# 9 Fourier Transform Properties

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 9 Fourier Transform Properties The Fourier transform is a major cornerstone in the analysis and representation of signals and linear, time-invariant systems, and its elegance and importance cannot be overemphasized. Much of its usefulness stems directly from the properties of the Fourier transform, which we discuss for the continuoustime case in this lecture. Many of the Fourier transform properties might at first appear to be simple (or perhaps not so simple) mathematical manipulations of the Fourier transform analysis and synthesis equations. However, in this and later lectures, as we discuss issues such as filtering, modulation, and sampling, it should become increasingly clear that these properties all have important interpretations and meaning in the context of signals and signal processing. The first property that we introduce in this lecture is the symmetry property, specifically the fact that for time functions that are real-valued, the Fourier transform is conjugate symmetric, i.e., X( - o) = X*(w). From this it follows that the real part and the magnitude of the Fourier transform of realvalued time functions are even functions of frequency and that the imaginary part and phase are odd functions of frequency. Because of this property of corjugate symmetry, in displaying or specifying the Fourier transform of a real-valued time function it is necessary to display the transform only for positive values of w. A second important property is that of time and frequency scaling, specifically that a linear expansion (or contraction) of the time axis in the time domain has the effect in the frequency domain of a linear contraction (expansion). In other words, linear scaling in time is reflected in an inverse scaling in frequency. As we discuss and demonstrate in the lecture, we are all likely to be somewhat familiar with this property from the shift in frequencies that occurs when we slow down or speed up a tape recording. More generally, this is one aspect of a broader set of issues relating to important trade-offs between the time domain and frequency domain. As we will see in later lectures, for example, it is often desirable to design signals that are both narrow in time and narrow in frequency. The relationship between time and frequency scaling is one indication that these are competing requirements; i.e., attempting

2 Signals and Systems 9-2 to make a signal narrower in time will typically have the effect of broadening its Fourier transform. Duality between the time and frequency domains is another important property of Fourier transforms. This property relates to the fact that the analysis equation and synthesis equation look almost identical except for a factor of 1/ 2 7r and the difference of a minus sign in the exponential in the integral. As a consequence, if we know the Fourier transform of a specified time function, then we also know the Fourier transform of a signal whose functional form is the same as the form of this Fourier transform. Said another way, the Fourier transform of the Fourier transform is proportional to the original signal reversed in time. One consequence of this is that whenever we evaluate one transform pair we have another one for free. As another consequence, if we have an effective and efficient algorithm or procedure for implementing or calculating the Fourier transform of a signal, then exactly the same procedure with only minor modification can be used to implement the inverse Fourier transform. This is in fact very heavily exploited in discrete-time signal analysis and processing, where explicit computation of the Fourier transform and its inverse play an important role. There are many other important properties of the Fourier transform, such as Parseval's relation, the time-shifting property, and the effects on the Fourier transform of differentiation and integration in the time domain. The time-shifting property identifies the fact that a linear displacement in time corresponds to a linear phase factor in the frequency domain. This becomes useful and important when we discuss filtering and the effects of the phase characteristics of a filter in the time domain. The differentiation property for Fourier transforms is very useful, as we see in this lecture, for analyzing systems represented by linear constant-coefficient differential equations. Also, we should recognize from the differentiation property that differentiating in the time domain has the effect of emphasizing high frequencies in the Fourier transform. We recall in the discussion of the Fourier series that higher frequencies tend to be associated with abrupt changes (for example, the step discontinuity in the square wave). In the time domain we recognize that differentiation will emphasize these abrupt changes, and the differentiation property states that, consistent with this result, the high frequencies are amplified in relation to the low frequencies. Two major properties that form the basis for a wide array of signal processing systems are the convolution and modulation properties. According to the convolution property, the Fourier transform maps convolution to multiplication; that is, the Fourier transform of the convolution of two time functions is the product of their corresponding Fourier transforms. For the analysis of linear, time-invariant systems, this is particularly useful because through the use of the Fourier transform we can map the sometimes difficult problem of evaluating a convolution to a simpler algebraic operation, namely multiplication. Furthermore, the convolution property highlights the fact that by decomposing a signal into a linear combination of complex exponentials, which the Fourier transform does, we can interpret the effect of a linear, timeinvariant system as simply scaling the (complex) amplitudes of each of these exponentials by a scale factor that is characteristic of the system. This "spectrum" of scale factors which the system applies is in fact the Fourier transform of the system impulse response. This is the underlying basis for the concept and implementation of filtering. The final property that we present in this lecture is the modulation property, which is the dual of the convolution property. According to the modulation property, the Fourier transform of the product of two time functions is

