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 information Analog systems represent information using physical quantitiesusing physical quantities– Voltage on a wire, magnetic field strengthVoltage on a wire, magnetic field strength

• Digital systems represent information Digital systems represent information using binary digits, or using binary digits, or bitsbits– 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 for Two 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 for We 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 (0-9)means we have 10 digits (0-9)• Decimal example:Decimal example:

103210321010 = 1 = 1x10x1033 + 0 + 0x10x1022 + 3 + 3x10x1011 + 2 + 2x10x1000

• HexadecimalHexadecimal, base 16, means we have 16 digits (0-9, A-F), base 16, means we have 16 digits (0-9, A-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° to ° to

100° F100° F– Each 1/10Each 1/10thth volt represents 1 degree volt represents 1 degree

• Digital thermometerDigital thermometer– 7-bit binary number could be used to 7-bit binary number could be used to

represent represent 00° to 127° F° to 127° F– Seven bits can be used to represent the Seven 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 the We must either add bits or decrease the rangerange

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Digital PrecisionDigital Precision

• 9-bit thermometer, 09-bit thermometer, 0° to 127.75° F° to 127.75° F– Each discrete number increase represents 0.25Each discrete number increase represents 0.25°° F F– 10.510.5° F ° F 10.5/0.25 = 42 = 101010 10.5/0.25 = 42 = 10101022

• 7-bit thermometer, 07-bit thermometer, 0° to 12.7° F° to 12.7° F– Each discrete number represents 12.7°/127 = 0.1° FEach discrete number represents 12.7°/127 = 0.1° F– 10.510.5° F ° F 10.5/0.1 = 105 = 1101001 10.5/0.1 = 105 = 110100122

• It is not possible to represent It is not possible to represent allall values exactly values exactly using digital representationusing digital representation– Example: 1/3 can’t be represented in binary, just like it Example: 1/3 can’t be represented in binary, just like it

can’t be represented in decimalcan’t be represented in decimal

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Example: Analog PhotographyExample: Analog Photography

• An analog camera uses a chemical An 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 related The amount of exposure is directly related to 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 light-A digital camera uses an array of light-sensitive receptors that measure the light sensitive receptors that measure the light as a binary numberas a binary number

• Image quality is determined mostly by two Image 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 to Analog 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 don’t “fade” like analogDigital storage mediums don’t “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 is A .jpg image taken 10 years ago is exactlyexactly the the

same todaysame today

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Analog vs. Digital StorageAnalog vs. Digital Storage

• Making an analog copy implies measuring the Making 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 0’s Making digital copies implies distinguishing 0’s from 1’s so copies are exactfrom 1’s 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 circuits Modern computers and digital circuits make it easy to do extremely complex make it easy to do extremely complex processingprocessing

• Digital processing allows precision and Digital 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 and Digital circuits consist of binary inputs and outputsoutputs

• In combinational circuits, the output is a direct In combinational circuits, the output is a direct function of its inputsfunction of its inputs

• In sequential circuits the output depends on the In sequential circuits the output depends on the current 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

ABCD

XY

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Combinational vs. SequentialCombinational vs. Sequential

• Combinational example: Clock chimeCombinational example: Clock chime– Inputs: Inputs: secondsseconds and and minutesminutes– Output: Output: chimechime– Behavior: Output Behavior: Output chimechime is 1 if and only if is 1 if and only if secondsseconds is is

0 and 0 and minutesminutes is 0, 15, 30, or 45. is 0, 15, 30, or 45.

• Sequential example: CounterSequential example: Counter– Inputs: Inputs: incinc and and clkclk– Output: Output: countcount– Behavior: Increment count on Behavior: Increment count on clkclk edge if and only if edge if and only if incinc is 1 is 1