The Continuous-time Fourier
|
|
- Noah Simmons
- 6 years ago
- Views:
Transcription
1 The Continuous-time Fourier Transform Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it
2 Outline Representation of Aperiodic signals: The continuous-time Fourier Transform The Fourier transform for periodic signals Properties of the continuous-time Fourier transform The convolution/multiplication property System characterization by linear constant- coefficient 2
3 4.1 The continuous-time Fourier transform Extend the Fourier series representation to Aperiodic signals Almost all of the signals, including all signals with finite energy can be represented by a linear combination of complex exponentials The representation of in terms of a linear combination takes a form of an integral (rather than a sum) Fourier transform: the resulting spectrum of coefficients in the representation Inverse Fourier transform: use these coefficients to represent the signal as a linear combination of complex exponentials 3
4 4.1 The continuous-time Fourier transform Development of Fourier transform representation of an aperiodic signal Consider a periodic square wave 4
5 4.1 The continuous-time Fourier transform Alternative interpreting 5
6 4.1 The continuous-time Fourier transform As T -> infinite The original periodic signals becomes a rectangular pulse The Fourier series coefficients become more and more closely spaced samples approached by a continuous envelop function For aperiodic signal We think of an aperiodic signal as the periodic signal with arbitrary large period 6
7 4.1 The continuous-time Fourier transform An example: 7
8 4.1 The continuous-time Fourier transform The Fourier series representation: Therefore, define the envelop as We have 8
9 4.1 The continuous-time Fourier transform Combining them together, we have As, we have Fourier Transform Inverse Fourier Transform 9
10 4.1 The continuous-time Fourier transform We call X(jw) as spectrum of x(t) It provides us with the information needed for describing x(t) as a linear combination of the exponential signals with different frequencies Some conclusions: For a periodic signal, the Fourier coefficients can be expressed in terms of equally spaced samples of the Fourier transform of one period Let be a finite-duration signal in one period 10
11 4.1 The continuous-time Fourier transform Convergence of Fourier transform: following the Dirichlet conditions X(t) is absolutely integrable X(t) have a finite number of maxima and minima within any finite interval X(t) has a finite number of discontinuities within any finite interval and each of these discontinuities is finite 11
12 4.1 The continuous-time Fourier transform Example 1: Consider the signal Determine the Fourier transform Solution: 12
13 4.1 The continuous-time Fourier transform Example 2: Consider the signal Determine the Fourier transform Solution: 13
14 4.1 The continuous-time Fourier transform Example 3: Consider the signal Determine the Fourier transform Solution: 14
15 4.1 The continuous-time Fourier transform Example 4: Consider the signal Determine the Fourier transform Solution: 15
16 4.1 The continuous-time Fourier transform 16
17 4.1 The continuous-time Fourier transform Example 5: Consider the signal x(t) whose Fourier transform is Determine x(t) Solution: 17
18 4.1 The continuous-time Fourier transform Sinc function: 18
19 4.1 The continuous-time Fourier transform Some property for sinc function: As W increases, X(jw) becomes broader Main peak of x(t) becomes higher The width of the first lobe ( ) becomes narrower 19
20 4.2 The Fourier transform for periodic signal The Fourier transform of periodic signal Consists of a train of impulses in the frequency domain So we have 20
21 4.2 The Fourier transform for periodic signal Example 1: Consider periodic square wave Determine its Fourier transform Solution: 21
22 4.2 The Fourier transform for periodic signal 22
23 4.2 The Fourier transform for periodic signal Example 2: Consider periodic signals Determine its Fourier transform Solution: 23
24 4.2 The Fourier transform for periodic signal 24
25 4.2 The Fourier transform for periodic signal Example 2: Consider the impulse train Determine its Fourier transform Solution: 25
26 4.3 Properties Linearity: Time shifting: Proof: 26
27 4.2 The Fourier transform for periodic signal Example 1: Consider the signal Determine its Fourier transform Solution: 27
28 4.3 Properties Conjugation: Proof: For real signal: 28
29 4.3 Properties For a real and even signal, is real and even Proof: 29
30 4.3 Properties By decompose the signal as even and odd parts 30
31 4.2 The Fourier transform for periodic signal Example 2: Consider the signal Determine its Fourier transform Solution: 31
32 4.3 Properties Differentiation and integration: Proof: 32
