Linear Filters and Convolution. Ahmed Ashraf
|
|
|
- Ella Ramsey
- 6 years ago
- Views:
Transcription
1 Linear Filters and Convolution Ahmed Ashraf
2 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 (LTI) systems. Traditionally they have been studied for signals that vary with time. But since we are currently dealing with single images, which vary with space, such filters for images are termed as Linear Shift Invariant (LSI) Filters.
3 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 (LTI) systems. Traditionally they have been studied for signals that vary with time. But since we are currently dealing with single images, which vary with space, such filters for images are termed as Linear Shift Invariant (LSI) Filters. A Linear Time Invariant system has two properties: 1. Time Invariance. 2. Linearity
4 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 (LTI) systems. Traditionally they have been studied for signals that vary with time. But since we are currently dealing with single images, which vary with space, such filters for images are termed as Linear Shift Invariant (LSI) Filters. A Linear Time Invariant system has two properties: 1. Time Invariance. 2. Linearity Time Invariance: Same inputs produce the same outputs at all times. In other words if an input f(t) produces g(t) today, f(t) will also produce g(t) tomorrow. Or f(t-a) would produce g(t-a)
5 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 (LTI) systems. Traditionally they have been studied for signals that vary with time. But since we are currently dealing with single images, which vary with space, such filters for images are termed as Linear Shift Invariant (LSI) Filters. A Linear Time Invariant system has two i.e. if properties: 1. Time Invariance. 2. Linearity LTI Time Invariance: Same inputs produce the same outputs at all times. In other words if an input f(t) produces g(t) today, f(t) will also produce g(t) tomorrow. Or f(t-a) would produce g(t-a)
6 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 (LTI) systems. Traditionally they have been studied for signals that vary with time. But since we are currently dealing with single images, which vary with space, such filters for images are termed as Linear Shift Invariant (LSI) Filters. A Linear Time Invariant system has two i.e. if properties: 1. Time Invariance. 2. Linearity LTI Time Invariance: Same inputs produce the same outputs at all times. In other words if an input f(t) produces g(t) today, f(t) will also produce g(t) tomorrow. Or f(t-a) would produce g(t-a) Then, LTI
7 Linear Time(Shift) Invariant (LTI) Systems Linearity: Linearity is a combination of two properties in turn:
8 Linear Time(Shift) Invariant (LTI) Systems Linearity: Linearity is a combination of two properties in turn: i) If f(t) produces an output g(t), then for a linear system, a.f(t) produces an output a.g(t) (This property is called HOMOGENEITY)
9 Linear Time(Shift) Invariant (LTI) Systems Linearity: Linearity is a combination of two properties in turn: i) If f(t) produces an output g(t), then for a linear system, a.f(t) produces an output a.g(t) (This property is called HOMOGENEITY) i) If f 1 (t) produces g 1 (t), AND f 2 (t) produces g 2 (t), then for a linear system, f 1 (t)+ f 2 (t) produces g 1 (t)+ g 2 (t). (This property is called SUPERPOSITION)
10 Impulse Centered at the Origin Impulse Functions
11 Impulse Functions Impulse Centered at the Origin Shifted Impulse (Right Shifted)
12 Impulse Functions Impulse Centered at the Origin Shifted Impulse (Right Shifted) Shifted Impulse (Left Shifted)
13 Impulse Functions Impulse Centered at the Origin Shifted Impulse (Right Shifted) Shifted Impulse (Left Shifted) To figure out the direction of the shift, imagine this as a shift in the origin, and see where the argument of the impulse function is zero, i.e t-3 is zero for t=+3, and t+3 is zero for t=-3
14 Arbitrary Function as a weighted sum of shifted impulses t 0 t 1 t 2 t 3..
15 Impulse Response An LTI system is usually described in terms of its impulse response (i.e. output in response to an impulse input)
16 Impulse Response An LTI system is usually described in terms of its impulse response (i.e. output in response to an impulse input) This is because the impulse response captures the complete behavior of the system.
17 Impulse Response An LTI system is usually described in terms of its impulse response (i.e. output in response to an impulse input) This is because the impulse response captures the complete behavior of the system. Why?
