Lec 2: Gates and Logic


 Tracey Manning
 2 years ago
 Views:
Transcription
1 Lec 2: Gtes nd Logic Kvit Bl CS 34, Fll 28 Computer Science Cornell University Announcements Clss newsgroup creted Posted on wepge Use it for prtner finding First ssignment is to find prtners Due this Fridy Sections strt this week
2 A switch A switch is simple device tht cn ct s conductor or isoltor Cn e used for mzing things Trnsistors Solidstte switch The most mzing invention of the 9s  + collector P P emitter N se PNP nd NPN se collector emitter PNP se emitter collector +  N P N NPN 2
3 P nd N Trnsistors PNP Trnsistor B E NPN Trnsistor B E C Connect E to C when se = C Connect E to C when se = Inverter in out Function: NOT Clled n inverter Symol: in out In Out Truth tle Useful for tking the inverse of n input CMOS: complementrysymmetry metl oxide semiconductor 3
4 NAND Gte Function: NAND Symol: out A B out +Vdd NOR Gte out Function: NOR Symol: Vss A B out out 4
5 NOT: Building Functions AND: OR: NAND nd NOR re universl Cn implement ny function with NAND or just NOR gtes useful for mnufcturing NOT: Building Functions AND: OR: NAND nd NOR re universl Cn implement ny function with NAND or just NOR gtes useful for mnufcturing 5
6 Logic Equtions AND out = = & = OR out = + = = NOT out = =! = Identities Identities useful for mnipulting logic equtions For optimiztion & ese of implementtion + = + = + = = = = (+c) = + c ( + ) = ( ) = + + = 6
7 Logic Mnipultion Cn specify functions y descriing gtes, truth tles or logic equtions Cn mnipulte logic equtions lgericlly Cn lso use truth tle to prove equivlence Exmple: (+)(+c) = + c c + +c LHS c RHS (+)(+c) = + + c + c = + (+c) + c = ( + (+c)) + c = + c Logic Minimiztion A common prolem is how to implement desired function most efficiently One cn derive the eqution from the truth tle c minterm for ll outputs c tht re, c c tke the corresponding c minterm c Otin the result in c sum of products form c c How does one find the most efficient eqution? Mnipulte lgericlly until stisfied Use Krnugh mps (or K mps) 7
8 Multiplexer A multiplexer selects etween multiple inputs out =, if d = out =, if d = d Build truth tle Minimize digrm Derive logic digrm Multiplexer Implementtion d d out Build truth tle = d + d + d + d = d + d 8
9 Multiplexer Implementtion Drw the circuit d d out out = d + d d out Logic Gtes One cn uy gtes seprtely ex. 74xxx series of integrted circuits cost ~$ per chip, mostly for pckging nd testing Cumersome, ut possile to uild devices using gtes put together mnully 9
10 Integrted Circuits Or one cn mnufcture complete design using custom msk Intel Pentium hs pproximtely 25 million trnsistors Voting mchine Build something interesting A voting mchine Elections re coming up! Assume: A vote is recorded on piece of pper, y punching out hole, there re t most 7 choices we will not worry out hnging chds or invlids
11 Voting mchine For now, let s just disply the numericl identifier to the llot supervisor we won t do counting yet, just decoding we cn use four photosensitive trnsistors to find out which hole is punched out A photosensitive trnsistor detects the presence of light Photosensitive mteril triggers the gte Bllot Reding Input: pper with hole in it Out: numer the llot supervisor cn record Bllots The 34 vote recording mchine
12 Encoders c 2 3 o d 4 o e 5 o 2.. A 3it encoder (7to3) (5 inputs shown) N sensors in row Wnt to distinguish which of the N sensors hs fired Wnt to represent the firing sensor numer in compct form N might e lrge Only one wire is on t ny time Silly to route N wires everywhere, etter to encode in log N wires Numer Representtions 37 Deciml numers re written in se 3 x + 7 x = 37 Just s esily use other ses Bse 2  Binry Bse 8  Octl Bse 6 Hexdeciml 2
13 Numer Representtions 37 Bse conversion vi repetitive division Divide y se, write reminder, move left with quotient Snity check with 37 nd Binry Representtion 37 =
