1. ECE 331 – Digital System Design Basic Logic Operations, Boolean Expressions and Truth Tables, and Standard Logic Gates The slides included herein were taken from the materials accompanying Fundamentals of Logic Design, 6 th Edition, by Roth and Kinney, and were used with permission from Cengage Learning.

2. Fall 2010ECE 331 - Digital System Design2 Basic Logic Operations

3. Fall 2010ECE 331 - Digital System Design3 Basic Logic Operations AND  Logical multiplication (product)  A. B = AB = “A and B” OR  Logical addition (sum)  A + B = “A or B” NOT  Logical complement  A' = “not A” Order of Precedence: 1. NOT 2. AND 3. OR - can be modified by parenthesis

4. Fall 2010ECE 331 - Digital System Design4 Truth Tables ABA. BA + BA'B' 000011 010110 100101 111100 ANDORNOT

5. Fall 2010ECE 331 - Digital System Design5 Logic Gates NOT AND OR A A'

6. Fall 2010ECE 331 - Digital System Design6 Additional Logic Operations NAND  F = (A. B)'  F = not (A and B)  Compound logic function NOR  F = (A + B)'  F = not (A or B)  Compound logic function Not the same as F = A'. B' Not the same as F = A' + B'

7. Fall 2010ECE 331 - Digital System Design7 Truth Tables Exercise: Derive the Truth Table for the NAND and NOR gates.

8. Fall 2010ECE 331 - Digital System Design8 Logic Gates shorthand for a NOT gate

9. Fall 2010ECE 331 - Digital System Design9 Additional Logic Operations Exclusive-OR  F = (A.B') + (A'.B)  Output = 1 iff one of the inputs is 1 (but not both).  Inclusive-OR: F = A + B Exclusive-NOR  F = (A.B) + (A'.B')  Output = 1 iff both input are 0 or both inputs are 1.  Also known as the Equivalence function.

10. Fall 2010ECE 331 - Digital System Design10 Truth Tables Exercise: Derive the Truth Table for the XOR and XNOR gates.

11. Fall 2010ECE 331 - Digital System Design11 Logic Gates Exclusive-OR Equivalence Exclusive-NOR

12. Fall 2010ECE 331 - Digital System Design12 Truth Tables

13. Fall 2010ECE 331 - Digital System Design13 Truth Tables Used to describe the functional behavior of a Boolean expression and/or Logic circuit. Each row in the truth table represents a unique combination of the input variables.  For n input variables, there are 2 n rows. The output of the logic function is defined for each row. Each row is assigned a numerical value, with the rows listed in ascending order. The order of the input variables defined in the logic function is important.

14. Fall 2010ECE 331 - Digital System Design14 3-input Truth Table #ABCF(A,B,C) 0000 1001 2010 3011 4100 5101 6110 7111

15. Fall 2010ECE 331 - Digital System Design15 4-input Truth Table #ABCDF(A,B,C,D) 00000 10001 20010 30011 40100 50101 60110 … 121100 131101 141110 151111

16. Fall 2010ECE 331 - Digital System Design16 Boolean Expressions

17. Fall 2010ECE 331 - Digital System Design17 Boolean Expressions Boolean expressions are composed of Literals – variables and their complements Logical operations Examples F = A.B'.C + A'.B.C' + A.B.C + A'.B'.C' F = (A+B+C').(A'+B'+C).(A+B+C) F = A.B'.C' + A.(B.C' + B'.C) literals logic operations

18. Fall 2010ECE 331 - Digital System Design18 Boolean Expressions Boolean expressions are realized using a network (or combination) of logic gates.  Each logic gate implements one of the logic operations in the Boolean expression  Each input to a logic gate represents one of the literals in the Boolean expression logic operations literals A B F

19. Fall 2010ECE 331 - Digital System Design19 Boolean Expressions Boolean expressions are evaluated by Substituting a 0 or 1 for each literal Calculating the logical value of the expression A Truth Table specifies the value of the Boolean expression for every combination of the variables in the Boolean expression. For an n-variable Boolean expression, the truth table has 2 n rows (one for each combination).

20. Fall 2010ECE 331 - Digital System Design20 Boolean Expressions Example: Evaluate the following Boolean expression, for all combination of inputs, using a Truth table. F(A,B,C) = A'.B'.C + A.B'.C'

21. Fall 2010ECE 331 - Digital System Design21 Boolean Expressions Two Boolean expressions are equivalent if they have the same value for each combination of the variables in the Boolean expression.  F 1 = (A + B)'  F 2 = A'.B' How do you prove that two Boolean expressions are equivalent?  Truth table  Boolean Algebra

22. Fall 2010ECE 331 - Digital System Design22 Boolean Expressions Example: Using a Truth table, prove that the following two Boolean expressions are equivalent. F 1 = (A + B)' F 2 = A'.B'

23. Fall 2010ECE 331 - Digital System Design23 Standard Logic Gates

24. Fall 2010ECE 331 - Digital System Design24 Standard Logic Gates Discrete components used to build logic circuits.  74xx08AND gate  74xx32OR gate  74xx04NOT gate  74xx00NAND gate  74xx02NOR gate  74xx86XOR gate Logic Families (TTL, LS-TTL, F, HC, …)

25. Fall 2010ECE 331 - Digital System Design25 Standard Logic Gates: Data sheets Data sheets provide essential information:  Logic Function  Truth Table  Pin-out  Electrical Characteristics  Timing Characteristics  Package Description(s) This information is necessary when building logic circuits from discrete components. Each Logic Family has a unique set of characteristics.

26. Fall 2010ECE 331 - Digital System Design26 Logic Families TTL  Low-Power TTL (“L”)  High-Speed TTL (“H”)  Schottky (“S”)  Low-Power Schottky (“LS”)  Advanced Schottky (“AS”)  Advanced Low-Power Schottky (“ALS”)  Fast (“F”) CMOS  High-Speed CMOS (“HC”)  Advanced CMOS (“AC”)

27. Fall 2010ECE 331 - Digital System Design27 Comparison of Logic Families

28. Fall 2010ECE 331 - Digital System Design28 Combinational Logic Circuits

29. Fall 2010ECE 331 - Digital System Design29 Combinational Logic Circuits Composed of an interconnected set of logic gates. Also known as Switching Circuits Logic circuits can be designed from  Truth tables  Boolean expressions Logic circuits realized through  Interconnection of discrete components  Synthesis from a Hardware Description Language

30. Fall 2010ECE 331 - Digital System Design30 Example: Derive the Boolean expression and draw the circuit diagram for the following truth table: ABF(A,B) 001 010 101 111

31. Fall 2010ECE 331 - Digital System Design31 Exercise: Derive the Boolean expression and draw the circuit diagram for the following truth table: ABCF(A,B,C) 0000 0011 0101 0110 1001 1010 1100 1111

32. Fall 2010ECE 331 - Digital System Design32 Wiring Diagrams

33. Fall 2010ECE 331 - Digital System Design33 Standard Logic Gate: Pin-out

34. Fall 2010ECE 331 - Digital System Design34 Standard Logic Gate: Pin-out

35. Fall 2010ECE 331 - Digital System Design35 Standard Logic Gate: Pin-out

36. Fall 2010ECE 331 - Digital System Design36 Example: Draw the circuit diagram and the wiring diagram for the following Boolean expression: F = B'.C + A.B

37. Fall 2010ECE 331 - Digital System Design37 Circuit Diagram

38. Fall 2010ECE 331 - Digital System Design38 Wiring Diagram