18 Impulse Response An LTI system is usually described in terms of its impulse response (i.e. output in response to an impulse input) This is because the impulse response captures the complete behavior of the system. Why? Because, if we know the response to an impulse, we can use shift invariance, and linearity (homogeneity and superposition) to compute the output to arbitrary inputs.
19 Impulse Response An LTI system is usually described in terms of its impulse response (i.e. output in response to an impulse input) This is because the impulse response captures the complete behavior of the system. Why? Because, if we know the response to an impulse, we can use shift invariance, and linearity (homogeneity and superposition) to compute the output to arbitrary inputs. Impulse LTI Impulse Response
20 Impulse Response An LTI system is usually described in terms of its impulse response (i.e. output in response to an impulse input) This is because the impulse response captures the complete behavior of the system. Why? Because, if we know the response to an impulse, we can use shift invariance, and linearity (homogeneity and superposition) to compute the output to arbitrary inputs. Impulse LTI Impulse Response A Scaled and Shifted Impulse LTI A Scaled and Shifted version of the Impulse Response
21 Given Impulse Response, Computing output to arbitrary function If the impulse response of a system (filter) is h(t), i.e. it produces an output h(t) in response to d(t), what would be its output in response to the following function as an input:
22 Given Impulse Response, Computing output to arbitrary function If the impulse response of a system (filter) is h(t), i.e. it produces an output h(t) in response to d(t), what would be its output in response to the following function as an input: Given:
23 Given Impulse Response, Computing output to arbitrary function If the impulse response of a system (filter) is h(t), i.e. it produces an output h(t) in response to d(t), what would be its output in response to the following function as an input: Given:
24 Given Impulse Response, Computing output to arbitrary function If the impulse response of a system (filter) is h(t), i.e. it produces an output h(t) in response to d(t), what would be its output in response to the following function as an input: Given:
25 Given Impulse Response, Computing output to arbitrary function If the impulse response of a system (filter) is h(t), i.e. it produces an output h(t) in response to d(t), what would be its output in response to the following function as an input: Given:
26 Given Impulse Response, Computing output to arbitrary function If the impulse response of a system (filter) is h(t), i.e. it produces an output h(t) in response to d(t), what would be its output in response to the following function as an input: Given: This sum is the convolution of f(t) and h(t), written as f(t)*h(t), and IS the output of the filter, i.e. the output is the correlation of the flipped h(t) with f(t)
27 Convolution between f(t) and h(t) i.e., the concepts of convolution, flipping one signal, and then taking its correlation with the input to get the output are NOT handed to us by fiat. Rather they naturally emerge from the properties of Linearity and Time/Shift Invariance. Homework: Extend these concepts for 2D images and material covered in Lecture 3.
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
Math Fall Linear Filters
Math 658-6 Fall 212 Linear Filters 1. Convolutions and filters. A filter is a black box that takes an input signal, processes it, and then returns an output signal that in some way modifies the input.
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]
ECE-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
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
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
Linear Filters. L[e iωt ] = 2π ĥ(ω)eiωt. Proof: Let L[e iωt ] = ẽ ω (t). Because L is time-invariant, we have that. L[e iω(t a) ] = ẽ ω (t a).
![Linear Filters. L[e iωt ] = 2π ĥ(ω)eiωt. Proof: Let L[e iωt ] = ẽ ω (t). Because L is time-invariant, we have that. L[e iω(t a) ] = ẽ ω (t a).](/thumbs/75/71827976.jpg "Linear Filters. L[e iωt ] = 2π ĥ(ω)eiωt. Proof: Let L[e iωt ] = ẽ ω (t). Because L is time-invariant, we have that. L[e iω(t a) ] = ẽ ω (t a).")
