Linear Systems. ! Textbook: Strum, Contemporary Linear Systems using MATLAB.
|
|
- Liliana Walters
- 5 years ago
- Views:
Transcription
1 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 Systems c. Continuous-Time Fourier Series and Transforms d. State-Space Topics for Continuous Systems 3. Discrete Systems a. z Transforms and Applications b. Frequency Response of Discrete Systems c. Discrete Fourier Series and Transforms d. State-Space Topics for Discrete Systems
2 Chap. 1 Basic Concepts LS 2 Continus V. S. Discrete Continuous System Discrete System
3 LS 3 Unit Impulse 1. A virtual function. 2. Any arbitrary analog signal can be decomposed by impulses. 3. The response of any analog signal can be decomposed by impulse responses. Properties: (Sifting Property) Example:
4 LS 4 Unit Step Function Representing Signals a. b.
5 Conversion between Continuous and Discrete Signals LS 5 Sampling Theorem: Let : the maximum frequency component of the signal, then : the sampling frequency. if the signal can be uniquely represented by the samples, or the signal can fully recovered from the samples.
6 Chap. 2 Continuous Systems LS 6 Basic Concept Linearity: A system is linear if and only if it satisfies the principle of homogeneity and additivity. 1. Homogeneity 2. Additivity 3. Homogeneity and additivity
7 Time Invariance: The same input applied at different times will produce the same output except shifted in time. That is, for arbitrary LS 7 Linear Time-Invariant Systems (LTI): Linear + time-invariant. 1. Can be analyzed by Laplace and Fourier transforms. 2. In time domain, described by linear differential equations with constant coefficients. 3. In transform domains, described by linear algebraic equation. Causality: Output depends on only previous inputs. That is, only depends on.
8 LS 8 Stability: if the input is bounded, the output is also bounded (bounded-input-bounded-output BIBO). Problems 2.1. Discuss the properties of the following continuous systems. a.. (i) Linear? (ii) Time invariant? b.. Causal or noncausal? c. Nth-Order Differential Equation Model In general, a single-input-single-output LTI system can be modeled by Nth-order differential equation as follows, Example: Initial Condition Solution of a Differential Equation 1. Find the homogeneous solution, i.e.
9 LS 9 Let, substitute this to the above equation. Then, which is a polynomial called characteristic equation. Let be the roots of the characteristic equation, then Note: if stable, the real parts of the roots must be negative. The initial condition solution is in the form where initial conditions are unknown coefficients to be determined by the. Problem 2.2 Let a. Find the system s characteristic equation. b. Find the root. c. Is the system stable? d. Find the general form of the homogeneous solution. e. Find the initial solution for and. The Unit Impulse Response Model Let the input be, and output, then is called impulse response. For any input, we have
10 LS 10 Sifting property: Stability Definition: Problem 2.3 a. Find impulse response. b. Is the system stable? Convolution Let,, then Let, then
11 LS 11 A continuous-time function can be broken up into a summation of shifted impulse functions. Suppose an LTI continous system has the response of impulse 1.Time Invariant 2.Homogeneity 3.Additivity
12 LS 12 4.Let, then (Convolution Integral) where * is the convolution symbol. Alternative form: Example:
13 LS 13
14 Example: LS 14
15 LS 15 Problem 2.4,, find by MATLAB. Analytic solution: Problem 2.5 Convolution with Impulse functions.
16 LS 16 Sinusoidal Steady-State Response What is the output when the input is an AC signal, that is, a sinusoidal signal? Since let, then, Let then,
17 LS 17 Conclusion: in general, when a sinusoidal signal is applied to a stable LTI system, the output is. That is, the output is also a sinusoidal signal. Problem 2.6 A highpass filter has impulse response. If Find. First, compute. The input has two frequency components 1. :, no output.
18 LS :, the output is Alternative Path to Problem 2.7 A bandpass analog filter is described by where is the output and is input. a. Determine the frequency response. b. Find the output of
19 LS 19 State-Space Model If there is no derivative term in the input, that is, we can define Then we can turn the original equation to a set of first-order differential equations:. We call state variables and the following state vector:. Then, the set of equations become in matrix form
20 LS 20 A common definition for the state of a system is as follows: The state of a system is a minimum set of quantities which if known at are uniquely determined for by specifying the inputs to the system for. Why? The M ouputs of a system are related to the states and a single input by the output equation where is an M by N matrix and is an M by 1 vector. Problem 2.8 a. Describe the system in state variable form with and. b. Find the output equation if and.
