Tables of Common Transform Pairs


 Frederick McDaniel
 1 years ago
 Views:
Transcription
1 ble of Common rnform Pir 0 by Mrc Ph. Stoecklin mrc toecklin.net verion v.5.3 Engineer nd tudent in communiction nd mthemtic re confronted with tion uch the rnform, the ourier, or the plce. Often it i quite hrd to quickly find the pproprite in book or the Internet, much le to hve comprehenive overview of tion pir nd correponding propertie. In thi document I compiled hndy collection of the mot common pir nd propertie of the continuoutime frequency ourier (πf), continuoutime pultion ourier (), rnform, dicretetime ourier D, nd plce. Plee note tht, before including tion pir in the tble, I verified it correctne. Neverthele, it i till poible tht you my find error or typo. I m very grteful to everyone dropping me line nd pointing out ny concern or typo. Nottion, Convention, nd Ueful ormul Imginry unit j = Complex conjugte = + jb = jb Rel prt Re {f(t)} = [f(t) + f (t)] Imginry prt Im {f(t)} = [f(t) f (t)] j { Dirc delt/unit impule δ[n] =, n = 0 0, n 0 Heviide tep/unit tep u[n] = {, n 0 0, n < 0 Sine/Coine Sinc function in (x) = ejx e jx j inc (x) in(x) x co (x) = ejx +e jx (unnormlied) { Rectngulr function rect( t ) = if t 0 if t > ringulr function tring ( ) t = rect( t ) rect( t ) = t t 0 t > Convolution continuoutime: (f g)(t) = + f(τ) g (t τ)dτ Prevl theorem Geometric erie dicretetime: generl ttement: continuoutime: dicretetime: k=0 xk = x (u v)[n] = m= u[m] v [n m] + f(t)g (t)dt = + (f)g (f)df + f(t) dt = + (f) df + n= x[n] = +π π π X(ej ) d n k=0 xk = xn+ x in generl: n k=m xk = xm x n+ x
2 Mrc Ph. Stoecklin ABES O RANSORM PAIRS v.5.3 ble of Continuoutime requency ourier rnform Pir f(t) = { (f)} = + f(t)ejπft df (f) = {f(t)} = + f(t)e jπft dt time reverl complex conjugtion revered conjugtion even/ymmetry odd/ntiymmetry f(t) f( t) f (t) f ( t) f(t) i purely rel f(t) i purely imginry f(t) = f ( t) f(t) = f ( t) (f) ( f) frequency reverl ( f) revered conjugtion (f) complex conjugtion (f) = ( f) even/ymmetry (f) = ( f) odd/ntiymmetry (f) i purely rel (f) i purely imginry time hifting f(t t 0 ) time cling linerity time multipliction delt function f(t)e jπf 0t f (f) f f f(t) + bg(t) f(t)g(t) f(t) g(t) δ(t) hifted delt function δ(t t 0 ) e jπf 0t twoided exponentil decy e t > 0 e πt e jπt ine in (πf 0 t + φ) coine co (πf 0 t + φ) ine modultion f(t) in (πf 0 t) coine modultion f(t) co (πf 0 t) qured ine qured coine in (t) co (t) (f)e jπft 0 (f ( f 0 ) frequency hifting f (f) frequency cling (f) + bg(t) (f) G(f) (f)g(f) frequency multipliction e jπft 0 δ(f) delt function δ(f f 0 ) hifted delt function +4π f e πf e jπ( 4 f ) j [ e jφ δ (f + f 0 ) e jφ δ (f f 0 ) ] [ e jφ δ (f + f 0 ) + e jφ δ (f f 0 ) ] j [ (f + f 0) (f f 0 )] [ (f + f 0) + (f f 0 )] [ ] 4 δ(f) δ f π δ f + π [ ] 4 δ(f) + δ f π + δ f + π rectngulr rect ( t t = 0 t > tringulr tring ( t = t t 0 t > t 0 tep u(t) = [0,+ ] (t) = 0 t < 0 t 0 ignum gn (t) = t < 0 inc qured inc nth time derivtive nth frequency derivtive inc (Bt) inc (Bt) d n dt n f(t) t n f(t) +t inc f inc f jπf + δ(f) jπf B rect ( f B B tring ( f B (jπf) n (f) ( jπ) n dn df n (f) πe π f ) = B [ ) B,+ B ](f)