14 Hexdeciml Representtion deciml = (25) 6 Convention Bse 6 is written with leding x 37 = x25 Need extr digits!,, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Binry to hexdeciml is esy Divide into groups of 4, trnslte groupwise into hex digits Encoder Truth Tle c d o2 o o 2 o o c 3 o 2 d 4 A 3it encoder with 4 inputs o2 = cd o = cd + cd o = cd + cd for simplicity 4
15 Bllot Reding Ok, we uilt first hlf of the mchine Need to disply the result Bllots The 34 voting mchine 7Segment LED Decoder 4 inputs encoded in inry 8 outputs, ech driving n independent, rectngulr LED Cn disply numers Just simple logic circuit Write the truth tle 5
16 7Segment LED Decoder 4 inputs encoded in inry 8 outputs, ech driving n independent, rectngulr LED Cn disply numers 7Segment LED Decoder 4 inputs encoded in inry 8 outputs, ech driving n independent, rectngulr LED Cn disply numers 6
17 7 7Segment Decoder Truth Tle o 5 i 2 o o 2 o 6 o 4 o 3 o i i i 3 o o o 2 o 3 o 4 o 5 o 6 Exercise: find the error(s) in this truth tle 7Segment Decoder Truth Tle o 5 i 2 o o 2 o 6 o 4 o 3 o i i i 3 o o o 2 o 3 o 4 o 5 o 6
18 Bllot Reding Done! Bllots The 34 voting mchine Summry We cn now implement ny logic circuit Cn do it efficiently, using Krnugh mps to find the miniml terms required Cn use either NAND or NOR gtes to implement the logic circuit Cn use P nd Ntrnsistors to implement NAND or NOR gtes 8
CS 316: Gates and Logic
CS 36: Gtes nd Logi Kvit Bl Fll 27 Computer Siene Cornell University Announements Clss newsgroup reted Posted on wepge Use it for prtner finding First ssignment is to find prtners P nd N Trnsistors PNP
More informationCS99S Laboratory 2 Preparation Copyright W. J. Dally 2001 October 1, 2001
CS99S Lortory 2 Preprtion Copyright W. J. Dlly 2 Octoer, 2 Ojectives:. Understnd the principle of sttic CMOS gte circuits 2. Build simple logic gtes from MOS trnsistors 3. Evlute these gtes to oserve logic
More informationBasics of Logic Design: Boolean Algebra, Logic Gates. Administrative
Bsics of Logic Design: Boolen Alger, Logic Gtes Computer Science 104 Administrtive Homework #3 Due Sundy Midterm I Mondy in clss, closed ook, closed notes Ø Will provide IA32 instruction set hndout Ø Lst
More informationTwo hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. Date: Friday 16 th May 2008. Time: 14:00 16:00
COMP20212 Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE Digitl Design Techniques Dte: Fridy 16 th My 2008 Time: 14:00 16:00 Plese nswer ny THREE Questions from the FOUR questions provided
More information! Transistors MOSFET. " Model. ! Zeroth order transistor model. " Good enough for [what?] ! How to construct static CMOS gates
ESE370: CircuitLevel Modeling, Design, nd Optimiztion or Digitl Systems Lec 2: Septemer 2, 2016 Trnsistor Introduction nd Gtes rom Trnsistors Tody! Trnsistors MOSFET " Model! Zeroth order trnsistor model
More informationBinary Representation of Numbers Autar Kaw
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse rel number to its binry representtion,. convert binry number to n equivlent bse number. In everydy
More informationGates and Logic: From switches to Transistors, Logic Gates and Logic Circuits
Gates and Logic: From switches to Transistors, Logic Gates and Logic Circuits Hakim Weatherspoon CS 3410, Spring 2013 Computer Science Cornell University See: P&H ppendix C.2 and C.3 (lso, see C.0 and
More informationLearning Outcomes. Computer Systems  Architecture Lecture 4  Boolean Logic. What is Logic? Boolean Logic 10/28/2010
/28/2 Lerning Outcomes At the end of this lecture you should: Computer Systems  Architecture Lecture 4  Boolen Logic Eddie Edwrds eedwrds@doc.ic.c.uk http://www.doc.ic.c.uk/~eedwrds/compsys (Hevily sed
More informationa. Implement the function using a minimal network of 2:4 decoders and OR gates.
