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![Page 1: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/1.jpg)
1
ECE2030 ECE2030 Introduction to Computer Introduction to Computer EngineeringEngineering
Lecture 8: Quine-McCluskey MethodLecture 8: Quine-McCluskey Method
Prof. Hsien-Hsin Sean LeeProf. Hsien-Hsin Sean Lee
School of ECESchool of ECE
Georgia Institute of Georgia Institute of TechnologyTechnology
![Page 2: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/2.jpg)
H.-H. S. LeeH.-H. S. Lee2
Quine-McCluskey Method A systematic solution to K-Map when
more complex function with more literals is given
In principle, can be applied to an arbitrary large number of inputs, i.e. works for BBn n
where nn can be arbitrarily large One can translate Quine-McCluskey
method into a computer program to perform minimization
![Page 3: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/3.jpg)
H.-H. S. LeeH.-H. S. Lee3
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
![Page 4: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/4.jpg)
H.-H. S. LeeH.-H. S. Lee4
Q-M Method (I)
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
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”
![Page 5: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/5.jpg)
H.-H. S. LeeH.-H. S. Lee5
Q-M Method: Grouping minterms
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
(29) 1 1 1 0 1(30) 1 1 1 1 0
(2) 0 0 0 1 0(4) 0 0 1 0 0(8) 0 1 0 0 0(16) 1 0 0 0 0
A B C D E(0) 0 0 0 0 0
(6) 0 0 1 1 0(10) 0 1 0 1 0(12) 0 1 1 0 0(18) 1 0 0 1 0(7) 0 0 1 1 1(11) 0 1 0 1 1(13) 0 1 1 0 1(14) 0 1 1 1 0(19) 1 0 0 1 1
![Page 6: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/6.jpg)
H.-H. S. LeeH.-H. S. Lee6
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
![Page 7: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/7.jpg)
H.-H. S. LeeH.-H. S. Lee7
Q-M Method: Grouping minterms
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
(29) 1 1 1 0 1(30) 1 1 1 1 0
(2) 0 0 0 1 0(4) 0 0 1 0 0(8) 0 1 0 0 0(16) 1 0 0 0 0
A B C D E(0) 0 0 0 0 0
(6) 0 0 1 1 0(10) 0 1 0 1 0(12) 0 1 1 0 0(18) 1 0 0 1 0
(7) 0 0 1 1 1(11) 0 1 0 1 1(13) 0 1 1 0 1(14) 0 1 1 1 0(19) 1 0 0 1 1
A B C D E
(0,2) 0 0 0 – 0
(0,4) 0 0 - 0 0
(0,8) 0 - 0 0 0
(0,16) - 0 0 0 0
(2,6) 0 0 - 1 0
(2,10) 0 - 0 1 0
(2,18) - 0 0 1 0
(4,6) 0 0 1 - 0(4,12) 0 - 1 0 0(8,10) 0 1 0 - 0(8,12) 0 1 - 0 0(16,18) 1 0 0 - 0
(6,7) 0 0 1 1 -
(6,14) 0 - 1 1 0(10,11) 0 1 0 1 -
(10,14) 0 1 - 1 0(12,13) 0 1 1 0 -
(12,14) 0 1 1 - 0(18,19) 1 0 0 1 -
A B C D E(13,29) - 1 1 0 1
(14,30) - 1 1 1 0
![Page 8: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/8.jpg)
H.-H. S. LeeH.-H. S. Lee8
Q-M Method: Grouping minterms
A B C D E
(0,2) 0 0 0 – 0
(0,4) 0 0 - 0 0(0,8) 0 - 0 0 0
(0,16) - 0 0 0 0
(2,6) 0 0 - 1 0 (2,10) 0 - 0 1 0
(2,18) - 0 0 1 0(4,6) 0 0 1 - 0(4,12) 0 - 1 0 0(8,10) 0 1 0 - 0(8,12) 0 1 - 0 0(16,18) 1 0 0 - 0
(6,7) 0 0 1 1 -(6,14) 0 - 1 1 0(10,11) 0 1 0 1 -(10,14) 0 1 - 1 0(12,13) 0 1 1 0 -(12,14) 0 1 1 - 0(18,19) 1 0 0 1 -
(13,29) - 1 1 0 1(14,30) - 1 1 1 0
A B C D E(0,2,4,6) 0 0 - – 0
(0,2,8,10) 0 - 0 – 0
(0,2,16,18) - 0 0 – 0
(0,4,8,12) 0 - - 0 0
(2,6,10,14) 0 - - 1 0
(4,6,12,14) 0 - 1 - 0
(8,10,12,14) 0 1 - - 0
