# 1 COMP541 Combinational Logic Montek Singh Jan 16, 2007.

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1 COMP541 Combinational Logic Montek Singh Jan 16, 2007

2Today  Basics of digital logic (review) Basic functions Basic functions Boolean algebra Boolean algebra Gates to implement Boolean functions Gates to implement Boolean functions  Identities and Simplification (review?)

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

4AND  Symbol is dot C = A · B C = A · B  Or no symbol C = AB C = AB  Truth table ->  C is 1 only if Both A and B are 1 Both A and B are 1

5OR  Symbol is + Not addition Not addition C = A + B C = A + B  Truth table ->  C is 1 if either 1 Or both! Or both!

6NOT  Unary  Symbol is bar C = Ā C = Ā  Truth table ->  Inversion

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

8 AND Gate Timing Diagrams

9 OR Gate

10Inverter

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

12 Representation: Schematic  Schematic = circuit diagram

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

14 Representation: Truth Table  2 n rows where n = # of variables

15Functions  Can get same truth table with different functions  Usually want ‘simplest’ Fewest gates, or using only particular types of gates Fewest gates, or using only particular types of gates More on this later More on this later

16Identities  Use identities to manipulate functions  I used distributive law … … to transform from … to transform from to

17 Table of Identities

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

19 Single Variable Identities

20Commutativity  Operation is independent of order of variables

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

22Distributivity  Can substitute arbitrarily large algebraic expressions for the variables Distribute an operation over the entire expression Distribute an operation over the entire expression

23 DeMorgan’s Theorem  Used a lot  NOR  invert, then AND  NAND  invert, then OR

24 Truth Tables for DeMorgan’s

25 Algebraic Manipulation  Consider function

26 Simplify Function Apply

27 Fewer Gates

28 Consensus Theorem  The third term is redundant Can just drop Can just drop  Proof in book, but in summary: For third term to be true, Y & Z both must be 1 For third term to be true, Y & Z both must be 1 Then one of the first two terms must be 1! Then one of the first two terms must be 1!

29 Complement of a Function  Definition: 1s & 0s swapped in truth table  Mechanical way to derive algebraic form Take the dual Take the dual  Recall: Interchange AND and OR, and 1s & 0s Complement each literal Complement each literal

30 Mechanically Go From Truth Table to Function

31 From Truth Table to Func  Consider a truth table  Can implement F by taking OR of all terms that are 1

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

33 Definition: Minterm  Product term in which all variables appear once (complemented or not)

34 Number of Minterms  For n variables, there will be 2 n minterms  Like binary numbers from 0 to 2 n -1  In book, numbered same way (with decimal conversion)

35Maxterms  Sum term in which all variables appear once (complemented or not)

36 Minterm related to Maxterm  Minterm and maxterm with same subscripts are complements  Example

37 Sum of Minterms  Like the introductory slide  OR all of the minterms of truth table row with a 1

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

39 Product of Maxterms  Recall that maxterm is true except for its own case  So M1 is only false for 001

40 Product of Maxterms  Can express F as AND of all rows that should evaluate to 0 or

41Recap  Working (so far) with AND, OR, and NOT  Algebraic identities  Algebraic simplification  Minterms and maxterms  Can now synthesize function (and gates) from truth table