信號與系統 Signals and Systems

Size: px
Start display at page:

Download "信號與系統 Signals and Systems"

Transcription

1 Spring 2010 信號與系統 Signals and Systems Chapter SS-2 Linear Time-Invariant Systems Feng-Li Lian NTU-EE Feb10 Jun10 Figures and images used in these lecture notes are adopted from Signals & Systems by Alan V. Oppenheim and Alan S. Willsky, 1997

2 Outline NTUEE-SS2-LTI-2 Discrete-Time Linear Time-Invariant Systems The convolution sum Continuous-Time Linear Time-Invariant Systems The convolution integral Properties of Linear Time-Invariant Systems Causal Linear Time-Invariant Systems Described by Differential & Difference Equations Singularity Functions h[n] h(t)

3 Chapter 1 and Chapter 2 NTUEE-SS2-LTI-3 h[n] h(t) Signals Systems

4 In Section 1.5, We Introduced Unit Impulse Functions NTUEE-SS2-LTI-4 Sample by Unit Impulse For x[n] More generally,

5 DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-5 Representation of DT Signals by Impulses:

6 DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-6 Representation of DT Signals by Impulses: More generally, The sifting property of the DT unit impulse x[n] = a superposition of scaled versions of shifted unit impulses [n-k]

7 DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-7 DT Unit Impulse Response & Convolution Sum: Linear System Linear System Linear System Linear System Linear System

8 DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-8 DT Unit Impulse Response & Convolution Sum: Linear System Linear System Linear System Linear System Linear System

9 DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-9

10 DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-10 Linear System

11 DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-11 If the linear system (L) is also time-invariant (TI) Then, Hence, for an LTI system, Known as the convolution of x[n] & h[n] Referred as the convolution sum or superposition sum Symbolically,

12 DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-12 Example 2.1:

13 DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-13 Example 2.2:

14 DT LTI Systems: Convolution Sum Example 2.2: NTUEE-SS2-LTI-14

15 DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-15 Example 2.1:

16 DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-16 Example 2.2:

17 DT LTI Systems: Convolution Sum Example 2.2: NTUEE-SS2-LTI-17

18 DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-18 Example 2.2:

19 DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-19 Example 2.3:

20 DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-20 Example 2.3:

21 DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-21 Example 2.3:

22 DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-22 Example 2.4:

23 DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-23

24 DT LTI Systems: Convolution Sum NTUEE-SS2-LTI-24

25 DT LTI Systems: Convolution Sum Example 2.5: NTUEE-SS2-LTI-25

26 Outline NTUEE-SS2-LTI-26 Discrete-Time Linear Time-Invariant Systems The convolution sum Continuous-Time Linear Time-Invariant Systems The convolution integral Properties of Linear Time-Invariant Systems Causal Linear Time-Invariant Systems Described by Differential & Difference Equations Singularity Functions

27 CT LTI Systems: Convolution Integral Representation of CT Signals by Impulses: NTUEE-SS2-LTI-27

28 CT LTI Systems: Convolution Integral NTUEE-SS2-LTI-28 Representation of CT Signals by Impulses:

29 CT LTI Systems: Convolution Integral NTUEE-SS2-LTI-29 Graphical interpretation:

30 CT LTI Systems: Convolution Integral NTUEE-SS2-LTI-30 CT Impulse Response & Convolution Integral: Linear System Linear System Linear System Linear System Linear System

31 CT LTI Systems: Convolution Integral NTUEE-SS2-LTI-31 CT Impulse Response & Convolution Integral: Linear System Linear System Linear System Linear System Linear System

32 CT LTI Systems: Convolution Integral NTUEE-SS2-LTI-32 Linear System

33 CT LTI Systems: Convolution Integral NTUEE-SS2-LTI-33 CT Unit Impulse Response & Convolution Integral: Linear System Linear System

34 CT LTI Systems: Convolution Integral NTUEE-SS2-LTI-34 If the linear system (L) is also time-invariant (TI) Then, Hence, for an LTI system, Known as the convolution of x(t) & h(t) Referred as the convolution integral or the superposition integral Symbolically,

35 CT LTI Systems: Convolution Integral NTUEE-SS2-LTI-35 Example 2.6:

36 CT LTI Systems: Convolution Integral NTUEE-SS2-LTI-36 Example 2.7:

37 CT LTI Systems: Convolution Integral NTUEE-SS2-LTI-37 Example 2.8:

38 Convolution Sum and Integral NTUEE-SS2-LTI-38 Signal and System: h(t)

39 Outline NTUEE-SS2-LTI-39 Discrete-Time Linear Time-Invariant Systems The convolution sum Continuous-Time Linear Time-Invariant Systems The convolution integral Properties of Linear Time-Invariant Systems Causal Linear Time-Invariant Systems Described by Differential & Difference Equations Singularity Functions

