A Fourier Transform Model in Excel #1
|
|
- Pauline Lloyd
- 6 years ago
- Views:
Transcription
1 A Fourier Transorm Model in Ecel # -This is a tutorial about the implementation o a Fourier transorm in Ecel. This irst part goes over adjustments in the general Fourier transorm ormula to be applicable on real time sampled signals with a inite number o known samples. - This is not an introduction to comple unctions or Fourier transorm. In order to ollow this it s ideal to have minimal knowledge o basic comple number theory. I you learned Fourier analysis in school a couple o years or a couple o decades back and you are vaguely aware o the Fourier transorm theory, it is enough. You could stop and go back and read some theory at any time. < <ecelunusual.com> by George Lungu
2 < Introduction: - The deinition o the Fourier transorm o a temporal signal G( is: - The deinition o the inverse Fourier transorm o a requency unction G() is: G( ) t t e G( ) e j t j t dt d - t and are time and requency respectively, they are real numbers. - j is the imaginary symbol, i is some times used to denote it. - The Euler Formula is: e jz cos( z) j sin( z) - Using Euler s ormula and assuming in our particular case that is a real unction we can rewrite the Fourier transorm as: G( ) cos( dt j sin( dt
3 Let s see how we can apply the previous ormula in practice to get a reasonable approimation o the Fourier transorm: g We used the notation nh) = g n g g g g - g - In practice we usually have a limited number o equally spaced time samples (+) o a continuous unction contained in a table. - In practice the samples usually start at an arbitrary time zero. Minus ininity or plus ininity are uneasible so we will do the integration on the available period o time [, T]. - We can approimate the integral o a unction in numerical ashion by using a sum o its samples multiplied by the length o the time interval between the sample h. h h h (-)h h = T tt dt h n t n nh) n G( ) h nh) cos( nh) j h nh) sin( nh) n -- Though very similar, the ormula above is not the standard DFT (Discrete Fourier Transorm) ormula but something improvised ad hoc based on the ull ormula o the transorm and numerical approimations. Since we sum rom to not rom / to / the ormula above is an approimation o the Fourier transorm o t+/) rather then The irst term is the real part o the transorm and the second term (ater j ) the imaginary part < n n
4 G( ) h nh) cos( nh) j h nh) sin( nh) A visualization: Real part - Re(G()) Imaginary part - Im(G()) - I we have a saw-tooth unction and a cosine unction o requency. Calculating the sampled Fourier transorm or requency would mean multiplying all red and blue point values situated on the same vertical grid line and adding all the products together cos(pi** Detailed visualization or calculating the real part o the Fourier transorm at requency : h + cos(pi** h RFT - Redneck Fourier Transorm < 4 h nh) cos( nh)
5 < 5 - In order to calculate the real part o G() we need to do the operation demonstrated in the previous page or every requency that we want to calculate G() or. This implies multiplication o with a cosine o that requency or time samples. - In order to calculate the imaginary part o G() we need to do the operation demonstrated in the previous page (with a sine instead o a cosine) or every requency that we want to calculate G() or. This implies multiplication o with a sine o that requency or time samples. - The charts the right show the saw-tooth unction and a cosine o our dierent requencies used to calculate the real part o the Fourier transorm or our dierent requencies..5 cos(pi** cos(pi** cos(pi** cos(pi**
6 < 6 Overview o the Fourier transorm components: - I we write the Fourier transorm o a real value time signal we can see that it has a real part and an imaginary part: G( ) h nh) cos( nh) j h nh) sin( nh) Real part - Re(G()) Imaginary part - Im(G()) - Which can be written in short like this: G( ) j ImG ( ) G( ) Re - Instead o writing as real and imaginary, the Fourier transorm is most o the times epressed as Amplitude and Phase: G( ) ImG ( ) G( ) Re Phase G( ) Im G( a tan Re G( ) )
This is only a list of questions use a separate sheet to work out the problems. 1. (1.2 and 1.4) Use the given graph to answer each question.
