ELEG 3124 SYSTEMS AND SIGNALS Ch. 5 Fourier Transform

Size: px
Start display at page:

Download "ELEG 3124 SYSTEMS AND SIGNALS Ch. 5 Fourier Transform"

Transcription

1 Department of Electrical Engineering University of Arkansas ELEG 3124 SYSTEMS AND SIGNALS Ch. 5 Fourier Transform Dr. Jingxian Wu wuj@uark.edu

2 OUTLINE 2 Introduction Fourier Transform Properties of Fourier Transform Applications of Fourier Transform

3 INTRODUCTION: MOTIVATION 3 Motivation: Fourier series: periodic signals can be decomposed as the summation of orthogonal complex exponential signals x( cn exp jn0t 1 T cn x( exp jn0t n T 0 each harmonic contains a unique frequency: n/t dt time domain frequency domain How about aperiodic signals T?

4 INTRODUCTION: TRANSFER FUNCTION 4 System transfer function j t e ) h(t e jt H() H( ) h( exp System with periodic inputs e 0 jn t h( e jt dt H( n 0) jn 0 t jn0t cne n h( n c n e H( n ) 0 jn 0 t x( h( cn n e H( n ) 0 jn 0 t

5 OUTLINE 5 Introduction Fourier Transform Properties of Fourier Transform Applications of Fourier Transform

6 FOURIER TRANSFORM Fourier Transform 6 X ( ) x( e given x(, we can find its Fourier transform Inverse Fourier Transform x( 1 2 X ( ) e jt given X (), we can find the time domain signal x( signal is decomposed into the weighted summation of complex exponential functions. (integration is the extreme case of summation) dt j t d x ( X () X ()

7 FOURIER TRANSFORM 7 Example x( rect ( t / Find the Fourier transform of )

8 FOURIER TRANSFORM 8 Example Find the Fourier transform of x( exp( a t ) a 0

9 FOURIER TRANSFORM 9 Example Find the Fourier transform of x( exp( a u( a 0

10 FOURIER TRANSFORM 10 Example Find the Fourier transform of x( ( t a)

11 FOURIER TRANSFORM: TABLE 11 Department of Engineering Science Sonoma State University

12 FOURIER TRANSFORM The existence of Fourier transform Not all signals have Fourier transform If a signal have Fourier transform, it must satisfy the following two conditions Example 1. x( is absolutely integrable x ( dt x( exp( u( 2. x( is well behaved The signal has finite number of discontinuities, minima, and maxima within any finite interval of time. 12

13 OUTLINE 13 Introduction Fourier Transform Properties of Fourier Transform Applications of Fourier Transform

14 PROPERTIES: LINEARITY Linearity If x X ( ) x X ( ) 1( 1 2( 2 14 then ax ( bx2( ax1( ) bx 2( 1 ) Example Find the Fourier transform of x( 2rect ( t / ) 3exp( 2 u( 4 (

15 PROPERTY: TIME-SHIFT 15 Time shift If x( X ( ) Then x( t t0) X ( )exp[ jt0 ] phase shift Review: complex number j c c e c cos( ) j c sin( ) a a c cos c b b c sin a tan( b / a 2 2 a ) Phase shift of a complex number c by : 0 time shift in time domain frequency shift in frequency domain jb c j ) c exp j( ) exp( 0 0

16 PROPERTY: TIME SHIFT 16 Example: Find the Fourier transform of x( rect t 2

17 PROPERTY: TIME SCALING 17 Time scaling If x( X ( ) Then x( a 1 a X a Example Let X ( ) rect 1 / 2, find the Fourier transform of x( 2t 4)

18 PROPERTY: SYMMETRY 18 Symmetry x( If x( X ( ), and is a real-valued time signal Then X ( ) X * ( )

19 PROPERTY: DIFFERENTIATION 19 Differentiation If x( X ( ) Then dx( dt ) jx ( ) n d x( n dt n j X ( Example Let X ( ) rect 1 / 2, find the Fourier transform of dx ( dt

20 PROPERTY: DIFFERENTIATION 20 Example Find the Fourier transform of x( sgn( d 1 (Hint: sgn( ( ) dt 2

21 PROPERTY: CONVOLUTION 21 Convolution If x( X ( ), h( H( ) Then x( h( X ( ) H( ) x( h( x( h( X () H() X ( ) H( ) time domain frequency domain

22 PROPERTY: CONVOLUTION 22 Example An LTI system has impulse response h( exp at u( ) If the input is x( ( a b)exp bt u( ( c a)exp( c u( t Find the output ( a 0, b 0, c 0)

23 PROPERTY: MULTIPLICATION 23 Multiplication If x( X ( ), m( M( ) Then 1 x( m( X ( ) M ( ) 2

24 PROPERTY: DUALITY 24 Duality If g( G( ) Then G( 2g ( )

25 PROPERTY: DUALITY 25 Example Find the Fourier transform of h( Sa t 2 (recall: rect ( t / ) sinc ) 2

26 PROPERTY: DUALITY 26 Example Find the Fourier transform of x( 1 Find the Fourier transform of x 0 ( e j t

27 PROPERTY: SUMMARY 27

28 PROPERTY: EXAMPLES 28 Examples 1. Find the Fourier transform of x ( t ) cos( t 0 ) 2. Find the Fourier transform of x( u( 1 2 u( sgn( 1 sgn( 2 j

