ECE 301 Division 1, Fall 2006 Instructor: Mimi Boutin Final Examination
|
|
- Erin Clarke
- 5 years ago
- Views:
Transcription
1 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 info. 2. You have two hours to answer the 8 questions contained in this exam, for a total of up to 47 points. When the end of the exam is announced, you must stop writing immediately. Anyone caught writing after the exam is over will get a grade of zero. 3. This booklet contains 9 pages. The last pages contain a table of formulas and properties (5 pages) as well as some scratch paper (4 pages). You may detach the scratch paper and the formula pages from the booklet once the exam begins. Each transform and each property is labeled with a number. To save time, you may use these numbers to specify which transform/property you are using when justifying your answer. In general, if you use a fact which is not contained in this table, you must explain why it is true in order to get full credit. The only exception are the properties of the ROC, which you can use without justification. 4. This is a closed book exam. The only personal items allowed are pens/pencils, erasers and something to drink. Anything else is strictly forbidden. That includes cell phones, ipods, and PDAs. 5. You must keep your eyes on your exam at all times. Looking around is strictly forbidden. Name: Signature:
2 (5 pts). An LTI system has unit impulse ( ) response h[n] = u[n+2]. Compute the n system s response to the input x[n] = +j 3 u[n]. (Simplify your answer until all signs disappear.) 2
3 2. A discrete-time system is such that when the input is one of the signals in the left column, then the output is the corresponding signal in the right column: input output x 0 [n] = δ[n] y 0 [n] = δ[n ], x [n] = δ[n ] y [n] = 4δ[n 2], x 2 [n] = δ[n 2] y 2 [n] = 9δ[n 3], x 3 [n] = δ[n 3] y 3 [n] = 6δ[n 4],. x k [n] = δ[n k] y k [n] = (k + ) 2 δ[n (k + )] for any integer k. (0 pts) a) Can this system be time-invariant? Explain. (0 pts) b) Assuming that this system is linear, what input x[n] would yield the output y[n] = u[n ]? 3
4 (22 pts) 3. Let x(t) and y(t) be the input and the output of a discrete-time system, respectively. Answer each of the questions below with either yes or no (no justification needed). Yes No If y(t) = x(2t), is the system causal? If y(t) = (t + 2)x(t), is the system causal? If y(t) = x( t 2 ), is the system causal? If y(t) = x(t) + t, is the system memoryless? If y(t) = x(t 2 ), is the system memoryless? If y(t) = x(t/3), is the system stable? If y(t) = tx(t/3), is the system stable? If y(t) = t x(τ)dτ, is the system stable? If y(t) = sin (x(t)), is the system time invariant? If y(t) = u(t) x(t), is the system LTI? If y(t) = (tu(t)) x(t), is the system linear? 4
5 (5 pts) 4 Compute the energy and the power of the signal x(t) = 3ejt +j. 5
6 (5 pts) 5. Compute the coefficients a k of the ourier series of the signal x(t) = k= (Simplify your answer as much as possible.) 3 (u(t k) u(t 2 ) + 2k). 6
7 (5 pts) 6. The Laplace transform of the unit impulse response of a system is H(s) =,Re(s) > 2. s + 2 Determine the response y(t) of the system when the input is x(t) = e 3 t. 7
8 7. The block diagram below shows a system consisting of a continuous-time LTI system followed by a sampler, conversion to a sequence, and an LTI discretetime system. The continuous-time LTI system is causal and satisfies the linear, constant-coefficient differential equation dy c (t) dt The input x c (t) is a unit impulse δ(t). + y c (t) = x c (t). (0 pts) a) Determine the input y c (t). (Problem 7 continues on the next page.) 8
