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Combinational Logic 1. 2 Topics Basics of digital logic Basic functions ♦ Boolean algebra ♦ Gates to implement Boolean functions Identities and Simplification.

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Presentation on theme: "Combinational Logic 1. 2 Topics Basics of digital logic Basic functions ♦ Boolean algebra ♦ Gates to implement Boolean functions Identities and Simplification."— Presentation transcript:

1 Combinational Logic 1

2 2 Topics Basics of digital logic Basic functions ♦ Boolean algebra ♦ Gates to implement Boolean functions Identities and Simplification

3 3 Binary Logic Binary variables ♦ Can be 0 or 1 (T or F, low or high) ♦ Variables named with single letters in examples ♦ Use words when designing circuits Basic Functions ♦ AND ♦ OR ♦ NOT

4 4 AND Operator Symbol is dot ♦ Z = X · Y Or no symbol ♦ Z = XY Truth table -> Z is 1 only if ♦ Both X and Y are 1

5 5 OR Operator Symbol is + ♦ Not addition ♦ Z = X + Y Truth table -> Z is 1 if either 1 ♦ Or both!

6 6 NOT Operator Unary Symbol is bar (or ’) ♦ Z = X’ Truth table -> Inversion

7 7 Gates Circuit diagrams are traditionally used to document circuits Remember that 0 and 1 are represented by voltages

8 8 AND Gate Timing Diagrams

9 9 OR Gate

10 10 Inverter

11 11 More Inputs Work same way What’s output?

12 12 Digital Circuit Representation: Schematic

13 13 Digital Circuit Representation: Boolean Algebra For now equations with operators AND, OR, and NOT Can evaluate terms, then final OR Alternate representations next

14 14 Digital Circuit Representation: Truth Table 2 n rows where n # of variables

15 15 Functions Can get same truth table with different functions Usually want simplest function ♦ Fewest gates or using particular types of gates ♦ More on this later

16 16 Identities Use identities to manipulate functions On previous slide, I used distributive law to transform from to

17 17 Table of Identities

18 18 Duals Left and right columns are duals Replace AND with OR, 0s with 1s

19 19 Single Variable Identities

20 20 Commutative Order independent

21 21 Associative Independent of order in which we group So can also be written as and

22 22 Distributive Can substitute arbitrarily large algebraic expressions for the variables

23 23 DeMorgan’s Theorem Used a lot NOR equals invert AND NAND equals invert OR

24 24 Truth Tables for DeMorgan’s

25 25 Algebraic Manipulation Consider function

26 26 Simplify Function Apply

27 27 Fewer Gates

28 28 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!

29 29 Complement of a Function Definition: 1s & 0s swapped in truth table

30 30 Truth Table of the Complement of a Function XYZ F = X + Y ’ ZF’F’ 00001 00110 01001 01101 10010 10110 11010 11110

31 31 Algebraic Form for Complement Mechanical way to derive algebraic form for the complement of a function 1.Take the dual Recall: Interchange AND & OR, and 1s & 0s 2.Complement each literal (a literal is a variable complemented or not; e.g. x, x’, y, y’ each is a literal)

32 32 Example: Algebraic form for the complement of a function F = X + Y’Z To get the complement F’ 1.Take dual of right hand side X. (Y’ + Z) 2.Complement each literal: X’. (Y + Z’) F’ = X’. (Y + Z’)

33 Mechanically Go From Truth Table to Function

34 34 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

35 35 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

36 36 Definition: Minterm Product term in which all variables appear once (complemented or not) For the variables X, Y and Z example minterms : X’Y’Z’, X’Y’Z, X’YZ’, …., XYZ

37 37 Definition: Minterm (continued) Min Term Each minterm represents exactly one combination of the binary variables in a truth table.

38 38 Truth Tables of Minterms

39 39 Number of Minterms For n variables, there will be 2 n minterms Minterms are labeled from minterm 0, to minterm 2 n -1 ♦ m 0, m 1, m 2, …, m 2 n -2, m 2 n -1 For n = 3, we have ♦ m 0, m 1, m 2, m 3, m 4, m 5, m 6, m 7

40 40 Definition: Maxterm Sum term in which all variables appear once (complemented or not) For the variables X, Y and Z the maxterms are: X+Y+Z, X+Y+Z’ …., X’+Y’+Z’

41 41 Definition: Maxterms (continued) mmmmmmmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmm,m xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx,mmmmmmmmmmmmmmmmmmmmmmm mmmmmmmmmmmmmmmmmmmmmmm Maxterm

42 42 Truth Tables of Maxterms

43 43 Minterm related to Maxterm Minterms and maxterms with same subscripts are complements Example

44 44 Standard Form of F: Sum of Minterms OR all of the minterms of truth table for which the function value is 1 F = m 0 + m 2 + m 5 + m 7

45 45 Complement of F Not surprisingly, just sum of the other minterms In this case F’ = m 1 + m 3 + m 4 + m 6

46 46 Product of Maxterms Recall that maxterm is true except for its own row So M1 is only false for 001

47 47 Product of Maxterms F = m 0 + m 2 + m 5 + m 7 Remember: ♦ M1 is only false for 001 ♦ M3 is only false for 011 ♦ M4 is only false for 100 ♦ M6 is only false for 110 Can express F as AND of M1, M3, M4, M6 or

48 48 Recap Working (so far) with AND, OR, and NOT Algebraic identities Algebraic simplification Minterms and maxterms Can now synthesize function from truth table


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