29 PROPERTY: EXAMPLES 29 Examples 3. A LTI system with impulse response Find the output when input is at u( ) h( exp t x( u( 4. If x( X ( ), find the Fourier transform of t x( ) d t (Hint: x( ) d x( u( )

30 PROPERTY: EXAMPLES 30 Example 5. (Modulation) If x( X ( ), m ( t ) cos( t 0 ) Find the Fourier transform of x( m( 6. If 1 X ( ), find x( a j

31 PROPERTY: DIFFERENTIATION IN FREQ. DOMAIN 31 Differentiation in frequency domain If: Then: x( X ( ) n ( j x( n d X () n d

32 PROPERTY: DIFFERENTIATION IN FREQ. DOMAIN 32 Example Find the Fourier transform of t exp( a u(, a 0

33 PROPERTY: FREQUENCY SHIFT 33 Frequency shift If: Then: x( X ( ) x( exp( j0 X ( 0) Example If X ( ) rect 1 / 2, find the Fourier transform x( exp( j2

34 PROPERTY: PARSAVAL S THEOREM 34 Review: signal energy E Parsaval s theorem x( 2 dt x( dt X ( ) 2 d

35 PROPERTY: PARSAVAL S THEOREM 35 Example: Find the energy of the signal x( exp( 2 u(

36 PROPERTY: PERIODIC SIGNAL 36 Fourier transform of periodic signal Periodic signal can be written as Fourier series n x( c exp jn t n 0 Perform Fourier transform on both sides n X ( ) 2 cn ( n0)

37 OUTLINE 37 Introduction Fourier Transform Properties of Fourier Transform Applications of Fourier Transform

38 APPLICATIONS: FILTERING 38 Filtering Filtering is the process by which the essential and useful part of a signal is separated from undesirable components. Passing a signal through a filter (system). At the output of the filter, some undesired part of the signal (e.g. noise) is removed. Based on the convolution property, we can design filter that only allow signal within a certain frequency range to pass through. x( h( x( h( X () H() X ( ) H( ) filter filter time domain frequency domain

39 APPLICATIONS: FILTERING 39 Classifications of filters Low pass filter High pass filter Band pass filter Band stop (Notch) filter

40 APPLICATIONS: FILTERING 40 Practical filters (non-ideal filters)

41 APPLICATION: FILTERING 41 A filtering example A demo of a notch filter X () H() X ( ) H( )

42 APPLICATIONS: FILTERING 42 Example Find out the frequency response of the RC circuit What kind of filters it is?

43 APPLICATION: SAMPLING THEOREM 43 Sampling theorem: time domain Sampling: convert the continuous-time signal to discrete-time signal. x( n p ( ( t nt) sampling period x s ( x( p(

44 APPLICATION: SAMPLING THEOREM 44 Sampling theorem: frequency domain Fourier transform of the impulse train impulse train is periodic p( n Fourier series 1 jnst ( t nts ) 1e T s n Find Fourier transform on both sides 2 P ( ) ( ns ) T s n Time domain multiplication Frequency domain convolution 1 x( p( X ( ) P( ) 2 1 x ( p( X ( ns ) T s n s 2 T s

45 APPLICATION: SAMPLING THEOREM 45 Sampling theorem: frequency domain Sampling in time domain Repetition in frequency domain

46 APPLICATION: SAMPLING THEOREM 46 Sampling theorem If the sampling rate is twice of the bandwidth, then the original signal can be perfectly reconstructed from the samples. 2 s B s 2 B s 2 B Department of Engineering Science Sonoma State University s 2 B

47 APPLICATION: AMPLITUDE MODULATION 47 What is modulation? The process by which some characteristic of a carrier wave is varied in accordance with an information-bearing signal Information bearing signal Three signals: modulation Carrier wave Information bearing signal (modulating signal) Usually at low frequency (baseband) E.g. speech signal: 20Hz 20KHz Carrier wave Usually a high frequency sinusoidal (passband) Modulated signal E.g. AM radio station (1050KHz) FM radio station (100.1MHz), 2.4GHz, etc. Modulated signal: passband signal

48 APPLICATION: AMPLITUDE MODULATION 48 Amplitude Modulation (AM) s( A m( cos(2 f c A direct product between message signal and carrier signal c m( Mixer s( A c Local Oscillator cos( 2f c

49 APPLICATION: AMPLITUDE MODULATION 49 Amplitude Modulation (AM) Ac S( f ) M ( f fc) M ( f fc) 2

The Continuous-time Fourier

The Continuous-time Fourier The Continuous-time Fourier Transform Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it Email: ruiwang@tongji.edu.cn Outline Representation of Aperiodic signals:

More information

ENSC327 Communications Systems 2: Fourier Representations. Jie Liang School of Engineering Science Simon Fraser University

ENSC327 Communications Systems 2: Fourier Representations. Jie Liang School of Engineering Science Simon Fraser University ENSC327 Communications Systems 2: Fourier Representations Jie Liang School of Engineering Science Simon Fraser University 1 Outline Chap 2.1 2.5: Signal Classifications Fourier Transform Dirac Delta Function

More information

CHAPTER 4 FOURIER SERIES S A B A R I N A I S M A I L

CHAPTER 4 FOURIER SERIES S A B A R I N A I S M A I L CHAPTER 4 FOURIER SERIES 1 S A B A R I N A I S M A I L Outline Introduction of the Fourier series. The properties of the Fourier series. Symmetry consideration Application of the Fourier series to circuit

