Combinational Logic 1

1. CombinationalLogic 1

2. Today • Basics of digital logic • Basic functions • Boolean algebra • Gates to implement Boolean functions • Identities and Simplification Digital Logic, Fall 2005Digital Logic

3. Digital circuits are a hardware that manipulate binary info. • Each basic unit in a circuit is called: Gate. • Boolean Algebra: a mathematical system that describe the binary logic system. Digital Logic, Fall 2005Digital Logic

4. Binary Logic • Deals with binary variables and mathematical logic. • Binary variables • Can be 0 or 1 (T or F, low or high) • Variables named with single letters in examples • Really use words when designing circuits Basic Functions • AND • OR • NOT Digital Logic, Fall 2005Digital Logic

5. AND • Symbol is dot • Z = A · B • Or no symbol • Z = AB • Truth table -> • Z is 1 only if • Both A and B are 1 • Truth table: a table of combinations of the binary variables showing the relationship between the values of variables and the result. Digital Logic, Fall 2005Digital Logic

6. OR • Symbol is + • Not addition • Z = A + B • Truth table -> • Z is 1 if either 1 • Or both! Digital Logic, Fall 2005Digital Logic

7. NOT • Unary • Symbol is bar • Z = Ā • Truth table -> • Inversion Digital Logic, Fall 2005Digital Logic

8. Gates • Electronic circuits that operate on one or more inputs to produce an output. • Remember that 0 and 1 are represented by voltages Digital Logic, Fall 2005Digital Logic

9. AND Gate Timing Diagrams Digital Logic, Fall 2005Digital Logic

10. OR Gate Digital Logic, Fall 2005Digital Logic

11. Inverter Digital Logic, Fall 2005Digital Logic

12. More Inputs • Work same way • What’s output? Digital Logic, Fall 2005Digital Logic

13. Representation: Schematic Digital Logic, Fall 2005Digital Logic

14. Representation: Boolean Algebra • Deals with binary variables and logic operations. • For now equations with operators AND, OR, and NOT • Boolean function: described by Boolean equation. • Boolean equation: express logical relationship between binary variables Term Boolean function Digital Logic, Fall 2005Digital Logic

15. Representation: Truth Table • 2n rows where n # of variables Digital Logic, Fall 2005Digital Logic

16. Functions • Can get same truth table with different functions • Usually want simplest • Fewest gates or using particular types of gates • More on this later Digital Logic, Fall 2005Digital Logic

17. Combinational Logic Circuits: Circuit gates interconnected by wires that carry logic signals. Digital Logic, Fall 2005Digital Logic

18. Identities • Use identities to manipulate functions • I used distributive law to transform from to Digital Logic, Fall 2005Digital Logic

19. Table of Identities Digital Logic, Fall 2005Digital Logic

20. Duals • Left and right columns are duals • Replace AND with OR, 0s with 1s Digital Logic, Fall 2005Digital Logic

21. Single Variable Identities Digital Logic, Fall 2005Digital Logic

22. Commutative • Order independent Digital Logic, Fall 2005Digital Logic

23. Associative • Independent of order in which we group • So can also be written as and Digital Logic, Fall 2005Digital Logic

24. Distributive • Can substitute arbitrarily large algebraic expressions for the variables Digital Logic, Fall 2005Digital Logic

25. DeMorgan’s Theorem • Used a lot • NOR equals invert AND • NAND equals invert OR Digital Logic, Fall 2005Digital Logic

26. Truth Tables for DeMorgan’s Digital Logic, Fall 2005Digital Logic

27. Algebraic Manipulation • Consider function Digital Logic, Fall 2005Digital Logic

28. Simplify Function Apply Apply Apply Digital Logic, Fall 2005Digital Logic

29. Fewer Gates Digital Logic, Fall 2005Digital Logic

30. Consensus Theorem • The third term is redundant • Can just drop • Proof in book, but in summary • For third term to be true, Y & Z both 1 • Then one of the first two terms must be 1! Digital Logic, Fall 2005Digital Logic

31. Complement of a Function • Definition: 1s & 0s swapped in truth table • Mechanical way to derive algebraic form • Take the dual • Recall: Interchange AND & OR, and 1s & 0s • Complement each literal Digital Logic, Fall 2005Digital Logic

32. Complement of a Function • Definition: 1s & 0s swapped in truth table • Mechanical way to derive algebraic form for the complement of a function • Take the dual • Recall: Interchange AND & OR, and 1s & 0s • Complement each literal (a literal is a variable complemented or not; e.g. x , x’ , y, y’ each is a literal) Digital Logic, Fall 2005Digital Logic

33. Truth Table of the Complement of a Function Digital Logic, Fall 2005Digital Logic

34. Algebraic form for the Complement of a Function • F = X + Y’Z • Take dual of right hand side to get the complement F’ • F’ = X’ . (Y + Z’) Digital Logic, Fall 2005Digital Logic

35. Mechanically Go From Truth Table to Function

36. From Truth Table to Function • Consider a truth table • Can implement F by taking OR of all terms that correspond to rows for which F is 1 • “Standard Form” of the function Digital Logic, Fall 2005Digital Logic

37. Standard Forms • Not necessarily simplest F • But it’s mechanical way to go from truth table to function • Definitions: • Product terms – AND  ĀBZ • Sum terms – OR  X + Ā • This is logical product and sum, not arithmetic Digital Logic, Fall 2005Digital Logic

38. Definition: Minterm • Product term in which all variables appear once (complemented or not) • Represents exactly one combination of the binary variables in a truth table. Its value is 1 only for that combination Digital Logic, Fall 2005Digital Logic

39. Number of Minterms • For n variables, there will be 2n minterms • Minterms are labeled from minterm 0, m0 to to minterm 2n-1, m2n-1 Digital Logic, Fall 2005Digital Logic

40. Definition: Maxterms • Sum term in which all variables appear once (complemented or not) Digital Logic, Fall 2005Digital Logic

41. Minterm related to Maxterm • Minterm and maxterm with same subscripts are complements • Example Digital Logic, Fall 2005Digital Logic

42. Standard Form of F:Sum of Minterms • OR all of the minterms of truth table for which the function value is 1 • F = m0 + m2 + m5 + m7 • A function that includes all the minterms is equal to logic 1 Ex: G(X,Y)=Σm(0,1,2,3)=1 Digital Logic, Fall 2005Digital Logic

43. Complement of F • Not surprisingly, just sum of the other minterms • In this case F’ = m1 + m3 + m4 + m6 Digital Logic, Fall 2005Digital Logic

44. Product of Maxterms • Recall that maxterm is true except for its own row • So M1 is only false for 001 Digital Logic, Fall 2005Digital Logic

45. Product of Maxterms • Can express F as AND of all Maxterms of rows that should evaluate to 0 or Digital Logic, Fall 2005Digital Logic

46. Recap • Working (so far) with AND, OR, and NOT • Algebraic identities • Algebraic simplification • Minterms and maxterms • Can now synthesize function (and gates) from truth table Digital Logic, Fall 2005Digital Logic