CSE4 Eercise Solutions I. Given threeinput Boolen function f(,, c) = m(, 2, 4, 6, 7) + d().. Implement the function using miniml network of 2:4 decoders nd OR gtes. Stndrd solution: Use s the enle signl,
More informationAssuming all values are initially zero, what are the values of A and B after executing this Verilog code inside an always block? C=1; A <= C; B = C;
B26 Appendix B The Bsics of Logic Design Check Yourself ALU n [Arthritic Logic Unit or (rre) Arithmetic Logic Unit] A rndomnumer genertor supplied s stndrd with ll computer systems Stn KellyBootle,
More informationQuadratic Equations  1
Alger Module A60 Qudrtic Equtions  1 Copyright This puliction The Northern Alert Institute of Technology 00. All Rights Reserved. LAST REVISED Novemer, 008 Qudrtic Equtions  1 Sttement of Prerequisite
More informationSquare Roots Teacher Notes
Henri Picciotto Squre Roots Techer Notes This unit is intended to help students develop n understnding of squre roots from visul / geometric point of view, nd lso to develop their numer sense round this
More informationHomework 3 Solutions
CS 341: Foundtions of Computer Science II Prof. Mrvin Nkym Homework 3 Solutions 1. Give NFAs with the specified numer of sttes recognizing ech of the following lnguges. In ll cses, the lphet is Σ = {,1}.
More informationChapter 1 Introduction to CMOS Circuit Design
Chpter 1 Introduction to CMOS Circuit Design JinFu Li Advnced Relile Systems (ARES) L. Deprtment of Electricl Engineering Ntionl Centrl University Jhongli, Tiwn Outline Introduction MOS Trnsistor Switches
More informationLecture 15  Curve Fitting Techniques
Lecture 15  Curve Fitting Techniques Topics curve fitting motivtion liner regression Curve fitting  motivtion For root finding, we used given function to identify where it crossed zero where does fx
More informationReasoning to Solve Equations and Inequalities
Lesson4 Resoning to Solve Equtions nd Inequlities In erlier work in this unit, you modeled situtions with severl vriles nd equtions. For exmple, suppose you were given usiness plns for concert showing
More information10.5 Graphing Quadratic Functions
0.5 Grphing Qudrtic Functions Now tht we cn solve qudrtic equtions, we wnt to lern how to grph the function ssocited with the qudrtic eqution. We cll this the qudrtic function. Grphs of Qudrtic Functions
More informationUse Geometry Expressions to create a more complex locus of points. Find evidence for equivalence using Geometry Expressions.