9 (5 pts) b) Determine the frequency response H(e jω ) and the unit impulse response h[n] such that w[n] = δ[n]. 9
10 8. A commonly used system to maintain privacy in voice communication is a speech scrambler. The input of the system is a normal speech signal x(t) and the output is the scrambled version y(t). The signal y(t) is transmitted and then unscrambled at the receiver. We assume that all inputs to the scrambler are real and band limited to the frequency ω M ; that is, X(ω) = 0 for ω > ω M. Given any such input, our proposed scrambler permutes different bands of the spectrum of the input signal. In addition, the output is real and band limited to the same frequency band: that is Y(ω) = 0 for ω > ω M. The specific algorithm for the scrambler is Y(ω) = X(ω ω M ), when ω > 0, Y(ω) = X(ω + ω M ), when ω < 0. (5 pts) a) Assuming that X(ω) = ω 2 (u(ω + 3) u(ω 3)), sketch the graph of Y(ω). (0 pts) b) Draw a block diagram for such an ideal scrambler. (0 pts) c) Draw a block diagram for the associated unscrambler. 0
11 Table DT Signal Energy and Power E = CT Signal Energy and Power n= P = lim N 2N + x[n] 2 () N n= N x[n] 2 (2) E = P = lim T 2T x(t) 2 dt (3) T T x(t) 2 dt (4) ourier Series of CT Periodic Signals with period T x(t) = a k = X a k e jk 2πT t k= Z T x(t)e jk 2πT t dt (6) T 0 (5) ourier Series of DT Periodic Signals with period N x[n] = a k = N X k=0 N n=0 a k e jk 2πN n N X x[n]e jk 2πN n (7) (8)
12 CT ourier Transform.T. : X(ω) = Inverse.T.: x(t) = Z x(t)e jωt dt (9) Z X(ω)e jωt dω (0) 2π Properties of CT ourier Transform Let x(t) be a continuous-time signal and denote by X(ω) its ourier transform. Let y(t) be another continuous-time signal and denote by Y (ω) its ourier transform. Signal Linearity: ax(t) + by(t) ax(ω) + by (ω) () Time Shifting: x(t t 0 ) e jωt0 X(ω) (2) requency Shifting: e jω 0 t x(t) X(ω ω 0 ) (3) «ω Time and requency Scaling: x(at) a X (4) a Multiplication: x(t)y(t) T X(ω) Y (ω) (5) 2π Convolution: x(t) y(t) X(ω)Y (ω) (6) Differentiation in Time: d x(t) jωx(ω) (7) dt Some CT ourier Transform Pairs e jω 0 t 2πδ(ω ω 0 ) (8) sin Wt πt u(t + T ) u(t T ) δ(t) u(t) e at u(t), Re{a} > 0 te at u(t), Re{a} > 0 2πδ(ω) (9) u(ω + W) u(ω W) (20) 2 sin(ωt ) ω (2) (22) + πδ(ω) (23) jω a + jω (24) (a + jω) 2 (25) 2
13 DT ourier Transform Let x[n] be a discrete-time signal and denote by X(ω) its ourier transform..t.:x(ω) = Inverse.T.: x[n] = X x[n]e jωn (26) n= Z X(ω)e jωn dω (27) 2π 2π Properties of DT ourier Transform Let x(t) be a signal and denote by X(ω) its ourier transform. Let y(t) be another signal and denote by Y (ω) its ourier transform. Signal.T. Linearity: ax[n] + by[n] ax(ω) + by (ω) (28) Time Shifting: x[n n 0 ] e jωn0 X(ω) (29) requency Shifting: e jω 0 n x[n] X(ω ω 0 ) (30) Time Reversal: x[ n] X( ω) (3) Time Exp.: j x[ n x k [n] = k ], if k divides n 0, else. X(ω) (32) Multiplication: x[n]y[n] X(ω) Y (ω) 2π (33) Convolution: x[n] y[n] X(ω)Y (ω) (34) Differencing in Time: x[n] x[n ] ( e jω )X(ω) (35) Some DT ourier Transform Pairs N X k=0 sin Wn a k e jk 2πN n 2π X a k δ(ω 2πk N ) (36) k= e jω 0 n X 2π δ(ω ω 0 2πl) (37) l= X 2π δ(ω 2πl) (38) l= πn,0 < W < π u(ω + W) u(ω W)X(ω) = δ[n] u[n] α n u[n], α < (n + )α n u[n], α < X(ω)periodic with period 2π j, 0 ω < W 0, π ω > W (39) (40) X e jω + π δ(ω 2πk) (4) k= αe jω (42) ( αe jω ) 2 (43) 3
