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Chapter 3: 1 / 50 Karnaugh Map A djacent Squares Number of squares = number of combinations E ach square represents a minterm 2 Variables 4 squares 3.

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Presentation on theme: "Chapter 3: 1 / 50 Karnaugh Map A djacent Squares Number of squares = number of combinations E ach square represents a minterm 2 Variables 4 squares 3."— Presentation transcript:

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2 Chapter 3:

3 1 / 50 Karnaugh Map A djacent Squares Number of squares = number of combinations E ach square represents a minterm 2 Variables 4 squares 3 Variables 8 squares 4 Variables 16 squares Each two adjacent squares differ in one variable T wo adjacent minterms can be combined together Example:F = x y + x y = x ( y + y ) = x

4 2 / 50 Two-variable Map m0m0 m1m1 m2m2 m3m3 x yMinterm 00 m0m0 10 1m1m1 21 0m2m2 31 m3m3 y x01 0 1 Note: adjacent squares horizontally and vertically NOT diagonally

5 3 / 50 Two-variable Map Example x yFMinterm 00 0m0m0 10 10m1m1 21 00m2m2 31 1m3m3 y x01 0 1 m0m0 m1m1 m2m2 m3m3 y 00 x01

6 4 / 50 Two-variable Map Example x yFMinterm 00 0m0m0 10 11m1m1 21 01m2m2 31 1m3m3 y x01 0 1 m0m0 m1m1 m2m2 m3m3 y 01 x11

7 5 / 50 Three-variable Map m0m0 m1m1 m3m3 m2m2 m4m4 m5m5 m7m7 m6m6 x y zMinterm 00 0 0m0m0 10 0 1m1m1 20 1 0m2m2 30 1 1m3m3 41 0 0m4m4 51 0 1m5m5 61 1 0m6m6 71 1 1m7m7 y z x00011110 0 1

8 6 / 50 Three-variable Map m0m0 m1m1 m3m3 m2m2 m4m4 m5m5 m7m7 m6m6 x y zFMinterm 00 0 00m0m0 10 0 10m1m1 20 1 01m2m2 30 1 11m3m3 41 0 01m4m4 51 0 11m5m5 61 1 00m6m6 71 1 10m7m7 y z x00011110 0 1 Example y 0011 x1100 z

9 7 / 50 Three-variable Map m0m0 m1m1 m3m3 m2m2 m4m4 m5m5 m7m7 m6m6 x y zFMinterm 00 0 00m0m0 10 0 10m1m1 20 1 00m2m2 30 1 11m3m3 41 0 01m4m4 51 0 10m5m5 61 1 01m6m6 71 1 11m7m7 y z x00011110 0 1 Example y 0010 x1011 z Extra

10 8 / 50 Three-variable Map x y zFMinterm 00 0 00m0m0 10 0 11m1m1 20 1 00m2m2 30 1 11m3m3 41 0 00m4m4 51 0 11m5m5 61 1 00m6m6 71 1 11m7m7 Example y 0110 x0110 z y 0110 x0110 z

11 9 / 50 Three-variable Map m0m0 m1m1 m3m3 m2m2 m4m4 m5m5 m7m7 m6m6 x y zFMinterm 00 0 01m0m0 10 0 10m1m1 20 1 01m2m2 30 1 10m3m3 41 0 01m4m4 51 0 11m5m5 61 1 01m6m6 71 1 10m7m7 y z x00011110 0 1 Example y 1001 x1101 z

12 10 / 50 Four-variable Map m0m0 m1m1 m3m3 m2m2 m4m4 m5m5 m7m7 m6m6 m 12 m 13 m 15 m 14 m8m8 m9m9 m 11 m 10 w x y zMinterm 00 0 m0m0 10 0 0 1 m1m1 20 0 1 0 m2m2 30 0 1 1 m3m3 40 1 0 0 m4m4 50 1 m5m5 60 1 1 0 m6m6 70 1 1 1 m7m7 81 0 0 0 m8m8 91 0 0 1 m9m9 10 m 10 111 0 1 1 m 11 121 1 0 0 m 12 131 1 0 1 m 13 141 1 1 0 m 14 151 1 m 15 y z wx00011110 00 01 11 10