More information

Topic 3: Fourier Series (FS)

Topic 3: Fourier Series (FS) ELEC264: Signals And Systems Topic 3: Fourier Series (FS) o o o o Introduction to frequency analysis of signals CT FS Fourier series of CT periodic signals Signal Symmetry and CT Fourier Series Properties

More information

Chapter 5 Frequency Domain Analysis of Systems

Chapter 5 Frequency Domain Analysis of Systems Chapter 5 Frequency Domain Analysis of Systems CT, LTI Systems Consider the following CT LTI system: xt () ht () yt () Assumption: the impulse response h(t) is absolutely integrable, i.e., ht ( ) dt< (this

More information

Chapter 5 Frequency Domain Analysis of Systems

Chapter 5 Frequency Domain Analysis of Systems Chapter 5 Frequency Domain Analysis of Systems CT, LTI Systems Consider the following CT LTI system: xt () ht () yt () Assumption: the impulse response h(t) is absolutely integrable, i.e., ht ( ) dt< (this

More information

Fourier Series Representation of

Fourier Series Representation of Fourier Series Representation of Periodic Signals Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it Email: ruiwang@tongji.edu.cn Outline The response of LIT system

More information

信號與系統 Signals and Systems

信號與系統 Signals and Systems Spring 2011 信號與系統 Signals and Systems Chapter SS-4 The Continuous-Time Fourier Transform Feng-Li Lian NTU-EE Feb11 Jun11 Figures and images used in these lecture notes are adopted from Signals & Systems

More information

3.2 Complex Sinusoids and Frequency Response of LTI Systems

3.2 Complex Sinusoids and Frequency Response of LTI Systems 3. Introduction. A signal can be represented as a weighted superposition of complex sinusoids. x(t) or x[n]. LTI system: LTI System Output = A weighted superposition of the system response to each complex

More information

The Continuous Time Fourier Transform

The Continuous Time Fourier Transform COMM 401: Signals & Systems Theory Lecture 8 The Continuous Time Fourier Transform Fourier Transform Continuous time CT signals Discrete time DT signals Aperiodic signals nonperiodic periodic signals Aperiodic

More information

4 The Continuous Time Fourier Transform

4 The Continuous Time Fourier Transform 96 4 The Continuous Time ourier Transform ourier (or frequency domain) analysis turns out to be a tool of even greater usefulness Extension of ourier series representation to aperiodic signals oundation

More information

Representation of Signals & Systems

Representation of Signals & Systems Representation of Signals & Systems Reference: Chapter 2,Communication Systems, Simon Haykin. Hilbert Transform Fourier transform frequency content of a signal (frequency selectivity designing frequency-selective

More information

3. Frequency-Domain Analysis of Continuous- Time Signals and Systems

3. Frequency-Domain Analysis of Continuous- Time Signals and Systems 3. Frequency-Domain Analysis of Continuous- ime Signals and Systems 3.. Definition of Continuous-ime Fourier Series (3.3-3.4) 3.2. Properties of Continuous-ime Fourier Series (3.5) 3.3. Definition of Continuous-ime

More information

ELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process

ELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process Department of Electrical Engineering University of Arkansas ELEG 3143 Probability & Stochastic Process Ch. 6 Stochastic Process Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Definition of stochastic process (random

More information

Frequency Response & Filter Analysis. Week Date Lecture Title. 4-Mar Introduction & Systems Overview 6-Mar [Linear Dynamical Systems] 2

Frequency Response & Filter Analysis. Week Date Lecture Title. 4-Mar Introduction & Systems Overview 6-Mar [Linear Dynamical Systems] 2 http://elec34.com Frequency Response & Filter Analysis 4 School of Information Technology and Electrical Engineering at The University of Queensland Lecture Schedule: Week Date Lecture Title 4-Mar Introduction

More information

ω 0 = 2π/T 0 is called the fundamental angular frequency and ω 2 = 2ω 0 is called the

ω 0 = 2π/T 0 is called the fundamental angular frequency and ω 2 = 2ω 0 is called the he ime-frequency Concept []. Review of Fourier Series Consider the following set of time functions {3A sin t, A sin t}. We can represent these functions in different ways by plotting the amplitude versus

More information

Continuous-Time Fourier Transform

Continuous-Time Fourier Transform Signals and Systems Continuous-Time Fourier Transform Chang-Su Kim continuous time discrete time periodic (series) CTFS DTFS aperiodic (transform) CTFT DTFT Lowpass Filtering Blurring or Smoothing Original

More information

Continuous Time Signal Analysis: the Fourier Transform. Lathi Chapter 4

Continuous Time Signal Analysis: the Fourier Transform. Lathi Chapter 4 Continuous Time Signal Analysis: the Fourier Transform Lathi Chapter 4 Topics Aperiodic signal representation by the Fourier integral (CTFT) Continuous-time Fourier transform Transforms of some useful

More information

Review: Continuous Fourier Transform

Review: Continuous Fourier Transform Review: Continuous Fourier Transform Review: convolution x t h t = x τ h(t τ)dτ Convolution in time domain Derivation Convolution Property Interchange the order of integrals Let Convolution Property By

