MAE143A Signals & Systems - Homework 1, Winter 2014 due by the end of class Thursday January 16, 2014.
|
|
- Caren Little
- 5 years ago
- Views:
Transcription
1 MAE43A Signals & Systems - Homework, Winter 4 due by the end of class Thursday January 6, 4. Question Time shifting [Chaparro Question.5] Consider a finite-support signal and zero everywhere else. Part (a): Carefully plot x(t + ). Part (b): Carefully plot x( t + ). x(t) t, t, Part(c): Add the two signals above to get a new signal y(t). To verify your results, represent each signal above analytically and then show that the resulting signal is correct. [That is, use a formula for x(t) and then for the transformed versions in parts (a) and (b). Compare this to your sum from the two plots.] Part (a): x(t + ) Part (b): x( t + ) t +, t + t +, t else else t +, t + t +, t else else Plot of x(t+) Plot of x( t+) Figure : Part (a) and (b) t +, t < t + t +, t Part (c): y(t) x(t + ) + x( t + ) t +, < t else Plot of Part (c) is the sum of Part (a) and (b) plots.
2 Plot of x(t+)+x( t+) Figure : Part (c) Matlab code: clear all close all %Qa&b x -:.:; ft zeros(size(x)); ft zeros(size(x)); q x > - & x < ; ft(q) x(q)+; p x > & x < ; ft(p) -x(p)+; figure() plot(x,ft) title( Plot of x(t+) ) xlabel( ) ylabel( ) figure() plot(x,ft) title( Plot of x(-t+) ) xlabel( ) ylabel( ) %Qc y ft + ft; k find(x); y(k) ; figure(3) plot(x,y) hold on
3 plot([ ],[ ]) title( Plot of x(t+)+x(-t+) ) xlabel( ) ylabel( ) Question Even and odd functions [Chaparro Question.6] According to Euler s identity the sine and cosine are defined in terms of complex exponentials. You would then ask what if instead of complex exponentials you were to use real exponentials. This then yields the hyperbolic functions defined on < t < : cosh(ω t) eωt + e Ωt, sinh(ω t) eωt e Ωt. Part (a): Let Ω rad/sec. Use the definitions to plot cosh(t) and sinh(t). Part (b): Is cosh(t) even or odd? Part (c): Is sinh(t) even or odd? Part (d): Obtain an expression for x(t) e t u(t) in terms of hyperbolic functions [and u(t)]. Use symbolic matlab to plot x(t) e t u(t) and then to plot your expression in terms of hyperbolic functions. Compare them. Part (a): Plot of cosh(t) 5 Plot of sinh(t) Figure 3: cosh(t) (left) and sinh(t) (right) plots Part (b): cosh( t) e t +e t cosh(t), cosh(t) is even.
4 Part (c): sinh( t) e t e t sinh(t), sinh(t) is odd. Part (d): Note that u(t) is a unit step signal cosh(t) sinh(t) e t, so, x(t) [cosh(t) sinh(t)]u(t). Plot of e t u(t) Plot of (cosh(t) sinh(t))u(t) Figure 4: e t u(t) (left) and [cosh(t) sinh(t)]u(t) (right) plots Matlab code: clear all close all t -3:.:3; cosh.5*(exp(t) + exp(-t)); sinh.5*(exp(t) - exp(-t)); figure() plot(t,cosh) title( Plot of cosh(t) ) xlabel( ) ylabel( ) figure() plot(t,sinh) title( Plot of sinh(t) ) xlabel( ) ylabel( ) %Part d clear all syms t e exp(-t);%creating exp(-t) fuction %figure %ezplot(e,[-,6,,3])%plot only exp(-t) hyp cosh(t) - sinh(t);%equivalent of exp(-t) u heaviside(t);%unit step function
5 figure(3) ezplot(e*u,[-,6,,]) title( Plot of eˆ-t}u(t) ) xlabel( ) ylabel( ) figure(4) ezplot(hyp*u,[-,6,,]) title( Plot of (cosh(t)-sinh(t))u(t) ) xlabel( ) ylabel( ) Question 3 Impulse and unit-step functions [Chaparro Question.8] By introducing the impulse δ(t) or unit-step u(t) signals, we expand the conventional calculus. One of the advantages of having the δ(t) function is that we are now able to find the derivative of discontinuous signals. Let us illustrate this advantage. Consider a periodic sinusoid defined for all times. and a causal sinusoid defined as x(t) cos(ω t), < t <, x (t) cos(ω t)u(t), where the unit-step function indicates that the function has a discontinuity at zero, since for t + the function is close to, and for t the function is zero. Part (a): Find the derivative y(t) dx(t)/dt and plot it. Part (b): Find the derivate z(t) dx (t)/dt [treat x (t) as the product of two functions cos(ω t) and u(t)] and plot it. Express z(t) in terms on y(t). Part (c): Verify that the integral of z(t) gives you back x (t). Part (a): y(t) d dt x(t) d dt cos(ω t) Ω sin(ω t). Set Ω rad/s. Part (b): z(t) d dt x (t) d dt cos(ω t)u(t) δ(t) cos(ω t) Ω sin(ω t)u(t). So, z(t) δ(t)x(t) + y(t)u(t) δ(t) + y(t)u(t) since x(t) at t. Use symbolic matlab to plot z(t). Part (c): t z()d t δ() Ω sin(ω )d, t < + cos(ω ) t, t, t < cos(ω t), t cos(ω t)u(t) x (t), t < t [δ() Ω sin(ω )] d, t, t < + cos(ω t), t Matlab code:
6 Plot of y(t) Figure 5: Plot of y(t) Plot of z(t) Figure 6: Plot of z(t) clear all close all %Part(a) t -pi:(pi/):3.4; figure() plot(t,-sin(t)) axis([-3.5,pi,-,]) title( Plot of y(t) ) xlabel( ) ylabel( ) %Part (b) clear t syms t
7 z dirac(t)*cos(t) - sin(t)*heaviside(t); %Note that Matlab s Dirac is not exactly what it should be figure() ezplot(z,[-3.5,6.3,-.,.]) hold on plot([ ],[,])%to add a spike at t title( Plot of z(t) ) xlabel( ) ylabel( ) Question 4 Ramp in terms of unit-step signals [Chaparro Question.] A ramp, r(t) tu(t), can be expressed as r(t) Part (a): Show that the above expression for r(t) is equivalent to r(t) Part (b): Compute the derivative of r(t) to show that u(t) t u()u(t ) d. d tu(t). u()u(t ) d, u()δ(t ) d. Part (a): Values of u() and u(t ) and their product depending on where t is on the real line can be sketched as follows
8 u() t t< u(t ) t t u()u(t ) t t t< So, r(t) tu(t) u()u(t )d t d t, t, t < Part (b): d dt r(t) d dt u(t) u()u(t )d u() d u(t )d dt u(t), t u()δ(t )d, by the sampling property of δ(t), t < As an alternative for the last step of the above calculation, d dt r(t) d dt tu(t) u(t) + tδ(t) u(t) + u(t); therefore u(t) u()δ(t )d.
MAE143A Signals & Systems - Homework 5, Winter 2013 due by the end of class Tuesday February 12, 2013.
MAE43A Signals & Systems - Homework 5, Winter 23 due by the end of class Tuesday February 2, 23. If left under my door, then straight to the recycling bin with it. This week s homework will be a refresher
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 information7. Find the Fourier transform of f (t)=2 cos(2π t)[u (t) u(t 1)]. 8. (a) Show that a periodic signal with exponential Fourier series f (t)= δ (ω nω 0
Fourier Transform Problems 1. Find the Fourier transform of the following signals: a) f 1 (t )=e 3 t sin(10 t)u (t) b) f 1 (t )=e 4 t cos(10 t)u (t) 2. Find the Fourier transform of the following signals:
More information16.362: Signals and Systems: 1.0
16.362: Signals and Systems: 1.0 Prof. K. Chandra ECE, UMASS Lowell September 1, 2016 1 Background The pre-requisites for this course are Calculus II and Differential Equations. A basic understanding of
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 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 informationVariation of Parameters
Variation of Parameters James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University April 13, 218 Outline Variation of Parameters Example One We eventually
More informationSection 6: Summary Section 7
Section 6: Summary Section 7 a n = 2 ( 2πnx cos b n = 2 ( 2πnx sin d = f(x dx f(x = d + n= [ a n cos f(x dx f(x dx ( ( ] 2πnx 2πnx + b n sin You can sometimes combine multiple integrals using symmetry
More informationMAE143A Signals & Systems?!?!?
