二進(jìn)制和中國古代八卦圖萊布尼茨自己的說法

1Spring2023ZDMC–Lec.#1數(shù)字系統(tǒng)設(shè)計(jì)

DigitalSystemDesign王維東WeidongWang浙江大學(xué)信息與電子工程系Dept.ofInformationScience&ElectronicEngineering信息與通信工程爭論所ZhejiangUniversity2Spring2023ZDMC–Lec.#1任課教師王維東浙江大學(xué)信息與電子工程學(xué)系,信電樓306郵箱：DepartmentofInformationScienceandElectronicEngineeringZhejiangUniversityHangzhou,310027Tel:86-571-87953170(O)TA:黃露LuHUANG，6719473;;沈翰祺HanqiSHEN;；Tuesday&Fairday14:00-16:30PMOfficeHours：玉泉信電樓308〔可以短信郵件聯(lián)系〕.Prerequisites預(yù)修課程電子電路根底電子線路C語言HowtolearnthisCourse?Notonlylistening,thinkingandwaiting….ButExercise,Simulation,Practice!課程簡介課程代碼：111C0120參考書閻石,數(shù)字電子技術(shù)根底,第5版,高等教育出版社,2023.王金明著，數(shù)字系統(tǒng)設(shè)計(jì)與VerilogHDL，電子工業(yè)出版社，第5版補(bǔ)充講義/期中考試前預(yù)備Stanford大學(xué)108A課程notes.,G.Borriello,ContemporaryLogicDesign,secondedition,電子工業(yè)出版社,2023.,數(shù)字設(shè)計(jì)(第四版),電子工業(yè)出版社,2023.RonaldJ.Tocci,etc.DigitalSystems:PrinciplesandApplications(10thed),PearsonEducation,Inc,2023.4Spring2023ZDMC–Lec.#1OtherCourseInfoWebsite::///wdwd/教學(xué)工作/數(shù)字系統(tǒng)設(shè)計(jì)/2023/Checkfrequently答疑玉泉信電樓308室/周五下午2:30-5:00上課課間、課后Email，短信，微信群Grading(考核)Finalgradeswillbecomputedapproximatelyasfollows:課程作業(yè)+小測驗(yàn)+上課出勤率+期中考試+Project-30%ClassRoomCheckHomeworkSets作業(yè)在上交截止期內(nèi)有效Project2projects(1or2membersteam)Project-2可選〔在30%范圍內(nèi)加分3~5分〕FinialExam期末閉卷考試-70%授課時(shí)間和地點(diǎn)：2023年春夏學(xué)期，周二上午，第3、4節(jié)(9:50-11:25)星期五上午，第1、2節(jié)(08:00-09:35)地點(diǎn)：紫金港西1-207(多)數(shù)字系統(tǒng)設(shè)計(jì)/202378Spring2023ZDMC–Lec.#1課程構(gòu)造

根底理論學(xué)問(必備)數(shù)字系統(tǒng)和編碼、規(guī)律代數(shù)、門電路數(shù)字電路分析與設(shè)計(jì)組合規(guī)律電路觸發(fā)器、半導(dǎo)體存貯器、可編程器件時(shí)序規(guī)律電路脈沖電路掌握器與數(shù)字系統(tǒng)狀態(tài)機(jī)掌握器微碼掌握器測試和驗(yàn)證微處理器----函數(shù)化簡與數(shù)制碼制

910Winter2023ZDMC–Lec.#1–44規(guī)律函數(shù)的化簡法公式化簡法反復(fù)應(yīng)用根本公式和常用公式，消去多余的乘積項(xiàng)和多余的因子?？ㄖZ圖化簡法將規(guī)律函數(shù)的最小項(xiàng)之和的以圖形的方式表示出來。11Winter2023ZDMC–Lec.#1–44規(guī)律函數(shù)的化簡法規(guī)律函數(shù)的最簡形式最簡與或------包含的乘積項(xiàng)已經(jīng)最少，每個(gè)乘積項(xiàng)的因子也最少，稱為最簡的與-或規(guī)律式。12Winter2023ZDMC–Lec.#1–44卡諾圖化簡法規(guī)律函數(shù)的卡諾圖表示法實(shí)質(zhì)：將規(guī)律函數(shù)的最小項(xiàng)之和的以圖形的方式表示出來以2n個(gè)小方塊分別代表n變量的全部最小項(xiàng)，并將它們排列成矩陣，而且使幾何位置相鄰的兩個(gè)最小項(xiàng)在規(guī)律上也是相鄰的〔只有一個(gè)變量不同〕，就得到表示n變量全部最小項(xiàng)的卡諾圖。13Winter2023ZDMC–Lec.#1–44表示最小項(xiàng)的卡諾圖二變量卡諾圖

