Finite-Difference Time-Domain Method (FDTD)

Size: px
Start display at page:

Download "Finite-Difference Time-Domain Method (FDTD)"

Transcription

1 Computatioal Photoics, Summer Term 04, Abbe School of Photoics, FSU Jea, Prof. Thomas Pertsch Computatioal Photoics Semiar 05, 0 Jue 04 Fiite-Differece Time-Domai Method (FDTD) Lear how to implemet a D versio of FDTD xted the code to 3D problems (volutar) lear how to save simulatio results i movie format

2 Computatioal Photoics, Summer Term 04, Abbe School of Photoics, FSU Jea, Prof. Thomas Pertsch D FDTD: Yee Grid for & compoets chagig of idex otatio to iteger idices t().5 t() x(i) t t j i i i i i 0i x 0i t i i i i 0 x 3 x(i) t t j i i i i i 0i x 0i t i i i x 0 i

3 Computatioal Photoics, Summer Term 04, Abbe School of Photoics, FSU Jea, Prof. Thomas Pertsch 3D FDTD: lectric field compoets chagig of idex otatio to iteger idices (k) x x x (j) x x(i),,,, t i j k i j k i, j, k i, j, k x i, j, k x j i, j, k x i, j, k 0 i, j, k j t x,, x,, i j k i j k i, j, k i, j, k i, j, k i, j, k i, j, k 0i, j, k x,,,, x t i j k i j k i, j, k x i, j, k i, j, k j i, j, k i, j, k 0 i, j, k x

4 Computatioal Photoics, Summer Term 04, Abbe School of Photoics, FSU Jea, Prof. Thomas Pertsch 3D FDTD: Magetic field compoets chagig of idex otatio to iteger idices (k) x x x (j) x x(i) x t,,,, i j k i j k i, j, k i, j, k i, j, k x i, j, k 0 t,,,, x i j k i j k i, j, k x i, j, k i, j, k i, j, k 0 x,,,, t x i j k x i j k i, j, k i, j, k i, j, k i, j, k 0 x

5 Computatioal Photoics, Summer Term 04, Abbe School of Photoics, FSU Jea, Prof. Thomas Pertsch 3D FDTD: Field discretiatio ad boudar coditios Number of grid poits i Yee grid:,,,,,,,,,,,, x x x x N N N N N N x x N N N N N N x x N N N N N N Boudar coditios for ud fields: x x x x x(i) (k) (j) x :,,: 0, :,:, 0 :, N,: 0, :,:, N 0 x x x,:,: 0, :,:, 0 N,:,: 0, :,:, N 0 x,:,: 0, :,,: 0 N,:,: 0, :, N,: 0 x

6 Computatioal Photoics, Summer Term 04, Abbe School of Photoics, FSU Jea, Prof. Thomas Pertsch Task I: Implemetatio of D FDTD method Phsical problem: Simulate propagatio of a ultrashort pulse i a dispersio-free dielectric medium (x) See what happes whe the pulse hits the iterface betwee two differet dielectric media xcitatio: curret source (A/m ) with frequec 5*0 5 (red light) with delta-shaped spatial profile ad Gauss-shaped temporal pulse profile of width w 0 = fs Simulatio grid: spatial sie of 8 µm with discretiatio x=30m ad metallic walls ( =0 at the boudaries) temporal sie of 60 fs with discretiatio t=x/(c) Result output: graphical presetatio ad for ever 5 th calculatio step Useful Matlab fuctios: roud, drawow, subplot c= *0 8, µ 0 =4p*0-7, 0 =/(c *µ 0 )

7 Computatioal Photoics, Summer Term 04, Abbe School of Photoics, FSU Jea, Prof. Thomas Pertsch Task I: Implemetatio of D FDTD method fuctio [,,X,T] = fdtd_d(eps_rel, grid_sie, time_spa,... source_freuqec, source_positio, source_pulse_legth) % FUNCTION CALL: % [,,X,T] = fdtd_d(eps_rel,grid_sie, time_spa,... source_freuqec, source_positio, source_pulse_legth) % VARIABLS: % eps_rel epsilo distributio i the simulatio area % grid_sie spatial sie of simulatio grid % time_spa time spa of simulatio % source_frequec frequec of curret source % source_positio spatial positio of curret source % source_pulse_legth temporal width of Gauss-shaped source(w0) %

8 Computatioal Photoics, Summer Term 04, Abbe School of Photoics, FSU Jea, Prof. Thomas Pertsch Task II: Implemetatio of 3D FDTD method Phsical problem: Watch radiatio characteristics of pulsed ertia dipole Simulatio grid: spatial sie of 0xx3 grid poits with discretiatio x===30m metallic wall boudar coditios temporal sie of 0 fs with discretiatio t=x/(c) xcitatio: curret source (A/m ) with frequec 5*0 5 (cw red light) delta-shaped spatial profile i ceter of computatio grid Result output: graphical represetatio of i the x- plae cetered i the middle alog the directio (plot ever 5 th calculatio step) Possibl useful Matlab fuctios: pcolor, mod

