2019姜书艳数字逻辑设计及应用课件

数字逻辑设计及应用Chapter2

NumberSystemsandcodes(数系与编码)介绍在数字逻辑体系中信号的表达方式、类型，不同表达方式之间的转换，运算的规则等。DigitalLogicDesignandApplication(数字逻辑设计及应用)1数字逻辑设计及应用Chapter2NumberSysReviewofChapter1(第一章内容回顾)AnalogversusDigital(模拟与数字)DigitalDevices(数字器件)：

Gates（门电路）、Flip-flops（触发器）ElectronicandSoftwareAspectsofDigitalDesign

(数字设计的电子技术和软件技术)DigitalLogicDesignandApplication(数字逻辑设计及应用)2ReviewofChapter1(第一章内容回顾)AReviewofChapter1(第一章内容回顾)IntegratedCircuit（IC，集成电路）ProgrammableLogicDevices（PLA、PLD、CPLD、FPGA,可编程逻辑器件）Application-SpecificICs（ASIC,专用集成电路）Printed-CircuitBoards(PCB,印制电路板)DigitalLogicDesignandApplication(数字逻辑设计及应用)3ReviewofChapter1(第一章内容回顾)IChapter2

NumberSystemsandcodes

(数系与编码)TwokindsofInformation(信息主要有两类)：

NumericData(数值信息)

NonnumericData(非数值信息)DigitalLogicDesignandApplication(数字逻辑设计及应用)—NumberSystemsandtheirConversions(数制及其转换)--NonnumericDataRepresentation–Codes(非数值信息的表征--编码)4Chapter2NumberSystemsandChapter2

NumberSystemsandcodes

(数系与编码)数字系统只处理数字信号0,1；需要将任意信息用(0,1)表达；用（0，1）表达数量：数制二进制用（0，1）表达不同对象：符号编码DigitalLogicDesignandApplication(数字逻辑设计及应用)5Chapter2NumberSystemsand6HowtoEncodeText:ASCII,UnicodeASCII:7-(or8-)bitencodingofeachletter,number,orsymbolSampleASCIIencodings……66HowtoEncodeText:ASCII,Un7HowtoEncodeText:ASCII,UnicodeUnicode:Increasinglypopular16-bitencodingEncodescharactersfromvariousworldlanguages……77HowtoEncodeText:ASCII,Un8HowtoEncodeNumbers:BinaryNumbersBaseten(decimal)Tensymbols:0,1,2,...,8,and9Morethan9--nextpositionSoeachpositionpower（幂）of10Nothingspecialaboutbase10--usedbecausewehave10fingers104103102523101100Eachpositionrepresentsabasequantity;symbolinpositionmeanshowmanyofthatquantity88HowtoEncodeNumbers:Binary9HowtoEncodeNumbers:BinaryNumbersBasetwo(binary)Twosymbols:0and1Morethan1--nextpositionSoeachpositionpower（幂）of22423221012120Q:Howmuch?

+

=

4

15

+

=aEachpositionrepresentsabasequantity;symbolinpositionmeanshowmanyofthatquantityThereareonly10typesofpeopleintheworld:thosewhounderstandbinary,andthosewhodon't.99HowtoEncodeNumbers:Binary10Usefultoknowpowersof2:2423222120292827262516842151225612864321684215122561286432Practicecountingupbypowersof2:1010Usefultoknowpowersof2:22.1PositionalNumberSystem

(按位计数制)AnyDecimalNumberDCanBeRepresentedastheFollowing(任意十进制数D可表示如下):D

=dp-1

dp-2...d1

d0

.

d-1

d-2...d-n推广：D2=∑di×2i

D16=∑di×16i

Weightofibit;BaseorRadixofrNumberSystem(第i位的权；r计数制的基数)DigitalLogicDesignandApplication(数字逻辑设计及应用)112.1PositionalNumberSystem2.1PositionalNumberSystem

(按位计数制)按位计数制的特点1)采用基数（BaseorRadix），R进制的基数是R2)基数确定数符的个数如十进制的数符为：0、1、2、3、4、5、6、7、8、9，个数为10