3 Fourier Transform Properties 9-3 proportional to the convolution of their Fourier transforms. As we will see in a later lecture, this simple property provides the basis for the understanding and interpretation of amplitude modulation which is widely used in communication systems. Amplitude modulation also provides the basis for sampling, which is the major bridge between continuous-time and discrete-time signal processing and the foundation for many modern signal processing systems using digital and other discrete-time technologies. We will spend several lectures exploring further the ideas of filtering, modulation, and sampling. Before doing so, however, we will first develop in Lectures 10 and 11 the ideas of Fourier series and the Fourier transform for the discrete-time case so that when we discuss filtering, modulation, and sampling we can blend ideas and issues for both classes of signals and systems. Suggested Reading Section 4.6, Properties of the Continuous-Time Fourier Transform, pages Section 4.7, The Convolution Property, pages Section 6.0, Introduction, pages Section 4.8, The Modulation Property, pages Section 4.9, Tables of Fourier Properties and of Basic Fourier Transform and Fourier Series Pairs, pages Section 4.10, The Polar Representation of Continuous-Time Fourier Transforms, pages Section , Calculation of Frequency and Impulse Responses for LTI Systems Characterized by Differential Equations, pages

4 Signals and Systems CONTINUOUS - TIME FOURIER TRANSFORM 9.1 Analysis and synthesis equations for the continuous-time Fourier transform. X(t) =1 +00 X(G)= f X(co) e jot dco x(t) e~jot dt synthesis analysis x(t) +->. X(W) X(w) = Re IX(w) = IX(eo)ej x(" + j Im (j)[ PROPERTIES OF THE FOURIER TRANSFORM 9.2 Symmetry properties of the Fourier transform. Symmetry: X(t) X(j) x(t) real => X(-w) = X*() Re X(o) IX(co)I Im X(o) '4X(o) = Re X(-o) = IX(-w)I = -Im X(-4) even odd

5 Fourier Transform Properties 9-5 Example 4.7: eat u(t) a+jcw a > o IxMe) 9.3 1/a The Fourier transform 1/a/2 for an exponential - -- time function illustrating the property that the Fourier transform magnitude is even and 1/a -a a the phase is odd. 7r/4-7/4 IT/2 Time and frequency scaling: 1 \9.4 x(at) -+ X The property of time and frequency scaling Example: for the Fourier transform. &>tut)~1 _1 e- at Ut L -I- - 1 a+jo a 1+ j (J 1/a x(t) e t C

6 Signals and Systems DEMONSTRATION 9.1 Time and frequency scaling. A glockenspiel note is recorded and then replayed at twice and half speed. I 9.5 The property of duality.

7 Fourier Transform Properties Example 4.11: 2sin wt 2 2rt X(W) 9.6 Illustration of the duality property. w w Example 4.10: X((o) 2 co Parseval's relation: +00 f 00 Ix(t) 2 dt 1 27r +00) f-00 IX(c) 2 do 9.7 Parseval's relation for the Fourier transform and the Fourier series. if 0O I'(t)I1 2 dt +00 k=- 00 lak12

8 Signals and Systems 9-8 Time shifting: 9.8 Some additional properties of the Fourier transform. Differentiation: x(t-t ).- e-jwto X(o) dx(t) dt jw X(W) Integration: f00 x(t)dr X(W) + 7r X(O) b(w) Jco f t Linearity: ax (t) + bx 2 (t) -. ax 1 (w) + bx 2 (w) 9.9 Transparencies 9.9 and 9.10 illustrate the convolution property and its interpretation for LTI systems. This transparency indicates the response to an impulse.

9 Fourier Transform Properties CONVOLUTION PROPERTY h(t) * x(t) H (w) X( ) 9.10 Response to a complex exponential. x(t) h(t) h(t) * x(t) X(W) H(w) H(w) X(W) e*jcot H(w o ) H(w) = frequency response FILTERING Ideal lowpass filter: -I Co Co 9.11 Filtering as an illustration of the interpretation of the convolution property for LTI systems. Differentiator: dx(t) y(t) =t => H(co) = jco IH(M)I Oppo

10 Signals and Systems 9-10 DEMONSTRATION 9.2 Lowpass filtering of an image. DEMONSTRATION 9.3 Effect of differentiating an image.

