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Chapter 1Chapter 1
Introduction to Digital Systems DesignIntroduction to Digital Systems Design
ECEn 224ECEn 224
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Digital vs. AnalogDigital vs. Analog
Analog systems represent informationAnalog systems represent information
using physical quantitiesusing physical quantities
Voltage on a wire, magnetic field strengthVoltage on a wire, magnetic field strength Digital systems represent informationDigital systems represent information
using binary digits, orusing binary digits, orbitsbits
1 or 0, high or low, on or off1 or 0, high or low, on or off
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Positional Number SystemsPositional Number Systems
Two discrete values are insufficient forTwo discrete values are insufficient for
most applicationsmost applications
We combine bits to represent more valuesWe combine bits to represent more values We use a positional number system forWe use a positional number system for
binary just like we do in decimalbinary just like we do in decimal
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Positional Number SystemsPositional Number Systems
DecimalDecimal, base 10, base 10,, means we have 10 digits (0means we have 10 digits (0--9)9)
Decimal example:Decimal example:
103210321010 = 1= 1x10x1033 + 0+ 0x10x1022 +3+3x10x1011 + 2+ 2x10x1000
HexadecimalHexadecimal, base 16, means we have 16 digits (0, base 16, means we have 16 digits (0--9, A9, A--F)F)
Hexadecimal example:Hexadecimal example:
2A52A51616 = 2= 2x16x1622 + 10+ 10x16x1611 + 5+ 5x16x1600 = 512 + 160 + 5 = 677= 512 + 160 + 5 = 677
BinaryBinary, base 2, follows the same pattern, base 2, follows the same pattern
Binary example:Binary example:
1011101122 = 1= 1x2x233
+ 0+ 0x2x222 + 1+ 1x2x211 + 1+ 1x2x200 = 8 + 0 + 2 + 1 = 11= 8 + 0 + 2 + 1 = 11
Counting in any base is analogous to counting in decimalCounting in any base is analogous to counting in decimal
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Digital vs. AnalogDigital vs. Analog
Analog thermometerAnalog thermometer
0V to 10V, could be used to represent 00V to 10V, could be used to represent 0 toto
100100 FF
Each 1/10Each 1/10thth volt represents 1 degreevolt represents 1 degree
Digital thermometerDigital thermometer
77--bit binary number could be used tobit binary number could be used to
representrepresent 00 to 127to 127 FF
Seven bits can be used to represent theSeven bits can be used to represent the
numbers 0 to 127numbers 0 to 127
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Digital PrecisionDigital Precision
How would you represent 10.5How would you represent 10.5 F?F?
Analog example: 1.05VAnalog example: 1.05V
Digital example: ????Digital example: ???? 0001010000101022 = 10= 101010
0001011000101122 = 11= 111010
We must either add bits or decrease theWe must either add bits or decrease therangerange
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Digital PrecisionDigital Precision
99--bit thermometer, 0bit thermometer, 0 to 127.75to 127.75 FF Each discrete number increase represents 0.25Each discrete number increase represents 0.25 FF
10.510.5 FF 10.5/0.25 = 42 = 10101010.5/0.25 = 42 = 10101022
77--bit thermometer, 0bit thermometer, 0 to 12.7to 12.7 FF Each discrete number represents 12.7Each discrete number represents 12.7/127 = 0.1/127 = 0.1 FF
10.510.5 FF 10.5/0.1 = 105 = 110100110.5/0.1 = 105 = 110100122
It is not possible to representIt is not possible to represent allallvalues exactlyvalues exactly
using digital representationusing digital representation Example: 1/3 cant be represented in binary, just likeExample: 1/3 cant be represented in binary, just like
it cant be represented in decimalit cant be represented in decimal
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Example: Analog PhotographyExample: Analog Photography
An analog camera uses a chemicalAn analog camera uses a chemical
reaction in the film when exposed to lightreaction in the film when exposed to light
The amount of exposure is directly relatedThe amount of exposure is directly relatedto the amount of light that hits the filmto the amount of light that hits the film
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Example: Digital PhotographyExample: Digital Photography
A digital camera uses an array of lightA digital camera uses an array of light--
sensitive receptors that measure the lightsensitive receptors that measure the light
as a binary numberas a binary number
Image quality is determined mostly by twoImage quality is determined mostly by two
factors:factors:
The number of bits per pixelThe number of bits per pixel
The number of pixels per imageThe number of pixels per image
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Example: Digital PhotographyExample: Digital Photography
1284x897 pixels, 24-bit color
1284x897 pixels, 6-bit color
100x70 pixels, 24-bit color
100x70 pixels, 6-bit color
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Analog vs. Digital StorageAnalog vs. Digital Storage
Analog storage mediums fade over time due toAnalog storage mediums fade over time due to
gradual physical degradationgradual physical degradation
Photos turn yellow with timePhotos turn yellow with time
Cassette audio tapes lose their clarityCassette audio tapes lose their clarity
Digital storage mediums dont fade like analogDigital storage mediums dont fade like analog
If a 0 or 1 fades it will still be a 0 or 1If a 0 or 1 fades it will still be a 0 or 1
A .jpg image taken 10 years ago isA .jpg image taken 10 years ago is exactlyexactlythethesame todaysame today
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Analog vs. Digital StorageAnalog vs. Digital Storage
Making an analog copy implies measuring theMaking an analog copy implies measuring the
storage mediumstorage medium
Always introduces some errorsAlways introduces some errors
Copies of copies are even worseCopies of copies are even worse
Making digital copies implies distinguishing 0sMaking digital copies implies distinguishing 0s
from 1s so copies are exactfrom 1s so copies are exact
Copies can be made without any errorCopies can be made without any error Copies of copies are identicalCopies of copies are identical
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Analog vs. Digital ProcessingAnalog vs. Digital Processing
Modern computers and digital circuitsModern computers and digital circuits
make it easy to do extremely complexmake it easy to do extremely complex
processingprocessing
Digital processing allows precision andDigital processing allows precision and
error to be exactly predictederror to be exactly predicted
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Combinational vs. Sequential CircuitsCombinational vs. Sequential Circuits
Digital circuits consist of binary inputs andDigital circuits consist of binary inputs andoutputsoutputs
In combinational circuits, the output is a directIn combinational circuits, the output is a directfunction of its inputsfunction of its inputs
In sequential circuits the output depends on theIn sequential circuits the output depends on thecurrent input and previous inputscurrent input and previous inputs Sequential circuits contain memory that tracks stateSequential circuits contain memory that tracks state
A clock is used to signal when to change statesA clock is used to signal when to change states
AB
CD
XY
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Combinational vs. SequentialCombinational vs. Sequential
Combinational example: Clock chimeCombinational example: Clock chime
Inputs:Inputs: secondsseconds andand minutesminutes
Output:Output: chimechime
Behavior: OutputBehavior: Output chimechime is 1 if and only ifis 1 if and only ifsecondsseconds isis0 and0 and minutesminutes is 0, 15, 30, or45.is 0, 15, 30, or45.
Sequential example: CounterSequential example: Counter
Inputs:Inputs: incinc andand clkclk
Output:Output: countcount
Behavior: Increment count onBehavior: Increment count on clkclk edge if and only ifedge if and only if
incinc is 1is 1