二进制的数符为：0、1，个数为23）逢基数进一DigitalLogicDesignandApplication(数字逻辑设计及应用)122.1PositionalNumberSystem2.1PositionalNumberSystem

(按位计数制)DigitalLogicDesignandApplication(数字逻辑设计及应用)DecimalandBinaryDecimalsystem:baseis10,thedigitmaybe0to9Binarysystem:baseis2,thedigitmaybe0or1bit:onedigitinbinarysystem;

132.1PositionalNumberSystem14UsingDigitalDatainaDigitalSystemtemperaturesensor00110000"33"AtemperaturesensoroutputstemperatureinbinaryThesystemreadsthetemperature,outputsASCIIcode:“F”forfreezing(0-32)“B”forboiling(212ormore)“N”fornormal1414UsingDigitalDatainaDigi15UsingDigitalDatainaDigitalSystemtemperaturesensor00110000"33"DigitalSystemdisplayN"N"1000111if(input<="00100000")//"32"

output="1000110"//"F"elseif(input>="11010100")//"212"

output="1000010"//"B"else

output="1001110"//"N"AdisplayconvertsitsASCIIinputtothecorrespondingletter1515UsingDigitalDatainaDigi2.2OctalandHexadecimalNumbers

(八进制和十六进制)

基数

数码

特性

OctalNumber

(八进制)

8

0~7

逢八进一

BinaryNumber

(二进制)

2

0,1

逢二进一

HexadecimalNumber(十六进制)

16

0~9,A~F

逢十六进一

DigitalLogicDesignandApplication(数字逻辑设计及应用)162.2OctalandHexadecimalNu说明DigitalLogicDesignandApplication(数字逻辑设计及应用)选择什么数制来表示信息，对数字系统的成本和性能影响很大，在数字电路中多使用二进制.MostSignificantBit（MSB,最高有效位）LeastSignificantBit（LSB,最低有效位）1011100010112MSBLSB17说明DigitalLogicDesignandDigitalLogicDesignandApplication(数字逻辑设计及应用)表2.1十进制、二进制、八进制与十六进制数十进制二进制八进制十六进制

00000

00

10001

11

20010

22

30011

33

40100

44

50101

55

60110

66

70111

77

81000108

9100111910101012A11101113B12110014C13110115D14111016E15111117F18DigitalLogicDesignandAppli19BaseSixteen:AnotherBaseUsedbyDesignersNicebecauseeachpositionrepresentsfourbase-twopositionsCompactwaytowritebinarynumbersKnownashexadecimal,orjusthexQ:Write11110000inhexF0Q:ConverthexA01tobinary101000000001A:A:1919BaseSixteen:AnotherBaseU二进制与八进制和十六进制之间的转换位数替换法：保持小数点不变，每位八进制数对应3位二进制数;

每位十六进制数对应4位二进制数;二进制转换时，从小数点开始向左右分组，在MSB前面和LSB后面可以加0；转换为二进制时，MSB前面和LSB后面的0不写；例：1011100010112=56138=B8B1610111000.10112=207.548=B8.B161000110010012=()8

=()16DigitalLogicDesignandApplication(数字逻辑设计及应用)20二进制与八进制和十六进制之间的转换位数替换法：保持小数点不变21HexExample:RFIDTagBatteryless（无电池）

tagpowered（功率）byradiofieldTransmits（发送）uniqueidentification（鉴定）

numberExample:32bitid（身份证明）8-bitprovincenumber,8-bitcitynumber,16-bitanimalnumberTagcontentsareinbinaryButprogrammersusehexwhenwriting/readingRFIDtagProvince#City#Animal#Province:

7City:

160Animal:

51310100000000001110000001000000001A0070201

TagIDinhex:07A00201(a)(b)(c)(d)(f)(e)2121HexExample:RFIDTagBattery2.3GeneralPositional-Number-SystemConversion(常用按位计数制的转换)DigitalLogicDesignandApplication(数字逻辑设计及应用)ANumberinanyRadixtoRadix10

(任意进制数十进制数)Method:Expandingtheformulausingradix-10arithmetic(方法：利用位权展开)Example1：(101.01)2=()10(7F.8)16=()10Moreeasyway(更简便的方法)？