9 Computatioal Photoics, Summer Term 04, Abbe School of Photoics, FSU Jea, Prof. Thomas Pertsch Task II: Implemetatio of 3D FDTD method fuctio fdtd_3d(eps_rel,dr,time_spa,source_frequec,source_pulse_legth,jx,j,j,... plot_field_compoet, plotlaer) % FUNCTION CALL % fdtd_3d(eps_rel,dr,time_spa,source_frequec,source_pulse_legth,jx,j,j,... % plot_field_compoet, plotlaer) % VARIABLS: % eps_rel epsilo distributio i the simulatio area (3D arra) % dr spatial discretiatio of simulatio grid % time_spa time spa of simulatio % source_frequec frequec of curret source % source_pulse_legth temporal width of Gaussia pulse % jx,j,j compoets of source curret (3D arra) % plot_field_compoet field compoet to be plot % plotlaer idex of slice, i which result is plotted i D

10 Computatioal Photoics, Summer Term 04, Abbe School of Photoics, FSU Jea, Prof. Thomas Pertsch Task III*: FDTD movie volutar task Output the result of the FDTD as a plaable movie file! Remark: This fuctioalit will deped o the video codecs istalled o the specific sstem! Possibl useful MATLAB fuctios: switch, avifile, getframe, addframe, close

11 Computatioal Photoics, Summer Term 04, Abbe School of Photoics, FSU Jea, Prof. Thomas Pertsch omework 4 ( Jue 04) Solve at least tasks I & II. Prepare a oe page report about our solutio with a figure of some calculated example. Submit our m-files of our program together with our oe page report electroicall to teachig-aooptics@ui-jea.de b 3 Jue 0. Please put everthig together i oe sigle which cotais our ame (FAMILY NAM, Give Name) ad matriculatio umber. Late submissios will ot be accepted! 4 Jue the solutios of the tasks will be available olie at the lectures homepage >>> Computatioal Photoics. You are expected to solve the task ourself ad a declaratio of idepedet work must be siged b ever studet at the ed of the semester.

Finite-Difference Time-Domain Method (FDTD)

Finite-Difference Time-Domain Method (FDTD) Computatioal Photoics, Summer Term 0, Abbe School of Photoics, FSU Jea, Prof. Thomas Pertsch Computatioal Photoics Semiar 06, 8 Jue 0 Fiite-Differece Time-Domai Method (FDTD) Lear how to implemet a oe-dimesioal

More information

Finite-Difference Time-Domain Method (FDTD)

Finite-Difference Time-Domain Method (FDTD) Computatioal Photoics Semiar 05 Fiite-Differece Time-Domai Method (FDTD) Lear how to implemet a D versio of the FDTD method ted the code to 3D problems D FDTD: Yee Grid for & Compoets Chagig of ide otatio

More information

Ray Optics Theory and Mode Theory. Dr. Mohammad Faisal Dept. of EEE, BUET

Ray Optics Theory and Mode Theory. Dr. Mohammad Faisal Dept. of EEE, BUET Ray Optics Theory ad Mode Theory Dr. Mohammad Faisal Dept. of, BUT Optical Fiber WG For light to be trasmitted through fiber core, i.e., for total iteral reflectio i medium, > Ray Theory Trasmissio Ray

More information

FFTs in Graphics and Vision. The Fast Fourier Transform

FFTs in Graphics and Vision. The Fast Fourier Transform FFTs i Graphics ad Visio The Fast Fourier Trasform 1 Outlie The FFT Algorithm Applicatios i 1D Multi-Dimesioal FFTs More Applicatios Real FFTs 2 Computatioal Complexity To compute the movig dot-product

More information

Analysis Methods for Slab Waveguides

Analysis Methods for Slab Waveguides Aalsis Methods for Slab Waveguides Maxwell s Equatios ad Wave Equatios Aaltical Methods for Waveguide Aalsis: Marcatilis Method Simple Effective Idex Method Numerical Methods for Waveguide Aalsis: Fiite-Elemet

More information

Lecture 2: Finite Difference Methods in Heat Transfer

Lecture 2: Finite Difference Methods in Heat Transfer Lecture 2: Fiite Differece Methods i Heat Trasfer V.Vuorie Aalto Uiversity School of Egieerig Heat ad Mass Trasfer Course, Autum 2016 November 1 st 2017, Otaiemi ville.vuorie@aalto.fi Overview Part 1 (

More information

Signals & Systems Chapter3

Signals & Systems Chapter3 Sigals & Systems Chapter3 1.2 Discrete-Time (D-T) Sigals Electroic systems do most of the processig of a sigal usig a computer. A computer ca t directly process a C-T sigal but istead eeds a stream of

More information

G-2 Applied Computational Aerodynamics. G-1 Program THINFOIL

G-2 Applied Computational Aerodynamics. G-1 Program THINFOIL G-2 Applied Computatioal Aerodyamics G-1 Program THINFOIL THINFOIL solves Laplace s Equatio by fiite differeces usig a variety of iteratio methods. It was writte by Valery Razgoyaev. The iteratio optios

More information

GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING

GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING ECE 06 Summer 07 Problem Set #5 Assiged: Jue 3, 07 Due Date: Jue 30, 07 Readig: Chapter 5 o FIR Filters. PROBLEM 5..* (The

More information

EE692 Applied EM- FDTD Method Chapter 3 Introduction to Maxwell s Equations and the Yee Algorithm. ds dl ds

EE692 Applied EM- FDTD Method Chapter 3 Introduction to Maxwell s Equations and the Yee Algorithm. ds dl ds C:\tom\classes_grad\ee692_FDTD\otes\chap_03\ee692_FDTD_chap_03.doc Page 1 of 38 EE692 Applied EM- FDTD Method Chapter 3 Itroductio to Mawell s Equatios ad the Yee Algorithm 3.2 Mawell s Equatios i Three

More information

} is said to be a Cauchy sequence provided the following condition is true.