3 Mrc Ph. Stoecklin ABES O RANSORM PAIRS v ble of Continuoutime Pultion ourier rnform Pir x(t) = {X()} = + x(t)ejt d X() = {x(t)} = + x(t)e jt dt time reverl complex conjugtion revered conjugtion even/ymmetry odd/ntiymmetry x(t) x( t) x (t) x ( t) x(t) i purely rel x(t) i purely imginry x(t) = x ( t) x(t) = x ( t) X() X( ) frequency reverl X ( ) revered conjugtion X () complex conjugtion X(f) = X ( ) even/ymmetry X(f) = X ( ) odd/ntiymmetry X() i purely rel X() i purely imginry time hifting x(t t 0 ) time cling x(t)e j 0t x (f) x f X()e jt 0 X( 0 ) frequency hifting X X() frequency cling linerity time multipliction x (t) + bx (t) x (t)x (t) x (t) x (t) X () + bx () π X () X () X ()X () frequency multipliction delt function δ(t) hifted delt function δ(t t 0 ) e j 0t twoided exponentil decy e t > 0 exponentil decy e t u(t) R{} > 0 revered exponentil decy e t u( t) R{} > 0 e t σ ine in ( 0 t + φ) coine co ( 0 t + φ) ine modultion x(t) in ( 0 t) coine modultion x(t) co ( 0 t) qured ine in ( 0 t) qured coine co ( 0 t) rectngulr rect ( t t = 0 t > tringulr tring ( t = t t 0 t > t 0 tep u(t) = [0,+ ] (t) = 0 t < 0 t 0 ignum gn (t) = t < 0 inc inc ( t) qured inc inc ( t) e jt 0 πδ() delt function πδ( 0 ) hifted delt function + +j j σ πe σ [ jπ e jφ δ ( + 0 ) e jφ δ ( 0 ) ] [ π e jφ δ ( + 0 ) + e jφ δ ( 0 ) ] j [X ( + 0) X ( 0 )] [X ( + 0) + X ( 0 )] π [δ(f) δ ( 0 ) δ ( + 0 )] π [δ() + δ ( 0 ) + δ ( + 0 )] inc inc πδ(f) + j j rect π = [ π,+π ](f) tring π nth time derivtive nth frequency derivtive time invere d n dt n f(t) t n f(t) t (j) n X() j n d n df n X() jπgn()
4 Mrc Ph. Stoecklin ABES O RANSORM PAIRS v ble of rnform Pir x[n] = {X()} = πj X() n d X() = {x[n]} = + n= x[n] n ROC time reverl complex conjugtion revered conjugtion rel prt imginry prt x[n] x[ n] x [n] x [ n] Re{x[n]} Im{x[n]} X() R x X( ) R x X ( ) R x X ( ) R x [X() + X ( )] R x j [X() X ( )] R x time hifting x[n n 0 ] n 0X() R x cling in n x[n] X R x downmpling by N x[nn], N N 0 N N k=0 X WN k N W N = e j N R x linerity time multipliction x [n] + bx [n] x [n]x [n] x [n] x [n] X () + bx () R x R y ( X (u)x ) πj u u du R x R y X ()X (t) R x R y delt function δ[n] hifted delt function δ[n n 0 ] n 0 tep rmp u[n] u[ n ] nu[n] n u[n] n u[ n ] n 3 u[n] n 3 u[ n ] ( ) n > < ( ) > (+) ( ) 3 > (+) ( ) 3 < ( +4+) ( ) 4 > ( +4+) ( ) 4 < + < exponentil exp. intervl n u[n] n u[ n ] n u[n ] n n u[n] n n u[n] e n u[n] { n n = 0,..., N 0 otherwie > < > ( ) > (+ ( ) 3 > e > e N N > 0 ine coine in ( 0 n) u[n] co ( 0 n) u[n] n in ( 0 n) u[n] n co ( 0 n) u[n] in( 0 ) co( 0 )+ ( co( 0 )) co( 0 )+ in( 0 ) co( 0 )+ ( co( 0 )) co( 0 )+ > > > > differentition in integrtion in mi= (n i+) m m! nx[n] x[n] n m u[n] dx() R d x X() 0 d R x ( ) m+ Note: =
5 Mrc Ph. Stoecklin ABES O RANSORM PAIRS v ble of Common Dicrete ime ourier rnform (D) Pir x[n] = +π π π X(ej )e jn d D X(e j ) = + n= x[n]e jn time reverl complex conjugtion revered conjugtion even/ymmetry odd/ntiymmetry x[n] x[ n] x [n] x [ n] x[n] i purely rel x[n] i purely imginry x[n] = x [ n] x[n] = x [ n] D X(e j ) D X(e j ) D X (e j ) D X (e j ) D X(e j ) = X (e j ) even/ymmetry D X(e j ) = X (e j ) odd/ntiymmetry D D X(e j ) i purely rel X(e j ) i purely imginry time hifting x[n n 0 ] x[n]e j 0n D X(e j )e jn 0 D X(e j( 0) ) frequency hifting D downmpling by N x[nn] N N 0 x [ ] n n = kn upmpling by N N 0 otherwie N D X(e jn ) N πk k=0 X(ej N ) linerity time multipliction x [n] + bx [n] x [n]x [n] x [n] x [n] D X (e j ) + bx (e j ) D X (e j ) X (e j ) = +π π π X (e j( σ) )X (e jσ )dσ D X (e j )X (e j ) frequency multipliction delt function δ[n] hifted delt function δ[n n 0 ] e j 0n D D e jn 0 D δ() delt function D δ( 0 ) hifted delt function ine in ( 0 n + φ) coine co ( 0 n + φ) D D j [e jφ δ ( πk) e +jφ δ ( 0 + πk)] [e jφ δ ( πk) + e +jφ δ ( 0 + πk)] rectngulr tep rect ( n n M M = 0 otherwie u[n] decying tep n u[n] ( < ) pecil decying tep (n + ) n u[n] ( < ) inc MA MA derivtion in( cn) πn D D D D = c π inc (cn) D rect ( n M 0 n M = 0 otherwie rect n M 0 n M = 0 otherwie nx[n] difference x[n] x[n ] n in[ 0 (n+)] u[n] < in 0 D D in[(m+ )] in(/) e j + δ() e j ( e j ) ( rect < = c c 0 c < < π in[(m+)/] e in(/) jm/ in[m/] in(/) e j(m )/ D j d d X(ej ) D ( e j )X(e j ) D co( 0 e j )+ e j Note: Prevl: δ() = + n= + k= x[n] = π δ( + πk) +π X(e j ) d π rect() = + k= rect( + πk)
6 Mrc Ph. Stoecklin ABES O RANSORM PAIRS v ble of plce rnform Pir f(t) = { ()} = πj lim c+j c j ()et d () = {f(t)} = + f(t)e t dt complex conjugtion f(t) f (t) () ( ) time hifting f(t ) t > 0 e t f(t) time cling f(t) linerity f (t) + bf (t) time multipliction f (t)f (t) time convolution f (t) f (t) () ( + ) frequency hifting ( ) () + b () () () () () frequency product delt function δ(t) hifted delt function δ(t ) e exponentil decy unit tep u(t) rmp tu(t) prbol t u(t) 3 nth power t n n! n+ exponentil decy e t twoided exponentil decy e t te t ( t)e t exponentil pproch e t ine coine hyperbolic ine hyperbolic coine exponentilly decying ine exponentilly decying coine in (t) co (t) inh (t) coh (t) e t in (t) e t co (t) + (+) (+) (+) + + (+) + + (+) + frequency differentition frequency nth differentition tf(t) t n f(t) () ( ) n (n) () time differentition f (t) = d dt f(t) () f(0) time nd differentition f (t) = d dt f(t) () f(0) f (0) time nth differentition f (n) (t) = dn dt n f(t) n () n f(0)... f (n ) (0) time integrtion frequency integrtion t 0 f(τ)dτ = (u f)(t) t f(t) () (u)du time invere time differentition f (t) f n (t) () f () n + f (0) n + f (0) n f n (0)
EE 179 April 21, 2014 Digital and Analog Communication Systems Handout #16 Homework #2 Solutions
EE 79 April, 04 Digital and Analog Communication Systems Handout #6 Homework # Solutions. Operations on signals (Lathi& Ding.33). For the signal g(t) shown below, sketch: a. g(t 4); b. g(t/.5); c. g(t
More informationLecture 8 ELE 301: Signals and Systems
Lecture 8 ELE 3: Signals and Systems Prof. Paul Cuff Princeton University Fall 22 Cuff (Lecture 7) ELE 3: Signals and Systems Fall 22 / 37 Properties of the Fourier Transform Properties of the Fourier
More informationM5A42 APPLIED STOCHASTIC PROCESSES PROBLEM SHEET 1 SOLUTIONS Term 1 20102011
M5A42 APPLIED STOCHASTIC PROCESSES PROBLEM SHEET 1 SOLUTIONS Term 1 21211 1. Clculte the men, vrince nd chrcteristic function of the following probbility density functions. ) The exponentil distribution
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
