ME 452 Fourier Series and Fourier Transform
|
|
- Ira Robinson
- 6 years ago
- Views:
Transcription
1 ME 452 Fourier Series and Fourier ransform Fourier series From Joseph Fourier in 87 as a resul of his sudy on he flow of hea. If f() is almos any periodic funcion i can be wrien as an infinie sum of sines and cosines. a f ( ) = + a Cos( ) + b Sin( ) ω ω 2 = = where 2 2 = f Cos d ( ) ( ω ) and b = f Sin d ( ) ( ω ) a Find he Fourier series for he square wave shown below. Period is 2 and frequency is /(2). Figure A square wave wih a period of 2. Soluion: o find he coefficiens a and b, we need o inegrae over one period. For his problem we will ae he period from o 2. he equaion for f() is: f ( ) = + 2 he equaions for a and b can be evaluaed as: 2 a = + Cos d + Cos d = ( ) ( ) ( ) ( ) 2 even b = + Sin d + Sin d = ( ) ( ) ( ) ( ) 4 odd We can hen wrie f() as a Fourier series. f ( ) = 4 Sin( ) which can be wrien as = odd --
2 4 4 4 f ( ) = sin( ) + sin(3) + sin(5) In general a sinusoid can be wrien as f ( ) = Asin(2f) where A is he ampliude and f is he frequency in cycles per second or Herz. In our case A = and f = /(2). We can use MALAB o plo a few cycles. %PloSine.m f = /(2*pi); = -4*pi:pi/:4*pi; f = (4/pi)*sin(2*pi*f*); %fundamenal f = (4/(3*pi))*sin(2*pi*3*f*); %3rd harmonic f2 = (4/(5*pi))*sin(2*pi*5*f*); %5h harmonic f3 = (4/(7*pi))*sin(2*pi*7*f*); %7h harmonic figure();clf; plo(, f, ''); hold on; plo(, f, 'b'); plo(, f2, 'r'); plo(, f3, 'g'); axis([-pi pi -.4.4]); Our equaion f ( ) = 4 Sin( ) = odd Figure 2 he fundamenal sinusoid and he firs hree odd harmonics. says ha we need o add all of hese sinusoids ogeher and if we do we will ge a square wave. -2-
3 %Fourier.m % his program plos erms of he Fourier series for a square wave. erms = [ ]; %Vecor of number of erms in each plo figure();clf; for indx = :6 %6 plos sum = zeros(,24); %24 erms = linspace(-2*pi,2*pi,24); % goes from -2Pi o +2Pi for =:2:2*erms(indx) sum = sum + 4/(*pi)*sin(*); end subplo(3, 2, indx); plo(, sum, 'LineWidh', ); hold on; plo(, square(), ''); xlabel('ime'); ylabel(''); sile = ['Fourier Series wih ' in2sr(erms(indx)) ' erms']; ile(sile); axis([-2*pi 2*pi -.4.4]) end Fourier Series wih erms Fourier Series wih 3 erms ime Fourier Series wih 5 erms ime Fourier Series wih erms ime Fourier Series wih 5 erms ime Fourier Series wih erms ime ime Figure 3 Fourier series for a square wave for various number of erms. So a square wave can be hough of as an infinie sum of sinusoids a odd harmonics of he fundamenal and whose ampliude geomerically decreases wih frequency. -3-
4 Since we have come o regard a sinusoid as a "pure frequency" we say he square wave has a frequency conen consising of he odd harmonics of is fundamenal. he figure below shows a graph of he ampliudes of he sinusoids a he various frequencies for he square wave..4 Frequency plo of a square wave.2.8 gain Figure 4 Frequency plo of he odd harmonic frequencies in a square wave. o ge he exponenial form of he series we use Euler s ideniy: ± jx e = Cos( x) ± jsin( x) Using his equaion we can wrie he cosine and sine in erms of he naural log base, e. jx jx e + e jx jx e e Cos( x) = Sin( x) = 2 2 j Subsiue hese equaions for cos and sin ino he Fourier series a f ( ) = + a Cos( ) + b Sin( ) ω ω 2 = = his gives he exponenial form of he Fourier series which can be wrien as: jω jω jθ 2 f ( ) = C e where C = f e d = C e = ( ) and ω = If we wrie he square wave in he exponenial forma hen, in Figure 4, he "Gain" number is C Fourier ransform 5 5 Frequency/(2*pi) he Fourier series applies o periodic funcions such as he square wave. If he funcion is no periodic he Fourier series no longer applies direcly insead we use he Fourier ransform. he exponenial form of he Fourier series is: f ( ) = jω C e = where C jω jθ = f e d = C e ( ) 2 and ω = -4-
5 he Fourier ransform is: F{ f ( )} jω = F( ω ) = f ( ) e d Noe ha he Fourier ransform is similar o he definiion of C in he Fourier series excep he inegral is from - o + and he variable ωo has become a coninuous variable ω. Lie he Fourier series he Fourier ransform is easier o undersand if we apply Euler's ideniy and wrie i in erms of sines and cosines. + F { f ( )} = F( ω ) = f ( ) cos( ω) d j f ( )sin( ω) d + In plain English wha we are doing is muliplying he funcion imes he cosine funcion and he sine funcion and finding he area under he resuling curve. he Fourier ransform can hus be viewed as a correlaion beween sines and cosines and a given signal. In MALAB we can do he Fourier ransform using he command ff which sands for Fas Fourier ransform. he FF is a fas algorihm for calculaing he Fourier ransform. % % %Do real ff fs = ; = /fs; fsig = ; = ::.-; %ime o. seconds x = cos(2*pi*fsig* - pi/6); %Signal a Hz x(lengh()/2:lengh()) = ; figure();clf; sem(, x, 'MarerSize', ); axis([ ]); ile('signal'); xlabel('ime in seconds'); ylabel(''); XFF = ff(x); XFF = XFF/max(abs(XFF)); figure(2);clf; L = lengh(x); = :L-; delaf = fs/l; f = *delaf; plo(f, abs(xff)); axis([ fs/2 ]); xlabel('frequency in Hz'); ylabel('gain'); ile('fourier ransform in MALAB'); -5-