Mth Calculus Practice Eam Questions NOTE: These questions should not be taken as a complete list o possible problems. The are merel intended to be eamples o the diicult level o the regular eam questions.
More information( x) f = where P and Q are polynomials.
9.8 Graphing Rational Functions Lets begin with a deinition. Deinition: Rational Function A rational unction is a unction o the orm ( ) ( ) ( ) P where P and Q are polynomials. Q An eample o a simple rational
More informationThe Fourier Transform
The Fourier Transorm Fourier Series Fourier Transorm The Basic Theorems and Applications Sampling Bracewell, R. The Fourier Transorm and Its Applications, 3rd ed. New York: McGraw-Hill, 2. Eric W. Weisstein.
More information! " k x 2k$1 # $ k x 2k. " # p $ 1! px! p " p 1 # !"#$%&'"()'*"+$",&-('./&-/. !"#$%&'()"*#%+!'",' -./#")'.,&'+.0#.1)2,'!%)2%! !"#$%&'"%(")*$+&#,*$,#
"#$%&'()"*#%+'",' -./#")'.,&'+.0#.1)2,' %)2% "#$%&'"()'*"+$",&-('./&-/. Taylor Series o a unction at x a is " # a k " # " x a# k k0 k It is a Power Series centered at a. Maclaurin Series o a unction is
More informationNumerical Methods - Lecture 2. Numerical Methods. Lecture 2. Analysis of errors in numerical methods
Numerical Methods - Lecture 1 Numerical Methods Lecture. Analysis o errors in numerical methods Numerical Methods - Lecture Why represent numbers in loating point ormat? Eample 1. How a number 56.78 can
More informationENSC327 Communications Systems 2: Fourier Representations. School of Engineering Science Simon Fraser University
ENSC37 Communications Systems : Fourier Representations School o Engineering Science Simon Fraser University Outline Chap..5: Signal Classiications Fourier Transorm Dirac Delta Function Unit Impulse Fourier
More informationIn many diverse fields physical data is collected or analysed as Fourier components.
1. Fourier Methods In many diverse ields physical data is collected or analysed as Fourier components. In this section we briely discuss the mathematics o Fourier series and Fourier transorms. 1. Fourier
More informationTLT-5200/5206 COMMUNICATION THEORY, Exercise 3, Fall TLT-5200/5206 COMMUNICATION THEORY, Exercise 3, Fall Problem 1.
TLT-5/56 COMMUNICATION THEORY, Exercise 3, Fall Problem. The "random walk" was modelled as a random sequence [ n] where W[i] are binary i.i.d. random variables with P[W[i] = s] = p (orward step with probability
More informationRoberto s Notes on Differential Calculus Chapter 8: Graphical analysis Section 1. Extreme points
Roberto s Notes on Dierential Calculus Chapter 8: Graphical analysis Section 1 Extreme points What you need to know already: How to solve basic algebraic and trigonometric equations. All basic techniques
More informationAsymptote. 2 Problems 2 Methods
Asymptote Problems Methods Problems Assume we have the ollowing transer unction which has a zero at =, a pole at = and a pole at =. We are going to look at two problems: problem is where >> and problem
More informationDiscrete-Time Fourier Transform (DTFT)
Connexions module: m047 Discrete-Time Fourier Transorm DTFT) Don Johnson This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License Abstract Discussion
More informationLab on Taylor Polynomials. This Lab is accompanied by an Answer Sheet that you are to complete and turn in to your instructor.
Lab on Taylor Polynomials This Lab is accompanied by an Answer Sheet that you are to complete and turn in to your instructor. In this Lab we will approimate complicated unctions by simple unctions. The
More informationSIO 211B, Rudnick. We start with a definition of the Fourier transform! ĝ f of a time series! ( )
SIO B, Rudnick! XVIII.Wavelets The goal o a wavelet transorm is a description o a time series that is both requency and time selective. The wavelet transorm can be contrasted with the well-known and very
More informationForced Response - Particular Solution x p (t)
Governing Equation 1.003J/1.053J Dynamics and Control I, Spring 007 Proessor Peacoc 5/7/007 Lecture 1 Vibrations: Second Order Systems - Forced Response Governing Equation Figure 1: Cart attached to spring
More informationMHF 4U Unit 7: Combining Functions May 29, Review Solutions
MHF 4U Unit 7: Combining Functions May 9, 008. Review Solutions Use the ollowing unctions to answer questions 5, ( ) g( ), h( ) sin, w {(, ), (3, ), (4, 7)}, r, and l ) log ( ) + (, ) Determine: a) + w
More informationDefinition: Let f(x) be a function of one variable with continuous derivatives of all orders at a the point x 0, then the series.