(F1AC)16=(((F16)+1)16+A)16+C222.3GeneralPositional-Number2.3GeneralPositional-Number-SystemConversion(常用按位计数制的转换)DigitalLogicDesignandApplication(数字逻辑设计及应用)ANumberinRadix10toanyRadix

(十进制其它进制)Method：RadixMultiplicationorDivision(基数乘除法)

IntegerParts(整数部分)：除r取余，逆序排列

Example2：(156)10=()2

DecimalFractionParts(小数部分)：乘r取整，顺序排列

Example3：(0.37)10=()2232.3GeneralPositional-Number2.3GeneralPositional-Number-SystemConversion(常用按位计数制的转换)DigitalLogicDesignandApplication(数字逻辑设计及应用)Example4：Require<10-2

，Completethefollowingconversion(617.28)10=()22-n<=<10-2思考：任意两种进位计数制之间的转换以十进制（二进制）作为桥梁n=7242.3GeneralPositional-Number25ConvertingTo/FromBinarybyHand:Summary2525ConvertingTo/FromBinaryby26Divide-By-2MethodCommoninAutomaticConversionRepeatedlydividedecimalnumberby2,placeremainderincurrentbinarydigit(startingfrom1scolumn)Note:WorksforanybaseN—justdividebyNinstead2626Divide-By-2MethodCommonin27Bytes,Kilobytes,Megabytes,andMoreByte:8bitsCommonmetricprefixes:kilo(thousand,or103),mega(million,or106),giga(billion,千兆or109),andtera(trillion,万亿or1012),e.g.,kilobyte,orKByte2727Bytes,Kilobytes,Megabytes,28Bytes,Kilobytes,Megabytes,andMoreBUT,metricprefixesalsocommonlyusedinaccurately216=65536commonlywrittenas“64Kbyte”TypicalwhendescribingmemorysizesAlsowatchoutfor“KB”forkilobytevs.“Kb”forkilobit2828Bytes,Kilobytes,Megabytes,29Example:DIP-SwitchControlledChannelCeilingfanreceivershouldbesetinfactorytorespondtochannel“73”Convert73tobinary,setDIP（指拨）switchaccordinglyDesiredvalue:73400211160813206411280647273sum:(b)(a)Q:2929Example:DIP-SwitchControllExample:DIP-SwitchControlledChannelchannelreceiverCeilingfanmodule0if(InA=InB)

Out=1else

Out=001010100"73"DIPswitch1000100001InAInBOut"34"(c)30Example:DIP-SwitchControlled2.4AdditionandSubtractionofNondecimalNumbers(非十进制数的加法和减法)DigitalLogicDesignandApplication(数字逻辑设计及应用)TwoBinaryNumberArithmetic

(两个二进制数的算术运算)Addition(加法)：

Carry(进位)1+1=10Subtraction(减法)：

Borrows(借位)10–1=1312.4AdditionandSubtractionCarryin(进位输入):Cin(P.32)

Carryout(进位输出)Cout

Sum(本位和):S2.4AdditionandSubtractionofNon-decimalNumbers(非十进制数的加法和减法)DigitalLogicDesignandApplication(数字逻辑设计及应用)

10111110+1000110132Carryin(进位输入):Cin(P.32)2DigitalLogicDesignandApplication(数字逻辑设计及应用)表2.3.1二进制加法真值表输

入输

出被加数X加数Y输入进位Cin和S进位输出Cout000000011001010011011001010101110011111133DigitalLogicDesignandAppli34AdderExample:DIP-Switch-BasedAddingCalculatorGoal:Createcalculatorthataddstwo8-bitbinarynumbers,specifiedusingDIPswitchesDIPswitch:Dual-In-linePackageswitch,moveeachswitchupordownSolution:Use8-bitadder3434AdderExample:DIP-Switch-BaAdderExample:DIP-Switch-BasedAddingCalculatorSolution:Use8-bitadderDIPswitches10a7……..a0b7……..b0s7…………s08-bitcarry-rippleaddercoci0CALCLEDs35AdderExample:DIP-Switch-Base36AdderExample:DIP-Switch-BasedAddingCalculatorTopreventspurious(假的）valuesfromappearingatoutput,canplaceregisteratoutputActually,thelightflickers(闪烁）fromspuriousvalueswouldbetoofastforhumanstodetect—buttheprincipleofregisteringoutputstoavoidspuriousvaluesbeingreadbyexternaldevices(whichnormallyaren’thumans)applieshere.3636AdderExample:DIP-Switch-BaAdderExample:DIP-Switch-BasedAddingCalculatorDIPswitches10a7…….a0b7……b0s7………s08-bitadder8-bitregistercoci0CALCLEDseclkld37AdderExample:DIP-Switch-BaseBorrowin(借位输入):BinBorrowout(借位输出):Bout

