Lecture 01 Number Systems and Conversion PDF
| Title | Lecture 01 Number Systems and Conversion |
|---|---|
| Author | Samuel Okei |
| Course | Modern Digital System Design |
| Institution | Texas Tech University |
| Pages | 47 |
| File Size | 3.6 MB |
| File Type | |
| Total Downloads | 102 |
| Total Views | 144 |
Summary
Download Lecture 01 Number Systems and Conversion PDF
Description
WELCOMETO ModernDigitalSystemDesign ECE 2372 / Fall TexasTechUniversity Dr.Tooraj Nikoubin
2018 / Lecture 01 Introduction, NumberSystemsandConversion
Grading Scheme Course Requirements and Corresponding Weight 1 Test # 1
15%
2 Test # 2 3 Final exam
15% 30%
4 Project 5 Homework and Quiz
20% 20% Bonus?%
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Grading and Scheme
Bonus
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Class attending policy
1. 2. 3. 4. 5. 6. 7.
Efficientstudyandclassattention Cellphone&Laptop? Classactivities(Quiz,Present&absence) Politeorshy? Challengeorstress? WhoisBrave? SampleofTests
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Main Sources • 1-M.M. Mano and C.R. Kime,
"Logic and Computer Design Fundamentals" 4 th Edition, Pearson -Prentice Hall.
• 2-Charles H.Roth, Jr. and Larry L. Kinney,
" FundamentalsofLogicDesign"
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Other References 1. Victor P. Nelson, H. Troy Nagle, Bill D. Carroll, David Irwin “DigitalLogicCircuitAnalysisandDesign ” 2. M. Mano, “DigitalDesign”, 3rd Edition, Prentice Hall, UpperSaddle River, New Jersey, 2002 3. Nazeib M. Botros, “HDLprogrammingFundamentalsVHDLandVerilog” 4. Stephen Brown and Zvonko Vranesic, “FundamentalsofDigitalLogicwith Verilog Design” , McGraw-Hill, 2003 Dr.ToorajNikoubin/Fall2018/Lecture1/Introduction
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Main Sources for the test 1-PowerPointslides 2-Charles H.Roth, Jr. and Larry L. Kinney, " FundamentalsofLogicDesign" 3-M.M. Mano and C.R. Kime, "Logic and Computer Design Fundamentals" 4 th Edition, Pearson -Prentice Hall. --------------------------------4.Homework --------------------------------5.Quiz --------------------------------6.Sampleoftest
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Topic Lectures Introduction,NumberSystemsandConversion 1L LogicGatesandBooleanAlgebra 1L ApplicationsofBooleanAlgebra,MintermandMaxtermExpansions 2L Multi‐ LevelGateCircuits,NANDandNORGates 1L KarnaughMapsandQuine‐McCluskyMethodforsimplification 1L
1L CombinationalCircuitDesign, 1L Multiplexers,Decoders,ROMandProgrammablelogicDevices 2L AddandSub,Adders,SubtractorsandComparators 1L Coding 1L Hardwaredescriptionlanguageforcombinationalcircuits 1L
1L LatchandFlip‐Flops 2L RegistersandBuffers 1L CountersandCountercircuitdesign 2L AnalysisofclockedSequentialCircuits 2L SequentialCircuitDesign 2L Statemachinedesign 2L ArithmeticCircuits 1L Hardwaredescriptionlanguageforsequentialcircuits 1L
1L L=75minTotal 28L
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ECE 2372 ( Modern Digital System Design ) TA: TBA Tutors: TBA
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Monday Tuesday Wednesday
Email: Office Hours:
Thursday Friday Saturday Sunday
Office : ECE Computer Lab Dr.ToorajNikoubin/Fall2018/Lecture1/Introduction
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DigitalandAnalogSignals Analogsignal WithInfinitepossiblevalues voltage on a wire created by microphone
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Digitalsignal WithFinitepossiblevalues button pressed a keypad
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ExampleofDigitizationBenefit • Analogsignal(e.g.,audio)maylosequality • Voltagelevelsnotsaved/copied/transmittedperfectly
• Digitizedversionenables near‐perfectsave/copy/turn.
