Fourier Analysis and its applications

Save this PDF as:
Size: px
Start display at page:

Download "Fourier Analysis and its applications"

Transcription

1 Fourier Analysis and its applications Fourier analysis originated from the study of heat conduction: Jean Baptiste Joseph Fourier ( ) Fourier analysis enables a function (signal) to be decomposed into its frequency components. Wide range of applications: Spectrum analysis Digital filtering (e.g. electronics, image processing), Deconvolution Audio compression (MP3 etc.) Solving differential equations

2 Definition of Fourier series Consider a periodic function f (t) with period T = such that f (t + ) = f (t), (1) Any periodic function can be expressed as an infinite Fourier series of sin and cos functions with frequencies that are integer multiples of ω = /T = 1, the frequency of the function: f (t) = a (a n cos(nt) + b n sin(nt)), (2) n=1 where a 0, a 1,... b 1,..., b n are the Fourier coefficients.

3 Convergence Fourier Analysis Theorem Dirichlet s theorem If f (t) is periodic of period, if for π < t < π the function f (t) has a finite number of maximum and minimum values and a finite number of discontinuities, and if π π f (t)dt is finite, then the Fourier series converges to f (t) at all points where f (t) is continuous, and at discontinuities it converges to the arithmetic mean of the right-hand and left-hand limits of the function. For all practical purposes, these conditions hold and the series converges.

4 Fourier series as an expansion of orthogonal functions can be viewed as an expansion of orthogonal functions that form a complete set over any interval: π { πδm,n if m 0 sin(mt) sin(nt)dt = (3) π 0 if m = 0 π { πδm,n if m 0 cos(mt) cos(nt)dt = (4) if m = n = 0 π π π sin(mt) cos(nt)dt = 0 all integral m and n (5) From these orthogonality relations we can derive the Fourier coefficients a n and b n

5 Fourier coefficients Fourier Analysis a n = 1 π b n = 1 π π π π π f (t) cos(nt)dt (6) f (t) sin(nt)dt (7) You can prove this by substituting the Fourier series for f (t) (eqn. (2)) in to the above relations and use the orthogonality equations (eqns. (3,4,5).

6 Example - the square wave Consider the unit step function (i.e square wave): f (t) = { 1 for π < t < 0 +1 for 0 < t < π (8) Since the function is odd, all a n s are zero. The b n s are given by b n = 1 π 0 π( 1) sin(nt)dt + 1 π π = 2 π 0 = 2 [1 cos(nπ)] nπ = π sin(nt)dt = 2 nπ [ cos(nt)]π 0 { 0, n = 2, 4, 6,... 4/nπ, n = 1, 3, 5,... 0 (+1) sin(nt)dt

7 Fourier series of square wave

8 Fourier series of square wave: N=1

9 Fourier series of square wave: N=5

10 Fourier series of square wave: N=50

11 Remarks Fourier Analysis The amplitude of the high frequency sine waves falls of 1/n. Therefore convergence is slow. In general: For discontinuous functions, like the square wave, the coefficients decrease as 1/n. For continuous functions with discontinuous slopes (e.g. full wave rectifier), coefficients typically decrease as 1/n 2. Overshoot at discontinuities is called Gibbs phenomenon. In electronics, square wave pulses are common. If apparatus does not pass the high frequencies, square wave will be rounded off.

12 Fourier series of complex functions Definition of Fourier series can easily be extended to complex functions. Using the identity e ix = cos(x) + i sin(x) we can rewrite Fourier series (eqn.(2)) as f (t) = n= c n e int, (9) where (a n ib n )/2, n > 0, c n = a 0 /2 n = 0, (a n + ib n )/2 n < 0 which can be obtained by integration (10) c n = 1 π π f (t)e int dt (11)

13 Fourier series of functions with arbitrary period So far, considered functions periodic on a interval. Can easily extend to functions with any period T : Consider a periodic function f (t) with period T, and corresponding angular frequency ω = /T. Let t ωt = t/t. Then the complex Fourier series (eqn. (9)) becomes f (t) = c n e inωt, (12) where, c n = 1 T n= T /2 T /2 f (t)e inωt dt (13)

14 Fourier series of functions with arbitrary period Similarly for the Fourier series of real functions (eqn. (2)) becomes: where, f (t) = a (a n cos(nωt) + b n sin(nωt)), (14) n=1 a n = 2 T b n = 2 T T /2 T /2 T /2 T /2 f (t) cos(nωt)dt (15) f (t) sin(nωt)dt (16) Function is expressed as a series of sin and cos functions with frequencies that are integer multiples of ω, the frequency of the function.