6 Signal Fourier ransform in MALAB gain Noe ha he Fourier ransform is given by: Figure 5 A sampled cosine wave and is ff F( ω ) ( ) jω = f e d When we ae he Fourier ransform of a signal we deermine wha sinusoidal frequencies ha signal conains. We refer o his as he frequency conen of he signal. Unforunaely, he Fourier series does no converges and herefore does no exis for many funcions. If however, we replace ime in seconds j e ω by s e where s σ + jω = we ge F( s) s = f ( ) e d frequency in Hz Which is he Laplace ransform of he signal. he Laplace ransform converges for many more signal since i muliplies he funcion, f() by e σ where σ is real and he exponenial forces a convergence. If we consider he jω erm as a se of purely imaginary numbers and he s = σ + jω as an arbirary complex number han can be real, imaginary, or complex, we see ha he Laplace ransform is a generalizaion of h Fourier ransform. his also means ha if we have a ransfer funcion of a sable sysem in s we can ge is frequency response funcion by subsiuing s = jω effecively changing he Laplace ransform funcion o he Fourier ransform funcion. -6-
Chapter 2 : Fourier Series. Chapter 3 : Fourier Series
Chaper 2 : Fourier Series.0 Inroducion Fourier Series : represenaion of periodic signals as weighed sums of harmonically relaed frequencies. If a signal x() is periodic signal, hen x() can be represened
More information23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More information6.003 Homework #8 Solutions
6.003 Homework #8 Soluions Problems. Fourier Series Deermine he Fourier series coefficiens a k for x () shown below. x ()= x ( + 0) 0 a 0 = 0 a k = e /0 sin(/0) for k 0 a k = π x()e k d = 0 0 π e 0 k d
More informationEECS 2602 Winter Laboratory 3 Fourier series, Fourier transform and Bode Plots in MATLAB
EECS 6 Winer 7 Laboraory 3 Fourier series, Fourier ransform and Bode Plos in MATLAB Inroducion: The objecives of his lab are o use MATLAB:. To plo periodic signals wih Fourier series represenaion. To obain
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signals & Sysems Prof. Mar Fowler Noe Se #1 C-T Signals: Circuis wih Periodic Sources 1/1 Solving Circuis wih Periodic Sources FS maes i easy o find he response of an RLC circui o a periodic source!
More information23.5. Half-Range Series. Introduction. Prerequisites. Learning Outcomes
Half-Range Series 2.5 Inroducion In his Secion we address he following problem: Can we find a Fourier series expansion of a funcion defined over a finie inerval? Of course we recognise ha such a funcion
More information6.003 Homework #9 Solutions
6.00 Homework #9 Soluions Problems. Fourier varieies a. Deermine he Fourier series coefficiens of he following signal, which is periodic in 0. x () 0 0 a 0 5 a k sin πk 5 sin πk 5 πk for k 0 a k 0 πk j
More information6.003 Homework #9 Solutions
6.003 Homework #9 Soluions Problems. Fourier varieies a. Deermine he Fourier series coefficiens of he following signal, which is periodic in 0. x () 0 3 0 a 0 5 a k a k 0 πk j3 e 0 e j πk 0 jπk πk e 0
More informationMon Apr 9 EP 7.6 Convolutions and Laplace transforms. Announcements: Warm-up Exercise:
Mah 225-4 Week 3 April 9-3 EP 7.6 - convoluions; 6.-6.2 - eigenvalues, eigenvecors and diagonalizabiliy; 7. - sysems of differenial equaions. Mon Apr 9 EP 7.6 Convoluions and Laplace ransforms. Announcemens:
More informationEchocardiography Project and Finite Fourier Series
Echocardiography Projec and Finie Fourier Series 1 U M An echocardiagram is a plo of how a porion of he hear moves as he funcion of ime over he one or more hearbea cycles If he hearbea repeas iself every
More informationChapter 2. First Order Scalar Equations
Chaper. Firs Order Scalar Equaions We sar our sudy of differenial equaions in he same way he pioneers in his field did. We show paricular echniques o solve paricular ypes of firs order differenial equaions.
