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COMP541 Combinational Logic - II

COMP541 Combinational Logic - II. Montek Singh Jan 19, 2010. Today. Basics of Boolean Algebra (review) Identities and Simplification Basics of Logic Implementation Minterms and maxterms Going from truth table to logic implementation. Identities. Use identities to manipulate functions

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COMP541 Combinational Logic - II

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  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

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