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

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

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

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

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

3-8 Decoder 6

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

Construct minterms for nonzero outputs 8

OR them together 9

Minterms 10

Sum-of-products expressions 11

Realization of any Boolean functions using a decoder 12

The Seven Segment Display Binary to base 10 display

The Seven Segment Display Binary to base 10 display

The Seven Segment Display Binary to the base 10 display

Seven Segment Display with Each Controlling Function 16

f5 17

f5 18

Problem 1 19

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

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

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

Ex. XOR Realizing any Boolean functions with maxterms 23

Ex. XOR Realizing any Boolean functions with minterms 24

Ex. Equivalent Realizations of XOR 25

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

The Three Input Majority Voters 27

The Three Input Majority Voters 28

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

Problem 3 30

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