More informationMATH 31B: MIDTERM 2 REVIEW. x 2 e x2 2x dx = 1. ue u du 2. x 2 e x2 e x2] + C 2. dx = x ln(x) 2 2. ln x dx = x ln x x + C. 2, or dx = 2u du.
MATH 3B: MIDTERM REVIEW JOE HUGHES. Inegraion by Pars. Evaluae 3 e. Soluion: Firs make he subsiuion u =. Then =, hence 3 e = e = ue u Now inegrae by pars o ge ue u = ue u e u + C and subsiue he definiion
More informationSome Basic Information about M-S-D Systems
Some Basic Informaion abou M-S-D Sysems 1 Inroducion We wan o give some summary of he facs concerning unforced (homogeneous) and forced (non-homogeneous) models for linear oscillaors governed by second-order,
More informationRepresenting a Signal. Continuous-Time Fourier Methods. Linearity and Superposition. Real and Complex Sinusoids. Jean Baptiste Joseph Fourier
Represening a Signal Coninuous-ime ourier Mehods he convoluion mehod for finding he response of a sysem o an exciaion aes advanage of he lineariy and imeinvariance of he sysem and represens he exciaion
More informationChapter 4 The Fourier Series and Fourier Transform
Represenaion of Signals in Terms of Frequency Componens Chaper 4 The Fourier Series and Fourier Transform Consider he CT signal defined by x () = Acos( ω + θ ), = The frequencies `presen in he signal are
More informationEECS20n, Solution to Midterm 2, 11/17/00
EECS20n, Soluion o Miderm 2, /7/00. 0 poins Wrie he following in Caresian coordinaes (i.e. in he form x + jy) (a) 2 poins j 3 j 2 + j += j ++j +=2 (b) 2 poins ( j)/( + j) = j (c) 2 poins cos π/4+jsin π/4
More informationf t te e = possesses a Laplace transform. Exercises for Module-III (Transform Calculus)
Exercises for Module-III (Transform Calculus) ) Discuss he piecewise coninuiy of he following funcions: =,, +, > c) e,, = d) sin,, = ) Show ha he funcion ( ) sin ( ) f e e = possesses a Laplace ransform.
More informationChallenge Problems. DIS 203 and 210. March 6, (e 2) k. k(k + 2). k=1. f(x) = k(k + 2) = 1 x k
Challenge Problems DIS 03 and 0 March 6, 05 Choose one of he following problems, and work on i in your group. Your goal is o convince me ha your answer is correc. Even if your answer isn compleely correc,
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More informationTraveling Waves. Chapter Introduction
Chaper 4 Traveling Waves 4.1 Inroducion To dae, we have considered oscillaions, i.e., periodic, ofen harmonic, variaions of a physical characerisic of a sysem. The sysem a one ime is indisinguishable from
More information6.302 Feedback Systems Recitation 4: Complex Variables and the s-plane Prof. Joel L. Dawson
Number 1 quesion: Why deal wih imaginary and complex numbers a all? One answer is ha, as an analyical echnique, hey make our lives easier. Consider passing a cosine hrough an LTI filer wih impulse response
More informationKEY. Math 334 Midterm III Winter 2008 section 002 Instructor: Scott Glasgow
KEY Mah 334 Miderm III Winer 008 secion 00 Insrucor: Sco Glasgow Please do NOT wrie on his exam. No credi will be given for such work. Raher wrie in a blue book, or on your own paper, preferably engineering
More information10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
More informationSystem of Linear Differential Equations
Sysem of Linear Differenial Equaions In "Ordinary Differenial Equaions" we've learned how o solve a differenial equaion for a variable, such as: y'k5$e K2$x =0 solve DE yx = K 5 2 ek2 x C_C1 2$y''C7$y
More informationTwo Coupled Oscillators / Normal Modes
Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own
More informationThe following report makes use of the process from Chapter 2 in Dr. Cumming s thesis.