More information

06EC44-Signals and System Chapter Fourier Representation for four Signal Classes

06EC44-Signals and System Chapter Fourier Representation for four Signal Classes Chapter 5.1 Fourier Representation for four Signal Classes 5.1.1Mathematical Development of Fourier Transform If the period is stretched without limit, the periodic signal no longer remains periodic but

More information

Ch 4: The Continuous-Time Fourier Transform

Ch 4: The Continuous-Time Fourier Transform Ch 4: The Continuous-Time Fourier Transform Fourier Transform of x(t) Inverse Fourier Transform jt X ( j) x ( t ) e dt jt x ( t ) X ( j) e d 2 Ghulam Muhammad, King Saud University Continuous-time aperiodic

More information

E2.5 Signals & Linear Systems. Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & 2)

E2.5 Signals & Linear Systems. Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & 2) E.5 Signals & Linear Systems Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & ) 1. Sketch each of the following continuous-time signals, specify if the signal is periodic/non-periodic,

More information

Review of Analog Signal Analysis

Review of Analog Signal Analysis Review of Analog Signal Analysis Chapter Intended Learning Outcomes: (i) Review of Fourier series which is used to analyze continuous-time periodic signals (ii) Review of Fourier transform which is used

More information

Frequency Response and Continuous-time Fourier Series

Frequency Response and Continuous-time Fourier Series Frequency Response and Continuous-time Fourier Series Recall course objectives Main Course Objective: Fundamentals of systems/signals interaction (we d like to understand how systems transform or affect

More information

ELEG 3143 Probability & Stochastic Process Ch. 4 Multiple Random Variables

ELEG 3143 Probability & Stochastic Process Ch. 4 Multiple Random Variables Department o Electrical Engineering University o Arkansas ELEG 3143 Probability & Stochastic Process Ch. 4 Multiple Random Variables Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Two discrete random variables

More information

6.003: Signals and Systems. CT Fourier Transform

6.003: Signals and Systems. CT Fourier Transform 6.003: Signals and Systems CT Fourier Transform April 8, 200 CT Fourier Transform Representing signals by their frequency content. X(jω)= x(t)e jωt dt ( analysis equation) x(t)= X(jω)e jωt dω ( synthesis

More information

SEISMIC WAVE PROPAGATION. Lecture 2: Fourier Analysis

SEISMIC WAVE PROPAGATION. Lecture 2: Fourier Analysis SEISMIC WAVE PROPAGATION Lecture 2: Fourier Analysis Fourier Series & Fourier Transforms Fourier Series Review of trigonometric identities Analysing the square wave Fourier Transform Transforms of some

More information

ECE 451 Black Box Modeling

ECE 451 Black Box Modeling Black Box Modeling Jose E. Schutt-Aine Electrical & Computer Engineering University of Illinois jose@emlab.uiuc.edu Simulation for Digital Design Nonlinear Network Nonlinear Network 3 Linear N-Port with

More information

3 Fourier Series Representation of Periodic Signals

3 Fourier Series Representation of Periodic Signals 65 66 3 Fourier Series Representation of Periodic Signals Fourier (or frequency domain) analysis constitutes a tool of great usefulness Accomplishes decomposition of broad classes of signals using complex

More information

Review of Fourier Transform

Review of Fourier Transform Review of Fourier Transform Fourier series works for periodic signals only. What s about aperiodic signals? This is very large & important class of signals Aperiodic signal can be considered as periodic

More information

ENSC327 Communications Systems 2: Fourier Representations. School of Engineering Science Simon Fraser University

ENSC327 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 information

Lab10: FM Spectra and VCO

Lab10: FM Spectra and VCO Lab10: FM Spectra and VCO Prepared by: Keyur Desai Dept. of Electrical Engineering Michigan State University ECE458 Lab 10 What is FM? A type of analog modulation Remember a common strategy in analog modulation?

More information

Discrete Time Signals and Systems Time-frequency Analysis. Gloria Menegaz

Discrete Time Signals and Systems Time-frequency Analysis. Gloria Menegaz Discrete Time Signals and Systems Time-frequency Analysis Gloria Menegaz Time-frequency Analysis Fourier transform (1D and 2D) Reference textbook: Discrete time signal processing, A.W. Oppenheim and R.W.

More information

ANALOG AND DIGITAL SIGNAL PROCESSING CHAPTER 3 : LINEAR SYSTEM RESPONSE (GENERAL CASE)

ANALOG AND DIGITAL SIGNAL PROCESSING CHAPTER 3 : LINEAR SYSTEM RESPONSE (GENERAL CASE) 3. Linear System Response (general case) 3. INTRODUCTION In chapter 2, we determined that : a) If the system is linear (or operate in a linear domain) b) If the input signal can be assumed as periodic

More information

Chapter 4 The Fourier Series and Fourier Transform

Chapter 4 The Fourier Series and Fourier Transform Chapter 4 The Fourier Series and Fourier Transform Representation of Signals in Terms of Frequency Components Consider the CT signal defined by N xt () = Acos( ω t+ θ ), t k = 1 k k k The frequencies `present

More information

6.003: Signals and Systems. CT Fourier Transform

6.003: Signals and Systems. CT Fourier Transform 6.003: Signals and Systems CT Fourier Transform April 8, 200 CT Fourier Transform Representing signals by their frequency content. X(jω)= x(t)e jωt dt ( analysis equation) x(t)= 2π X(jω)e jωt dω ( synthesis

