ECE301 Fall, 2006 Exam 1 Soluation October 7, Name: Score: / Consider the system described by the differential equation

Size: px
Start display at page:

Download "ECE301 Fall, 2006 Exam 1 Soluation October 7, Name: Score: / Consider the system described by the differential equation"

Transcription

1 ECE301 Fall, 2006 Exam 1 Soluation October 7, Name: Score: /100 You must show all of your work for full credit. Calculators may NOT be used. 1. Consider the system described by the differential equation with α β + 1. (D 2 + αd + βy(t (D + αx(t (1 (a (5 points Find the eigenvalues and modes. Solution: The eigenvalues are the solutions of the characteristic equation s 2 + αs + β s 2 + (β + 1s + β (s + 1(s + β 0. (2 These are λ 1 and λ β, which have corresponding modes c 1 e t and c 2 e βt. (b (5 points Is the system stable? (Justify your answer. Solution: The system is stable because all eigenvalues are in the left half-plane (because the given value of β was positive. (c (10 points Find the zero input response y 0 (t corresponding to the initial conditions y(0 1, ẏ(0 1. Solution: Solving the system of linear equations y(0 1 c 1 + c 2 (3 ẏ(0 1 c c 2 (4 for the coefficients c 1 and c 2 yields c 2 2/( and c 1 (1 + β/( so the ZIR is ( ( 1 + β 2 y(t e t + e βt (5 (d (10 points Find the impulse response h(t. Solution: The impulse response is given by h(t b 0 δ(t + [P(Dy n (t] u(t. (6 In (1, b 0 0 and P(D D + α. The y n (t is the ZIR corresponding to initial conditions y(0 0 and ẏ(0 1 (see (2.24a, (2.24b on page 167 of the textbook. Solving the system of linear equations y(0 0 c 1 + c 2 (7 ẏ(0 1 c c 2 (8 for the coefficients c 1 and c 2 yields c 2 1/( and c 1 1/( so y n (t ( 1 e t + ( 1 e βt (9

2 ECE301 Fall, 2006 Exam 1 Soluation October 7, and h(t [( 1 α e t + (e (5 points Find the transfer function H(s. Solution: The transfer function is given by ( ] [( ( ] α β β 1 e βt u(t e t + e βt u(t. (10 H(s P(s Q(s Checking for consistency, we calculate L {h(t} s + α s 2 + αs + β s + β + 1 (s + 1(s + β. (11 β/( 1/( + s + 1 s + β βs β 2 + s + 1 ((s + 1(s + β (s + ((1 + β ((s + 1(s + β s β (s + 1(s + β so our answers to parts (c and (e are consistent. (f (5 points Find the ZIR using Laplace Transforms. Solution: Taking the Laplace Transform of (1 with x(t 0 we obtain ( s 2 Y (s sy(0 ẏ(0 + (β + 1(sY (s y(0 + βy (s 0. (16 Moving the initial conditions to the right hand side we have ( s 2 Y (s + (β + 1sY (s + β (s + 1y(0 + ẏ(0. (17 Solving for Y (s and substituting for the initial conditions from (c, we obtain so Y (s y(t which matches the result of part (c. 2. For the system described by the difference equation with α β. s + β + 2 s 2 + (β + 1s + β A s B s + β β + 1 (β 1(s (β 1(s + β (12 (13 (14 (15 (18 (19 (20 ( ( 1 + β 2 e t + e βt (21 y[n] + 2αy[n 1] + βy[n 2] x[n 1] (22