2.4 Local properties o unctions o several variables In this section we will learn how to address three kinds o problems which are o great importance in the ield o applied mathematics: how to obtain the
More information( ) x y z. 3 Functions 36. SECTION D Composite Functions
3 Functions 36 SECTION D Composite Functions By the end o this section you will be able to understand what is meant by a composite unction ind composition o unctions combine unctions by addition, subtraction,
More information5.2. November 30, 2012 Mrs. Poland. Verifying Trigonometric Identities
5.2 Verifying Trigonometric Identities Verifying Identities by Working With One Side Verifying Identities by Working With Both Sides November 30, 2012 Mrs. Poland Objective #4: Students will be able to
More informationMathematical methods and its applications Dr. S. K. Gupta Department of Mathematics Indian Institute of Technology, Roorkee
Mathematical methods and its applications Dr. S. K. Gupta Department of Mathematics Indian Institute of Technology, Roorkee Lecture - 56 Fourier sine and cosine transforms Welcome to lecture series on
More information3.5 Graphs of Rational Functions
Math 30 www.timetodare.com Eample Graph the reciprocal unction ( ) 3.5 Graphs o Rational Functions Answer the ollowing questions: a) What is the domain o the unction? b) What is the range o the unction?
More informationNew Functions from Old Functions
.3 New Functions rom Old Functions In this section we start with the basic unctions we discussed in Section. and obtain new unctions b shiting, stretching, and relecting their graphs. We also show how
More information8. THEOREM If the partial derivatives f x. and f y exist near (a, b) and are continuous at (a, b), then f is differentiable at (a, b).
8. THEOREM I the partial derivatives and eist near (a b) and are continuous at (a b) then is dierentiable at (a b). For a dierentiable unction o two variables z= ( ) we deine the dierentials d and d to
More informationFigure 3.1 Effect on frequency spectrum of increasing period T 0. Consider the amplitude spectrum of a periodic waveform as shown in Figure 3.2.
3. Fourier ransorm From Fourier Series to Fourier ransorm [, 2] In communication systems, we oten deal with non-periodic signals. An extension o the time-requency relationship to a non-periodic signal
More information2 Frequency-Domain Analysis
2 requency-domain Analysis Electrical engineers live in the two worlds, so to speak, o time and requency. requency-domain analysis is an extremely valuable tool to the communications engineer, more so
More informationStanding Waves If the same type of waves move through a common region and their frequencies, f, are the same then so are their wavelengths, λ.