33 4.2 The Fourier transform for periodic signal Example 3: Consider the signal Determine its Fourier transform Solution: 33
34 4.2 The Fourier transform for periodic signal Example 4: Consider the signal Determine its Fourier transform Solution: 34
35 4.3 Properties Time and frequency scaling: Proof: 35
36 4.3 Properties Duality: 36
37 4.3 Properties 37
38 4.3 Properties Example 5: Consider the signal Determine its Fourier transform Solution: 38
39 4.3 Properties Parseval s s relation: Proof: 39
40 4.3 Properties Example 6: Evaluate the following time-domain expressions: 40
41 4.3 Properties Solution: 1 2 5/8 E X ( j ) d 2b 1 1 D g (0) G ( j ) d D j X( j ) d
42 4.4 The convolution property Linear convolution in time domain product in frequency domain Proof: 42
43 4.4 The convolution property In Using Fourier analysis to study LTI system, we require The unit impulse function of the LTI system has Fourier transform Using transform techniques to examine unstable LTI system Use the Laplace transform (in Chapter 9) 43
44 4.4 The convolution property Example 1: Check the time-shifting with unit impulse transform as follows by using Fourier transform Solution: 44
45 4.4 The convolution property Example 2: Determine the Fourier transform of a differentiator Solution: 45
46 4.4 The convolution property Example 3: Determine the Fourier transform of a integrator Solution: the unit impulse response is a unit step function 46
47 4.4 The convolution property Example 4: Accomplish a frequency-selective filter using LTI system Solution: 47
48 4.4 The convolution property Example 5: Consider a system with input and unit impulse function given as follows. Determine the output by using Fourier transform. 48
49 4.4 The convolution property Solution: for a!= b 49
50 4.4 The convolution property Solution: for a = b 50
51 4.4 The convolution property Example 6: Consider a system with input and unit impulse function given as follows. Determine the output by using Fourier transform. 51
52 4.4 The convolution property Solution: 52
53 4.5 The multiplication property The multiplication in the time domain corresponds to the convolution in the frequency domain Amplitude modulation: multiplication of one signal by another Using one signal to scale or modulate the amplitude of the other 53
54 4.5 The multiplication property Example 1: Let s(t) be a signal with spectrum given as Fig. (a). Also, consider the signal of Then 54
55 4.5 The multiplication property 55
56 4.5 The multiplication property Example 2: Determine the Fourier transform of the signal Solution: 56
57 4.5 The multiplication property Consider a system given as follows: 57
58 4.5 The multiplication property Consider a system given as follows: 58
59 4.6 Summary of the properties X(t), y(t) Linearity Time shifting Frequency shifting Conjugation Time reversal e Time Scaling (Period ) Convolution 59
60 4.6 Summary of the properties Multiplication Differentiation in time Integration Differentiation in frequency Conjugate Symmetry for Real Signals real 60
61 4.6 Summary of the properties Symmetry for Real and Even Signals Real and Odd Signals Even-odd Decomposition of Real Signal [x(t) real] Parseval s Relation for Periodic Signals X(t) real and even X(t) real and odd real and even imaginary and odd 61
62 4.6 Summary of the properties 62
63 4.6 Summary of the properties 63
64 4.7 System characterization by linear constant-coefficient t i t differential equations Determine the frequency response of the LTI system of For a LTI system, we have 64
65 4.7 System characterization by linear constant-coefficient t i t differential equations Taking Fourier transform on both sides, we have Apply linearity Apply differentiation property, we have 65
66 4.7 System characterization by linear constant-coefficient t i t differential equations Or equivalently, We thus have 66
67 4.7 System characterization by linear constant-coefficient t i t differential equations Example 1: consider a LTI system (a>0) The frequency response is 67
68 4.7 System characterization by linear constant-coefficient t i t differential equations Example 2: consider a LTI system (a>0) The frequency response is 68
69 4.7 System characterization by linear constant-coefficient t i t differential equations Example 3: consider a LTI system (a>0) The frequency response is 69
70 4.7 System characterization by linear constant-coefficient t i t differential equations Then we have 70
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 informationFourier 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
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 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 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 informationModeling and Analysis of Systems Lecture #8 - Transfer Function. Guillaume Drion Academic year