Zaleski 1 Joseph Zaleski Mah 451H Final Repor Conformal Mapping Mehods and ZST Hele Shaw Flow Inroducion The Hele Shaw problem has been sudied using linear sabiliy analysis and numerical mehods, bu a novel
More informationLaplace transfom: t-translation rule , Haynes Miller and Jeremy Orloff
Laplace ransfom: -ranslaion rule 8.03, Haynes Miller and Jeremy Orloff Inroducory example Consider he sysem ẋ + 3x = f(, where f is he inpu and x he response. We know is uni impulse response is 0 for
More informationContinuous Time. Time-Domain System Analysis. Impulse Response. Impulse Response. Impulse Response. Impulse Response. ( t) + b 0.
Time-Domain Sysem Analysis Coninuous Time. J. Robers - All Righs Reserved. Edied by Dr. Rober Akl 1. J. Robers - All Righs Reserved. Edied by Dr. Rober Akl 2 Le a sysem be described by a 2 y ( ) + a 1
More informationLAPLACE TRANSFORM AND TRANSFER FUNCTION
CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dep. of Chemical and Biological Engineering 5-1 Road Map of he Lecure V Laplace Transform and Transfer funcions
More informationSection 7.4 Modeling Changing Amplitude and Midline
488 Chaper 7 Secion 7.4 Modeling Changing Ampliude and Midline While sinusoidal funcions can model a variey of behaviors, i is ofen necessary o combine sinusoidal funcions wih linear and exponenial curves
More informationSolutions to Assignment 1
MA 2326 Differenial Equaions Insrucor: Peronela Radu Friday, February 8, 203 Soluions o Assignmen. Find he general soluions of he following ODEs: (a) 2 x = an x Soluion: I is a separable equaion as we
More informationSection 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients
Secion 3.5 Nonhomogeneous Equaions; Mehod of Undeermined Coefficiens Key Terms/Ideas: Linear Differenial operaor Nonlinear operaor Second order homogeneous DE Second order nonhomogeneous DE Soluion o homogeneous
More informationConvolution. Lecture #6 2CT.3 8. BME 333 Biomedical Signals and Systems - J.Schesser
Convoluion Lecure #6 C.3 8 Deiniion When we compue he ollowing inegral or τ and τ we say ha he we are convoluing wih g d his says: ae τ, lip i convolve in ime -τ, hen displace i in ime by seconds -τ, and
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 3 Signals & Sysems Prof. Mark Fowler Noe Se #2 Wha are Coninuous-Time Signals??? Reading Assignmen: Secion. of Kamen and Heck /22 Course Flow Diagram The arrows here show concepual flow beween ideas.
More informationMath 334 Fall 2011 Homework 11 Solutions
Dec. 2, 2 Mah 334 Fall 2 Homework Soluions Basic Problem. Transform he following iniial value problem ino an iniial value problem for a sysem: u + p()u + q() u g(), u() u, u () v. () Soluion. Le v u. Then
More information6.003: Signals and Systems. Fourier Representations
6.003: Signals and Sysems Fourier Represenaions Ocober 27, 20 Fourier Represenaions Fourier series represen signals in erms of sinusoids. leads o a new represenaion for sysems as filers. Fourier Series
More information( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is
UNIT IMPULSE RESPONSE, UNIT STEP RESPONSE, STABILITY. Uni impulse funcion (Dirac dela funcion, dela funcion) rigorously defined is no sricly a funcion, bu disribuion (or measure), precise reamen requires
More informationMath Final Exam Solutions
Mah 246 - Final Exam Soluions Friday, July h, 204 () Find explici soluions and give he inerval of definiion o he following iniial value problems (a) ( + 2 )y + 2y = e, y(0) = 0 Soluion: In normal form,
More information6.003: Signal Processing
6.003: Signal Processing Coninuous-Time Fourier Transform Definiion Examples Properies Relaion o Fourier Series Sepember 5, 08 Quiz Thursday, Ocober 4, from 3pm o 5pm. No lecure on Ocober 4. The exam is
More informationDifferential Equations
Mah 21 (Fall 29) Differenial Equaions Soluion #3 1. Find he paricular soluion of he following differenial equaion by variaion of parameer (a) y + y = csc (b) 2 y + y y = ln, > Soluion: (a) The corresponding
More informationdt = C exp (3 ln t 4 ). t 4 W = C exp ( ln(4 t) 3) = C(4 t) 3.