15 Fourier series - summary Fourier series decomposes periodic signal into its frequency components. Even discontinuous functions or functions with discontinuous slopes (where Taylor expansion fails) can be expressed in a converging Fourier series. Solving differential equations. e.g. harmonic oscillator with some periodic driving force.

16 From Fourier series to Fourier transform Concept of Fourier series can be expanded to non-periodic functions. Consider the Fourier series for functions with period T in complex representation: f (t) = n= c n e in ωt, (17) with c n = ω T /2 T /2 f (t)e in ωt dt, (18) where we have written ω = ω, since the discrete frequencies nω are separated by ω = /T.

17 From Fourier series to Fourier transform Now define c n = ω g(n ω), (19) so that g(n ω) = f (t) = 1 T /2 n= T /2 f (t)e in ωt dt, (20) ωg(n ω)e in ωt. (21) Take the limit as T. Then, n ω becomes the continuous variable ω and the summation becomes an integral as ω = /T dω.

18 Fourier Analysis time domain f (t) = 1 frequency domain g(ω) = 1 We define g(ω) to be the Fourier transform of f (t): g(ω)e iωt dω, (22) f (t)e iωt dt (23) Fourier transform F[f (t)] = g(ω) (24) inverse transform F 1 [g(ω)] = f (t) (25)

19 Remarks: Fourier Analysis As with the Fourier series, the Fourier transform is used to obtain information on the frequency spectrum. However, F[f (t)] = g(ω) is complex in general, even if f (t) is real. Fourier transform is often represented in terms of its magnitude and phase: g(ω) = g(ω) e iθ(ω), where θ = tan 1 (Im(g(ω)/Re(g(ω))). Prefactor (1/ ) of the transforms depends on convention. Here we chose them to be distributed symmetrically. Often, the Fourier transform is used to go from the time domain to the frequency domain. However, mathematically, the original transform could be considered the inverse and vice versa. Therefore, the sign in the exponential also depends on the convention.

20 The Dirac delta function If Fourier transform are consistent, then inserting f (t) into g(ω) should lead to an identity: 1 ( 1 ) g(ω) = g(ω )e iω t dω e iωt dt (26) } {{ } = f (t) [ 1 ] e i(ω ω)t dt g(ω )dω (27) The term in the square brackets [...] has to be the Dirac delta function: δ(ω ω) = 1 e i(ω ω)t dt (28)

21 The Dirac delta function Consider δ(ω ω) τ = 1 τ τ e i(ω ω)t dt = sin((ω ω)τ) π(ω ω)

22 The Dirac delta function Eqns.(27) and (28) have to hold for any function g(ω). Set g(ω) = 1: 1 = However, according to eqn. (28), δ(0) =. δ(ω ω)dω (29) Delta function is a distribution, not a function, and has only a meaning inside an integrand!

23 Properties of the Fourier transform Linearity: F[f 1 (t) + f 2 (t)] = F[f 1 (t)] + F[f 2 (t)]. Shifting: F[f (t t 0 )] = = 1 1 = e iωt 0 1 = e iωt 0 g(ω), f (t t 0 )e iωt dt f (τ)e iω(τ+t 0) dτ f (τ)e iωτ dτ where τ = t t 0. Similarly, F 1 [g(ω ω 0 )] = e iω 0t f (t)

24 Properties of the Fourier transform Scaling: Let F[f (t)] = g(ω) and α > 0. Then, F[f (αt)] = = α = 1 α g(ω/α), f (αt)e iωt dt f (t )e iωt /α dt where substitution t = αt was made. For α < 0, we obtain F[f (αt)] = 1 α g(ω/α)

25 Properties of the Fourier transform Therefore, F[f (αt)] = 1 α g(ω/α) Similarly, F 1 [g(βt)] = 1 β f (t/β). This is a crucial relation!. As function f (t) becomes more localized, its Fourier transform becomes broader in frequency domain.

26 Parseval s theorem Fourier Analysis Consider the integral I = f 1(t) f 2 (t)dt. Substitute Fourier transforms of f 1 (t) and f 2 (t): [ 1 ] [ 1 ] I = g 1 (ω)e iωt dω g 2 (ω )e iω t dω dt = = = g 1 (ω)g 2 (ω ) [ 1 ] e i(ω ω)t dt dωdω g 1 (ω)g 2 (ω )δ(ω ω)dωdω g 1 (ω)g 2 (ω)dω For f 1 (t) = f 2 (t), we obtain Parseval s theorem: f (t) 2 dt = } {{ } Power g(ω) 2 } {{ } Power spectrum dω (30)

27 Higher dimensions Fourier Analysis So far considered 1D Fourier transform. Generalization to higher dimensions as follows: 2D: 3D: F[f (x, y)] = g(k x, k y ) = 1 F[f (x, y, z)] = g(k x, k y, k z ) = ( ) = g( k) = ( 1 ) 3 2 f (x, y)e i(kx x+ky y) dxdy f (x, y, z)e i(kx x+ky y+kz z) dxdydz f ( r)e i k r d r

FOURIER ANALYSIS. Lucas Illing. 1 Fourier Series 2 1.1 General Introduction... 2 1.2 Discontinuous Functions... 5 1.3 Complex Fourier Series...

