1 COMP541 Combinational Logic - II Montek Singh Jan 18, 2012

2Today  Basics of Boolean Algebra (review) Identities and Simplification Identities and Simplification  Basics of Logic Implementation Minterms and maxterms Minterms and maxterms Going from truth table to logic implementation Going from truth table to logic implementation

3Identities  Use identities to manipulate functions  You can use distributive law … … to transform from … to transform from to

4 Table of Identities

5Duals  Left and right columns are duals  Replace AND and OR, 0s and 1s

6 Single Variable Identities

7Commutativity  Operation is independent of order of variables

8Associativity  Independent of order in which we group  So can also be written as and

9Distributivity  Can substitute arbitrarily large algebraic expressions for the variables Distribute an operation over the entire expression Distribute an operation over the entire expression

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

11 Truth Tables for DeMorgan’s

12 Algebraic Manipulation  Consider function

13 Simplify Function Apply

14 Fewer Gates

15 Consensus Theorem  The third term is redundant Can just drop Can just drop  Proof 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 is already 1! Then one of the first two terms is already 1!

16 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

17 Mechanically Go From Truth Table to Function

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

19 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

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

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

22Maxterms  Sum term in which all variables appear once (complemented or not)

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

24 Sum of Minterms  Like Slide 18  OR all of the minterms of truth table row with a 1 “ON-set minterms” “ON-set minterms”

25 Sum of Products  Simplifying sum-of-minterms can yield a sum of products  Difference is each term need not be a minterm i.e., terms do not need to have all variables i.e., terms do not need to have all variables  A bunch of ANDs and one OR

26 Two-Level Implementation  Sum of products has 2 levels of gates

27 More Levels of Gates?  What’s best? Hard to answer Hard to answer More gate delays (more on this later) More gate delays (more on this later) But maybe we only have 2-input gates But maybe we only have 2-input gates  So multi-input ANDs and ORs have to be decomposed

28 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

29 Complement of F  Not surprisingly, just sum of the other minterms “OFF-set minterms” “OFF-set minterms”  In this case m 1 + m 3 + m 4 + m 6

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

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

32 Product of Sums  Result: another standard form  ORs followed by AND Terms do not have to be maxterms Terms do not have to be maxterms

33Recap  Working (so far) with AND, OR, and NOT  Algebraic identities  Algebraic simplification  Minterms and maxterms  Can now synthesize function (and gates) from truth table

34 Next Time  Lab on Fri, 1/20: Demo lab software Demo lab software Do 1 st lab assignment Do 1 st lab assignment  download software: see website for link  couple of simple Verilog programming problems  Next Week: More on combinational logic