14 Laplace Transform X(s) = Z x(t)e st dt (44) Properties of Laplace Transform Let x(t), x (t) and x 2 (t) be three CT signals and denote by X(s), X (s) and X 2 (s) their respective Laplace transform. Let R be the ROC of X(s), let R be the ROC of X (z) and let R 2 be the ROC of X 2 (s). Signal L.T. ROC Linearity: ax (t) + bx 2 (t) ax (s) + bx 2 (s) At least R R 2 (45) Time Shifting: x(t t 0 ) e st0 X(s) R (46) Shifting in s: e s 0 t x(t) X(s s 0 ) R + s 0 (47) Conjugation: x (t) X (s ) R (48) «s Time Scaling: x(at) a X ar (49) a Convolution: x (t) x 2 (t) X (s)x 2 (s) At least R R 2 (50) Differentiation in Time: Differentiation in s: Integration : d x(t) sx(s) At least R (5) dt dx(s) tx(t) R (52) ds Z t x(τ)dτ X(s) At least R Re{s} > 0 (53) s Some Laplace Transform Pairs Signal LT ROC δ(t) all s (54) u(t) s Re{s} > 0 (55) s u(t) cos(ω 0 t) Re{s} > 0 (56) s 2 + ω0 2 e αt u(t) s + α Re{s} > α (57) e αt u( t) s + α Re{s} < α (58) (59) 4
15 z-transform X(z) = X n= x[n]z n (60) Properties of z-transform Let x[n], x [n] and x 2 [n] be three DT signals and denote by X(z), X (z) and X 2 (z) their respective z-transform. Let R be the ROC of X(z), let R be the ROC of X (z) and let R 2 be the ROC of X 2 (z). Signal z-t. ROC Linearity: ax [n] + bx 2 [n] ax (z) + bx 2 (z) At least R R 2 (6) Time Shifting: x[n n 0 ] z n0 X(z) R,but perhaps adding/deleting z = 0 (62) Time Shifting: x[ n] X(z ) R (63) Scaling in z: e jω 0 n x[n] X(e jω0 z) R (64) Conjugation: x (t) X (z ) R (65) Convolution: x [n] x 2 [n] X (z)x 2 (z) At least R R 2 (66) Some z-transform Pairs Signal LT ROC u[n] z z > (67) u( n ) z z < (68) α n u[n] αz z > α (69) α n u[ n ] αz z < α (70) δ[n] all z (7) 5
16 -SCRATCH - (will not be graded) 6
17 -SCRATCH - (will not be graded) 7
18 -SCRATCH - (will not be graded) 8
19 -SCRATCH - (will not be graded) 9
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 informationECE 301. Division 3, Fall 2007 Instructor: Mimi Boutin Midterm Examination 3
ECE 30 Division 3, all 2007 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 informationECE 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 informationTable 1: Properties of the Continuous-Time Fourier Series. Property Periodic Signal Fourier Series Coefficients
able : Properties of the Continuous-ime Fourier Series x(t = a k e jkω0t = a k = x(te jkω0t dt = a k e jk(/t x(te jk(/t dt Property Periodic Signal Fourier Series Coefficients x(t y(t } Periodic with period
More informationTable 1: Properties of the Continuous-Time Fourier Series. Property Periodic Signal Fourier Series Coefficients
able : Properties of the Continuous-ime Fourier Series x(t = e jkω0t = = x(te jkω0t dt = e jk(/t x(te jk(/t dt Property Periodic Signal Fourier Series Coefficients x(t y(t } Periodic with period and fundamental
More informationELEN 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 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 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 informationFinal 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 informationEC Signals and Systems
UNIT I CLASSIFICATION OF SIGNALS AND SYSTEMS Continuous time signals (CT signals), discrete time signals (DT signals) Step, Ramp, Pulse, Impulse, Exponential 1. Define Unit Impulse Signal [M/J 1], [M/J
More informationFinal 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 informationDigital 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 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 informationNew 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 informationECE : Linear Circuit Analysis II
Purdue University School of Electrical and Computer Engineering ECE 20200 : Linear Circuit Analysis II Summer 2014 Instructor: Aung Kyi San Instructions: Midterm Examination I July 2, 2014 1. Wait for
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 informationYour solutions for time-domain waveforms should all be expressed as real-valued functions.