A B C D E(0,2,4,6 0 - - - 0 8,10,12,14)
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
![Page 9: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/9.jpg)
H.-H. S. LeeH.-H. S. Lee9
Prime Implicants 30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
(6,7) 0 0 1 1 -
(10,11) 0 1 0 1 -
(12,13) 0 1 1 0 -
(18,19) 1 0 0 1 -
(13,29) - 1 1 0 1
(14,30) - 1 1 1 0(0,2,16,18) - 0 0 – 0
(0,2,4,6 0 - - - 0 8,10,12,14)
A B C D E
• Unchecked terms are prime implicants
![Page 10: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/10.jpg)
H.-H. S. LeeH.-H. S. Lee10
Prime Implicants 30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
CDBADCBA
(6,7) 0 0 1 1 -
(10,11) 0 1 0 1 -
(12,13) 0 1 1 0 -
(18,19) 1 0 0 1 -
(13,29) - 1 1 0 1
(14,30) - 1 1 1 0(0,2,16,18) - 0 0 – 0
(0,2,4,6 0 - - - 0 8,10,12,14)
A B C D E
• Unchecked terms are prime implicants
DBCADCBAEDBCEBCD
ECBEA
![Page 11: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/11.jpg)
H.-H. S. LeeH.-H. S. Lee11
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)
![Page 12: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/12.jpg)
H.-H. S. LeeH.-H. S. Lee12
Q-M Method: Prime Implicant Table
0 2 4 6 7 8 10
11
12
13
14
16
18
19
29
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
(0,2,4,6,8,10,12,14)
XX XX XX XX XX XX XX XX
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
![Page 13: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/13.jpg)
H.-H. S. LeeH.-H. S. Lee13
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
![Page 14: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/14.jpg)
H.-H. S. LeeH.-H. S. Lee14
Q-M Method
0 2 4 6 7 8 10
11
12
13
14
16
18
19
29
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
(0,2,4,6,8,10,12,14)
XX XX XX XX XX XX XX XX
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
![Page 15: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/15.jpg)
H.-H. S. LeeH.-H. S. Lee15
Q-M Method
0 2 4 6 7 8 10
11
12
13
14
16
18
19
29
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
(0,2,4,6,8,10,12,14)
XX XX XX XX XX XX XX XX
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
• Select (0,2,4,6,8,10,12,14)
![Page 16: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/16.jpg)
H.-H. S. LeeH.-H. S. Lee16
Q-M Method
0 2 4 6 7 8 10
11
12
13
14
16
18
19
29
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
(0,2,4,6,8,10,12,14)
XX XX XX XX XX XX XX XX
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
• Select (0,2,4,6,8,10,12,14)
![Page 17: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/17.jpg)
H.-H. S. LeeH.-H. S. Lee17
Q-M Method
0 2 4 6 7 8 10
11
12
13
14
16
18
19
29
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
(0,2,4,6,8,10,12,14)
XX XX XX XX XX XX XX XX
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
• Select (0,2,4,6,8,10,12,14), (6,7)
![Page 18: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/18.jpg)
H.-H. S. LeeH.-H. S. Lee18
Q-M Method
0 2 4 6 7 8 10
11
12
13
14
16
18
19
29
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
(0,2,4,6,8,10,12,14)
XX XX XX XX XX XX XX XX
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
• Select (0,2,4,6,8,10,12,14), (6,7)
