Introduction to Digital Logic Missouri S&T University CPE 2210 Boolean Representations

Transcription

1 Introduction to Digital Logic Missouri S&T University CPE 2210 Egemen K. Çetinkaya Egemen K. Çetinkaya Department of Electrical & Computer Engineering Missouri University of Science and Technology 7 September 2016 rev Egemen K. Çetinkaya

2 Introduction Outline Boolean representations Summary 2

3 Logic Gates AND Gate Output is 1 if only both inputs are 1 Symbol: Truth table: x y AND F x y F

4 Logic Gates OR Gate Output is 1 if one input is 1 Symbol: Truth table: x y OR F x y F

5 Also known as inverter Symbol: Truth table: x Logic Gates NOT Gate N O T F x F

6 Boolean Algebra Operators Basic operators (more later) Name Symbol Example How to say not or a or ā a prime complement of a inverse of a and or a b a b product of a and b ab or + a + b sum of a and b a or b Note that expression in () must be done first 6

7 Boolean Algebra Properties Understand the important Boolean algebra properties commutativity distributivity associativity identity complementery null elements idempotent law involution law DeMorgan s law absorption law 7

8 Equations Mappings allows simplification e.g. F = x Circuits using logic gates, represents actual physical implementation e.g. Truth tables represents all possible combinations of input values one function has only one truth table representation e.g. x x F F 8

9 Mapping Conversion Equations to circuits Circuits to equations Equations to truth tables Truth tables to equations Circuits to truth tables Truth tables to circuits 9

10 Equation: Equations to Circuits mathematical statement Different equations can represent same function We have done previous examples 10

11 Circuits to Equations A circuit is interconnection of components Boolean functions can be represented via logic gates Different circuits can represent same function Conversion process: start from the circuit s inputs write the output of each gate as an expression We have done examples 11

12 Equations to Truth Tables Equations to truth tables Truth tables show all input variable combinations A Boolean function has only one truth table Conversion process: create truth table structure fill the output using Boolean algebra if necessary create intermediate columns E.g.: Find the truth table for F=a b+ac+bc 12

13 Truth Tables to Equations Conversion process: first create a product term for each 1 in the output column then OR all the product terms E.g.: What is the F function? x y F

14 Circuits to Truth Tables Conversion process: first convert circuit to an equation then convert equation to truth table E.g.: Convert the following circuit to truth table? a b F c 14

15 Truth Tables to Circuits Conversion process: first truth table to an equation then convert equation to circuit E.g.: Convert the truth table to circuit? x y z F

16 Terminology Example F(a,b,c)=ab+c+ac What are the variables? What are the literals? What are the product terms? 16

17 Terminology Example F(a,b,c)=ab+c+ac Variables: a, b, c Literals: a, b, c, a, c Product terms: ab, c, ac 17

18 Sum-of-Products Form Equation written as OR of product terms Examples: Following are in sum-of-products form ab + a c a + b + cde Following are not in sum-of-products form (a+b)c (a ) + b variable should be complemented or uncomplemented form 18

19 Product-of-Sums Form Equation written as AND of sum terms Examples: Following are in product-of-sums form a (a + c) (a + b) a b (c + d + e) Following are not in sum-of-products form (a b)+c (a ) b variable should be complemented or uncomplemented form 19

20 Normal Term A product or a sum term no variable appears more than once A nonnormal term can be simplified to a constant or a normal term Nonnormal terms example: abbc, a+a+c, aa b Normal terms example: abc, a +b+c 20

21 Minterm A product term in which literals include every variable only once, either true or complemented form Example: What are the minterms of the function: F(a,b,c)=ab+abc +ac +c 21

22 Minterm A product term in which literals include every variable only once, either true or complemented form Example: What are the minterms of the function: F(a,b,c)=ab+abc +ac +c Answer: abc An equation is sum-of-minterms form if: every product term is a minterm Example: F(a,b,c)=abc+ab c+a b c 22

23 Maxterm A sum term in which literals include every variable only once, either true or complemented form Example: What are the maxterms of the function: F(a,b,c)=(a+b) (a+b+c ) (a+c ) c 23

24 Maxterm A sum term in which literals include every variable only once, either true or complemented form Example: What are the maxterms of the function: F(a,b,c)=(a+b) (a+b+c ) (a+c ) c Answer: a+b+c An equation is product-of-maxterms form if: every sum term is a maxterm Example: F(a,b,c)=(a+b+c) (a+b +c) (a +b +c ) 24

25 Minterms and Maxterms Minterm is a product term that is 1 Maxterm is a sum term that is 0 row x y z F minterm maxterm F(0,0,0)?? F(0,0,1)?? F(0,1,0)?? F(0,1,1)?? F(1,0,0)?? F(1,0,1)?? F(1,1,0)?? F(1,1,1)?? 25

26 Minterms and Maxterms Minterm is a product term that is 1 Maxterm is a sum term that is 0 row x y z F minterm maxterm F(0,0,0) x y z x+y+z F(0,0,1) x y z x+y+z F(0,1,0) x yz x+y +z F(0,1,1) x yz x+y +z F(1,0,0) xy z x +y+z F(1,0,1) xy z x +y+z F(1,1,0) xyz x +y +z F(1,1,1) xyz x +y +z 26

27 Minterms and Maxterms Example Minterm is a product term that is 1 Maxterm is a sum term that is 0 What is the canonical sum? What is the canonical product? What is the minterm list? What is the maxterm list? row x y F

28 Minterms and Maxterms Example Minterm is a product term that is 1 Maxterm is a sum term that is 0 Canonical sum: F=x y +x y Canonical product: F=(x +y) (x +y ) Minterm list: x,y (0,1) row x y F note that book shows m Maxterm list: x,y (2,3) 28

29 Canonical Form Conversion How do you convert any equation to canonical form? E.g. F(a,b)=ab+b 29

30 Canonical Form Conversion How do you convert any equation to canonical form? E.g. F(a,b)=ab+b 1. Convert equation to its sum-of-products form 2. Expand each term until every term is a minterm F(a,b)=ab+b (a+a )=ab+ab +a b 30

31 Multiple Output Multiple output example: can you simplify it? Egemen K. Çetinkaya a b c F G 31

32 Example Egemen K. Çetinkaya 32

33 Summary Three Boolean representations: circuits equations truth tables Important terminology variables/literals product term/sum term/sum-of-products/product-of-sums normal term/minterm/maxterm canonical sum (sum-of-minterms) canonical product (product-of-maxterms) 33

34 References and Further Reading [V2011] Frank Vahid, Digital Design with RTL Design, VHDL, and Verilog, 2nd edition, Wiley, [BV2009] Stephen Brown and Zvonko Vranesic, Fundamentals of Digital Logic with VHDL Design, 3rd edition, McGraw-Hill, [MKM2016] M. Morris Mano, Charles R. Kime, Tom Martin, Logic and Computer Design Fundamentals, 5th edition, Pearson, [W2006] John F. Wakerly, Digital Design Principles and Practices, 4th edition, Prentice Hall,

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