ECE-486 Test 2, Feb 23, 2017 2 Hours; Closed book; Allowed calculator models: (a) Casio fx-115 models (b) HP33s and HP 35s (c) TI-30X and TI-36X models. Calculators not included in this list are not permitted.
More informationFinal Exam of ECE301, Prof. Wang s section 8 10am Tuesday, May 6, 2014, EE 129.
Final Exam of ECE301, Prof. Wang s section 8 10am Tuesday, May 6, 2014, EE 129. 1. Please make sure that it is your name printed on the exam booklet. Enter your student ID number, e-mail address, and signature
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 informationThe 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 informationFinal 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 informationSchool of Information Technology and Electrical Engineering EXAMINATION. ELEC3004 Signals, Systems & Control
This exam paper must not be removed from the venue Venue Seat Number Student Number Family Name First Name School of Information Technology and Electrical Engineering EXAMINATION Semester One Final Examinations,
More informationEE 224 Signals and Systems I Review 1/10
EE 224 Signals and Systems I Review 1/10 Class Contents Signals and Systems Continuous-Time and Discrete-Time Time-Domain and Frequency Domain (all these dimensions are tightly coupled) SIGNALS SYSTEMS
More informationSignals 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 informationDefinition of Discrete-Time Fourier Transform (DTFT)
Definition of Discrete-Time ourier Transform (DTT) {x[n]} = X(e jω ) + n= {X(e jω )} = x[n] x[n]e jωn Why use the above awkward notation for the transform? X(e jω )e jωn dω Answer: It is consistent with
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 informationDiscrete Time Fourier Transform
Discrete Time Fourier Transform Recall that we wrote the sampled signal x s (t) = x(kt)δ(t kt). We calculate its Fourier Transform. We do the following: Ex. Find the Continuous Time Fourier Transform of
More informationNew 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 informationReview of Discrete-Time System
Review of Discrete-Time System Electrical & Computer Engineering University of Maryland, College Park Acknowledgment: ENEE630 slides were based on class notes developed by Profs. K.J. Ray Liu and Min Wu.
More 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: 09-Dec-13 COURSE: ECE 3084A (Prof. Michaels) NAME: STUDENT #: LAST, FIRST Write your name on the front page
More informationNAME: 23 February 2017 EE301 Signals and Systems Exam 1 Cover Sheet
NAME: 23 February 2017 EE301 Signals and Systems Exam 1 Cover Sheet Test Duration: 75 minutes Coverage: Chaps 1,2 Open Book but Closed Notes One 85 in x 11 in crib sheet Calculators NOT allowed DO NOT
More informationNAME: 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 informationGrades 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 informationSchool of Information Technology and Electrical Engineering EXAMINATION. ELEC3004 Signals, Systems & Control
This exam paper must not be removed from the venue Venue Seat Number Student Number Family Name First Name School of Information Technology and Electrical Engineering EXAMINATION Semester One Final Examinations,
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 informationDigital Signal Processing Lecture 3 - Discrete-Time Systems
Digital Signal Processing - Discrete-Time Systems Electrical Engineering and Computer Science University of Tennessee, Knoxville August 25, 2015 Overview 1 2 3 4 5 6 7 8 Introduction Three components of
More informationEE 637 Final April 30, Spring Each problem is worth 20 points for a total score of 100 points
EE 637 Final April 30, Spring 2018 Name: Instructions: This is a 120 minute exam containing five problems. Each problem is worth 20 points for a total score of 100 points You may only use your brain and
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 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 informationGrades 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 informationChapter 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 informationECE-314 Fall 2012 Review Questions for Midterm Examination II
ECE-314 Fall 2012 Review Questions for Midterm Examination II First, make sure you study all the problems and their solutions from homework sets 4-7. Then work on the following additional problems. Problem
More 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 informationECE 413 Digital Signal Processing Midterm Exam, Spring Instructions:
University of Waterloo Department of Electrical and Computer Engineering ECE 4 Digital Signal Processing Midterm Exam, Spring 00 June 0th, 00, 5:0-6:50 PM Instructor: Dr. Oleg Michailovich Student s name:
More informationSignals and Systems Spring 2004 Lecture #9
Signals and Systems Spring 2004 Lecture #9 (3/4/04). The convolution Property of the CTFT 2. Frequency Response and LTI Systems Revisited 3. Multiplication Property and Parseval s Relation 4. The DT Fourier
More informationDigital Signal Processing Lecture 10 - Discrete Fourier Transform
Digital Signal Processing - Discrete Fourier Transform Electrical Engineering and Computer Science University of Tennessee, Knoxville November 12, 2015 Overview 1 2 3 4 Review - 1 Introduction Discrete-time
More informationECSE 512 Digital Signal Processing I Fall 2010 FINAL EXAMINATION
FINAL EXAMINATION 9:00 am 12:00 pm, December 20, 2010 Duration: 180 minutes Examiner: Prof. M. Vu Assoc. Examiner: Prof. B. Champagne There are 6 questions for a total of 120 points. This is a closed book
More informationDiscrete-time Fourier transform (DTFT) representation of DT aperiodic signals Section The (DT) Fourier transform (or spectrum) of x[n] is
Discrete-time Fourier transform (DTFT) representation of DT aperiodic signals Section 5. 3 The (DT) Fourier transform (or spectrum) of x[n] is X ( e jω) = n= x[n]e jωn x[n] can be reconstructed from its
More informationlog dx a u = log a e du
Formuls from Trigonometry: sin A cos A = cosa ± B) = cos A cos B sin A sin B sin A = sin A cos A tn A = tn A tn A sina ± B) = sin A cos B ± cos A sin B tn A±tn B tna ± B) = tn A tn B cos A = cos A sin
More informationModule 4. Related web links and videos. 1. FT and ZT
Module 4 Laplace transforms, ROC, rational systems, Z transform, properties of LT and ZT, rational functions, system properties from ROC, inverse transforms Related web links and videos Sl no Web link
More informationFinal 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 informationEach 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 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 informationNew 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 informationGood Luck. EE 637 Final May 4, Spring Name: Instructions: This is a 120 minute exam containing five problems.
EE 637 Final May 4, Spring 200 Name: Instructions: This is a 20 minute exam containing five problems. Each problem is worth 20 points for a total score of 00 points You may only use your brain and a pencil
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 informationDiscrete Time Signals and Systems Time-frequency Analysis. Gloria Menegaz
Discrete Time Signals and Systems Time-frequency Analysis Gloria Menegaz Time-frequency Analysis Fourier transform (1D and 2D) Reference textbook: Discrete time signal processing, A.W. Oppenheim and R.W.
More informationNAME: 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 informationLTI Systems (Continuous & Discrete) - Basics
LTI Systems (Continuous & Discrete) - Basics 1. A system with an input x(t) and output y(t) is described by the relation: y(t) = t. x(t). This system is (a) linear and time-invariant (b) linear and time-varying
More informationHomework 7 Solution EE235, Spring Find the Fourier transform of the following signals using tables: te t u(t) h(t) = sin(2πt)e t u(t) (2)
Homework 7 Solution EE35, Spring. Find the Fourier transform of the following signals using tables: (a) te t u(t) h(t) H(jω) te t u(t) ( + jω) (b) sin(πt)e t u(t) h(t) sin(πt)e t u(t) () h(t) ( ejπt e
More informationUNIVERSITY OF OSLO. Please make sure that your copy of the problem set is complete before you attempt to answer anything.
UNIVERSITY OF OSLO Faculty of mathematics and natural sciences Examination in INF3470/4470 Digital signal processing Day of examination: December 9th, 011 Examination hours: 14.30 18.30 This problem set
More informationSignals & Systems. Lecture 5 Continuous-Time Fourier Transform. Alp Ertürk
Signals & Systems Lecture 5 Continuous-Time Fourier Transform Alp Ertürk alp.erturk@kocaeli.edu.tr Fourier Series Representation of Continuous-Time Periodic Signals Synthesis equation: x t = a k e jkω
More informationQuestion 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 informationFourier series for continuous and discrete time signals
8-9 Signals and Systems Fall 5 Fourier series for continuous and discrete time signals The road to Fourier : Two weeks ago you saw that if we give a complex exponential as an input to a system, the output
More informationProfessor Fearing EECS120/Problem Set 2 v 1.01 Fall 2016 Due at 4 pm, Fri. Sep. 9 in HW box under stairs (1st floor Cory) Reading: O&W Ch 1, Ch2.