More information

Representing a Signal

Representing a Signal The Fourier Series Representing a Signal The convolution method for finding the response of a system to an excitation takes advantage of the linearity and timeinvariance of the system and represents the

More information

Chapter 2: Problem Solutions

Chapter 2: Problem Solutions Chapter 2: Problem Solutions Discrete Time Processing of Continuous Time Signals Sampling à Problem 2.1. Problem: Consider a sinusoidal signal and let us sample it at a frequency F s 2kHz. xt 3cos1000t

More information

EE123 Digital Signal Processing

EE123 Digital Signal Processing EE123 Digital Signal Processing Lecture 2B D. T. Fourier Transform M. Lustig, EECS UC Berkeley Something Fun gotenna http://www.gotenna.com/# Text messaging radio Bluetooth phone interface MURS VHF radio

More information

Lecture 27 Frequency Response 2

Lecture 27 Frequency Response 2 Lecture 27 Frequency Response 2 Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/6/12 1 Application of Ideal Filters Suppose we can generate a square wave with a fundamental period

More information

Basic Electronics. Introductory Lecture Course for. Technology and Instrumentation in Particle Physics Chicago, Illinois June 9-14, 2011

Basic Electronics. Introductory Lecture Course for. Technology and Instrumentation in Particle Physics Chicago, Illinois June 9-14, 2011 Basic Electronics Introductory Lecture Course for Technology and Instrumentation in Particle Physics 2011 Chicago, Illinois June 9-14, 2011 Presented By Gary Drake Argonne National Laboratory Session 2

More information

Fourier Transform for Continuous Functions

Fourier Transform for Continuous Functions Fourier Transform for Continuous Functions Central goal: representing a signal by a set of orthogonal bases that are corresponding to frequencies or spectrum. Fourier series allows to find the spectrum

More information

Chapter 4 The Fourier Series and Fourier Transform

Chapter 4 The Fourier Series and Fourier Transform Chapter 4 The Fourier Series and Fourier Transform Fourier Series Representation of Periodic Signals Let x(t) be a CT periodic signal with period T, i.e., xt ( + T) = xt ( ), t R Example: the rectangular

More information

2A1H Time-Frequency Analysis II

2A1H Time-Frequency Analysis II 2AH Time-Frequency Analysis II Bugs/queries to david.murray@eng.ox.ac.uk HT 209 For any corrections see the course page DW Murray at www.robots.ox.ac.uk/ dwm/courses/2tf. (a) A signal g(t) with period

More information

Communication Signals (Haykin Sec. 2.4 and Ziemer Sec Sec. 2.4) KECE321 Communication Systems I

Communication Signals (Haykin Sec. 2.4 and Ziemer Sec Sec. 2.4) KECE321 Communication Systems I Communication Signals (Haykin Sec..4 and iemer Sec...4-Sec..4) KECE3 Communication Systems I Lecture #3, March, 0 Prof. Young-Chai Ko 년 3 월 일일요일 Review Signal classification Phasor signal and spectra Representation

More information

LOPE3202: Communication Systems 10/18/2017 2

LOPE3202: Communication Systems 10/18/2017 2 By Lecturer Ahmed Wael Academic Year 2017-2018 LOPE3202: Communication Systems 10/18/2017 We need tools to build any communication system. Mathematics is our premium tool to do work with signals and systems.

More information

Notes 07 largely plagiarized by %khc

Notes 07 largely plagiarized by %khc Notes 07 largely plagiarized by %khc Warning This set of notes covers the Fourier transform. However, i probably won t talk about everything here in section; instead i will highlight important properties

More information

FOURIER TRANSFORMS. At, is sometimes taken as 0.5 or it may not have any specific value. Shifting at

FOURIER TRANSFORMS. At, is sometimes taken as 0.5 or it may not have any specific value. Shifting at Chapter 2 FOURIER TRANSFORMS 2.1 Introduction The Fourier series expresses any periodic function into a sum of sinusoids. The Fourier transform is the extension of this idea to non-periodic functions by

More information

Continuous-time Fourier Methods

Continuous-time Fourier Methods ELEC 321-001 SIGNALS and SYSTEMS Continuous-time Fourier Methods Chapter 6 1 Representing a Signal The convolution method for finding the response of a system to an excitation takes advantage of the linearity

More information

The Discrete-time Fourier Transform

The Discrete-time Fourier Transform The Discrete-time Fourier Transform Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it Email: ruiwang@tongji.edu.cn Outline Representation of Aperiodic signals: The

More information

Signals, Instruments, and Systems W5. Introduction to Signal Processing Sampling, Reconstruction, and Filters

Signals, Instruments, and Systems W5. Introduction to Signal Processing Sampling, Reconstruction, and Filters Signals, Instruments, and Systems W5 Introduction to Signal Processing Sampling, Reconstruction, and Filters Acknowledgments Recapitulation of Key Concepts from the Last Lecture Dirac delta function (

More information

EE123 Digital Signal Processing

EE123 Digital Signal Processing EE123 Digital Signal Processing Lecture 2A D.T Systems D. T. Fourier Transform A couple of things Read Ch 2 2.0-2.9 It s OK to use 2nd edition My office hours: posted on-line W 4-5pm Cory 506 ham radio

More information

6.003 Homework #10 Solutions

6.003 Homework #10 Solutions 6.3 Homework # Solutions Problems. DT Fourier Series Determine the Fourier Series coefficients for each of the following DT signals, which are periodic in N = 8. x [n] / n x [n] n x 3 [n] n x 4 [n] / n