三變量的卡諾圖4變量的卡諾圖Y14Winter2023ZDMC–Lec.#1–44表示最小項(xiàng)的卡諾圖二變量卡諾圖

三變量的卡諾圖4變量的卡諾圖YY15Winter2023ZDMC–Lec.#1–44表示最小項(xiàng)的卡諾圖二變量卡諾圖三變量的卡諾圖4變量的卡諾圖Y1Y3Y216Winter2023ZDMC–Lec.#1–44五變量的卡諾圖Y17Winter2023ZDMC–Lec.#1–44用卡諾圖表示規(guī)律函數(shù)將函數(shù)表示為最小項(xiàng)之和的形式。在卡諾圖上與這些最小項(xiàng)對應(yīng)的位置上填入1，其余地方填0。18Winter2023ZDMC–Lec.#1–44用卡諾圖表示規(guī)律函數(shù)例：19Winter2023ZDMC–Lec.#1–44用卡諾圖表示規(guī)律函數(shù)Y20Winter2023ZDMC–Lec.#1–44用卡諾圖化簡函數(shù)依據(jù)：具有相鄰性的最小項(xiàng)可合并，消去不同因子。

在卡諾圖中，最小項(xiàng)的相鄰性可以從圖形中直觀地反映出來。21Winter2023ZDMC–Lec.#1–44合并最小項(xiàng)的原則：兩個(gè)相鄰最小項(xiàng)可合并為一項(xiàng)，消去一對因子四個(gè)排成矩形的相鄰最小項(xiàng)可合并為一項(xiàng)，消去兩對因子八個(gè)相鄰最小項(xiàng)可合并為一項(xiàng)，消去三對因子22Winter2023ZDMC–Lec.#1–44兩個(gè)相鄰最小項(xiàng)可合并為一項(xiàng)，

消去一對因子YY23Winter2023ZDMC–Lec.#1–44化簡步驟：------用卡諾圖表示規(guī)律函數(shù)------找出可合并的最小項(xiàng)------化簡后的乘積項(xiàng)相加 〔項(xiàng)數(shù)最少，每項(xiàng)因子最少〕

用卡諾圖化簡函數(shù)24Winter2023ZDMC–Lec.#1–44卡諾圖化簡的原則化簡后的乘積項(xiàng)應(yīng)包含函數(shù)式的全部最小項(xiàng)， 即掩蓋圖中全部的1。乘積項(xiàng)的數(shù)目最少，即圈成的矩形最少。每個(gè)乘積項(xiàng)因子最少，即圈成的矩形最大。25Winter2023ZDMC–Lec.#1–44例：

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1001ABCY26Winter2023ZDMC–Lec.#1–44例：

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100011111101ABCY27Winter2023ZDMC–Lec.#1–44例：

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100011111101ABCY28Winter2023ZDMC–Lec.#1–44例：化簡結(jié)果不唯一YY29Winter2023ZDMC–Lec.#1–44例：0001111000011110ABCDY30Winter2023ZDMC–Lec.#1–44例：00011110001001011001111111101111ABCDY31Winter2023ZDMC–Lec.#1–44約束項(xiàng)任意項(xiàng)規(guī)律函數(shù)中的無關(guān)項(xiàng)：約束項(xiàng)和任意項(xiàng)可以寫入函數(shù)式，也可不包含在函數(shù)式中，因此統(tǒng)稱為無關(guān)項(xiàng)。在規(guī)律函數(shù)中，對輸入變量取值的限制，在這些取值下為1的最小項(xiàng)稱為約束項(xiàng)在輸入變量某些取值下，函數(shù)值為1或?yàn)?不影響規(guī)律電路的功能，在這些取值下為1的最小項(xiàng)稱為任意項(xiàng)具有無關(guān)項(xiàng)的規(guī)律函數(shù)及其化簡