13 11 / 50 Four-variable Map w x y zFMinterm 00 0 1 m0m0 10 0 0 1 1 m1m1 20 0 1 0 1 m2m2 30 0 1 1 0 m3m3 40 1 0 0 1 m4m4 50 1 1 m5m5 60 1 1 0 1 m6m6 70 1 1 1 0 m7m7 81 0 0 0 1 m8m8 91 0 0 1 1 m9m9 10 0 m 10 111 0 1 1 0 m 11 121 1 0 0 1 m 12 131 1 0 1 1 m 13 141 1 1 0 1 m 14 151 1 0 m 15 y z wx00011110 00 01 11 10 Example y 1101 1101 x w 1101 1100 z

14 12 / 50 Four-variable Map Example Simplify: F = A B C + B C D + A B C D + A B C C B A D

15 13 / 50 Four-variable Map Example Simplify: F = A B C + B C D + A B C D + A B C C B A D 11

16 14 / 50 Four-variable Map Example Simplify: F = A B C + B C D + A B C D + A B C C B A D 1 1

17 15 / 50 Four-variable Map Example Simplify: F = A B C + B C D + A B C D + A B C C B A D 1

18 16 / 50 Four-variable Map Example Simplify: F = A B C + B C D + A B C D + A B C C B A D 11

19 17 / 50 Four-variable Map Example Simplify: F = A B C + B C D + A B C D + A B C C 111 1 B A 111 D

20 18 / 50 Five-variable Map DED BC00011110 00m0m0 m1m1 m3m3 m2m2 01m4m4 m5m5 m7m7 m6m6 C B 11m 12 m 13 m 15 m 14 10m8m8 m9m9 m 11 m 10 E A = 0 DED BC00011110 00m 16 m 17 m 19 m 18 01m 20 m 21 m 23 m 22 C B 11m 28 m 29 m 31 m 30 10m 24 m 25 m 27 m 26 E A = 1

21 19 / 50 Five-variable Map A = 0 A = 1

22 20 / 50 Implicants C 1 111 B A 111 1 D Implicant: Gives F = 1

23 21 / 50 Prime Implicants C 1 111 B A 111 1 D Prime Implicant: Cant grow beyond this size

24 22 / 50 Essential Prime Implicants C 1 111 B A 111 1 D Essential Prime Implicant: No other choice Not essential 8 Implicants 5 Prime implicants 4 Essential prime implicants

25 23 / 50 Product of Sums Simplification w x y zFF 00 0 10 10 0 0 1 10 20 0 1 0 10 30 0 1 1 01 40 1 0 0 10 50 1 10 60 1 1 0 10 70 1 1 1 01 81 0 0 0 10 91 0 0 1 10 10 01 111 0 1 1 01 121 1 0 0 10 131 1 0 1 10 141 1 1 0 10 151 1 01 y 1101 1101 x w 1101 1100 z y 1101 1101 x w 1101 1100 z

26 24 / 50 Dont-Care Condition Example A B C$ Value 0 0 0$ 0.00 0 0 1$ 0.05 0 1 0$ 0.10 0 1 1Not possible 1 0 0$ 0.25 1 0 1Not possible 1 1 0Not possible 1 1 1Not possible You can only drop one coin at a time. Used as dont care

27 25 / 50 Dont-Care Condition Example A B CF 0 0 00 0 0 11 0 1 00 0 1 1x 1 0 01 1 0 1x 1 1 0x 1 1 1x F Dont care what value F may take Logic Circuit A B C

28 26 / 50 Dont-Care Condition Example F B 01x0 A1xxx C A B C

29 27 / 50 Dont-Care Condition Example y x11x x1 x w 1 1 z F (w, x, y, z) = (1, 3, 7, 11, 15) d (w, x, y, z) = (0, 2, 5) x = 0 x = 1 y xx 0x0 x w 000 000 z x = 0x = 1

30 28 / 50 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 Demorgans NOR Gate NOT-OR functions (AND by Demorgans)