FOURIER ANALYSIS. Lucas Illing. 1 Fourier Series 2 1.1 General Introduction... 2 1.2 Discontinuous Functions... 5 1.3 Complex Fourier Series... FOURIER ANALYSIS Lucas Illing 2008 Contents Fourier Series 2. General Introduction........................ 2.2 Discontinuous Functions...................... 5.3 Complex Fourier Series.......................

More information

Introduction to Frequency Domain Processing 1

Introduction to Frequency Domain Processing 1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.151 Advanced System Dynamics and Control Introduction to Frequency Domain Processing 1 1 Introduction - Superposition In this

More information

Fourier Analysis. u m, a n u n = am um, u m

Fourier Analysis. u m, a n u n = am um, u m Fourier Analysis Fourier series allow you to expand a function on a finite interval as an infinite series of trigonometric functions. What if the interval is infinite? That s the subject of this chapter.

More information

The continuous and discrete Fourier transforms

The continuous and discrete Fourier transforms FYSA21 Mathematical Tools in Science The continuous and discrete Fourier transforms Lennart Lindegren Lund Observatory (Department of Astronomy, Lund University) 1 The continuous Fourier transform 1.1

More information

y cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx y 1 u 2 du u 1 3u 3 C

y cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx y 1 u 2 du u 1 3u 3 C Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.

More information

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

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

More information

ENCOURAGING THE INTEGRATION OF COMPLEX NUMBERS IN UNDERGRADUATE ORDINARY DIFFERENTIAL EQUATIONS

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

More information

Analog and Digital Signals, Time and Frequency Representation of Signals

Analog and Digital Signals, Time and Frequency Representation of Signals 1 Analog and Digital Signals, Time and Frequency Representation of Signals Required reading: Garcia 3.1, 3.2 CSE 3213, Fall 2010 Instructor: N. Vlajic 2 Data vs. Signal Analog vs. Digital Analog Signals

More information

054414 PROCESS CONTROL SYSTEM DESIGN. 054414 Process Control System Design LECTURE 2: FREQUENCY DOMAIN ANALYSIS

054414 PROCESS CONTROL SYSTEM DESIGN. 054414 Process Control System Design LECTURE 2: FREQUENCY DOMAIN ANALYSIS 05444 PROCESS CONTROL SYSTEM DESIGN 05444 Process Control System Design LECTURE : FREQUENCY DOMAIN ANALYSIS Daniel R. Lewin Department of Chemical Engineering Technion, Haifa, Israel - Objectives On completing

More information

Chapter 3. Integral Transforms

Chapter 3. Integral Transforms Chapter 3 Integral Transforms This part of the course introduces two extremely powerful methods to solving differential equations: the Fourier and the Laplace transforms. Beside its practical use, the

More information

Mechanical Vibrations

Mechanical Vibrations Mechanical Vibrations A mass m is suspended at the end of a spring, its weight stretches the spring by a length L to reach a static state (the equilibrium position of the system). Let u(t) denote the displacement,

More information

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

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

More information

2.2 Magic with complex exponentials

2.2 Magic with complex exponentials 2.2. MAGIC WITH COMPLEX EXPONENTIALS 97 2.2 Magic with complex exponentials We don t really know what aspects of complex variables you learned about in high school, so the goal here is to start more or

More information

y cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx

y cos 3 x dx y cos 2 x cos x dx y 1 sin 2 x cos x dx Trigonometric Integrals In this section we use trigonometric identities to integrate certain combinations of trigonometric functions. We start with powers of sine and cosine. EXAMPLE Evaluate cos 3 x dx.

