1. Karnaugh Mapping Digital Electronics

2. Karnaugh Mapping or K-Mapping This presentation will demonstrate how to Create and label two, three, & four variable K-Maps. Use the K-Mapping technique to simplify logic designs with two, three, and four variables. Use the K-Mapping technique to simplify logic design containing don’t care conditions. Boolean Algebra Simplification K-Mapping Simplification ≡ 2

3. Karnaugh Map Technique K-Maps are a graphical technique used to simplify a logic equation. K-Maps are procedural and much cleaner than Boolean simplification. K-Maps can be used for any number of input variables, BUT are only practical for two, three, and four variables. 3

4. K-Map Format Each minterm in a truth table corresponds to a cell in the K-Map. K-Map cells are labeled such that both horizontal and vertical movement differ only by one variable. Since the adjacent cells differ by only one variable, they can be grouped to create simpler terms in the sum-of- products expression. The sum-of-products expression for the logic function can be obtained by OR-ing together the cells or group of cells that contain 1s. 4

5. Adjacent Cells = Simplification V V 10 10 V 10 10 5

6. Truth Table to K-Map Mapping V WXF WX Minterm – 0001 Minterm – 1010 Minterm – 2101 Minterm – 3110 V 01 23 1 0 1 0 Two Variable K-Map 6

7. V 00 00 Groups of One – 4 Two Variable K-Map Groupings 1 1 1 1 7

8. V 00 00 Groups of Two – 4 Two Variable K-Map Groupings 1111 1 1 1 1 8

9. V 11 11 Group of Four – 1 Two Variable K-Map Groupings 9

10. K-Map Simplification Process 1. Construct a label for the K-Map. Place 1s in cells corresponding to the 1s in the truth table. Place 0s in the other cells. 2. Identify and group all isolated 1’s. Isolated 1’s are ones that cannot be grouped with any other one, or can only be grouped with one other adjacent one. 3. Group any hex. 4. Group any octet, even if it contains some 1s already grouped but not enclosed in a hex. 5. Group any quad, even if it contains some 1s already grouped but not enclosed in a hex or octet. 6. Group any pair, even if it contains some 1s already grouped but not enclosed in a hex, octet, or quad. 7. OR together all terms to generate the SOP equation. 10

11. Example #1: 2 Variable K-Map Example: After labeling and transferring the truth table data into the K-Map, write the simplified sum-of-products (SOP) logic expression for the logic function F 1. V JKF1F1 001 011 100 110 11

12. Example #1: 2 Variable K-Map Example: After labeling and transferring the truth table data into the K-Map, write the simplified sum-of-products (SOP) logic expression for the logic function F 1. V 11 00 Solution: JKF1F1 001 011 100 110 12

13. Truth Table to K-Map Mapping WXYF WXY Minterm – 00001 Minterm – 10010 Minterm – 20100 Minterm – 30110 Minterm – 41000 Minterm – 51011 Minterm – 61101 Minterm – 71110 V 01 23 67 45 1 Three Variable K-Map 0 0 0 0 1 10 Only one variable changes for every row change 13

14. Three Variable K-Map Groupings V 00 00 00 00 11111111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 Groups of One – 8 (not shown) Groups of Two – 12 14

15. Groups of Four – 6 Three Variable K-Map Groupings V 00 00 00 00 1 1 1 1 1 1 1 1 11 1 1 11 1 1 11 1 1 11 11 15

16. Group of Eight - 1 Three Variable K-Map Groupings V 11 11 11 11 16

17. Example #2: 3 Variable K-Map Example: After labeling and transferring the truth table data into the K-Map, write the simplified sum-of-products (SOP) logic expression for the logic function F 2. EFGF2F2 0000 0011 0101 0110 1000 1011 1101 1110 17

18. Example #2: 3 Variable K-Map Example: After labeling and transferring the truth table data into the K-Map, write the simplified sum-of-products (SOP) logic expression for the logic function F 2. V 01 10 10 01 Solution: EFGF2F2 0000 0011 0101 0110 1000 1011 1101 1110 18

19. Truth Table to K-Map Mapping Four Variable K-Map WXYZF WXYZ Minterm – 000000 Minterm – 100011 Minterm – 200101 Minterm – 300110 Minterm – 401001 Minterm – 501011 Minterm – 601100 Minterm – 701111 Minterm – 810000 Minterm – 910010 Minterm – 1010101 Minterm – 1110110 Minterm – 1211001 Minterm – 1311010 Minterm – 1411101 Minterm – 1511111 V 0132 4576 12131514 891110 1011 1101 0100 0110 Only one variable changes for every row change Only one variable changes for every column change 19

20. Four Variable K-Map Groupings V 0000 0000 0000 0000 Groups of One – 16 (not shown) Groups of Two – 32 (not shown) Groups of Four – 24 (seven shown) 11 11 11 11 1 1 1 1 11 11 1 1 1 1 1 1 1 1 1 1 1 1 20

21. Four Variable K-Map Groupings V 0000 0000 0000 0000 Groups of Eight – 8 (two shown) 11 11 11 11 1 1 1 1 1 1 1 1 21

22. Four Variable K-Map Groupings V 1111 1111 1111 1111 Group of Sixteen – 1 22

23. Example #3: 4 Variable K-Map Example: After labeling and transferring the truth table data into the K-Map, write the simplified sum-of-products (SOP) logic expression for the logic function F 3. RSTUF3F3 00000 00011 00100 00111 01000 01011 01101 01111 10000 10011 10100 10110 11001 11010 11101 11111 V 23

24. Example #3 : 4 Variable K-Map Example: After labeling and transferring the truth-table data into the K-Map, write the simplified sum-of-products (SOP) logic expression for the logic function F 3. Solution: RSTUF3F3 00000 00011 00100 00111 01000 01011 01101 01111 10000 10011 10100 10110 11001 11010 11101 11111 V 0110 0111 1011 0100 24

25. Don’t Care Conditions A don’t care condition, marked by (X) in the truth table, indicates a condition where the design doesn’t care if the output is a (0) or a (1). A don’t care condition can be treated as a (0) or a (1) in a K-Map. Treating a don’t care as a (0) means that you do not need to group it. Treating a don’t care as a (1) allows you to make a grouping larger, resulting in a simpler term in the SOP equation. 25

26. Some You Group, Some You Don’t V X0 10 00 X0 This don’t care condition was treated as a (1). This allowed the grouping of a single one to become a grouping of two, resulting in a simpler term. There was no advantage in treating this don’t care condition as a (1), thus it was treated as a (0) and not grouped. 26

27. Example #4: Don’t Care Conditions Example: After labeling and transferring the truth table data into the K-Map, write the simplified sum-of-products (SOP) logic expression for the logic function F 4. Be sure to take advantage of the don’t care conditions. RSTUF4F4 0000X 00010 00101 0011X 01000 0101X 0110X 01111 10001 10011 10101 1011X 1100X 11010 11100 11110 V 27

28. Example #4: Don’t Care Conditions Example: After labeling and transferring the truth table data into the K-Map, write the simplified sum-of-products (SOP) logic expression for the logic function F 4. Be sure to take advantage of the don’t care conditions. Solution: RSTUF4F4 0000X 00010 00101 0011X 01000 0101X 0110X 01111 10001 10011 10101 1011X 1100X 11010 11100 11110 V X0X1 0X1X X000 11X1 28