Mah Rahman Exam Review Soluions () Consider he IVP: ( 4)y 3y + 4y = ; y(3) = 0, y (3) =. (a) Please deermine he longes inerval for which he IVP is guaraneed o have a unique soluion. Soluion: The disconinuiies
More informationDEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2008
[E5] IMPERIAL COLLEGE LONDON DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 008 EEE/ISE PART II MEng BEng and ACGI SIGNALS AND LINEAR SYSTEMS Time allowed: :00 hours There are FOUR quesions
More informationSignal and System (Chapter 3. Continuous-Time Systems)
Signal and Sysem (Chaper 3. Coninuous-Time Sysems) Prof. Kwang-Chun Ho kwangho@hansung.ac.kr Tel: 0-760-453 Fax:0-760-4435 1 Dep. Elecronics and Informaion Eng. 1 Nodes, Branches, Loops A nework wih b
More informationShort Introduction to Fractional Calculus
. Shor Inroducion o Fracional Calculus Mauro Bologna Deparameno de Física, Faculad de Ciencias Universidad de Tarapacá, Arica, Chile email: mbologna@ua.cl Absrac In he pas few years fracional calculus
More informationMatlab and Python programming: how to get started
Malab and Pyhon programming: how o ge sared Equipping readers he skills o wrie programs o explore complex sysems and discover ineresing paerns from big daa is one of he main goals of his book. In his chaper,
More informationMath 333 Problem Set #2 Solution 14 February 2003
Mah 333 Problem Se #2 Soluion 14 February 2003 A1. Solve he iniial value problem dy dx = x2 + e 3x ; 2y 4 y(0) = 1. Soluion: This is separable; we wrie 2y 4 dy = x 2 + e x dx and inegrae o ge The iniial
More informationSOLUTIONS TO ECE 3084
SOLUTIONS TO ECE 384 PROBLEM 2.. For each sysem below, specify wheher or no i is: (i) memoryless; (ii) causal; (iii) inverible; (iv) linear; (v) ime invarian; Explain your reasoning. If he propery is no
More information6.003: Signals and Systems. Relations among Fourier Representations
6.003: Signals and Sysems Relaions among Fourier Represenaions April 22, 200 Mid-erm Examinaion #3 W ednesday, April 28, 7:30-9:30pm. No reciaions on he day of he exam. Coverage: Lecures 20 Reciaions 20
More informationENGI 9420 Engineering Analysis Assignment 2 Solutions
ENGI 940 Engineering Analysis Assignmen Soluions 0 Fall [Second order ODEs, Laplace ransforms; Secions.0-.09]. Use Laplace ransforms o solve he iniial value problem [0] dy y, y( 0) 4 d + [This was Quesion
More informationChapter 6. Systems of First Order Linear Differential Equations
Chaper 6 Sysems of Firs Order Linear Differenial Equaions We will only discuss firs order sysems However higher order sysems may be made ino firs order sysems by a rick shown below We will have a sligh
More informationADDITIONAL PROBLEMS (a) Find the Fourier transform of the half-cosine pulse shown in Fig. 2.40(a). Additional Problems 91
ddiional Problems 9 n inverse relaionship exiss beween he ime-domain and freuency-domain descripions of a signal. Whenever an operaion is performed on he waveform of a signal in he ime domain, a corresponding
More informationQ1) [20 points] answer for the following questions (ON THIS SHEET):
Dr. Anas Al Tarabsheh The Hashemie Universiy Elecrical and Compuer Engineering Deparmen (Makeup Exam) Signals and Sysems Firs Semeser 011/01 Final Exam Dae: 1/06/01 Exam Duraion: hours Noe: means convoluion
More information+ γ3 A = I + A 1 (1 + γ + γ2. = I + A 1 ( t H O M E W O R K # 4 Sebastian A. Nugroho October 5, 2017
THE UNIVERSITY OF TEXAS AT SAN ANTONIO EE 543 LINEAR SYSTEMS AND CONTROL H O M E W O R K # 4 Sebasian A. Nugroho Ocober 5, 27 The objecive of his homework is o es your undersanding of he conen of Module
More informationNotes 04 largely plagiarized by %khc
Noes 04 largely plagiarized by %khc Convoluion Recap Some ricks: x() () =x() x() (, 0 )=x(, 0 ) R ț x() u() = x( )d x() () =ẋ() This hen ells us ha an inegraor has impulse response h() =u(), and ha a differeniaor
More informationDelhi Noida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubaneswar Kolkata Patna Web: Ph:
Serial : 0. ND_NW_EE_Signal & Sysems_4068 Delhi Noida Bhopal Hyderabad Jaipur Lucknow Indore Pune Bhubaneswar Kolkaa Pana Web: E-mail: info@madeeasy.in Ph: 0-4546 CLASS TEST 08-9 ELECTRICAL ENGINEERING
More informationLaplace Transforms. Examples. Is this equation differential? y 2 2y + 1 = 0, y 2 2y + 1 = 0, (y ) 2 2y + 1 = cos x,
Laplace Transforms Definiion. An ordinary differenial equaion is an equaion ha conains one or several derivaives of an unknown funcion which we call y and which we wan o deermine from he equaion. The equaion
More informationMath 2142 Exam 1 Review Problems. x 2 + f (0) 3! for the 3rd Taylor polynomial at x = 0. To calculate the various quantities:
Mah 4 Eam Review Problems Problem. Calculae he 3rd Taylor polynomial for arcsin a =. Soluion. Le f() = arcsin. For his problem, we use he formula f() + f () + f ()! + f () 3! for he 3rd Taylor polynomial
More informationLaplace Transform and its Relation to Fourier Transform
Chaper 6 Laplace Transform and is Relaion o Fourier Transform (A Brief Summary) Gis of he Maer 2 Domains of Represenaion Represenaion of signals and sysems Time Domain Coninuous Discree Time Time () [n]
More informationIII-A. Fourier Series Expansion
Summer 28 Signals & Sysems S.F. Hsieh III-A. Fourier Series Expansion Inroducion. Divide and conquer Signals can be decomposed as linear combinaions of: (a) shifed impulses: (sifing propery) Why? x() x()δ(
More informationHint: There's a table of particular solutions at the end of today's notes.