11 Fourier Transform Properties 9-11 MODULATION PROPERTY s(t) p(t) 1 -- [S(o) * P(w)] 27r 9.12 The modulation property for the Fourier transform. Modulation: s(t) p(t) -- [S(w) * P(co)] 27r Convolution: s(t) * p(t) S(co) P(W) MARKERBOARD 9.1 -t %~, %~' I Y~wj~ jq4i I _+)+ Gt (t) =At (7): (V j. J)j &o,e - I I =,ji342. Jh~+I t ~

12 MIT OpenCourseWare Resource: Signals and Systems Professor Alan V. Oppenheim The following may not correspond to a particular course on MIT OpenCourseWare, but has been provided by the author as an individual learning resource. For information about citing these materials or our Terms of Use, visit:

### Lecture 8 ELE 301: Signals and Systems

Lecture 8 ELE 3: Signals and Systems Prof. Paul Cuff Princeton University Fall 2-2 Cuff (Lecture 7) ELE 3: Signals and Systems Fall 2-2 / 37 Properties of the Fourier Transform Properties of the Fourier

### UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 101 - Fall 2010 Linear Systems Fundamentals

UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 101 - Fall 2010 Linear Systems Fundamentals FINAL EXAM WITH SOLUTIONS (YOURS!) You are allowed one 2-sided sheet of

### Frequency Response of FIR Filters

Frequency Response of FIR Filters Chapter 6 This chapter continues the study of FIR filters from Chapter 5, but the emphasis is frequency response, which relates to how the filter responds to an input

### Frequency Domain and Fourier Transforms

Chapter 4 Frequency Domain and Fourier Transforms Frequency domain analysis and Fourier transforms are a cornerstone of signal and system analysis. These ideas are also one of the conceptual pillars within

### UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 101 - Fall 2009 Linear Systems Fundamentals

UNIVERSITY OF CALIFORNIA, SAN DIEGO Electrical & Computer Engineering Department ECE 101 - Fall 2009 Linear Systems Fundamentals MIDTERM EXAM You are allowed one 2-sided sheet of notes. No books, no other

### SGN-1158 Introduction to Signal Processing Test. Solutions

SGN-1158 Introduction to Signal Processing Test. Solutions 1. Convolve the function ( ) with itself and show that the Fourier transform of the result is the square of the Fourier transform of ( ). (Hints:

### Review of Fourier series formulas. Representation of nonperiodic functions. ECE 3640 Lecture 5 Fourier Transforms and their properties

ECE 3640 Lecture 5 Fourier Transforms and their properties Objective: To learn about Fourier transforms, which are a representation of nonperiodic functions in terms of trigonometric functions. Also, to

### Lecture 18: The Time-Bandwidth Product

WAVELETS AND MULTIRATE DIGITAL SIGNAL PROCESSING Lecture 18: The Time-Bandwih Product Prof.Prof.V.M.Gadre, EE, IIT Bombay 1 Introduction In this lecture, our aim is to define the time Bandwih Product,

### Analysis/resynthesis with the short time Fourier transform

Analysis/resynthesis with the short time Fourier transform summer 2006 lecture on analysis, modeling and transformation of audio signals Axel Röbel Institute of communication science TU-Berlin IRCAM Analysis/Synthesis

### TCOM 370 NOTES 99-4 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS

TCOM 370 NOTES 99-4 BANDWIDTH, FREQUENCY RESPONSE, AND CAPACITY OF COMMUNICATION LINKS 1. Bandwidth: The bandwidth of a communication link, or in general any system, was loosely defined as the width of

### ENCOURAGING THE INTEGRATION OF COMPLEX NUMBERS IN UNDERGRADUATE ORDINARY DIFFERENTIAL EQUATIONS

Texas College Mathematics Journal Volume 6, Number 2, Pages 18 24 S applied for(xx)0000-0 Article electronically published on September 23, 2009 ENCOURAGING THE INTEGRATION OF COMPLEX NUMBERS IN UNDERGRADUATE

### Class Note for Signals and Systems. Stanley Chan University of California, San Diego

Class Note for Signals and Systems Stanley Chan University of California, San Diego 2 Acknowledgement This class note is prepared for ECE 101: Linear Systems Fundamentals at the University of California,

### San José State University Department of Electrical Engineering EE 112, Linear Systems, Spring 2010

San José State University Department of Electrical Engineering EE 112, Linear Systems, Spring 2010 Instructor: Robert H. Morelos-Zaragoza Office Location: ENGR 373 Telephone: (408) 924-3879 Email: robert.morelos-zaragoza@sjsu.edu

### CONTROLLABILITY. Chapter 2. 2.1 Reachable Set and Controllability. Suppose we have a linear system described by the state equation

Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in

### Basic Op Amp Circuits

Basic Op Amp ircuits Manuel Toledo INEL 5205 Instrumentation August 3, 2008 Introduction The operational amplifier (op amp or OA for short) is perhaps the most important building block for the design of

### 2 The wireless channel

CHAPTER The wireless channel A good understanding of the wireless channel, its key physical parameters and the modeling issues, lays the foundation for the rest of the book. This is the goal of this chapter.

