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Presentation transcript:

1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology

H.-H. S. Lee 2 Quine-McCluskey Method A systematic solution to K-Map when more complex function with more literals is given B n n In principle, can be applied to an arbitrary large number of inputs, i.e. works for B n where n can be arbitrarily large One can translate Quine-McCluskey method into a computer program to perform minimization

H.-H. S. Lee 3 Quine-McCluskey Method Two basic steps Finding all prime implicants of a given Boolean function Select a minimal set of prime implicants that cover this function

H.-H. S. Lee 4 Q-M Method (I) Transform the given Boolean function into a canonical SOP function Convert each Minterm into binary format Arrange each binary minterm in groups All the minterms in one group contain the same number of “1”

H.-H. S. Lee 5 Q-M Method: Grouping minterms (29) (30) (2) (4) (8) (16) A B C D E (0) (6) (10) (12) (18) (7) (11) (13) (14) (19)

H.-H. S. Lee 6 Q-M Method (II) Combine terms with Hamming distance=1 from adjacent groups  Check (  ) the terms being combined The checked terms are “covered” by the combined new term Keep doing this till no combination is possible between adjacent groups

H.-H. S. Lee 7 Q-M Method: Grouping minterms (29) (30) (2) (4) (8) (16) A B C D E (0) (6) (10) (12) (18) (7) (11) (13) (14) (19) A B C D E   (0,2) – 0  (0,4)  (0,8)  (0,16)  (2,6)  (2,10)  (2,18)  (4,6) (4,12) (8,10) (8,12) (16,18) (6,7)  (6,14)  (10,11)  (10,14) (12,13)  (12,14) (18,19)  A B C D E (13,29)  (14,30) 

H.-H. S. Lee 8 Q-M Method: Grouping minterms A B C D E   (0,2) – 0  (0,4) (0,8)  (0,16)  (2,6)  (2,10)  (2,18) (4,6) (4,12) (8,10) (8,12) (16,18) (6,7) (6,14) (10,11) (10,14) (12,13) (12,14) (18,19) (13,29) (14,30) A B C D E (0,2,4,6) – 0  (0,2,8,10) – 0  (0,2,16,18) – 0  (0,4,8,12)   (2,6,10,14)  (4,6,12,14)  (8,10,12,14)  A B C D E (0,2,4, ,10,12,14)      

H.-H. S. Lee 9 Prime Implicants (6,7) (10,11) (12,13) (18,19) (13,29) (14,30) (0,2,16,18) – 0 (0,2,4, ,10,12,14) A B C D E Unchecked terms are prime implicants

H.-H. S. Lee 10 Prime Implicants (6,7) (10,11) (12,13) (18,19) (13,29) (14,30) (0,2,16,18) – 0 (0,2,4, ,10,12,14) A B C D E Unchecked terms are prime implicants

H.-H. S. Lee 11 Q-M Method (III) Form a Prime Implicant Table X-axis: the minterm Y-axis: prime implicants  An  is placed at the intersection of a row and column if the corresponding prime implicant includes the corresponding product (term)

H.-H. S. Lee 12 Q-M Method: Prime Implicant Table (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX

H.-H. S. Lee 13 Q-M Method (IV) Locate the essential row from the table These are essential prime implicants The row consists of minterms covered by a single “  ” Mark all minterms covered by the essential prime implicants Find non-essential prime implicants to cover the rest of minterms Form the SOP function with the prime implicants selected, which is the minimal representation

H.-H. S. Lee 14 Q-M Method (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX

H.-H. S. Lee 15 Q-M Method (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14)

H.-H. S. Lee 16 Q-M Method (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14)

H.-H. S. Lee 17 Q-M Method (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7)

H.-H. S. Lee 18 Q-M Method (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7)

H.-H. S. Lee 19 Q-M Method (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7), (10,11)

H.-H. S. Lee 20 Q-M Method (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7), (10,11)

H.-H. S. Lee 21 Q-M Method (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18)

H.-H. S. Lee 22 Q-M Method (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18)

H.-H. S. Lee 23 Q-M Method (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19)

H.-H. S. Lee 24 Q-M Method (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19)

H.-H. S. Lee 25 Q-M Method (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19), (13,29)

H.-H. S. Lee 26 Q-M Method (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19), (13,29)

H.-H. S. Lee 27 Q-M Method (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19), (13,29), (14,30) Now all the minterms are covered by selected prime implicants !