![Page 19: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/19.jpg)
H.-H. S. LeeH.-H. S. Lee19
Q-M Method
0 2 4 6 7 8 10
11
12
13
14
16
18
19
29
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
(0,2,4,6,8,10,12,14)
XX XX XX XX XX XX XX XX
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
• Select (0,2,4,6,8,10,12,14), (6,7), (10,11)
![Page 20: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/20.jpg)
H.-H. S. LeeH.-H. S. Lee20
Q-M Method
0 2 4 6 7 8 10
11
12
13
14
16
18
19
29
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
(0,2,4,6,8,10,12,14)
XX XX XX XX XX XX XX XX
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
• Select (0,2,4,6,8,10,12,14), (6,7), (10,11)
![Page 21: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/21.jpg)
H.-H. S. LeeH.-H. S. Lee21
Q-M Method
0 2 4 6 7 8 10
11
12
13
14
16
18
19
29
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
(0,2,4,6,8,10,12,14)
XX XX XX XX XX XX XX XX
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
• Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18)
![Page 22: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/22.jpg)
H.-H. S. LeeH.-H. S. Lee22
Q-M Method
0 2 4 6 7 8 10
11
12
13
14
16
18
19
29
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
(0,2,4,6,8,10,12,14)
XX XX XX XX XX XX XX XX
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
• Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18)
![Page 23: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/23.jpg)
H.-H. S. LeeH.-H. S. Lee23
Q-M Method
0 2 4 6 7 8 10
11
12
13
14
16
18
19
29
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
(0,2,4,6,8,10,12,14)
XX XX XX XX XX XX XX XX
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
• Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19)
![Page 24: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/24.jpg)
H.-H. S. LeeH.-H. S. Lee24
Q-M Method
0 2 4 6 7 8 10
11
12
13
14
16
18
19
29
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
(0,2,4,6,8,10,12,14)
XX XX XX XX XX XX XX XX
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
• Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19)
![Page 25: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/25.jpg)
H.-H. S. LeeH.-H. S. Lee25
Q-M Method
0 2 4 6 7 8 10
11
12
13
14
16
18
19
29
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
(0,2,4,6,8,10,12,14)
XX XX XX XX XX XX XX XX
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
• Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19), (13,29)
![Page 26: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/26.jpg)
H.-H. S. LeeH.-H. S. Lee26
Q-M Method
0 2 4 6 7 8 10
11
12
13
14
16
18
19
29
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
(0,2,4,6,8,10,12,14)
XX XX XX XX XX XX XX XX
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
• Select (0,2,4,6,8,10,12,14), (6,7), (10,11), (0,2,16,18), (18,19), (13,29)
![Page 27: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/27.jpg)
H.-H. S. LeeH.-H. S. Lee27
Q-M Method
0 2 4 6 7 8 10
11
12
13
14
16
18
19
29
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
(0,2,4,6,8,10,12,14)
XX XX XX XX XX XX XX XX
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
• 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 !