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 informationHarvard College. Math 21a: Multivariable Calculus Formula and Theorem Review
Hrvrd College Mth 21: Multivrible Clculus Formul nd Theorem Review Tommy McWillim, 13 tmcwillim@college.hrvrd.edu December 15, 2009 1 Contents Tble of Contents 4 9 Vectors nd the Geometry of Spce 5 9.1
More informationClass Note for Signals and Systems. Stanley Chan University of California, San Diego
Class Note for Signals and Systems Stanley Chan University of California, San Diego 2 Acknowledgement This class note is prepared for ECE 101: Linear Systems Fundamentals at the University of California,
More informationSolution to Problem Set 1
CSE 5: Introduction to the Theory o Computtion, Winter A. Hevi nd J. Mo Solution to Prolem Set Jnury, Solution to Prolem Set.4 ). L = {w w egin with nd end with }. q q q q, d). L = {w w h length t let
More informationA new algorithm for generating Pythagorean triples
A new lgorithm for generting Pythgoren triples RH Dye 1 nd RWD Nicklls 2 The Mthemticl Gzette (1998); 82 (Mrch, No. 493), p. 86 91 (JSTOR rchive) http://www.nicklls.org/dick/ppers/mths/pythgtriples1998.pdf
More informationReview of Fourier series formulas. Representation of nonperiodic functions. ECE 3640 Lecture 5 Fourier Transforms and their properties
ECE 3640 Lecture 5 Fourier Transforms and their properties Objective: To learn about Fourier transforms, which are a representation of nonperiodic functions in terms of trigonometric functions. Also, to
More informationBasically, logarithmic transformations ask, a number, to what power equals another number?
Wht i logrithm? To nwer thi, firt try to nwer the following: wht i x in thi eqution? 9 = 3 x wht i x in thi eqution? 8 = 2 x Biclly, logrithmic trnformtion k, number, to wht power equl nother number? In
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationFactoring Polynomials
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
More informationSection 74 Translation of Axes
62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 74 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 D E F 0 Finding Equtions of Conics In the
More information5 Signal Design for Bandlimited Channels
225 5 Signal Design for Bandlimited Channels So far, we have not imposed any bandwidth constraints on the transmitted passband signal, or equivalently, on the transmitted baseband signal s b (t) I[k]g
More informationLectures 8 and 9 1 Rectangular waveguides
1 Lectures 8 nd 9 1 Rectngulr wveguides y b x z Consider rectngulr wveguide with 0 < x b. There re two types of wves in hollow wveguide with only one conductor; Trnsverse electric wves
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 informationDifference Equations
Difference Equations Andrew W H House 10 June 004 1 The Basics of Difference Equations Recall that in a previous section we saw that IIR systems cannot be evaluated using the convolution sum because it
More informationMath 22B, Homework #8 1. y 5y + 6y = 2e t
Math 22B, Homework #8 3.7 Problem # We find a particular olution of the ODE y 5y + 6y 2e t uing the method of variation of parameter and then verify the olution uing the method of undetermined coefficient.