### Methods for Vibration Analysis

. 17 Methods for Vibration Analysis 17 1 Chapter 17: METHODS FOR VIBRATION ANALYSIS 17 2 17.1 PROBLEM CLASSIFICATION According to S. H. Krandall (1956), engineering problems can be classified into three

### Teaching Signals and Systems through Portfolios, Writing, and Independent Learning

Session 2632 Teaching Signals and Systems through Portfolios, Writing, and Independent Learning Richard Vaz, Nicholas Arcolano WPI I. Introduction This paper describes an integrated approach to outcome-driven

### Conceptual similarity to linear algebra

Modern approach to packing more carrier frequencies within agivenfrequencyband orthogonal FDM Conceptual similarity to linear algebra 3-D space: Given two vectors x =(x 1,x 2,x 3 )andy = (y 1,y 2,y 3 ),

### Agilent PN 89400-13 Extending Vector Signal Analysis to 26.5 GHz with 20 MHz Information Bandwidth

Agilent PN 89400-13 Extending Vector Signal Analysis to 26.5 GHz with 20 MHz Information Bandwidth Product Note The Agilent Technologies 89400 series vector signal analyzers provide unmatched signal analysis

### Lecture 13 Linear quadratic Lyapunov theory

EE363 Winter 28-9 Lecture 13 Linear quadratic Lyapunov theory the Lyapunov equation Lyapunov stability conditions the Lyapunov operator and integral evaluating quadratic integrals analysis of ARE discrete-time

### BSEE Degree Plan Bachelor of Science in Electrical Engineering: 2015-16

BSEE Degree Plan Bachelor of Science in Electrical Engineering: 2015-16 Freshman Year ENG 1003 Composition I 3 ENG 1013 Composition II 3 ENGR 1402 Concepts of Engineering 2 PHYS 2034 University Physics

### (Refer Slide Time: 01:11-01:27)

Digital Signal Processing Prof. S. C. Dutta Roy Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 6 Digital systems (contd.); inverse systems, stability, FIR and IIR,

### Discrete-Time Signals and Systems

2 Discrete-Time Signals and Systems 2.0 INTRODUCTION The term signal is generally applied to something that conveys information. Signals may, for example, convey information about the state or behavior

### General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1

A B I L E N E C H R I S T I A N U N I V E R S I T Y Department of Mathematics General Theory of Differential Equations Sections 2.8, 3.1-3.2, 4.1 Dr. John Ehrke Department of Mathematics Fall 2012 Questions

### chapter Introduction to Digital Signal Processing and Digital Filtering 1.1 Introduction 1.2 Historical Perspective

Introduction to Digital Signal Processing and Digital Filtering chapter 1 Introduction to Digital Signal Processing and Digital Filtering 1.1 Introduction Digital signal processing (DSP) refers to anything

### EE 179 April 21, 2014 Digital and Analog Communication Systems Handout #16 Homework #2 Solutions

EE 79 April, 04 Digital and Analog Communication Systems Handout #6 Homework # Solutions. Operations on signals (Lathi& Ding.3-3). For the signal g(t) shown below, sketch: a. g(t 4); b. g(t/.5); c. g(t

### South Carolina College- and Career-Ready (SCCCR) Pre-Calculus

South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know

### INTRUSION PREVENTION AND EXPERT SYSTEMS

INTRUSION PREVENTION AND EXPERT SYSTEMS By Avi Chesla avic@v-secure.com Introduction Over the past few years, the market has developed new expectations from the security industry, especially from the intrusion

### ELECTRICAL ENGINEERING

EE ELECTRICAL ENGINEERING See beginning of Section H for abbreviations, course numbers and coding. The * denotes labs which are held on alternate weeks. A minimum grade of C is required for all prerequisite

### S. Boyd EE102. Lecture 1 Signals. notation and meaning. common signals. size of a signal. qualitative properties of signals.