Mah 8- Fri Apr 4, Finish Wednesday's noes firs Then 94 Forced oscillaion problems via Fourier Series Today we will revisi he forced oscillaion problems of las Friday, where we prediced wheher or no resonance
More informationKEY. Math 334 Midterm III Fall 2008 sections 001 and 003 Instructor: Scott Glasgow
KEY Mah 334 Miderm III Fall 28 secions and 3 Insrucor: Sco Glasgow Please do NOT wrie on his exam. No credi will be given for such work. Raher wrie in a blue book, or on your own paper, preferably engineering
More informationSystem Processes input signal (excitation) and produces output signal (response)
Signal A funcion of ime Sysem Processes inpu signal (exciaion) and produces oupu signal (response) Exciaion Inpu Sysem Oupu Response 1. Types of signals 2. Going from analog o digial world 3. An example
More informationd 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3
and d = c b - b c c d = c b - b c c This process is coninued unil he nh row has been compleed. The complee array of coefficiens is riangular. Noe ha in developing he array an enire row may be divided or
More informationTHE 2-BODY PROBLEM. FIGURE 1. A pair of ellipses sharing a common focus. (c,b) c+a ROBERT J. VANDERBEI
THE 2-BODY PROBLEM ROBERT J. VANDERBEI ABSTRACT. In his shor noe, we show ha a pair of ellipses wih a common focus is a soluion o he 2-body problem. INTRODUCTION. Solving he 2-body problem from scrach
More information1 1 + x 2 dx. tan 1 (2) = ] ] x 3. Solution: Recall that the given integral is improper because. x 3. 1 x 3. dx = lim dx.
. Use Simpson s rule wih n 4 o esimae an () +. Soluion: Since we are using 4 seps, 4 Thus we have [ ( ) f() + 4f + f() + 4f 3 [ + 4 4 6 5 + + 4 4 3 + ] 5 [ + 6 6 5 + + 6 3 + ]. 5. Our funcion is f() +.
More informationLet us start with a two dimensional case. We consider a vector ( x,
Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our
More information6.003: Signals and Systems
6.003: Signals and Sysems Relaions among Fourier Represenaions November 5, 20 Mid-erm Examinaion #3 Wednesday, November 6, 7:30-9:30pm, No reciaions on he day of he exam. Coverage: Lecures 8 Reciaions
More informationThe equation to any straight line can be expressed in the form:
Sring Graphs Par 1 Answers 1 TI-Nspire Invesigaion Suden min Aims Deermine a series of equaions of sraigh lines o form a paern similar o ha formed by he cables on he Jerusalem Chords Bridge. Deermine he
More informationSINUSOIDAL WAVEFORMS
SINUSOIDAL WAVEFORMS The sinusoidal waveform is he only waveform whose shape is no affeced by he response characerisics of R, L, and C elemens. Enzo Paerno CIRCUIT ELEMENTS R [ Ω ] Resisance: Ω: Ohms Georg
More informationRadical Expressions. Terminology: A radical will have the following; a radical sign, a radicand, and an index.