} is said to be a Cauchy sequence provided the following condition is true. Math 4200, Fial Exam Review I. Itroductio to Proofs 1. Prove the Pythagorea theorem. 2. Show that 43 is a irratioal umber. II. Itroductio to Logic 1. Costruct a truth table for the statemet ( p ad ~ r

More information

Question 1: The magnetic case

Question 1: The magnetic case September 6, 018 Corell Uiversity, Departmet of Physics PHYS 337, Advace E&M, HW # 4, due: 9/19/018, 11:15 AM Questio 1: The magetic case I class, we skipped over some details, so here you are asked to

More information

Finite-Difference Time-Domain Simulation of Light Propagation in 2D Periodic and Quasi-Periodic Photonic Structures

Finite-Difference Time-Domain Simulation of Light Propagation in 2D Periodic and Quasi-Periodic Photonic Structures JNS 3 (213) 359-364 Fiite-Differece Time-Domai Simulatio of Light Propagatio i 2D Periodic ad Quasi-Periodic Photoic Structures N. Dadashadeh a,b*, O.G. Romaov b a Islamic Aad Uiversity of Hadishahr, Hadishahr,

More information

PC5215 Numerical Recipes with Applications - Review Problems

PC5215 Numerical Recipes with Applications - Review Problems PC55 Numerical Recipes with Applicatios - Review Problems Give the IEEE 754 sigle precisio bit patter (biary or he format) of the followig umbers: 0 0 05 00 0 00 Note that it has 8 bits for the epoet,

More information

Math 5311 Problem Set #5 Solutions

Math 5311 Problem Set #5 Solutions Math 5311 Problem Set #5 Solutios March 9, 009 Problem 1 O&S 11.1.3 Part (a) Solve with boudary coditios u = 1 0 x < L/ 1 L/ < x L u (0) = u (L) = 0. Let s refer to [0, L/) as regio 1 ad (L/, L] as regio.

More information

Lecture 33: Bootstrap

Lecture 33: Bootstrap Lecture 33: ootstrap Motivatio To evaluate ad compare differet estimators, we eed cosistet estimators of variaces or asymptotic variaces of estimators. This is also importat for hypothesis testig ad cofidece

More information

FIR Filters. Lecture #7 Chapter 5. BME 310 Biomedical Computing - J.Schesser

FIR Filters. Lecture #7 Chapter 5. BME 310 Biomedical Computing - J.Schesser FIR Filters Lecture #7 Chapter 5 8 What Is this Course All About? To Gai a Appreciatio of the Various Types of Sigals ad Systems To Aalyze The Various Types of Systems To Lear the Skills ad Tools eeded

More information

ADVANCED DIGITAL SIGNAL PROCESSING

ADVANCED DIGITAL SIGNAL PROCESSING ADVANCED DIGITAL SIGNAL PROCESSING PROF. S. C. CHAN (email : sccha@eee.hku.hk, Rm. CYC-702) DISCRETE-TIME SIGNALS AND SYSTEMS MULTI-DIMENSIONAL SIGNALS AND SYSTEMS RANDOM PROCESSES AND APPLICATIONS ADAPTIVE

More information

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

More information

Fall 2018 Exam 3 HAND IN PART 0 10 PIN: 17 INSTRUCTIONS

Fall 2018 Exam 3 HAND IN PART 0 10 PIN: 17 INSTRUCTIONS MARK BOX problem poits HAND IN PART 0 10 1 10 2 5 NAME: Solutios 3 10 PIN: 17 4 16 65=13x5 % 100 INSTRUCTIONS This exam comes i two parts. (1) HAND IN PART. Had i oly this part. (2) STATEMENT OF MULTIPLE

More information

Infinite Sequences and Series

Infinite Sequences and Series Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet

More information

Lecture 8: Solving the Heat, Laplace and Wave equations using finite difference methods

Lecture 8: Solving the Heat, Laplace and Wave equations using finite difference methods Itroductory lecture otes o Partial Differetial Equatios - c Athoy Peirce. Not to be copied, used, or revised without explicit writte permissio from the copyright ower. 1 Lecture 8: Solvig the Heat, Laplace

More information

It is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function.

It is always the case that unions, intersections, complements, and set differences are preserved by the inverse image of a function. MATH 532 Measurable Fuctios Dr. Neal, WKU Throughout, let ( X, F, µ) be a measure space ad let (!, F, P ) deote the special case of a probability space. We shall ow begi to study real-valued fuctios defied

More information

REFLECTION AND REFRACTION

REFLECTION AND REFRACTION REFLECTION AND REFRACTION REFLECTION AND TRANSMISSION FOR NORMAL INCIDENCE ON A DIELECTRIC MEDIUM Assumptios: No-magetic media which meas that B H. No dampig, purely dielectric media. No free surface charges.

More information

Data Description. Measure of Central Tendency. Data Description. Chapter x i

Data Description. Measure of Central Tendency. Data Description. Chapter x i Data Descriptio Describe Distributio with Numbers Example: Birth weights (i lb) of 5 babies bor from two groups of wome uder differet care programs. Group : 7, 6, 8, 7, 7 Group : 3, 4, 8, 9, Chapter 3

More information

Signal Processing. Lecture 02: Discrete Time Signals and Systems. Ahmet Taha Koru, Ph. D. Yildiz Technical University.