More information

TMA4213/4215 Matematikk 4M/N Vår 2013

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

More information

Oscillations. Vern Lindberg. June 10, 2010

Oscillations. Vern Lindberg. June 10, 2010 Oscillations Vern Lindberg June 10, 2010 You have discussed oscillations in Vibs and Waves: we will therefore touch lightly on Chapter 3, mainly trying to refresh your memory and extend the concepts. 1

More information

Introduction to Complex Fourier Series

Introduction to Complex Fourier Series Introduction to Complex Fourier Series Nathan Pflueger 1 December 2014 Fourier series come in two flavors. What we have studied so far are called real Fourier series: these decompose a given periodic function

More information

Matter Waves. Home Work Solutions

Matter Waves. Home Work Solutions Chapter 5 Matter Waves. Home Work s 5.1 Problem 5.10 (In the text book) An electron has a de Broglie wavelength equal to the diameter of the hydrogen atom. What is the kinetic energy of the electron? How

More information

Second Order Systems

Second Order Systems Second Order Systems Second Order Equations Standard Form G () s = τ s K + ζτs + 1 K = Gain τ = Natural Period of Oscillation ζ = Damping Factor (zeta) Note: this has to be 1.0!!! Corresponding Differential

More information

Time domain modeling

Time domain modeling Time domain modeling Equationof motion of a WEC Frequency domain: Ok if all effects/forces are linear M+ A ω X && % ω = F% ω K + K X% ω B ω + B X% & ω ( ) H PTO PTO + others Time domain: Must be linear

More information

Introduction to Complex Numbers in Physics/Engineering

Introduction to Complex Numbers in Physics/Engineering Introduction to Complex Numbers in Physics/Engineering ference: Mary L. Boas, Mathematical Methods in the Physical Sciences Chapter 2 & 14 George Arfken, Mathematical Methods for Physicists Chapter 6 The

More information

L and C connected together. To be able: To analyse some basic circuits.

L and C connected together. To be able: To analyse some basic circuits. circuits: Sinusoidal Voltages and urrents Aims: To appreciate: Similarities between oscillation in circuit and mechanical pendulum. Role of energy loss mechanisms in damping. Why we study sinusoidal signals

More information

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.

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

More information

Let s first see how precession works in quantitative detail. The system is illustrated below: ...

Let s first see how precession works in quantitative detail. The system is illustrated below: ... lecture 20 Topics: Precession of tops Nutation Vectors in the body frame The free symmetric top in the body frame Euler s equations The free symmetric top ala Euler s The tennis racket theorem As you know,

More information

Homework # 3 Solutions

Homework # 3 Solutions Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8

More information

Solution: F = kx is Hooke s law for a mass and spring system. Angular frequency of this system is: k m therefore, k

Solution: F = kx is Hooke s law for a mass and spring system. Angular frequency of this system is: k m therefore, k Physics 1C Midterm 1 Summer Session II, 2011 Solutions 1. If F = kx, then k m is (a) A (b) ω (c) ω 2 (d) Aω (e) A 2 ω Solution: F = kx is Hooke s law for a mass and spring system. Angular frequency of

More information

Taylor and Maclaurin Series

Taylor and Maclaurin Series Taylor and Maclaurin Series In the preceding section we were able to find power series representations for a certain restricted class of functions. Here we investigate more general problems: Which functions

More information

Appendix D Digital Modulation and GMSK

Appendix D Digital Modulation and GMSK D1 Appendix D Digital Modulation and GMSK A brief introduction to digital modulation schemes is given, showing the logical development of GMSK from simpler schemes. GMSK is of interest since it is used

More information

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. 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

More information

2.6 The driven oscillator

2.6 The driven oscillator 2.6. THE DRIVEN OSCILLATOR 131 2.6 The driven oscillator We would like to understand what happens when we apply forces to the harmonic oscillator. That is, we want to solve the equation M d2 x(t) 2 + γ

More information

TOPIC 4: DERIVATIVES

TOPIC 4: DERIVATIVES TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the

More information

Derive 5: The Easiest... Just Got Better!

Derive 5: The Easiest... Just Got Better! Liverpool John Moores University, 1-15 July 000 Derive 5: The Easiest... Just Got Better! Michel Beaudin École de Technologie Supérieure, Canada Email; mbeaudin@seg.etsmtl.ca 1. Introduction Engineering

More information

Using the Impedance Method

Using the Impedance Method Using the Impedance Method The impedance method allows us to completely eliminate the differential equation approach for the determination of the response of circuits. In fact the impedance method even

More information

RAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS PART A

RAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS PART A RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:

More information

An Introduction to Partial Differential Equations in the Undergraduate Curriculum

An Introduction to Partial Differential Equations in the Undergraduate Curriculum An Introduction to Partial Differential Equations in the Undergraduate Curriculum J. Tolosa & M. Vajiac LECTURE 11 Laplace s Equation in a Disk 11.1. Outline of Lecture The Laplacian in Polar Coordinates

More information

Fourier Transforms. Figure 7-1:Fourier transform from time domain to frequency domain

Fourier Transforms. Figure 7-1:Fourier transform from time domain to frequency domain Fourier Transforms The whole is the sum of the parts Euclid Fourier Series As nmr spectroscopists working with modern digital equipment we routinely acquire data in the form of the free induction decay.