![Page 28: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/28.jpg)
H.-H. S. LeeH.-H. S. Lee28
Q-M Method
0 2 4 6 7 8 10
11
12
13
14
16
18
19
29
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
(0,2,4,6,8,10,12,14)
XX XX XX XX XX XX XX XX
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
• 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
![Page 29: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/29.jpg)
H.-H. S. LeeH.-H. S. Lee29
Q-M Method Result0 2 4 6 7 8 1
011
12
13
14
16
18
19
29
30
(6,7) XX XX
(10,11) XX XX
(12,13) XX XX
(18,19) XX XX
(13,29) XX XX
(14,30) XX XX
(0,2,16,18) XX XX XX XX
(0,2,4,6,8,10,12,14)
XX XX XX XX XX XX XX XX
EAECBEBCDEDBCDCBADCBACDBA
,10,12,14)(0,2,4,6,8
)(0,2,16,18(14,30)(13,29)(18,19)(10,11)(6,7)
30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0,E)D,C,B,F(A,
![Page 30: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/30.jpg)
H.-H. S. LeeH.-H. S. Lee30
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
d(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F
00 01 11 10
00 1 1 0 0
01 1 X X 1
11 1 0 0 X
10 1 1 0 1
ABCD
![Page 31: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/31.jpg)
H.-H. S. LeeH.-H. S. Lee31
Grouping minterms
(1) 0 0 0 1(4) 0 1 0 0(8) 1 0 0 0
A B C D (0) 0 0 0 0
(5) 0 1 0 1(6) 0 1 1 0(9) 1 0 0 1(10) 1 0 1 0(12) 1 1 0 0
(7) 0 1 1 1(14) 1 1 1 0
d(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F A B C D
(0,1) 0 0 0 -(0,4) 0 - 0 0(0,8) - 0 0 0
(1,5) 0 - 0 1(1,9) - 0 0 1(4,5) 0 1 0 –(4,6) 0 1 – 0(4,12) – 1 0 0 (8,9) 1 0 0 –(8,10) 1 0 – 0(8,12) 1 – 0 0
(5,7) 0 1 - 1(6,7) 0 1 1 -(6,14) - 1 1 0(10,14) 1 – 1 0(12,14) 1 1 - 0
A B C D
(4,5,6,7) 0 1 - -(4,6,12,14) - 1 - 0(8,10,12,14) 1 - - 0
(0,1,4,5) 0 - 0 -(0,1,8,9) - 0 0 –(0,4,8,12) - - 0 0
![Page 32: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/32.jpg)
H.-H. S. LeeH.-H. S. Lee32
Prime Implicantsd(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F
A B C D
(4,5,6,7) 0 1 - -(4,6,12,14) - 1 - 0(8,10,12,14) 1 - - 0
(0,1,4,5) 0 - 0 -(0,1,8,9) - 0 0 –(0,4,8,12) - - 0 0
0 1 4 6 8 9 10 12 5 7 14
(0,1,4,5) XX XX XX XX
(0,1,8,9) XX XX XX XX
(0,4,8,12) XX XX XX XX
(4,5,6,7) XX XX XX XX
(4,6,12,14) XX XX XX XX
(8,10,12,14) XX XX XX XX
Don’t Care
![Page 33: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/33.jpg)
H.-H. S. LeeH.-H. S. Lee33
Prime Implicantsd(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F
A B C D
(4,5,6,7) 0 1 - -(4,6,12,14) - 1 - 0(8,10,12,14) 1 - - 0
(0,1,4,5) 0 - 0 -(0,1,8,9) - 0 0 –(0,4,8,12) - - 0 0
0 1 4 6 8 9 10 12 5 7 14
(0,1,4,5) XX XX XX XX
(0,1,8,9) XX XX XX XX
(0,4,8,12) XX XX XX XX
(4,5,6,7) XX XX XX XX
(4,6,12,14) XX XX XX XX
(8,10,12,14) XX XX XX XX
Don’t Care
![Page 34: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/34.jpg)
H.-H. S. LeeH.-H. S. Lee34
Prime Implicantsd(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F
A B C D
(4,5,6,7) 0 1 - -(4,6,12,14) - 1 - 0(8,10,12,14) 1 - - 0
(0,1,4,5) 0 - 0 -(0,1,8,9) - 0 0 –(0,1,8,9) - 0 0 –(0,4,8,12) - - 0 0
0 1 4 6 8 9 10 12 5 7 14
(0,1,4,5) XX XX XX XX
(0,1,8,9) XX XX XX XX
(0,4,8,12) XX XX XX XX
(4,5,6,7) XX XX XX XX
(4,6,12,14) XX XX XX XX
(8,10,12,14) XX XX XX XX
Don’t Care