Signal Processing. Lecture 02: Discrete Time Signals and Systems. Ahmet Taha Koru, Ph. D. Yildiz Technical University. Sigal Processig Lecture 02: Discrete Time Sigals ad Systems Ahmet Taha Koru, Ph. D. Yildiz Techical Uiversity 2017-2018 Fall ATK (YTU) Sigal Processig 2017-2018 Fall 1 / 51 Discrete Time Sigals Discrete

More information

Student Supplement for Optoelectronics and Photonics: Principles and Practices Second Edition A Complete Course in Power Point.

Student Supplement for Optoelectronics and Photonics: Principles and Practices Second Edition A Complete Course in Power Point. Studet Suppleet for Optoelectroics ad Photoics: Priciples ad Practices Secod Editio A Coplete Course i Power Poit Chapter [ February 04] Copyright Iforatio ad Perissio: Part I This Power Poit presetatio

More information

Math 257: Finite difference methods

Math 257: Finite difference methods Math 257: Fiite differece methods 1 Fiite Differeces Remember the defiitio of a derivative f f(x + ) f(x) (x) = lim 0 Also recall Taylor s formula: (1) f(x + ) = f(x) + f (x) + 2 f (x) + 3 f (3) (x) +...

More information

Department of Informatics Prof. Dr. Michael Böhlen Binzmühlestrasse Zurich Phone:

Department of Informatics Prof. Dr. Michael Böhlen Binzmühlestrasse Zurich Phone: Departmet of Iformatics Prof. Dr. Michael Böhle Bizmühlestrasse 14 8050 Zurich Phoe: +41 44 635 4333 Email: boehle@ifi.uzh.ch Iformatik II Midterm1 Sprig 018 3.03.018 Advice You have 90 miutes to complete

More information

PHYS-3301 Lecture 10. Wave Packet Envelope Wave Properties of Matter and Quantum Mechanics I CHAPTER 5. Announcement. Sep.

PHYS-3301 Lecture 10. Wave Packet Envelope Wave Properties of Matter and Quantum Mechanics I CHAPTER 5. Announcement. Sep. Aoucemet Course webpage http://www.phys.ttu.edu/~slee/3301/ PHYS-3301 Lecture 10 HW3 (due 10/4) Chapter 5 4, 8, 11, 15, 22, 27, 36, 40, 42 Sep. 27, 2018 Exam 1 (10/4) Chapters 3, 4, & 5 CHAPTER 5 Wave

More information

CHAPTER XI DATAPATH ELEMENTS

CHAPTER XI DATAPATH ELEMENTS CHAPTER XI- CHAPTER XI CHAPTER XI READ REE-DOC ON COURSE WEBPAGE CHAPTER XI-2 INTRODUCTION -INTRODUCTION So far we have discussed may small compoets ad buildig blocks. Oe fial step i our buildig blocks

More information

MATH spring 2008 lecture 3 Answers to selected problems. 0 sin14 xdx = x dx. ; (iv) x +

MATH spring 2008 lecture 3 Answers to selected problems. 0 sin14 xdx = x dx. ; (iv) x + MATH - sprig 008 lecture Aswers to selected problems INTEGRALS. f =? For atiderivatives i geeral see the itegrals website at http://itegrals.wolfram.com. (5-vi (0 i ( ( i ( π ; (v π a. This is example

More information

Problem Set # 5 Solutions

Problem Set # 5 Solutions MIT./8.4/6.898/8.435 Quatum Iformatio Sciece I Fall, 00 Sam Ocko October 5, 00 Problem Set # 5 Solutios. Most uitar trasforms are hard to approimate. (a) We are dealig with boolea fuctios that take bits

More information

Expectation-Maximization Algorithm.

Expectation-Maximization Algorithm. Expectatio-Maximizatio Algorithm. Petr Pošík Czech Techical Uiversity i Prague Faculty of Electrical Egieerig Dept. of Cyberetics MLE 2 Likelihood.........................................................................................................

More information

Finite Difference Derivations for Spreadsheet Modeling John C. Walton Modified: November 15, 2007 jcw

Finite Difference Derivations for Spreadsheet Modeling John C. Walton Modified: November 15, 2007 jcw Fiite Differece Derivatios for Spreadsheet Modelig Joh C. Walto Modified: November 15, 2007 jcw Figure 1. Suset with 11 swas o Little Platte Lake, Michiga. Page 1 Modificatio Date: November 15, 2007 Review

More information

Annotations to the assignments and the solution sheet. Note the following points

Annotations to the assignments and the solution sheet. Note the following points WS 26/7 Trial Exam: Fudametals of Computer Egieerig Seite: Aotatios to the assigmets ad the solutio sheet This is a multiple choice examiatio, that meas: Solutio approaches are ot assessed. For each sub-task

More information

Lecture 1 Probability and Statistics

Lecture 1 Probability and Statistics Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark

More information

Mini Lecture 10.1 Radical Expressions and Functions. 81x d. x 4x 4

Mini Lecture 10.1 Radical Expressions and Functions. 81x d. x 4x 4 Mii Lecture 0. Radical Expressios ad Fuctios Learig Objectives:. Evaluate square roots.. Evaluate square root fuctios.. Fid the domai of square root fuctios.. Use models that are square root fuctios. 5.

