Chapter2NumberSystemsandcodes(數(shù)系與編碼)NumericData－NumberSystemsandtheirConversions

（數(shù)值信息——數(shù)制及其轉(zhuǎn)換）

NonnumericData－Codes

（非數(shù)值信息——編碼）DigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)1Chapter2NumberSystemsandReviewofChapter2(第二章內(nèi)容回顧)Binary,Octal,andHexadecimalNumbers(二進(jìn)制、八進(jìn)制、十六進(jìn)制)PositionalNumberSystem(按位計數(shù)制)DigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)2ReviewofChapter2(第二章內(nèi)容回顧)BReviewofChapter2(第二章內(nèi)容回顧)GeneralPositional-Number-SystemConversion(常用按位計數(shù)制的轉(zhuǎn)換)ANumberinanyRadixtoRadix10:Expandingtheformulausingradix-10arithmetic(任意進(jìn)制數(shù)

十進(jìn)制數(shù)：利用位權(quán)展開)DigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)3ReviewofChapter2(第二章內(nèi)容回顧)GReviewofChapter2(第二章內(nèi)容回顧)GeneralPositional-Number-SystemConversion(常用按位計數(shù)制的轉(zhuǎn)換)ANumberinRadix10toanyRadix:RadixMultiplicationorDivision(十進(jìn)制

其它進(jìn)制：基數(shù)乘除法)Note:DecimalFractionPartsConversion[注意：小數(shù)部分的轉(zhuǎn)換（誤差）]DigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)4ReviewofChapter2(第二章內(nèi)容回顧)GReviewofChapter2(第二章內(nèi)容回顧)AdditionandSubtractionofNondecimalNumbers(非十進(jìn)制的加法和減法)（Table2-3)

進(jìn)位輸入Cin、進(jìn)位輸出Cout、本位和S

借位輸入Bin、借位輸出Bout、本位差DDigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)5ReviewofChapter2(第二章內(nèi)容回顧)AReviewofChapter2(第二章內(nèi)容回顧)RepresentationofNegativeNumbers

(負(fù)數(shù)的表示)Signed-Magnitude[符號－數(shù)值（原碼）]ComplementNumberSystems(補(bǔ)碼數(shù)制)Radix–Complement(基數(shù)補(bǔ)碼)DiminishedRadix–Complement[基數(shù)減1補(bǔ)碼（基數(shù)反碼）]DigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)6ReviewofChapter2(第二章內(nèi)容回顧)RReviewofChapter2(第二章內(nèi)容回顧)BinarySigned-Magnitude,Ones’–Complement,andTwo’s–ComplementRepresentation

(二進(jìn)制的原碼、反碼、補(bǔ)碼)正數(shù)的原碼、反碼、補(bǔ)碼表示相同負(fù)數(shù)的原碼表示：符號位為1負(fù)數(shù)的反碼表示：符號位不變，其余在原碼基礎(chǔ)上按位取反在|D|的原碼基礎(chǔ)上按位取反（包括符號位）負(fù)數(shù)的補(bǔ)碼表示：反碼+1DigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)7ReviewofChapter2(第二章內(nèi)容回顧)B2.5.4Two’s–ComplementRepresentation

(二進(jìn)制補(bǔ)碼表示法)Ann-bitTwo’s-Complementrangeis(n位二進(jìn)制補(bǔ)碼表示范圍)：

–2n-1~+(2n-1–1)

OnlyonerepresentationsofZero(零只有一種表示)ObtainaTwo’s-Complement(二進(jìn)制補(bǔ)碼的求取)：

Ones’–Complement(反碼)+1（為什么？？）ExpandingtheSignBit(符號位擴(kuò)展)DigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)82.5.4Two’s–ComplementRepr2.5RepresentationofNegativeNumbers(負(fù)數(shù)的表示)Example2.5.2：Writethe8-bitsigned-magnitude,two’s-complementforeachofthesebinarynumbers.(分別寫出下面二進(jìn)制數(shù)的8位符號–數(shù)值碼、補(bǔ)碼)(–1101)2(–0.1101)2DigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)92.5RepresentationofNegati2.5RepresentationofNegativeNumbers(負(fù)數(shù)的表示)DigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)

