4.3 Analysis of Non-periodic Con6nuous-Time Signals. We view this non-periodic signal as a periodic signal with period as infinite large.
|
|
- Jeffry Chase
- 6 years ago
- Views:
Transcription
1 We view this non-periodic signal as a periodic signal with period as infinite large.
2 For periodic signal: c k = 1 T 0 T 0 /2 T 0 /2 x(t)e jkω 0t dt For non-periodic signal: we make T 0 -> c k T 0 = T 0 /2 T 0 /2 X(ω) = lim[c k T 0 ] = lim T 0 > T 0 > x(t)e jkω 0t dt T 0 /2 T 0 /2 = x(t)e jωt dt where w = kw 0 x(t)e jkω 0t dt
3 Shorthand nota<on: X(ω) = {x(t)} x(t) = 1 {X(w)} x(t) X(w)
4 x(t) : waveform in 6me domain X(f) : same waveform in frequency domain
5 How to understand Fourier transform? X( f ) = x(t)e j2π ft dt For X(f), at any frequency f, all the x(t) ( t from - to ) has contribu<on. x(t) = X( f )e j2π ft dt Similar For x(t), at any frequency t, all the X(f) ( f from - to ) has contribu<on.
6 Time Domain How to understand Fourier Series and Fourier Transform?!x (1) (t) = b 1 sin(ω 0 t)!x(t)!x (1) (t) Time t is from - to we call it <me domain b 1 = 4A π Only one frequency is used in this example: ω 0 = 2π T 0 or f 0 = 1/T 0
7 Time Domain How to understand Fourier Series and Fourier Transform?!x (2) (t) = b 1 sin(ω 0 t)+ b 2 sin(2ω 0 t)!x(t)!x (2) (t) Time t is from - to we call it <me domain b 1 = 4A π b 2 = 0 two frequencies are used in this example: ω 0 = 2π T 0 and 2ω 0
8 Time Domain How to understand Fourier Series and Fourier Transform?!x (3) (t) = b 1 sin(ω 0 t)+ b 2 sin(2ω 0 t)+ b 3 sin(3ω 0 t)!x(t)!x (3) (t) Time t is from - to we call it <me domain three frequencies are used in this example: ω 0 = 2π T 0 b 1 = 4A π b 2 = 0 b 3 = 4A 3π and 2ω 0 3ω 0
9 4.2.1 Approxima6ng a periodic signal with trigonometric func6ons Let s try a 15-frequency approxima<on to error can be reduced.!x(t) and see if the approximate!x (15) (t) = b 1 sin(ω 0 t)+ b 2 sin(2ω 0 t) b 15 sin(15ω 0 t)!ε 15 (t) =!x(t)!x (15) (t)!x(t)!x (15) (t)!ε 15 (t) =!x(t)!x (15) (t) A A -A T 0 -A
10 Time Domain How to understand Fourier Series and Fourier Transform?!x (15) (t) = b 1 sin(ω 0 t)+ b 2 sin(2ω 0 t) b 15 sin(15ω 0 t) Time t is from - to we call it <me domain 15 frequencies are used in this example: ω 0 = 2π T 0 b 1 = 4A π b 2 = 0 b 3 = 4A 3π b 4 = 0 b 5 = 4A 5π and 2ω 0 3ω 0,.