3 ECE301 Fall, 2006 Exam 1 Soluation October 7, (a (5 points Find the eigenvalues and modes. Solution: The eigenvalues are the roots of γ 2 + 2αγ + β γ βγ + β 0 (23 so the eigenvalues are both γ β α. The corresponding modes are c 1 ( α n and c 2 n(α n. (b (5 points Is the system stable? (Justify your answer. Solution: The system is unstable because it has either eigenvalues outside the unit circle or a repeated eigenvalue on the unit circle, depending on the given value of α. (c (10 points Find the unit impulse response h[n]. Solution: The unit impulse response will have the form h[n] b N a N δ[n] + y c [n]u[n]. (24 In this problem, b N 0 and y c [n] c 1 ( α n + c 2 n( α n with coefficients satisfying the solution obtained by iterative calculation using h[n] 0 n < 0. We find so solving the system of linear equations h[0] 2αh[ 1] βh[ 2] + δ[ 1] 0 (25 h[1] 2αh[0] βh[ 1] + δ[0] 1 (26 h[0] 0 c 1 (27 h[1] 1 c 1 ( α + c 2 ( α (28 for the coefficients yields c 1 0 and c 2 1/α. Thus h[n] ( n( αn u[n] n( α n 1 u[n]. (29 α (d (10 points Use convolution to find an expression for the zero state response corresponding to the input x[n] 3 n u[n]. (30 You need not actually evaluate the sum, but should simplify as much as reasonably possible. Solution: The formula for the ZSR is y ZSR x[m]h[n m] (31 so we have y ZSR m 3 m u[m](n m( α n m 1 u[n m] (32 3 m (n m( α n m 1 (33 n n( α n 1 3 m ( α m ( α n 1 n 3 m m( α m (34 ( α (n n 1 ( 3α m m( 3α m (35

4 ECE301 Fall, 2006 Exam 1 Soluation October 7, (e (Extra Credit 10 points Evaluate the sum. Solution: The first sum above is (using the formula from Chapter B, given on the formula sheet, ( 3α m ( 3α (n+1 1 ( 3α 1 1 ( 3α n ( 3α 1 ( 3α (36 (37 ( 3α n + 3α. ( α The second sum is (using another formula from Chapter B, given on the formula sheet, m( 3α m ( 3α 1 + [n(( 3α 1 1 1]( 3α (n+1 ( 3α 1 2 (39 ( 3α 1 ( 1 + [n(( 3α 1 1 1]( 3α n (3α (40 Substituting these into (35 yields something rather complicated in terms of α. It would not be difficult (only tedious to compute for a particular value of α, but the general solution is rather messy so I will not provide it here. 3. Consider the transfer function H(s β(s + 2 (s s + α. (41 (a (5 points Determine corner frequencies that you would use in sketching the magnitude plot. Solution: To determine the corner frequencies we must first determine whether the denominator has two real roots or a pair of complex conjugate roots. The roots will be complex if 25 2 < 4α. this was not true for any of the given values of α so the corner frequencies are 2 and 25/2 ± 625 4α 2 /2. The exact corner frequencies depended upon the given value of α, but in all cases, rough estimates would place them somewhere between ω 1 rad/s and ω 25 rad/s. (b (5 points Determine the value of the magnitude (in db at ω 1 rad/s. You need not evaluate the square roots. Solution: We calculate as follows. H(j β(j + 2 j j + α β (α 1 2 (42 so in db we have 20log 10 H(j 20log 10 β + 20log log (α 1 2 (43 20log 10 β + 10log log 10 (625 + (α 1 2. (44