21 LS 21 System Simulation (Numerical Solution Using MATLAB) Consider Let then, Case 1: initial condition... A=[0 1 ; -2-3]; B=[0;1/3]; C=[1 0]; D=0; v0=[1;0]; tspan=[0 5]; x=@(t) 0; df=@(t,v) A*v+B*x(t); [t vv]=ode45(df,tspan,v0); plot(t,vv(:,1)); xlabel( t ); ylabel( theta ); Case 2: initial condition...
22 LS 22 v0=[0;0]; 1; A*v+B*x(t); [t vv]=ode45(df,tspan,v0); plot(t,vv(:,1)); xlabel( t ); ylabel( theta ); Example 2.1: An Oscillatory System a. Find the characteristic equation and roots. b. Solve for initial conditions of and. c. Write the state-space equation and use MATLAB to solve the equation. Assume. Plot the result with (b) to compare.
23 LS 23 Example 2.2: Second-Order Systems where a. From is the natural frequency., find b. Find the characteristic equation in terms of RLC circuit parameters and. c. Let L and C fixed and, find the range of R that will yield (i) Purely imaginary roots. (ii) Complex roots. (iii) Real roots., or d. Find the state-space equation. e. Plot for the following cases:,,,,. wn=10; zeta=2.3; K=100; A=[0 1 ; -wn^2-2*zeta*wn]; B=[0;K]; v0=[0;0]; tspan=[0 2]; x=@(t) 1;
24 A*v+B*x(t); [t vv]=ode45(df,tspan,v0); plot(t,vv(:,1)); xlabel( t ); ylabel( theta ); LS 24
25 LS 25 Example 2.3 Unit Impulse Response of a Lowpass Filter A lowpass RC filter can be modeled by a. Find the characteristic equation. b. Find for and. c. Find for and. d. Find for and. e. Let, show that delta=1; A=-1; B=1; v0=0; tspan=[0 4]; x=@(t) (t<=delta && t>=0)/delta; df=@(t,v) A*v+B*x(t); [t vv]=ode45(df,tspan,v0); plot(t,vv); xlabel( t ); ylabel( theta );
26 LS 26 Example 2.4 Convolution a. An LTI causal system is modeled by unit impulse response. Prove that for, b. A finite duration integrator can be modeled by the unit impulse response. If the input is
27 LS 27, Find the output by graphic method. c. Find the output by analytical method. d. If the hypothetical integrator is modeld by the noncausal unit impulse response, find.
Chap. 3 Laplace Transforms and Applications
Chap 3 Laplace Transforms and Applications LS 1 Basic Concepts Bilateral Laplace Transform: where is a complex variable Region of Convergence (ROC): The region of s for which the integral converges Transform
More informationResponses of Digital Filters Chapter Intended Learning Outcomes:
Responses of Digital Filters Chapter Intended Learning Outcomes: (i) Understanding the relationships between impulse response, frequency response, difference equation and transfer function in characterizing
More informationSignals 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 informationSolution 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 informationThe 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 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 informationChap 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 informationEEL3135: 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 informationChapter 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 informationIntroduction to DSP Time Domain Representation of Signals and Systems
Introduction to DSP Time Domain Representation of Signals and Systems Dr. Waleed Al-Hanafy waleed alhanafy@yahoo.com Faculty of Electronic Engineering, Menoufia Univ., Egypt Digital Signal Processing (ECE407)
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 information信號與系統 Signals and Systems
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
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 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 informationFILTER DESIGN FOR SIGNAL PROCESSING USING MATLAB AND MATHEMATICAL
FILTER DESIGN FOR SIGNAL PROCESSING USING MATLAB AND MATHEMATICAL Miroslav D. Lutovac The University of Belgrade Belgrade, Yugoslavia Dejan V. Tosic The University of Belgrade Belgrade, Yugoslavia Brian
More informationLinear Convolution Using FFT
Linear Convolution Using FFT Another useful property is that we can perform circular convolution and see how many points remain the same as those of linear convolution. When P < L and an L-point circular
More informationCosc 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 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(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 informationChapter 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 informationChapter 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 informationContinuous and Discrete Time Signals and Systems
Continuous and Discrete Time Signals and Systems Mrinal Mandal University of Alberta, Edmonton, Canada and Amir Asif York University, Toronto, Canada CAMBRIDGE UNIVERSITY PRESS Contents Preface Parti Introduction
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 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 informationZ - Transform. It offers the techniques for digital filter design and frequency analysis of digital signals.