More information

SIGNAL PROCESSING & SIMULATION NEWSLETTER

SIGNAL PROCESSING & SIMULATION NEWSLETTER 1 of 10 1/25/2008 3:38 AM SIGNAL PROCESSING & SIMULATION NEWSLETTER Note: This is not a particularly interesting topic for anyone other than those who ar e involved in simulation. So if you have difficulty

More information

Wavelet analysis. Wavelet requirements. Example signals. Stationary signal 2 Hz + 10 Hz + 20Hz. Zero mean, oscillatory (wave) Fast decay (let)

Wavelet analysis. Wavelet requirements. Example signals. Stationary signal 2 Hz + 10 Hz + 20Hz. Zero mean, oscillatory (wave) Fast decay (let) Wavelet analysis In the case of Fourier series, the orthonormal basis is generated by integral dilation of a single function e jx Every 2π-periodic square-integrable function is generated by a superposition

More information

Chapter 8 - Power Density Spectrum

Chapter 8 - Power Density Spectrum EE385 Class Notes 8/8/03 John Stensby Chapter 8 - Power Density Spectrum Let X(t) be a WSS random process. X(t) has an average power, given in watts, of E[X(t) ], a constant. his total average power is

More information

f x a 0 n 1 a 0 a 1 cos x a 2 cos 2x a 3 cos 3x b 1 sin x b 2 sin 2x b 3 sin 3x a n cos nx b n sin nx n 1 f x dx y

f x a 0 n 1 a 0 a 1 cos x a 2 cos 2x a 3 cos 3x b 1 sin x b 2 sin 2x b 3 sin 3x a n cos nx b n sin nx n 1 f x dx y Fourier Series When the French mathematician Joseph Fourier (768 83) was tring to solve a problem in heat conduction, he needed to epress a function f as an infinite series of sine and cosine functions:

More information

Mathematical Methods for Computer Science

Mathematical Methods for Computer Science Mathematical Methods for Computer Science lectures on Fourier and related methods Professor J. Daugman Computer Laboratory University of Cambridge Computer Science Tripos, Part IB Michaelmas Term 2015/16

More information

Section 2.7 One-to-One Functions and Their Inverses

Section 2.7 One-to-One Functions and Their Inverses Section. One-to-One Functions and Their Inverses One-to-One Functions HORIZONTAL LINE TEST: A function is one-to-one if and only if no horizontal line intersects its graph more than once. EXAMPLES: 1.

More information

The Heat Equation. Lectures INF2320 p. 1/88

The Heat Equation. Lectures INF2320 p. 1/88 The Heat Equation Lectures INF232 p. 1/88 Lectures INF232 p. 2/88 The Heat Equation We study the heat equation: u t = u xx for x (,1), t >, (1) u(,t) = u(1,t) = for t >, (2) u(x,) = f(x) for x (,1), (3)

More information

Convolution, Correlation, & Fourier Transforms. James R. Graham 10/25/2005

Convolution, Correlation, & Fourier Transforms. James R. Graham 10/25/2005 Convolution, Correlation, & Fourier Transforms James R. Graham 10/25/2005 Introduction A large class of signal processing techniques fall under the category of Fourier transform methods These methods fall

More information

Inner Product Spaces

Inner Product Spaces Math 571 Inner Product Spaces 1. Preliminaries An inner product space is a vector space V along with a function, called an inner product which associates each pair of vectors u, v with a scalar u, v, and

More information

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

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

More information

9 Fourier Transform Properties

9 Fourier Transform Properties 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.

More information

Lecture L5 - Other Coordinate Systems

Lecture L5 - Other Coordinate Systems S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L5 - Other Coordinate Systems In this lecture, we will look at some other common systems of coordinates. We will present polar coordinates

More information

12.4 UNDRIVEN, PARALLEL RLC CIRCUIT*

12.4 UNDRIVEN, PARALLEL RLC CIRCUIT* + v C C R L - v i L FIGURE 12.24 The parallel second-order RLC circuit shown in Figure 2.14a. 12.4 UNDRIVEN, PARALLEL RLC CIRCUIT* We will now analyze the undriven parallel RLC circuit shown in Figure

More information

Differentiation and Integration

Differentiation and Integration This material is a supplement to Appendix G of Stewart. You should read the appendix, except the last section on complex exponentials, before this material. Differentiation and Integration Suppose we have

More information

Unified Lecture # 4 Vectors

Unified Lecture # 4 Vectors Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,

More information

Simple Harmonic Motion

Simple Harmonic Motion Simple Harmonic Motion 1 Object To determine the period of motion of objects that are executing simple harmonic motion and to check the theoretical prediction of such periods. 2 Apparatus Assorted weights