5 ECE301 Fall, 2006 Exam 1 Soluation October 7, (c (5 points Determine the approximate slope of the magnitude (in db and the approximate value of the phase (in deg at ω 10 3 rad/s. Solution: ω 10 3 rad/s is much smaller than any of the corner frequencies so, since there are neither zeros nor poles at the origin, the slope of the magnitude is 0 db/dec and the phase is 0 degrees. (d (5 points Determine the approximate slope of the magnitude (in db and the approximate value of the phase (in deg at ω 10 4 rad/s. Solution: ω 10 4 rad/s is much larger than any of the corner frequencies so the slope of the magnitude is approximately 20( db/dec and the phase is approximately 90( degrees. 4. (10 points Indicate whether the following system is causal, invertible, linear, memoryless, and/or time invariant by circling the correct options. (The system may have more than one of these properties. Justify your answer. y(t x(2t + 3 (causal, invertible, linear, memoryless, time invariant Solution: Causality When t 1 we have y(1 x(2 + 3, so the output depends on a future value of the input. Thus we have shown the system is noncausal. Invertibility The system is invertible by taking x(t y(t/2 3. Linearity We show that the system is nonlinear by defining output y 1 (t in terms of input x 1 (t and output y 2 (t in terms of input x 2 (t, calculating the output y(t corresponding to the input x(t αx 1 (t + βx 2 (t and determining whether y(t y 1 (t + y 2 (t. We calculate as follows. y 1 (t x 1 (2t + 3 (45 y 2 (t x 2 (2t + 3 (46 x(t αx 1 (t + βx 2 (t (47 x(t (αx 1 + βx 2 (t (48 y(t (αx 1 + βx 2 (2t + 3 (49 αx 1 (2t + βx 2 (2t + 3 (50 αx 1 (2t βx 2 (2t + 3 (51 y 1 (t + y 2 (t (52 Memorylessness When t 1 we have y( 1 x( 2 + 3, so the output depends on a past value of the input. Thus we have shown the system is not memoryless. Time Invariance To show that the system is time invariant, we define a new function x a (t x(t a, then compute showing that the system is time invariant. y a (t x a (2t + 3 (53 x(2(t a + 3 (54 y(t a + 3, (55

x(t) = t[u(t 1) u(t 2)] + 1[u(t 2) u(t 3)]

x(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 information

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

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

More information

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

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

More information

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

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

More information

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

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

More information

NAME: ht () 1 2π. Hj0 ( ) dω Find the value of BW for the system having the following impulse response.

NAME: 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 information

LTI Systems (Continuous & Discrete) - Basics

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

More information

ECE 350 Signals and Systems Spring 2011 Final Exam - Solutions. Three 8 ½ x 11 sheets of notes, and a calculator are allowed during the exam.

ECE 350 Signals and Systems Spring 2011 Final Exam - Solutions. Three 8 ½ x 11 sheets of notes, and a calculator are allowed during the exam. ECE 35 Spring - Final Exam 9 May ECE 35 Signals and Systems Spring Final Exam - Solutions Three 8 ½ x sheets of notes, and a calculator are allowed during the exam Write all answers neatly and show your

More information

ECE 301 Fall 2011 Division 1 Homework 5 Solutions

ECE 301 Fall 2011 Division 1 Homework 5 Solutions ECE 301 Fall 2011 ivision 1 Homework 5 Solutions Reading: Sections 2.4, 3.1, and 3.2 in the textbook. Problem 1. Suppose system S is initially at rest and satisfies the following input-output difference

More information

The Johns Hopkins University Department of Electrical and Computer Engineering Introduction to Linear Systems Fall 2002.

The Johns Hopkins University Department of Electrical and Computer Engineering Introduction to Linear Systems Fall 2002. The Johns Hopkins University Department of Electrical and Computer Engineering 505.460 Introduction to Linear Systems Fall 2002 Final exam Name: You are allowed to use: 1. Table 3.1 (page 206) & Table

More information

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

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

More information

Module 4 : Laplace and Z Transform Problem Set 4

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

More information

6.003 (Fall 2011) Quiz #3 November 16, 2011

6.003 (Fall 2011) Quiz #3 November 16, 2011 6.003 (Fall 2011) Quiz #3 November 16, 2011 Name: Kerberos Username: Please circle your section number: Section Time 2 11 am 3 1 pm 4 2 pm Grades will be determined by the correctness of your answers (explanations

More information

EE 210. Signals and Systems Solutions of homework 2

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

More information

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

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

More information

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

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

More information

GEORGIA 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. 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 information

Problem Weight Score Total 100

Problem 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 information

GEORGIA 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. COURSE: ECE 3084A (Prof. Michaels) GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM DATE: 09-Dec-13 COURSE: ECE 3084A (Prof. Michaels) NAME: STUDENT #: LAST, FIRST Write your name on the front page

More information

ECE-314 Fall 2012 Review Questions for Midterm Examination II

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

More information

Grades will be determined by the correctness of your answers (explanations are not required).