11 Time Domain b 4 in(4ω 0 t)!x (4) (t) b 3 in(3ω 0 t) b 2 sin(2ω 0 t) b 1 sin(ω 0 t)
12 Time Domain and Frequency Domain Frequency Domain
13 Time Domain and Frequency Domain sin(x) func<on can be viewed as a circle projected onto a line
14 Time Domain and Frequency Domain
15 6me domain x(t) frequency domain X(f) Example of Music
16 4.3.2 Existence of Fourier Transform s it always possible to determine the Fourier series coefficients? Dirichlet Condi;on 3F ² Finite absolute value: x(t) dt < ² Finite number of discon<nui<es in!x(t) ² Finite number of minima and maxima in one period
17 4.3.2 Existence of Fourier Transform Example 4.12 Fourier Transform of a Rectangular Pulse Using the forward Fourier transform integral, find the Fourier transform of the isolated rectangular pulse signal x(t) = A t τ
18 4.3.2 Existence of Fourier Transform Example 4.12 Fourier Transform of a Rectangular Pulse Frequency Domain
19 4.3.2 Existence of Fourier Transform Example 4.12 Fourier Transform of a Rectangular Pulse
20 4.3.5 Proper6es of Fourier Transform Linearity: x 1 (t) X 1 (w) and x 2 (t) X 2 (w) α 1 x 1 (t)+α 2 x 2 (t) α 1 X 1 (w)+α 2 X 2 (w) Where a 1 and a 2 are any two constants Duality: x(t) X(w) X(t) 2π x( w) x(t) X( f ) X(t) x( f )
21 4.3.5 Proper6es of Fourier Transform Symmetry of Fourier Transform: x(t): Real, m{x(t)} = 0 X * (w) = X( w) x(t): mage, Re{x(t)} = 0 X * (w) = X( w) Time ShiTing: x(t) X(w) x(t τ ) X(w)e jwτ Frequency ShiTing: x(t) X(w) x(t)e jw 0t X(w w 0 )
22 4.3.5 Proper6es of Fourier Transform Modula6on Property: x(t) X(w) x(t)cos(w 0 t) 1 2 X w w 0 ( ) + X ( w + w 0 ) Or x(t)cos(w 0 t) 1 2 X f f 0 ( ) + X ( f + f 0 ) x(t)sin(w 0 t) 1 2 X ( w w 0)e jπ /2 + X ( w + w 0 )e More general format x(t)cos(w 0 t +θ) 1 2 e jθ X f f 0 jπ /2 ( ) + e jθ X ( f + f 0 )
23 4.3.5 Proper6es of Fourier Transform Modula6on Property: Find the Fourier Transform of the modulated pulse given by cos(2π f 0 t), t < τ x(t) = 0, t < τ
24 4.3.5 Proper6es of Fourier Transform Modula6on Property:
25 4.3.5 Proper6es of Fourier Transform Convolu6on Property: x 1 (t) X 1 (w) x 2 (t) X 2 (w) x 1 (t)* x 2 (t) X 1 (w)x 2 (w) X 1 (w)* X 2 (w) x 1 (t)x 2 (t)
Chapter 4. Fourier Analysis for Con5nuous-Time Signals and Systems Chapter Objec5ves
Chapter 4. Fourier Analysis for Con5nuous-Time Signals and Systems Chapter Objec5ves 1. Learn techniques for represen3ng con$nuous-$me periodic signals using orthogonal sets of periodic basis func3ons.
More informationSignals & Systems. Lecture 5 Continuous-Time Fourier Transform. Alp Ertürk
Signals & Systems Lecture 5 Continuous-Time Fourier Transform Alp Ertürk alp.erturk@kocaeli.edu.tr Fourier Series Representation of Continuous-Time Periodic Signals Synthesis equation: x t = a k e jkω
More informationChapter 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 informationChapter 5. Fourier Analysis for Discrete-Time Signals and Systems Chapter
Chapter 5. Fourier Analysis for Discrete-Time Signals and Systems Chapter Objec@ves 1. Learn techniques for represen3ng discrete-)me periodic signals using orthogonal sets of periodic basis func3ons. 2.
More informationChapter 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 informationEC Signals and Systems
UNIT I CLASSIFICATION OF SIGNALS AND SYSTEMS Continuous time signals (CT signals), discrete time signals (DT signals) Step, Ramp, Pulse, Impulse, Exponential 1. Define Unit Impulse Signal [M/J 1], [M/J
More informationENSC327 Communications Systems 2: Fourier Representations. School of Engineering Science Simon Fraser University
ENSC37 Communications Systems : Fourier Representations School o Engineering Science Simon Fraser University Outline Chap..5: Signal Classiications Fourier Transorm Dirac Delta Function Unit Impulse Fourier
More informationFourier Series and Fourier Transforms
Fourier Series and Fourier Transforms EECS2 (6.082), MIT Fall 2006 Lectures 2 and 3 Fourier Series From your differential equations course, 18.03, you know Fourier s expression representing a T -periodic
More information2.1 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 informationEE 3054: Signals, Systems, and Transforms Summer It is observed of some continuous-time LTI system that the input signal.