40 Properties of LTI Systems NTUEE-SS2-LTI-40 Convolution Sum & Integral of LTI Systems: h[n] h(t)

41 Properties of LTI Systems NTUEE-SS2-LTI-41 Properties of LTI Systems 1. Commutative property 2. Distributive property 3. Associative property 4. With or without memory 5. Invertibility 6. Causality h[n] h(t) 7. Stability 8. Unit step response

42 Properties of LTI Systems NTUEE-SS2-LTI-42 Commutative Property:

43 Properties of LTI Systems NTUEE-SS2-LTI-43 Distributive Property: h 1 [n] h 1 [n]+h 2 [n] + h 2 [n]

44 Properties of LTI Systems NTUEE-SS2-LTI-44 Distributive Property: h[n] + h[n] + h[n]

45 Properties of LTI Systems NTUEE-SS2-LTI-45 Example 2.10

46 Properties of LTI Systems NTUEE-SS2-LTI-46 Associative Property:

47 In Section 1.6.1: Basic System Properties NTUEE-SS2-LTI-47 Systems with or without memory Memoryless systems Output depends only on the input at that same time (resistor) Systems with memory (accumulator) (delay)

48 Properties of LTI Systems NTUEE-SS2-LTI-48 Memoryless: A DT LTI system is memoryless if The impulse response: The convolution sum: Similarly, for CT LTI system:

49 In Section 1.6.2: Basic System Properties NTUEE-SS2-LTI-49 Invertibility & Inverse Systems Invertible systems Distinct inputs lead to distinct outputs

50 Properties of LTI Systems NTUEE-SS2-LTI-50 Invertibility: h1(t) h2(t) Identity System (t)

51 Properties of LTI Systems NTUEE-SS2-LTI-51 Example 2.11: Pure time shift h1(t) h2(t)

52 Properties of LTI Systems NTUEE-SS2-LTI-52 Example 2.12 h1[n] h2[n]

53 Properties of LTI Systems Causality: NTUEE-SS2-LTI-53 The output of a causal system depends only on the present and past values of the input to the system Specifically, y[n] must not depend on x[k], for k > n It implies that the system is initially rest A CT LTI system is causal if

54 Properties of LTI Systems NTUEE-SS2-LTI-54 Convolution Sum & Integral

55 In Section 1.6.4: Basic System Properties NTUEE-SS2-LTI-55 Stability Stable systems Small inputs lead to responses that do not diverge Every bounded input excites a bounded output Bounded-input bounded-output stable (BIBO stable) For all x(t) < a, then y(t) < b, for all t Balance in a bank account?

56 Properties of LTI Systems NTUEE-SS2-LTI-56 Stability: A system is stable if every bounded input produces a bounded output Stable LTI

57 Properties of LTI Systems NTUEE-SS2-LTI-57 Stability: For CT LTI stable system: Stable LTI

58 Properties of LTI Systems NTUEE-SS2-LTI-58 Example 2.13: Pure time shift

59 Properties of LTI Systems NTUEE-SS2-LTI-59 Example 2.13: Accumulator

60 Properties of LTI Systems NTUEE-SS2-LTI-60 Unit Step Response: For an LTI system, its impulse response is: DT LTI CT LTI Its unit step response is: DT LTI CT LTI

61 Outline NTUEE-SS2-LTI-61 Discrete-Time Linear Time-Invariant Systems The convolution sum Continuous-Time Linear Time-Invariant Systems The convolution integral Properties of Linear Time-Invariant Systems 1. Commutative property 2. Distributive property 3. Associative property 4. With or without memory 5. Invertibility 6. Causality 7. Stability 8. Unit step response Causal Linear Time-Invariant Systems Described by Differential & Difference Equations Singularity Functions

62 Singularity Functions NTUEE-SS2-LTI-62 Singularity Functions CT unit impulse function is one of singularity functions

63 Singularity Functions NTUEE-SS2-LTI-63 Singularity Functions

64 Singularity Functions NTUEE-SS2-LTI-64 Example 2.16

65 Singularity Functions NTUEE-SS2-LTI-65 Example 2.16

66 Singularity Functions NTUEE-SS2-LTI-66 Defining the Unit Impulse through Convolution:

67 Singularity Functions NTUEE-SS2-LTI-67 Defining Unit Impulse through Convolution:

68 Singularity Functions NTUEE-SS2-LTI-68 Defining Unit Impulse through Convolution:

69 Singularity Functions NTUEE-SS2-LTI-69 Unit Doublets of Derivative Operation:

70 Singularity Functions NTUEE-SS2-LTI-70 Unit Doublets of Derivative Operation:

71 Singularity Functions NTUEE-SS2-LTI-71 Unit Doublets of Integration Operation:

72 Singularity Functions NTUEE-SS2-LTI-72 Unit Doublets of Integration Operation:

73 Singularity Functions NTUEE-SS2-LTI-73 Unit Doublets of Integration Operation:

74 Singularity Functions NTUEE-SS2-LTI-74 In Summary

75 Outline NTUEE-SS2-LTI-75 Discrete-Time Linear Time-Invariant Systems The convolution sum Continuous-Time Linear Time-Invariant Systems The convolution integral Properties of Linear Time-Invariant Systems 1. Commutative property 2. Distributive property 3. Associative property 4. With or without memory 5. Invertibility 6. Causality 7. Stability 8. Unit step response Causal Linear Time-Invariant Systems Described by Differential & Difference Equations Singularity Functions

