1. Discrete Systems I Lecture 09 Minterms and Decoders Profs. Koike and Yukita

2. Realizing arbitrary Boolean functions We can write the truth table for any Boolean function. We will discover ways to derive a Boolean function for any truth table. 2

3. Minterms A minterm is a special Boolean function: – for which one and only one combination of input bits yields a 1, – and all other combinations yield a 0. Example  3

4. Find the Boolean expression for this special kind of a truth table 1.Write out all the combinations of function’s input values. 2.Draw NOT bar over those variables that are 0 in that combination of bits that makes the function 1. 4

5. Decoder A decoder is a circuit that has n input lines and 2 n output lines. Realized as a collection of minterms. 5

6. 3-8 Decoder 6

7. Realizing any Boolean using minterms We will give a definition later. 7

8. Construct minterms for nonzero outputs 8

9. OR them together 9

10. Minterms 10

11. Sum-of-products expressions 11

12. Realization of any Boolean functions using a decoder 12

13. The Seven Segment Display Binary to base 10 display 13 000000010010 0011

14. The Seven Segment Display Binary to base 10 display 14 010001010110 0111

15. The Seven Segment Display Binary to the base 10 display 15 1000 1001

16. Seven Segment Display with Each Controlling Function 16

17. f5 17

18. f5 18

19. Problem 1 19

20. Problem 2 Write the minterm realizations of the remaining six functions needed to implement a seven segment display. 20

21. Maxterms A maxterm is a special Boolean function: – for which one and only one combination of input bits yields a 0, – and all other combinations yield a 1. Example  21

22. Find the Boolean expression for this special kind of a truth table 1.Write out all the combinations of function’s input values. 2.Draw NOT bar over those variables that are 1 in that combination of bits that makes the function 0. 22

23. Ex. XOR Realizing any Boolean functions with maxterms 23

24. Ex. XOR Realizing any Boolean functions with minterms 24

25. Ex. Equivalent Realizations of XOR 25

26. The Three Input Majority Voter A three input Boolean function that counts its input bits. If there are more 1’s than 0’s, the function is to produce a 1 as output, and if there are more 0’s than 1’s, the function is to produce a 0. It is clear that there can never be a tie. 26

27. The Three Input Majority Voters 27

28. The Three Input Majority Voters 28

29. Problem 3 Write the minterm and maxtermrealizations of the following: (1) NAND (2) NOR (3) The implication function (4) AND 29

30. Problem 3 30

31. Problem 4-5 Problem 4: Write the minterm realization of the three-input majority voters. Problem 5: Write the maxterm realization of each of the seven functions needed to implement a full seven-segment display. 31