Radical Epressions Wha are Radical Epressions? A radical epression is an algebraic epression ha conains a radical. The following are eamples of radical epressions + a Terminology: A radical will have he
More informationEE363 homework 1 solutions
EE363 Prof. S. Boyd EE363 homework 1 soluions 1. LQR for a riple accumulaor. We consider he sysem x +1 = Ax + Bu, y = Cx, wih 1 1 A = 1 1, B =, C = [ 1 ]. 1 1 This sysem has ransfer funcion H(z) = (z 1)
More informationLinear Time-invariant systems, Convolution, and Cross-correlation
Linear Time-invarian sysems, Convoluion, and Cross-correlaion (1) Linear Time-invarian (LTI) sysem A sysem akes in an inpu funcion and reurns an oupu funcion. x() T y() Inpu Sysem Oupu y() = T[x()] An
More informationThe Laplace Transform
The Laplace Transform Previous basis funcions: 1, x, cosx, sinx, exp(jw). New basis funcion for he LT => complex exponenial funcions LT provides a broader characerisics of CT signals and CT LTI sysems
More informationConcourse Math Spring 2012 Worked Examples: Matrix Methods for Solving Systems of 1st Order Linear Differential Equations
Concourse Mah 80 Spring 0 Worked Examples: Marix Mehods for Solving Sysems of s Order Linear Differenial Equaions The Main Idea: Given a sysem of s order linear differenial equaions d x d Ax wih iniial
More information6.003 Homework #13 Solutions
6.003 Homework #3 Soluions Problems. Transformaion Consider he following ransformaion from x() o y(): x() w () w () w 3 () + y() p() cos() where p() = δ( k). Deermine an expression for y() when x() = sin(/)/().
More informationChapter One Fourier Series and Fourier Transform
Chaper One I. Fourier Series Represenaion of Periodic Signals -Trigonomeric Fourier Series: The rigonomeric Fourier series represenaion of a periodic signal x() x( + T0 ) wih fundamenal period T0 is given
More informationSolutions of Sample Problems for Third In-Class Exam Math 246, Spring 2011, Professor David Levermore
Soluions of Sample Problems for Third In-Class Exam Mah 6, Spring, Professor David Levermore Compue he Laplace ransform of f e from is definiion Soluion The definiion of he Laplace ransform gives L[f]s
More information2. Nonlinear Conservation Law Equations
. Nonlinear Conservaion Law Equaions One of he clear lessons learned over recen years in sudying nonlinear parial differenial equaions is ha i is generally no wise o ry o aack a general class of nonlinear
More informationTHE BERNOULLI NUMBERS. t k. = lim. = lim = 1, d t B 1 = lim. 1+e t te t = lim t 0 (e t 1) 2. = lim = 1 2.
THE BERNOULLI NUMBERS The Bernoulli numbers are defined here by he exponenial generaing funcion ( e The firs one is easy o compue: (2 and (3 B 0 lim 0 e lim, 0 e ( d B lim 0 d e +e e lim 0 (e 2 lim 0 2(e
More informationChapter 8 The Complete Response of RL and RC Circuits
Chaper 8 The Complee Response of RL and RC Circuis Seoul Naional Universiy Deparmen of Elecrical and Compuer Engineering Wha is Firs Order Circuis? Circuis ha conain only one inducor or only one capacior
More informationLinear Surface Gravity Waves 3., Dispersion, Group Velocity, and Energy Propagation
Chaper 4 Linear Surface Graviy Waves 3., Dispersion, Group Velociy, and Energy Propagaion 4. Descripion In many aspecs of wave evoluion, he concep of group velociy plays a cenral role. Mos people now i
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 3 Signals & Sysems Prof. Mark Fowler Noe Se # Wha are Coninuous-Time Signals??? /6 Coninuous-Time Signal Coninuous Time (C-T) Signal: A C-T signal is defined on he coninuum of ime values. Tha is:
More information( ) ( ) ( ) () () Signals And Systems Exam#1. 1. Given x(t) and y(t) below: x(t) y(t) (A) Give the expression of x(t) in terms of step functions.
Signals And Sysems Exam#. Given x() and y() below: x() y() 4 4 (A) Give he expression of x() in erms of sep funcions. (%) x () = q() q( ) + q( 4) (B) Plo x(.5). (%) x() g() = x( ) h() = g(. 5) = x(. 5)
More informationFrom Complex Fourier Series to Fourier Transforms
Topic From Complex Fourier Series o Fourier Transforms. Inroducion In he previous lecure you saw ha complex Fourier Series and is coeciens were dened by as f ( = n= C ne in! where C n = T T = T = f (e
More informationY 0.4Y 0.45Y Y to a proper ARMA specification.