7//4 MAE43A Signals & Systems 24 Winter MAE43A Signals & Systems http://oodgeroo.ucsd.edu/~bob/signals Tu, Th 8:-9:2 Pepper Canyon 6 Mo 6:-6:5 Pepper Canyon 6 Professor Bob Bitmead rbitmead@ucsd.edu Jacobs
More informationMath 308 Week 8 Solutions
Math 38 Week 8 Solutions There is a solution manual to Chapter 4 online: www.pearsoncustom.com/tamu math/. This online solutions manual contains solutions to some of the suggested problems. Here are solutions
More informationSection Kamen and Heck And Harman. Fourier Transform
s Section 3.4-3.7 Kamen and Heck And Harman 1 3.4 Definition (Equation 3.30) Exists if integral converges (Equation 3.31) Example 3.7 Constant Signal Does not have a Fourier transform in the ordinary sense.
More informationENGIN 211, Engineering Math. Laplace Transforms
ENGIN 211, Engineering Math Laplace Transforms 1 Why Laplace Transform? Laplace transform converts a function in the time domain to its frequency domain. It is a powerful, systematic method in solving
More informationEE 3054: Signals, Systems, and Transforms Summer It is observed of some continuous-time LTI system that the input signal.
EE 34: Signals, Systems, and Transforms Summer 7 Test No notes, closed book. Show your work. Simplify your answers. 3. It is observed of some continuous-time LTI system that the input signal = 3 u(t) produces
More 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 informationFourier series: Additional notes
Fourier series: Additional notes Linking Fourier series representations for signals Rectangular waveform Require FS expansion of signal y(t) below: 1 y(t) 1 4 4 8 12 t (seconds) Period T = 8, so ω = 2π/T
More informationSolutions to Homework 3
Solutions to Homework 3 Section 3.4, Repeated Roots; Reduction of Order Q 1). Find the general solution to 2y + y = 0. Answer: The charactertic equation : r 2 2r + 1 = 0, solving it we get r = 1 as a repeated
More informationHomework 6 Solutions
8-290 Signals and Systems Profs. Byron Yu and Pulkit Grover Fall 208 Homework 6 Solutions. Part One. (2 points) Consider an LTI system with impulse response h(t) e αt u(t), (a) Compute the frequency response
More information6.003 Homework #10 Solutions
6.3 Homework # Solutions Problems. DT Fourier Series Determine the Fourier Series coefficients for each of the following DT signals, which are periodic in N = 8. x [n] / n x [n] n x 3 [n] n x 4 [n] / n
More informationAssignment 4 Solutions Continuous-Time Fourier Transform
Assignment 4 Solutions Continuous-Time Fourier Transform ECE 3 Signals and Systems II Version 1.01 Spring 006 1. Properties of complex numbers. Let c 1 α 1 + jβ 1 and c α + jβ be two complex numbers. a.