約束項(xiàng)、任意項(xiàng)和規(guī)律函數(shù)式中的無關(guān)項(xiàng)32Winter2023ZDMC–Lec.#1–44無關(guān)項(xiàng)在化簡規(guī)律函數(shù)中的應(yīng)用合理地利用無關(guān)項(xiàng)，可得更簡潔的化簡結(jié)果。參加〔或去掉〕無關(guān)項(xiàng)，應(yīng)使化簡后的項(xiàng)數(shù)最少，每項(xiàng)因子最少······從卡諾圖上直觀地看，參加無關(guān)項(xiàng)的目的是為矩形圈最大，矩形組合數(shù)最少。33Winter2023ZDMC–Lec.#1–440001111000101111101ABCDY34Winter2023ZDMC–Lec.#1–44000111100001x0010x1011x0xx101x0xABCDY35Winter2023ZDMC–Lec.#1–44000111100001x0010x1011x0xx101x0xABCDY36Winter2023ZDMC–Lec.#1–44例：00011110000001011x0111xxxx1010xxABCDY原碼、反碼與補(bǔ)碼原碼符號位0為+1為-00011010101011011000補(bǔ)碼N=N正數(shù)N=2n-N負(fù)數(shù)反碼加10001101011010011100037Winter2023ZDMC–Lec.#1–44反碼N=N正數(shù)N=(2n-1)-N負(fù)數(shù)每位取反，1的補(bǔ)碼00011010110100101111數(shù)制與編碼〔復(fù)習(xí)〕

NUMBERSYSTEMSANDCODES二進(jìn)-十進(jìn)制轉(zhuǎn)換BINARY-TO-DECIMALCONVERSIONS例1例238Winter2023ZDMC–Lec.#1–44二進(jìn)制和中國古代八卦圖萊布尼茨自己的說法18世紀(jì)初，萊布尼茨收到耶酥會士白晉所寄的伏羲八卦圖，正式爭論八卦符號，以此完善了自己的二進(jìn)制體系，然后寫了論文《二進(jìn)位算術(shù)的闡述—關(guān)于只用0和1兼論其用處及伏羲氏所用數(shù)字的意義》，發(fā)表在法國《皇家科學(xué)院院刊》上。二進(jìn)制就是這樣誕生的.39八卦圖衍生自漢族古代的《河圖》與《洛書》，傳為伏羲所作。其中《河圖》演化為先天八卦，《洛書》演化為后天八卦。八卦各有三爻，“乾、坤、震、巽、坎、離、艮、兌”分立八方，象征“天、地、雷、風(fēng)、水、火、山、澤”八種性質(zhì)與自然現(xiàn)象，象征世界的變化與循環(huán)，分類方法猶如五行，世間萬物皆可分類歸至八卦之中。太極八卦太極八卦即是說明宇宙從無極而太極，以至萬物化生的過程。其中的太極即為天地未開、混沌未分陰陽之前的狀態(tài)。兩儀即為太極的陰、陽二儀?！断缔o》又說：“兩儀生四象，四象生八卦”。40數(shù)制十進(jìn)-二進(jìn)制轉(zhuǎn)換DECIMAL-TO-BINARYCONVERSIONS例1例241Winter2023ZDMC–Lec.#1–44十六進(jìn)制

HEXADECIMALNUMBERSYSTEM16位數(shù)字0,1,2,3,4,5,6,7,8,9,A=10,B=11,C=12,D=13,E=14,F=1542Winter2023ZDMC–Lec.#1–44十六進(jìn)制轉(zhuǎn)換Hex-to-DecimalConversion43Winter2023ZDMC–Lec.#1–44十六進(jìn)制轉(zhuǎn)換Decimal-to-HexConversion44Winter2023ZDMC–Lec.#1–44十六進(jìn)制轉(zhuǎn)換Hex-to-BinaryConversionBinary-to-HexConversion45Winter2023ZDMC–Lec.#1–44十六進(jìn)制CountinginHexadecimalWithNhexdigitpositions,wecancountfromdecimal0to16N-1,foratotalof16N

differentvalues.Forexample,withthreehexdigits,wecancountfrom00016toFFF16,whichis010to409510,foratotalof4096=163differentvalues.46Winter2023ZDMC–Lec.#1–44碼制

