COMP541 Combinational Logic - II

1. COMP541Combinational Logic - II Montek Singh Jan 19, 2010

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

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

4. Table of Identities

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

6. Single Variable Identities

7. Commutativity • Operation is independent of order of variables

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

9. Distributivity • Can substitute arbitrarily large algebraic expressions for the variables • 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 Apply Apply

14. Fewer Gates

15. Consensus Theorem • The third term is redundant • Can just drop • Proof summary: • For third term to be true, Y & Z both must be 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 • Recall: Interchange AND and OR, and 1s & 0s • 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 • Sum terms – OR  X + Ā • 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 2n minterms • Like binary numbers from 0 to 2n-1 • Often numbered same way (with decimal conversion)

22. Maxterms • 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”

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 • 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 • More gate delays (more on this later) • 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 • Recall: Interchange AND and OR, and 1s & 0s • Complement each literal

29. Complement of F • Not surprisingly, just sum of the other minterms • “OFF-set minterms” • In this case m1 + m3 + m4 + m6

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

33. 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

34. Next Time • Lab Prep • Demo lab software • Talk about FPGA internals • Overview of components on board • Downloading and testing • More on combinational logic