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
More informationGates, Circuits, and Boolean Algebra
Gates, Circuits, and Boolean Algebra Computers and Electricity A gate is a device that performs a basic operation on electrical signals Gates are combined into circuits to perform more complicated tasks
More informationBinary Golomb Codes. CSE 589 Applied Algorithms Autumn Constructing a Binary Golomb Code. Example. Example. Comparison of GC with Entropy
CSE 589 pplied lgorithms utumn Golom Coding rithmetic Coding LZW Sequitur Binry Golom Codes Binry source with s much more frequent thn s. Vriletovrile length code Prefix code. Golom code of order 4 input
More informationEquations between labeled directed graphs
Equtions etween leled directed grphs Existence of solutions GrretFontelles A., Misnikov A., Ventur E. My 2013 Motivtionl prolem H 1 nd H 2 two sugroups of the free group generted y X A, F (X, A). H 1
More informationVariable Dry Run (for Python)
Vrile Dr Run (for Pthon) Age group: Ailities ssumed: Time: Size of group: Focus Vriles Assignment Sequencing Progrmming 7 dult Ver simple progrmming, sic understnding of ssignment nd vriles 2050 minutes
More informationGeneralized Inverses: How to Invert a NonInvertible Matrix
Generlized Inverses: How to Invert NonInvertible Mtrix S. Swyer September 7, 2006 rev August 6, 2008. Introduction nd Definition. Let A be generl m n mtrix. Then nturl question is when we cn solve Ax
More informationContent Objectives: After completing the activity, students will gain experience of informally proving Pythagoras Theorem
Pythgors Theorem S Topic 1 Level: Key Stge 3 Dimension: Mesures, Shpe nd Spce Module: Lerning Geometry through Deductive Approch Unit: Pythgors Theorem Student ility: Averge Content Ojectives: After completing
More informationAppendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
More information4 Geometry: Shapes. 4.1 Circumference and area of a circle. FM Functional Maths AU (AO2) Assessing Understanding PS (AO3) Problem Solving HOMEWORK 4A
Geometry: Shpes. Circumference nd re of circle HOMEWORK D C 3 5 6 7 8 9 0 3 U Find the circumference of ech of the following circles, round off your nswers to dp. Dimeter 3 cm Rdius c Rdius 8 m d Dimeter
More informationWarmup for Differential Calculus
Summer Assignment Wrmup for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
More informationChapter 9: Quadratic Equations
Chpter 9: Qudrtic Equtions QUADRATIC EQUATIONS DEFINITION + + c = 0,, c re constnts (generlly integers) ROOTS Synonyms: Solutions or Zeros Cn hve 0, 1, or rel roots Consider the grph of qudrtic equtions.
More informationLogical Design. Design with Basic Logic Gates
Logicl Design Zvi Kohvi nd Nirj K. Jh Design ith Bsic Logic Gtes Logic gtes: perform logicl opertions on input signls Positive (negtive) logic polrit: constnt () denotes high voltge nd constnt lo (high)
More information5.2. LINE INTEGRALS 265. Let us quickly review the kind of integrals we have studied so far before we introduce a new one.
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous relvlued
More informationWords Symbols Diagram. abcde. a + b + c + d + e
Logi Gtes nd Properties We will e using logil opertions to uild mhines tht n do rithmeti lultions. It s useful to think of these opertions s si omponents tht n e hooked together into omplex networks. To
More informationSirindhorn International Institute of Technology Thammasat University at Rangsit
Sirindhorn Interntionl Institute of Technology Thmmst University t Rngsit School of Informtion, Computer nd Communiction Technology COURSE : ECS 204 Bsic Electricl Engineering L INSTRUCTOR : Asst. Prof.
More informationDigital Electronics Basics: Combinational Logic
Digitl Eletronis Bsis: for Bsi Eletronis http://ktse.eie.polyu.edu.hk/eie29 by Prof. Mihel Tse Jnury 25 Digitl versus nlog So fr, our disussion bout eletronis hs been predominntly nlog, whih is onerned
More informationIn the following there are presented four different kinds of simulation games for a given Büchi automaton A = :
Simultion Gmes Motivtion There re t lest two distinct purposes for which it is useful to compute simultion reltionships etween the sttes of utomt. Firstly, with the use of simultion reltions it is possile
More informationDIGITAL CIRCUITS EXAMPLES (questions and solutions) Dr. N. AYDIN
DIGITAL CIRCUITS EXAMPLES (questions nd solutions) Dr. N. AYDIN nydin@yildiz.edu.tr Emple. Assume tht the inverter in the network elow hs propgtion dely of 5 ns nd the AND gte hs propgtion dely of ns.