• “Sample”voltageatparticularrate, savesampleusingbitencoding
• Voltagelevelsstillnot keptperfectlyButwe candistinguish0s from1s
Letbitencodingbe: 1V:“01” 2V:“10” 3V:“11” Dr.ToorajNikoubin/Fall2018/Lecture1/Introduction
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ASampleofDigitalBoard
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ASampleofDigitalBoard
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DigitalDesign • Whatisdigital? – Digital camera, Digital TV, Digital Watch, Digital Radio, Digital City (e-city), Digital Photo Frame …etc – Which gives the things in countable form – Scene (analog) to Image (digital)
• Whydigital? – Countable form, makes easy to manage – Easy management makes more useful and versatile
• Whatdigit? – How to count: Decimal digit: 0 to 9 Dr.ToorajNikoubin/Fall2018/Lecture1/Introduction
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WhatDigit?=>NumberSystem • Famous Number System: Dec, Rom, Bin • Decimal System: 0 -9 – May evolves: because human have 10 finger • Roman System – May evolves to make easy to look and feel – Pre/Post Concept: (IV, V & VI) is (5-1, 5 & 5+1) • Binary System, Others (Oct, Hex) – One can cut an apple in to two Dr.ToorajNikoubin/Fall2018/Lecture1/Introduction
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Design&LogicDesign • Whatisdesign?
– Given problem spec, solve it with available components – While meeting quantitative (size, cost, power) and qualitative (beauty, elegance)
• Whatislogicdesign? – Choose digital logic components to perform specified control, data manipulation, or communication function and their interconnection – Which logic components to choose? Many implementation technologies (fixed-function components, programmabledevices, individual transistors on a chip, etc.) – Design optimized/transformed to meet design constraints Dr.ToorajNikoubin/Fall2018/Lecture1/Introduction
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Digital Signals with Only Two Values: Binary • Binary digital signal -- only two possible values – Typically represented as 0 and 1 One Binary digit is BIT value – We’ll only consider binarydigital signals – Binary is popular because • Transistors, the basic digital electric component, operate at two states (switch on and switch off ) • Storing/transmitting one of twovalues is easier than three or more (e.g., loud beep or quiet beep, reflection or no reflection) Dr.ToorajNikoubin/Fall2018/Lecture1/Introduction
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Number Systems and Conversion
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Decimal
Binary
Octal
Hexadecimal
0
00000
0
0
1
00001
1
1
2
00010
2
2
3
00011
3
3
4
00100
4
4
5
00101
5
5
6
00110
6
6
7
00111
7
7
8
01000
10
8
9
01001
11
9
10
01010
12
A
11
01011
13
B
12
01100
14
C
13
01101
15
D
14
01110
16
E
15
01111
17
F
16
10000
20
10
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Outline • Number System Decimal, Binary, Octal, Hex • Conversion (one to another) Decimal to Binary, Octal, Hex & Vice Versa Binary to HEX & vice versa • Other representation Signed, Unsigned, Complement
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Significant Digits
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Decimal (base 10)
• Uses positional representation • Each digit corresponds to a power of 10 based on its position in the number • The powers of 10 increment from 0, 1, 2, etc. as you move right to left Dr.ToorajNikoubin/Fall2018/Lecture1/Introduction
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Binary ( Base 2) • Two digits: 0, 1 • To make the binary numbers more readable, the digits are often put in groups of 4
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Binary to Decimal
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How to Encode Numbers: Binary Numbers Working with binary numbers In base ten, helps to know powers of 10 one, ten, hundred, thousand, ten thousand, ...
