Logic Design...Digital Systems • Digital discipline –Discrete voltages instead of continuous...
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Number Systems Logic Design
Transcript of Logic Design...Digital Systems • Digital discipline –Discrete voltages instead of continuous...
NumberSystems
LogicDesign
DigitalSystems
• Digitaldiscipline– Discretevoltagesinsteadofcontinuous– Simplertodesignthananalogcircuits• canbuildmoresophisticatedsystems
– Digitalsystemsreplacinganalogpredecessors• i.e.,digitalcameras,digitaltelevision,cellphones,CDs
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TheDigitalAbstraction
• Mostphysicalvariablesarecontinuous– Voltageonawire– Frequencyofanoscillation– Positionofamass
• Digitalabstractionconsidersdiscretesubsetofvalues
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TheAnalyticalEngine
• DesignedbyCharlesBabbagefrom1834–1871– Consideredtobethefirstdigitalcomputer
• Builtfrommechanicalgears,whereeachgearrepresentedadiscretevalue(0-9)– Babbagediedbeforeitwasfinished
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DigitalDiscipline:BinaryValues
• Twodiscretevalues:– 1’sand0’s– 1,TRUE,HIGH– 0,FALSE,LOW
• 1and0:voltagelevels,rotatinggears,fluidlevels,etc.• Digitalcircuitsusevoltagelevelstorepresent1and0• Bit:Binarydigit
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• Introducedbinaryvariables• Introducedthethreefundamentallogic operations: AND, OR, andNOT
GeorgeBoole,1815-1864
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NumberSystems
• Decimalnumbers
• Binarynumbers
537410 = 5 × 103 + 3 × 102 + 7 × 101 + 4 × 100
fivethousands
10's column
100's column
1000's column
threehundreds
seventens
fourones
1's column
11012 = 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20 = 1310oneeight
2's column
4's column
8's column
onefour
notwo
oneone
1's column
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• 20=1• 21=2• 22=4• 23=8• 24=16• 25=32• 26=64• 27=128
• 28=256• 29=512• 210=1024• 211=2048• 212=4096• 213=8192• 214=16384• 215=32768
PowersofTwo
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• Binarytodecimalconversion:– Convert100112todecimal
• Decimaltobinaryconversion:– Convert4710tobinary
NumberConversion
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• Binarytodecimalconversion:– Convert100112todecimal– 16×1+8×0+4×0+2×1+1×1=1910
• Decimaltobinaryconversion:– Convert4710tobinary– 32×1+16×0+8×1+4×1+2×1+1×1=1011112
NumberConversion
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• Twomethods:o Method1:Findthelargestpowerof2thatfits,
subtractandrepeato Method2:Repeatedlydivideby2,remaindergoes
innextmostsignificantbit
DecimaltoBinaryConversion
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Method1:Findthelargestpowerof2thatfits,subtractandrepeat5310
Method2:Repeatedlydivideby2,remaindergoesinnextmostsignificantbit5310
DecimaltoBinaryConversion
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Method1:Findthelargestpowerof2thatfits,subtractandrepeat5310 32×153-32=21 16×121-16=5 4×15-4=1 1×1
=1101012Method2:Repeatedlydivideby2,remaindergoesinnextmostsignificantbit
5310= 53/2=26R1 26/2=13R0
13/2=6R1 6/2=3R0 3/2=1R1 1/2=0R1 =1101012
DecimaltoBinaryConversion
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Anotherexample:Convert7510tobinary.7510=64+8+2+1=10010112
or
75/2 =37R137/2 =18 R118/2 =9 R09/2 =4 R14/2 =2 R02/2 =1 R01/2 =0 R1
DecimaltoBinaryConversion
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BinaryValuesandRange
• N-digitdecimalnumber– Howmanyvalues?– Range?– Example:3-digitdecimalnumber:
• N-bitbinarynumber– Howmanyvalues?– Range:– Example:3-digitbinarynumber:
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BinaryValuesandRange
• N-digitdecimalnumber– Howmanyvalues?10N– Range?[0,10N-1]– Example:3-digitdecimalnumber:• 103=1000possiblevalues• Range:[0,999]
• N-bitbinarynumber– Howmanyvalues?2N– Range:[0,2N-1]– Example:3-digitbinarynumber:• 23=8possiblevalues• Range:[0,7]=[0002to1112]
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HexDigit DecimalEquivalent BinaryEquivalent0 0 0000
1 1 0001
2 2 0010
3 3 0011
4 4 0100
5 5 0101
6 6 0110
7 7 0111
8 8 1000
9 9 1001
A 10 1010
B 11 1011
C 12 1100
D 13 1101
E 14 1110
F 15 1111
HexadecimalNumbers
Shorthandforbinary(Base16)
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• Hexadecimaltobinaryconversion:– Convert4AF16(alsowritten0x4AF)tobinary
• Hexadecimaltodecimalconversion:– Convert0x4AFtodecimal
HexadecimaltoBinaryConversion
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• Hexadecimaltobinaryconversion:– Convert4AF16(alsowritten0x4AF)tobinary– 0100101011112
• Hexadecimaltodecimalconversion:– Convert4AF16todecimal– 162×4+161×10+160×15=119910
HexadecimaltoBinaryConversion
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Bits,Bytes,Nibbles…
• Bits• Bytes&Nibbles• Bytes
10010110nibble
byte
CEBF9AD7least
significantbyte
mostsignificantbyte
10010110least
significantbit
mostsignificant
bit
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LargePowersofTwo
• 210=1kilo ≈1000(1024)• 220=1mega ≈1million(1,048,576)• 230=1giga≈1billion(1,073,741,824)
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EstimatingPowersofTwo
• Whatisthevalueof224?• Howmanyvaluescana32-bitvariablerepresent?