More information

LECTURE 12 Sections Introduction to the Fourier series of periodic signals

LECTURE 12 Sections Introduction to the Fourier series of periodic signals Signals and Systems I Wednesday, February 11, 29 LECURE 12 Sections 3.1-3.3 Introduction to the Fourier series of periodic signals Chapter 3: Fourier Series of periodic signals 3. Introduction 3.1 Historical

More information

Representation of Signals and Systems. Lecturer: David Shiung

Representation of Signals and Systems. Lecturer: David Shiung Representation of Signals and Systems Lecturer: David Shiung 1 Abstract (1/2) Fourier analysis Properties of the Fourier transform Dirac delta function Fourier transform of periodic signals Fourier-transform

More information

Discrete-time Signals and Systems in

Discrete-time Signals and Systems in Discrete-time Signals and Systems in the Frequency Domain Chapter 3, Sections 3.1-39 3.9 Chapter 4, Sections 4.8-4.9 Dr. Iyad Jafar Outline Introduction The Continuous-Time FourierTransform (CTFT) The

More information

MEDE2500 Tutorial Nov-7

MEDE2500 Tutorial Nov-7 (updated 2016-Nov-4,7:40pm) MEDE2500 (2016-2017) Tutorial 3 MEDE2500 Tutorial 3 2016-Nov-7 Content 1. The Dirac Delta Function, singularity functions, even and odd functions 2. The sampling process and

More information

EA2.3 - Electronics 2 1

EA2.3 - 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 information

E : Lecture 1 Introduction

E : Lecture 1 Introduction E85.2607: Lecture 1 Introduction 1 Administrivia 2 DSP review 3 Fun with Matlab E85.2607: Lecture 1 Introduction 2010-01-21 1 / 24 Course overview Advanced Digital Signal Theory Design, analysis, and implementation

More information

Digital Band-pass Modulation PROF. MICHAEL TSAI 2011/11/10

Digital Band-pass Modulation PROF. MICHAEL TSAI 2011/11/10 Digital Band-pass Modulation PROF. MICHAEL TSAI 211/11/1 Band-pass Signal Representation a t g t General form: 2πf c t + φ t g t = a t cos 2πf c t + φ t Envelope Phase Envelope is always non-negative,

More information

L6: Short-time Fourier analysis and synthesis

L6: Short-time Fourier analysis and synthesis L6: Short-time Fourier analysis and synthesis Overview Analysis: Fourier-transform view Analysis: filtering view Synthesis: filter bank summation (FBS) method Synthesis: overlap-add (OLA) method STFT magnitude

More information

Fourier analysis of discrete-time signals. (Lathi Chapt. 10 and these slides)

Fourier analysis of discrete-time signals. (Lathi Chapt. 10 and these slides) Fourier analysis of discrete-time signals (Lathi Chapt. 10 and these slides) Towards the discrete-time Fourier transform How we will get there? Periodic discrete-time signal representation by Discrete-time

More information

EE303: Communication Systems

EE303: Communication Systems EE303: Communication Systems Professor A. Manikas Chair of Communications and Array Processing Imperial College London Introductory Concepts Prof. A. Manikas (Imperial College) EE303: Introductory Concepts

More information

ECE 3084 QUIZ 2 SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY APRIL 2, Name:

ECE 3084 QUIZ 2 SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY APRIL 2, Name: ECE 3084 QUIZ 2 SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY APRIL 2, 205 Name:. The quiz is closed book, except for one 2-sided sheet of handwritten notes. 2. Turn off

More information

The Hilbert Transform

The Hilbert Transform The Hilbert Transform David Hilbert 1 ABSTRACT: In this presentation, the basic theoretical background of the Hilbert Transform is introduced. Using this transform, normal real-valued time domain functions

More information

Discrete Systems & Z-Transforms. Week Date Lecture Title. 9-Mar Signals as Vectors & Systems as Maps 10-Mar [Signals] 3

Discrete Systems & Z-Transforms. Week Date Lecture Title. 9-Mar Signals as Vectors & Systems as Maps 10-Mar [Signals] 3 http:elec34.org Discrete Systems & Z-Transforms 4 School of Information Technology and Electrical Engineering at The University of Queensland Lecture Schedule: eek Date Lecture Title -Mar Introduction

More information

The Johns Hopkins University Department of Electrical and Computer Engineering Introduction to Linear Systems Fall 2002.

The Johns Hopkins University Department of Electrical and Computer Engineering Introduction to Linear Systems Fall 2002. The Johns Hopkins University Department of Electrical and Computer Engineering 505.460 Introduction to Linear Systems Fall 2002 Final exam Name: You are allowed to use: 1. Table 3.1 (page 206) & Table

More information

DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2010

DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2010 [E2.5] IMPERIAL COLLEGE LONDON DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2010 EEE/ISE PART II MEng. BEng and ACGI SIGNALS AND LINEAR SYSTEMS Time allowed: 2:00 hours There are FOUR

More information

Signals & Systems. Chapter 7: Sampling. Adapted from: Lecture notes from MIT, Binghamton University, and Purdue. Dr. Hamid R.