More information

(# x) 2 n. (" x) 2 = 30 2 = 900. = sum. " x 2 = =174. " x. Chapter 12. Quick math overview. #(x " x ) 2 = # x 2 "

(# x) 2 n. ( x) 2 = 30 2 = 900. = sum.  x 2 = =174.  x. Chapter 12. Quick math overview. #(x  x ) 2 = # x 2 Chapter 12 Describig Distributios with Numbers Chapter 12 1 Quick math overview = sum These expressios are algebraically equivalet #(x " x ) 2 = # x 2 " (# x) 2 Examples x :{ 2,3,5,6,6,8 } " x = 2 + 3+

More information

A Review of the Total Field/Scattered Field Technique for the FDTD Method

A Review of the Total Field/Scattered Field Technique for the FDTD Method Forum for Electromagetic Research Methods ad Applicatios Techologies (FERMAT) A Review of the / Techique for the FDTD Method Mike Potter * ad Jea-Pierre Béreger ** * Electrical ad Computer Egieerig Uiversit

More information

Math 132, Fall 2009 Exam 2: Solutions

Math 132, Fall 2009 Exam 2: Solutions Math 3, Fall 009 Exam : Solutios () a) ( poits) Determie for which positive real umbers p, is the followig improper itegral coverget, ad for which it is diverget. Evaluate the itegral for each value of

More information

Find a formula for the exponential function whose graph is given , 1 2,16 1, 6

Find a formula for the exponential function whose graph is given , 1 2,16 1, 6 Math 4 Activity (Due by EOC Apr. ) Graph the followig epoetial fuctios by modifyig the graph of f. Fid the rage of each fuctio.. g. g. g 4. g. g 6. g Fid a formula for the epoetial fuctio whose graph is

More information

Lecture 1 Probability and Statistics

Lecture 1 Probability and Statistics Wikipedia: Lecture 1 Probability ad Statistics Bejami Disraeli, British statesma ad literary figure (1804 1881): There are three kids of lies: lies, damed lies, ad statistics. popularized i US by Mark

More information

Finite Dierence Schemes

Finite Dierence Schemes MATH-459 Numerical Methods for Coservatio Laws by Prof. Ja S. Hesthave Solutio set 2: Fiite Dierece Schemes Exercise 2. Cosistecy A method is cosistet if its local trucatio error T k satises T k (x, t)

More information

(4 pts.) (4 pts.) (4 pts.) b) y(x,t) = 1/(ax 2 +b) This function has no time dependence, so cannot be a wave.

(4 pts.) (4 pts.) (4 pts.) b) y(x,t) = 1/(ax 2 +b) This function has no time dependence, so cannot be a wave. 12. For each of the possible wave forms below, idicate which satisf the wave equatio, ad which represet reasoable waveforms for actual waves o a strig. For those which do represet waves, fid the speed

More information

Lecture 7: Polar representation of complex numbers

Lecture 7: Polar representation of complex numbers Lecture 7: Polar represetatio of comple umbers See FLAP Module M3.1 Sectio.7 ad M3. Sectios 1 ad. 7.1 The Argad diagram I two dimesioal Cartesia coordiates (,), we are used to plottig the fuctio ( ) with

More information

STAT Homework 1 - Solutions

STAT Homework 1 - Solutions STAT-36700 Homework 1 - Solutios Fall 018 September 11, 018 This cotais solutios for Homework 1. Please ote that we have icluded several additioal commets ad approaches to the problems to give you better

More information

Let us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f.

Let us give one more example of MLE. Example 3. The uniform distribution U[0, θ] on the interval [0, θ] has p.d.f. Lecture 5 Let us give oe more example of MLE. Example 3. The uiform distributio U[0, ] o the iterval [0, ] has p.d.f. { 1 f(x =, 0 x, 0, otherwise The likelihood fuctio ϕ( = f(x i = 1 I(X 1,..., X [0,

More information

Chapter 2 Feedback Control Theory Continued

Chapter 2 Feedback Control Theory Continued Chapter Feedback Cotrol Theor Cotiued. Itroductio I the previous chapter, the respose characteristic of simple first ad secod order trasfer fuctios were studied. It was show that first order trasfer fuctio,

More information

Some Basic Probability Concepts. 2.1 Experiments, Outcomes and Random Variables

Some Basic Probability Concepts. 2.1 Experiments, Outcomes and Random Variables Some Basic Probability Cocepts 2. Experimets, Outcomes ad Radom Variables A radom variable is a variable whose value is ukow util it is observed. The value of a radom variable results from a experimet;

More information

Hashing and Amortization

Hashing and Amortization Lecture Hashig ad Amortizatio Supplemetal readig i CLRS: Chapter ; Chapter 7 itro; Sectio 7.. Arrays ad Hashig Arrays are very useful. The items i a array are statically addressed, so that isertig, deletig,

More information

Random Models. Tusheng Zhang. February 14, 2013

Random Models. Tusheng Zhang. February 14, 2013 Radom Models Tusheg Zhag February 14, 013 1 Radom Walks Let me describe the model. Radom walks are used to describe the motio of a movig particle (object). Suppose that a particle (object) moves alog the

More information

Introduction to Signals and Systems, Part V: Lecture Summary

Introduction to Signals and Systems, Part V: Lecture Summary EEL33: Discrete-Time Sigals ad Systems Itroductio to Sigals ad Systems, Part V: Lecture Summary Itroductio to Sigals ad Systems, Part V: Lecture Summary So far we have oly looked at examples of o-recursive