More informationSeries FOURIER SERIES. Graham S McDonald. A self-contained Tutorial Module for learning the technique of Fourier series analysis
Series FOURIER SERIES Graham S McDonald A self-contained Tutorial Module for learning the technique of Fourier series analysis Table of contents Begin Tutorial c 24 g.s.mcdonald@salford.ac.uk 1. Theory
More informationMAE143A Signals & Systems
MAE43A Signals & Systems Winter 206 MAE43A Signals & Systems http://numbat.ucsd.edu/~bob/signals Tu, Th: 09:30-0:50 Pepper Canyon 06 We 09:00-09:50 Pepper Canyon 09 Professor Bob Bitmead rbitmead@ucsd.edu
More informationSpecial Mathematics Laplace Transform
Special Mathematics Laplace Transform March 28 ii Nature laughs at the difficulties of integration. Pierre-Simon Laplace 4 Laplace Transform Motivation Properties of the Laplace transform the Laplace transform
More informationGeneralized sources (Sect. 6.5). The Dirac delta generalized function. Definition Consider the sequence of functions for n 1, Remarks:
Generalized sources (Sect. 6.5). The Dirac delta generalized function. Definition Consider the sequence of functions for n, d n, t < δ n (t) = n, t 3 d3 d n, t > n. d t The Dirac delta generalized function
More information6.7 Hyperbolic Functions
6.7 6.7 Hyperbolic Functions Even and Odd Parts of an Exponential Function We recall that a function f is called even if f( x) = f(x). f is called odd if f( x) = f(x). The sine function is odd while the
More informationPractice Problems For Test 3
Practice Problems For Test 3 Power Series Preliminary Material. Find the interval of convergence of the following. Be sure to determine the convergence at the endpoints. (a) ( ) k (x ) k (x 3) k= k (b)
More informationEE Homework 12 - Solutions. 1. The transfer function of the system is given to be H(s) = s j j
EE3054 - Homework 2 - Solutions. The transfer function of the system is given to be H(s) = s 2 +3s+3. Decomposing into partial fractions, H(s) = 0.5774j s +.5 0.866j + 0.5774j s +.5 + 0.866j. () (a) The
More informationUnit 2: Modeling in the Frequency Domain Part 2: The Laplace Transform. The Laplace Transform. The need for Laplace
Unit : Modeling in the Frequency Domain Part : Engineering 81: Control Systems I Faculty of Engineering & Applied Science Memorial University of Newfoundland January 1, 010 1 Pair Table Unit, Part : Unit,
More informationCHEE 319 Tutorial 3 Solutions. 1. Using partial fraction expansions, find the causal function f whose Laplace transform. F (s) F (s) = C 1 s + C 2
CHEE 39 Tutorial 3 Solutions. Using partial fraction expansions, find the causal function f whose Laplace transform is given by: F (s) 0 f(t)e st dt (.) F (s) = s(s+) ; Solution: Note that the polynomial
More informationECE 3084 QUIZ 2 SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY APRIL 2, Name:
ECE 3084 QUIZ 2 SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING GEORGIA INSTITUTE OF TECHNOLOGY APRIL 2, 205 Name:. The quiz is closed book, except for one 2-sided sheet of handwritten notes. 2. Turn off
More informationTime Response of Systems
Chapter 0 Time Response of Systems 0. Some Standard Time Responses Let us try to get some impulse time responses just by inspection: Poles F (s) f(t) s-plane Time response p =0 s p =0,p 2 =0 s 2 t p =
More informationECE 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 informationEE 313 Linear Systems and Signals The University of Texas at Austin. Solution Set for Homework #1 on Sinusoidal Signals
Solution Set for Homework #1 on Sinusoidal Signals By Mr. Houshang Salimian and Prof. Brian L. Evans September 7, 2018 1. Prologue: This problem helps you to identify the points of interest in a sinusoidal
More informationECE 3793 Matlab Project 3 Solution
ECE 3793 Matlab Project 3 Solution Spring 27 Dr. Havlicek. (a) In text problem 9.22(d), we are given X(s) = s + 2 s 2 + 7s + 2 4 < Re {s} < 3. The following Matlab statements determine the partial fraction
More informationCHAPTER 4 FOURIER SERIES S A B A R I N A I S M A I L
CHAPTER 4 FOURIER SERIES 1 S A B A R I N A I S M A I L Outline Introduction of the Fourier series. The properties of the Fourier series. Symmetry consideration Application of the Fourier series to circuit
More informationFourier Series, Fourier Transforms, and Periodic Response to Periodic Forcing
Fourier Series, Fourier ransforms, and Periodic Response to Periodic Forcing CEE 54. Structural Dynamics Department of Civil and Environmental Engineering Duke University Henri P. Gavin Fall, 26 his document
More informationSOLVING DIFFERENTIAL EQUATIONS. Amir Asif. Department of Computer Science and Engineering York University, Toronto, ON M3J1P3
SOLVING DIFFERENTIAL EQUATIONS Amir Asif Department of Computer Science and Engineering York University, Toronto, ON M3J1P3 ABSTRACT This article reviews a direct method for solving linear, constant-coefficient
More informationDefinition of the Laplace transform. 0 x(t)e st dt
Definition of the Laplace transform Bilateral Laplace Transform: X(s) = x(t)e st dt Unilateral (or one-sided) Laplace Transform: X(s) = 0 x(t)e st dt ECE352 1 Definition of the Laplace transform (cont.)