1、(–1101)22、(–0.1101)21、5位二進(jìn)制表示：原碼反碼補(bǔ)碼1110110010100112、8位二進(jìn)制表示：原碼反碼補(bǔ)碼100011011111001011110011[[D]反]反=D[[D]補(bǔ)]補(bǔ)=D102.5RepresentationofNegati2.6Two’s–ComplementAdditionandSubtraction(二進(jìn)制補(bǔ)碼的加法和減法)AdditionRules:Addedbyordinarybinaryaddition(加法：按普通二進(jìn)制加法相加)P.39SubtractionRules:Takingitstwo’scomplement,thenadd(減法：將減數(shù)求補(bǔ)，再相加)DigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)112.6Two’s–ComplementAdditi2.6Two’s–ComplementAdditionandSubtraction(二進(jìn)制補(bǔ)碼的加法和減法)DigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)

＋20010－31101

＋

＋5

＋

0101

＋

－5

＋

1011

＋70111－811000

＋70111＋10001

＋

－4

＋

1100

＋

－6

＋

1010

＋310011－51011122.6Two’s–ComplementAdditi13Adder/SubtractorExample:CalculatorPreviouscalculatorusedseparateadderandsubtractorDIPswitches108-bitregisterCALCLEDsefclkld88800888882x10110wiciAABBSScowo8-bitadder8-bitsubtractor1313Adder/SubtractorExample:Ca14Adder/SubtractorExample:CalculatorImprovebyusingadder/subtractor,andtwo’scomplementnumbersDIPswitches108-bitregister8-bitadder/subtractorsubCALCLEDseSABfclkld1088881414Adder/SubtractorExample:Ca2.6Two’s–ComplementAdditionandSubtraction(二進(jìn)制補(bǔ)碼的加法和減法)Overflow（溢出）如果加法運(yùn)算產(chǎn)生的和超出了數(shù)制表示的范圍，則結(jié)果發(fā)生了溢出（Overflow）。

對于二進(jìn)制補(bǔ)碼，加數(shù)的符號相同，和的符號與加數(shù)的符號不同。（或者，Cin

與Cout不同）P.41對于無符號二進(jìn)制數(shù)，若最高有效位上發(fā)生進(jìn)位或借位，就指示結(jié)果超出范圍。

－51011＋70111

＋

－6

＋

1010

＋

＋3

＋

0011

－1110101＝

＋5＋101010＝

－6

DigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)152.6Two’s–ComplementAdditi16OverflowSometimesresultcan’tberepresentedwithgivennumberofbitsEithertoolargemagnitudeofpositiveornegativeEx.4-bittwo’scomplementadditionof0111+0001(7+1=8).But4-bittwo’scomplementcan’trepresentnumber>70111+0001=1000WRONGanswer,1000intwo’scomplementis-8,not+8Adder/subtractorshouldindicatewhenoverflowhasoccurred,soresultcanbediscarded1616OverflowSometimesresultcan17DetectingOverflow:Method1Fortwo’scomplementnumbers,overflowoccurswhenthetwonumbers’signbitsarethesamebutdifferfromtheresult’ssignbitIfthetwonumbers’signbitsareinitiallydifferent,overflowisimpossibleAddingpositiveandnegativecan’texceedlargestmagnitudepositiveornegative01111000+0001signbitsoverflow(a)11110111+0100overflow(b)10001111+1011nooverflow(c)Ifthenumbers’signbitshavethesamevalue,whichdiffersfromtheresult’ssignbit,overflowhasoccurred.1717DetectingOverflow:Method118DetectingOverflow:Method2Evensimplermethod:Detectdifferencebetweencarry-intosignbitandcarry-outfromsignbit01111111001000+0001overflow(a)11100010111+0100overflow(b)10000001111+1011nooverflow(c)Ifthecarryintothesignbitcolumndiffersfromthecarryoutofthatcolumn,overflowhasoccurred.1818DetectingOverflow:Method22.10BinaryCodesforDecimalNumbers

(十進(jìn)制數(shù)的二進(jìn)制編碼)DigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)Asetofn-bitstringsinwhichdifferentbitstringsRepresentdifferentnumbersorotherthings.

(用于表示不同數(shù)或其它事件的一組n位二進(jìn)制碼的集合)192.10BinaryCodesforDecima2.10BinaryCodesforDecimalNumbers

(十進(jìn)制數(shù)的二進(jìn)制編碼)Howtorepresenta1-bitDecimalnumberwitha4-bitBinarycode(如何用4位二進(jìn)制碼表示1位十進(jìn)制碼)？——BinaryCodedDecimal(BCD碼）DigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)202.10BinaryCodesforDecimal2.10BinaryCodesforDecimalNumbers