More information

Chapter 6 Circular Motion

Chapter 6 Circular Motion Chapter 6 Circular Motion 6.1 Introduction... 1 6.2 Cylindrical Coordinate System... 2 6.2.1 Unit Vectors... 3 6.2.2 Infinitesimal Line, Area, and Volume Elements in Cylindrical Coordinates... 4 Example

More information

Lecture 7 ELE 301: Signals and Systems

Lecture 7 ELE 301: Signals and Systems Lecture 7 ELE 3: Signals and Systems Prof. Paul Cuff Princeton University Fall 2-2 Cuff (Lecture 7) ELE 3: Signals and Systems Fall 2-2 / 22 Introduction to Fourier Transforms Fourier transform as a limit

More information

Parabolic Equations. Chapter 5. Contents. 5.1.2 Well-Posed Initial-Boundary Value Problem. 5.1.3 Time Irreversibility of the Heat Equation

Parabolic Equations. Chapter 5. Contents. 5.1.2 Well-Posed Initial-Boundary Value Problem. 5.1.3 Time Irreversibility of the Heat Equation 7 5.1 Definitions Properties Chapter 5 Parabolic Equations Note that we require the solution u(, t bounded in R n for all t. In particular we assume that the boundedness of the smooth function u at infinity

More information

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

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

More information

Georgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1

Georgia Department of Education Kathy Cox, State Superintendent of Schools 7/19/2005 All Rights Reserved 1 Accelerated Mathematics 3 This is a course in precalculus and statistics, designed to prepare students to take AB or BC Advanced Placement Calculus. It includes rational, circular trigonometric, and inverse

More information

Lecture L3 - Vectors, Matrices and Coordinate Transformations

Lecture L3 - Vectors, Matrices and Coordinate Transformations S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between

More information

2.2 Separable Equations

2.2 Separable Equations 2.2 Separable Equations 73 2.2 Separable Equations An equation y = f(x, y) is called separable provided algebraic operations, usually multiplication, division and factorization, allow it to be written

More information

Differentiation of vectors

Differentiation of vectors Chapter 4 Differentiation of vectors 4.1 Vector-valued functions In the previous chapters we have considered real functions of several (usually two) variables f : D R, where D is a subset of R n, where

More information

7. Beats. sin( + λ) + sin( λ) = 2 cos(λ) sin( )

7. Beats. sin( + λ) + sin( λ) = 2 cos(λ) sin( ) 34 7. Beats 7.1. What beats are. Musicians tune their instruments using beats. Beats occur when two very nearby pitches are sounded simultaneously. We ll make a mathematical study of this effect, using

More information

1.5 / 1 -- Communication Networks II (Görg) -- www.comnets.uni-bremen.de. 1.5 Transforms

1.5 / 1 -- Communication Networks II (Görg) -- www.comnets.uni-bremen.de. 1.5 Transforms .5 / -- Communication Networks II (Görg) -- www.comnets.uni-bremen.de.5 Transforms Using different summation and integral transformations pmf, pdf and cdf/ccdf can be transformed in such a way, that even

More information

Second Order Linear Partial Differential Equations. Part I

Second Order Linear Partial Differential Equations. Part I Second Order Linear Partial Differential Equations Part I Second linear partial differential equations; Separation of Variables; - point boundary value problems; Eigenvalues and Eigenfunctions Introduction

More information

Correlation and Convolution Class Notes for CMSC 426, Fall 2005 David Jacobs

Correlation and Convolution Class Notes for CMSC 426, Fall 2005 David Jacobs Correlation and Convolution Class otes for CMSC 46, Fall 5 David Jacobs Introduction Correlation and Convolution are basic operations that we will perform to extract information from images. They are in

More information

The Method of Least Squares. Lectures INF2320 p. 1/80

The Method of Least Squares. Lectures INF2320 p. 1/80 The Method of Least Squares Lectures INF2320 p. 1/80 Lectures INF2320 p. 2/80 The method of least squares We study the following problem: Given n points (t i,y i ) for i = 1,...,n in the (t,y)-plane. How

More information

Lecture 8 ELE 301: Signals and Systems

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

More information

www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates

www.mathsbox.org.uk ab = c a If the coefficients a,b and c are real then either α and β are real or α and β are complex conjugates Further Pure Summary Notes. Roots of Quadratic Equations For a quadratic equation ax + bx + c = 0 with roots α and β Sum of the roots Product of roots a + b = b a ab = c a If the coefficients a,b and c

More information

Lectures 5-6: Taylor Series

Lectures 5-6: Taylor Series Math 1d Instructor: Padraic Bartlett Lectures 5-: Taylor Series Weeks 5- Caltech 213 1 Taylor Polynomials and Series As we saw in week 4, power series are remarkably nice objects to work with. In particular,