In base two, helps to know powers of 2 one, two, four, eight, sixteen, thirty two, sixty four, one hundred twenty eight - Count up by powers of two
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Octal (base 8) • Shorter & easier to read than binary • 8 digits: 0, 1, 2, 3, 4, 5, 6, 7, • Octal numbers
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Hexadecimal (base 16) • Shorter & easier to read than binary • 16 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F • “0x” often precedes hexadecimal numbers
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Fractional Number Point: Decimal Point, Binary Point, Hexadecimal point Decimal 247.75 = 2x102 +4x10 +7x100 +7x10−1 +5x10−2 Binary 10.101= 1x21 +0x20 +1x2−1 +0x2−2 +1x2−3 Hexadecimal 6A.7D=6x161 +10x160 +7x16−1 +Dx16−2 Dr.ToorajNikoubin/Fall2018/Lecture1/Introduction
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Converting To and From Decimal
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Decimal ↔ Binary Successive Division
Decimal(Base10)
Binary(Base2)
a) Divide the decimal number by 2; the remainder is the LSB of the binarynumber. b) If the quotation is zero, the conversion is complete. Otherwise repeat step (a) using the quotation as the decimal number. The new remainder is the next most significant bit of the binary number.
Binary(Base2)
Weighted Multiplication
Decimal(Base10)
a) Multiply each bit of the binary number by its corresponding bit- Multiplication weighting factor (i.e., Bit-0→2 =1; Bit-1→𝟐𝟏 =2; Bit-2→𝟐𝟐 =4; etc). b) Sum up all of the products in step (a) to get the decimal number.
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A3 A2 A1 + B3 B2 B1
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Decimal to Binary : Subtraction Method
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Decimal to Binary : Subtraction Method Examples: 39, 27, 18, 7
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Decimal to Binary : Division Method • Good for computer: Divide decimal number by 2 and insert remainder into new binary number. – Continue dividing quotient by 2 until the quotient is 0. • Example: Convert decimal number 12 to binary 12 div 2 = ( Quo=6 , Rem=0) LSB 6 div 2 = (Quo=3, Rem=0) 3 div 2 = (Quo=1,Rem=1) 1 div 2 = ( Quo=0, Rem=1) MSB 12 = 1100 Dr.ToorajNikoubin/Fall2018/Lecture1/Introduction
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Conversions Process Decimal ↔ Base (n) Decimal(Base10)
Successive Division
AnyBase(Basen)
a) Divide the decimal number by n; the remainder is the LSB of the anybasenumber. b) If the quotation is zero, the conversion is complete. Otherwise repeat step (a) using the quotation as the decimal number. The new remainder is the next most significant bit of the anybase number. Weighted
AnyBase(Basen)Multiplication
Decimal(Base10)
a) Multiply each bit of the anybase number by its corresponding bit- Multiplication weighting factor (i.e., Bit-0→𝑛 =1; Bit-1→𝒏𝟏 =n; Bit-2→𝒏𝟐 =4; etc). b) Sum up all of the products in step (a) to get the decimal number. Dr.ToorajNikoubin/Fall2018/Lecture1/Introduction
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Decimal ↔ Octal Conversion The Process: Successive Division • Divide number by 8; R is the LSB of the octalnumber • While Q is 0 • Using the Q as the decimal number. • New remainder is MSB of the octalnumber.
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Decimal ↔ Hexadecimal Conversion The Process: Successive Division • Divide number by 16; R is the LSB of the hexnumber • While Q is 0 • Using the Q as the decimal number. • New remainder is MSB of the hexnumber.
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Example: Hex → Octal Example: Convertthehexadecimalnumber5AHintoitsoctalequivalent. Solution: Firstconvertthehexadecimalnumberintoitsdecimalequivalentthen convertthedecimalnumberintoitsoctalequivalent.
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Example:Octal→Binary Example: Convert the octal number 132 into its binary equivalent. Solution: First convert the octal number into its decimal equivalent, then convert the decimal number into its binary equivalent.
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Binary ↔ Octal ↔ Hex Shortcut • Relation • Binary, octal, and hex number systems • All powers of two • Exploit (This Relation) •Make conversion easier.
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Substitution Code
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Substitution Code
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Other Representation Signed&UnsignedNumber • Signednumberlastbit(oneMSB)issignedbit Assume: 8 bit number Unsigned 12 : 0000 1100 Signed +12 : 0000 1100 Signed -12 : 1000 1100 • Complementnumber Unsigned binary 12 = 00001100 1’s Complement of 12 = 1111 0011 Dr.ToorajNikoubin/Fall2018/Lecture1/Introduction
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Example # 1: Convert
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to decimal.
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Example # 2:
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ThankYou...
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