EE 34: Signals, Systems, and Transforms Summer 7 Test No notes, closed book. Show your work. Simplify your answers. 3. It is observed of some continuous-time LTI system that the input signal = 3 u(t) produces
More information5. THE CLASSES OF FOURIER TRANSFORMS
5. THE CLASSES OF FOURIER TRANSFORMS There are four classes of Fourier transform, which are represented in the following table. So far, we have concentrated on the discrete Fourier transform. Table 1.
More informationThe 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 informationThe Fourier Transform (and more )
The Fourier Transform (and more ) imrod Peleg ov. 5 Outline Introduce Fourier series and transforms Introduce Discrete Time Fourier Transforms, (DTFT) Introduce Discrete Fourier Transforms (DFT) Consider
More informationENSC327 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 informationFourier 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 informationCMPT 318: Lecture 5 Complex Exponentials, Spectrum Representation
CMPT 318: Lecture 5 Complex Exponentials, Spectrum Representation Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 23, 2006 1 Exponentials The exponential is
More informationChapter 6 THE SAMPLING PROCESS 6.1 Introduction 6.2 Fourier Transform Revisited
Chapter 6 THE SAMPLING PROCESS 6.1 Introduction 6.2 Fourier Transform Revisited Copyright c 2005 Andreas Antoniou Victoria, BC, Canada Email: aantoniou@ieee.org July 14, 2018 Frame # 1 Slide # 1 A. Antoniou
More informationLecture 7 ELE 301: Signals and Systems
Lecture 7 ELE 30: Signals and Systems Prof. Paul Cuff Princeton University Fall 20-2 Cuff (Lecture 7) ELE 30: Signals and Systems Fall 20-2 / 22 Introduction to Fourier Transforms Fourier transform as
More informationTLT-5200/5206 COMMUNICATION THEORY, Exercise 1, Fall 2012
Problem 1. a) Derivation By definition, we can write the inverse Fourier transform of the derivative of x(t) as d d jπ X( f ) { e } df d jπ x() t X( f ) e df jπ j π fx( f ) e df Again, by definition of
More informationChapter 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 information7. Find the Fourier transform of f (t)=2 cos(2π t)[u (t) u(t 1)]. 8. (a) Show that a periodic signal with exponential Fourier series f (t)= δ (ω nω 0
Fourier Transform Problems 1. Find the Fourier transform of the following signals: a) f 1 (t )=e 3 t sin(10 t)u (t) b) f 1 (t )=e 4 t cos(10 t)u (t) 2. Find the Fourier transform of the following signals:
More informationECE 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 informationSignals and Systems I Have Known and Loved. (Lecture notes for CSE 3451) Andrew W. Eckford
Signals and Systems I Have Known and Loved (Lecture notes for CSE 3451) Andrew W. Eckford Department of Electrical Engineering and Computer Science York University, Toronto, Ontario, Canada Version: December
More informationEA2.3 - Electronics 2 1
In the previous lecture, I talked about the idea of complex frequency s, where s = σ + jω. Using such concept of complex frequency allows us to analyse signals and systems with better generality. In this
More informationReview: 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 informationFigure 3.1 Effect on frequency spectrum of increasing period T 0. Consider the amplitude spectrum of a periodic waveform as shown in Figure 3.2.