S. Boyd EE102 Lecture 1 Signals notation and meaning common signals size of a signal qualitative properties of signals impulsive signals 1 1 Signals a signal is a function of time, e.g., f is the force

### Department of Electrical and Computer Engineering Ben-Gurion University of the Negev. LAB 1 - Introduction to USRP

Department of Electrical and Computer Engineering Ben-Gurion University of the Negev LAB 1 - Introduction to USRP - 1-1 Introduction In this lab you will use software reconfigurable RF hardware from National

### Lecture Notes on Polynomials

Lecture Notes on Polynomials Arne Jensen Department of Mathematical Sciences Aalborg University c 008 Introduction These lecture notes give a very short introduction to polynomials with real and complex

### Admin stuff. 4 Image Pyramids. Spatial Domain. Projects. Fourier domain 2/26/2008. Fourier as a change of basis

Admin stuff 4 Image Pyramids Change of office hours on Wed 4 th April Mon 3 st March 9.3.3pm (right after class) Change of time/date t of last class Currently Mon 5 th May What about Thursday 8 th May?

### Nyquist Sampling Theorem. By: Arnold Evia

Nyquist Sampling Theorem By: Arnold Evia Table of Contents What is the Nyquist Sampling Theorem? Bandwidth Sampling Impulse Response Train Fourier Transform of Impulse Response Train Sampling in the Fourier

### CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA

We Can Early Learning Curriculum PreK Grades 8 12 INSIDE ALGEBRA, GRADES 8 12 CORRELATED TO THE SOUTH CAROLINA COLLEGE AND CAREER-READY FOUNDATIONS IN ALGEBRA April 2016 www.voyagersopris.com Mathematical

### From Fundamentals of Digital Communication Copyright by Upamanyu Madhow, 2003-2006

Chapter Introduction to Modulation From Fundamentals of Digital Communication Copyright by Upamanyu Madhow, 003-006 Modulation refers to the representation of digital information in terms of analog waveforms

### Computer Networks and Internets, 5e Chapter 6 Information Sources and Signals. Introduction

Computer Networks and Internets, 5e Chapter 6 Information Sources and Signals Modified from the lecture slides of Lami Kaya (LKaya@ieee.org) for use CECS 474, Fall 2008. 2009 Pearson Education Inc., Upper

### RF Measurements Using a Modular Digitizer

RF Measurements Using a Modular Digitizer Modern modular digitizers, like the Spectrum M4i series PCIe digitizers, offer greater bandwidth and higher resolution at any given bandwidth than ever before.

### Low Pass Filter Rise Time vs Bandwidth

AN121 Dataforth Corporation Page 1 of 7 DID YOU KNOW? The number googol is ten raised to the hundredth power or 1 followed by 100 zeros. Edward Kasner (1878-1955) a noted mathematician is best remembered

### Room Acoustic Reproduction by Spatial Room Response

Room Acoustic Reproduction by Spatial Room Response Rendering Hoda Nasereddin 1, Mohammad Asgari 2 and Ayoub Banoushi 3 Audio Engineer, Broadcast engineering department, IRIB university, Tehran, Iran,

### TMA4213/4215 Matematikk 4M/N Vår 2013

Norges teknisk naturvitenskapelige universitet Institutt for matematiske fag TMA43/45 Matematikk 4M/N Vår 3 Løsningsforslag Øving a) The Fourier series of the signal is f(x) =.4 cos ( 4 L x) +cos ( 5 L

### The Fast Fourier Transform (FFT) and MATLAB Examples

The Fast Fourier Transform (FFT) and MATLAB Examples Learning Objectives Discrete Fourier transforms (DFTs) and their relationship to the Fourier transforms Implementation issues with the DFT via the FFT

### FOURIER TRANSFORM BASED SIMPLE CHORD ANALYSIS. UIUC Physics 193 POM

FOURIER TRANSFORM BASED SIMPLE CHORD ANALYSIS Fanbo Xiang UIUC Physics 193 POM Professor Steven M. Errede Fall 2014 1 Introduction Chords, an essential part of music, have long been analyzed. Different

### Systems with Persistent Memory: the Observation Inequality Problems and Solutions

Chapter 6 Systems with Persistent Memory: the Observation Inequality Problems and Solutions Facts that are recalled in the problems wt) = ut) + 1 c A 1 s ] R c t s)) hws) + Ks r)wr)dr ds. 6.1) w = w +

### Big Ideas in Mathematics

Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards

### NRZ Bandwidth - HF Cutoff vs. SNR

Application Note: HFAN-09.0. Rev.2; 04/08 NRZ Bandwidth - HF Cutoff vs. SNR Functional Diagrams Pin Configurations appear at end of data sheet. Functional Diagrams continued at end of data sheet. UCSP

### Networks, instruments and data centres: what does your seismic (meta-) data really mean? Joachim Wassermann (LMU Munich)

Networks, instruments and data centres: what does your seismic (meta-) data really mean? Joachim Wassermann (LMU Munich) Quest Workshop, Hveragerdi 2011 Why an issue? Seismic Networks: Data Exchange many