Linear Filters 1. Convolutions and filters. A filter is a black box that takes an input signal, processes it, and then returns an output signal that in some way modifies the input. For example, if the
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
Lecture 3 Matlab Simulink Minimum Phase, Maximum Phase and Linear Phase Systems
Lecture 3 Matlab Simulink Minimum Phase, Maximum Phase and Linear Phase Systems Lester Liu October 31, 2012 Minimum Phase, Maximum Phase and Linear Phase LTI Systems In this section, we will explore the
Convolution and Linear Systems
CS 450: Introduction to Digital Signal and Image Processing Bryan Morse BYU Computer Science Introduction Analyzing Systems Goal: analyze a device that turns one signal into another. Notation: f (t) g(t)
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
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.
ECE 602 Solution to Homework Assignment 1
ECE 6 Solution to Assignment 1 December 1, 7 1 ECE 6 Solution to Homework Assignment 1 1. For a function to be linear, it must satisfy y = f() = in addition to y = ax, so only the first graph represents
Spatial-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
Representation of Signals and Systems. Lecturer: David Shiung
Representation of Signals and Systems Lecturer: David Shiung 1 Abstract (1/2) Fourier analysis Properties of the Fourier transform Dirac delta function Fourier transform of periodic signals Fourier-transform
Dr. 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
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
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
Chapter 2: Linear systems & sinusoids OVE EDFORS DEPT. OF EIT, LUND UNIVERSITY
Chapter 2: Linear systems & sinusoids OVE EDFORS DEPT. OF EIT, LUND UNIVERSITY Learning outcomes After this lecture, the student should understand what a linear system is, including linearity conditions,
Lecture 29. Convolution Integrals and Their Applications
Math 245 - Mathematics of Physics and Engineering I Lecture 29. Convolution Integrals and Their Applications March 3, 212 Konstantin Zuev (USC) Math 245, Lecture 29 March 3, 212 1 / 13 Agenda Convolution
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
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
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
Lecture 31. Basic Theory of First Order Linear Systems
Math 245 - Mathematics of Physics and Engineering I Lecture 31. Basic Theory of First Order Linear Systems April 4, 2012 Konstantin Zuev (USC) Math 245, Lecture 31 April 4, 2012 1 / 10 Agenda Existence
Solutions to Homework 5 - Math 3410
Solutions to Homework 5 - Math 34 (Page 57: # 489) Determine whether the following vectors in R 4 are linearly dependent or independent: (a) (, 2, 3, ), (3, 7,, 2), (, 3, 7, 4) Solution From x(, 2, 3,
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
2 The Fourier Transform
2 The Fourier Transform The Fourier transform decomposes a signal defined on an infinite time interval into a λ-frequency component, where λ can be any real(or even complex) number. 2.1 Informal Development
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
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:
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
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
Introduction to Linear Time-Invariant Systems
Introduction to Linear Time-Invariant Systems Marco Cagnazzo Multimedia Networking 1 Signal spaces Discrete-time signals (sequences) u: n Z u n C If n N, u n R, then u is referred to as a real sequence
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.
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
Lecture 1 January 5, 2016
MATH 262/CME 372: Applied Fourier Analysis and Winter 26 Elements of Modern Signal Processing Lecture January 5, 26 Prof. Emmanuel Candes Scribe: Carlos A. Sing-Long; Edited by E. Candes & E. Bates Outline
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
Communication 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/
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
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
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
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)
Background LTI Systems (4A) Young Won Lim 4/20/15
Background LTI Systems (4A) Copyright (c) 2014-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2
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
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.
