1. SYEN Digital Systems
Chapter 2 – Part 6
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2. Table Methods for PI Generation
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3. An Example: F(x,y,z)= m(2,3,6,7)
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4. Results - Step 2
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5. Step 3
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6. The Results of Step 3
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7. Computational Complexity Issues
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8. Q-M on F(x,y,z)= m(2,3,6,7)
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9. The Result of Step 3
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10. Result of Step 4
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11. Review of Boolean Logic
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12. Canonical Forms
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13. Minimum Literal SOP Form
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14. Tabular Method to Find a Cover
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15. Table Method Example
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16. Select Essential Prime Implicants
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17. Select Essential Prime Implicants
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18. Less Than Prime Implicants
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19. Secondary Essential PIs
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20. Cyclic Structures
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21. Cyclic Structure: Pick One
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22. Less Thans
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23. Secondary Essential PIs
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24. Now Go Back and Try Again
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25. Finish Up
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26. Quine-McCluskey (tabular) method
1. Arrange all minterms in group such that all terms in the same group have the same # of 1’s in their binary representation.
2. Compare every term of the lowest-index group with each term in the successive group. Whenever possible, combine two terms being compared by means of
gxi+gxi´=g(xi+xi´)=g.
Two terms from adjacent groups are combinable if their binary representation differ by just a single digit in the same position.
3. The process continues until no further combinations are possible. The remaining unchecked terms constitute the set of PI.
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27. Using prime implicant chart, we can find essential PI
Ex) f(x1,x2,x3,x4) = (0,1,2,5,6,7,8,9,10,13,15) 15
(5,7) (5,13) (6,7) (9,13)
7 13
(1,5) (1,9) (2,6) (2,10) (8,9) (8,10)
(13,15)
(7,15) 
(0,1,8,9) (0,2,8,10) (1,5,9,13) (5,7,13,15)
(0,1)
(0,2)
(0,8)
x1,x2,x3,x4 #
Using prime implicant chart, we can find essential PI
(2,6) (6,7) (0,1,8,9) (0,2,8,10) (1,5,9,13) (5,7,13,15)                    
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28. If we choose p1 first, then p3, p5 are next.
The essential PI’s are (0,2,8,10) and (5,7,13,15) . So, f(x1,x2,x3,x4) = (0,2,7,8) + (5,7,13,15) + PI’s
Here are 4 different choices (2,6) + (0,1,8,9), (2,6) + (1,5,9,13) (6,7) + (0,1,8,9), or (6,7) + (1,5,9,13)
The reduced PI chart
A PI pj dominates PI pk iff every minterm covered by pk is also covered by pj.
pj pk
m1 m2 m3 m4 
(can remove)
Branching method
p p p p p5
m1 m2 m3 m4 m5
If we choose p1 first, then p3, p5 are next. p1 p4 p3 p5 p2
Quine – McCluskey method (no limitation of the # of variables)
(2,6)
(6,7)
(0,1,8,9)
(1,5,9,13)
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29. Quine-McCluskey example
F(A,B,C,D) =  (3,9,11,12,13,14,15) d (1,4,6)
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30. Ex) f(A,B,C,D) = (3,9,11,12,13,14,15) + d (1,4,6) PI chart:
(1,3, 9, 11) (4, 6,12,14) (9,13,11,15)
(12,13,14,15)             
Reduced PI chart:
(4, 6,12,14) (9,13,11,15)
(12,13,14,15)        
Result: (1,3,9,11) + (12,13,14,15)
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31. Minimum SOP to Minimum POS
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32. Minimum POS Example Given g(w,x,y,z):
Form the Complement (Circle Zeros):
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33. Table Method Minimum SOP
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