Standing Waves I the same type o waves move through a common region and their requencies,, are the same then so are their wavelengths,. This ollows rom: v=. Since the waves move through a common region,
More informationIntroduction to Analog And Digital Communications
Introduction to Analog And Digital Communications Second Edition Simon Haykin, Michael Moher Chapter Fourier Representation o Signals and Systems.1 The Fourier Transorm. Properties o the Fourier Transorm.3
More informationLab 3: The FFT and Digital Filtering. Slides prepared by: Chun-Te (Randy) Chu
Lab 3: The FFT and Digital Filtering Slides prepared by: Chun-Te (Randy) Chu Lab 3: The FFT and Digital Filtering Assignment 1 Assignment 2 Assignment 3 Assignment 4 Assignment 5 What you will learn in
More informationAP Calculus AB Summer Assignment
AP Calculus AB Summer Assignment Name: When you come back to school, it is my epectation that you will have this packet completed. You will be way behind at the beginning of the year if you haven t attempted
More informationMath 2412 Activity 1(Due by EOC Sep. 17)
Math 4 Activity (Due by EOC Sep. 7) Determine whether each relation is a unction.(indicate why or why not.) Find the domain and range o each relation.. 4,5, 6,7, 8,8. 5,6, 5,7, 6,6, 6,7 Determine whether
More informationSec 3.1. lim and lim e 0. Exponential Functions. f x 9, write the equation of the graph that results from: A. Limit Rules
Sec 3. Eponential Functions A. Limit Rules. r lim a a r. I a, then lim a and lim a 0 3. I 0 a, then lim a 0 and lim a 4. lim e 0 5. e lim and lim e 0 Eamples:. Starting with the graph o a.) Shiting 9 units
More informationNumerical Solution of Ordinary Differential Equations in Fluctuationlessness Theorem Perspective
Numerical Solution o Ordinary Dierential Equations in Fluctuationlessness Theorem Perspective NEJLA ALTAY Bahçeşehir University Faculty o Arts and Sciences Beşiktaş, İstanbul TÜRKİYE TURKEY METİN DEMİRALP
More informationBasic properties of limits
Roberto s Notes on Dierential Calculus Chapter : Limits and continuity Section Basic properties o its What you need to know already: The basic concepts, notation and terminology related to its. What you
More informationChapter 4 Image Enhancement in the Frequency Domain
Chapter 4 Image Enhancement in the Frequency Domain 3. Fourier transorm -D Let be a unction o real variable,the ourier transorm o is F { } F u ep jπu d j F { F u } F u ep[ jπ u ] du F u R u + ji u or F
More informationLecture 8 Optimization
4/9/015 Lecture 8 Optimization EE 4386/5301 Computational Methods in EE Spring 015 Optimization 1 Outline Introduction 1D Optimization Parabolic interpolation Golden section search Newton s method Multidimensional
More informationCOMPOSITE AND INVERSE FUNCTIONS & PIECEWISE FUNCTIONS
Functions Modeling Change: A Preparation or Calculus, 4th Edition, 2011, Connally 2.4 COMPOSITE AND INVERSE FUNCTIONS & PIECEWISE FUNCTIONS Functions Modeling Change: A Preparation or Calculus, 4th Edition,
More informationMath Review and Lessons in Calculus
Math Review and Lessons in Calculus Agenda Rules o Eponents Functions Inverses Limits Calculus Rules o Eponents 0 Zero Eponent Rule a * b ab Product Rule * 3 5 a / b a-b Quotient Rule 5 / 3 -a / a Negative
More informationy2 = 0. Show that u = e2xsin(2y) satisfies Laplace's equation.
Review 1 1) State the largest possible domain o deinition or the unction (, ) = 3 - ) Determine the largest set o points in the -plane on which (, ) = sin-1( - ) deines a continuous unction 3) Find the
More informationLecture 13: Applications of Fourier transforms (Recipes, Chapter 13)
Lecture 13: Applications o Fourier transorms (Recipes, Chapter 13 There are many applications o FT, some o which involve the convolution theorem (Recipes 13.1: The convolution o h(t and r(t is deined by
More informationComputational Methods for Domains with! Complex Boundaries-I!
http://www.nd.edu/~gtrggva/cfd-course/ Computational Methods or Domains with Comple Boundaries-I Grétar Trggvason Spring For most engineering problems it is necessar to deal with comple geometries, consisting
More informationPhysics 5153 Classical Mechanics. Solution by Quadrature-1
October 14, 003 11:47:49 1 Introduction Physics 5153 Classical Mechanics Solution by Quadrature In the previous lectures, we have reduced the number o eective degrees o reedom that are needed to solve
More information23.4. Convergence. Introduction. Prerequisites. Learning Outcomes
Convergence 3.4 Introduction In this Section we examine, briefly, the convergence characteristics of a Fourier series. We have seen that a Fourier series can be found for functions which are not necessarily
More informationCategories and Natural Transformations
Categories and Natural Transormations Ethan Jerzak 17 August 2007 1 Introduction The motivation or studying Category Theory is to ormalise the underlying similarities between a broad range o mathematical
More informationAP Calculus AB Summer Assignment
AP Calculus AB Summer Assignment As Advanced placement students, our irst assignment or the 07-08 school ear is to come to class the ver irst da in top mathematical orm. Calculus is a world o change. While
More informationAP Calculus AB Summer Assignment
AP Calculus AB Summer Assignment Name: When you come back to school, you will be epected to have attempted every problem. These skills are all different tools that you will pull out of your toolbo this
More information1/100 Range: 1/10 1/ 2. 1) Constant: choose a value for the constant that can be graphed on the coordinate grid below.