3. Fourier ransorm From Fourier Series to Fourier ransorm [, 2] In communication systems, we oten deal with non-periodic signals. An extension o the time-requency relationship to a non-periodic signal
More informationHomework 7 Solution EE235, Spring Find the Fourier transform of the following signals using tables: te t u(t) h(t) = sin(2πt)e t u(t) (2)
Homework 7 Solution EE35, Spring. Find the Fourier transform of the following signals using tables: (a) te t u(t) h(t) H(jω) te t u(t) ( + jω) (b) sin(πt)e t u(t) h(t) sin(πt)e t u(t) () h(t) ( ejπt e
More informationω 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 informationMusic 270a: Complex Exponentials and Spectrum Representation
Music 270a: Complex Exponentials and Spectrum Representation Tamara Smyth, trsmyth@ucsd.edu Department of Music, University of California, San Diego (UCSD) October 24, 2016 1 Exponentials The exponential
More informationCore Concepts Review. Orthogonality of Complex Sinusoids Consider two (possibly non-harmonic) complex sinusoids
Overview of Continuous-Time Fourier Transform Topics Definition Compare & contrast with Laplace transform Conditions for existence Relationship to LTI systems Examples Ideal lowpass filters Relationship
More informationIntroduction to Fourier Transforms. Lecture 7 ELE 301: Signals and Systems. Fourier Series. Rect Example
Introduction to Fourier ransforms Lecture 7 ELE 3: Signals and Systems Fourier transform as a limit of the Fourier series Inverse Fourier transform: he Fourier integral theorem Prof. Paul Cuff Princeton
More informationLine Spectra and their Applications
In [ ]: cd matlab pwd Line Spectra and their Applications Scope and Background Reading This session concludes our introduction to Fourier Series. Last time (http://nbviewer.jupyter.org/github/cpjobling/eg-47-
More informationReview 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 informationEE 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 informationLECTURE 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 informationAssignment 3 Solutions
Assignment Solutions Networks and systems August 8, 7. Consider an LTI system with transfer function H(jw) = input is sin(t + π 4 ), what is the output? +jw. If the Solution : C For an LTI system with
More informationLOPE3202: 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 informationQuestion Paper Code : AEC11T02
Hall Ticket No Question Paper Code : AEC11T02 VARDHAMAN COLLEGE OF ENGINEERING (AUTONOMOUS) Affiliated to JNTUH, Hyderabad Four Year B. Tech III Semester Tutorial Question Bank 2013-14 (Regulations: VCE-R11)
More informationComplex symmetry Signals and Systems Fall 2015
18-90 Signals and Systems Fall 015 Complex symmetry 1. Complex symmetry This section deals with the complex symmetry property. As an example I will use the DTFT for a aperiodic discrete-time signal. The
More informationVer 3808 E1.10 Fourier Series and Transforms (2014) E1.10 Fourier Series and Transforms. Problem Sheet 1 (Lecture 1)
Ver 88 E. Fourier Series and Transforms 4 Key: [A] easy... [E]hard Questions from RBH textbook: 4., 4.8. E. Fourier Series and Transforms Problem Sheet Lecture. [B] Using the geometric progression formula,
More informationTable 1: Properties of the Continuous-Time Fourier Series. Property Periodic Signal Fourier Series Coefficients
able : Properties of the Continuous-ime Fourier Series x(t = e jkω0t = = x(te jkω0t dt = e jk(/t x(te jk(/t dt Property Periodic Signal Fourier Series Coefficients x(t y(t } Periodic with period and fundamental
More informationSignals 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 informationEE301 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 informationLecture 8 ELE 301: Signals and Systems
Lecture 8 ELE 30: Signals and Systems Prof. Paul Cuff Princeton University Fall 20-2 Cuff (Lecture 7) ELE 30: Signals and Systems Fall 20-2 / 37 Properties of the Fourier Transform Properties of the Fourier
More information4 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 informationSeries 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 24 g.s.mcdonald@salford.ac.uk 1. Theory
More informationDiscrete Fourier Transform
Discrete Fourier Transform Valentina Hubeika, Jan Černocký DCGM FIT BUT Brno, {ihubeika,cernocky}@fit.vutbr.cz Diskrete Fourier transform (DFT) We have just one problem with DFS that needs to be solved.