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EstimatingPowersofTwo
• Whatisthevalueof224?24×220≈16million• Howmanyvaluescana32-bitvariablerepresent?22×230≈4billion
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Addition
• Decimal
• Binary
37345168+
10110011+
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Addition
• Decimal
• Binary
37345168+8902
carries 11
10110011+1110
11 carries
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BinaryAdditionExamples
• Addthefollowing4-bitbinarynumbers
• Addthefollowing4-bitbinarynumbers
10010101+
10110110+
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BinaryAdditionExamples
• Addthefollowing4-bitbinarynumbers
• Addthefollowing4-bitbinarynumbers
10010101+1110
1
10110110+10001
111
Overflow!27
• Digitalsystemsoperateonafixednumberofbits
• Overflow:whenresultistoobigtofitintheavailablenumberofbits
Overflow
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• Wehavetworepresentation:• Sign/MagnitudeNumbers• Two’sComplementNumbers
SignedBinaryNumbers
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• 1signbit,N-1magnitudebits
• Signbitisthemostsignificant(left-most)bit– Positivenumber:signbit=0– Negativenumber:signbit=1
Sign/MagnitudeNumbers
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• Example,4-bitsign/magrepresentationsof±6:+6=0110
-6=1110• RangeofanN-bitsign/magnitudenumber: [-(2N-1-1),2N-1-1]
Sign/MagnitudeNumbers
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Problems:• Additiondoesn’twork,forexample-6+6:
1110+011010100(wrong!)
• Tworepresentationsof0(±0):
10000000
Sign/MagnitudeNumbers
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• Don’thavesameproblemsassign/magnitudenumbers:– Additionworks– Singlerepresentationfor0
Two’sComplementNumbers
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• msbhasvalueof-2N-1
• Mostpositive4-bitnumber:0111• Mostnegative4-bitnumber:1000• Themostsignificantbitstillindicatesthesign(1=negative,0=positive)
• RangeofanN-bittwo’scomplementnumber:[-(2N-1),2N-1-1]
+
Two’sComplementNumbers
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• “TakingtheTwo’scomplement”flipsthesignofatwo’scomplementnumber
• Method:1. Invertthebits2. Add1
• Example:Flipthesignof310=001121. 11002. +11101=-310
“TakingtheTwo’sComplement”
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• Takethetwo’scomplementof610=01102
• Whatisthedecimalvalueofthetwo’scomplementnumber10012?
Two’sComplementExamples
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• Takethetwo’scomplementof610=011021. 10012. +110102=-610
• Whatisthedecimalvalueofthetwo’scomplementnumber10012?
1. 01102. +101112=710,so10012=-710
Two’sComplementExamples
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• Add6+(-6)usingtwo’scomplementnumbers
• Add-2+3usingtwo’scomplementnumbers
Two’sComplementAddition
+01101010
+11100011
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• Add6+(-6)usingtwo’scomplementnumbers
• Add-2+3usingtwo’scomplementnumbers
Two’sComplementAddition
+0110101010000
111
+1110001110001
111
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IncreasingBitWidth
ExtendnumberfromNtoMbits(M>N):– Sign-extension– Zero-extension
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• Signbitcopiedtomsb’s• Numbervalueissame
• Example1:– 4-bitrepresentationof3=0011– 8-bitsign-extendedvalue:00000011
• Example2:– 4-bitrepresentationof-5=1011– 8-bitsign-extendedvalue:11111011
Sign-Extension
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• Zeroscopiedtomsb’s• Valuechangesfornegativenumbers
• Example1:– 4-bitvalue=0011=310– 8-bitzero-extendedvalue:00000011=310
• Example2:– 4-bitvalue=1011=-510– 8-bitzero-extendedvalue:00001011=1110
Zero-Extension
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NumberSystemComparison
NumberSystem RangeUnsigned [0,2N-1]
Sign/Magnitude [-(2N-1-1),2N-1-1]
Two’sComplement [-2N-1,2N-1-1]
Forexample,4-bitrepresentation:
-8
1000 1001
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1010 1011 1100 1101 1110 1111 0000 0001 0010 0011 0100 0101 0110 0111 Two's Complement
10001001101010111100110111101111
00000001 0010 0011 0100 0101 0110 0111
1000 1001 1010 1011 1100 1101 1110 11110000 0001 0010 0011 0100 0101 0110 0111
Sign/Magnitude
Unsigned
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