More informationUnion, Intersection and Complement. Formal Foundations Computer Theory
Union, Intersection nd Complement FAs Union, Intersection nd Complement FAs Forml Foundtions Computer Theory Ferury 21, 2013 This hndout shows (y exmples) how to construct FAs for the union, intersection
More informationRegular Sets and Expressions
Regulr Sets nd Expressions Finite utomt re importnt in science, mthemtics, nd engineering. Engineers like them ecuse they re super models for circuits (And, since the dvent of VLSI systems sometimes finite
More informationPROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
More information1. True or False? A voltage level in the range 0 to 2 volts is interpreted as a binary 1.
File: chap04, Chapter 04 1. True or False? A voltage level in the range 0 to 2 volts is interpreted as a binary 1. 2. True or False? A gate is a device that accepts a single input signal and produces one
More informationOne Minute To Learn Programming: Finite Automata
Gret Theoreticl Ides In Computer Science Steven Rudich CS 15251 Spring 2005 Lecture 9 Fe 8 2005 Crnegie Mellon University One Minute To Lern Progrmming: Finite Automt Let me tech you progrmming lnguge
More informationVersion 001 CIRCUITS holland (1290) 1
Version CRCUTS hollnd (9) This printout should hve questions Multiplechoice questions my continue on the next column or pge find ll choices efore nswering AP M 99 MC points The power dissipted in wire
More informationExample A rectangular box without lid is to be made from a square cardboard of sides 18 cm by cutting equal squares from each corner and then folding
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
More informationCMOS Logic Building Blocks
Chpter 5 CMOS Logic Building Blocks In this chpter we discuss structures, lyout nd trnsient properties of sic CMOS logic uilding locks. These locks come into two groups referred to s gte logic nd switch
More informationSection 54 Trigonometric Functions
5 Trigonometric Functions Section 5 Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
More informationStart Here. IMPORTANT: To ensure that the software is installed correctly, do not connect the USB cable until step 17. Remove tape and cardboard
Strt Here 1 IMPORTANT: To ensure tht the softwre is instlled correctly, do not connect the USB cle until step 17. Follow the steps in order. If you hve prolems during setup, see Trouleshooting in the lst
More informationGeometry 71 Geometric Mean and the Pythagorean Theorem
Geometry 71 Geometric Men nd the Pythgoren Theorem. Geometric Men 1. Def: The geometric men etween two positive numers nd is the positive numer x where: = x. x Ex 1: Find the geometric men etween the
More informationBoolean Logic. Building a Modern Computer From First Principles.
Boolen Logic Building Modern Computer From First Principles www.nnd2tetris.org Elements of Computing Systems, Nisn & Schocken, MIT Press, www.nnd2tetris.org, Chpter 1: Boolen Logic slide 1 Boolen lger
More informationEQUATIONS OF LINES AND PLANES
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in pointdirection nd twopoint
More informationSect 8.3 Triangles and Hexagons
13 Objective 1: Sect 8.3 Tringles nd Hexgons Understnding nd Clssifying Different Types of Polygons. A Polygon is closed twodimensionl geometric figure consisting of t lest three line segments for its
More informationExperiment 6: Friction
Experiment 6: Friction In previous lbs we studied Newton s lws in n idel setting, tht is, one where friction nd ir resistnce were ignored. However, from our everydy experience with motion, we know tht
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationIntroduction to Logic Gates. ESDI Lesson 2. Logic Gates. Logic Gates: The Inverter. Logic Gates. Gate Symbols. The Inverter
Introduction to Logic Gtes SI Lesson 2 Logic Gtes Logic gtes rwing Logic ircuit nlzing Logic ircuit oolen lger Universl Gtes: NN nd NOR Implementtion using NN or NOR Gtes Positive nd Negtive Logic Implementtion
More informationIn this section make precise the idea of a matrix inverse and develop a method to find the inverse of a given square matrix when it exists.
