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ECE/CS 352 Digital Systems Fundamentals

ECE/CS 352 Digital Systems Fundamentals. Spring 2001 Chapter 2 Part 3. Tom Kaminski & Charles R. Kime. Boolean Operator Precedence. Review: Duality Principle. Duality In Proofs. Useful Theorems. Proof of Simplification. Proof of Concensus. Proof of DeMorgan’s Law.

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ECE/CS 352 Digital Systems Fundamentals

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  1. ECE/CS 352 Digital Systems Fundamentals Spring 2001 Chapter 2 Part 3 Tom Kaminski& Charles R. Kime ECE/CS 352 Digital System Fundamentals

  2. Boolean Operator Precedence ECE/CS 352 Digital System Fundamentals

  3. Review: Duality Principle ECE/CS 352 Digital System Fundamentals

  4. Duality In Proofs ECE/CS 352 Digital System Fundamentals

  5. Useful Theorems ECE/CS 352 Digital System Fundamentals

  6. Proof of Simplification ECE/CS 352 Digital System Fundamentals

  7. Proof of Concensus ECE/CS 352 Digital System Fundamentals

  8. Proof of DeMorgan’s Law ECE/CS 352 Digital System Fundamentals

  9. Boolean Function Evaluation ECE/CS 352 Digital System Fundamentals

  10. Expression Simplification • Simplify to contain the smallest number of literals (complemented and uncomplemented variables): ECE/CS 352 Digital System Fundamentals

  11. Complementing Functions • This generate a lot of terms. You might want to simplify the expression first. ECE/CS 352 Digital System Fundamentals

  12. Canonical Forms • It is useful to specify Boolean functions of n variables in a manner that is easy to compare. • Two such Canonical Forms are in common usage: • Sum of Minterms • Product of Maxterms ECE/CS 352 Digital System Fundamentals

  13. Minterms ECE/CS 352 Digital System Fundamentals

  14. Maxterms ECE/CS 352 Digital System Fundamentals

  15. Maxterms and Minterms The index above is important for describing which variables in the terms are true and which are complemented. ECE/CS 352 Digital System Fundamentals

  16. Standard Order ECE/CS 352 Digital System Fundamentals

  17. Purpose of the Index • The index for the minterm or maxterm, expressed as a binary number, is used to determine whether the variable is shown in the true form or complemented form. ECE/CS 352 Digital System Fundamentals

  18. Index Example in Three Variables ECE/CS 352 Digital System Fundamentals

  19. Four Variables, Index 0-7 ECE/CS 352 Digital System Fundamentals

  20. Four Variables, Index 8-15 ECE/CS 352 Digital System Fundamentals

  21. Minterm and Maxterm Relationship ECE/CS 352 Digital System Fundamentals

  22. Function Tables for Both Minterms of two variables Maxterms of two variables ECE/CS 352 Digital System Fundamentals

  23. Observations ECE/CS 352 Digital System Fundamentals

  24. Minterm Function Example ECE/CS 352 Digital System Fundamentals

  25. Minterm Function Example • F(A, B, C, D, E) = m2 + m9 + m17 + m23 ECE/CS 352 Digital System Fundamentals

  26. Maxterm Function Example ECE/CS 352 Digital System Fundamentals

  27. Maxterm Function Example ECE/CS 352 Digital System Fundamentals

  28. Cannonical Sum of Minterms ECE/CS 352 Digital System Fundamentals

  29. Another SOM Example ECE/CS 352 Digital System Fundamentals

  30. Shorthand SOM Form Note that we explicitly show the standard variables in order and drop the “m” designators. ECE/CS 352 Digital System Fundamentals

  31. Canonical Product of Maxterms ECE/CS 352 Digital System Fundamentals

  32. Product of Maxterm Example ECE/CS 352 Digital System Fundamentals

  33. Function Complements Then: Or alternately: ECE/CS 352 Digital System Fundamentals

  34. Conversion Between Forms ECE/CS 352 Digital System Fundamentals

  35. Review of Canonical Forms ECE/CS 352 Digital System Fundamentals

  36. Review: Indices ECE/CS 352 Digital System Fundamentals

  37. Forms of Terms, Complements ECE/CS 352 Digital System Fundamentals

  38. Review: Sum of Minterms Form ECE/CS 352 Digital System Fundamentals

  39. Review: Product of Maxterms ECE/CS 352 Digital System Fundamentals

  40. Review: Complements, Conversions ECE/CS 352 Digital System Fundamentals

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