lecture (09) quine mccluskey method -...
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Lecture (09)Quine‐McCluskey
MethodBy:
Dr. Ahmed ElShafee
Dr. Ahmed ElShafee, ACU : Spring 2018, CSE202 Logic Design I١
Quine‐McCluskeyMethodDetermination of Prime Implicants• the function must be given as a sum of minterms.
• all of the prime implicants of a function are systematically formed by combining minterms
To reduce the required number of comparisons, the binary minterms are sorted into groups according to the number of 1’s in each term
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Dr. Ahmed ElShafee, ACU : Spring 2018, CSE202 Logic Design I٢
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Dr. Ahmed ElShafee, ACU : Spring 2018, CSE202 Logic Design I٣
• Two terms in any two groups can be combined as they differ in exactly one variable.
• First, we will compare the term in group 0 with all of the terms in group 1.
• Terms 0000 and 0001 can be combined to eliminate the fourth variable, which yields 000–.
• Similarly, 0 and 2 combine to form 00–0 (a′b′d′), and 0 and 8 combine to form –000 (b′c′d′). The resulting terms are listed in Column II
• the corresponding decimal numbers differ by a power of 2 (1, 2, 4, 8, etc.).
• A term may be used more than once because X + X = X.Dr. Ahmed ElShafee, ACU : Spring 2016, Logic Design٤
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Example
• Find all of the prime implicants of the function
• f (a,b,c,d) = Σm(0,1,2,5,6,7,8,9,10,14)
Dr. Ahmed ElShafee, ACU : Spring 2018, CSE202 Logic Design I٥
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Dr. Ahmed ElShafee, ACU : Spring 2016, Logic Design٦
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Dr. Ahmed ElShafee, ACU : Spring 2016, Logic Design٧
Dr. Ahmed ElShafee, ACU : Spring 2016, Logic Design٨
![Page 5: Lecture (09) Quine McCluskey Method - draelshafee.netdraelshafee.net/Spring2018/cse202-logic-design-i---lecture-09.pdf · Lecture (09) Quine‐McCluskey Method By: Dr. Ahmed ElShafee](https://reader030.vdocuments.site/reader030/viewer/2022013016/5baa066209d3f2f51d8daf59/html5/thumbnails/5.jpg)
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Dr. Ahmed ElShafee, ACU : Spring 2016, Logic Design٩
The terms which have not been checked off because they cannot be combined with other terms are called prime implicants.
Dr. Ahmed ElShafee, ACU : Spring 2018, CSE202 Logic Design I١٠
![Page 6: Lecture (09) Quine McCluskey Method - draelshafee.netdraelshafee.net/Spring2018/cse202-logic-design-i---lecture-09.pdf · Lecture (09) Quine‐McCluskey Method By: Dr. Ahmed ElShafee](https://reader030.vdocuments.site/reader030/viewer/2022013016/5baa066209d3f2f51d8daf59/html5/thumbnails/6.jpg)
• All prim implicante
• Minimum form ???
Dr. Ahmed ElShafee, ACU : Spring 2018, CSE202 Logic Design I١١
The Prime Implicant Chart
• The minterms of the function are listed across the top of the chart, and the prime implicants are listed down the side.
• If a prime implicant covers a given minterm, an X is placed at the intersection of the corresponding row and column.
• If a minterm is covered by only one prime implicant, then that prime implicant is called an essential prime implicant
Dr. Ahmed ElShafee, ACU : Spring 2018, CSE202 Logic Design I١٢
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Dr. Ahmed ElShafee, ACU : Spring 2018, CSE202 Logic Design I١٣
• A minimum set of prime implicants must now be chosen to cover the remaining columns
Dr. Ahmed ElShafee, ACU : Spring 2016, Logic Design١٤
![Page 8: Lecture (09) Quine McCluskey Method - draelshafee.netdraelshafee.net/Spring2018/cse202-logic-design-i---lecture-09.pdf · Lecture (09) Quine‐McCluskey Method By: Dr. Ahmed ElShafee](https://reader030.vdocuments.site/reader030/viewer/2022013016/5baa066209d3f2f51d8daf59/html5/thumbnails/8.jpg)
• A minimum set of prime implicants must now be chosen to cover the remaining columns
Dr. Ahmed ElShafee, ACU : Spring 2016, Logic Design١٥
Example
• A prime implicant chart which has two or more X’s in every column is called a cyclic
• prime implicant chart. The following function has such a chart
Dr. Ahmed ElShafee, ACU : Spring 2016, Logic Design١٦
![Page 9: Lecture (09) Quine McCluskey Method - draelshafee.netdraelshafee.net/Spring2018/cse202-logic-design-i---lecture-09.pdf · Lecture (09) Quine‐McCluskey Method By: Dr. Ahmed ElShafee](https://reader030.vdocuments.site/reader030/viewer/2022013016/5baa066209d3f2f51d8daf59/html5/thumbnails/9.jpg)
• Derivation of prime implicants:
Dr. Ahmed ElShafee, ACU : Spring 2016, Logic Design١٧
• Select P1 first.
Dr. Ahmed ElShafee, ACU : Spring 2016, Logic Design١٨
![Page 10: Lecture (09) Quine McCluskey Method - draelshafee.netdraelshafee.net/Spring2018/cse202-logic-design-i---lecture-09.pdf · Lecture (09) Quine‐McCluskey Method By: Dr. Ahmed ElShafee](https://reader030.vdocuments.site/reader030/viewer/2022013016/5baa066209d3f2f51d8daf59/html5/thumbnails/10.jpg)
• Select P2 first.
Dr. Ahmed ElShafee, ACU : Spring 2018, CSE202 Logic Design I١٩
Simplification of IncompletelySpecified Functions•
Dr. Ahmed ElShafee, ACU : Spring 2018, CSE202 Logic Design I٢٠
![Page 11: Lecture (09) Quine McCluskey Method - draelshafee.netdraelshafee.net/Spring2018/cse202-logic-design-i---lecture-09.pdf · Lecture (09) Quine‐McCluskey Method By: Dr. Ahmed ElShafee](https://reader030.vdocuments.site/reader030/viewer/2022013016/5baa066209d3f2f51d8daf59/html5/thumbnails/11.jpg)
Example
• Simplify
• F(A,B,C,D) = Σm(2,3,7,9,11,13) + Σ d(1,10,15)
Dr. Ahmed ElShafee, ACU : Spring 2018, CSE202 Logic Design I٢١
• Treat the don’t cares (1,10,15) as required minterms
Dr. Ahmed ElShafee, ACU : Spring 2018, CSE202 Logic Design I٢٢
F(A,B,C,D) = Σm(2,3,7,9,11,13) + Σ d(1,10,15)
![Page 12: Lecture (09) Quine McCluskey Method - draelshafee.netdraelshafee.net/Spring2018/cse202-logic-design-i---lecture-09.pdf · Lecture (09) Quine‐McCluskey Method By: Dr. Ahmed ElShafee](https://reader030.vdocuments.site/reader030/viewer/2022013016/5baa066209d3f2f51d8daf59/html5/thumbnails/12.jpg)
• The don’t cares are not list at the top of the table
Dr. Ahmed ElShafee, ACU : Spring 2018, CSE202 Logic Design I٢٣
F(A,B,C,D) = Σm(2,3,7,9,11,13) + Σ d(1,10,15)
Thanks,..
Dr. Ahmed ElShafee, ACU : Spring 2018, CSE202 Logic Design I
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