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

2. Standard Forms
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3. Standard Sum-of-Products (SOP)
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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
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6. Standard Product-of-Sums (POS)
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7. Standard Product-of-Sums (POS)
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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.
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9. OR/AND Two-level Implementation
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10. SOP and POS Observations
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11. Equivalent Cost Circuits
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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.
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13. Karnaugh Maps (K-map)
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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
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15. Two Variable Maps A Two variable Karnaugh Map:
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16. K-Map and Function Tables
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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 x y
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18. Three Variable Maps
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19. Example Functions
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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:
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21. Combining Squares Example
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22. Alternate K-Map Diagramx y z m0 m1 m3 m2 m4 m5 m7 m6 yz 1 00 01 11 10
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