Modeling and Analysis of Systems Lecture #8 - Transfer Function Guillaume Drion Academic year 2015-2016 1 Input-output representation of LTI systems Can we mathematically describe a LTI system using the
More informationReview 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 information6.003 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 informationFOURIER 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 informationThe 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 informationLecture 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 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 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 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 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 informationELEG 3124 SYSTEMS AND SIGNALS Ch. 5 Fourier Transform
Department of Electrical Engineering University of Arkansas ELEG 3124 SYSTEMS AND SIGNALS Ch. 5 Fourier Transform Dr. Jingxian Wu wuj@uark.edu OUTLINE 2 Introduction Fourier Transform Properties of Fourier
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 informationENT 315 Medical Signal Processing CHAPTER 2 DISCRETE FOURIER TRANSFORM. Dr. Lim Chee Chin
ENT 315 Medical Signal Processing CHAPTER 2 DISCRETE FOURIER TRANSFORM Dr. Lim Chee Chin Outline Introduction Discrete Fourier Series Properties of Discrete Fourier Series Time domain aliasing due to frequency
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 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 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 informationTherefore the new Fourier coefficients are. Module 2 : Signals in Frequency Domain Problem Set 2. Problem 1
Module 2 : Signals in Frequency Domain Problem Set 2 Problem 1 Let be a periodic signal with fundamental period T and Fourier series coefficients. Derive the Fourier series coefficients of each of the
More informationCh 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 informationTheory 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 information3 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 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 informationContinuous 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 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 informationSignals and systems Lecture (S3) Square Wave Example, Signal Power and Properties of Fourier Series March 18, 2008
Signals and systems Lecture (S3) Square Wave Example, Signal Power and Properties of Fourier Series March 18, 2008 Today s Topics 1. Derivation of a Fourier series representation of a square wave signal
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 informationDiscrete-Time Fourier Transform
Discrete-Time Fourier Transform Chapter Intended Learning Outcomes: (i) (ii) (iii) Represent discrete-time signals using discrete-time Fourier transform Understand the properties of discrete-time Fourier
More information(i) Represent continuous-time periodic signals using Fourier series
Fourier Series Chapter Intended Learning Outcomes: (i) Represent continuous-time periodic signals using Fourier series (ii) (iii) Understand the properties of Fourier series Understand the relationship
More information(i) Understanding the characteristics and properties of DTFT
Discrete-Time Fourier Transform (DTFT) Chapter Intended Learning Outcomes: (i) Understanding the characteristics and properties of DTFT (ii) Ability to perform discrete-time signal conversion between the
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 informationUniversity Question Paper Solution
Unit 1: Introduction University Question Paper Solution 1. Determine whether the following systems are: i) Memoryless, ii) Stable iii) Causal iv) Linear and v) Time-invariant. i) y(n)= nx(n) ii) y(t)=
More informationRepresentation 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 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 informationMITOCW MITRES_6-007S11lec09_300k.mp4
MITOCW MITRES_6-007S11lec09_300k.mp4 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for
More information06EC44-Signals and System Chapter EC44 Signals and Systems (Chapter 4 )
06EC44 Signals and Systems (Chapter 4 ) Aurthored By: Prof. krupa Rasane Asst.Prof E&C Dept. KLE Society s College of Engineering and Technology Belgaum CONTENT Fourier Series Representation 1.1.1 Introduction
More informationQUESTION BANK SIGNALS AND SYSTEMS (4 th SEM ECE)
QUESTION BANK SIGNALS AND SYSTEMS (4 th SEM ECE) 1. For the signal shown in Fig. 1, find x(2t + 3). i. Fig. 1 2. What is the classification of the systems? 3. What are the Dirichlet s conditions of Fourier
More informationRui Wang, Assistant professor Dept. of Information and Communication Tongji University.