Grades will be determined by the correctness of your answers (explanations are not required). 6.00 (Fall 20) Final Examination December 9, 20 Name: Kerberos Username: Please circle your section number: Section Time 2 am pm 4 2 pm Grades will be determined by the correctness of your answers (explanations

More information

EE Homework 12 - Solutions. 1. The transfer function of the system is given to be H(s) = s j j

EE 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 information

Problem Value

Problem 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 information

Grades will be determined by the correctness of your answers (explanations are not required).

Grades will be determined by the correctness of your answers (explanations are not required). 6.00 (Fall 2011) Final Examination December 19, 2011 Name: Kerberos Username: Please circle your section number: Section Time 2 11 am 1 pm 4 2 pm Grades will be determined by the correctness of your answers

More information

MAE143 A - Signals and Systems - Winter 11 Midterm, February 2nd

MAE143 A - Signals and Systems - Winter 11 Midterm, February 2nd MAE43 A - Signals and Systems - Winter Midterm, February 2nd Instructions (i) This exam is open book. You may use whatever written materials you choose, including your class notes and textbook. You may

More information

2 Classification of Continuous-Time Systems

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

More information

ECE 301 Division 1, Fall 2006 Instructor: Mimi Boutin Final Examination

ECE 301 Division 1, Fall 2006 Instructor: Mimi Boutin Final Examination ECE 30 Division, all 2006 Instructor: Mimi Boutin inal Examination Instructions:. Wait for the BEGIN signal before opening this booklet. In the meantime, read the instructions below and fill out the requested

More information

Problem Value

Problem 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 information

GATE EE Topic wise Questions SIGNALS & SYSTEMS

GATE EE Topic wise Questions SIGNALS & SYSTEMS www.gatehelp.com GATE EE Topic wise Questions YEAR 010 ONE MARK Question. 1 For the system /( s + 1), the approximate time taken for a step response to reach 98% of the final value is (A) 1 s (B) s (C)

More information

Ch 2: Linear Time-Invariant System

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

More information

QUESTION BANK SIGNALS AND SYSTEMS (4 th SEM ECE)

QUESTION BANK SIGNALS AND SYSTEMS (4 th SEM ECE) QUESTION BANK SIGNALS AND SYSTEMS (4 th SEM ECE) 1. For the signal shown in Fig. 1, find x(2t + 3). i. Fig. 1 2. What is the classification of the systems? 3. What are the Dirichlet s conditions of Fourier

More information

Signals and Systems Profs. Byron Yu and Pulkit Grover Fall Midterm 2 Solutions

Signals and Systems Profs. Byron Yu and Pulkit Grover Fall Midterm 2 Solutions 8-90 Signals and Systems Profs. Byron Yu and Pulkit Grover Fall 08 Midterm Solutions Name: Andrew ID: Problem Score Max 8 5 3 6 4 7 5 8 6 7 6 8 6 9 0 0 Total 00 Midterm Solutions. (8 points) Indicate whether

More information

EECE 3620: Linear Time-Invariant Systems: Chapter 2

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

More information

Problem Value Score No/Wrong Rec 3

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

More information

Problem Value

Problem Value GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING FINAL EXAM DATE: 2-May-05 COURSE: ECE-2025 NAME: GT #: LAST, FIRST (ex: gtz123a) Recitation Section: Circle the date & time when

More information

ECE 301 Division 1, Fall 2008 Instructor: Mimi Boutin Final Examination Instructions:

ECE 301 Division 1, Fall 2008 Instructor: Mimi Boutin Final Examination Instructions: ECE 30 Division, all 2008 Instructor: Mimi Boutin inal Examination Instructions:. Wait for the BEGIN signal before opening this booklet. In the meantime, read the instructions below and fill out the requested

More information

Final Exam of ECE301, Section 1 (Prof. Chih-Chun Wang) 1 3pm, Friday, December 13, 2016, EE 129.