Continuous-Time Fourier Transform
Signals and Systems Continuous-Time Fourier Transform Chang-Su Kim continuous time discrete time periodic (series) CTFS DTFS aperiodic (transform) CTFT DTFT Lowpass Filtering Blurring or Smoothing Original
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
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 [
6.02 Fall 2012 Lecture #10
6.02 Fall 2012 Lecture #10 Linear time-invariant (LTI) models Convolution 6.02 Fall 2012 Lecture 10, Slide #1 Modeling Channel Behavior codeword bits in generate x[n] 1001110101 digitized modulate DAC
ENSC327 Communications Systems 2: Fourier Representations. Jie Liang School of Engineering Science Simon Fraser University
ENSC327 Communications Systems 2: Fourier Representations Jie Liang School of Engineering Science Simon Fraser University 1 Outline Chap 2.1 2.5: Signal Classifications Fourier Transform Dirac Delta Function
E2.5 Signals & Linear Systems. Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & 2)
E.5 Signals & Linear Systems Tutorial Sheet 1 Introduction to Signals & Systems (Lectures 1 & ) 1. Sketch each of the following continuous-time signals, specify if the signal is periodic/non-periodic,
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
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
Module 1: Introduction to Digital Control Lecture Note 4
Digital Control Module Lecture 4 Module : Introduction to Digital Control Lecture Note 4 Data Reconstruction Most of the control systems have analog controlled processes which are inherently driven by
Introduction ODEs and Linear Systems
BENG 221 Mathematical Methods in Bioengineering ODEs and Linear Systems Gert Cauwenberghs Department of Bioengineering UC San Diego 1.1 Course Objectives 1. Acquire methods for quantitative analysis and
6 The Fourier transform
6 The Fourier transform In this presentation we assume that the reader is already familiar with the Fourier transform. This means that we will not make a complete overview of its properties and applications.
The spatial frequency domain
The spatial frequenc /3/5 wk9-a- Recall: plane wave propagation λ z= θ path dela increases linearl with E ep i2π sinθ + λ z i2π cosθ λ z plane of observation /3/5 wk9-a-2 Spatial frequenc angle of propagation?
MIT 2.71/2.710 Optics 10/31/05 wk9-a-1. The spatial frequency domain
10/31/05 wk9-a-1 The spatial frequency domain Recall: plane wave propagation x path delay increases linearly with x λ z=0 θ E 0 x exp i2π sinθ + λ z i2π cosθ λ z plane of observation 10/31/05 wk9-a-2 Spatial
DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2010
[E2.5] IMPERIAL COLLEGE LONDON DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2010 EEE/ISE PART II MEng. BEng and ACGI SIGNALS AND LINEAR SYSTEMS Time allowed: 2:00 hours There are FOUR
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
2. Time-Domain Analysis of Continuous- Time Signals and Systems
2. Time-Domain Analysis of Continuous- Time Signals and Systems 2.1. Continuous-Time Impulse Function (1.4.2) 2.2. Convolution Integral (2.2) 2.3. Continuous-Time Impulse Response (2.2) 2.4. Classification
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
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
Probability 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
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]
CLTI Differential Equations (3A) Young Won Lim 6/4/15
CLTI Differential Equations (3A) Copyright (c) 2011-2015 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version
Lecture 16. Theory of Second Order Linear Homogeneous ODEs
Math 245 - Mathematics of Physics and Engineering I Lecture 16. Theory of Second Order Linear Homogeneous ODEs February 17, 2012 Konstantin Zuev (USC) Math 245, Lecture 16 February 17, 2012 1 / 12 Agenda
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
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
Module 3. Convolution. Aim
Module Convolution Digital Signal Processing. Slide 4. Aim How to perform convolution in real-time systems efficiently? Is convolution in time domain equivalent to multiplication of the transformed sequence?
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
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/
Fourier Transforms 1D
Fourier Transforms 1D 3D Image Processing Alireza Ghane 1 Overview Recap Intuitions Function representations shift-invariant spaces linear, time-invariant (LTI) systems complex numbers Fourier Transforms
Linear Independence and the Wronskian
Linear Independence and the Wronskian MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Operator Notation Let functions p(t) and q(t) be continuous functions
9.2 The Input-Output Description of a System
Lecture Notes on Control Systems/D. Ghose/212 22 9.2 The Input-Output Description of a System The input-output description of a system can be obtained by first defining a delta function, the representation
L2 gains and system approximation quality 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.242, Fall 24: MODEL REDUCTION L2 gains and system approximation quality 1 This lecture discusses the utility
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?.............................
ECEN 420 LINEAR CONTROL SYSTEMS. Lecture 2 Laplace Transform I 1/52
1/52 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 2 Laplace Transform I Linear Time Invariant Systems A general LTI system may be described by the linear constant coefficient differential equation: a n d n
LTI H. the system H when xn [ ] is the input.