HG Jan 04 ECON 50 Exercises II - 0 Feb 04 (wih answers Exercise. Read secion 8 in lecure noes 3 (LN3 on he common facor problem in ARMA-processes. Consider he following process Y 0.4Y 0.45Y 0.5 ( where
More informationSignals and Systems Profs. Byron Yu and Pulkit Grover Fall Midterm 1 Solutions
8-90 Signals and Sysems Profs. Byron Yu and Pulki Grover Fall 07 Miderm Soluions Name: Andrew ID: Problem Score Max 0 8 4 6 5 0 6 0 7 8 9 0 6 Toal 00 Miderm Soluions. (0 poins) Deermine wheher he following
More informationWeek 1 Lecture 2 Problems 2, 5. What if something oscillates with no obvious spring? What is ω? (problem set problem)
Week 1 Lecure Problems, 5 Wha if somehing oscillaes wih no obvious spring? Wha is ω? (problem se problem) Sar wih Try and ge o SHM form E. Full beer can in lake, oscillaing F = m & = ge rearrange: F =
More informationSample Final Exam (finals03) Covering Chapters 1-9 of Fundamentals of Signals & Systems
Sample Final Exam Covering Chaper 9 (final04) Sample Final Exam (final03) Covering Chaper 9 of Fundamenal of Signal & Syem Problem (0 mar) Conider he caual opamp circui iniially a re depiced below. I LI
More informationMath Week 15: Section 7.4, mass-spring systems. These are notes for Monday. There will also be course review notes for Tuesday, posted later.
Mah 50-004 Week 5: Secion 7.4, mass-spring sysems. These are noes for Monday. There will also be course review noes for Tuesday, posed laer. Mon Apr 3 7.4 mass-spring sysems. Announcemens: Warm up exercise:
More informationMATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence
MATH 433/533, Fourier Analysis Secion 6, Proof of Fourier s Theorem for Poinwise Convergence Firs, some commens abou inegraing periodic funcions. If g is a periodic funcion, g(x + ) g(x) for all real x,
More informationCHAPTER 12 DIRECT CURRENT CIRCUITS
CHAPTER 12 DIRECT CURRENT CIUITS DIRECT CURRENT CIUITS 257 12.1 RESISTORS IN SERIES AND IN PARALLEL When wo resisors are conneced ogeher as shown in Figure 12.1 we said ha hey are conneced in series. As
More informationGuest Lectures for Dr. MacFarlane s EE3350 Part Deux
Gues Lecures for Dr. MacFarlane s EE3350 Par Deux Michael Plane Mon., 08-30-2010 Wrie name in corner. Poin ou his is a review, so I will go faser. Remind hem o go lisen o online lecure abou geing an A
More informationSolution of Integro-Differential Equations by Using ELzaki Transform
Global Journal of Mahemaical Sciences: Theory and Pracical. Volume, Number (), pp. - Inernaional Research Publicaion House hp://www.irphouse.com Soluion of Inegro-Differenial Equaions by Using ELzaki Transform
More informationVoltage/current relationship Stored Energy. RL / RC circuits Steady State / Transient response Natural / Step response
Review Capaciors/Inducors Volage/curren relaionship Sored Energy s Order Circuis RL / RC circuis Seady Sae / Transien response Naural / Sep response EE4 Summer 5: Lecure 5 Insrucor: Ocavian Florescu Lecure
More informationLecture #6: Continuous-Time Signals
EEL5: Discree-Time Signals and Sysems Lecure #6: Coninuous-Time Signals Lecure #6: Coninuous-Time Signals. Inroducion In his lecure, we discussed he ollowing opics:. Mahemaical represenaion and ransormaions
More informationChapter 7 Response of First-order RL and RC Circuits
Chaper 7 Response of Firs-order RL and RC Circuis 7.- The Naural Response of RL and RC Circuis 7.3 The Sep Response of RL and RC Circuis 7.4 A General Soluion for Sep and Naural Responses 7.5 Sequenial
More information2.7. Some common engineering functions. Introduction. Prerequisites. Learning Outcomes
Some common engineering funcions 2.7 Inroducion This secion provides a caalogue of some common funcions ofen used in Science and Engineering. These include polynomials, raional funcions, he modulus funcion
More informationSpectral Analysis. Joseph Fourier The two representations of a signal are connected via the Fourier transform. Z x(t)exp( j2πft)dt
Specral Analysis Asignalx may be represened as a funcion of ime as x() or as a funcion of frequency X(f). This is due o relaionships developed by a French mahemaician, physicis, and Egypologis, Joseph
More informationINDEX. Transient analysis 1 Initial Conditions 1
INDEX Secion Page Transien analysis 1 Iniial Condiions 1 Please inform me of your opinion of he relaive emphasis of he review maerial by simply making commens on his page and sending i o me a: Frank Mera
More informationClass Meeting # 10: Introduction to the Wave Equation
MATH 8.5 COURSE NOTES - CLASS MEETING # 0 8.5 Inroducion o PDEs, Fall 0 Professor: Jared Speck Class Meeing # 0: Inroducion o he Wave Equaion. Wha is he wave equaion? The sandard wave equaion for a funcion
More information