### 6.1 Add & Subtract Polynomial Expression & Functions

6.1 Add & Subtract Polynomial Expression & Functions Objectives 1. Know the meaning of the words term, monomial, binomial, trinomial, polynomial, degree, coefficient, like terms, polynomial funciton, quardrtic

### This document is downloaded from DR-NTU, Nanyang Technological University Library, Singapore.

This document is downloaded from DR-NTU, Nanyang Technological University Library, Singapore. Title Transcription of polyphonic signals using fast filter bank( Accepted version ) Author(s) Foo, Say Wei;

### Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

### F en = mω 0 2 x. We should regard this as a model of the response of an atom, rather than a classical model of the atom itself.

The Electron Oscillator/Lorentz Atom Consider a simple model of a classical atom, in which the electron is harmonically bound to the nucleus n x e F en = mω 0 2 x origin resonance frequency Note: We should

### Dear Accelerated Pre-Calculus Student:

Dear Accelerated Pre-Calculus Student: I am very excited that you have decided to take this course in the upcoming school year! This is a fastpaced, college-preparatory mathematics course that will also

### MATHEMATICAL SURVEY AND APPLICATION OF THE CROSS-AMBIGUITY FUNCTION

MATHEMATICAL SURVEY AND APPLICATION OF THE CROSS-AMBIGUITY FUNCTION Dennis Vandenberg Department of Mathematical Sciences Indiana University South Bend E-mail Address: djvanden@iusb.edu Submitted to the

### 2013 MBA Jump Start Program

2013 MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Algebra Review Calculus Permutations and Combinations [Online Appendix: Basic Mathematical Concepts] 2 1 Equation of

### Digital Transmission (Line Coding)

Digital Transmission (Line Coding) Pulse Transmission Source Multiplexer Line Coder Line Coding: Output of the multiplexer (TDM) is coded into electrical pulses or waveforms for the purpose of transmission

### Optimizing and Modeling Dynamics in Networks

Optimizing and Modeling Dynamics in Networks Ibrahim Matta 1 Introduction The Internet has grown very large. No one knows exactly how large, but rough estimates indicate billions of users (around 1.8B

### Algebra 2 Year-at-a-Glance Leander ISD 2007-08. 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks

Algebra 2 Year-at-a-Glance Leander ISD 2007-08 1st Six Weeks 2nd Six Weeks 3rd Six Weeks 4th Six Weeks 5th Six Weeks 6th Six Weeks Essential Unit of Study 6 weeks 3 weeks 3 weeks 6 weeks 3 weeks 3 weeks

### COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS

COMPLEX NUMBERS AND DIFFERENTIAL EQUATIONS BORIS HASSELBLATT CONTENTS. Introduction. Why complex numbers were introduced 3. Complex numbers, Euler s formula 3 4. Homogeneous differential equations 8 5.

### Grade 4 - Module 5: Fraction Equivalence, Ordering, and Operations

Grade 4 - Module 5: Fraction Equivalence, Ordering, and Operations Benchmark (standard or reference point by which something is measured) Common denominator (when two or more fractions have the same denominator)

### ANALYZER BASICS WHAT IS AN FFT SPECTRUM ANALYZER? 2-1

WHAT IS AN FFT SPECTRUM ANALYZER? ANALYZER BASICS The SR760 FFT Spectrum Analyzer takes a time varying input signal, like you would see on an oscilloscope trace, and computes its frequency spectrum. Fourier's

### x1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.

Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability

### The Fourier Analysis Tool in Microsoft Excel

The Fourier Analysis Tool in Microsoft Excel Douglas A. Kerr Issue March 4, 2009 ABSTRACT AD ITRODUCTIO The spreadsheet application Microsoft Excel includes a tool that will calculate the discrete Fourier

### Similar matrices and Jordan form

Similar matrices and Jordan form We ve nearly covered the entire heart of linear algebra once we ve finished singular value decompositions we ll have seen all the most central topics. A T A is positive

### CONVOLUTION Digital Signal Processing

CONVOLUTION Digital Signal Processing Introduction As digital signal processing continues to emerge as a major discipline in the field of electrical engineering an even greater demand has evolved to understand

### WAVEFORM DICTIONARIES AS APPLIED TO THE AUSTRALIAN EXCHANGE RATE

Sunway Academic Journal 3, 87 98 (26) WAVEFORM DICTIONARIES AS APPLIED TO THE AUSTRALIAN EXCHANGE RATE SHIRLEY WONG a RAY ANDERSON Victoria University, Footscray Park Campus, Australia ABSTRACT This paper