Name 1) Constant: choose a value or the constant that can be graphed on the coordinate grid below a y Toolkit Functions Lab Worksheet thru inverse trig ) Identity: y ) Reciprocal: 1 ( ) y / 1/ 1/1 1/ 1
More informationThe concept of limit
Roberto s Notes on Dierential Calculus Chapter 1: Limits and continuity Section 1 The concept o limit What you need to know already: All basic concepts about unctions. What you can learn here: What limits
More informationProblem Set. Problems on Unordered Summation. Math 5323, Fall Februray 15, 2001 ANSWERS
Problem Set Problems on Unordered Summation Math 5323, Fall 2001 Februray 15, 2001 ANSWERS i 1 Unordered Sums o Real Terms In calculus and real analysis, one deines the convergence o an ininite series
More informationFluctuationlessness Theorem and its Application to Boundary Value Problems of ODEs
Fluctuationlessness Theorem and its Application to Boundary Value Problems o ODEs NEJLA ALTAY İstanbul Technical University Inormatics Institute Maslak, 34469, İstanbul TÜRKİYE TURKEY) nejla@be.itu.edu.tr
More informationLecture : Feedback Linearization
ecture : Feedbac inearization Niola Misovic, dipl ing and Pro Zoran Vuic June 29 Summary: This document ollows the lectures on eedbac linearization tought at the University o Zagreb, Faculty o Electrical
More informationSection 1.2 Domain and Range
Section 1. Domain and Range 1 Section 1. Domain and Range One o our main goals in mathematics is to model the real world with mathematical unctions. In doing so, it is important to keep in mind the limitations
More information0,0 B 5,0 C 0, 4 3,5. y x. Recitation Worksheet 1A. 1. Plot these points in the xy plane: A
Math 13 Recitation Worksheet 1A 1 Plot these points in the y plane: A 0,0 B 5,0 C 0, 4 D 3,5 Without using a calculator, sketch a graph o each o these in the y plane: A y B 3 Consider the unction a Evaluate
More informationm f f unchanged under the field redefinition (1), the complex mass matrix m should transform into
PHY 396 T: SUSY Solutions or problem set #8. Problem (a): To keep the net quark mass term L QCD L mass = ψ α c m ψ c α + hermitian conjugate (S.) unchanged under the ield redeinition (), the complex mass
More information3 What You Should Know About Complex Numbers
3 What You Should Know About Complex Numbers Life is complex it has a real part, and an imaginary part Andrew Koenig. Complex numbers are an extension of the more familiar world of real numbers that make
More informationSection 6. Object-Image Relationships
6-1 Section 6 Object-Image elationships Object-Image elationships The purpose o this study is to examine the imaging properties o the general system that has been deined by its Gaussian properties and
More informationObjectives. By the time the student is finished with this section of the workbook, he/she should be able
FUNCTIONS Quadratic Functions......8 Absolute Value Functions.....48 Translations o Functions..57 Radical Functions...61 Eponential Functions...7 Logarithmic Functions......8 Cubic Functions......91 Piece-Wise
More informationEDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 4 - CALCULUS TUTORIAL 2 - INTEGRATION
EDEXCEL NATIONAL CERTIFICATE UNIT 8 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME - CALCULUS TUTORIAL - INTEGRATION CONTENTS Be able to apply calculus Differentiation: review of standard derivatives, differentiation
More informationDifferentiation. The main problem of differential calculus deals with finding the slope of the tangent line at a point on a curve.