Final Exam of ECE301, Section 1 (Prof. Chih-Chun Wang) 1 3pm, Friday, December 13, 2016, EE 129. Final Exam of ECE301, Section 1 (Prof. Chih-Chun Wang) 1 3pm, Friday, December 13, 2016, EE 129. 1. Please make sure that it is your name printed on the exam booklet. Enter your student ID number, and

More information

ECE 314 Signals and Systems Fall 2012

ECE 314 Signals and Systems Fall 2012 ECE 31 ignals and ystems Fall 01 olutions to Homework 5 Problem.51 Determine the impulse response of the system described by y(n) = x(n) + ax(n k). Replace x by δ to obtain the impulse response: h(n) =

More information

Z Transform (Part - II)

Z Transform (Part - II) Z Transform (Part - II). The Z Transform of the following real exponential sequence x(nt) = a n, nt 0 = 0, nt < 0, a > 0 (a) ; z > (c) for all z z (b) ; z (d) ; z < a > a az az Soln. The given sequence

More information

ECE 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, 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 information

Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam!

Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella 5. 2. 2 Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid -

More information

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

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

More information

Like bilateral Laplace transforms, ROC must be used to determine a unique inverse z-transform.

Like bilateral Laplace transforms, ROC must be used to determine a unique inverse z-transform. Inversion of the z-transform Focus on rational z-transform of z 1. Apply partial fraction expansion. Like bilateral Laplace transforms, ROC must be used to determine a unique inverse z-transform. Let X(z)

More information

This homework will not be collected or graded. It is intended to help you practice for the final exam. Solutions will be posted.

This homework will not be collected or graded. It is intended to help you practice for the final exam. Solutions will be posted. 6.003 Homework #14 This homework will not be collected or graded. It is intended to help you practice for the final exam. Solutions will be posted. Problems 1. Neural signals The following figure illustrates

More information

Lecture 9 Infinite Impulse Response Filters

Lecture 9 Infinite Impulse Response Filters Lecture 9 Infinite Impulse Response Filters Outline 9 Infinite Impulse Response Filters 9 First-Order Low-Pass Filter 93 IIR Filter Design 5 93 CT Butterworth filter design 5 93 Bilinear transform 7 9

More information

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

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

More information

ECGR4124 Digital Signal Processing Exam 2 Spring 2017

ECGR4124 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

Chapter Intended Learning Outcomes: (i) Understanding the relationship between transform and the Fourier transform for discrete-time signals

Chapter 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 information

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

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

More information

ECE 301. Division 2, Fall 2006 Instructor: Mimi Boutin Midterm Examination 3

ECE 301. Division 2, Fall 2006 Instructor: Mimi Boutin Midterm Examination 3 ECE 30 Division 2, Fall 2006 Instructor: Mimi Boutin Midterm Examination 3 Instructions:. Wait for the BEGIN signal before opening this booklet. In the meantime, read the instructions below and fill out

More information

Signals and Systems. Problem Set: The z-transform and DT Fourier Transform

Signals and Systems. Problem Set: The z-transform and DT Fourier Transform Signals and Systems Problem Set: The z-transform and DT Fourier Transform Updated: October 9, 7 Problem Set Problem - Transfer functions in MATLAB A discrete-time, causal LTI system is described by the

More information

EEL3135: Homework #4

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

More information

Final Exam of ECE301, Section 3 (CRN ) 8 10am, Wednesday, December 13, 2017, Hiler Thtr.