More informationPractice Problems For Test 3
Practice Problems For Test 3 Power Series Preliminary Material. Find the interval of convergence of the following. Be sure to determine the convergence at the endpoints. (a) ( ) k (x ) k (x 3) k= k (b)
More information2 Background: Fourier Series Analysis and Synthesis
Signal Processing First Lab 15: Fourier Series Pre-Lab and Warm-Up: You should read at least the Pre-Lab and Warm-up sections of this lab assignment and go over all exercises in the Pre-Lab section before
More informationHomework 6. April 11, H(s) = Y (s) X(s) = s 1. s 2 + 3s + 2
Homework 6 April, 6 Problem Steady state of LTI systems The transfer function of an LTI system is H(s) = Y (s) X(s) = s s + 3s + If the input to this system is x(t) = + cos(t + π/4), < t
More information2.161 Signal Processing: Continuous and Discrete Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 2.6 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 informatione jωt y (t) = ω 2 Ke jωt K =
BME 171, Sec 2: Homework 2 Solutions due Tue, Sep 16 by 5pm 1. Consider a system governed by the second-order differential equation a d2 y(t) + b dy(t) where a, b and c are nonnegative real numbers. (a)
More informationFourier Transform. Find the Fourier series for a periodic waveform Determine the output of a filter when the input is a periodic function
Objectives: Be able to Fourier Transform Find the Fourier series for a periodic waveform Determine the output of a filter when the input is a periodic function Filters with a Single Sinusoidal Input: Suppose
More informationAPPM 2360: Midterm 3 July 12, 2013.
APPM 2360: Midterm 3 July 12, 2013. ON THE FRONT OF YOUR BLUEBOOK write: (1) your name, (2) your instructor s name, (3) your recitation section number and (4) a grading table. Text books, class notes,
More informationDigital Signal Processing: Signal Transforms
Digital Signal Processing: Signal Transforms Aishy Amer, Mohammed Ghazal January 19, 1 Instructions: 1. This tutorial introduces frequency analysis in Matlab using the Fourier and z transforms.. More Matlab
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 informationChapter 4 The Fourier Series and Fourier Transform
Chapter 4 The Fourier Series and Fourier Transform Fourier Series Representation of Periodic Signals Let x(t) be a CT periodic signal with period T, i.e., xt ( + T) = xt ( ), t R Example: the rectangular
More informationFrequency Analysis: The Fourier
CHAPTER 4 Frequency Analysis: The Fourier Series A Mathematician is a device for turning coffee into theorems. Paul Erdos (93 996) mathematician 4. INTRODUCTION In this chapter and the next we consider
More informationProblem Weight Score Total 100
EE 350 EXAM IV 15 December 2010 Last Name (Print): First Name (Print): ID number (Last 4 digits): Section: DO NOT TURN THIS PAGE UNTIL YOU ARE TOLD TO DO SO Problem Weight Score 1 25 2 25 3 25 4 25 Total
More informationChapter 4 The Fourier Series and Fourier Transform
Chapter 4 The Fourier Series and Fourier Transform Representation of Signals in Terms of Frequency Components Consider the CT signal defined by N xt () = Acos( ω t+ θ ), t k = 1 k k k The frequencies `present
More information/ \ ( )-----/\/\/\/ \ / In Lecture 3 we offered this as an example of a first order LTI system.