76 Causal LTI Systems by Difference & Differential Equations NTUEE-SS2-LTI-76 Linear Constant-Coefficient Differential Equations e.x., RC circuit RC Circuit Provide an implicit specification of the system You have learned how to solve the equation in Diff Eqn

77 Causal LTI Systems by Difference & Differential Equations NTUEE-SS2-LTI-77 Linear Constant-Coefficient Differential Equations For a general CT LTI system, with N-th order, CT LTI

78 Causal LTI Systems by Difference & Differential Equations NTUEE-SS2-LTI-78 Linear Constant-Coefficient Difference Equations For a general DT LTI system, with N-th order, DT LTI

79 Causal LTI Systems by Difference & Differential Equations NTUEE-SS2-LTI-79 Recursive Equation:

80 Causal LTI Systems by Difference & Differential Equations Recursive Equation: For example, (Example 2.15) LTI NTUEE-SS2-LTI-80

81 Causal LTI Systems by Difference & Differential Equations NTUEE-SS2-LTI-81 Nonrecursive Equation: When N = 0,

82 Causal LTI Systems by Difference & Differential Equations NTUEE-SS2-LTI-82 Block Diagram Representations:

83 Causal LTI Systems by Difference & Differential Equations NTUEE-SS2-LTI-83 Block Diagram Representations:

84 Causal LTI Systems by Difference & Differential Equations NTUEE-SS2-LTI-84 Block Diagram Representations:

85 Causal LTI Systems by Difference & Differential Equations NTUEE-SS2-LTI-85 Block Diagram Representations: Example 9.30 (pp.711)

86 Causal LTI Systems by Difference & Differential Equations NTUEE-SS2-LTI-86 Block Diagram Representations: Example (pp.786)

87 Chapter 2: Linear Time-Invariant Systems Discrete-Time Linear Time-Invariant Systems The convolution sum Continuous-Time Linear Time-Invariant Systems The convolution integral NTUEE-SS2-LTI-87 Properties of Linear Time-Invariant Systems 1. Commutative property 2. Distributive property 3. Associative property 4. With or without memory 5. Invertibility 6. Causality 7. Stability 8. Unit step response Causal Linear Time-Invariant Systems Described by Differential & Difference Equations Singularity Functions

88 Flowchart Signals & Systems (Chap 1) LTI & Convolution (Chap 2) NTUEE-SS2-LTI-88 Bounded/Convergent Periodic Aperiodic FS (Chap 3) CT DT FT CT (Chap 4) DT (Chap 5) Unbounded/Non-convergent LT zt CT DT (Chap 9) (Chap 10) Time-Frequency (Chap 6) Communication (Chap 8) CT-DT (Chap 7) Control (Chap 11)

信號與系統 Signals and Systems

信號與系統 Signals and Systems Spring 2015 信號與系統 Signals and Systems Chapter SS-2 Linear Time-Invariant Systems Feng-Li Lian NTU-EE Feb15 Jun15 Figures and images used in these lecture notes are adopted from Signals & Systems by Alan

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

信號與系統 Signals and Systems

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 information

Ch 2: Linear Time-Invariant System

Ch 2: Linear Time-Invariant System Ch 2: Linear Time-Invariant System A system is said to be Linear Time-Invariant (LTI) if it possesses the basic system properties of linearity and time-invariance. Consider a system with an output signal

More information

University Question Paper Solution

University 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 information

NAME: 23 February 2017 EE301 Signals and Systems Exam 1 Cover Sheet

NAME: 23 February 2017 EE301 Signals and Systems Exam 1 Cover Sheet NAME: 23 February 2017 EE301 Signals and Systems Exam 1 Cover Sheet Test Duration: 75 minutes Coverage: Chaps 1,2 Open Book but Closed Notes One 85 in x 11 in crib sheet Calculators NOT allowed DO NOT

More information

Chap 2. Discrete-Time Signals and Systems

Chap 2. Discrete-Time Signals and Systems Digital Signal Processing Chap 2. Discrete-Time Signals and Systems Chang-Su Kim Discrete-Time Signals CT Signal DT Signal Representation 0 4 1 1 1 2 3 Functional representation 1, n 1,3 x[ n] 4, n 2 0,

More information

ELEG 305: Digital Signal Processing

ELEG 305: Digital Signal Processing ELEG 305: Digital Signal Processing Lecture 1: Course Overview; Discrete-Time Signals & Systems Kenneth E. Barner Department of Electrical and Computer Engineering University of Delaware Fall 2008 K. E.