Final Exam of ECE301, Section 3 (CRN ) 8 10am, Wednesday, December 13, 2017, Hiler Thtr. Final Exam of ECE301, Section 3 (CRN 17101-003) 8 10am, Wednesday, December 13, 2017, Hiler Thtr. 1. Please make sure that it is your name printed on the exam booklet. Enter your student ID number, and

More information

EE361: Signals and System II

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

More information

信號與系統 Signals and Systems

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

More information

(i) Represent discrete-time signals using transform. (ii) Understand the relationship between transform and discrete-time Fourier transform

(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 information

Lecture 2. Introduction to Systems (Lathi )

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

More information

ECE 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. 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 information

The stability of linear time-invariant feedback systems

The stability of linear time-invariant feedback systems The stability of linear time-invariant feedbac systems A. Theory The system is atrictly stable if The degree of the numerator of H(s) (H(z)) the degree of the denominator of H(s) (H(z)) and/or The poles

More information

Ch. 7: Z-transform Reading

Ch. 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 information

Prüfung Regelungstechnik I (Control Systems I) Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam!

Prüfung Regelungstechnik I (Control Systems I) Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella 29. 8. 2 Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid

More information

III. Time Domain Analysis of systems

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

More information

Chapter 7: The z-transform

Chapter 7: The z-transform Chapter 7: The -Transform ECE352 1 The -Transform - definition Continuous-time systems: e st H(s) y(t) = e st H(s) e st is an eigenfunction of the LTI system h(t), and H(s) is the corresponding eigenvalue.

More information

Digital Signal Processing. Midterm 1 Solution

Digital Signal Processing. Midterm 1 Solution EE 123 University of California, Berkeley Anant Sahai February 15, 27 Digital Signal Processing Instructions Midterm 1 Solution Total time allowed for the exam is 8 minutes Some useful formulas: Discrete

More information

Discrete-time signals and systems

Discrete-time signals and systems Discrete-time signals and systems 1 DISCRETE-TIME DYNAMICAL SYSTEMS x(t) G y(t) Linear system: Output y(n) is a linear function of the inputs sequence: y(n) = k= h(k)x(n k) h(k): impulse response of the

More information

Quiz. Good luck! Signals & Systems ( ) Number of Problems: 19. Number of Points: 19. Permitted aids:

Quiz. Good luck! Signals & Systems ( ) Number of Problems: 19. Number of Points: 19. Permitted aids: Quiz November th, Signals & Systems (--) Prof. R. D Andrea Quiz Exam Duration: Min Number of Problems: Number of Points: Permitted aids: Important: None Questions must be answered on the provided answer

More information

12/20/2017. Lectures on Signals & systems Engineering. Designed and Presented by Dr. Ayman Elshenawy Elsefy

12/20/2017. Lectures on Signals & systems Engineering. Designed and Presented by Dr. Ayman Elshenawy Elsefy //7 ectures on Signals & systems Engineering Designed and Presented by Dr. Ayman Elshenawy Elsefy Dept. of Systems & Computer Eng. Al-Azhar University Email : eaymanelshenawy@yahoo.com aplace Transform

More information

University Question Paper Solution

University Question Paper Solution Unit 1: Introduction University Question Paper Solution 1. Determine whether the following systems are: i) Memoryless, ii) Stable iii) Causal iv) Linear and v) Time-invariant. i) y(n)= nx(n) ii) y(t)=

More information

Time Response Analysis (Part II)

Time Response Analysis (Part II) Time Response Analysis (Part II). A critically damped, continuous-time, second order system, when sampled, will have (in Z domain) (a) A simple pole (b) Double pole on real axis (c) Double pole on imaginary

More information

ELEN 4810 Midterm Exam

ELEN 4810 Midterm Exam ELEN 4810 Midterm Exam Wednesday, October 26, 2016, 10:10-11:25 AM. One sheet of handwritten notes is allowed. No electronics of any kind are allowed. Please record your answers in the exam booklet. Raise

More information

6.003 Homework #10 Solutions

6.003 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 information

UNIT-II Z-TRANSFORM. This expression is also called a one sided z-transform. This non causal sequence produces positive powers of z in X (z).