More informationCT Rectangular Function Pairs (5B)
C Rectangular Function Pairs (5B) Continuous ime Rect Function Pairs Copyright (c) 009-013 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU
More informationChapter 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 information06EC44-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 informationFourier Representations of Signals & LTI Systems
3. Introduction. A signal can be represented as a weighted superposition of complex sinusoids. x(t) or x[n] 2. LTI system: LTI System Output = A weighted superposition of the system response to each complex
More informationGATE EE Topic wise Questions SIGNALS & SYSTEMS
www.gatehelp.com GATE EE Topic wise Questions YEAR 010 ONE MARK Question. 1 For the system /( s + 1), the approximate time taken for a step response to reach 98% of the final value is (A) 1 s (B) s (C)
More informationTable 1: Properties of the Continuous-Time Fourier Series. Property Periodic Signal Fourier Series Coefficients
able : Properties of the Continuous-ime Fourier Series x(t = a k e jkω0t = a k = x(te jkω0t dt = a k e jk(/t x(te jk(/t dt Property Periodic Signal Fourier Series Coefficients x(t y(t } Periodic with period
More informationFourier transform representation of CT aperiodic signals Section 4.1
Fourier transform representation of CT aperiodic signals Section 4. A large class of aperiodic CT signals can be represented by the CT Fourier transform (CTFT). The (CT) Fourier transform (or spectrum)
More informationThe Laplace Transform
The Laplace Transform Introduction There are two common approaches to the developing and understanding the Laplace transform It can be viewed as a generalization of the CTFT to include some signals with
More informationCH.4 Continuous-Time Fourier Series
CH.4 Continuous-Time Fourier Series First step to Fourier analysis. My mathematical model is killing me! The difference between mathematicians and engineers is mathematicians develop mathematical tools
More informationCourse Notes for Signals and Systems. Krishna R Narayanan
Course Notes for Signals and Systems Krishna R Narayanan May 7, 018 Contents 1 Math Review 5 1.1 Trigonometric Identities............................. 5 1. Complex Numbers................................
More informationHomework 6 EE235, Spring 2011
Homework 6 EE235, Spring 211 1. Fourier Series. Determine w and the non-zero Fourier series coefficients for the following functions: (a 2 cos(3πt + sin(1πt + π 3 w π e j3πt + e j3πt + 1 j2 [ej(1πt+ π
More informationSIGNALS AND SYSTEMS: PAPER 3C1 HANDOUT 6a. Dr David Corrigan 1. Electronic and Electrical Engineering Dept.
SIGNALS AND SYSTEMS: PAPER 3C HANDOUT 6a. Dr David Corrigan. Electronic and Electrical Engineering Dept. corrigad@tcd.ie www.mee.tcd.ie/ corrigad FOURIER SERIES Have seen how the behaviour of systems can
More informationAssignment 4 Solutions Continuous-Time Fourier Transform
Assignment 4 Solutions Continuous-Time Fourier Transform ECE 3 Signals and Systems II Version 1.01 Spring 006 1. Properties of complex numbers. Let c 1 α 1 + jβ 1 and c α + jβ be two complex numbers. a.
More information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 informationFourier Analysis and Power Spectral Density
Chapter 4 Fourier Analysis and Power Spectral Density 4. Fourier Series and ransforms Recall Fourier series for periodic functions for x(t + ) = x(t), where x(t) = 2 a + a = 2 a n = 2 b n = 2 n= a n cos
More information09/29/2009 Reading: Hambley Chapter 5 and Appendix A
EE40 Lec 10 Complex Numbers and Phasors Prof. Nathan Cheung 09/29/2009 Reading: Hambley Chapter 5 and Appendix A Slide 1 OUTLINE Phasors as notation for Sinusoids Arithmetic with Complex Numbers Complex
More informationECE Unit 4. Realizable system used to approximate the ideal system is shown below: Figure 4.47 (b) Digital Processing of Analog Signals
ECE 8440 - Unit 4 Digital Processing of Analog Signals- - Non- Ideal Case (See sec8on 4.8) Before considering the non- ideal case, recall the ideal case: 1 Assump8ons involved in ideal case: - no aliasing
More informationThen r (t) can be expanded into a linear combination of the complex exponential signals ( e j2π(kf 0)t ) k= as. c k e j2π(kf0)t + c k e j2π(kf 0)t
.3 ourier Series Definition.37. Exponential ourier series: Let the real or complex signal r t be a periodic signal with period. Suppose the following Dirichlet conditions are satisfied: a r t is absolutely
More informationSection Kamen and Heck And Harman. Fourier Transform
s Section 3.4-3.7 Kamen and Heck And Harman 1 3.4 Definition (Equation 3.30) Exists if integral converges (Equation 3.31) Example 3.7 Constant Signal Does not have a Fourier transform in the ordinary sense.