More information

Solutions to Homework 7

Solutions to Homework 7 Solutios to Homework 7 Due Wedesday, August 4, 004. Chapter 4.1) 3, 4, 9, 0, 7, 30. Chapter 4.) 4, 9, 10, 11, 1. Chapter 4.1. Solutio to problem 3. The sum has the form a 1 a + a 3 with a k = 1/k. Sice

More information

Scattering at an Interface:

Scattering at an Interface: 8/9/08 Course Istructor Dr. Raymod C. Rumpf Office: A 337 Phoe: (95) 747 6958 E Mail: rcrumpf@utep.edu EE 4347 Applied Electromagetics Topic 3h Scatterig at a Iterface: Phase Matchig & Special Agles Phase

More information

10-701/ Machine Learning Mid-term Exam Solution

10-701/ Machine Learning Mid-term Exam Solution 0-70/5-78 Machie Learig Mid-term Exam Solutio Your Name: Your Adrew ID: True or False (Give oe setece explaatio) (20%). (F) For a cotiuous radom variable x ad its probability distributio fuctio p(x), it

More information

6.3 Testing Series With Positive Terms

6.3 Testing Series With Positive Terms 6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial

More information

Question1 Multiple choices (circle the most appropriate one):

Question1 Multiple choices (circle the most appropriate one): Philadelphia Uiversity Studet Name: Faculty of Egieerig Studet Number: Dept. of Computer Egieerig Fial Exam, First Semester: 2014/2015 Course Title: Digital Sigal Aalysis ad Processig Date: 01/02/2015

More information

Solutions. Number of Problems: 4. None. Use only the prepared sheets for your solutions. Additional paper is available from the supervisors.

Solutions. Number of Problems: 4. None. Use only the prepared sheets for your solutions. Additional paper is available from the supervisors. Quiz November 4th, 23 Sigals & Systems (5-575-) P. Reist & Prof. R. D Adrea Solutios Exam Duratio: 4 miutes Number of Problems: 4 Permitted aids: Noe. Use oly the prepared sheets for your solutios. Additioal

More information

Chapter 4. Fourier Series

Chapter 4. Fourier Series Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,

More information

A Lattice Green Function Introduction. Abstract

A Lattice Green Function Introduction. Abstract August 5, 25 A Lattice Gree Fuctio Itroductio Stefa Hollos Exstrom Laboratories LLC, 662 Nelso Park Dr, Logmot, Colorado 853, USA Abstract We preset a itroductio to lattice Gree fuctios. Electroic address:

More information

Lecture 4. We also define the set of possible values for the random walk as the set of all x R d such that P(S n = x) > 0 for some n.

Lecture 4. We also define the set of possible values for the random walk as the set of all x R d such that P(S n = x) > 0 for some n. Radom Walks ad Browia Motio Tel Aviv Uiversity Sprig 20 Lecture date: Mar 2, 20 Lecture 4 Istructor: Ro Peled Scribe: Lira Rotem This lecture deals primarily with recurrece for geeral radom walks. We preset

More information

Numerical Methods for Ordinary Differential Equations

Numerical Methods for Ordinary Differential Equations Numerical Methods for Ordiary Differetial Equatios Braislav K. Nikolić Departmet of Physics ad Astroomy, Uiversity of Delaware, U.S.A. PHYS 460/660: Computatioal Methods of Physics http://www.physics.udel.edu/~bikolic/teachig/phys660/phys660.html

More information

Signal Processing in Mechatronics. Lecture 3, Convolution, Fourier Series and Fourier Transform

Signal Processing in Mechatronics. Lecture 3, Convolution, Fourier Series and Fourier Transform Sigal Processig i Mechatroics Summer semester, 1 Lecture 3, Covolutio, Fourier Series ad Fourier rasform Dr. Zhu K.P. AIS, UM 1 1. Covolutio Covolutio Descriptio of LI Systems he mai premise is that the

More information

SUMMARY OF SEQUENCES AND SERIES

SUMMARY OF SEQUENCES AND SERIES SUMMARY OF SEQUENCES AND SERIES Importat Defiitios, Results ad Theorems for Sequeces ad Series Defiitio. A sequece {a } has a limit L ad we write lim a = L if for every ɛ > 0, there is a correspodig iteger

More information

CALCULUS BASIC SUMMER REVIEW

CALCULUS BASIC SUMMER REVIEW CALCULUS BASIC SUMMER REVIEW NAME rise y y y Slope of a o vertical lie: m ru Poit Slope Equatio: y y m( ) The slope is m ad a poit o your lie is, ). ( y Slope-Itercept Equatio: y m b slope= m y-itercept=

More information

6. Uniform distribution mod 1

6. Uniform distribution mod 1 6. Uiform distributio mod 1 6.1 Uiform distributio ad Weyl s criterio Let x be a seuece of real umbers. We may decompose x as the sum of its iteger part [x ] = sup{m Z m x } (i.e. the largest iteger which

More information

Lecture III-2: Light propagation in nonmagnetic

Lecture III-2: Light propagation in nonmagnetic A. La Rosa Lecture Notes ALIED OTIC Lecture III2: Light propagatio i omagetic materials 2.1 urface ( ), volume ( ), ad curret ( j ) desities produced by arizatio charges The objective i this sectio is