Linear Time Invariant (LTI) Systems Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it Email: ruiwang@tongji.edu.cn Outline Discrete-time LTI system: The convolution
More information3.2 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 informationModule 4 : Laplace and Z Transform Problem Set 4
Module 4 : Laplace and Z Transform Problem Set 4 Problem 1 The input x(t) and output y(t) of a causal LTI system are related to the block diagram representation shown in the figure. (a) Determine a differential
More informationChapter Intended Learning Outcomes: (i) Understanding the relationship between transform and the Fourier transform for discrete-time signals
z Transform Chapter Intended Learning Outcomes: (i) Understanding the relationship between transform and the Fourier transform for discrete-time signals (ii) Understanding the characteristics and properties
More informationLecture 28 Continuous-Time Fourier Transform 2
Lecture 28 Continuous-Time Fourier Transform 2 Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/6/14 1 Limit of the Fourier Series Rewrite (11.9) and (11.10) as As, the fundamental
More informationIntroduction to Signals and Systems Lecture #4 - Input-output Representation of LTI Systems Guillaume Drion Academic year
Introduction to Signals and Systems Lecture #4 - Input-output Representation of LTI Systems Guillaume Drion Academic year 2017-2018 1 Outline Systems modeling: input/output approach of LTI systems. Convolution
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 informationEE301 Signals and Systems Spring 2016 Exam 2 Thursday, Mar. 31, Cover Sheet
EE301 Signals and Systems Spring 2016 Exam 2 Thursday, Mar. 31, 2016 Cover Sheet Test Duration: 75 minutes. Coverage: Chapter 4, Hmwks 6-7 Open Book but Closed Notes. One 8.5 in. x 11 in. crib sheet Calculators
More informationChapter 8 The Discrete Fourier Transform
Chapter 8 The Discrete Fourier Transform Introduction Representation of periodic sequences: the discrete Fourier series Properties of the DFS The Fourier transform of periodic signals Sampling the Fourier
More informationFrequency 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 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 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 informationHST.582J / 6.555J / J Biomedical Signal and Image Processing Spring 2007
MIT OpenCourseare http://ocw.mit.edu HST.58J / 6.555J / 16.56J Biomedical Signal and Image Processing Spring 7 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationINSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad
INSTITUTE OF AERONAUTICAL ENGINEERING (Autonomous) Dundigal, Hyderabad - 500 043 ELECTRONICS AND COMMUNICATION ENGINEERING TUTORIAL BANK Name : SIGNALS AND SYSTEMS Code : A30406 Class : II B. Tech I Semester
More informationSimon Fraser University School of Engineering Science ENSC Linear Systems Spring Instructor Jim Cavers ASB
Simon Fraser University School of Engineering Science ENSC 380-3 Linear Systems Spring 2000 This course covers the modeling and analysis of continuous and discrete signals and systems using linear techniques.