More information1. Nature of Impulse Response - Pole on Real Axis. z y(n) = r n. z r
. Nature of Impulse Respose - Pole o Real Axis Causal system trasfer fuctio: Hz) = z yz) = z r z z r y) = r r > : the respose grows mootoically > r > : y decays to zero mootoically r > : oscillatory, decayig
More informationPhasor Young Won Lim 05/19/2015
Phasor Copyright (c) 2009-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version
More information3. 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 information2.161 Signal Processing: Continuous and Discrete
MI OpenCourseWare http://ocw.mit.edu.6 Signal Processing: Continuous and Discrete Fall 8 For information about citing these materials or our erms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSES INSIUE
More informationEE 224 Signals and Systems I Complex numbers sinusodal signals Complex exponentials e jωt phasor addition
EE 224 Signals and Sysems I Complex numbers sinusodal signals Complex exponenials e jω phasor addiion 1/28 Complex Numbers Recangular Polar y z r z θ x Good for addiion/subracion Good for muliplicaion/division
More informationGeorge Mason University Signals and Systems I Spring 2016
George Mason University Signals and Systems I Spring 206 Problem Set #6 Assigned: March, 206 Due Date: March 5, 206 Reading: This problem set is on Fourier series representations of periodic signals. The
More informationFourier transforms. Definition F(ω) = - should know these! f(t).e -jωt.dt. ω = 2πf. other definitions exist. f(t) = - F(ω).e jωt.
Fourier transforms This is intended to be a practical exposition, not fully mathematically rigorous ref The Fourier Transform and its Applications R. Bracewell (McGraw Hill) Definition F(ω) = - f(t).e
More informationContinuous-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 informationTopic 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 informationMathematical Review for Signal and Systems
Mathematical Review for Signal and Systems 1 Trigonometric Identities It will be useful to memorize sin θ, cos θ, tan θ values for θ = 0, π/3, π/4, π/ and π ±θ, π θ for the above values of θ. The following
More informationSolutions to Problems in Chapter 4
Solutions to Problems in Chapter 4 Problems with Solutions Problem 4. Fourier Series of the Output Voltage of an Ideal Full-Wave Diode Bridge Rectifier he nonlinear circuit in Figure 4. is a full-wave
More informationSIGNALS AND SYSTEMS: PAPER 3C1 HANDOUT 6. Dr Anil Kokaram Electronic and Electrical Engineering Dept.
SIGNALS AND SYSTEMS: PAPER 3C HANDOUT 6. Dr Anil Kokaram Electronic and Electrical Engineering Dept. anil.kokaram@tcd.ie www.mee.tcd.ie/ sigmedia FOURIER ANALYSIS Have seen how the behaviour of systems
More informationFrequency Domain Representations of Sampled and Wrapped Signals
Frequency Domain Representations of Sampled and Wrapped Signals Peter Kabal Department of Electrical & Computer Engineering McGill University Montreal, Canada v1.5 March 2011 c 2011 Peter Kabal 2011/03/11
More informationFourier series for continuous and discrete time signals
8-9 Signals and Systems Fall 5 Fourier series for continuous and discrete time signals The road to Fourier : Two weeks ago you saw that if we give a complex exponential as an input to a system, the output
More informationLINEAR SYSTEMS. J. Elder PSYC 6256 Principles of Neural Coding
LINEAR SYSTEMS Linear Systems 2 Neural coding and cognitive neuroscience in general concerns input-output relationships. Inputs Light intensity Pre-synaptic action potentials Number of items in display
More informationSignal and systems. Linear Systems. Luigi Palopoli. Signal and systems p. 1/5
Signal and systems p. 1/5 Signal and systems Linear Systems Luigi Palopoli palopoli@dit.unitn.it Wrap-Up Signal and systems p. 2/5 Signal and systems p. 3/5 Fourier Series We have see that is a signal
More informationLecture 13: Discrete Time Fourier Transform (DTFT)
Lecture 13: Discrete Time Fourier Transform (DTFT) ECE 401: Signal and Image Analysis University of Illinois 3/9/2017 1 Sampled Systems Review 2 DTFT and Convolution 3 Inverse DTFT 4 Ideal Lowpass Filter