18.03 Class 17, March 12, 2010 Linearity and time invariance [1] RLC [2] Superposition III [3] Time invariance [4] Review of solution methods [1] We've spent a lot of time with mx" + bx' + cx = q(t). There
More informationEECS 20N: Midterm 2 Solutions
EECS 0N: Midterm Solutions (a) The LTI system is not causal because its impulse response isn t zero for all time less than zero. See Figure. Figure : The system s impulse response in (a). (b) Recall that
More informationProblem Value
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM DATE: 30-Apr-04 COURSE: ECE-2025 NAME: GT #: LAST, FIRST Recitation Section: Circle the date & time when your Recitation
More informationDate: 1 April (1) The only reference material you may use is one 8½x11 crib sheet and a calculator.
PH1140: Oscillations and Waves Name: Solutions Conference: Date: 1 April 2005 EXAM #1: D2005 INSTRUCTIONS: (1) The only reference material you may use is one 8½x11 crib sheet and a calculator. (2) Show
More informationDate: 31 March (1) The only reference material you may use is one 8½x11 crib sheet and a calculator.
PH1140: Oscillations and Waves Name: SOLUTIONS AT END Conference: Date: 31 March 2005 EXAM #1: D2006 INSTRUCTIONS: (1) The only reference material you may use is one 8½x11 crib sheet and a calculator.
More informationHow many initial conditions are required to fully determine the general solution to a 2nd order linear differential equation?
How many initial conditions are required to fully determine the general solution to a 2nd order linear differential equation? (A) 0 (B) 1 (C) 2 (D) more than 2 (E) it depends or don t know How many of
More informationECE 301 Fall 2011 Division 1. Homework 1 Solutions.
ECE 3 Fall 2 Division. Homework Solutions. Reading: Course information handout on the course website; textbook sections.,.,.2,.3,.4; online review notes on complex numbers. Problem. For each discrete-time
More informationBasic Procedures for Common Problems
Basic Procedures for Common Problems ECHE 550, Fall 2002 Steady State Multivariable Modeling and Control 1 Determine what variables are available to manipulate (inputs, u) and what variables are available
More informationFourier transform representation of CT aperiodic signals Section 4.1
Fourier transform representation of CT aperiodic signals Section 4. A large class of aperiodic CT signals can be represented by the CT Fourier transform (CTFT). The (CT) Fourier transform (or spectrum)
More informationMath 221 Topics since the second exam
Laplace Transforms. Math 1 Topics since the second exam There is a whole different set of techniques for solving n-th order linear equations, which are based on the Laplace transform of a function. For
More informationProblem Value
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM DATE: 30-Apr-04 COURSE: ECE-2025 NAME: GT #: LAST, FIRST Recitation Section: Circle the date & time when your Recitation
More informationECE 308 SIGNALS AND SYSTEMS SPRING 2013 Examination #2 14 March 2013
ECE 308 SIGNALS AND SYSTEMS SPRING 2013 Examination #2 14 March 2013 Name: Instructions: The examination lasts for 75 minutes and is closed book, closed notes. No electronic devices are permitted, including
More informationSolutions of the Sample Problems for the Third In-Class Exam Math 246, Fall 2017, Professor David Levermore
Solutions of the Sample Problems for the Third In-Class Exam Math 6 Fall 07 Professor David Levermore Compute the Laplace transform of ft t e t ut from its definition Solution The definition of the Laplace
More informationSolutions - Homework # 3
ECE-34: Signals and Systems Summer 23 PROBLEM One period of the DTFS coefficients is given by: X[] = (/3) 2, 8. Solutions - Homewor # 3 a) What is the fundamental period 'N' of the time-domain signal x[n]?