More information(i) Represent discrete-time signals using transform. (ii) Understand the relationship between transform and discrete-time Fourier transform
z Transform Chapter Intended Learning Outcomes: (i) Represent discrete-time signals using transform (ii) Understand the relationship between transform and discrete-time Fourier transform (iii) Understand
More informationFlash File. Module 3 : Sampling and Reconstruction Lecture 28 : Discrete time Fourier transform and its Properties. Objectives: Scope of this Lecture:
Module 3 : Sampling and Reconstruction Lecture 28 : Discrete time Fourier transform and its Properties Objectives: Scope of this Lecture: In the previous lecture we defined digital signal processing and
More informationCh.11 The Discrete-Time Fourier Transform (DTFT)
EE2S11 Signals and Systems, part 2 Ch.11 The Discrete-Time Fourier Transform (DTFT Contents definition of the DTFT relation to the -transform, region of convergence, stability frequency plots convolution
More informationModule 1: Signals & System
Module 1: Signals & System Lecture 6: Basic Signals in Detail Basic Signals in detail We now introduce formally some of the basic signals namely 1) The Unit Impulse function. 2) The Unit Step function
More informationLECTURE NOTES ON SIGNALS AND SYSTEMS (15A04303) II B.TECH I SEMESTER ECE (JNTUA R15)
LECTURE NOTES ON SIGNALS AND SYSTEMS (15A04303) II B.TECH I SEMESTER ECE (JNTUA R15) PREPARED BY MS. G. DILLIRANI M.TECH, M.I.S.T.E, (PH.D) ASSISTANT PROFESSOR DEPARTMENT OF ELECTRONICS AND COMMUNICATION
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 informationHow to manipulate Frequencies in Discrete-time Domain? Two Main Approaches
How to manipulate Frequencies in Discrete-time Domain? Two Main Approaches Difference Equations (an LTI system) x[n]: input, y[n]: output That is, building a system that maes use of the current and previous
More informationDiscrete Time Fourier Transform (DTFT)
Discrete Time Fourier Transform (DTFT) 1 Discrete Time Fourier Transform (DTFT) The DTFT is the Fourier transform of choice for analyzing infinite-length signals and systems Useful for conceptual, pencil-and-paper
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 information1 Mathematical Preliminaries
Mathematical Preliminaries We shall go through in this first chapter all of the mathematics needed for reading the rest of this book. The reader is expected to have taken a one year course in differential
More informationCommunication 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 informationProperties of Fourier Series - GATE Study Material in PDF
Properties of Fourier Series - GAE Study Material in PDF In the previous article, we learnt the Basics of Fourier Series, the different types and all about the different Fourier Series spectrums. Now,
More informationCommunication Theory II
Communication Theory II Lecture 4: Review on Fourier analysis and probabilty theory Ahmed Elnakib, PhD Assistant Professor, Mansoura University, Egypt Febraury 19 th, 2015 1 Course Website o http://lms.mans.edu.eg/eng/
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 information6.003: Signal Processing
6.003: Signal Processing Discrete Fourier Transform Discrete Fourier Transform (DFT) Relations to Discrete-Time Fourier Transform (DTFT) Relations to Discrete-Time Fourier Series (DTFS) October 16, 2018
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 informationChapter 3 Fourier Representations of Signals and Linear Time-Invariant Systems
Chapter 3 Fourier Representations of Signals and Linear Time-Invariant Systems Introduction Complex Sinusoids and Frequency Response of LTI Systems. Fourier Representations for Four Classes of Signals
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 informationLinear Systems. ! Textbook: Strum, Contemporary Linear Systems using MATLAB.