Z - Transform The z-transform is a very important tool in describing and analyzing digital systems. It offers the techniques for digital filter design and frequency analysis of digital signals. Definition
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 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 informationModeling 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 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 information06/12/ 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 informationSolution 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 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 informationReview of Linear Time-Invariant Network Analysis
D1 APPENDIX D Review of Linear Time-Invariant Network Analysis Consider a network with input x(t) and output y(t) as shown in Figure D-1. If an input x 1 (t) produces an output y 1 (t), and an input x
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 informationSolving 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 informationLet H(z) = P(z)/Q(z) be the system function of a rational form. Let us represent both P(z) and Q(z) as polynomials of z (not z -1 )
Review: Poles and Zeros of Fractional Form Let H() = P()/Q() be the system function of a rational form. Let us represent both P() and Q() as polynomials of (not - ) Then Poles: the roots of Q()=0 Zeros:
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 informationCh. 7: Z-transform Reading
c J. Fessler, June 9, 3, 6:3 (student version) 7. Ch. 7: Z-transform Definition Properties linearity / superposition time shift convolution: y[n] =h[n] x[n] Y (z) =H(z) X(z) Inverse z-transform by coefficient
More informationChapter 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 informationLTI 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 informationChap 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 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 informationInterconnection 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 informationECE-314 Fall 2012 Review Questions for Midterm Examination II
ECE-314 Fall 2012 Review Questions for Midterm Examination II First, make sure you study all the problems and their solutions from homework sets 4-7. Then work on the following additional problems. Problem
More informationEE482: Digital Signal Processing Applications
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu EE482: Digital Signal Processing Applications Spring 2014 TTh 14:30-15:45 CBC C222 Lecture 02 DSP Fundamentals 14/01/21 http://www.ee.unlv.edu/~b1morris/ee482/
More informationLecture 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 informationSignals and Systems Laboratory with MATLAB
Signals and Systems Laboratory with MATLAB Alex Palamides Anastasia Veloni @ CRC Press Taylor &. Francis Group Boca Raton London NewYork CRC Press is an imprint of the Taylor & Francis Group, an informa
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 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 information# FIR. [ ] = b k. # [ ]x[ n " k] [ ] = h k. x[ n] = Ae j" e j# ˆ n Complex exponential input. [ ]Ae j" e j ˆ. ˆ )Ae j# e j ˆ. y n. y n.
[ ] = h k M [ ] = b k x[ n " k] FIR k= M [ ]x[ n " k] convolution k= x[ n] = Ae j" e j ˆ n Complex exponential input [ ] = h k M % k= [ ]Ae j" e j ˆ % M = ' h[ k]e " j ˆ & k= k = H (" ˆ )Ae j e j ˆ ( )
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 informationMultidimensional digital signal processing
PSfrag replacements Two-dimensional discrete signals N 1 A 2-D discrete signal (also N called a sequence or array) is a function 2 defined over thex(n set 1 of, n 2 ordered ) pairs of integers: y(nx 1,
More informationLaplace Transform Part 1: Introduction (I&N Chap 13)
Laplace Transform Part 1: Introduction (I&N Chap 13) Definition of the L.T. L.T. of Singularity Functions L.T. Pairs Properties of the L.T. Inverse L.T. Convolution IVT(initial value theorem) & FVT (final
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 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 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 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 informationDescription of Systems
Description of Systems Systems Broadly speaking, a system is anything that responds when stimulated or excited The systems most commonly analyzed by engineers are artificial systems designed and built
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 informationECE 301 Division 1 Final Exam Solutions, 12/12/2011, 3:20-5:20pm in PHYS 114.
ECE 301 Division 1 Final Exam Solutions, 12/12/2011, 3:20-5:20pm in PHYS 114. The exam for both sections of ECE 301 is conducted in the same room, but the problems are completely different. Your ID will
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 informationECGR4124 Digital Signal Processing Exam 2 Spring 2017
ECGR4124 Digital Signal Processing Exam 2 Spring 2017 Name: LAST 4 NUMBERS of Student Number: Do NOT begin until told to do so Make sure that you have all pages before starting NO TEXTBOOK, NO CALCULATOR,
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 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 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 informationPoles, Zeros and System Response
Time Response After the engineer obtains a mathematical representation of a subsystem, the subsystem is analyzed for its transient and steady state responses to see if these characteristics yield the desired
More informationDIGITAL 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 informationDynamic Response. Assoc. Prof. Enver Tatlicioglu. Department of Electrical & Electronics Engineering Izmir Institute of Technology.