More informationA3. Frequency Representation of Continuous Time and Discrete Time Signals
A3. Frequency Representation of Continuous Time and Discrete Time Signals Objectives Define the magnitude and phase plots of continuous time sinusoidal signals Extend the magnitude and phase plots to discrete
More information23.4. Convergence. Introduction. Prerequisites. Learning Outcomes
Convergence 3.4 Introduction In this Section we examine, briefly, the convergence characteristics of a Fourier series. We have seen that a Fourier series can be found for functions which are not necessarily
More informationRepresenting 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 informationFourier series. XE31EO2 - Pavel Máša. Electrical Circuits 2 Lecture1. XE31EO2 - Pavel Máša - Fourier Series
Fourier series Electrical Circuits Lecture - Fourier Series Filtr RLC defibrillator MOTIVATION WHAT WE CAN'T EXPLAIN YET Source voltage rectangular waveform Resistor voltage sinusoidal waveform - Fourier
More informationX. Chen More on Sampling
X. Chen More on Sampling 9 More on Sampling 9.1 Notations denotes the sampling time in second. Ω s = 2π/ and Ω s /2 are, respectively, the sampling frequency and Nyquist frequency in rad/sec. Ω and ω denote,
More informationFourier transform. XE31EO2 - Pavel Máša. EO2 Lecture 2. XE31EO2 - Pavel Máša - Fourier Transform
Fourier transform EO2 Lecture 2 Pavel Máša - Fourier Transform INTRODUCTION We already know complex form of Fourier series f(t) = 1X k= 1 A k e jk! t A k = 1 T Series frequency spectra is discrete Circuits
More informationNotes 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 informationDiscrete Time Fourier Transform
Discrete Time Fourier Transform Recall that we wrote the sampled signal x s (t) = x(kt)δ(t kt). We calculate its Fourier Transform. We do the following: Ex. Find the Continuous Time Fourier Transform of
More information8 Continuous-Time Fourier Transform
8 Continuous-Time Fourier Transform Recommended Problems P8. Consider the signal x(t), which consists of a single rectangular pulse of unit height, is symmetric about the origin, and has a total width
More informationLaplace Transforms and use in Automatic Control
Laplace Transforms and use in Automatic Control P.S. Gandhi Mechanical Engineering IIT Bombay Acknowledgements: P.Santosh Krishna, SYSCON Recap Fourier series Fourier transform: aperiodic Convolution integral
More informationFourier 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 informationI. Signals & Sinusoids
I. Signals & Sinusoids [p. 3] Signal definition Sinusoidal signal Plotting a sinusoid [p. 12] Signal operations Time shifting Time scaling Time reversal Combining time shifting & scaling [p. 17] Trigonometric
More informationENGIN 211, Engineering Math. Fourier Series and Transform
ENGIN 11, Engineering Math Fourier Series and ransform 1 Periodic Functions and Harmonics f(t) Period: a a+ t Frequency: f = 1 Angular velocity (or angular frequency): ω = ππ = π Such a periodic function
More information2 Frequency-Domain Analysis
2 requency-domain Analysis Electrical engineers live in the two worlds, so to speak, o time and requency. requency-domain analysis is an extremely valuable tool to the communications engineer, more so
More informationSignals and Systems I Have Known and Loved. Andrew W. Eckford
Signals and Systems I Have Known and Loved Andrew W. Eckford Department of Electrical Engineering and Computer Science York University, oronto, Ontario, Canada Version: September 2, 216 Copyright c 215
More informationHomework 3 Solutions
EECS Signals & Systems University of California, Berkeley: Fall 7 Ramchandran September, 7 Homework 3 Solutions (Send your grades to ee.gsi@gmail.com. Check the course website for details) Review Problem
More informationReview of Discrete-Time System
Review of Discrete-Time System Electrical & Computer Engineering University of Maryland, College Park Acknowledgment: ENEE630 slides were based on class notes developed by Profs. K.J. Ray Liu and Min Wu.
More information