More informationMODULE 3. 0, y = 0 for all y
Topics: Inner products MOULE 3 The inner product of two vectors: The inner product of two vectors x, y V, denoted by x, y is (in generl) complex vlued function which hs the following four properties: i)
More informationTRANSFORM AND ITS APPLICATION
LAPLACE TRANSFORM AND ITS APPLICATION IN CIRCUIT ANALYSIS C.T. Pan. Definition of the Laplace Tranform. Ueful Laplace Tranform Pair.3 Circuit Analyi in S Domain.4 The Tranfer Function and the Convolution
More informationThe Chain Rule. rf dx. t t lim " (x) dt " (0) dx. df dt = df. dt dt. f (r) = rf v (1) df dx
The Chin Rule The Chin Rule In this section, we generlize the chin rule to functions of more thn one vrible. In prticulr, we will show tht the product in the singlevrible chin rule extends to n inner
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 informationSUBSTITUTION I.. f(ax + b)
Integrtion SUBSTITUTION I.. f(x + b) Grhm S McDonld nd Silvi C Dll A Tutoril Module for prctising the integrtion of expressions of the form f(x + b) Tble of contents Begin Tutoril c 004 g.s.mcdonld@slford.c.uk
More information15.6. The mean value and the rootmeansquare value of a function. Introduction. Prerequisites. Learning Outcomes. Learning Style
The men vlue nd the rootmensqure vlue of function 5.6 Introduction Currents nd voltges often vry with time nd engineers my wish to know the verge vlue of such current or voltge over some prticulr time
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 information2.4 Circular Waveguide
.4 Circulr Wveguide y x Figure.5: A circulr wveguide of rdius. For circulr wveguide of rdius (Fig..5, we cn perform the sme sequence of steps in cylindricl coordintes s we did in rectngulr coordintes to
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 informationLaplace Transforms: Theory, Problems, and Solutions. Marcel B. Finan Arkansas Tech University c All Rights Reserved
Laplace Tranform: Theory, Problem, and Solution Marcel B. Finan Arkana Tech Univerity c All Right Reerved Content 43 The Laplace Tranform: Baic Definition and Reult 3 44 Further Studie of Laplace Tranform
More informationTaylor Polynomials and Taylor Series Math 126
Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will
More informationDouble Integrals over General Regions
Double Integrls over Generl egions. Let be the region in the plne bounded b the lines, x, nd x. Evlute the double integrl x dx d. Solution. We cn either slice the region verticll or horizontll. ( x x Slicing
More information4: RIEMANN SUMS, RIEMANN INTEGRALS, FUNDAMENTAL THEOREM OF CALCULUS
4: RIEMA SUMS, RIEMA ITEGRALS, FUDAMETAL THEOREM OF CALCULUS STEVE HEILMA Contents 1. Review 1 2. Riemnn Sums 2 3. Riemnn Integrl 3 4. Fundmentl Theorem of Clculus 7 5. Appendix: ottion 10 1. Review Theorem
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 informationThe invention of line integrals is motivated by solving problems in fluid flow, forces, electricity and magnetism.
Instrutor: Longfei Li Mth 43 Leture Notes 16. Line Integrls The invention of line integrls is motivted by solving problems in fluid flow, fores, eletriity nd mgnetism. Line Integrls of Funtion We n integrte
More informationActuarial Science with
Actuarial Science with 1. life insurance & actuarial notations Arthur Charpentier joint work with Christophe Dutang & Vincent Goulet and Giorgio Alfredo Spedicato s lifecontingencies package Meielisalp
More information9 CONTINUOUS DISTRIBUTIONS
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
More informationActuarial mathematics 2
Actuarial mathematics 2 Life insurance contracts Edward Furman Department of Mathematics and Statistics York University January 3, 212 Edward Furman Actuarial mathematics MATH 328 1 / 45 Definition.1 (Life
More informationReview Problems for the Final of Math 121, Fall 2014
Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since
More informationIntegration. 148 Chapter 7 Integration
48 Chpter 7 Integrtion 7 Integrtion t ech, by supposing tht during ech tenth of second the object is going t constnt speed Since the object initilly hs speed, we gin suppose it mintins this speed, but
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 informationFrequency Response and Continuoustime Fourier Transform
Frequency Response and Continuoustime Fourier Transform Goals Signals and Systems in the FDpart II I. (Finiteenergy) signals in the Frequency Domain  The Fourier Transform of a signal  Classification
More informationCalculus of variations with fractional derivatives and fractional integrals
Anis do CNMAC v.2 ISSN 1984820X Clculus of vritions with frctionl derivtives nd frctionl integrls Ricrdo Almeid, Delfim F. M. Torres Deprtment of Mthemtics, University of Aveiro 3810193 Aveiro, Portugl
More informationRepresentation of functions as power series
Representation of functions as power series Dr. Philippe B. Laval Kennesaw State University November 9, 008 Abstract This document is a summary of the theory and techniques used to represent functions
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 informationFrequency Domain Characterization of Signals. Yao Wang Polytechnic University, Brooklyn, NY11201 http: //eeweb.poly.edu/~yao
Frequency Domain Characterization of Signals Yao Wang Polytechnic University, Brooklyn, NY1121 http: //eeweb.poly.edu/~yao Signal Representation What is a signal Timedomain description Waveform representation
More informationReview Solutions MAT V1102. 1. (a) If u = 4 x, then du = dx. Hence, substitution implies 1. dx = du = 2 u + C = 2 4 x + C.