UNIT-II Z-TRANSFORM. This expression is also called a one sided z-transform. This non causal sequence produces positive powers of z in X (z). Page no: 1 UNIT-II Z-TRANSFORM The Z-Transform The direct -transform, properties of the -transform, rational -transforms, inversion of the transform, analysis of linear time-invariant systems in the -

More information

University of Illinois at Urbana-Champaign ECE 310: Digital Signal Processing

University of Illinois at Urbana-Champaign ECE 310: Digital Signal Processing University of Illinois at Urbana-Champaign ECE 0: Digital Signal Processing Chandra Radhakrishnan PROBLEM SET : SOLUTIONS Peter Kairouz Problem. Hz z 7 z +/9, causal ROC z > contains the unit circle BIBO

More information

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

More information

Control Systems. Frequency domain analysis. L. Lanari

Control Systems. Frequency domain analysis. L. Lanari Control Systems m i l e r p r a in r e v y n is o Frequency domain analysis L. Lanari outline introduce the Laplace unilateral transform define its properties show its advantages in turning ODEs to algebraic

More information

EE538 Final Exam Fall :20 pm -5:20 pm PHYS 223 Dec. 17, Cover Sheet

EE538 Final Exam Fall :20 pm -5:20 pm PHYS 223 Dec. 17, Cover Sheet EE538 Final Exam Fall 005 3:0 pm -5:0 pm PHYS 3 Dec. 17, 005 Cover Sheet Test Duration: 10 minutes. Open Book but Closed Notes. Calculators ARE allowed!! This test contains five problems. Each of the five

More information

Chapter 3 Convolution Representation

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

More information

Properties of LTI Systems

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

More information

One-Sided Laplace Transform and Differential Equations

One-Sided Laplace Transform and Differential Equations One-Sided Laplace Transform and Differential Equations As in the dcrete-time case, the one-sided transform allows us to take initial conditions into account. Preliminaries The one-sided Laplace transform

More information

Laplace Transforms and use in Automatic Control

Laplace Transforms and use in Automatic Control Laplace Transforms and use in Automatic Control P.S. Gandhi Mechanical Engineering IIT Bombay Acknowledgements: P.Santosh Krishna, SYSCON Recap Fourier series Fourier transform: aperiodic Convolution integral

More information

Question Paper Code : AEC11T02

Question Paper Code : AEC11T02 Hall Ticket No Question Paper Code : AEC11T02 VARDHAMAN COLLEGE OF ENGINEERING (AUTONOMOUS) Affiliated to JNTUH, Hyderabad Four Year B. Tech III Semester Tutorial Question Bank 2013-14 (Regulations: VCE-R11)

More information

ELEG 305: Digital Signal Processing

ELEG 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 information

SIDDHARTH GROUP OF INSTITUTIONS:: PUTTUR Siddharth Nagar, Narayanavanam Road QUESTION BANK (DESCRIPTIVE)

SIDDHARTH GROUP OF INSTITUTIONS:: PUTTUR Siddharth Nagar, Narayanavanam Road QUESTION BANK (DESCRIPTIVE) SIDDHARTH GROUP OF INSTITUTIONS:: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) Subject with Code : Digital Signal Processing(16EC422) Year & Sem: III-B.Tech & II-Sem Course

More information

Background LTI Systems (4A) Young Won Lim 4/20/15

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

More information

Final Exam of ECE301, Prof. Wang s section 1 3pm Tuesday, December 11, 2012, Lily 1105.

Final Exam of ECE301, Prof. Wang s section 1 3pm Tuesday, December 11, 2012, Lily 1105. Final Exam of ECE301, Prof. Wang s section 1 3pm Tuesday, December 11, 2012, Lily 1105. 1. Please make sure that it is your name printed on the exam booklet. Enter your student ID number, e-mail address,

More information

EE Homework 5 - Solutions

EE Homework 5 - Solutions EE054 - Homework 5 - Solutions 1. We know the general result that the -transform of α n 1 u[n] is with 1 α 1 ROC α < < and the -transform of α n 1 u[ n 1] is 1 α 1 with ROC 0 < α. Using this result, the

More information

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

Therefore the new Fourier coefficients are. Module 2 : Signals in Frequency Domain Problem Set 2. Problem 1 Module 2 : Signals in Frequency Domain Problem Set 2 Problem 1 Let be a periodic signal with fundamental period T and Fourier series coefficients. Derive the Fourier series coefficients of each of the

