1ECE2030 ECE2030 Introduction to Computer Introduction to Computer EngineeringEngineeringLecture 8: Quine-McCluskey MethodLecture 8: Quine-McCluskey MethodProf.Hsien-Hsin Sean LeeProf.Hsien-Hsin Sean LeeSchool of ECESchool of ECEGeorgia Institute of Georgia Institute of TechnologyTechnology H.-H. S. Lee H.-H. S. Lee2Quine-McCluskey MethodA systematic solution to -Map !hen more comple" #unction !ith more literals is gi$enIn principle% can &e applied to an ar&itrary large num&er o# inputs% i'e' !orks #or ((nn !here nn can &e ar&itrarily large)ne can translate Quine-McCluskey method into a computer program to per#orm minimi*ation H.-H. S. Lee H.-H. S. Lee+Quine-McCluskey Method ,!o &asic steps-inding all prime implicants o# a gi$en (oolean #unction.elect a minimal set o# prime implicants that co$er this #unction H.-H. S. Lee H.-H. S. Lee/Q-M Method 0I1= 30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0, E) D, C, B, F(A,,rans#orm the gi$en (oolean #unction into a canonical .)2 #unctionCon$ert each Minterm into &inary #ormatArrange each &inary minterm in groupsAll the minterms in one group contain the same num&er o# 314 H.-H. S. Lee H.-H. S. Lee5Q-M Method: 6rouping minterms= 30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0, E) D, C, B, F(A,(29) 1 1 1 0 1(30) 1 1 1 1 0(2)0 0 0 1 0(4)0 0 1 0 0(8)0 1 0 0 0(16) 1 0 0 0 0 A B C D E(0)0 0 0 0 0(6)0 0 1 1 0(10) 0 1 0 1 0(12) 0 1 1 0 0(18) 1 0 0 1 0(7)0 0 1 1 1(11) 0 1 0 1 1(13) 0 1 1 0 1(14) 0 1 1 1 0(19) 1 0 0 1 1 H.-H. S. Lee H.-H. S. Lee7Q-M Method 0II1Com&ine terms !ith 8amming distance91 #rom ad:acent groupsCheck 01 the terms &eing com&ined,he checked terms are 3co$ered4 &y the com&ined ne! termeep doing this till no com&ination is possi&le &et!een ad:acent groups H.-H. S. Lee H.-H. S. Lee;Q-M Method: 6rouping 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 H.-H. S. Lee H.-H. S. Lee8Q-M Method: 6rouping mintermsA 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 - - - 08,10,12,14) = 30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0, E) D, C, B, F(A, H.-H. S. Lee H.-H. S. Leenchecked terms are prime implicants H.-H. S. Lee H.-H. S. Lee1?2rime Implicants= 30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0, E) D, C, B, F(A,CD B A =D C B 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=>nchecked terms are prime implicantsD BC A =D C B A =E D BC =E BCD =E C B =E A = H.-H. S. Lee H.-H. S. Lee11Q-M Method 0III1-orm a 2rime Implicant ,a&le @-a"is: the mintermA-a"is: prime implicantsAn is placed at the intersection o# a ro! and column i# the corresponding prime implicant includes the corresponding product 0term1 H.-H. S. Lee H.-H. S. Lee12Q-M Method: 2rime Implicant ,a&le0 2 4! " #0###2#3#4#

#"#$2$30%&!' @ @ @ @%#0#' @ @ @ @%#2' @ @ @ @%#"$' @ @ @ @%#3&2$' @ @ @ @%#4&30' @ @ @ @%0&2"' @ @ @ @ @ @ @ @%0&2&4&&"'@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @= 30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0, E) D, C, B, F(A, H.-H. S. Lee H.-H. S. Lee1+Q-M Method 0IB1Locate the essential ro! #rom the ta&le,hese are essential prime implicants,he ro! consists o# minterms co$ered &y a single ()Mark all minterms co$ered &y the essential prime implicants-ind non-essential prime implicants to co$er the rest o# minterms-orm the .)2 #unction !ith the prime implicants selected% !hich is the minimal representation H.-H. S. Lee H.-H. S. Lee1/Q-M Method0 2 4! " #0###2#3#4#

#"#$2$30%&!' @ @ @ @%#0#' @ @ @ @%#2' @ @ @ @%#"$' @ @ @ @%#3&2$' @ @ @ @%#4&30' @ @ @ @%0&2"' @ @ @ @ @ @ @ @%0&2&4&&"'@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @= 30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0, E) D, C, B, F(A, H.-H. S. Lee H.-H. S. Lee15Q-M Method0 2 4! " #0###2#3#4#

#"#$2$30%&!' @ @ @ @%#0#' @ @ @ @%#2' @ @ @ @%#"$' @ @ @ @%#3&2$' @ @ @ @%#4&30' @ @ @ @%0&2"' @ @ @ @ @ @ @ @%0&2&4&&"'@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @= 30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0, E) D, C, B, F(A,= .elect 0?%2%/%7%8%1?%12%1/1 H.-H. S. Lee H.-H. S. Lee17Q-M Method0 2 4! " #0###2#3#4#

#"#$2$30%&!' @ @ @ @%#0#' @ @ @ @%#2' @ @ @ @%#"$' @ @ @ @%#3&2$' @ @ @ @%#4&30' @ @ @ @%0&2"' @ @ @ @ @ @ @ @%0&2&4&&"'@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @= 30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0, E) D, C, B, F(A,= .elect 0?%2%/%7%8%1?%12%1/1 H.-H. S. Lee H.-H. S. Lee1;Q-M Method0 2 4! " #0###2#3#4#

#"#$2$30%&!' @ @ @ @%#0#' @ @ @ @%#2' @ @ @ @%#"$' @ @ @ @%#3&2$' @ @ @ @%#4&30' @ @ @ @%0&2"' @ @ @ @ @ @ @ @%0&2&4&&"'@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @= 30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0, E) D, C, B, F(A,= .elect 0?%2%/%7%8%1?%12%1/1% 07%;1 H.-H. S. Lee H.-H. S. Lee18Q-M Method0 2 4! " #0###2#3#4#

#"#$2$30%&!' @ @ @ @%#0#' @ @ @ @%#2' @ @ @ @%#"$' @ @ @ @%#3&2$' @ @ @ @%#4&30' @ @ @ @%0&2"' @ @ @ @ @ @ @ @%0&2&4&&"'@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @= 30) 29, 19, 18, 16, 14, 13, 12, 11, 10, 8, 7, 6, 4, 2, m(0, E) D, C, B, F(A,= .elect 0?%2%/%7%8%1?%12%1/1% 07%;1 H.-H. S. Lee H.-H. S. Lee1