Review Solutions MAT V. (a) If u 4 x, then du dx. Hence, substitution implies dx du u + C 4 x + C. 4 x u (b) If u e t + e t, then du (e t e t )dt. Thus, by substitution, we have e t e t dt e t + e t u
More informationModule Summary Sheets. C3, Methods for Advanced Mathematics (Version B reference to new book) Topic 2: Natural Logarithms and Exponentials
MEI Mthemtics in Ection nd Instry Topic : Proof MEI Structured Mthemtics Mole Summry Sheets C, Methods for Anced Mthemtics (Version B reference to new book) Topic : Nturl Logrithms nd Eponentils Topic
More informationExam 1 Study Guide. Differentiation and Antidifferentiation Rules from Calculus I
Exm Stuy Guie Mth 2020  Clculus II, Winter 204 The following is list of importnt concepts from ech section tht will be teste on exm. This is not complete list of the mteril tht you shoul know for the
More informationApplication of Fourier Transform to PDE (I) Fourier Sine Transform (application to PDEs defined on a semiinfinite domain)
Application of Fourier Transform to PDE (I) Fourier Sine Transform (application to PDEs defined on a semiinfinite domain) The Fourier Sine Transform pair are F. T. : U = 2/ u x sin x dx, denoted as U
More informationFUNDAMENTALS OF ENGINEERING (FE) EXAMINATION REVIEW
FE: Electric Circuits C.A. Gross EE11 FUNDAMENTALS OF ENGINEERING (FE) EXAMINATION REIEW ELECTRICAL ENGINEERING Charles A. Gross, Professor Emeritus Electrical and Comp Engineering Auburn University Broun
More informationManual for SOA Exam MLC.
Chapter 5 Life annuities Extract from: Arcones Manual for the SOA Exam MLC Fall 2009 Edition available at http://wwwactexmadrivercom/ 1/94 Due n year temporary annuity Definition 1 A due n year term annuity
More informationCHAPTER 9: Moments of Inertia
HPTER 9: Moments of nerti! Moment of nerti of res! Second Moment, or Moment of nerti, of n re! Prllelis Theorem! Rdius of Grtion of n re! Determintion of the Moment of nerti of n re ntegrtion! Moments
More informationExamination paper for Solutions to Matematikk 4M and 4N
Department of Mathematical Sciences Examination paper for Solutions to Matematikk 4M and 4N Academic contact during examination: Trygve K. Karper Phone: 99 63 9 5 Examination date:. mai 04 Examination
More informationThe Exponential Distribution
21 The Exponential Distribution From DiscreteTime to ContinuousTime: In Chapter 6 of the text we will be considering Markov processes in continuous time. In a sense, we already have a very good understanding
More information200506 Second Term MAT2060B 1. Supplementary Notes 3 Interchange of Differentiation and Integration
Source: http://www.mth.cuhk.edu.hk/~mt26/mt26b/notes/notes3.pdf 256 Second Term MAT26B 1 Supplementry Notes 3 Interchnge of Differentition nd Integrtion The theme of this course is bout vrious limiting
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 informationg(y(a), y(b)) = o, B a y(a)+b b y(b)=c, Boundary Value Problems Lecture Notes to Accompany
Lecture Notes to Accompny Scientific Computing An Introductory Survey Second Edition by Michel T Heth Boundry Vlue Problems Side conditions prescribing solution or derivtive vlues t specified points required
More information1 Numerical Solution to Quadratic Equations
cs42: introduction to numericl nlysis 09/4/0 Lecture 2: Introduction Prt II nd Solving Equtions Instructor: Professor Amos Ron Scribes: Yunpeng Li, Mrk Cowlishw Numericl Solution to Qudrtic Equtions Recll
More informationDerivatives and Rates of Change
Section 2.1 Derivtives nd Rtes of Cnge 2010 Kiryl Tsiscnk Derivtives nd Rtes of Cnge Te Tngent Problem EXAMPLE: Grp te prbol y = x 2 nd te tngent line t te point P(1,1). Solution: We ve: DEFINITION: Te
More informationManual for SOA Exam MLC.
Chapter 4. Life Insurance. c 29. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam MLC. Fall 29 Edition. available at http://www.actexmadriver.com/ c 29. Miguel A. Arcones.