More information

Understanding Poles and Zeros

Understanding Poles and Zeros MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2.14 Analysis and Design of Feedback Control Systems Understanding Poles and Zeros 1 System Poles and Zeros The transfer function

More information

The one dimensional heat equation: Neumann and Robin boundary conditions

The one dimensional heat equation: Neumann and Robin boundary conditions The one dimensional heat equation: Neumann and Robin boundary conditions Ryan C. Trinity University Partial Differential Equations February 28, 2012 with Neumann boundary conditions Our goal is to solve:

More information

Positive Feedback and Oscillators

Positive Feedback and Oscillators Physics 3330 Experiment #6 Fall 1999 Positive Feedback and Oscillators Purpose In this experiment we will study how spontaneous oscillations may be caused by positive feedback. You will construct an active

More information

Estimated Pre Calculus Pacing Timeline

Estimated Pre Calculus Pacing Timeline Estimated Pre Calculus Pacing Timeline 2010-2011 School Year The timeframes listed on this calendar are estimates based on a fifty-minute class period. You may need to adjust some of them from time to

More information

Design of FIR Filters

Design of FIR Filters Design of FIR Filters Elena Punskaya www-sigproc.eng.cam.ac.uk/~op205 Some material adapted from courses by Prof. Simon Godsill, Dr. Arnaud Doucet, Dr. Malcolm Macleod and Prof. Peter Rayner 68 FIR as

More information

Let s examine the response of the circuit shown on Figure 1. The form of the source voltage Vs is shown on Figure 2. R. Figure 1.

Let s examine the response of the circuit shown on Figure 1. The form of the source voltage Vs is shown on Figure 2. R. Figure 1. Examples of Transient and RL Circuits. The Series RLC Circuit Impulse response of Circuit. Let s examine the response of the circuit shown on Figure 1. The form of the source voltage Vs is shown on Figure.

More information

Aliasing, Image Sampling and Reconstruction

Aliasing, Image Sampling and Reconstruction Aliasing, Image Sampling and Reconstruction Recall: a pixel is a point It is NOT a box, disc or teeny wee light It has no dimension It occupies no area It can have a coordinate More than a point, it is

More information

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

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

More information

Columbia University Department of Physics QUALIFYING EXAMINATION

Columbia University Department of Physics QUALIFYING EXAMINATION Columbia University Department of Physics QUALIFYING EXAMINATION Monday, January 13, 2014 1:00PM to 3:00PM Classical Physics Section 1. Classical Mechanics Two hours are permitted for the completion of

More information

x a x 2 (1 + x 2 ) n.

x a x 2 (1 + x 2 ) n. Limits and continuity Suppose that we have a function f : R R. Let a R. We say that f(x) tends to the limit l as x tends to a; lim f(x) = l ; x a if, given any real number ɛ > 0, there exists a real number

More information

5.3 Improper Integrals Involving Rational and Exponential Functions

5.3 Improper Integrals Involving Rational and Exponential Functions Section 5.3 Improper Integrals Involving Rational and Exponential Functions 99.. 3. 4. dθ +a cos θ =, < a

More information

G. GRAPHING FUNCTIONS

G. GRAPHING FUNCTIONS G. GRAPHING FUNCTIONS To get a quick insight int o how the graph of a function looks, it is very helpful to know how certain simple operations on the graph are related to the way the function epression

More information

LABORATORY 10 TIME AVERAGES, RMS VALUES AND THE BRIDGE RECTIFIER. Bridge Rectifier

LABORATORY 10 TIME AVERAGES, RMS VALUES AND THE BRIDGE RECTIFIER. Bridge Rectifier LABORATORY 10 TIME AVERAGES, RMS VALUES AND THE BRIDGE RECTIFIER Full-wave Rectification: Bridge Rectifier For many electronic circuits, DC supply voltages are required but only AC voltages are available.

More information

Controller Design in Frequency Domain

Controller Design in Frequency Domain ECSE 4440 Control System Engineering Fall 2001 Project 3 Controller Design in Frequency Domain TA 1. Abstract 2. Introduction 3. Controller design in Frequency domain 4. Experiment 5. Colclusion 1. Abstract

More information

4.5 Chebyshev Polynomials

4.5 Chebyshev Polynomials 230 CHAP. 4 INTERPOLATION AND POLYNOMIAL APPROXIMATION 4.5 Chebyshev Polynomials We now turn our attention to polynomial interpolation for f (x) over [ 1, 1] based on the nodes 1 x 0 < x 1 < < x N 1. Both

More information

Analysis/resynthesis with the short time Fourier transform

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

More information

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.

a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given

More information

10 Discrete-Time Fourier Series

10 Discrete-Time Fourier Series 10 Discrete-Time Fourier Series In this and the next lecture we parallel for discrete time the discussion of the last three lectures for continuous time. Specifically, we consider the representation of

More information

Series FOURIER SERIES. Graham S McDonald. A self-contained Tutorial Module for learning the technique of Fourier series analysis

Series FOURIER SERIES. Graham S McDonald. A self-contained Tutorial Module for learning the technique of Fourier series analysis Series FOURIER SERIES Graham S McDonald A self-contained Tutorial Module for learning the technique of Fourier series analysis Table of contents Begin Tutorial c 004 g.s.mcdonald@salford.ac.uk 1. Theory.

