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COMP541 Combinational Logic

2. Today. Basics of digital logic (review)Basic functionsBoolean algebraGates to implement Boolean functionsIdentities and Simplification (review?). 3. Binary Logic. Binary variablesCan be 0 or 1 (T or F, low or high)Variables named with single letters in examplesReally use words when designing circuitsBasic FunctionsANDORNOT.

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COMP541 Combinational Logic

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

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

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

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

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

    6. 6 NOT Unary Symbol is bar C = A Truth table -> Inversion

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

    8. 8 AND Gate

    9. 9 OR Gate

    10. 10 Inverter

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

    12. 12 Representation: Schematic Schematic = circuit diagram

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

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

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

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

    17. 17 Table of Identities

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

    19. 19 Single Variable Identities

    20. 20 Commutativity Operation is independent of order of variables

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

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

    23. 23 DeMorgan’s Theorem Used a lot NOR ? invert, then AND NAND ? invert, then OR

    24. 24 Truth Tables for DeMorgan’s

    25. 25 Algebraic Manipulation Consider function

    26. 26 Simplify Function

    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 must be 1 Then one of the first two terms must be 1!

    29. 29 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

    30. 30 Mechanically Go From Truth Table to Function

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

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

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

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

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

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

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

    38. 38 Complement of F Not surprisingly, just sum of the other minterms In this case m1 + m3 + m4 + m6

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

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

    41. 41 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

    42. 42 Next Time Lab Prep Demo lab software Talk about FPGA internals Overview of components on board Downloading and testing Karnaugh maps: mechanical synthesis approach (quick)

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