More information

ECE 669 Parallel Computer Architecture

ECE 669 Parallel Computer Architecture ECE 669 Parallel Computer Architecture Lecture 4 Parallel Applicatios Outlie Motivatig Problems (applicatio case studies) Classifyig problems Parallelizig applicatios Examiig tradeoffs Uderstadig commuicatio

More information

Journal of Computational Physics 149, (1999) Article ID jcph , available online at

Journal of Computational Physics 149, (1999) Article ID jcph , available online at Joural of Computatioal Physics 149, 418 422 (1999) Article ID jcph.1998.6131, available olie at http://www.idealibrary.com o NOTE Defiig Wave Amplitude i Characteristic Boudary Coditios Key Words: Euler

More information

Module 1 Fundamentals in statistics

Module 1 Fundamentals in statistics Normal Distributio Repeated observatios that differ because of experimetal error ofte vary about some cetral value i a roughly symmetrical distributio i which small deviatios occur much more frequetly

More information

CALCULATION OF FIBONACCI VECTORS

CALCULATION OF FIBONACCI VECTORS CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College

More information

Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation. Specialized values

Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation. Specialized values EulerPhi Notatios Traditioal ame Euler totiet fuctio Traditioal otatio Φ Mathematica StadardForm otatio EulerPhi Primary defiitio 3.06.02.000.0 Φ gcd,k, ; For oegative iteger, the Euler totiet fuctio Φ

More information

CHAPTER 10 INFINITE SEQUENCES AND SERIES

CHAPTER 10 INFINITE SEQUENCES AND SERIES CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece

More information

Unit 6: Sequences and Series

Unit 6: Sequences and Series AMHS Hoors Algebra 2 - Uit 6 Uit 6: Sequeces ad Series 26 Sequeces Defiitio: A sequece is a ordered list of umbers ad is formally defied as a fuctio whose domai is the set of positive itegers. It is commo

More information

L 5 & 6: RelHydro/Basel. f(x)= ( ) f( ) ( ) ( ) ( ) n! 1! 2! 3! If the TE of f(x)= sin(x) around x 0 is: sin(x) = x - 3! 5!

L 5 & 6: RelHydro/Basel. f(x)= ( ) f( ) ( ) ( ) ( ) n! 1! 2! 3! If the TE of f(x)= sin(x) around x 0 is: sin(x) = x - 3! 5! aylor epasio: Let ƒ() be a ifiitely differetiable real fuctio. At ay poit i the eighbourhood of =0, the fuctio ca be represeted as a power series of the followig form: X 0 f(a) f() ƒ() f()= ( ) f( ) (

More information

Fourier Series and the Wave Equation

Fourier Series and the Wave Equation Fourier Series ad the Wave Equatio We start with the oe-dimesioal wave equatio u u =, x u(, t) = u(, t) =, ux (,) = f( x), u ( x,) = This represets a vibratig strig, where u is the displacemet of the strig

More information

Notes on iteration and Newton s method. Iteration

Notes on iteration and Newton s method. Iteration Notes o iteratio ad Newto s method Iteratio Iteratio meas doig somethig over ad over. I our cotet, a iteratio is a sequece of umbers, vectors, fuctios, etc. geerated by a iteratio rule of the type 1 f

More information

STAT 350 Handout 19 Sampling Distribution, Central Limit Theorem (6.6)

STAT 350 Handout 19 Sampling Distribution, Central Limit Theorem (6.6) STAT 350 Hadout 9 Samplig Distributio, Cetral Limit Theorem (6.6) A radom sample is a sequece of radom variables X, X 2,, X that are idepedet ad idetically distributed. o This property is ofte abbreviated

More information

is also known as the general term of the sequence

is also known as the general term of the sequence Lesso : Sequeces ad Series Outlie Objectives: I ca determie whether a sequece has a patter. I ca determie whether a sequece ca be geeralized to fid a formula for the geeral term i the sequece. I ca determie

More information

Math 116 Practice for Exam 3

Math 116 Practice for Exam 3 Math 6 Practice for Eam 3 Geerated April 4, 26 Name: SOLUTIONS Istructor: Sectio Number:. This eam has questios. Note that the problems are ot of equal difficulty, so you may wat to skip over ad retur

More information

1 of 7 7/16/2009 6:06 AM Virtual Laboratories > 6. Radom Samples > 1 2 3 4 5 6 7 6. Order Statistics Defiitios Suppose agai that we have a basic radom experimet, ad that X is a real-valued radom variable

More information

Solutions - Homework # 1

Solutions - Homework # 1 ECE-4: Sigals ad Systems Summer Solutios - Homework # PROBLEM A cotiuous time sigal is show i the figure. Carefully sketch each of the followig sigals: x(t) a) x(t-) b) x(-t) c) x(t+) d) x( - t/) e) x(t)*(

More information

Characterization and Simulation of HF Propagation in a realistic disturbed and horizontally inhomogeneous Ionosphere

Characterization and Simulation of HF Propagation in a realistic disturbed and horizontally inhomogeneous Ionosphere Characterizatio ad Simulatio of HF Propagatio i a realistic disturbed ad horizotally ihomogeeous Ioosphere Vadim E. Gherm ad Nikolay N.Zerov Departmet of Radiophysics Uiversity of St.Petersburg St.Petersburg

More information

Sound II. Sound intensity level. Question. Energy and Intensity of sound waves

Sound II. Sound intensity level. Question. Energy and Intensity of sound waves Soud. Eergy ad tesity terferece of soud waes Stadig waes Complex soud waes Eergy ad tesity of soud waes power tesity eergy P time power P area A area A (uits W/m ) Soud itesity leel β 0log o o 0 - W/m