More informationSection 6.5 Impulse Functions
Section 6.5 Impulse Functions Key terms/ideas: Unit impulse function (technically a generalized function or distribution ) Dirac delta function Laplace transform of the Dirac delta function IVPs with forcing
More informationHomework Solutions:
Homework Solutions: 1.1-1.3 Section 1.1: 1. Problems 1, 3, 5 In these problems, we want to compare and contrast the direction fields for the given (autonomous) differential equations of the form y = ay
More information3. Frequency-Domain Analysis of Continuous- Time Signals and Systems
3. Frequency-Domain Analysis of Continuous- ime Signals and Systems 3.. Definition of Continuous-ime Fourier Series (3.3-3.4) 3.2. Properties of Continuous-ime Fourier Series (3.5) 3.3. Definition of Continuous-ime
More information2.161 Signal Processing: Continuous and Discrete
MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 28 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS
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 informationLINEAR SYSTEMS. J. Elder PSYC 6256 Principles of Neural Coding
LINEAR SYSTEMS Linear Systems 2 Neural coding and cognitive neuroscience in general concerns input-output relationships. Inputs Light intensity Pre-synaptic action potentials Number of items in display
More informationGeorge Mason University Signals and Systems I Spring 2016
George Mason University Signals and Systems I Spring 206 Problem Set #6 Assigned: March, 206 Due Date: March 5, 206 Reading: This problem set is on Fourier series representations of periodic signals. The
More informationJUST THE MATHS UNIT NUMBER LAPLACE TRANSFORMS 3 (Differential equations) A.J.Hobson
JUST THE MATHS UNIT NUMBER 16.3 LAPLACE TRANSFORMS 3 (Differential equations) by A.J.Hobson 16.3.1 Examples of solving differential equations 16.3.2 The general solution of a differential equation 16.3.3
More informationLecture 7 ELE 301: Signals and Systems
Lecture 7 ELE 30: Signals and Systems Prof. Paul Cuff Princeton University Fall 20-2 Cuff (Lecture 7) ELE 30: Signals and Systems Fall 20-2 / 22 Introduction to Fourier Transforms Fourier transform as
More informationFourier Series and Transforms. Revision Lecture
E. (5-6) : / 3 Periodic signals can be written as a sum of sine and cosine waves: u(t) u(t) = a + n= (a ncosπnft+b n sinπnft) T = + T/3 T/ T +.65sin(πFt) -.6sin(πFt) +.6sin(πFt) + -.3cos(πFt) + T/ Fundamental
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 informationHarman Outline 1A Calculus CENG 5131 PDF
Harman Outline 1A Calculus CENG 5131 PDF September 5, 2013 III. Review of Differentiation A.Basic Definitions Harman Ch6,P297 Approximations for the Derivative The expression for the derivative of a function
More informationLECTURE 12 Sections Introduction to the Fourier series of periodic signals
Signals and Systems I Wednesday, February 11, 29 LECURE 12 Sections 3.1-3.3 Introduction to the Fourier series of periodic signals Chapter 3: Fourier Series of periodic signals 3. Introduction 3.1 Historical
More informationHomework 3 Solutions
EECS Signals & Systems University of California, Berkeley: Fall 7 Ramchandran September, 7 Homework 3 Solutions (Send your grades to ee.gsi@gmail.com. Check the course website for details) Review Problem
More informationHigher Order Linear Equations
C H A P T E R 4 Higher Order Linear Equations 4.1 1. The differential equation is in standard form. Its coefficients, as well as the function g(t) = t, are continuous everywhere. Hence solutions are valid
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 informationCircuit Analysis Using Fourier and Laplace Transforms
EE2015: Electrical Circuits and Networks Nagendra Krishnapura https://wwweeiitmacin/ nagendra/ Department of Electrical Engineering Indian Institute of Technology, Madras Chennai, 600036, India July-November
More informationANALOG AND DIGITAL SIGNAL PROCESSING CHAPTER 3 : LINEAR SYSTEM RESPONSE (GENERAL CASE)
3. Linear System Response (general case) 3. INTRODUCTION In chapter 2, we determined that : a) If the system is linear (or operate in a linear domain) b) If the input signal can be assumed as periodic
More informationGEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM. COURSE: ECE 3084A (Prof. Michaels)
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM DATE: 30-Apr-14 COURSE: ECE 3084A (Prof. Michaels) NAME: STUDENT #: LAST, FIRST Write your name on the front page
More informationMA 266 Review Topics - Exam # 2 (updated)
MA 66 Reiew Topics - Exam # updated Spring First Order Differential Equations Separable, st Order Linear, Homogeneous, Exact Second Order Linear Homogeneous with Equations Constant Coefficients The differential
More informationSource-Free RC Circuit
First Order Circuits Source-Free RC Circuit Initial charge on capacitor q = Cv(0) so that voltage at time 0 is v(0). What is v(t)? Prof Carruthers (ECE @ BU) EK307 Notes Summer 2018 150 / 264 First Order
More informationSignals and Systems I Have Known and Loved. (Lecture notes for CSE 3451) Andrew W. Eckford
Signals and Systems I Have Known and Loved (Lecture notes for CSE 3451) Andrew W. Eckford Department of Electrical Engineering and Computer Science York University, Toronto, Ontario, Canada Version: December
More informationSignals and Systems: Introduction
Dependent variable Signals and Systems: Introduction What is a signal? Signals may describe a wide variety of physical phenomena. The information in a signal is contained in a pattern of variations of
More information1. SINGULARITY FUNCTIONS
1. SINGULARITY FUNCTIONS 1.0 INTRODUCTION Singularity functions are discontinuous functions or their derivatives are discontinuous. A singularity is a point at which a function does not possess a derivative.
More informationHomework 1 Solutions
18-9 Signals and Systems Profs. Byron Yu and Pulkit Grover Fall 18 Homework 1 Solutions Part One 1. (8 points) Consider the DT signal given by the algorithm: x[] = 1 x[1] = x[n] = x[n 1] x[n ] (a) Plot
More informationx(t) = t[u(t 1) u(t 2)] + 1[u(t 2) u(t 3)]
ECE30 Summer II, 2006 Exam, Blue Version July 2, 2006 Name: Solution Score: 00/00 You must show all of your work for full credit. Calculators may NOT be used.. (5 points) x(t) = tu(t ) + ( t)u(t 2) u(t
More informationAspects of Continuous- and Discrete-Time Signals and Systems
Aspects of Continuous- and Discrete-Time Signals and Systems C.S. Ramalingam Department of Electrical Engineering IIT Madras C.S. Ramalingam (EE Dept., IIT Madras) Networks and Systems 1 / 45 Scaling the
More informationAPPPHYS 217 Tuesday 6 April 2010
APPPHYS 7 Tuesday 6 April Stability and input-output performance: second-order systems Here we present a detailed example to draw connections between today s topics and our prior review of linear algebra
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 informationLOPE3202: Communication Systems 10/18/2017 2
By Lecturer Ahmed Wael Academic Year 2017-2018 LOPE3202: Communication Systems 10/18/2017 We need tools to build any communication system. Mathematics is our premium tool to do work with signals and systems.
More 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 informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 5: Calculating the Laplace Transform of a Signal Introduction In this Lecture, you will learn: Laplace Transform of Simple
More informationNAME: ht () 1 2π. Hj0 ( ) dω Find the value of BW for the system having the following impulse response.
University of California at Berkeley Department of Electrical Engineering and Computer Sciences Professor J. M. Kahn, EECS 120, Fall 1998 Final Examination, Wednesday, December 16, 1998, 5-8 pm NAME: 1.
More informationLecture 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 informationProblem Sheet 1 Examples of Random Processes
RANDOM'PROCESSES'AND'TIME'SERIES'ANALYSIS.'PART'II:'RANDOM'PROCESSES' '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''Problem'Sheets' Problem Sheet 1 Examples of Random Processes 1. Give
More informationMatlab Section. November 8, 2005
Matlab Section November 8, 2005 1 1 General commands Clear all variables from memory : clear all Close all figure windows : close all Save a variable in.mat format : save filename name of variable Load
More information