Dierentiation The main problem o dierential calculus deals with inding the slope o the tangent line at a point on a curve. deinition() : The slope o a curve at a point p is the slope, i it eists, o the
More informationC6-2 Differentiation 3
chain, product and quotient rules C6- Differentiation Pre-requisites: C6- Estimate Time: 8 hours Summary Learn Solve Revise Answers Summary The chain rule is used to differentiate a function of a function.
More informationReferences Ideal Nyquist Channel and Raised Cosine Spectrum Chapter 4.5, 4.11, S. Haykin, Communication Systems, Wiley.
Baseand Data Transmission III Reerences Ideal yquist Channel and Raised Cosine Spectrum Chapter 4.5, 4., S. Haykin, Communication Systems, iley. Equalization Chapter 9., F. G. Stremler, Communication Systems,
More informationAnswer Key-Math 11- Optional Review Homework For Exam 2
Answer Key-Math - Optional Review Homework For Eam 2. Compute the derivative or each o the ollowing unctions: Please do not simpliy your derivatives here. I simliied some, only in the case that you want
More informationAnalytic Trigonometry. Copyright Cengage Learning. All rights reserved.
Analytic Trigonometry Copyright Cengage Learning. All rights reserved. 7.1 Trigonometric Identities Copyright Cengage Learning. All rights reserved. Objectives Simplifying Trigonometric Expressions Proving
More informationIncreasing and Decreasing Functions and the First Derivative Test. Increasing and Decreasing Functions. Video
SECTION and Decreasing Functions and the First Derivative Test 79 Section and Decreasing Functions and the First Derivative Test Determine intervals on which a unction is increasing or decreasing Appl
More informationExponential and Logarithmic. Functions CHAPTER The Algebra of Functions; Composite
CHAPTER 9 Exponential and Logarithmic Functions 9. The Algebra o Functions; Composite Functions 9.2 Inverse Functions 9.3 Exponential Functions 9.4 Exponential Growth and Decay Functions 9.5 Logarithmic
More informationSpecial types of Riemann sums
Roberto s Notes on Subject Chapter 4: Deinite integrals and the FTC Section 3 Special types o Riemann sums What you need to know already: What a Riemann sum is. What you can learn here: The key types o
More informationOutline. Approximate sampling theorem (AST) recall Lecture 1. P. L. Butzer, G. Schmeisser, R. L. Stens
Outline Basic relations valid or the Bernstein space B and their extensions to unctions rom larger spaces in terms o their distances rom B Part 3: Distance unctional approach o Part applied to undamental
More informationFunctions. Essential Question What are some of the characteristics of the graph of a logarithmic function?
5. Logarithms and Logarithmic Functions Essential Question What are some o the characteristics o the graph o a logarithmic unction? Ever eponential unction o the orm () = b, where b is a positive real
More informationMATH1901 Differential Calculus (Advanced)
MATH1901 Dierential Calculus (Advanced) Capter 3: Functions Deinitions : A B A and B are sets assigns to eac element in A eactl one element in B A is te domain o te unction B is te codomain o te unction
More informationLeast-Squares Spectral Analysis Theory Summary
Least-Squares Spectral Analysis Theory Summary Reerence: Mtamakaya, J. D. (2012). Assessment o Atmospheric Pressure Loading on the International GNSS REPRO1 Solutions Periodic Signatures. Ph.D. dissertation,
More informationExercise Set 6.2: Double-Angle and Half-Angle Formulas
Exercise Set : Double-Angle and Half-Angle Formulas Answer the following π 1 (a Evaluate sin π (b Evaluate π π (c Is sin = (d Graph f ( x = sin ( x and g ( x = sin ( x on the same set of axes (e Is sin
More informationlecture 7: Trigonometric Interpolation
lecture : Trigonometric Interpolation 9 Trigonometric interpolation for periodic functions Thus far all our interpolation schemes have been based on polynomials However, if the function f is periodic,
More informationFourier Series and Fourier Transforms