More information

EE 341 Homework Chapter 2

EE 341 Homework Chapter 2 EE 341 Homework Chapter 2 2.1 The electrical circuit shown in Fig. P2.1 consists of two resistors R1 and R2 and a capacitor C. Determine the differential equation relating the input voltage v(t) to the

More information

Rui Wang, Assistant professor Dept. of Information and Communication Tongji University.

Rui 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 information

Cosc 3451 Signals and Systems. What is a system? Systems Terminology and Properties of Systems

Cosc 3451 Signals and Systems. What is a system? Systems Terminology and Properties of Systems Cosc 3451 Signals and Systems Systems Terminology and Properties of Systems What is a system? an entity that manipulates one or more signals to yield new signals (often to accomplish a function) can be

More information

Solution 10 July 2015 ECE301 Signals and Systems: Midterm. Cover Sheet

Solution 10 July 2015 ECE301 Signals and Systems: Midterm. Cover Sheet Solution 10 July 2015 ECE301 Signals and Systems: Midterm Cover Sheet Test Duration: 60 minutes Coverage: Chap. 1,2,3,4 One 8.5" x 11" crib sheet is allowed. Calculators, textbooks, notes are not allowed.

More information

New Mexico State University Klipsch School of Electrical Engineering. EE312 - Signals and Systems I Spring 2018 Exam #1

New Mexico State University Klipsch School of Electrical Engineering. EE312 - Signals and Systems I Spring 2018 Exam #1 New Mexico State University Klipsch School of Electrical Engineering EE312 - Signals and Systems I Spring 2018 Exam #1 Name: Prob. 1 Prob. 2 Prob. 3 Prob. 4 Total / 30 points / 20 points / 25 points /

More information

EE 210. Signals and Systems Solutions of homework 2

EE 210. Signals and Systems Solutions of homework 2 EE 2. Signals and Systems Solutions of homework 2 Spring 2 Exercise Due Date Week of 22 nd Feb. Problems Q Compute and sketch the output y[n] of each discrete-time LTI system below with impulse response

More information

Digital Signal Processing Lecture 4

Digital Signal Processing Lecture 4 Remote Sensing Laboratory Dept. of Information Engineering and Computer Science University of Trento Via Sommarive, 14, I-38123 Povo, Trento, Italy Digital Signal Processing Lecture 4 Begüm Demir E-mail:

More information

Chapter 2 Time-Domain Representations of LTI Systems

Chapter 2 Time-Domain Representations of LTI Systems Chapter 2 Time-Domain Representations of LTI Systems 1 Introduction Impulse responses of LTI systems Linear constant-coefficients differential or difference equations of LTI systems Block diagram representations

More information

Lecture 2 Discrete-Time LTI Systems: Introduction

Lecture 2 Discrete-Time LTI Systems: Introduction Lecture 2 Discrete-Time LTI Systems: Introduction Outline 2.1 Classification of Systems.............................. 1 2.1.1 Memoryless................................. 1 2.1.2 Causal....................................

More information

EE361: Signals and System II

EE361: Signals and System II Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu EE361: Signals and System II Introduction http://www.ee.unlv.edu/~b1morris/ee361/ 2 Class Website http://www.ee.unlv.edu/~b1morris/ee361/ This

More information

2. CONVOLUTION. Convolution sum. Response of d.t. LTI systems at a certain input signal

2. CONVOLUTION. Convolution sum. Response of d.t. LTI systems at a certain input signal 2. CONVOLUTION Convolution sum. Response of d.t. LTI systems at a certain input signal Any signal multiplied by the unit impulse = the unit impulse weighted by the value of the signal in 0: xn [ ] δ [

More information

Interconnection of LTI Systems

Interconnection of LTI Systems EENG226 Signals and Systems Chapter 2 Time-Domain Representations of Linear Time-Invariant Systems Interconnection of LTI Systems Prof. Dr. Hasan AMCA Electrical and Electronic Engineering Department (ee.emu.edu.tr)

More information

Properties of LTI Systems

Properties of LTI Systems Properties of LTI Systems Properties of Continuous Time LTI Systems Systems with or without memory: A system is memory less if its output at any time depends only on the value of the input at that same

More information

Module 1: Signals & System

Module 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 information

Signals and Systems Chapter 2

Signals and Systems Chapter 2 Signals and Systems Chapter 2 Continuous-Time Systems Prof. Yasser Mostafa Kadah Overview of Chapter 2 Systems and their classification Linear time-invariant systems System Concept Mathematical transformation

More information

Chapter 3 Convolution Representation

Chapter 3 Convolution Representation Chapter 3 Convolution Representation DT Unit-Impulse Response Consider the DT SISO system: xn [ ] System yn [ ] xn [ ] = δ[ n] If the input signal is and the system has no energy at n = 0, the output yn

More information

Differential and Difference LTI systems

Differential and Difference LTI systems Signals and Systems Lecture: 6 Differential and Difference LTI systems Differential and difference linear time-invariant (LTI) systems constitute an extremely important class of systems in engineering.