(十進(jìn)制數(shù)的二進(jìn)制編碼)HowtorepresentaNegativeBCDnumber(負(fù)的BCD數(shù)如何表示)？Signed-MagnitudeRepresentation:Encodingofthesignbitisarbitrary(符號－數(shù)值表示：符號位的編碼任意)10’s-complement:0000indicatesplus,1001indicatesminus.(十進(jìn)制補(bǔ)碼表示：0000正，1001負(fù))AdditionofBCDDigits(BCD數(shù)的加法)P.50DigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)212.10BinaryCodesforDecimalDigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)22DigitalLogicDesignandAppli2.10BinaryCodesforDecimalNumbers

(十進(jìn)制數(shù)的二進(jìn)制編碼)(Table2-9)DigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)BCDCode2421CodeExcess-3(余3碼)BiquinaryCode(二五混合碼)1-out-of-10(10中取1碼)

WeightedCode(加權(quán)碼)Self-ComplementCode自反碼232.10BinaryCodesforDecimaDigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)24DigitalLogicDesignandAppli8421codeNaturalcode,justlike4-bitbinarynumbers;Eachdigitisweighted;Ithas10validcodewordsand6invalidcodewords.BCDcodes258421codeBCDcodes25Eachdigitisweighted;Self-complementing;UseMSBtoexpresshigher/lowerpart;Ithas10validcodesand6invalidcodes.2421codesBCDcodes26Eachdigitisweighted;2421coBCDcodesExcess-3codeItsdigitisnotweighted;8421code+“0011”;Self-complementing.27BCDcodesExcess-3codeItsdigiExamples:useBCDcodefordecimalnumbersA=19468421code:A=00011001010001102421code:A=0001111101001100Excess-3code:A=0100110001111001BCDcodes28Examples:BCDcodes281-out-of-10codeOnehotcode:Itisveryusefulincontrolsystems.Onehotcodes291-out-of-10codeOnehotcode:OTwohotcodesBiquinarycode

7-bits;twohotcode;First2bitsisonehotcodeforhigher/lowerrange;Last5bitsisonehotcodeintherange.Error-detectingproperty!

30TwohotcodesBiquinarycode30Fromonecodetoitsneighbor,onlyonebitchanged,notransitionstate.Temperaturecode31Fromonecodetoitsneighbor,2.11Graycode（格雷碼）DigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)322.11Graycode（格雷碼）DigitalLo2.11Graycode（格雷碼）DigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)特點(diǎn)：任意相鄰碼字間只有一位數(shù)位變化最高位的0和1只改變一次最大數(shù)回到0也只有一位碼元不同332.11Graycode（格雷碼）DigitalLo2.11Graycode（格雷碼）DigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)構(gòu)造方法直接構(gòu)造

Thebitsofann-bitbinarycordwordarenumberedfromrighttoleft,from0ton-1.[對n

位二進(jìn)制的碼字從右到左編號（0~n-1）]BitiofaGray-codecodewordis0ifbitsiandi+1ofthecorrespondingbinarycodewordarethesame,elsebitiis1.(若二進(jìn)制碼字的第i位和第i+1位相同，則對應(yīng)的格雷碼碼字的第i

位為0，否則為1。)ReflectedCode（反射碼）342.11Graycode（格雷碼）DigitalLoGraycodesTarget:codeforcontinueschangednumbers(inbinarysystem)topreventwrongcodehappenedintransitiontime;Property:Ineachpairofsuccessivecodewords,onlyonebitchanges.35GraycodesTarget:codeforconGraycodesFrombinarynumbertoGraycodeThewidthissame,theMSBissame;Fromlefttoright,ifabitinbinarynumberissameasitsleftbit,thegraycodeis0,ifitisdifferent,thegraycodeis1.Examples:binarynumber:1001001001100011Graycodes:110110110101001036GraycodesFrombinarynumbertError-detectingcodeInformationword+checkingbit37Error-detectingcodeInformatioDigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)2.12CharacterCodes(字符編碼)

ASCII碼（P36表2-11）

ASCIIcode：128Keyboardsigns,7-bitUsedforkeyboardordisplaydevice38DigitalLogicDesignandAppliDigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)2.13CodesforActions,Conditions,andStates(動作、條件和狀態(tài)的編碼)

使用b位二進(jìn)制編碼來表示n個不同狀態(tài)Word:adigitalstringtorepresentanobject

Usenbits,wecanmake2ndifferentwords;Tomakenwords,youmustusebits.39DigitalLogicDesignandAppliDigitalLogicDesignandApplication(數(shù)字邏輯設(shè)計及應(yīng)用)2.16CodesforSerialDataTransmissionandStorage(用于串行數(shù)據(jù)傳輸與存儲的編碼)Parallelway

usen-linetotransmitann-bitscodewords;