1. SYEN 3330 Digital SystemsJung H. Kim Chapter 2-8 1 SYEN 3330 Digital Systems Chapter 2 -Part 8

2. SYEN 3330 Digital Systems Chapter 2-8 Page 2 Exclusive OR/ Exclusive NOR

3. SYEN 3330 Digital Systems Chapter 2-8 Page 3 Tables for EXOR/ EXNOR

4. SYEN 3330 Digital Systems Chapter 2-8 Page 4 EXOR/EXNOR Extensions

5. SYEN 3330 Digital Systems Chapter 2-8 Page 5 EXOR Implementations

6. SYEN 3330 Digital Systems Chapter 2-8 Page 6 EXOR Implementations (Cont.)

7. SYEN 3330 Digital Systems Chapter 2-8 Page 7 Odd Function

8. SYEN 3330 Digital Systems Chapter 2-8 Page 8 Odd Function Implementation

9. SYEN 3330 Digital Systems Chapter 2-8 Page 9 K-Maps of ODD and EVEN

10. SYEN 3330 Digital Systems Chapter 2-8 Page 10 Parity Generators/Checkers

11. SYEN 3330 Digital Systems Chapter 2-8 Page 11 Integrated Circuits

12. SYEN 3330 Digital Systems Chapter 2-8 Page 12 Digital Logic Families

13. SYEN 3330 Digital Systems Chapter 2-8 Page 13 Compatibility

14. SYEN 3330 Digital Systems Chapter 2-8 Page 14 Propagation Delay

15. SYEN 3330 Digital Systems Chapter 2-8 Page 15 Propagation Delay Example

16. SYEN 3330 Digital Systems Chapter 2-8 Page 16 Positive and Negative Logic

17. SYEN 3330 Digital Systems Chapter 2-8 Page 17 Positive and Negative Logic

18. SYEN 3330 Digital Systems Chapter 2-8 Page 18 Positive and Negative Logic (Cont.)

19. SYEN 3330 Digital Systems Chapter 2-8 Page 19 Logic Conventions

20. SYEN 3330 Digital Systems Chapter 2-8 Page 20 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  (from all 1-cube). 3. The process continues until no further combinations are possible. The remaining unchecked terms constitute the set of PI.

21. SYEN 3330 Digital Systems Chapter 2-8 Page 21 Ex) f(x 1,x 2,x 3,x 4 ) =  (0,1,2,5,6,7,8,9,10,13,15) Using prime implicant chart, we can find essential PI 1 1 15 (5,7) (5,13) (6,7) (9,13) 0 1 - 1 - 1 0 1 0 1 1 - 1 - 0 1 0 1 1 1 1 1 0 1 7 13 (1,5) (1,9) (2,6) (2,10) (8,9) (8,10) 0 1 0 1 0 1 1 0 1 0 0 1 1 0 5 6 9 10 0 - 0 1 - 0 0 1 0 - 1 0 - 0 1 0 1 0 0 - 1 0 - 0 - 1 1 1 1 1 - 1 (13,15) (7,15)                 128128 (0,1,8,9) (0,2,8,10) (1,5,9,13) (5,7,13,15) - 0 0 - - 0 - 0 - - 0 1 - 1 - 1 (0,1) (0,2) (0,8) 0 0 0 - 0 0 - 0 - 0 0 0 0 0 0 x 1,x 2,x 3,x 4 # 0 0 0 1 1 0 0 0 0 0 1 0         (2,6) (6,7) (0,1,8,9) (0,2,8,10) (1,5,9,13) (5,7,13,15) 0 1 2 5 6 7 8 9 10 13 15                

22. SYEN 3330 Digital Systems Chapter 2-8 Page 22 The essential PI’s are (0,2,8,10) and (5,7,13,15). So, f(x 1,x 2,x 3,x 4 ) = (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 p j dominates PI p k iff every minterm covered by p k is also covered by p j. pjpkpjpk m 1 m 2 m 3 m 4      (can remove) Branching method p1p2p3p4p5p1p2p3p4p5 m 1 m 2 m 3 m 4 m 5           If we choose p 1 first, then p 3, p 5 are next. p1p1 p4p4 p3p3 p5p5 p3p3 p2p2 Quine – McCluskey method (no limitation of the # of variables) (2,6) (6,7) (0,1,8,9) (1,5,9,13) 1 6 9    