More information

New Mexico State University Klipsch School of Electrical Engineering. EE312 - Signals and Systems I Fall 2017 Exam #1

New Mexico State University Klipsch School of Electrical Engineering. EE312 - Signals and Systems I Fall 2017 Exam #1 New Mexico State University Klipsch School of Electrical Engineering EE312 - Signals and Systems I Fall 2017 Exam #1 Name: Prob. 1 Prob. 2 Prob. 3 Prob. 4 Total / 30 points / 20 points / 25 points / 25

More information

NAME: 20 February 2014 EE301 Signals and Systems Exam 1 Cover Sheet

NAME: 20 February 2014 EE301 Signals and Systems Exam 1 Cover Sheet NAME: February 4 EE Signals and Systems Exam Cover Sheet Test Duration: 75 minutes. Coverage: Chaps., Open Book but Closed Notes. One 8.5 in. x in. crib sheet Calculators NOT allowed. This test contains

More information

NAME: 13 February 2013 EE301 Signals and Systems Exam 1 Cover Sheet

NAME: 13 February 2013 EE301 Signals and Systems Exam 1 Cover Sheet NAME: February EE Signals and Systems Exam Cover Sheet Test Duration: 75 minutes. Coverage: Chaps., Open Book but Closed Notes. One 8.5 in. x in. crib sheet Calculators NOT allowed. This test contains

More information

Lecture 2. Introduction to Systems (Lathi )

Lecture 2. Introduction to Systems (Lathi ) Lecture 2 Introduction to Systems (Lathi 1.6-1.8) Pier Luigi Dragotti Department of Electrical & Electronic Engineering Imperial College London URL: www.commsp.ee.ic.ac.uk/~pld/teaching/ E-mail: p.dragotti@imperial.ac.uk

More information

DIGITAL SIGNAL PROCESSING LECTURE 1

DIGITAL SIGNAL PROCESSING LECTURE 1 DIGITAL SIGNAL PROCESSING LECTURE 1 Fall 2010 2K8-5 th Semester Tahir Muhammad tmuhammad_07@yahoo.com Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, 1999-2000

More information

Lecture 1: Introduction Introduction

Lecture 1: Introduction Introduction Module 1: Signals in Natural Domain Lecture 1: Introduction Introduction The intent of this introduction is to give the reader an idea about Signals and Systems as a field of study and its applications.

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

Basic concepts in DT systems. Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 1

Basic concepts in DT systems. Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 1 Basic concepts in DT systems Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 4 1 Readings and homework For DT systems: Textbook: sections 1.5, 1.6 Suggested homework: pp. 57-58: 1.15 1.16 1.18 1.19

More information

06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 1

06/12/ rws/jMc- modif SuFY10 (MPF) - Textbook Section IX 1 IV. Continuous-Time Signals & LTI Systems [p. 3] Analog signal definition [p. 4] Periodic signal [p. 5] One-sided signal [p. 6] Finite length signal [p. 7] Impulse function [p. 9] Sampling property [p.11]

More information

The Convolution Sum for Discrete-Time LTI Systems

The Convolution Sum for Discrete-Time LTI Systems The Convolution Sum for Discrete-Time LTI Systems Andrew W. H. House 01 June 004 1 The Basics of the Convolution Sum Consider a DT LTI system, L. x(n) L y(n) DT convolution is based on an earlier result

More information

UNIT 1. SIGNALS AND SYSTEM

UNIT 1. SIGNALS AND SYSTEM Page no: 1 UNIT 1. SIGNALS AND SYSTEM INTRODUCTION A SIGNAL is defined as any physical quantity that changes with time, distance, speed, position, pressure, temperature or some other quantity. A SIGNAL

More information

III. Time Domain Analysis of systems

III. Time Domain Analysis of systems 1 III. Time Domain Analysis of systems Here, we adapt properties of continuous time systems to discrete time systems Section 2.2-2.5, pp 17-39 System Notation y(n) = T[ x(n) ] A. Types of Systems Memoryless

More information

Digital Signal Processing Lecture 5

Digital Signal Processing Lecture 5 Remote Sensing Laboratory Dept. of Information Engineering and Computer Science University of Trento Via Sommarive, 14, I-38123 Povo, Trento, Italy Digital Signal Processing Lecture 5 Begüm Demir E-mail:

More information

EE123 Digital Signal Processing

EE123 Digital Signal Processing EE123 Digital Signal Processing Lecture 2 Discrete Time Systems Today Last time: Administration Overview Announcement: HW1 will be out today Lab 0 out webcast out Today: Ch. 2 - Discrete-Time Signals and

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

The objective of this LabVIEW Mini Project was to understand the following concepts:

The objective of this LabVIEW Mini Project was to understand the following concepts: 1. Objective The objective of this LabVIEW Mini Project was to understand the following concepts: The convolution of two functions Creating LABVIEW Virtual Instruments see the visual representation of

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

QUESTION BANK SIGNALS AND SYSTEMS (4 th SEM ECE)

QUESTION 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 information

Fall 線性系統 Linear Systems. Chapter 08 State Feedback & State Estimators (SISO) Feng-Li Lian. NTU-EE Sep07 Jan08

Fall 線性系統 Linear Systems. Chapter 08 State Feedback & State Estimators (SISO) Feng-Li Lian. NTU-EE Sep07 Jan08 Fall 2007 線性系統 Linear Systems Chapter 08 State Feedback & State Estimators (SISO) Feng-Li Lian NTU-EE Sep07 Jan08 Materials used in these lecture notes are adopted from Linear System Theory & Design, 3rd.