More information

CLTI System Response (4A) Young Won Lim 4/11/15

CLTI System Response (4A) Young Won Lim 4/11/15 CLTI System Response (4A) 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 1.2

More information

5. Time-Domain Analysis of Discrete-Time Signals and Systems

5. Time-Domain Analysis of Discrete-Time Signals and Systems 5. Time-Domain Analysis of Discrete-Time Signals and Systems 5.1. Impulse Sequence (1.4.1) 5.2. Convolution Sum (2.1) 5.3. Discrete-Time Impulse Response (2.1) 5.4. Classification of a Linear Time-Invariant

More information

Each problem is worth 25 points, and you may solve the problems in any order.

Each problem is worth 25 points, and you may solve the problems in any order. EE 120: Signals & Systems Department of Electrical Engineering and Computer Sciences University of California, Berkeley Midterm Exam #2 April 11, 2016, 2:10-4:00pm Instructions: There are four questions

More information

Very useful for designing and analyzing signal processing systems

Very useful for designing and analyzing signal processing systems z-transform z-transform The z-transform generalizes the Discrete-Time Fourier Transform (DTFT) for analyzing infinite-length signals and systems Very useful for designing and analyzing signal processing

More information

DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2010

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

More information

Final Exam ECE301 Signals and Systems Friday, May 3, Cover Sheet

Final Exam ECE301 Signals and Systems Friday, May 3, Cover Sheet Name: Final Exam ECE3 Signals and Systems Friday, May 3, 3 Cover Sheet Write your name on this page and every page to be safe. Test Duration: minutes. Coverage: Comprehensive Open Book but Closed Notes.

More information

Automatique. A. Hably 1. Commande d un robot mobile. Automatique. A.Hably. Digital implementation

Automatique. A. Hably 1. Commande d un robot mobile. Automatique. A.Hably. Digital implementation A. Hably 1 1 Gipsa-lab, Grenoble-INP ahmad.hably@grenoble-inp.fr Commande d un robot mobile (Gipsa-lab (DA)) ASI 1 / 25 Outline 1 2 (Gipsa-lab (DA)) ASI 2 / 25 of controllers Signals must be sampled and

More information

A.1 THE SAMPLED TIME DOMAIN AND THE Z TRANSFORM. 0 δ(t)dt = 1, (A.1) δ(t)dt =

A.1 THE SAMPLED TIME DOMAIN AND THE Z TRANSFORM. 0 δ(t)dt = 1, (A.1) δ(t)dt = APPENDIX A THE Z TRANSFORM One of the most useful techniques in engineering or scientific analysis is transforming a problem from the time domain to the frequency domain ( 3). Using a Fourier or Laplace

More information

Final Exam December 20, 2011

Final Exam December 20, 2011 Final Exam December 20, 2011 Math 420 - Ordinary Differential Equations No credit will be given for answers without mathematical or logical justification. Simplify answers as much as possible. Leave solutions

More information

Problem Value Score No/Wrong Rec

Problem Value Score No/Wrong Rec GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL & COMPUTER ENGINEERING QUIZ #2 DATE: 14-Oct-11 COURSE: ECE-225 NAME: GT username: LAST, FIRST (ex: gpburdell3) 3 points 3 points 3 points Recitation

More information

Differential and Difference LTI systems

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

More information

6.003 Homework #6 Solutions

6.003 Homework #6 Solutions 6.3 Homework #6 Solutions Problems. Maximum gain For each of the following systems, find the frequency ω m for which the magnitude of the gain is greatest. a. + s + s ω m = w This system has poles at s

More information

Solutions. Number of Problems: 10

Solutions. Number of Problems: 10 Final Exam February 4th, 01 Signals & Systems (151-0575-01) Prof. R. D Andrea Solutions Exam Duration: 150 minutes Number of Problems: 10 Permitted aids: One double-sided A4 sheet. Questions can be answered

More information