More information

Chapter 12 Driven RLC Circuits

Chapter 12 Driven RLC Circuits hapter Driven ircuits. A Sources... -. A ircuits with a Source and One ircuit Element... -3.. Purely esistive oad... -3.. Purely Inductive oad... -6..3 Purely apacitive oad... -8.3 The Series ircuit...

More information

Review Solutions MAT V1102. 1. (a) If u = 4 x, then du = dx. Hence, substitution implies 1. dx = du = 2 u + C = 2 4 x + C.

Review Solutions MAT V1102. 1. (a) If u = 4 x, then du = dx. Hence, substitution implies 1. dx = du = 2 u + C = 2 4 x + C. Review Solutions MAT V. (a) If u 4 x, then du dx. Hence, substitution implies dx du u + C 4 x + C. 4 x u (b) If u e t + e t, then du (e t e t )dt. Thus, by substitution, we have e t e t dt e t + e t u

More information

CHAPTER 5 Round-off errors

CHAPTER 5 Round-off errors CHAPTER 5 Round-off errors In the two previous chapters we have seen how numbers can be represented in the binary numeral system and how this is the basis for representing numbers in computers. Since any

More information

5. Orthogonal matrices

5. Orthogonal matrices L Vandenberghe EE133A (Spring 2016) 5 Orthogonal matrices matrices with orthonormal columns orthogonal matrices tall matrices with orthonormal columns complex matrices with orthonormal columns 5-1 Orthonormal

More information

High Voltage Transient Analysis

High Voltage Transient Analysis High Voltage Transient Analysis 4. Surges on Transmission Lines Due to a variety of reasons, such as a direct stroke of lightning on the line, or by indirect strokes, or by switching operations or by faults,

More information

Calculus with Parametric Curves

Calculus with Parametric Curves Calculus with Parametric Curves Suppose f and g are differentiable functions and we want to find the tangent line at a point on the parametric curve x f(t), y g(t) where y is also a differentiable function

More information

5.4 The Heat Equation and Convection-Diffusion

5.4 The Heat Equation and Convection-Diffusion 5.4. THE HEAT EQUATION AND CONVECTION-DIFFUSION c 6 Gilbert Strang 5.4 The Heat Equation and Convection-Diffusion The wave equation conserves energy. The heat equation u t = u xx dissipates energy. The

More information

Dynamics at the Horsetooth, Volume 2A, Focussed Issue: Asymptotics and Perturbations

Dynamics at the Horsetooth, Volume 2A, Focussed Issue: Asymptotics and Perturbations Dynamics at the Horsetooth, Volume A, Focussed Issue: Asymptotics and Perturbations Asymptotic Expansion of Bessel Functions; Applications to Electromagnetics Department of Electrical Engineering Colorado

More information

Fourth-Order Compact Schemes of a Heat Conduction Problem with Neumann Boundary Conditions

Fourth-Order Compact Schemes of a Heat Conduction Problem with Neumann Boundary Conditions Fourth-Order Compact Schemes of a Heat Conduction Problem with Neumann Boundary Conditions Jennifer Zhao, 1 Weizhong Dai, Tianchan Niu 1 Department of Mathematics and Statistics, University of Michigan-Dearborn,

More information

The 1-D Wave Equation

The 1-D Wave Equation The -D Wave Equation 8.303 Linear Partial Differential Equations Matthew J. Hancock Fall 006 -D Wave Equation : Physical derivation Reference: Guenther & Lee., Myint-U & Debnath.-.4 [Oct. 3, 006] We consider

More information

ANALYTICAL METHODS FOR ENGINEERS

ANALYTICAL METHODS FOR ENGINEERS UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

More information

Second Order Linear Differential Equations

Second Order Linear Differential Equations CHAPTER 2 Second Order Linear Differential Equations 2.. Homogeneous Equations A differential equation is a relation involving variables x y y y. A solution is a function f x such that the substitution

More information

11. Rotation Translational Motion: Rotational Motion:

11. Rotation Translational Motion: Rotational Motion: 11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational

More information