数字电子技术(Floyd 第十版)课件Chapter 3.pptx
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数字电子技术(Floyd 第十版)课件Chapter 3.pptx
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,DigitalFundamentalsTenthEditionFloyd,Chapter2,2008PearsonEducation,Thepositionofeachdigitinaweightednumbersystemisassignedaweightbasedonthebaseorradixofthesystem.Theradixofdecimalnumbersisten,becauseonlytensymbols(0through9)areusedtorepresentanynumber.,Summary,Thecolumnweightsofdecimalnumbersarepowersoftenthatincreasefromrighttoleftbeginningwith100=1:
DecimalNumbers,105104103102101100.,Forfractionaldecimalnumbers,thecolumnweightsarenegativepowersoftenthatdecreasefromlefttoright:
102101100.10-110-210-310-4,Summary,DecimalNumbers,Expressthenumber480.52asthesumofvaluesofeachdigit.,Example,Solution,(9x103)+(2x102)+(4x101)+(0x100)or9x1,000+2x100+4x10+0x1,Decimalnumberscanbeexpressedasthesumoftheproductsofeachdigittimesthecolumnvalueforthatdigit.Thus,thenumber9240canbeexpressedas,480.52=(4x102)+(8x101)+(0x100)+(5x10-1)+(2x10-2),Summary,BinaryNumbers,Fordigitalsystems,thebinarynumbersystemisused.Binaryhasaradixoftwoandusesthedigits0and1torepresentquantities.,Thecolumnweightsofbinarynumbersarepowersoftwothatincreasefromrighttoleftbeginningwith20=1:
252423222120.,Forfractionalbinarynumbers,thecolumnweightsarenegativepowersoftwothatdecreasefromlefttoright:
222120.2-12-22-32-4,Summary,BinaryNumbers,Abinarycountingsequencefornumbersfromzerotofifteenisshown.,00000100012001030011401005010160110701118100091001101010111011121100131101141110151111,DecimalNumber,BinaryNumber,Noticethepatternofzerosandonesineachcolumn.,Digitalcountersfrequentlyhavethissamepatternofdigits:
Summary,BinaryConversions,Thedecimalequivalentofabinarynumbercanbedeterminedbyaddingthecolumnvaluesofallofthebitsthatare1anddiscardingallofthebitsthatare0.,Convertthebinarynumber100101.01todecimal.,Example,Solution,Startbywritingthecolumnweights;thenaddtheweightsthatcorrespondtoeach1inthenumber.,252423222120.2-12-2,32168421.,100101.01,32+4+1+=,37,Summary,BinaryConversions,Youcanconvertadecimalwholenumbertobinarybyreversingtheprocedure.Writethedecimalweightofeachcolumnandplace1sinthecolumnsthatsumtothedecimalnumber.,Convertthedecimalnumber49tobinary.,Example,Solution,Thecolumnweightsdoubleineachpositiontotheright.Writedowncolumnweightsuntilthelastnumberislargerthantheoneyouwanttoconvert.,26252423222120.,6432168421.,0110001.,Summary,Youcanconvertadecimalfractiontobinarybyrepeatedlymultiplyingthefractionalresultsofsuccessivemultiplicationsby2.Thecarriesformthebinarynumber.,Convertthedecimalfraction0.188tobinarybyrepeatedlymultiplyingthefractionalresultsby2.,Example,Solution,0.188x2=0.376carry=0,0.376x2=0.752carry=0,0.752x2=1.504carry=1,0.504x2=1.008carry=1,0.008x2=0.016carry=0,Answer=.00110(forfivesignificantdigits),MSB,BinaryConversions,1,0,0,1,1,0,Summary,Youcanconvertdecimaltoanyotherbasebyrepeatedlydividingbythebase.Forbinary,repeatedlydivideby2:
Convertthedecimalnumber49tobinarybyrepeatedlydividingby2.,Example,Solution,Youcandothisby“reversedivision”andtheanswerwillreadfromlefttoright.Putquotientstotheleftandremaindersontop.,24,12,6,3,1,0,Continueuntilthelastquotientis0,Answer:
BinaryConversions,Summary,BinaryAddition,Therulesforbinaryadditionare,0+0=0Sum=0,carry=0,0+1=0Sum=1,carry=0,1+0=0Sum=1,carry=0,1+1=10Sum=0,carry=1,Whenaninputcarry=1duetoapreviousresult,therulesare,1+0+0=01Sum=1,carry=0,1+0+1=10Sum=0,carry=1,1+1+0=10Sum=0,carry=1,1+1+1=10Sum=1,carry=1,Summary,BinaryAddition,Addthebinarynumbers00111and10101andshowtheequivalentdecimaladdition.,Example,Solution,001117,1010121,0,1,0,1,1,1,1,0,1,28,=,Summary,BinarySubtraction,Therulesforbinarysubtractionare,0-0=0,1-1=0,1-0=1,10-1=1withaborrowof1,Subtractthebinarynumber00111from10101andshowtheequivalentdecimalsubtraction.,Example,Solution,001117,1010121,0,/1,1,1,1,0,14,/1,/1,=,Summary,1sComplement,The1scomplementofabinarynumberisjusttheinverseofthedigits.Toformthe1scomplement,changeall0sto1sandall1sto0s.,Forexample,the1scomplementof11001010is,00110101,Indigitalcircuits,the1scomplementisformedbyusinginverters:
11001010,00110101,Summary,2sComplement,The2scomplementofabinarynumberisfoundbyadding1totheLSBofthe1scomplement.,Recallthatthe1scomplementof11001010is,00110101(1scomplement),Toformthe2scomplement,add1:
+1,00110110(2scomplement),11001010,00110101,1,00110110,Summary,SignedBinaryNumbers,Thereareseveralwaystorepresentsignedbinarynumbers.Inallcases,theMSBinasignednumberisthesignbit,thattellsyouifthenumberispositiveornegative.,Computersuseamodified2scomplementforsignednumbers.Positivenumbersarestoredintrueform(witha0forthesignbit)andnegativenumbersarestoredincomplementform(witha1forthesignbit).,Forexample,thepositivenumber58iswrittenusing8-bitsas00111010(trueform).,Signbit,Magnitudebits,Summary,SignedBinaryNumbers,Assumingthatthesignbit=-128,showthat11000110=-58asa2scomplementsignednumber:
Example,Solution,11000110,Columnweights:
-1286432168421.,-128+64+4+2=-58,Negativenumbersarewrittenasthe2scomplementofthecorrespondingpositivenumber.,-58=11000110(complementform),Aneasywaytoreadasignednumberthatusesthisnotationistoassignthesignbitacolumnweightof-128(foran8-bitnumber).Thenaddthecolumnweightsforthe1s.,Thenegativenumber-58iswrittenas:
Summary,FloatingPointNumbers,Expressthespeedoflight,c,insingleprecisionfloatingpointnotation.(c=0.2998x109),Example,Solution,Floatingpointnotationiscapableofrepresentingverylargeorsmallnumbersbyusingaformofscientificnotation.A32-bitsingleprecisionnumberisillustrated.,SE(8bits)F(23bits),Signbit,MagnitudewithMSBdropped,Biasedexponent(+127),Inscientificnotation,c=1.001110111101001010111000000x228.,Inbinary,c=000100011101111010010101110000002.,S=0becausethenumberispositive.E=28+127=15510=100110112.Fisthenext23bitsafterthefirst1isdropped.,Infloatingpointnotation,c=,Summary,ArithmeticOperationswithSignedNumbers,Usingthesignednumbernotationwithnegativenumbersin2scomplementformsimplifiesadditionandsubtractionofsignednumbers.,Rulesforaddition:
Addthetwosignednumbers.Discardanyfinalcarries.Theresultisinsignedform.Examples:
00011110=+3000001111=+15,00101101,=+45,00001110=+1411101111=-17,11111101,=-3,11111111=-111111000=-8,11110111,=-9,1,Summary,ArithmeticOperationswithSignedNumbers,01000000=+12801000001=+129,10000001,=-126,10000001=-12710000001=-127,100000010,=+2,Notethatifthenumberofbitsrequiredfortheanswerisexceeded,overflowwilloccur.Thisoccursonlyifbothnumbershavethesamesign.Theoverflowwillbeindicatedbyanincorrectsignbit.,Twoexamplesare:
Wrong!
Theanswerisincorrectandthesignbithaschanged.,Summary,ArithmeticOperationswithSignedNumbers,Rulesforsubtraction:
2scomplementthesubtrahendandaddthenumbers.Discardanyfinalcarries.Theresultisinsignedform.,00001111,=+15,1,2scomplementsubtrahendandadd:
00011110=+3011110001=-15,Repeattheexamplesdonepreviously,butsubtract:
00011111,=+31,00001110=+1400010001=+17,00000111,=+7,1,11111111=-100001000=+8,Summary,HexadecimalNumbers,Hexadecimalusessixteencharacterstorepresentnumbers:
thenumbers0through9andthealphabeticcharactersAthroughF.,0123456789101112131415,0123456789ABCDEF,0000000100100011010001010110011110001001101010111100110111101111,Decimal,Hexadecimal,Binary,Largebinarynumbercaneasilybeconvertedtohexadecimalbygroupingbits4atatimeandwritingtheequivalenthexadecimalcharacter.,Express10010110000011102inhexadecimal:
Example,Solution,Groupthebinarynumberby4-bitsstartingfromtheright.Thus,960E,Summary,HexadecimalNumbers,Hexadecimalisaweightednumbersystem.Thecolumnweightsarepowersof16,whichincreasefromrighttoleft.,.,1A2F16,670310,Columnweights,163162161160,4096256161,.,Express1A2F16indecimal.,Example,Solution,Startbywritingthecolumnweights:
4096256161,1(4096)+10(256)+2(16)+15
(1)=,0123456789101112131415,0123456789ABCDEF,0000000100100011010001010110011110001001101010111100110111101111,Decimal,Hexadecimal,Binary,Summary,OctalNumbers,Octaluseseightcharactersthenumbers0through7torepresentnumbers.Thereisno8or9characterinoctal.,0123456789101112131415,012345671011121314151617,0000000100100011010001010110011110001001101010111100110111101111,Decimal,Octal,Binary,Binarynumbercaneasilybeconvertedtooctalbygroupingbits3atatimeandwritingtheequivalentoctalcharacterforeachgroup.,Express10010110000011102inoctal:
Example,Solution,Groupthebinarynumberby3-bitsstartingfromtheright.Thus,1130168,Summary,OctalNumbers,Octalisalsoaweightednumbersystem.Thecolumnweightsarepowersof8,whichincreasefromrighttoleft.,.,37028,198610,Columnweights,83828180,5126481,.,Express37028indecimal.,Example,Solution,Startbywritingthecolumnweights:
5126481,3(512)+7(64)+0(8)+2
(1)=,0123456789101112131415,012345671011121314151617,0000000100100011010001010110011110001001101010111100110111101111,Decimal,Octal,Binary,Summary,BCD,Binarycodeddecimal(BCD)isaweightedcode
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