Professor Fearing EECS120/Problem Set 2 v 1.01 Fall 20 Due at 4 pm, Fri. Sep. 9 in HW box under stairs (1st floor Cory) Reading: O&W Ch 1, Ch2. Note: Π(t) = u(t + 1) u(t 1 ), and r(t) = tu(t) where u(t)
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 informationEE 438 Essential Definitions and Relations
May 2004 EE 438 Essential Definitions and Relations CT Metrics. Energy E x = x(t) 2 dt 2. Power P x = lim T 2T T / 2 T / 2 x(t) 2 dt 3. root mean squared value x rms = P x 4. Area A x = x(t) dt 5. Average
More informationFinal Exam 14 May LAST Name FIRST Name Lab Time
EECS 20n: Structure and Interpretation of Signals and Systems Department of Electrical Engineering and Computer Sciences UNIVERSITY OF CALIFORNIA BERKELEY Final Exam 14 May 2005 LAST Name FIRST Name Lab
More informationProblem 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 informationModule 4 : Laplace and Z Transform Problem Set 4
Module 4 : Laplace and Z Transform Problem Set 4 Problem 1 The input x(t) and output y(t) of a causal LTI system are related to the block diagram representation shown in the figure. (a) Determine a differential
More informationECE 3620: Laplace Transforms: Chapter 3:
ECE 3620: Laplace Transforms: Chapter 3: 3.1-3.4 Prof. K. Chandra ECE, UMASS Lowell September 21, 2016 1 Analysis of LTI Systems in the Frequency Domain Thus far we have understood the relationship between
More informationECE301 Fall, 2006 Exam 1 Soluation October 7, Name: Score: / Consider the system described by the differential equation
ECE301 Fall, 2006 Exam 1 Soluation October 7, 2006 1 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
More informationSchool of Information Technology and Electrical Engineering EXAMINATION. ELEC3004 Signals, Systems & Control
This exam paper must not be removed from the venue Venue Seat Number Student Number Family Name First Name School of Information Technology and Electrical Engineering EXAMINATION This paper is for St Lucia
More informationECE 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 informationDiscrete Time Systems
1 Discrete Time Systems {x[0], x[1], x[2], } H {y[0], y[1], y[2], } Example: y[n] = 2x[n] + 3x[n-1] + 4x[n-2] 2 FIR and IIR Systems FIR: Finite Impulse Response -- non-recursive y[n] = 2x[n] + 3x[n-1]
More information6.003 (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 informationCommunication Theory Summary of Important Definitions and Results
Signal and system theory Convolution of signals x(t) h(t) = y(t): Fourier Transform: Communication Theory Summary of Important Definitions and Results X(ω) = X(ω) = y(t) = X(ω) = j x(t) e jωt dt, 0 Properties
More informationQ1 Q2 Q3 Q4 Q5 Total
EE 120: Signals & Systems Department of Electrical Engineering and Computer Sciences University of California, Berkeley Midterm Exam #1 February 29, 2016, 2:10-4:00pm Instructions: There are five questions
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 informationECE 301 Fall 2010 Division 2 Homework 10 Solutions. { 1, if 2n t < 2n + 1, for any integer n, x(t) = 0, if 2n 1 t < 2n, for any integer n.
ECE 3 Fall Division Homework Solutions Problem. Reconstruction of a continuous-time signal from its samples. Consider the following periodic signal, depicted below: {, if n t < n +, for any integer n,
More informationZ-TRANSFORMS. Solution: Using the definition (5.1.2), we find: for case (b). y(n)= h(n) x(n) Y(z)= H(z)X(z) (convolution) (5.1.
