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SYEN 3330 Digital Systems

SYEN 3330 Digital Systems. Chapter 2 – Part 4. Standard Forms. Standard Sum-of-Products (SOP). Standard Sum-of-Products (SOP). The Canonical Sum-of-Minterms form has (5 * 3) = 15 literals and 5 terms. The reduced SOP form has 3 literals and 2 terms.

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SYEN 3330 Digital Systems

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  1. SYEN 3330 Digital Systems Chapter 2 – Part 4 SYEN 3330 Digital Systems

  2. Standard Forms SYEN 3330 Digital Systems

  3. Standard Sum-of-Products (SOP) SYEN 3330 Digital Systems

  4. Standard Sum-of-Products (SOP) The Canonical Sum-of-Minterms form has (5 * 3) = 15 literals and 5 terms. The reduced SOP form has 3 literals and 2 terms. SYEN 3330 Digital Systems

  5. AND/OR Two-level Implementation of SOP Expression SYEN 3330 Digital Systems

  6. Standard Product-of-Sums (POS) SYEN 3330 Digital Systems

  7. Standard Product-of-Sums (POS) SYEN 3330 Digital Systems

  8. Standard Product-of-Sums (POS) The Canonical Product-of-Maxterms form had (3 * 3) = 9 literals and 3 terms. The reduced POS form had 4 literals and 2 terms. SYEN 3330 Digital Systems

  9. OR/AND Two-level Implementation SYEN 3330 Digital Systems

  10. SOP and POS Observations SYEN 3330 Digital Systems

  11. Equivalent Cost Circuits SYEN 3330 Digital Systems

  12. Boolean Function Simplification • Reducing the literal cost of a Boolean Expression leads to simpler networks. • Simpler networks are less expensive to implement. • Boolean Algebra can help us minimize literal cost. • When do we stop trying to reduce the cost? • Do we know when we have a minimum? • We will introduce a systematic way to arrive a a minimum cost, two-level POS or SOP network. SYEN 3330 Digital Systems

  13. Karnaugh Maps (K-map) SYEN 3330 Digital Systems

  14. Uses of Karnaugh Maps • Provide a means for finding optimum: • Simple SOP and POS standard forms, and • Small two-level AND/OR and OR/AND circuits • Visualize concepts related to manipulating Boolean expressions • Demonstrate concepts used by computer-aided design programs to simplify large circuits SYEN 3330 Digital Systems

  15. Two Variable Maps A Two variable Karnaugh Map: SYEN 3330 Digital Systems

  16. K-Map and Function Tables Function Table K-Map SYEN 3330 Digital Systems

  17. K-Map Function Representations • For function F(x,y), the two adjacent cells containing 1’s can be combined using the Minimization Theorem: • For G(x,y), two pairs of adjacent cells containing 1’s can be combined using the Minimization Theorem: Duplicate xy SYEN 3330 Digital Systems

  18. Three Variable Maps SYEN 3330 Digital Systems

  19. Example Functions SYEN 3330 Digital Systems

  20. Combining Squares • By combining squares, we reduce the representation for a term, reducing the number of literals in the Boolean equation. • On a three-variable K-Map: SYEN 3330 Digital Systems

  21. Combining Squares Example SYEN 3330 Digital Systems

  22. yz y x 00 01 11 10 m0 m1 m3 m2 0 m0 m1 m3 m2 x m4 m5 m7 m6 1 m4 m5 m7 m6 z y yz x 00 01 11 10 0 m0 m1 m3 m2 1 x m4 m5 m7 m6 z Alternate K-Map Diagram SYEN 3330 Digital Systems

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