![LTI H. the system H when xn [ ] is the input.](/thumbs/81/84326358.jpg "LTI H. the system H when xn [ ] is the input.")
REVIEW OF 1D LTI SYSTEMS LTI xn [ ] y[ n] H Operator notation: y[ n] = H{ x[ n] } In English, this is read: yn [ ] is the output of the system H when xn [ ] is the input. 2 THE MOST IMPORTANT PROPERTIES
We begin exploring Euler s method by looking at direction fields. Consider the direction field below.
Emma Reid- MA 66, Lesson 9 (SU 17) Euler s Method (.7) So far in this course, we have seen some very special types of first order ODEs. We ve seen methods to solve linear, separable, homogeneous, Bernoulli,
CH.3 Continuous-Time Linear Time-Invariant System
CH.3 Continuous-Time Linear Time-Invariant System 1 LTI System Characterization 1.1 what does LTI mean? In Ch.2, the properties of the system are investigated. We are particularly interested in linear
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
S&S S&S S&S. Signals and Systems (18-396) Spring Semester, Department of Electrical and Computer Engineering
S&S S&S S&S Signals Systems (-96) Spring Semester, 2009 Department of Electrical Computer Engineering SOLUTION OF DIFFERENTIAL AND DIFFERENCE EQUATIONS Note: These notes summarize the comments from the
Section 3: Summary. Section 4. Evaluating Convolution Integrals. Convolution integral (memorise this):
Section 3: Summary Convolution integral (memorise this): f(t) input g(t) impulse response y(t) output y(t) g(t τ) f(τ) dτ Way to find the output of a linear system, described by a differential equation,
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
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
Homework 4. May An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt
Homework 4 May 2017 1. An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt Determine the impulse response of the system. Rewriting as y(t) = t e (t
1-D MATH REVIEW CONTINUOUS 1-D FUNCTIONS. Kronecker delta function and its relatives. x 0 = 0
-D MATH REVIEW CONTINUOUS -D FUNCTIONS Kronecker delta function and its relatives delta function δ ( 0 ) 0 = 0 NOTE: The delta function s amplitude is infinite and its area is. The amplitude is shown as
UCSD ECE153 Handout #40 Prof. Young-Han Kim Thursday, May 29, Homework Set #8 Due: Thursday, June 5, 2011
UCSD ECE53 Handout #40 Prof. Young-Han Kim Thursday, May 9, 04 Homework Set #8 Due: Thursday, June 5, 0. Discrete-time Wiener process. Let Z n, n 0 be a discrete time white Gaussian noise (WGN) process,
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
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.
REVIEW FOR MT3 ANSWER KEY MATH 2373, SPRING 2015
REVIEW FOR MT3 ANSWER KEY MATH 373 SPRING 15 PROF. YOICHIRO MORI This list of problems is not guaranteed to be an absolutel complete review. For completeness ou must also make sure that ou know how to
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)
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,
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
Convolution. 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
PART 1. Review of DSP. f (t)e iωt dt. F(ω) = f (t) = 1 2π. F(ω)e iωt dω. f (t) F (ω) The Fourier Transform. Fourier Transform.
PART 1 Review of DSP Mauricio Sacchi University of Alberta, Edmonton, AB, Canada The Fourier Transform F() = f (t) = 1 2π f (t)e it dt F()e it d Fourier Transform Inverse Transform f (t) F () Part 1 Review
Problem Value Score No/Wrong Rec 3
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING QUIZ #3 DATE: 21-Nov-11 COURSE: ECE-2025 NAME: GT username: LAST, FIRST (ex: gpburdell3) 3 points 3 points 3 points Recitation
Module 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
Frequency-Domain C/S of LTI Systems
Frequency-Domain C/S of LTI Systems x(n) LTI y(n) LTI: Linear Time-Invariant system h(n), the impulse response of an LTI systems describes the time domain c/s. H(ω), the frequency response describes the
Lecture 27 Frequency Response 2
Lecture 27 Frequency Response 2 Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/6/12 1 Application of Ideal Filters Suppose we can generate a square wave with a fundamental period