More information

Module 4. Related web links and videos. 1. FT and ZT

Module 4. Related web links and videos. 1.  FT and ZT Module 4 Laplace transforms, ROC, rational systems, Z transform, properties of LT and ZT, rational functions, system properties from ROC, inverse transforms Related web links and videos Sl no Web link

More information

Analog vs. discrete signals

Analog vs. discrete signals Analog vs. discrete signals Continuous-time signals are also known as analog signals because their amplitude is analogous (i.e., proportional) to the physical quantity they represent. Discrete-time signals

More information

New Mexico State University Klipsch School of Electrical Engineering EE312 - Signals and Systems I Fall 2015 Final Exam

New Mexico State University Klipsch School of Electrical Engineering EE312 - Signals and Systems I Fall 2015 Final Exam New Mexico State University Klipsch School of Electrical Engineering EE312 - Signals and Systems I Fall 2015 Name: Solve problems 1 3 and two from problems 4 7. Circle below which two of problems 4 7 you

More information

GATE EE Topic wise Questions SIGNALS & SYSTEMS

GATE 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 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

1.4 Unit Step & Unit Impulse Functions

1.4 Unit Step & Unit Impulse Functions 1.4 Unit Step & Unit Impulse Functions 1.4.1 The Discrete-Time Unit Impulse and Unit-Step Sequences Unit Impulse Function: δ n = ቊ 0, 1, n 0 n = 0 Figure 1.28: Discrete-time Unit Impulse (sample) 1 [n]

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

VU Signal and Image Processing

VU Signal and Image Processing 052600 VU Signal and Image Processing Torsten Möller + Hrvoje Bogunović + Raphael Sahann torsten.moeller@univie.ac.at hrvoje.bogunovic@meduniwien.ac.at raphael.sahann@univie.ac.at vda.cs.univie.ac.at/teaching/sip/18s/

More information

ECE 301 Division 1 Exam 1 Solutions, 10/6/2011, 8-9:45pm in ME 1061.

ECE 301 Division 1 Exam 1 Solutions, 10/6/2011, 8-9:45pm in ME 1061. ECE 301 Division 1 Exam 1 Solutions, 10/6/011, 8-9:45pm in ME 1061. Your ID will be checked during the exam. Please bring a No. pencil to fill out the answer sheet. This is a closed-book exam. No calculators

More information

Therefore the new Fourier coefficients are. Module 2 : Signals in Frequency Domain Problem Set 2. Problem 1

Therefore 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 information

Lecture 19 IIR Filters

Lecture 19 IIR Filters Lecture 19 IIR Filters Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/5/10 1 General IIR Difference Equation IIR system: infinite-impulse response system The most general class

More information

DIGITAL SIGNAL PROCESSING UNIT 1 SIGNALS AND SYSTEMS 1. What is a continuous and discrete time signal? Continuous time signal: A signal x(t) is said to be continuous if it is defined for all time t. Continuous

More information

Classification of Discrete-Time Systems. System Properties. Terminology: Implication. Terminology: Equivalence

Classification of Discrete-Time Systems. System Properties. Terminology: Implication. Terminology: Equivalence Classification of Discrete-Time Systems Professor Deepa Kundur University of Toronto Why is this so important? mathematical techniques developed to analyze systems are often contingent upon the general

More information

Lecture 11 FIR Filters

Lecture 11 FIR Filters Lecture 11 FIR Filters Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/4/12 1 The Unit Impulse Sequence Any sequence can be represented in this way. The equation is true if k ranges

More information

Chapter 1 Fundamental Concepts

Chapter 1 Fundamental Concepts Chapter 1 Fundamental Concepts 1 Signals A signal is a pattern of variation of a physical quantity, often as a function of time (but also space, distance, position, etc). These quantities are usually the

More information

EEL3135: Homework #4

EEL3135: Homework #4 EEL335: Homework #4 Problem : For each of the systems below, determine whether or not the system is () linear, () time-invariant, and (3) causal: (a) (b) (c) xn [ ] cos( 04πn) (d) xn [ ] xn [ ] xn [ 5]

More information

Digital Filters Ying Sun

Digital Filters Ying Sun Digital Filters Ying Sun Digital filters Finite impulse response (FIR filter: h[n] has a finite numbers of terms. Infinite impulse response (IIR filter: h[n] has infinite numbers of terms. Causal filter:

More information

Chap 4. State-Space Solutions and

Chap 4. State-Space Solutions and Chap 4. State-Space Solutions and Realizations Outlines 1. Introduction 2. Solution of LTI State Equation 3. Equivalent State Equations 4. Realizations 5. Solution of Linear Time-Varying (LTV) Equations

More information

Solution 7 August 2015 ECE301 Signals and Systems: Final Exam. Cover Sheet

Solution 7 August 2015 ECE301 Signals and Systems: Final Exam. Cover Sheet Solution 7 August 2015 ECE301 Signals and Systems: Final Exam Cover Sheet Test Duration: 120 minutes Coverage: Chap. 1, 2, 3, 4, 5, 7 One 8.5" x 11" crib sheet is allowed. Calculators, textbooks, notes

More information

Linear Systems. ! Textbook: Strum, Contemporary Linear Systems using MATLAB.

Linear 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 information

Material presented here is from the course 6.003, Signals & Systems offered by MIT faculty member, Prof. Alan Willsky, Copyright c 2003.

Material presented here is from the course 6.003, Signals & Systems offered by MIT faculty member, Prof. Alan Willsky, Copyright c 2003. EE-295 Image Processing, Spring 2008 Lecture 1 Material presented here is from the course 6.003, Signals & Systems offered by MIT faculty member, Prof. Alan Willsky, Copyright c 2003. This material is

More information

Lecture Schedule: Week Date Lecture Title

Lecture Schedule: Week Date Lecture Title http://elec34.org Sampling and CONVOLUTION 24 School of Information Technology and Electrical Engineering at The University of Queensland Lecture Schedule: Week Date Lecture Title 2-Mar Introduction 3-Mar

More information

Introduction 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 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 information

Question Paper Code : AEC11T02

Question 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 information

Linear Filters and Convolution. Ahmed Ashraf

Linear Filters and Convolution. Ahmed Ashraf Linear Filters and Convolution Ahmed Ashraf Linear Time(Shift) Invariant (LTI) Systems The Linear Filters that we are studying in the course belong to a class of systems known as Linear Time Invariant

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

2 Classification of Continuous-Time Systems

2 Classification of Continuous-Time Systems Continuous-Time Signals and Systems 1 Preliminaries Notation for a continuous-time signal: x(t) Notation: If x is the input to a system T and y the corresponding output, then we use one of the following

More information

2.161 Signal Processing: Continuous and Discrete Fall 2008

2.161 Signal Processing: Continuous and Discrete Fall 2008 MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Massachusetts

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

Modeling and Analysis of Systems Lecture #3 - Linear, Time-Invariant (LTI) Systems. Guillaume Drion Academic year

Modeling and Analysis of Systems Lecture #3 - Linear, Time-Invariant (LTI) Systems. Guillaume Drion Academic year Modeling and Analysis of Systems Lecture #3 - Linear, Time-Invariant (LTI) Systems Guillaume Drion Academic year 2015-2016 1 Outline Systems modeling: input/output approach and LTI systems. Convolution

More information

6.241 Dynamic Systems and Control

6.241 Dynamic Systems and Control 6.241 Dynamic Systems and Control Lecture 7: State-space Models Readings: DDV, Chapters 7,8 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology February 25, 2011 E. Frazzoli

More information

Laplace Transforms and use in Automatic Control

Laplace 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 information

Digital Signal Processing Lecture 3 - Discrete-Time Systems

Digital 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

Chapter 1 Fundamental Concepts

Chapter 1 Fundamental Concepts Chapter 1 Fundamental Concepts Signals A signal is a pattern of variation of a physical quantity as a function of time, space, distance, position, temperature, pressure, etc. These quantities are usually

More information

LTI Systems (Continuous & Discrete) - Basics

LTI Systems (Continuous & Discrete) - Basics LTI Systems (Continuous & Discrete) - Basics 1. A system with an input x(t) and output y(t) is described by the relation: y(t) = t. x(t). This system is (a) linear and time-invariant (b) linear and time-varying

More information

Solving a RLC Circuit using Convolution with DERIVE for Windows

Solving a RLC Circuit using Convolution with DERIVE for Windows Solving a RLC Circuit using Convolution with DERIVE for Windows Michel Beaudin École de technologie supérieure, rue Notre-Dame Ouest Montréal (Québec) Canada, H3C K3 mbeaudin@seg.etsmtl.ca - Introduction

More information

Full file at 2CT IMPULSE, IMPULSE RESPONSE, AND CONVOLUTION CHAPTER 2CT

Full file at   2CT IMPULSE, IMPULSE RESPONSE, AND CONVOLUTION CHAPTER 2CT 2CT.2 UNIT IMPULSE 2CT.2. CHAPTER 2CT 2CT.2.2 (d) We can see from plot (c) that the limit?7 Ä!, B :?7 9 œb 9$ 9. This result is consistent with the sampling property: B $ œb $ 9 9 9 9 2CT.2.3 # # $ sin

