3,4,5,7) X’Y F = XY’ + X’Y + Z y z 11 01 00 10 x 1 1 1 0 1 1 1 1 Z XY’

35. 3.7 NAND and NOR Implementation (8-15) NAND Implementation Multilevel Implementation EXAMPLE 1: F = A(CD + B) + BC’

36. 3.7 NAND and NOR Implementation (9-15) NAND Implementation Multilevel Implementation EXAMPLE 2: F = (AB’ + A’B).(C + D’)

37. 3.7 NAND and NOR Implementation (10-15) NOR Implementation • The NOR operation is the dual of the NAND operation. • The NOR gate is anothar universal gate to implement any Boolean function.

38. 3.7 NAND and NOR Implementation (11-15) NOR Implementation

39. 3.7 NAND and NOR Implementation (13-15) NOR Implementation Two-Level Implementation Example : F = (A+B).(C+D).E E

40. 3.7 NAND and NOR Implementation (15-15) NOR Implementation Multi-Level Implementation Example : F = (A B’ + A’B).(C+D’) A B’ A’ B

41. 3.8 Other Two-Level Implementations (1-7) Nondegeneratd forms Implementation • 16 possible combinations of two-level forms with 4 types of gates: AND, OR, NAND, and NOR • 8 are degenerate forms: degenerate to a single operation. • (AND-AND , AND-NAND, OR-OR , OR-NOR , NAND-NAND , NAND-NOR , NOR-AND , NOR-NAND) • 8 are generate forms: • NAND-AND = AND-NOR = AND-OR-INVERT • OR-NAND = NOR-OR = OR-AND-INVERT

42. 3.8 Other Two-Level Implementations (3-7) Nondegenerated forms Implementation Discussed before Degenerated forms Discuss now

43. 3.8 Other Two-Level Implementations (4-7) Nondegenerated forms Implementation • AND-NOR = NAND-AND = AND-OR-INVERT AND-OR-INVERT Example: F = (AB + CD + E) ‘

44. 3.8 Other Two-Level Implementations (5-7) Nondegenerated forms Implementation • OR-NAND = NOR-OR = OR-AND-INVERT OR-AND-INVERT Example: F = [(A+B) . (C+D) . E ] ‘

45. 3.8 Other Two-Level Implementations (6-7) Nondegenerated forms Implementation

46. 3.8 Other Two-Level Implementations (7-7) Nondegeneraetd forms Implementation Example (3.11) : F(x,y,z) = ∑ (0,7) AND-OR-INVERT: F’ = x’y + xy’ + z F = ( x’y + xy’ + z ) ‘ ----------------------------- OR-AND-INVERT: F = x’y’z’ + xyz’ F = [ (x’y’z’ + xyz’)’ ] ‘ F = [ (x+y+z) . (x’+y’+z) ] ‘

47. 3.9 Exclusive-OR Function (1-9) • Exclusive-OR (XOR) • denoted by the symbol : • x  y = xy‘ + x‘y • Exclusive-NOR (XNOR): • (x  y )‘ = xy + x‘y‘

48. 3.9 Exclusive-OR Function (2-9) • Exclusive-OR principles: • x  0 = x • x  1 = x‘ • x  x = 0 • x  x‘ = 1 • x  y‘ = x‘ y = (x  y)‘ • x  y = y  x • (x  y)  z = x  (y  z)

49. 3.9 Exclusive-OR Function (3-9) • Implementation Exclusive-OR with AND-OR-NOT: • x  y = xy‘ + x‘y • Implementation Exclusive-OR with NAND: • x  y = xy‘ + x‘y = x (x‘+y‘) + y (x‘+y‘) = x (xy)‘ + y (xy)‘ = [(x(xy)‘ + y(xy)‘)‘]‘ = [(x(xy)‘)‘ . (y(xy)‘)‘ ]‘

50. 3.9 Exclusive-OR Function (4-9) Odd Function: • Multiple-variable exclusive OR operation = odd function • (A  B  C) = (AB‘ + A‘B) C‘ + (A‘B‘ + AB) C = AB‘C‘ + A‘BC‘ + A‘B‘C + ABC = ∑ (1,2,4,7)