Linear Systems LS 1! Textbook: Strum, Contemporary Linear Systems using MATLAB.! Contents 1. Basic Concepts 2. Continuous Systems a. Laplace Transforms and Applications b. Frequency Response of Continuous
More informationSystems & Signals 315
1 / 15 Systems & Signals 315 Lecture 13: Signals and Linear Systems Dr. Herman A. Engelbrecht Stellenbosch University Dept. E & E Engineering 2 Maart 2009 Outline 2 / 15 1 Signal Transmission through a
More informationSYLLABUS. osmania university CHAPTER - 1 : TRANSIENT RESPONSE CHAPTER - 2 : LAPLACE TRANSFORM OF SIGNALS
i SYLLABUS osmania university UNIT - I CHAPTER - 1 : TRANSIENT RESPONSE Initial Conditions in Zero-Input Response of RC, RL and RLC Networks, Definitions of Unit Impulse, Unit Step and Ramp Functions,
More informationE2.5 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 informationThe Discrete-Time Fourier
Chapter 3 The Discrete-Time Fourier Transform 清大電機系林嘉文 cwlin@ee.nthu.edu.tw 03-5731152 Original PowerPoint slides prepared by S. K. Mitra 3-1-1 Continuous-Time Fourier Transform Definition The CTFT of
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 informationDigital Signal Processing Lecture 3 - Discrete-Time Systems
Digital Signal Processing - Discrete-Time Systems Electrical Engineering and Computer Science University of Tennessee, Knoxville August 25, 2015 Overview 1 2 3 4 5 6 7 8 Introduction Three components of
More information信號與系統 Signals and Systems
Spring 2013 Flowchart Introduction (Chap 1) LTI & Convolution (Chap 2) NTUEE-SS10-Z-2 信號與系統 Signals and Systems Chapter SS-10 The z-transform FS (Chap 3) Periodic Bounded/Convergent CT DT FT Aperiodic
More informationso mathematically we can say that x d [n] is a discrete-time signal. The output of the DT system is also discrete, denoted by y d [n].
ELEC 36 LECURE NOES WEEK 9: Chapters 7&9 Chapter 7 (cont d) Discrete-ime Processing of Continuous-ime Signals It is often advantageous to convert a continuous-time signal into a discrete-time signal so
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 informationContinuous-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/ (2π) X(e jω ) dω. 4. An 8 point sequence is given by x(n) = {2,2,2,2,1,1,1,1}. Compute 8 point DFT of x(n) by
Code No: RR320402 Set No. 1 III B.Tech II Semester Regular Examinations, Apr/May 2006 DIGITAL SIGNAL PROCESSING ( Common to Electronics & Communication Engineering, Electronics & Instrumentation Engineering,
More informationRepresentation 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 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 informationDiscrete-time Fourier transform (DTFT) representation of DT aperiodic signals Section The (DT) Fourier transform (or spectrum) of x[n] is
Discrete-time Fourier transform (DTFT) representation of DT aperiodic signals Section 5. 3 The (DT) Fourier transform (or spectrum) of x[n] is X ( e jω) = n= x[n]e jωn x[n] can be reconstructed from its
More informationHomework 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 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 informationIntroduction to DFT. Deployment of Telecommunication Infrastructures. Azadeh Faridi DTIC UPF, Spring 2009
Introduction to DFT Deployment of Telecommunication Infrastructures Azadeh Faridi DTIC UPF, Spring 2009 1 Review of Fourier Transform Many signals can be represented by a fourier integral of the following
More informationCHAPTER 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 informationTime Domain Analysis of Linear Systems Ch2. University of Central Oklahoma Dr. Mohamed Bingabr
Time Domain Analysis of Linear Systems Ch2 University of Central Oklahoma Dr. Mohamed Bingabr Outline Zero-input Response Impulse Response h(t) Convolution Zero-State Response System Stability System Response
More informationDEPARTMENT 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 informationChapter 6: The Laplace Transform. Chih-Wei Liu
Chapter 6: The Laplace Transform Chih-Wei Liu Outline Introduction The Laplace Transform The Unilateral Laplace Transform Properties of the Unilateral Laplace Transform Inversion of the Unilateral Laplace
More informationA system that is both linear and time-invariant is called linear time-invariant (LTI).
The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Lecture Notes: Time, Frequency & Transform Domains February 28, 2012 Signals & Systems Signals are mapped
More informationSpatial-Domain Convolution Filters
Spatial-Domain Filtering 9 Spatial-Domain Convolution Filters Consider a linear space-invariant (LSI) system as shown: The two separate inputs to the LSI system, x 1 (m) and x 2 (m), and their corresponding
More informationDSP-I DSP-I DSP-I DSP-I
NOTES FOR 8-79 LECTURES 3 and 4 Introduction to Discrete-Time Fourier Transforms (DTFTs Distributed: September 8, 2005 Notes: This handout contains in brief outline form the lecture notes used for 8-79
More informationDiscrete-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