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

Lecture 2 ELE 301: Signals and Systems

Lecture 2 ELE 301: Signals and Systems Lecture 2 ELE 301: Signals and Systems Prof. Paul Cuff Princeton University Fall 2011-12 Cuff (Lecture 2) ELE 301: Signals and Systems Fall 2011-12 1 / 70 Models of Continuous Time Signals Today s topics:

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

Introduction 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 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 information

06EC44-Signals and System Chapter EC44 Signals and Systems (Chapter 4 )

06EC44-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 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

Discrete-Time Systems

Discrete-Time Systems FIR Filters With this chapter we turn to systems as opposed to signals. The systems discussed in this chapter are finite impulse response (FIR) digital filters. The term digital filter arises because these

More information

Review of Discrete-Time System

Review 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

School of Engineering Faculty of Built Environment, Engineering, Technology & Design

School of Engineering Faculty of Built Environment, Engineering, Technology & Design Module Name and Code : ENG60803 Real Time Instrumentation Semester and Year : Semester 5/6, Year 3 Lecture Number/ Week : Lecture 3, Week 3 Learning Outcome (s) : LO5 Module Co-ordinator/Tutor : Dr. Phang

More information

16.30 Estimation and Control of Aerospace Systems

16.30 Estimation and Control of Aerospace Systems 16.30 Estimation and Control of Aerospace Systems Topic 5 addendum: Signals and Systems Aeronautics and Astronautics Massachusetts Institute of Technology Fall 2010 (MIT) Topic 5 addendum: Signals, Systems

More information

Chapter 2: Time-Domain Representations of Linear Time-Invariant Systems. Chih-Wei Liu

Chapter 2: Time-Domain Representations of Linear Time-Invariant Systems. Chih-Wei Liu Chapter : Time-Domain Representations of Linear Time-Invariant Systems Chih-Wei Liu Outline Characteristics of Systems Described by Differential and Difference Equations Block Diagram Representations State-Variable

More information

a_n x^(n) a_0 x = 0 with initial conditions x(0) =... = x^(n-2)(0) = 0, x^(n-1)(0) = 1/a_n.

a_n x^(n) a_0 x = 0 with initial conditions x(0) =... = x^(n-2)(0) = 0, x^(n-1)(0) = 1/a_n. 18.03 Class 25, April 7, 2010 Convolution, or, the superposition of impulse response 1. General equivalent initial conditions for delta response 2. Two implications of LTI 3. Superposition of unit impulse

More information

Signal Processing and Linear Systems1 Lecture 4: Characterizing Systems

Signal Processing and Linear Systems1 Lecture 4: Characterizing Systems Signal Processing and Linear Systems Lecture : Characterizing Systems Nicholas Dwork www.stanford.edu/~ndwork Our goal will be to develop a way to learn how the system behaves. In general, this is a very

More information

Analog Signals and Systems and their properties

Analog Signals and Systems and their properties Analog Signals and Systems and their properties Main Course Objective: Recall course objectives Understand the fundamentals of systems/signals interaction (know how systems can transform or filter signals)

More information

ELEN E4810: Digital Signal Processing Topic 2: Time domain

ELEN E4810: Digital Signal Processing Topic 2: Time domain ELEN E4810: Digital Signal Processing Topic 2: Time domain 1. Discrete-time systems 2. Convolution 3. Linear Constant-Coefficient Difference Equations (LCCDEs) 4. Correlation 1 1. Discrete-time systems

More information

Frequency Response (III) Lecture 26:

Frequency Response (III) Lecture 26: EECS 20 N March 21, 2001 Lecture 26: Frequency Response (III) Laurent El Ghaoui 1 outline reading assignment: Chapter 8 of Lee and Varaiya we ll concentrate on continuous-time systems: convolution integral

More information

EECE 3620: Linear Time-Invariant Systems: Chapter 2

EECE 3620: Linear Time-Invariant Systems: Chapter 2 EECE 3620: Linear Time-Invariant Systems: Chapter 2 Prof. K. Chandra ECE, UMASS Lowell September 7, 2016 1 Continuous Time Systems In the context of this course, a system can represent a simple or complex

More information

LECTURE NOTES DIGITAL SIGNAL PROCESSING III B.TECH II SEMESTER (JNTUK R 13)

LECTURE NOTES DIGITAL SIGNAL PROCESSING III B.TECH II SEMESTER (JNTUK R 13) LECTURE NOTES ON DIGITAL SIGNAL PROCESSING III B.TECH II SEMESTER (JNTUK R 13) FACULTY : B.V.S.RENUKA DEVI (Asst.Prof) / Dr. K. SRINIVASA RAO (Assoc. Prof) DEPARTMENT OF ELECTRONICS AND COMMUNICATIONS

More information

Lecture 1 From Continuous-Time to Discrete-Time

Lecture 1 From Continuous-Time to Discrete-Time Lecture From Continuous-Time to Discrete-Time Outline. Continuous and Discrete-Time Signals and Systems................. What is a signal?................................2 What is a system?.............................

More information