Fourier Series and Fourier Transforms EECS2 (6.082), MIT Fall 2006 Lectures 2 and 3 Fourier Series From your differential equations course, 18.03, you know Fourier s expression representing a T -periodic
More informationEA2.3 - Electronics 2 1
In the previous lecture, I talked about the idea of complex frequency s, where s = σ + jω. Using such concept of complex frequency allows us to analyse signals and systems with better generality. In this
More informationPre-Calculus and Trigonometry Capacity Matrix
Pre-Calculus and Capacity Matri Review Polynomials A1.1.4 A1.2.5 Add, subtract, multiply and simplify polynomials and rational epressions Solve polynomial equations and equations involving rational epressions
More informationThe Product and Quotient Rules
The Product and Quotient Rules In this section, you will learn how to find the derivative of a product of functions and the derivative of a quotient of functions. A function that is the product of functions
More informationPower Spectral Analysis of Elementary Cellular Automata
Power Spectral Analysis o Elementary Cellular Automata Shigeru Ninagawa Division o Inormation and Computer Science, Kanazawa Institute o Technology, 7- Ohgigaoka, Nonoichi, Ishikawa 92-850, Japan Spectral
More informationThe Poisson summation formula, the sampling theorem, and Dirac combs
The Poisson summation ormula, the sampling theorem, and Dirac combs Jordan Bell jordanbell@gmailcom Department o Mathematics, University o Toronto April 3, 24 Poisson summation ormula et S be the set o
More informationIntroduction to the Discrete Fourier Transform
Introduction to the Discrete ourier Transform Lucas J. van Vliet www.ph.tn.tudelft.nl/~lucas TNW: aculty of Applied Sciences IST: Imaging Science and technology PH: Linear Shift Invariant System A discrete
More informationShaker Rigs revisited
Shaker Rigs revisited One o the irst articles I ever wrote or Racecar Engineering was on 4 post and 7 post shaker rigs. That article was over 5 years a go and a lot o water has lowed under the bridge and
More informationGEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING ECE 2026 Summer 2018 Problem Set #1 Assigned: May 14, 2018 Due: May 22, 2018 Reading: Chapter 1; App. A on Complex Numbers,
More informationand ( x, y) in a domain D R a unique real number denoted x y and b) = x y = {(, ) + 36} that is all points inside and on
Mat 7 Calculus III Updated on 10/4/07 Dr. Firoz Chapter 14 Partial Derivatives Section 14.1 Functions o Several Variables Deinition: A unction o two variables is a rule that assigns to each ordered pair
More informationRelating axial motion of optical elements to focal shift
Relating aial motion o optical elements to ocal shit Katie Schwertz and J. H. Burge College o Optical Sciences, University o Arizona, Tucson AZ 857, USA katie.schwertz@gmail.com ABSTRACT In this paper,
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Processing Pro. Mark Fowler Note Set #10 Fourier Analysis or DT Signals eading Assignment: Sect. 4.2 & 4.4 o Proakis & Manolakis Much o Ch. 4 should be review so you are expected
More information6.5 Trigonometric Equations
6. Trigonometric Equations In this section, we discuss conditional trigonometric equations, that is, equations involving trigonometric functions that are satisfied only by some values of the variable (or
More informationChapter 4 Imaging. Lecture 21. d (110) Chem 793, Fall 2011, L. Ma
Chapter 4 Imaging Lecture 21 d (110) Imaging Imaging in the TEM Diraction Contrast in TEM Image HRTEM (High Resolution Transmission Electron Microscopy) Imaging or phase contrast imaging STEM imaging a
More informationCurve Sketching. The process of curve sketching can be performed in the following steps:
Curve Sketching So ar you have learned how to ind st and nd derivatives o unctions and use these derivatives to determine where a unction is:. Increasing/decreasing. Relative extrema 3. Concavity 4. Points
More information3 rd class Mech. Eng. Dept. hamdiahmed.weebly.com Fourier Series
Definition 1 Fourier Series A function f is said to be piecewise continuous on [a, b] if there exists finitely many points a = x 1 < x 2