Differencebit(本位差):D2.4AdditionandSubtractionofNon-decimalNumbers(非十进制数的加法和减法)DigitalLogicDesignandApplication(数字逻辑设计及应用)

10101010–0101010138Borrowin(借位输入):Bin2.4DigitalLogicDesignandApplication(数字逻辑设计及应用)表2.3.2二进制减法真值表输

入输

出被减数X减数Y输入借位Bin差D输出借位Bout000000011101011011011001010100110001111139DigitalLogicDesignandAppliSubtractorExample:DIP-SwitchBasedAdding/SubtractingCalculatorExtendearliercalculatorexampleSwitchfindicateswhetherwanttoadd(f=0)orsubtract(f=1)Usesubtractorand2x1mux40SubtractorExample:DIP-SwitchSubtractorExample:DIP-SwitchBasedAdding/SubtractingCalculatorDIPswitches108-bitregisterCALCLEDsefclkld88800888882x10110wiciAABBSScowo8-bitadder8-bitsubtractor41SubtractorExample:DIP-Switch2.5RepresentationofNegativeNumbers

(负数的表示)DigitalLogicDesignandApplication(数字逻辑设计及应用)2.5.1Signed-MagnitudeRepresentation[符号–数值表示法（原码）]MSBastheSignbit(0=plus,1=minus)[最高有效位表示符号位（0=正，1=负）]01111111＝＋12711111111＝－12700101110＝＋4610101110＝－4600000000＝＋010000000＝－0422.5RepresentationofNegati2.5RepresentationofNegativeNumbers

(负数的表示)DigitalLogicDesignandApplication(数字逻辑设计及应用)2.5.1Signed-MagnitudeRepresentation[符号–数值表示法（原码）]TwopossiblerepresentationsofZero[零有两种表示（+0、–0）]Ann-bitsigned-magnitudeintegerrangeis(n位二进制整数表示范围)：

–(2n-1

–1)~+(2n-1

–1)432.5RepresentationofNegati2.5RepresentationofNegativeNumbers(负数的表示)2.5.2ComplementNumberSystems(补码数制)Radix–Complement(基数补码)DiminishedRadix–Complement[基数减1补码(反码)]DigitalLogicDesignandApplication(数字逻辑设计及应用)442.5RepresentationofNegati2.5RepresentationofNegativeNumbers(负数的表示)2.5.3Radix–ComplementRepresentation(基数补码表示法)Thecomplementofann-digitnumberisobtainedbysubtractingitfromrn

(n位数的补码等于从rn

中减去该数)Example:Table2-4P.36DigitalLogicDesignandApplication(数字逻辑设计及应用)452.5RepresentationofNegati2.5RepresentationofNegativeNumbers(负数的表示)DiminishedRadix–ComplementRepresentation[基数减1补码表示法（反码）]：

TheDiminishedRadix–Complementofann-digitnumberisobtainedbysubtractingitfromrn-1

[n位数的反码等于从rn

–1

中减去该数]Example:Table2-42-5P.36DigitalLogicDesignandApplication(数字逻辑设计及应用)462.5RepresentationofNegati47Ten’sComplementBeforeintroducingtwo’scomplement,let’sconsiderten’scomplementBut,beawarethatcomputersDONOTUSETEN’SCOMPLEMENT.Introducedforintuition（直觉）only.Complementsforeachbasetennumbershowntoright.Complementisthenumberthatwhenaddedresultsin109876543211234567894747Ten’sComplementBeforeintro48Ten’sComplementNicefeatureoften’scomplementInsteadofsubtractinganumber,addingitscomplementresultsinanswerexactly10toomuchSojustdrop（丢下）the1–resultsinsubtractingusingadditiononly46107–4+601020313133010123456789987654321complements7–4=37+6=133Addingthecomplementresultsinananswerthatisexactly10toomuch–droppingthetenscolumngivestherightanswer.4848Ten’sComplementNicefeature2.5.4Two’s–ComplementRepresentation