31 29 / 50 Universal Gates NAND Gate NOT: AND: OR: DeMorgans

32 30 / 50 Universal Gates NOR Gate NOT: OR: AND: DeMorgans

33 31 / 50 NAND & NOR Implementation Two-Level Implementation

34 32 / 50 NAND & NOR Implementation Two-Level Implementation

35 33 / 50 NAND & NOR Implementation Multilevel NAND Implementation

36 34 / 50 NAND & NOR Implementation Multilevel NOR Implementation

37 35 / 50 Gate Shapes AND OR NAND NOR

38 36 / 50 Other Implementations AND-OR-Invert OR-AND-Invert

39 37 / 50 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

40 38 / 50 Exclusive-OR XOR F = x y = x y + x y XNOR F = x y = x y = x y + x y

41 39 / 50 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 xy XOR 000 011 101 110

42 40 / 50 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) xyz XORXNOR 00001 00110 01010 01101 10010 10101 11001 11110 y z x00011110 00101 11010

43 41 / 50 Parity 10101010 10101010 10001000 1010110101 1010110101 1000110001 Parity Generator Parity Checker

44 42 / 50 Parity Generator Odd Parity Even Parity 1010110101 Odd number of 1s 1010010100 Even number of 1s 10101010 10101010

45 43 / 50 Parity Checker Odd Parity Even Parity Error Check 1010110101 1010010100 Error Check

46 44 / 50 Homework Mano Chapter 3 3-1 3-3 3-5 3-7 3-9 3-15 3-16 3-18 3-22

47 45 / 50 Homework Mano 3-1Simplify the following Boolean functions, using three- variable maps: (a) F (x, y, z) = (0, 2, 6, 7) (b) F (A, B, C) = (0, 2, 3, 4, 6) (c) F (a, b, c) = (0, 1, 2, 3, 7) (d) F (x, y, z) = (3, 5, 6, 7)

48 46 / 50 Homework 3-3Simplify the following Boolean functions, using three- variable maps: (a) xy + xyz + xyz (b) xy + xz + xyz (c) AB + BC + BC 3-5Simplify the following Boolean functions, using four- variable maps : (a) F (w, x, y, z) = (1, 4, 5, 6, 12, 14, 15) (b) F (A, B, C, D) = (0, 1, 2, 4, 5, 7, 11, 15) (c) F (w, x, y, z) = (2, 3, 10, 11, 12, 13, 14, 15) (d) F (A, B, C, D) = (0, 2, 4, 5, 6, 7, 8, 10, 13, 15)

49 47 / 50 Homework 3-7Simplify the following Boolean functions, using four- variable maps: (a) wz + xz + xy + wxz (b) BD + ABC + ABC + ABC (c) ABC + BCD + BCD + ACD + ABC + ABCD (d) wxy + yz + xyz + xy 3-9Find the prime implicants for the following Boolean functions, and determine which are essential: (a) F (w, x, y, z) = (0, 2, 4, 5, 6, 7, 8, 10, 13, 15) (b) F (A, B, C, D) = (0, 2, 3, 5, 7, 8, 10, 11, 14, 15) (c) F (A, B, C, D) = (1, 3, 4, 5, 10, 11, 12, 13, 14, 15)

50 48 / 50 Homework 3-15Simplify the following Boolean function F, together with the dont-care conditions d, and then express the simplified function in sum of products: (a) F (x, y, z) = (0, 1, 2, 4, 5) d (x, y, z) = (3, 6, 7) (b) F (A, B, C, D) = (0, 6, 8, 13, 14) d (A, B, C, D) = (2, 4, 10) (c) F (A, B, C, D) = (1, 3, 5, 7, 9, 15) d (A, B, C, D) = (4, 6, 12, 13)

51 49 / 50 Homework 3-16Simplify the following expressions, and implement them with two-level NAND gate circuits: (a) AB + ABD + ABD + ACD + ABC (b) BD + BCD + ABCD 3-18Draw a logic diagram using only two-input NAND gates to implement the following expression: (AB + AB) (CD + CD)

52 50 / 50 Homework 3-22Convert the logic diagram of the circuit shown in Fig. 4-4 into a multiple-level NAND circuit.


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