Mth 52 Sec S060/S0602 Notes Mtrices IV 5 Inverse Mtrices 5 Introduction In our erlier work on mtrix multipliction, we sw the ide of the inverse of mtrix Tht is, for squre mtrix A, there my exist mtrix
More informationMaths Assessment Year 4: Number and Place Value
Nme: Mths Assessment Yer 4: Numer nd Plce Vlue 1. Count in multiples of 6, 7, 9, 25 nd 1 000; find 1 000 more or less thn given numer. 2. Find 1,000 more or less thn given numer. 3. Count ckwrds through
More informationChapter 4. Gates and Circuits. Chapter Goals. Chapter Goals. Computers and Electricity. Computers and Electricity. Gates
Chapter Goals Chapter 4 Gates and Circuits Identify the basic gates and describe the behavior of each Describe how gates are implemented using transistors Combine basic gates into circuits Describe the
More informationUniform convergence and its consequences
Uniform convergence nd its consequences The following issue is centrl in mthemtics: On some domin D, we hve sequence of functions {f n }. This mens tht we relly hve n uncountble set of ordinry sequences,
More informationSuffix Trees CMSC 423
Suffix Trees CMSC 423 Preprocessing Strings Over the next few lectures, we ll see severl methods for preprocessing string dt into dt structures tht mke mny questions (like serching) esy to nswer: Suffix
More informationBasic Math Review. Numbers. Important Properties. Absolute Value PROPERTIES OF ADDITION NATURAL NUMBERS {1, 2, 3, 4, 5, }
ƒ Bsic Mth Review Numers NATURAL NUMBERS {1,, 3, 4, 5, } WHOLE NUMBERS {0, 1,, 3, 4, } INTEGERS {, 3,, 1, 0, 1,, } The Numer Line 5 4 3 1 0 1 3 4 5 Negtive integers Positive integers RATIONAL NUMBERS All
More informationl What have discussed up until now & why: l C Programming language l More lowlevel then Java. l Better idea about what s really going on.
CS211 Computer Architecture l Topics Digital Logic l Transistors (Design & Types) l Logic Gates l Combinational Circuits l KMaps Class Checkpoint l What have discussed up until now & why: l C Programming
More information1.2 The Integers and Rational Numbers
.2. THE INTEGERS AND RATIONAL NUMBERS.2 The Integers n Rtionl Numers The elements of the set of integers: consist of three types of numers: Z {..., 5, 4, 3, 2,, 0,, 2, 3, 4, 5,...} I. The (positive) nturl
More informationRotational Equilibrium: A Question of Balance
Prt of the IEEE Techer InService Progrm  Lesson Focus Demonstrte the concept of rottionl equilirium. Lesson Synopsis The Rottionl Equilirium ctivity encourges students to explore the sic concepts of
More informationMA 15800 Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!
MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more
More informationAPPLICATION NOTE Revision 3.0 MTD/PS0534 August 13, 2008 KODAK IMAGE SENDORS COLOR CORRECTION FOR IMAGE SENSORS
APPLICATION NOTE Revision 3.0 MTD/PS0534 August 13, 2008 KODAK IMAGE SENDORS COLOR CORRECTION FOR IMAGE SENSORS TABLE OF FIGURES Figure 1: Spectrl Response of CMOS Imge Sensor...3 Figure 2: Byer CFA Ptterns...4
More informationPerfect competition model (PCM)
18/9/21 Consumers: Benefits, WT, nd Demnd roducers: Costs nd Supply Aggregting individul curves erfect competition model (CM) Key ehviourl ssumption Economic gents, whether they e consumers or producers,
More informationA Note on Complement of Trapezoidal Fuzzy Numbers Using the αcut Method
Interntionl Journl of Applictions of Fuzzy Sets nd Artificil Intelligence ISSN  Vol.  A Note on Complement of Trpezoidl Fuzzy Numers Using the αcut Method D. Stephen Dingr K. Jivgn PG nd Reserch Deprtment
More information. At first sight a! b seems an unwieldy formula but use of the following mnemonic will possibly help. a 1 a 2 a 3 a 1 a 2
7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6
More informationLines and Angles. 2. Straight line is a continuous set of points going on forever in both directions:
Lines nd Angles 1. Point shows position. A 2. Stright line is continuous set of points going on forever in oth directions: 3. Ry is line with one endpoint. The other goes on forever. G 4. Line segment
More informationVectors 2. 1. Recap of vectors
Vectors 2. Recp of vectors Vectors re directed line segments  they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
More information6.2 Volumes of Revolution: The Disk Method
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine soclled volumes of
More informationA Visual and Interactive Input abb Automata. Theory Course with JFLAP 4.0
Strt Puse Step Noninverted Tree A Visul nd Interctive Input Automt String ccepted! 5 nodes generted. Theory Course with JFLAP 4.0 q0 even 's, even 's q2 even 's, odd 's q1 odd 's, even 's q3 odd 's, odd
More informationMATH 150 HOMEWORK 4 SOLUTIONS
MATH 150 HOMEWORK 4 SOLUTIONS Section 1.8 Show tht the product of two of the numbers 65 1000 8 2001 + 3 177, 79 1212 9 2399 + 2 2001, nd 24 4493 5 8192 + 7 1777 is nonnegtive. Is your proof constructive
More informationA.7.1 Trigonometric interpretation of dot product... 324. A.7.2 Geometric interpretation of dot product... 324
A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................
More informationLINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES
LINEAR TRANSFORMATIONS AND THEIR REPRESENTING MATRICES DAVID WEBB CONTENTS Liner trnsformtions 2 The representing mtrix of liner trnsformtion 3 3 An ppliction: reflections in the plne 6 4 The lgebr of
More informationAlgorithms Chapter 4 Recurrences
Algorithms Chpter 4 Recurrences Outline The substitution method The recursion tree method The mster method Instructor: Ching Chi Lin 林清池助理教授 chingchilin@gmilcom Deprtment of Computer Science nd Engineering
More informationGraphs on Logarithmic and Semilogarithmic Paper
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
More informationFinite Automata. Informatics 2A: Lecture 3. John Longley. 25 September School of Informatics University of Edinburgh
Lnguges nd Automt Finite Automt Informtics 2A: Lecture 3 John Longley School of Informtics University of Edinburgh jrl@inf.ed.c.uk 25 September 2015 1 / 30 Lnguges nd Automt 1 Lnguges nd Automt Wht is
More informationFEASIBILITY OF USING PRESSED SUGAR CANE STALK FOR THE PRODUCTION OF CHARCOAL. D Ffoulkes, R Elliott and T R Preston
Trop Anim Prod 1980 5:2 125 FEASIBILITY OF USING PRESSED SUGAR CANE STALK FOR THE PRODUCTION OF CHARCOAL D Ffoulkes, R Elliott nd T R Preston Fcultd de Medicin Veterinri y Zootecni, University of Yuctn,
More informationLecture 5. Inner Product
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
More informationSolving Linear Equations  Formulas
1. Solving Liner Equtions  Formuls Ojective: Solve liner formuls for given vrile. Solving formuls is much like solving generl liner equtions. The only difference is we will hve severl vriles in the prolem
More informationQuadrilaterals Here are some examples using quadrilaterals
Qudrilterls Here re some exmples using qudrilterls Exmple 30: igonls of rhomus rhomus hs sides length nd one digonl length, wht is the length of the other digonl? 4  Exmple 31: igonls of prllelogrm Given
More information0.1 Basic Set Theory and Interval Notation
0.1 Bsic Set Theory nd Intervl Nottion 3 0.1 Bsic Set Theory nd Intervl Nottion 0.1.1 Some Bsic Set Theory Notions Like ll good Mth ooks, we egin with definition. Definition 0.1. A set is welldefined
More informationAREA OF A SURFACE OF REVOLUTION
AREA OF A SURFACE OF REVOLUTION h cut r πr h A surfce of revolution is formed when curve is rotted bout line. Such surfce is the lterl boundr of solid of revolution of the tpe discussed in Sections 7.
More informationVectors. The magnitude of a vector is its length, which can be determined by Pythagoras Theorem. The magnitude of a is written as a.
Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles
More information4.11 Inner Product Spaces
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
More informationBasic Analysis of Autarky and Free Trade Models
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
More informationIntroduction. Teacher s lesson notes The notes and examples are useful for new teachers and can form the basis of lesson plans.
Introduction Introduction The Key Stge 3 Mthemtics series covers the new Ntionl Curriculum for Mthemtics (SCAA: The Ntionl Curriculum Orders, DFE, Jnury 1995, 0 11 270894 3). Detiled curriculum references
More informationThe area of the larger square is: IF it s a right triangle, THEN + =
8.1 Pythgoren Theorem nd 2D Applitions The Pythgoren Theorem sttes tht IF tringle is right tringle, THEN the sum of the squres of the lengths of the legs equls the squre of the hypotenuse lengths. Tht
More informationLiveEngage Agent Guide. Version 1.0
Agent Guide Version 1.0 Tle of Contents Contents LOGGING IN... 3 NAVIGATING LIVEENGAGE... 4 Visitor List, Connection Are, Enggement Br & Going Online... 4 ENGAGEMENT OPTIONS... 5 Tking n Enggment... 5
More informationEquivalence Checking. Sean Weaver
Equivlene Cheking Sen Wever Equivlene Cheking Given two Boolen funtions, prove whether or not two they re funtionlly equivlent This tlk fouses speifilly on the mehnis of heking the equivlene of pirs of
More informationSCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics. Basic Algebra
SCHOOL OF ENGINEERING & BUILT ENVIRONMENT Mthemtics Bsic Alger. Opertions nd Epressions. Common Mistkes. Division of Algeric Epressions. Eponentil Functions nd Logrithms. Opertions nd their Inverses. Mnipulting
More informationand thus, they are similar. If k = 3 then the Jordan form of both matrices is
Homework ssignment 11 Section 7. pp. 24925 Exercise 1. Let N 1 nd N 2 be nilpotent mtrices over the field F. Prove tht N 1 nd N 2 re similr if nd only if they hve the sme miniml polynomil. Solution: If
More information5 a LAN 6 a gateway 7 a modem
STARTER With the help of this digrm, try to descrie the function of these components of typicl network system: 1 file server 2 ridge 3 router 4 ckone 5 LAN 6 gtewy 7 modem Another Novell LAN Router Internet
More informationwww.mathsbox.org.uk e.g. f(x) = x domain x 0 (cannot find the square root of negative values)
www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input
More informationDEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING
DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING Lab Manual Digital Electronics Laboratory (EC39) BACHELOR OF TECHNOLOGY Subject Code: EC 39 Subject Name: Digital Electronics Laboratory Teaching
More information11. Fourier series. sin mx cos nx dx = 0 for any m, n, sin 2 mx dx = π.
. Fourier series Summry of the bsic ides The following is quick summry of the introductory tretment of Fourier series in MATH. We consider function f with period π, tht is, stisfying f(x + π) = f(x) for
More informationBayesian Updating with Continuous Priors Class 13, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom
Byesin Updting with Continuous Priors Clss 3, 8.05, Spring 04 Jeremy Orloff nd Jonthn Bloom Lerning Gols. Understnd prmeterized fmily of distriutions s representing continuous rnge of hypotheses for the
More informationECE 109 Spring Launch Logisim by opening a terminal window and entering the command:
Problem Session 3 Overview Welcome to Logisim! Logisim is a logic simulator that allows you design and simulate digital circuit using a graphical user interface. It was written by Carl Burch. Logisim comes
More informationLecture 24: Laplace s Equation
Introductory lecture notes on Prtil Differentil Equtions  c Anthony Peirce. Not to e copied, used, or revised without explicit written permission from the copyright owner. 1 Lecture 24: Lplce s Eqution
More informationMathematics Higher Level
Mthemtics Higher Level Higher Mthemtics Exmintion Section : The Exmintion Mthemtics Higher Level. Structure of the exmintion pper The Higher Mthemtics Exmintion is divided into two ppers s detiled below:
More information