### FX 115 MS Training guide. FX 115 MS Calculator. Applicable activities. Quick Reference Guide (inside the calculator cover)

Tools FX 115 MS Calculator Handouts Other materials Applicable activities Quick Reference Guide (inside the calculator cover) Key Points/ Overview Advanced scientific calculator Two line display VPAM to

### 3 Orthogonal Vectors and Matrices

3 Orthogonal Vectors and Matrices The linear algebra portion of this course focuses on three matrix factorizations: QR factorization, singular valued decomposition (SVD), and LU factorization The first

### Periodic wave in spatial domain - length scale is wavelength Given symbol l y

1.4 Periodic Waves Often have situations where wave repeats at regular intervals Electromagnetic wave in optical fibre Sound from a guitar string. These regularly repeating waves are known as periodic

### Performance of Quasi-Constant Envelope Phase Modulation through Nonlinear Radio Channels

Performance of Quasi-Constant Envelope Phase Modulation through Nonlinear Radio Channels Qi Lu, Qingchong Liu Electrical and Systems Engineering Department Oakland University Rochester, MI 48309 USA E-mail:

### Engineering: Electrical Engineering

Engineering: Electrical Engineering CRAFTY Curriculum Foundations Project Clemson University, May 4 7, 2000 Ben Oni, Report Editor Kenneth Roby and Susan Ganter, Workshop Organizers Summary This report

### NETWORK STRUCTURES FOR IIR SYSTEMS. Solution 12.1

NETWORK STRUCTURES FOR IIR SYSTEMS Solution 12.1 (a) Direct Form I (text figure 6.10) corresponds to first implementing the right-hand side of the difference equation (i.e. the eros) followed by the left-hand

### The Fast Fourier Transform

The Fast Fourier Transform Chris Lomont, Jan 2010, http://www.lomont.org, updated Aug 2011 to include parameterized FFTs. This note derives the Fast Fourier Transform (FFT) algorithm and presents a small,

### 5 Signal Design for Bandlimited Channels

225 5 Signal Design for Bandlimited Channels So far, we have not imposed any bandwidth constraints on the transmitted passband signal, or equivalently, on the transmitted baseband signal s b (t) I[k]g

### a. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F

FINAL REVIEW WORKSHEET COLLEGE ALGEBRA Chapter 1. 1. Given the following equations, which are functions? (A) y 2 = 1 x 2 (B) y = 9 (C) y = x 3 5x (D) 5x + 2y = 10 (E) y = ± 1 2x (F) y = 3 x + 5 a. all

### DRAFT. Algebra 1 EOC Item Specifications

DRAFT Algebra 1 EOC Item Specifications The draft Florida Standards Assessment (FSA) Test Item Specifications (Specifications) are based upon the Florida Standards and the Florida Course Descriptions as

### Modulation and Demodulation

MIT 6.02 DRAFT Lecture Notes Last update: April 11, 2012 Comments, questions or bug reports? Please contact {hari, verghese} at mit.edu CHAPTER 14 Modulation and Demodulation This chapter describes the

### MATRIX TECHNICAL NOTES

200 WOOD AVENUE, MIDDLESEX, NJ 08846 PHONE (732) 469-9510 FAX (732) 469-0418 MATRIX TECHNICAL NOTES MTN-107 TEST SETUP FOR THE MEASUREMENT OF X-MOD, CTB, AND CSO USING A MEAN SQUARE CIRCUIT AS A DETECTOR

### Course overview Processamento de sinais 2009/10 LEA

Course overview Processamento de sinais 2009/10 LEA João Pedro Gomes jpg@isr.ist.utl.pt Instituto Superior Técnico Processamento de sinais MEAer (IST) Course overview 1 / 19 Course overview Motivation:

### 4ECE 320 Signals and Systems II Department of Electrical and Computer Engineering George Mason University Fall, 2015

ECE 320 1 Fall, 2015 4ECE 320 Signals and Systems II Department of Electrical and Computer Engineering George Mason University Fall, 2015 Class Meeting Information Day and Time: Tuesday and Thursday, 4:30

### Graduate Certificate in Systems Engineering

Graduate Certificate in Systems Engineering Systems Engineering is a multi-disciplinary field that aims at integrating the engineering and management functions in the development and creation of a product,

### Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two

### Vector Spaces; the Space R n

Vector Spaces; the Space R n Vector Spaces A vector space (over the real numbers) is a set V of mathematical entities, called vectors, U, V, W, etc, in which an addition operation + is defined and in which

### DEFINITION 5.1.1 A complex number is a matrix of the form. x y. , y x

Chapter 5 COMPLEX NUMBERS 5.1 Constructing the complex numbers One way of introducing the field C of complex numbers is via the arithmetic of matrices. DEFINITION 5.1.1 A complex number is a matrix of

### The Fourth International DERIVE-TI92/89 Conference Liverpool, U.K., 12-15 July 2000. Derive 5: The Easiest... Just Got Better!