Whennumbers,letters,orwordsarerepresentedbyaspecialgroupofsymbols,

wesaythattheyarebeingencoded,andthegroupofsymbolsiscalledacode.BCDCODEadecimalnumberisrepresentedbyitsequivalentbinarynumberstraightbinarycodingBinary-Coded-DecimalCodeDigitalsystemsallusesomeformofbinarynumbersfortheirinternaloperation,buttheexternalworldisdecimalinnature.eachdigitofthedecimalnumberbyafour-bitbinarynumber.TheBCDcodedoesnotusethenumbers1010,1011,1100,1101,1110,and1111.47Winter2023ZDMC–Lec.#1–44BCDCODEComparisonofBCDandBinaryThemainadvantageoftheBCDcodeistherelativeeaseofconvertingtoandfromdecimal.48Winter2023ZDMC–Lec.#1–44碼制THEGRAYCODEDigitalsystemsoperateatveryfastspeedsandrespondtochangesthatoccurinthedigitalinputs.Whenmultipleinputconditionsarechangingatthesametime,thesituationcanbemisinterpretedandcauseanerroneousreaction.Inordertoreducethelikelihoodofadigitalcircuitmisinterpretingachanginginput,theGraycodehasbeendevelopedasawaytorepresentasequenceofnumbers.49Winter2023ZDMC–Lec.#1–44GraycodetheGraycodeonlyonebiteverchangesbetweentwosuccessivenumbersinthesequence.ToconvertbinarytoGraythemostsignificantbituseastheGrayMSB.comparetheMSBbinarywiththenextbinarybit(B1).Iftheyarethesame,thenG1=0.Iftheyaredifferent,thenG1=1.G0canbefoundbycomparingB1withB0.50Winter2023ZDMC–Lec.#1–44PUTTINGITALLTOGETHER51Winter2023ZDMC–Lec.#1–44THEBYTE,NIBBLE,ANDWORDBytesbinarydataandinformationingroupsofeightbitsAbytealwaysconsistsofeightbitsitcanrepresentanyofnumeroustypesofdataorinformationNibblesBinarynumbersareoftenbrokendownintogroupsoffourbits,Becauseitishalfasbigasabyte,itwasnamedanibble.WordsThesizeoftheworddependsonthesizeofthedatapathwayinthesystemthatusestheinformationthecomputerinyourmicrowaveovencanprobablyhandleonlyonebyteatatime.Ithasawordsizeofeightbits.thepersonalcomputeronyourdeskcanhandleeightbytesatatime,soithasawordsizeof64bits.52Winter2023ZDMC–Lec.#1–44ALPHANUMERICCODESAcompletealphanumericcodewouldincludethe26lowercaseletters,26uppercaseletters,10numericdigits,7punctuationmarks,andanywherefrom20to40othercharacters,suchas,/,#,%,*,andsoon.ASCIICodeAmericanStandardCodeforInformationInterchange(ASCII).53Winter2023ZDMC–Lec.#1–44ASCIICodeTheASCIIcodepronounced“askee”isaseven-bitcodesoithas27=128possiblecodegroups54Winter2023ZDMC–Lec.#1–44PARITYMETHODFORERRORDETECTIONThemovementofbinarydataandcodesfromonelocationtoanotherasitdoesatpointx,thereceivermayincorrectlyinterpretthatbitasalogic1,manydigitalsystemsemploysomemethodfordetection(andsometimescorrection)oferrors.Oneofthesimplestandmostwidelyusedschemesforerrordetectionistheparitymethod.55Winter2023ZDMC–Lec.#1–44ParityBitanextrabitthatisattachedtoacodegroupTheparitybitismadeeither0or1,dependingonthenumberof1sthatarecontainedinthecodegroupeven-parity

methodtotalnumberof1sinthecodegroup(includingtheparitybit)isanevennumber.Thenewcodegroup,includingtheparitybitaddaparitybitof1tomakethetotalnumberof1sanevennumberERRORDETECTION56Winter2023ZDMC–Lec.#1–44ParityBitTheodd-paritymethodthetotalnumberof1s(includingtheparitybit)isanoddnumbertheparitybitbecomesanactualpartofthecodewordRegardlessofwhetherevenparityoroddparityisused,Forexampleaddingaparitybittotheseven-bitASCIIcodeproducesaneight-bitcode57Winter2023ZDMC–Lec.#1–44paritybitTheparitybitisissuedtodetectanysingle-biterrorsthatoccurduringthetransmissionofacodefromonelocationtoanother.ForexampleSupposethatthecharacter“A”isbeingtransmittedandoddparityisbeingused.ThetransmittedcodewouldbeWhenthereceivercircuitreceivesthiscode,itwillcheckthatthecodecontainsanoddnumberof1s(includingtheparitybit).Ifso,thereceiverwillassum