数字电子技术4.ppt

DigitalFundamentals CHAPTER4BooleanAlgebraandLogicSimplification 布尔代数和逻辑化简 马克思曾经说过 一门学科只有成功地运用数学语言进行描述时 才算达到了完善的地步 数学模型的意义 Logicsimplification Thepurposeofsimplification theleastcomponentsaspossibleaswecanusetoimplementthelogicfunction 4 1BooleanOperationsandExpressions BooleanOperationsandExpressions Addition0 0 00 1 11 0 11 1 1 Multiplication0 0 00 1 01 0 01 1 1 4 2LawsandRulesofBooleanAlgebra LawsBooleanAlgebra CommutativeLaws 交换律 AssociativeLaws 结合律 DistributiveLaws 分配律 LawsofBooleanAlgebra AssociativeLawofMultiplication A B C A B C LawsofBooleanAlgebra DistributiveLaw A B C AB AC applicableonlyinbooleanalgebra 求证 A BC A B A C 证明 右边 A B A C AA AB AC BC 分配律 A A B C BC 结合律 AA A A 1 B C BC 结合律 A 1 BC 1 B C 1 A BC A 1 1 左边 RulesofBooleanAlgebra Rules10 11 12 RulesofBooleanAlgebra Rule10 A AB A 原变量的吸收 RulesofBooleanAlgebra Rule11 反变量的被吸收 证明 例如 混合变量的被吸收 证明 例如 RulesofBooleanAlgebra Rule12 A B A C A BC 4 3DeMorgan sTheorem 德 摩根定理 DeMorgan sTheorems Theorem1 Theorem2 Remember Breakthebar changethesign 4 5SimplificationUsingBooleanAlgebra 利用布尔代数进行化简 Usuallywehavetoreduceaparticularexpressiontoitssimplestformorchangeitsformtoamoreconvenientonetoimplementtheexpressionmostefficiently Theapproachtakeninthissectionistousethebasiclaws rulesandtheoremsofbooleanalgebratosimplifyanexpression Minimumexpression Thefewestpossibleterms Fewestpossiblevariablesinaterm Ex2 德 摩根 conclusion XORgatecanbeimplementedbyusingfourNANDgates Ex3 prove 德 摩根 展开 异或门可以用4个与非门实现 Ex4 simplifytheexpressionY 利用德 摩根定理 Tosimplifysuccessfullyneedsmorepractice 4 8TheKarnaughMap 卡诺图 AKarnaughmapprovidesasystematicmethodforsimplifyingBooleanexpressionsand ifproperlyused willproducethesimplestlogicexpressionpossible knownastheminimumexpression Called cookbook methodforsimplification SomeDefinitions 1 Minterm最小项 Amintermisaproducttermthatcontainsalloftheinputvariables eachliteralnomorethanonce thatmakeupaBooleanexpression 2 Maxterm最大项 Amaxtermisasumtermthatcontainsalloftheinputvariables eachliteralnomorethanonce thatmakeupaBooleanexpression 3 Thesum of product SOP form积之和Example X AB CD EF 4 Theproductofsum POS form和之积Example X A B C D E F 5 standardSOPform containsallofthemintermsorrepresentedbytheminterms Minterm Mintermcannotbebrokenupbecauseitcontainsallofthevariables Wecanobtaintheexpressionfromthetruthtableaccordingtotheminterms Step1 findoutthe1soutput ExpressthecorrespondinginputsasmintermStep2 sumtheseminterms Celladjacency 逻辑相邻 thereisonlyasingle variablechangebetweenadjacencycells Adjacencycellscanbegrouped sothatwecouldreducesomevariables 将n个输入变量的全部最小项与输出量之间的关系用小方块阵列图表示 并且将逻辑相邻的最小项放在相邻的几何位置上 所得到的阵列图就是n变量的卡诺图 Examples 2 variable 3 variable 4 variable FormofKarnaughmap Example1 2 variableKarnaughmap variables Example2 3 variableKarnaughmap Tip 00and10areadjacencycells 00 01 11 10 Thecell0010 Thebinaryvaluecanbe0or1 thistermiscalleddon tcare Example3 4 variableKarnaughmap F A B C m1 m2 m4 m7 Cell1 2 4 7are1sandothersare0s Numberof4 variableKarnaughmap SimplificationUsingKarnaughMaps principle 1sadjacencycellscouldbegrouped sothatsomevariableswillbecancelled cookbook 卡诺图的几何相邻与逻辑相邻一致 可以直观的找出具有相邻性的最小项并将其合并 F AB BC Minimization 化简 Thismethodisapplicableonlyin3or4 variablecases andtheminimizationismoresimplethanothermethods Weshouldfollowthesesteps Step1 寻找可合并的单元格 aslargeaspossible 将其圈入一个卡诺圈 圈得一个乘积项 相邻单元的个数是2n个 并组成矩形时 可以合并 c 将所有卡诺圈表示的乘积项逐个加起来 即得该函数的最简 与或 表示式 b 依次将所有取值为1的单元格圈入卡诺圈中 直到所有为1的项都被使用后化简工作方算完成 各最小项可以重复使用 但每一次新的组合至少包含一个未使用过的项 Example1 F A B C D 0 2 3 5 6 8 9 10 11 12 13 14 15 Example2 Step1 logicexpression Karnaughmap 1 1 Tip1 有时可通过合并卡诺图中的0 先得到 再求反得Y 可能更快 Someusefultips Tip2 利用无关项 可以使结果大大简化 化简时可以将无关项当作1或0 目的是得到最简结果 F A Tip3 圈完全部的卡诺圈后要检查一下有没有冗余 Tip4 化简结果不唯一 SimplythefollowingKarnaughmaps Howmanymeanscouldbeusedtorepresentthelogicfunction TruthtableLogicexpression Booleanexpression Logiccircuit symbol KarnaughmapPulsedwaveform Thischapteryoushouldmaster Basiclawsandrulesincludingdemorgan stheoremTransfertherel