More informationFinal Exam Review Math Determine the derivative for each of the following: dy dx. dy dx. dy dx dy dx. dy dx dy dx. dy dx
Final Eam Review Math. Determine the derivative or each o the ollowing: a. y 6 b. y sec c. y ln d. y e. y e. y sin sin g. y cos h. i. y e y log j. k. l. 6 y y cosh y sin m. y ln n. y tan o. y arctan e
More informationTwo-step self-tuning phase-shifting interferometry
Two-step sel-tuning phase-shiting intererometry J. Vargas, 1,* J. Antonio Quiroga, T. Belenguer, 1 M. Servín, 3 J. C. Estrada 3 1 Laboratorio de Instrumentación Espacial, Instituto Nacional de Técnica
More informationComplex Numbers. Integers, Rationals, and Reals. The natural numbers: The integers:
Complex Numbers Integers, Rationals, and Reals The natural numbers: N {... 3, 2,, 0,, 2, 3...} The integers: Z {... 3, 2,, 0,, 2, 3...} Note that any two integers added, subtracted, or multiplied together
More information8.4 Inverse Functions
Section 8. Inverse Functions 803 8. Inverse Functions As we saw in the last section, in order to solve application problems involving eponential unctions, we will need to be able to solve eponential equations
More informationProducts and Convolutions of Gaussian Probability Density Functions
Tina Memo No. 003-003 Internal Report Products and Convolutions o Gaussian Probability Density Functions P.A. Bromiley Last updated / 9 / 03 Imaging Science and Biomedical Engineering Division, Medical
More informationComputer Problems for Fourier Series and Transforms
Computer Problems for Fourier Series and Transforms 1. Square waves are frequently used in electronics and signal processing. An example is shown below. 1 π < x < 0 1 0 < x < π y(x) = 1 π < x < 2π... and
More informationNAME DATE PERIOD. Trigonometric Identities. Review Vocabulary Complete each identity. (Lesson 4-1) 1 csc θ = 1. 1 tan θ = cos θ sin θ = 1
5-1 Trigonometric Identities What You ll Learn Scan the text under the Now heading. List two things that you will learn in the lesson. 1. 2. Lesson 5-1 Active Vocabulary Review Vocabulary Complete each
More informationAP Calculus BC Summer Assignment 2018
AP Calculus BC Summer Assignment 018 Name: When you come back to school, I will epect you to have attempted every problem. These skills are all different tools that we will pull out of our toolbo at different
More informationSESSION 6 Trig. Equations and Identities. Math 30-1 R 3. (Revisit, Review and Revive)
SESSION 6 Trig. Equations and Identities Math 30-1 R 3 (Revisit, Review and Revive) 1 P a g e 2 P a g e Mathematics 30-1 Learning Outcomes Specific Outcome 5: Solve, algebraically and graphically, first
More informationSyllabus Objective: 2.9 The student will sketch the graph of a polynomial, radical, or rational function.
Precalculus Notes: Unit Polynomial Functions Syllabus Objective:.9 The student will sketch the graph o a polynomial, radical, or rational unction. Polynomial Function: a unction that can be written in
More informationReview of Linear Systems Theory
Review of Linear Systems Theory The following is a (very) brief review of linear systems theory, convolution, and Fourier analysis. I work primarily with discrete signals, but each result developed in
More information9.3 Graphing Functions by Plotting Points, The Domain and Range of Functions
9. Graphing Functions by Plotting Points, The Domain and Range o Functions Now that we have a basic idea o what unctions are and how to deal with them, we would like to start talking about the graph o
More informationRelating axial motion of optical elements to focal shift
Relating aial motion o optical elements to ocal shit Katie Schwertz and J. H. Burge College o Optical Sciences, University o Arizona, Tucson AZ 857, USA katie.schwertz@gmail.com ABSTRACT In this paper,
More informationReview Algebra and Functions & Equations (10 questions)
Paper 1 Review No calculator allowed [ worked solutions included ] 1. Find the set of values of for which e e 3 e.. Given that 3 k 1 is positive for all values of, find the range of possible values for
More information