Signals & Systems. Chapter 7: Sampling. Adapted from: Lecture notes from MIT, Binghamton University, and Purdue. Dr. Hamid R. Signals & Systems Chapter 7: Sampling Adapted from: Lecture notes from MIT, Binghamton University, and Purdue Dr. Hamid R. Rabiee Fall 2013 Outline 1. The Concept and Representation of Periodic Sampling

More information

Review of Fundamentals of Digital Signal Processing

Review of Fundamentals of Digital Signal Processing Solution Manual for Theory and Applications of Digital Speech Processing by Lawrence Rabiner and Ronald Schafer Click here to Purchase full Solution Manual at http://solutionmanuals.info Link download

More information

EE123 Digital Signal Processing

EE123 Digital Signal Processing EE123 Digital Signal Processing Discrete Time Fourier Transform M. Lustig, EECS UC Berkeley A couple of things Read Ch 2 2.0-2.9 It s OK to use 2nd edition Class webcast in bcourses.berkeley.edu or linked

More information

Homework 4. May An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt

Homework 4. May An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt Homework 4 May 2017 1. An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt Determine the impulse response of the system. Rewriting as y(t) = t e (t

More information

EE301 Signals and Systems In-Class Exam Exam 3 Thursday, Apr. 19, Cover Sheet

EE301 Signals and Systems In-Class Exam Exam 3 Thursday, Apr. 19, Cover Sheet EE301 Signals and Systems In-Class Exam Exam 3 Thursday, Apr. 19, 2012 Cover Sheet Test Duration: 75 minutes. Coverage: Chaps. 5,7 Open Book but Closed Notes. One 8.5 in. x 11 in. crib sheet Calculators

More information

SIGNAL PROCESSING. B14 Option 4 lectures. Stephen Roberts

SIGNAL PROCESSING. B14 Option 4 lectures. Stephen Roberts SIGNAL PROCESSING B14 Option 4 lectures Stephen Roberts Recommended texts Lynn. An introduction to the analysis and processing of signals. Macmillan. Oppenhein & Shafer. Digital signal processing. Prentice

More information

ESS Finite Impulse Response Filters and the Z-transform

ESS Finite Impulse Response Filters and the Z-transform 9. Finite Impulse Response Filters and the Z-transform We are going to have two lectures on filters you can find much more material in Bob Crosson s notes. In the first lecture we will focus on some of

More information

Figure 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.

Figure 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 information

2.1 Basic Concepts Basic operations on signals Classication of signals

2.1 Basic Concepts Basic operations on signals Classication of signals Haberle³me Sistemlerine Giri³ (ELE 361) 9 Eylül 2017 TOBB Ekonomi ve Teknoloji Üniversitesi, Güz 2017-18 Dr. A. Melda Yüksel Turgut & Tolga Girici Lecture Notes Chapter 2 Signals and Linear Systems 2.1

More information

PS403 - Digital Signal processing

PS403 - Digital Signal processing PS403 - Digital Signal processing III. DSP - Digital Fourier Series and Transforms Key Text: Digital Signal Processing with Computer Applications (2 nd Ed.) Paul A Lynn and Wolfgang Fuerst, (Publisher:

More information

2. page 13*, righthand column, last line of text, change 3 to 2,... negative slope of 2 for 1 < t 2: Now... REFLECTOR MITTER. (0, d t ) (d r, d t )

2. page 13*, righthand column, last line of text, change 3 to 2,... negative slope of 2 for 1 < t 2: Now... REFLECTOR MITTER. (0, d t ) (d r, d t ) SP First ERRATA. These are mostly typos, but there are a few crucial mistakes in formulas. Underline is not used in the book, so I ve used it to denote changes. JHMcClellan, February 3, 00. page 0*, Figure

More information

6.003: Signals and Systems. Applications of Fourier Transforms

6.003: Signals and Systems. Applications of Fourier Transforms 6.003: Signals and Systems Applications of Fourier Transforms November 7, 20 Filtering Notion of a filter. LTI systems cannot create new frequencies. can only scale magnitudes and shift phases of existing

More information

The Hilbert Transform

The Hilbert Transform The Hilbert Transform Frank R. Kschischang The Edward S. Rogers Sr. Department of Electrical and Computer Engineering University of Toronto October 22, 2006; updated March 0, 205 Definition The Hilbert

More information

Continuous-time Signals. (AKA analog signals)

Continuous-time Signals. (AKA analog signals) Continuous-time Signals (AKA analog signals) I. Analog* Signals review Goals: - Common test signals used in system analysis - Signal operations and properties: scaling, shifting, periodicity, energy and

More information

Periodic (Uniform) Sampling ELEC364 & ELEC442

Periodic (Uniform) Sampling ELEC364 & ELEC442 M.A. Amer Concordia University Electrical and Computer Engineering Content and Figures are from: Periodic (Uniform) Sampling ELEC364 & ELEC442 Introduction to sampling Introduction to filter Ideal sampling:

More information

Theory and Problems of Signals and Systems

Theory and Problems of Signals and Systems SCHAUM'S OUTLINES OF Theory and Problems of Signals and Systems HWEI P. HSU is Professor of Electrical Engineering at Fairleigh Dickinson University. He received his B.S. from National Taiwan University

More information

Fourier Series and Transform KEEE343 Communication Theory Lecture #7, March 24, Prof. Young-Chai Ko

Fourier Series and Transform KEEE343 Communication Theory Lecture #7, March 24, Prof. Young-Chai Ko Fourier Series and Transform KEEE343 Communication Theory Lecture #7, March 24, 20 Prof. Young-Chai Ko koyc@korea.ac.kr Summary Fourier transform Properties Fourier transform of special function Fourier

More information

ECE 301 Fall 2010 Division 2 Homework 10 Solutions. { 1, if 2n t < 2n + 1, for any integer n, x(t) = 0, if 2n 1 t < 2n, for any integer n.