84 5. Z-TRANSFORMS 5 z-transforms Solution: Using the definition (5..2), we find: for case (a), and H(z) h 0 + h z + h 2 z 2 + h 3 z 3 2 + 3z + 5z 2 + 2z 3 H(z) h 0 + h z + h 2 z 2 + h 3 z 3 + h 4 z 4
More informationProblem 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 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 informationX. Chen More on Sampling
X. Chen More on Sampling 9 More on Sampling 9.1 Notations denotes the sampling time in second. Ω s = 2π/ and Ω s /2 are, respectively, the sampling frequency and Nyquist frequency in rad/sec. Ω and ω denote,
More informationChap 4. Sampling of Continuous-Time Signals
Digital Signal Processing Chap 4. Sampling of Continuous-Time Signals Chang-Su Kim Digital Processing of Continuous-Time Signals Digital processing of a CT signal involves three basic steps 1. Conversion
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 informationEEE4001F EXAM DIGITAL SIGNAL PROCESSING. University of Cape Town Department of Electrical Engineering PART A. June hours.
EEE400F EXAM DIGITAL SIGNAL PROCESSING PART A Basic digital signal processing theory.. A sequencex[n] has a zero-phase DTFT X(e jω ) given below: X(e jω ) University of Cape Town Department of Electrical
More informationso mathematically we can say that x d [n] is a discrete-time signal. The output of the DT system is also discrete, denoted by y d [n].
ELEC 36 LECURE NOES WEEK 9: Chapters 7&9 Chapter 7 (cont d) Discrete-ime Processing of Continuous-ime Signals It is often advantageous to convert a continuous-time signal into a discrete-time signal so
More information! Introduction. ! Discrete Time Signals & Systems. ! Z-Transform. ! Inverse Z-Transform. ! Sampling of Continuous Time Signals
ESE 531: Digital Signal Processing Lec 25: April 24, 2018 Review Course Content! Introduction! Discrete Time Signals & Systems! Discrete Time Fourier Transform! Z-Transform! Inverse Z-Transform! Sampling
More informationECE 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 informationSignals and Systems Lecture 8: Z Transform
Signals and Systems Lecture 8: Z Transform Farzaneh Abdollahi Department of Electrical Engineering Amirkabir University of Technology Winter 2012 Farzaneh Abdollahi Signal and Systems Lecture 8 1/29 Introduction
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 informationReview of Fundamentals of Digital Signal Processing
Solution Manual for Theory and Applications of Digital Speech Processing by Lawrence Rabiner and Ronald Schafer Click here to Purchase full Solution Manual at http://solutionmanuals.info Link download
More informationChap 2. Discrete-Time Signals and Systems
Digital Signal Processing Chap 2. Discrete-Time Signals and Systems Chang-Su Kim Discrete-Time Signals CT Signal DT Signal Representation 0 4 1 1 1 2 3 Functional representation 1, n 1,3 x[ n] 4, n 2 0,
More informationESE 531: Digital Signal Processing
ESE 531: Digital Signal Processing Lec 6: January 30, 2018 Inverse z-transform Lecture Outline! z-transform " Tie up loose ends " Regions of convergence properties! Inverse z-transform " Inspection " Partial
More informationSignals 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 informationlog dx a u = log a e du
Formuls from Trigonometry: sin A cos A = cosa ± B = cos A cos B sin A sin B sin A = sin A cos A tn A = tn A tn A sina ± B = sin A cos B ± cos A sin B tn A±tn B tna ± B = tn A tn B cos A = cos A sin A sin
More informationEECS 20N: Structure and Interpretation of Signals and Systems Final Exam Department of Electrical Engineering and Computer Sciences 13 December 2005
EECS 20N: Structure and Interpretation of Signals and Systems Final Exam Department of Electrical Engineering and Computer Sciences 13 December 2005 UNIVERSITY OF CALIFORNIA BERKELEY LAST Name FIRST Name
More informationHomework 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
More informationSignals & Systems Handout #4
Signals & Systems Handout #4 H-4. Elementary Discrete-Domain Functions (Sequences): Discrete-domain functions are defined for n Z. H-4.. Sequence Notation: We use the following notation to indicate the
More information