More information

Electromagnetic wave propagation in Particle-In-Cell codes

Electromagnetic wave propagation in Particle-In-Cell codes Electromagetic wave propagatio i Particle-I-Cell codes Remi Lehe Lawrece Berkeley Natioal Laboratory (LBNL) US Particle Accelerator School (USPAS) Summer Sessio Self-Cosistet Simulatios of Beam ad Plasma

More information

Chapter 18 Summary Sampling Distribution Models

Chapter 18 Summary Sampling Distribution Models Uit 5 Itroductio to Iferece Chapter 18 Summary Samplig Distributio Models What have we leared? Sample proportios ad meas will vary from sample to sample that s samplig error (samplig variability). Samplig

More information

Support vector machine revisited

Support vector machine revisited 6.867 Machie learig, lecture 8 (Jaakkola) 1 Lecture topics: Support vector machie ad kerels Kerel optimizatio, selectio Support vector machie revisited Our task here is to first tur the support vector

More information

Matriculation number: You have 90 minutes to complete the exam of InformatikIIb. The following rules apply:

Matriculation number: You have 90 minutes to complete the exam of InformatikIIb. The following rules apply: Departmet of Iformatics Prof. Dr. Michael Böhle Bizmühlestrasse 14 8050 Zurich Phoe: +41 44 635 4333 Email: boehle@ifi.uzh.ch AlgoDat Midterm1 Sprig 016 08.04.016 Name: Matriculatio umber: Advice You have

More information

SECTION 2 Electrostatics

SECTION 2 Electrostatics SECTION Electrostatics This sectio, based o Chapter of Griffiths, covers effects of electric fields ad forces i static (timeidepedet) situatios. The topics are: Electric field Gauss s Law Electric potetial

More information

Phys 6303 Final Exam Solutions December 19, 2012

Phys 6303 Final Exam Solutions December 19, 2012 Phys 633 Fial Exam s December 19, 212 You may NOT use ay book or otes other tha supplied with this test. You will have 3 hours to fiish. DO YOUR OWN WORK. Express your aswers clearly ad cocisely so that

More information

The Z-Transform. (t-t 0 ) Figure 1: Simplified graph of an impulse function. For an impulse, it can be shown that (1)

The Z-Transform. (t-t 0 ) Figure 1: Simplified graph of an impulse function. For an impulse, it can be shown that (1) The Z-Trasform Sampled Data The geeralied fuctio (t) (also kow as the impulse fuctio) is useful i the defiitio ad aalysis of sampled-data sigals. Figure below shows a simplified graph of a impulse. (t-t

More information

Chapter 7 z-transform

Chapter 7 z-transform Chapter 7 -Trasform Itroductio Trasform Uilateral Trasform Properties Uilateral Trasform Iversio of Uilateral Trasform Determiig the Frequecy Respose from Poles ad Zeros Itroductio Role i Discrete-Time

More information

REFLECTION AND REFRACTION

REFLECTION AND REFRACTION RFLCTON AND RFRACTON We ext ivestigate what happes whe a light ray movig i oe medium ecouters aother medium, i.e. the pheomea of reflectio ad refractio. We cosider a plae M wave strikig a plae iterface

More information

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 5: SINGULARITIES.

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 5: SINGULARITIES. ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 5: SINGULARITIES. ANDREW SALCH 1. The Jacobia criterio for osigularity. You have probably oticed by ow that some poits o varieties are smooth i a sese somethig

More information

n m CHAPTER 3 RATIONAL EXPONENTS AND RADICAL FUNCTIONS 3-1 Evaluate n th Roots and Use Rational Exponents Real nth Roots of a n th Root of a

n m CHAPTER 3 RATIONAL EXPONENTS AND RADICAL FUNCTIONS 3-1 Evaluate n th Roots and Use Rational Exponents Real nth Roots of a n th Root of a CHAPTER RATIONAL EXPONENTS AND RADICAL FUNCTIONS Big IDEAS: 1) Usig ratioal expoets ) Performig fuctio operatios ad fidig iverse fuctios ) Graphig radical fuctios ad solvig radical equatios Sectio: Essetial

More information

Algorithms for Clustering

Algorithms for Clustering CR2: Statistical Learig & Applicatios Algorithms for Clusterig Lecturer: J. Salmo Scribe: A. Alcolei Settig: give a data set X R p where is the umber of observatio ad p is the umber of features, we wat

More information

(7 One- and Two-Sample Estimation Problem )

(7 One- and Two-Sample Estimation Problem ) 34 Stat Lecture Notes (7 Oe- ad Two-Sample Estimatio Problem ) ( Book*: Chapter 8,pg65) Probability& Statistics for Egieers & Scietists By Walpole, Myers, Myers, Ye Estimatio 1 ) ( ˆ S P i i Poit estimate:

More information

TMA4205 Numerical Linear Algebra. The Poisson problem in R 2 : diagonalization methods

TMA4205 Numerical Linear Algebra. The Poisson problem in R 2 : diagonalization methods TMA4205 Numerical Liear Algebra The Poisso problem i R 2 : diagoalizatio methods September 3, 2007 c Eiar M Røquist Departmet of Mathematical Scieces NTNU, N-749 Trodheim, Norway All rights reserved A

More information

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense, 3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [

More information