![Page 35: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/35.jpg)
H.-H. S. LeeH.-H. S. Lee35
Prime Implicantsd(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F
A B C D
(4,5,6,7) 0 1 - -(4,6,12,14) - 1 - 0(8,10,12,14) 1 - - 0
(0,1,4,5) 0 - 0 -(0,1,8,9) - 0 0 –(0,1,8,9) - 0 0 –(0,4,8,12) - - 0 0
0 1 4 6 8 9 10 12 5 7 14
(0,1,4,5) XX XX XX XX
(0,1,8,9) XX XX XX XX
(0,4,8,12) XX XX XX XX
(4,5,6,7) XX XX XX XX
(4,6,12,14) XX XX XX XX
(8,10,12,14) XX XX XX XX
Don’t Care
![Page 36: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/36.jpg)
H.-H. S. LeeH.-H. S. Lee36
Prime Implicantsd(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F
A B C D
(4,5,6,7) 0 1 - -(4,6,12,14) - 1 - 0(8,10,12,14) 1 - - 0(8,10,12,14) 1 - - 0
(0,1,4,5) 0 - 0 -(0,1,8,9) - 0 0 –(0,1,8,9) - 0 0 –(0,4,8,12) - - 0 0
0 1 4 6 8 9 10 12 5 7 14
(0,1,4,5) XX XX XX XX
(0,1,8,9) XX XX XX XX
(0,4,8,12) XX XX XX XX
(4,5,6,7) XX XX XX XX
(4,6,12,14) XX XX XX XX
(8,10,12,14) XX XX XX XX
Don’t Care
![Page 37: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/37.jpg)
H.-H. S. LeeH.-H. S. Lee37
Prime Implicantsd(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F
A B C D
(4,5,6,7) 0 1 - -(4,5,6,7) 0 1 - -(4,6,12,14) - 1 - 0(8,10,12,14) 1 - - 0(8,10,12,14) 1 - - 0
(0,1,4,5) 0 - 0 -(0,1,8,9) - 0 0 –(0,1,8,9) - 0 0 –(0,4,8,12) - - 0 0
0 1 4 6 8 9 10 12 5 7 14
(0,1,4,5) XX XX XX XX
(0,1,8,9) XX XX XX XX
(0,4,8,12) XX XX XX XX
(4,5,6,7) XX XX XX XX
(4,6,12,14) XX XX XX XX
(8,10,12,14) XX XX XX XX
Don’t Care
Essential PI
Essential PI
Non-Essential PI
![Page 38: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/38.jpg)
H.-H. S. LeeH.-H. S. Lee38
Q-M Method Solution
0 1 4 6 8 9 10
12
5 7 14
(0,1,4,5) XX XX XX XX
(0,1,8,9) XX XX XX XX
(0,4,8,12) XX XX XX XX
(4,5,6,7) XX XX XX XX
(4,6,12,14) XX XX XX XX
(8,10,12,14) XX XX XX XX
A B C D
(4,5,6,7) 0 1 - -(4,5,6,7) 0 1 - -(4,6,12,14) - 1 - 0(8,10,12,14) 1 - - 0(8,10,12,14) 1 - - 0
(0,1,4,5) 0 - 0 -(0,1,8,9) - 0 0 –(0,1,8,9) - 0 0 –(0,4,8,12) - - 0 0
Don’t Care
BADA CB
d(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F
![Page 39: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/39.jpg)
H.-H. S. LeeH.-H. S. Lee39
Yet Another Q-M Method Solution
0 1 4 6 8 9 10
12
5 7 14
(0,1,4,5) XX XX XX XX
(0,1,8,9) XX XX XX XX
(0,4,8,12) XX XX XX XX
(4,5,6,7) XX XX XX XX
(4,6,12,14) XX XX XX XX
(8,10,12,14) XX XX XX XX
A B C D
(4,5,6,7) 0 1 - -(4,5,6,7) 0 1 - -(4,6,12,14) - 1 - 0(4,6,12,14) - 1 - 0(8,10,12,14) 1 - - 0(8,10,12,14) 1 - - 0
(0,1,4,5) 0 - 0 -(0,1,8,9) - 0 0 –(0,1,8,9) - 0 0 –(0,4,8,12) - - 0 0
Don’t Care
DB DA CB
d(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F
![Page 40: 1 ECE2030 Introduction to Computer Engineering Lecture 8: Quine-McCluskey Method Prof. Hsien-Hsin Sean Lee School of ECE Georgia Institute of Technology.](https://reader036.fdocuments.us/reader036/viewer/2022062721/56649f1e5503460f94c351fc/html5/thumbnails/40.jpg)
H.-H. S. LeeH.-H. S. Lee40
To Get the Same Answer w/ K-Map
BADA CB
d(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F
00 01 11 10
00 1 1 0 0
01 1 X X 1
11 1 0 0 X
10 1 1 0 1
ABCD
00 01 11 10
00 1 1 0 0
01 1 X X 1
11 1 0 0 X
10 1 1 0 1
ABCD
DB DA CB
d(5,7,14)12) 10, 9, 8, 6, 4, 1, m(0,F