Eng. Mohammed Timraz Electronics & Communication Engineer

1. University of Palestine Faculty of Engineering and Urban planning Software Engineering Department Digital Logic Design ESGD2201 Lecture 9 The Karnaugh Map. Eng. Mohammed Timraz Electronics & Communication Engineer Wednesday, 2nd July 2008

2. Agenda The Karnaugh Map 1. The two-variable Karnaugh map. 2. The three-variable Karnaugh map. 3. The four-variable Karnaugh map. 4. Karnaugh map SOP minimization. 5. Mapping a standard SOP expression. 6. Mapping a nonstandard SOP expression. 7. Karnaugh map simplification of SOP expression. 8. Mapping directly from a truth table. 9. “Don’t care “ conditions.

3. The Karnaugh Map • Introduction: • The Karnaugh map provides a systematic method for simplifying Boolean expression and, if properly used, will produce the simplest SOP or POS expression possible. • As you have seen, the effectiveness of algebraic simplification depends on your familiarity with all the laws, rules and theorems of Boolean algebra and on your ability to apply them. • The Karnaugh map on the other hand basically provides a “ cookbook “ method for simplification.

4. The Karnaugh Map • Introduction: After completing this section, you should be able to: • Construct a Karnaugh map for two, three or four variables. • Determine the binary value of each cell in a Karnaugh map. • Determine the standard product term represented by each cell in a Karnaugh map. • Explain cell adjacency and identify adjacent cells.

5. The Karnaugh Map • Introduction: • The number of cells in a Karnaugh map is equal to the total number of possible input variables combinations as is the number of rows in a truth table. • For two variables, the number of cells is 22=4. • For three variables, the number of cells is 23=8. • For four variables, the number of cells is 24=16.

6. The Karnaugh Map 1. The 2-Variables Karnaugh Map: It is an array of four cells (22=4). In this case A and B are used for the variables although other letters could be used. Binary values of A is along the left side and the values of B is across the top.

7. The Karnaugh Map 1. The 2-Variables Karnaugh Map: B 0 1 A 0 1

8. The Karnaugh Map 1. The 2-Variables Karnaugh Map:

9. The Karnaugh Map 1. The 2-Variables Karnaugh Map: B 0 1 A 0 1

10. The Karnaugh Map 1. The 2-Variables Karnaugh Map with truth table: Truth Table B 0 1 A 0 1

13. The Karnaugh Map 2. The 3-Variables Karnaugh Map: It is an array of eight cells (23=8). In this case A , B and C are used for the variables although other letters could be used. Binary values of A and B are along the left side and the values of C is across the top.

14. 000 001 010 011 110 111 100 101 The Karnaugh Map 2. The 3-Variables Karnaugh Map: C 0 1 AB 00 01 11 10

15. The Karnaugh Map 2. The 3-Variables Karnaugh Map:

16. 000 0 001 1 010 2 011 3 110 6 111 7 100 4 101 5 The Karnaugh Map 2. The 3-Variables Karnaugh Map: C 0 1 AB 00 01 11 10

17. The Karnaugh Map 2. The 3-Variables Karnaugh Map with truth table:

20. The Karnaugh Map 3. The 4-Variables Karnaugh Map: It is an array of sixteen cells (24=16). In this case A , B, C and D are used for the variables although other letters could be used. Binary values of A and B are along the left side and the values of C and D are across the top.

21. 0000 0011 0001 0010 0111 0100 0101 0110 1100 1111 1110 1101 1000 1011 1001 1010 The Karnaugh Map 3. The 4-Variables Karnaugh Map: CD 00 01 11 10 AB 00 01 11 10

22. The Karnaugh Map 3. The 4-Variables Karnaugh Map:

23. 0000 0 0011 3 0001 1 0010 2 0111 7 0100 4 0101 5 0110 6 1100 12 1111 15 1110 14 1101 13 1000 8 1011 11 1001 9 1010 10 The Karnaugh Map 3. The 4-Variables Karnaugh Map: CD 00 01 11 10 AB 00 01 11 10

24. The Karnaugh Map 3. The 4-Variables Karnaugh Map with truth table:

25. The Karnaugh Map 3. The 4-Variables Karnaugh Map with truth table:

28. The Karnaugh Map • 4. Karnaugh map SOP minimization. • The Karnaugh map is used for simplification Boolean expressions to their minimum form. • A minimized SOP expression contains the fewest possible terms with the fewest possible variables per term. • Generally; a minimum SOP expression can be implemented with fewer logic gates than a standard expression and this is the basic purpose in the simplification process.

29. The Karnaugh Map • 4. Karnaugh map SOP minimization. After completing this section, you should be able to: • Map a standard SOP expression on a Karnaugh map. • Combine the 1s on the map into maximum groups. • Determine the minimum product term for each group on the map. • Combine the minimum product terms to form a minimum SOP expression. • Convert a truth table into a Karnaugh map for simplification of the represented expression. • Use “ don’t care “ conditions on a Karnaugh map.

32. The Karnaugh Map • 5. Mapping a standard SOP expression. After completing this section, you should be able to: • Map a standard SOP expression on a Karnaugh map. • Combine the 1s on the map into maximum groups. • Determine the minimum product term for each group on the map. • Combine the minimum product terms to form a minimum SOP expression. • Convert a truth table into a Karnaugh map for simplification of the represented expression. • Use “ don’t care “ conditions on a Karnaugh map.