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 information4. Life Insurance. 4.1 Survival Distribution And Life Tables. Introduction. X, Ageatdeath. T (x), timeuntildeath
4. Life Insurance 4.1 Survival Distribution And Life Tables Introduction X, Ageatdeath T (x), timeuntildeath Life Table Engineers use life tables to study the reliability of complex mechanical and
More information1. Revision 2. Revision pv 3.  note that there are other equivalent formulae! 1 pv 16.5 4. A x A 1 x:n A 1
Tutorial 1 1. Revision 2. Revision pv 3.  note that there are other equivalent formulae! 1 pv 16.5 4. A x A 1 x:n A 1 x:n a x a x:n n a x 5. K x = int[t x ]  or, as an approximation: T x K x + 1 2 6.
More information6 Energy Methods And The Energy of Waves MATH 22C
6 Energy Methods And The Energy of Wves MATH 22C. Conservtion of Energy We discuss the principle of conservtion of energy for ODE s, derive the energy ssocited with the hrmonic oscilltor, nd then use this
More informationComplex Analysis I. All My Students Version 1
omplex Anlysis I All My Students Version 1 Fll, 009 of 84 ontents 1 OMPLEX NUMBERS 5 1.1 Sums nd Products......................................... 5 1. Bsic Algebric Properties.....................................
More informationCONVOLUTION Digital Signal Processing
CONVOLUTION Digital Signal Processing Introduction As digital signal processing continues to emerge as a major discipline in the field of electrical engineering an even greater demand has evolved to understand
More informationLectures 56: Taylor Series
Math 1d Instructor: Padraic Bartlett Lectures 5: Taylor Series Weeks 5 Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,
More informationExponentiation: Theorems, Proofs, Problems Pre/Calculus 11, Veritas Prep.
Exponentition: Theorems, Proofs, Problems Pre/Clculus, Verits Prep. Our Exponentition Theorems Theorem A: n+m = n m Theorem B: ( n ) m = nm Theorem C: (b) n = n b n ( ) n n Theorem D: = b b n Theorem E:
More informationProbability density function : An arbitrary continuous random variable X is similarly described by its probability density function f x = f X
Week 6 notes : Continuous random variables and their probability densities WEEK 6 page 1 uniform, normal, gamma, exponential,chisquared distributions, normal approx'n to the binomial Uniform [,1] random
More information7: Fourier Transforms: Convolution and Parseval s Theorem
Convolution Parseval s E. Fourier Series and Transforms (245559) Fourier Transform  Parseval and Convolution: 7 / Multiplication of Signals Question: What is the Fourier transform ofw(t) = u(t)v(t)?
More informationECG590I Asset Pricing. Lecture 2: Present Value 1
ECG59I Asset Pricing. Lecture 2: Present Value 1 2 Present Value If you have to decide between receiving 1$ now or 1$ one year from now, then you would rather have your money now. If you have to decide
More informationNull Similar Curves with Variable Transformations in Minkowski 3space
Null Similr Curves with Vrile Trnsformtions in Minkowski spce Mehmet Önder Cell Byr University, Fculty of Science nd Arts, Deprtment of Mthemtics, Murdiye Cmpus, 45047 Murdiye, Mnis, Turkey. mil: mehmet.onder@yr.edu.tr
More informationDistributions. (corresponding to the cumulative distribution function for the discrete case).
Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive
More informationThere are four common ways of finding the inverse ztransform:
Inverse ztransforms and Difference Equations Preliminaries We have seen that given any signal x[n], the twosided ztransform is given by n x[n]z n and X(z) converges in a region of the complex plane
More informationDefinition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) =
Vertical Asymptotes Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: lim f (x) = x a lim f (x) = lim x a lim f (x) = x a
More informationSurface Area and Volume
Surfce Are nd Volume Student Book  Series J Mthletics Instnt Workooks Copyright Surfce re nd volume Student Book  Series J Contents Topics Topic  Surfce re of right prism Topic 2  Surfce re of right
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 informationECE 5520: Digital Communications Lecture Notes Fall 2009
ECE 5520: Digital Communications Lecture Notes Fall 2009 Dr. Neal Patwari University of Utah Department of Electrical and Computer Engineering c 2006 ECE 5520 Fall 2009 2 Contents 1 Class Organization
More informationLaplace Transform. f(t)e st dt,
Chapter 7 Laplace Tranform The Laplace tranform can be ued to olve differential equation. Beide being a different and efficient alternative to variation of parameter and undetermined coefficient, the Laplace