(二进制补码表示法)

TheMSBofanumberinthissystemservesasthesignbit(最高有效位用做符号位);TheweightoftheMSBis-2n-1insteadof+2n-1(MSB的权是-2n-1

而不是+2n-1)ObtainaTwo’s-Complement

(二进制补码的求取)：

Ones’–Complement(反码)+1（Why？？）DigitalLogicDesignandApplication(数字逻辑设计及应用)492.5.4Two’s–ComplementRepr2.5.4Two’s–ComplementRepresentation

(二进制补码表示法)DigitalLogicDesignandApplication(数字逻辑设计及应用)例2.5.1若约定字长是一个字节，试求－119的补码。解：因－119的绝对值119＝01110111，则补码可以通过下式算法得到：全1码：11111111减去－119绝对值：01110111－119的反码：10001000加1：

1－119补码：10001001502.5.4Two’s–ComplementRepr2.5.4Two’s–ComplementRepresentation

(二进制补码表示法)OnlyonerepresentationsofZero

(零只有一种表示)0＝00000000逐位取反11111111

＋1约定8位00000000＝0DigitalLogicDesignandApplication(数字逻辑设计及应用)512.5.4Two’s–ComplementRepr2.5.4Two’s–ComplementRepresentation

(二进制补码表示法)Ann-bitTwo’s-Complementrangeis(n位二进制补码表示范围)：

–2n-1~+(2n-1

–1)

约定字长（8比特）后，补码表示数的范围－128～127

DigitalLogicDesignandApplication(数字逻辑设计及应用)522.5.4Two’s–ComplementRepre2.5.4Two’s–ComplementRepresentation

(二进制补码表示法)Positivenumberhasthesame：Sign-Magnitude,Ones’–Complement,andTwo’s-Complement

(正数的原码、反码、补码相同)DigitalLogicDesignandApplication(数字逻辑设计及应用)532.5.4Two’s–ComplementRepr表2-61位十进制数与4位二进制数十进制二进制原码二进制反码二进制补码-8————1000-7111110001001-6111010011010-5110110101011-4110010111100-3101111001101-2101011011110-110011110111101000或00001111或00000000100010001000120010001000103001100110011401000100010050101010101016011001100110701110111011154表2-61位十进制数与4位二进制数十进制二进制原码二进2.5RepresentationofNegativeNumbers(负数的表示)2.5.6Ones’–ComplementRepresentation(二进制反码表示法)TheSignBitdoesn’tchange,OtherBitsconversebasedontheSigned-Magnitude.

(符号位不变，其余在原码基础上按位取反)DigitalLogicDesignandApplication(数字逻辑设计及应用)552.5RepresentationofNegati56RepresentingNegativeNumbers:Two’sComplementBigadvantage:AllowsustoperformsubtractionusingadditionThus,onlyneedaddercomponent,noneedforseparatesubtractorcomponent5656RepresentingNegativeNumberTwo’sComplementSubtractorBuiltwithanAdderUsingtwo’scomplementA–B=A+(-B)=A+(two’scomplementofB)=A+invert_bits(B)+1SobuildsubtractorusingadderbyinvertingB’sbits,andsettingcarryinto157Two’sComplementSubtractorBuTwo’sComplementSubtractorBuiltwithanAdder1CinBAAdderSBAN-bit58Two’sComplementSubtractorBuSummaryofthisclass(本堂课小结)Binary,Octal,andHexadecimalNumbers(二进制、八进制、十六进制)PositionalNumberSystem(按位计数制)DigitalLogicDesignandApplication(数字逻辑设计及应用)59Summaryofthisclass(本堂课小结)BGeneralPosition