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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