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DIGITAL LOGIC DESIGN

DIGITAL LOGIC DESIGN. by Dr. Fenghui Yao Tennessee State University Department of Computer Science Nashville, TN. What is minimization?. Simplifying boolean expressions Algebraic manipulations is hard since there is not a uniform way of doing it

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DIGITAL LOGIC DESIGN

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  1. DIGITAL LOGIC DESIGN by Dr. Fenghui Yao Tennessee State University Department of Computer Science Nashville, TN Gate-Level Minimization

  2. What is minimization? • Simplifying boolean expressions • Algebraic manipulations is hard since there is not a uniform way of doing it • Karnaugh map or K-map techniques is very commonly used Gate-Level Minimization

  3. Two-Variable K-Map Gate-Level Minimization

  4. Example Gate-Level Minimization

  5. Example Example Gate-Level Minimization

  6. Three-Variable K-Map Gate-Level Minimization

  7. Three-Variable K-Map Gate-Level Minimization

  8. Example Gate-Level Minimization

  9. Note • In K-maps, you can have groups of 2, 4, 8, or 16 • You cannot have groups of other combinations such as a group of 6 Gate-Level Minimization

  10. Exercises Gate-Level Minimization

  11. Example • Represent F in the minimal format and draw the network diagram Gate-Level Minimization

  12. Example • Represent F in the minimal format and draw the network diagram Gate-Level Minimization

  13. Example • Represent F in the minimal format and draw the network diagram Gate-Level Minimization

  14. Four-Variable K-Map Gate-Level Minimization

  15. Four-Variable K-Map Gate-Level Minimization

  16. Example • Represent F in the minimal format and draw the network diagram Gate-Level Minimization

  17. Example • Represent F in the minimal format and draw the network diagram Gate-Level Minimization

  18. Example • Represent F in the minimal format and draw the network diagram Gate-Level Minimization

  19. Prime Implicants • You must cover all of the minterms • You must avoid redundancy • You must follow some rules • Prime Implicant • A product term that is generated by combining the maximum number of adjacent squares in the map • Essential Prime Implicant • A minterm that is covered by only one prime implicant Gate-Level Minimization

  20. Maxterm Simplification • Remember Gate-Level Minimization

  21. Example • Simplify F in product of sums Gate-Level Minimization

  22. Example (cont) • Step – 1 • Fill the K-map for F Gate-Level Minimization

  23. Example (cont) • Step – 1 • Fill the K-map for F Gate-Level Minimization

  24. Example (cont) • Step – 2 • Fill zeros in the rest of the squares Gate-Level Minimization

  25. Example (cont) • Step – 3 • Cover zeros. This is your F’ Gate-Level Minimization

  26. Important Gate-Level Minimization

  27. Don’t Care Conditions • A network is usually composed of sub-networks • Net-1 may not produce all combinations of A,B, and C • In this case, F don’t care about those combinations A B C Net-1 Net-2 F Gate-Level Minimization

  28. Don’t Care Conditions X can be considered as 0 or 1, whichever is more convenient 0 0 0 1 0 0 1 x 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 0 1 1 0 x 1 1 1 1 Gate-Level Minimization

  29. NAND/NOR Implementations • AND, OR, and NOT gates can be used to construct the digital systems • However, it is easier to fabricate NAND and NOR gates • So try to replace AND, OR, and NOT gates with NAND or NOR gates Gate-Level Minimization

  30. NAND Implementation • First implement with AND-OR • Put bubble at the output of each AND gate • Put bubbles at the inputs of each OR gate • Place necessary inverters Gate-Level Minimization

  31. Example Gate-Level Minimization

  32. Example Gate-Level Minimization

  33. Example Gate-Level Minimization

  34. NOR Implementation • First implement with AND-OR • Put bubble at the inputs of each AND gate • Put bubbles at the output of each OR gate • Place necessary inverters Gate-Level Minimization

  35. Example Gate-Level Minimization

  36. Study Problems • Course Book Chapter – 3 Problems • 3– 1 • 3 – 3 • 3 – 5 • 3 – 7 • 3 – 12 • 3 – 15 • 3 – 18 Gate-Level Minimization

  37. Questions Gate-Level Minimization

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