The Fourth International DERIVE-TI9/89 Conference Liverpool, U.K., -5 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de technologie supérieure 00, rue Notre-Dame Ouest Montréal

### System Modeling and Control for Mechanical Engineers

Session 1655 System Modeling and Control for Mechanical Engineers Hugh Jack, Associate Professor Padnos School of Engineering Grand Valley State University Grand Rapids, MI email: jackh@gvsu.edu Abstract

### STUDENT VERSION INSECT COLONY SURVIVAL OPTIMIZATION

STUDENT VERSION INSECT COLONY SURVIVAL OPTIMIZATION STATEMENT We model insect colony propagation or survival from nature using differential equations. We ask you to analyze and report on what is going

### Audio Engineering Society. Convention Paper. Presented at the 119th Convention 2005 October 7 10 New York, New York USA

Audio Engineering Society Convention Paper Presented at the 119th Convention 2005 October 7 10 New York, New York USA This convention paper has been reproduced from the author's advance manuscript, without

### The Membrane Equation

The Membrane Equation Professor David Heeger September 5, 2000 RC Circuits Figure 1A shows an RC (resistor, capacitor) equivalent circuit model for a patch of passive neural membrane. The capacitor represents

### Phone: (773) 481-8488 Spring 2015. Office hours: MW 7:30-8:20 and 11:00-12:20, T 7:30-7:50 and 9:55-12:15

Math 140 BDYR Prof. Hellen Colman Email: hcolman@ccc.edu Office: L309 College Algebra Hybrid Class Tuesdays 8:00AM 9:45AM Phone: (773) 481-8488 Spring 2015 Office hours: MW 7:30-8:20 and 11:00-12:20, T

### CITY UNIVERSITY LONDON. BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION

No: CITY UNIVERSITY LONDON BEng Degree in Computer Systems Engineering Part II BSc Degree in Computer Systems Engineering Part III PART 2 EXAMINATION ENGINEERING MATHEMATICS 2 (resit) EX2005 Date: August

### Shaft. Application of full spectrum to rotating machinery diagnostics. Centerlines. Paul Goldman, Ph.D. and Agnes Muszynska, Ph.D.

Shaft Centerlines Application of full spectrum to rotating machinery diagnostics Benefits of full spectrum plots Before we answer these questions, we d like to start with the following observation: The

### Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any.

Algebra 2 - Chapter Prerequisites Vocabulary Copy in your notebook: Add an example of each term with the symbols used in algebra 2 if there are any. P1 p. 1 1. counting(natural) numbers - {1,2,3,4,...}

### Recovery of Pyroshock Data From Distorted Acceleration Records

NASA Technical NASA-TP-2494 19850022828 Paper 2494 July 1985 Recovery of Pyroshock Data From Distorted Acceleration Records James Lee Smith J,. h_ctb ANGt. ll_' RI_SIrAt?CH C['.NTEI_ i..i}3.rar'," NAt';A

### (Refer Slide Time: 1:42)

Introduction to Computer Graphics Dr. Prem Kalra Department of Computer Science and Engineering Indian Institute of Technology, Delhi Lecture - 10 Curves So today we are going to have a new topic. So far

### Recommendations for TDR configuration for channel characterization by S-parameters. Pavel Zivny IEEE 802.3 100GCU Singapore, 2011/03 V1.

Recommendations for TDR configuration for channel characterization by S-parameters Pavel Zivny IEEE 802.3 100GCU Singapore, 2011/03 V1.0 Agenda TDR/TDT measurement setup TDR/TDT measurement flow DUT electrical

### Agilent Time Domain Analysis Using a Network Analyzer

Agilent Time Domain Analysis Using a Network Analyzer Application Note 1287-12 0.0 0.045 0.6 0.035 Cable S(1,1) 0.4 0.2 Cable S(1,1) 0.025 0.015 0.005 0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Frequency (GHz) 0.005

### Sampling Theorem Notes. Recall: That a time sampled signal is like taking a snap shot or picture of signal periodically.

Sampling Theorem We will show that a band limited signal can be reconstructed exactly from its discrete time samples. Recall: That a time sampled signal is like taking a snap shot or picture of signal