ECE 301 Fall 2010 Division 2 Homework 10 Solutions. { 1, if 2n t < 2n + 1, for any integer n, x(t) = 0, if 2n 1 t < 2n, for any integer n. ECE 3 Fall Division Homework Solutions Problem. Reconstruction of a continuous-time signal from its samples. Consider the following periodic signal, depicted below: {, if n t < n +, for any integer n,

More information

Copyright license. Exchanging Information with the Stars. The goal. Some challenges

Copyright license. Exchanging Information with the Stars. The goal. Some challenges Copyright license Exchanging Information with the Stars David G Messerschmitt Department of Electrical Engineering and Computer Sciences University of California at Berkeley messer@eecs.berkeley.edu Talk

More information

ELEN E4810: Digital Signal Processing Topic 11: Continuous Signals. 1. Sampling and Reconstruction 2. Quantization

ELEN E4810: Digital Signal Processing Topic 11: Continuous Signals. 1. Sampling and Reconstruction 2. Quantization ELEN E4810: Digital Signal Processing Topic 11: Continuous Signals 1. Sampling and Reconstruction 2. Quantization 1 1. Sampling & Reconstruction DSP must interact with an analog world: A to D D to A x(t)

More information

Signals and Systems Spring 2004 Lecture #9

Signals and Systems Spring 2004 Lecture #9 Signals and Systems Spring 2004 Lecture #9 (3/4/04). The convolution Property of the CTFT 2. Frequency Response and LTI Systems Revisited 3. Multiplication Property and Parseval s Relation 4. The DT Fourier

More information

3 What You Should Know About Complex Numbers

3 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 information

EE 224 Signals and Systems I Review 1/10

EE 224 Signals and Systems I Review 1/10 EE 224 Signals and Systems I Review 1/10 Class Contents Signals and Systems Continuous-Time and Discrete-Time Time-Domain and Frequency Domain (all these dimensions are tightly coupled) SIGNALS SYSTEMS

More information

A First Course in Digital Communications

A First Course in Digital Communications A First Course in Digital Communications Ha H. Nguyen and E. Shwedyk February 9 A First Course in Digital Communications 1/46 Introduction There are benefits to be gained when M-ary (M = 4 signaling methods

More information

Fourier Transforms For additional information, see the classic book The Fourier Transform and its Applications by Ronald N. Bracewell (which is on the shelves of most radio astronomers) and the Wikipedia

More information

Elec4621 Advanced Digital Signal Processing Chapter 11: Time-Frequency Analysis

Elec4621 Advanced Digital Signal Processing Chapter 11: Time-Frequency Analysis Elec461 Advanced Digital Signal Processing Chapter 11: Time-Frequency Analysis Dr. D. S. Taubman May 3, 011 In this last chapter of your notes, we are interested in the problem of nding the instantaneous

More information

Module 4. Signal Representation and Baseband Processing. Version 2 ECE IIT, Kharagpur

Module 4. Signal Representation and Baseband Processing. Version 2 ECE IIT, Kharagpur Module Signal Representation and Baseband Processing Version ECE II, Kharagpur Lesson 8 Response of Linear System to Random Processes Version ECE II, Kharagpur After reading this lesson, you will learn

More information

Aspects of Continuous- and Discrete-Time Signals and Systems

Aspects of Continuous- and Discrete-Time Signals and Systems Aspects of Continuous- and Discrete-Time Signals and Systems C.S. Ramalingam Department of Electrical Engineering IIT Madras C.S. Ramalingam (EE Dept., IIT Madras) Networks and Systems 1 / 45 Scaling the

More information

Central to this is two linear transformations: the Fourier Transform and the Laplace Transform. Both will be considered in later lectures.

Central to this is two linear transformations: the Fourier Transform and the Laplace Transform. Both will be considered in later lectures. In this second lecture, I will be considering signals from the frequency perspective. This is a complementary view of signals, that in the frequency domain, and is fundamental to the subject of signal

More information

Communication Systems Lecture 21, 22. Dong In Kim School of Information & Comm. Eng. Sungkyunkwan University

Communication Systems Lecture 21, 22. Dong In Kim School of Information & Comm. Eng. Sungkyunkwan University Communication Systems Lecture 1, Dong In Kim School of Information & Comm. Eng. Sungkyunkwan University 1 Outline Linear Systems with WSS Inputs Noise White noise, Gaussian noise, White Gaussian noise

More information

Fourier Series. Spectral Analysis of Periodic Signals

Fourier Series. Spectral Analysis of Periodic Signals Fourier Series. Spectral Analysis of Periodic Signals he response of continuous-time linear invariant systems to the complex exponential with unitary magnitude response of a continuous-time LI system at

More information

EE123 Digital Signal Processing

EE123 Digital Signal Processing EE123 Digital Signal Processing Lecture 1 Time-Dependent FT Announcements! Midterm: 2/22/216 Open everything... but cheat sheet recommended instead 1am-12pm How s the lab going? Frequency Analysis with

More information