More informationReview guide for the final exam in Math 233
Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered
More information1.1 DiscreteTime Fourier Transform
1.1 DiscreteTime Fourier Transform The discretetime Fourier transform has essentially the same properties as the continuoustime Fourier transform, and these properties play parallel roles in continuous
More informationUNIVERSITY OF OSLO FACULTY OF MATHEMATICS AND NATURAL SCIENCES
UNIVERSITY OF OSLO FACULTY OF MATHEMATICS AND NATURAL SCIENCES Solution to exm in: FYS30, Quntum mechnics Dy of exm: Nov. 30. 05 Permitted mteril: Approved clcultor, D.J. Griffiths: Introduction to Quntum
More informationOstrowski Type Inequalities and Applications in Numerical Integration. Edited By: Sever S. Dragomir. and. Themistocles M. Rassias
Ostrowski Type Inequlities nd Applictions in Numericl Integrtion Edited By: Sever S Drgomir nd Themistocles M Rssis SS Drgomir) School nd Communictions nd Informtics, Victori University of Technology,
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 informationSection 5.1 Continuous Random Variables: Introduction
Section 5. Continuous Random Variables: Introduction Not all random variables are discrete. For example:. Waiting times for anything (train, arrival of customer, production of mrna molecule from gene,
More informationMatrix Algebra CHAPTER 1 PREAMBLE 1.1 MATRIX ALGEBRA
CHAPTER 1 Mtrix Algebr PREAMBLE Tody, the importnce of mtrix lgebr is of utmost importnce in the field of physics nd engineering in more thn one wy, wheres before 1925, the mtrices were rrely used by the
More informationLecture 7 ELE 301: Signals and Systems
Lecture 7 ELE 3: Signals and Systems Prof. Paul Cuff Princeton University Fall 22 Cuff (Lecture 7) ELE 3: Signals and Systems Fall 22 / 22 Introduction to Fourier Transforms Fourier transform as a limit
More informationCHAPTER IV  BROWNIAN MOTION
CHAPTER IV  BROWNIAN MOTION JOSEPH G. CONLON 1. Construction of Brownian Motion There are two ways in which the idea of a Markov chain on a discrete state space can be generalized: (1) The discrete time
More informationOptimal Control. Palle Andersen. Aalborg University. Opt lecture 1 p. 1/2
Opt lecture 1 p. 1/2 Optimal Control Palle Andersen pa@control.aau.dk Aalborg University Opt lecture 1 p. 2/2 Optimal Control, course outline 1st lecture: Introduction to optimal control and quadratic
More informationFUNCTIONS AND EQUATIONS. xεs. The simplest way to represent a set is by listing its members. We use the notation
FUNCTIONS AND EQUATIONS. SETS AND SUBSETS.. Definition of set. A set is ny collection of objects which re clled its elements. If x is n element of the set S, we sy tht x belongs to S nd write If y does
More informationLecture 3 Gaussian Probability Distribution
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
More informationTaylor and Maclaurin Series
Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions
More informationFourier Analysis. u m, a n u n = am um, u m
Fourier Analysis Fourier series allow you to expand a function on a finite interval as an infinite series of trigonometric functions. What if the interval is infinite? That s the subject of this chapter.
More informationMathematics of Life Contingencies MATH 3281
Mathematics of Life Contingencies MATH 3281 Life annuities contracts Edward Furman Department of Mathematics and Statistics York University February 13, 2012 Edward Furman Mathematics of Life Contingencies
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 informationChapter 3 Transfer Functions Block Diagrams SignalFlow Graphs
Chapter 3 Tranfer Function Block Diagram SignalFlow Graph Tranfer Function Impule Repone δ(t) Linear TimeInvariant tem g(t) (t): impule repone T 0 t = δ τ g t τ dτ () = gt () Convolution!!! Tranfer
More informationMATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1
MATH3432: Green s Functions, Integrl Equtions nd the Clculus of Vritions Section 3 Integrl Equtions Integrl Opertors nd Liner Integrl Equtions As we sw in Section on opertor nottion, we work with functions
More informationDefinition 6.1.1. A r.v. X has a normal distribution with mean µ and variance σ 2, where µ R, and σ > 0, if its density is f(x) = 1. 2σ 2.
Chapter 6 Brownian Motion 6. Normal Distribution Definition 6... A r.v. X has a normal distribution with mean µ and variance σ, where µ R, and σ > 0, if its density is fx = πσ e x µ σ. The previous definition
More informationThe BlackScholesMerton Approach to Pricing Options
he BlackScholesMerton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the BlackScholesMerton approach to determining
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 information