H.-H. S. Lee 28 Q-M Method (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19), (13,29), (14,30) Now all the minterms are covered by selected prime implicants ! Note that (12,13), a non-essential prime implicant, is not needed

H.-H. S. Lee 29 Q-M Method Result (6,7)XX (10,11)XX (12,13)XX (18,19)XX (13,29)XX (14,30)XX (0,2,16,18)XXXX (0,2,4,6,8,10,12,14)XXXXXXXX

H.-H. S. Lee 30 Q-M Method Example 2 Sometimes, simplification by K-map method could be less than optimal due to human error Quine-McCluskey method can guarantee an optimal answer XX X AB CD

H.-H. S. Lee 31 Grouping minterms (1) (4) (8) A B C D (0) (5) (6) (9) (10) (12) (7) (14) A B C D (0,1) (0,4) (0,8)     (1,5) (1,9) (4,5) – (4,6) 0 1 – 0 (4,12) – (8,9) – (8,10) 1 0 – 0 (8,12) 1 – 0 0      (5,7) (6,7) (6,14) (10,14) 1 – 1 0 (12,14)   A B C D (4,5,6,7) (4,6,12,14) (8,10,12,14)        (0,1,4,5) (0,1,8,9) – (0,4,8,12)         

H.-H. S. Lee 32 Prime Implicants A B C D (4,5,6,7) (4,6,12,14) (8,10,12,14) (0,1,4,5) (0,1,8,9) – (0,4,8,12) (0,1,4,5)XXXX (0,1,8,9)XXXX (0,4,8,12)XXXX (4,5,6,7)XXXX (4,6,12,14)XXXX (8,10,12,14)XXXX Don’t Care

H.-H. S. Lee 33 Prime Implicants A B C D (4,5,6,7) (4,6,12,14) (8,10,12,14) (0,1,4,5) (0,1,8,9) – (0,4,8,12) (0,1,4,5)XXXX (0,1,8,9)XXXX (0,4,8,12)XXXX (4,5,6,7)XXXX (4,6,12,14)XXXX (8,10,12,14)XXXX Don’t Care

H.-H. S. Lee 34 Prime Implicants A B C D (4,5,6,7) (4,6,12,14) (8,10,12,14) (0,1,4,5) (0,1,8,9) – (0,4,8,12) (0,1,4,5)XXXX (0,1,8,9)XXXX (0,4,8,12)XXXX (4,5,6,7)XXXX (4,6,12,14)XXXX (8,10,12,14)XXXX Don’t Care

H.-H. S. Lee 35 Prime Implicants A B C D (4,5,6,7) (4,6,12,14) (8,10,12,14) (0,1,4,5) (0,1,8,9) – (0,4,8,12) (0,1,4,5)XXXX (0,1,8,9)XXXX (0,4,8,12)XXXX (4,5,6,7)XXXX (4,6,12,14)XXXX (8,10,12,14)XXXX Don’t Care

H.-H. S. Lee 36 Prime Implicants A B C D (4,5,6,7) (4,6,12,14) (8,10,12,14) (0,1,4,5) (0,1,8,9) – (0,4,8,12) (0,1,4,5)XXXX (0,1,8,9)XXXX (0,4,8,12)XXXX (4,5,6,7)XXXX (4,6,12,14)XXXX (8,10,12,14)XXXX Don’t Care

H.-H. S. Lee 37 Prime Implicants A B C D (4,5,6,7) (4,6,12,14) (8,10,12,14) (0,1,4,5) (0,1,8,9) – (0,4,8,12) (0,1,4,5)XXXX (0,1,8,9)XXXX (0,4,8,12)XXXX (4,5,6,7)XXXX (4,6,12,14)XXXX (8,10,12,14)XXXX Don’t Care Essential PI Non-Essential PI

H.-H. S. Lee 38 Q-M Method Solution (0,1,4,5)XXXX (0,1,8,9)XXXX (0,4,8,12)XXXX (4,5,6,7)XXXX (4,6,12,14)XXXX (8,10,12,14)XXXX A B C D (4,5,6,7) (4,6,12,14) (8,10,12,14) (0,1,4,5) (0,1,8,9) – (0,4,8,12) Don’t Care

H.-H. S. Lee 39 Yet Another Q-M Method Solution (0,1,4,5)XXXX (0,1,8,9)XXXX (0,4,8,12)XXXX (4,5,6,7)XXXX (4,6,12,14)XXXX (8,10,12,14)XXXX A B C D (4,5,6,7) (4,6,12,14) (8,10,12,14) (0,1,4,5) (0,1,8,9) – (0,4,8,12) Don’t Care

H.-H. S. Lee 40 To Get the Same Answer w/ K-Map XX X AB CD XX X AB CD