Dynamic Response Assoc. Prof. Enver Tatlicioglu Department of Electrical & Electronics Engineering Izmir Institute of Technology Chapter 3 Assoc. Prof. Enver Tatlicioglu (EEE@IYTE) EE362 Feedback Control
More informationZ-Transform. x (n) Sampler
Chapter Two A- Discrete Time Signals: The discrete time signal x(n) is obtained by taking samples of the analog signal xa (t) every Ts seconds as shown in Figure below. Analog signal Discrete time signal
More informationComputer Engineering 4TL4: Digital Signal Processing
Computer Engineering 4TL4: Digital Signal Processing Day Class Instructor: Dr. I. C. BRUCE Duration of Examination: 3 Hours McMaster University Final Examination December, 2003 This examination paper includes
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 informationDigital Signal Processing:
Digital Signal Processing: Mathematical and algorithmic manipulation of discretized and quantized or naturally digital signals in order to extract the most relevant and pertinent information that is carried
More informationDr. Ian R. Manchester
Dr Ian R. Manchester Week Content Notes 1 Introduction 2 Frequency Domain Modelling 3 Transient Performance and the s-plane 4 Block Diagrams 5 Feedback System Characteristics Assign 1 Due 6 Root Locus
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 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 informationLab 3: Poles, Zeros, and Time/Frequency Domain Response
ECEN 33 Linear Systems Spring 2 2-- P. Mathys Lab 3: Poles, Zeros, and Time/Frequency Domain Response of CT Systems Introduction Systems that are used for signal processing are very often characterized
More informationCh 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 informationEECE 2150 Circuits and Signals Final Exam Fall 2016 Dec 16
EECE 2150 Circuits and Signals Final Exam Fall 2016 Dec 16 Instructions: Write your name and section number on all pages Closed book, closed notes; Computers and cell phones are not allowed You can use
More informationDigital Signal Processing, Homework 1, Spring 2013, Prof. C.D. Chung
Digital Signal Processing, Homework, Spring 203, Prof. C.D. Chung. (0.5%) Page 99, Problem 2.2 (a) The impulse response h [n] of an LTI system is known to be zero, except in the interval N 0 n N. The input
More informationDiscrete-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 informationAnalog 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 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 informationNAME: 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 informationConvolution. Define a mathematical operation on discrete-time signals called convolution, represented by *. Given two discrete-time signals x 1, x 2,
Filters Filters So far: Sound signals, connection to Fourier Series, Introduction to Fourier Series and Transforms, Introduction to the FFT Today Filters Filters: Keep part of the signal we are interested
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 informationProbability and Statistics for Final Year Engineering Students
Probability and Statistics for Final Year Engineering Students By Yoni Nazarathy, Last Updated: May 24, 2011. Lecture 6p: Spectral Density, Passing Random Processes through LTI Systems, Filtering Terms
More informationStability Condition in Terms of the Pole Locations
Stability Condition in Terms of the Pole Locations A causal LTI digital filter is BIBO stable if and only if its impulse response h[n] is absolutely summable, i.e., 1 = S h [ n] < n= We now develop a stability
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 informationLinear 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 informationIT DIGITAL SIGNAL PROCESSING (2013 regulation) UNIT-1 SIGNALS AND SYSTEMS PART-A
DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING IT6502 - DIGITAL SIGNAL PROCESSING (2013 regulation) UNIT-1 SIGNALS AND SYSTEMS PART-A 1. What is a continuous and discrete time signal? Continuous
More informationExamples. 2-input, 1-output discrete-time systems: 1-input, 1-output discrete-time systems:
Discrete-Time s - I Time-Domain Representation CHAPTER 4 These lecture slides are based on "Digital Signal Processing: A Computer-Based Approach, 4th ed." textbook by S.K. Mitra and its instructor materials.
More information2. 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 information2 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 informationELEG 305: Digital Signal Processing
ELEG 305: Digital Signal Processing Lecture : Design of Digital IIR Filters (Part I) Kenneth E. Barner Department of Electrical and Computer Engineering University of Delaware Fall 008 K. E. Barner (Univ.
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 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 informationModule 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