EE 224 Signals and Systems I Review 1/10
|
|
- Ruth Burns
- 6 years ago
- Views:
Transcription
1 EE 224 Signals and Systems I Review 1/10
2 Class Contents Signals and Systems Continuous-Time and Discrete-Time Time-Domain and Frequency Domain (all these dimensions are tightly coupled) SIGNALS SYSTEMS TIME CT signals DT signals CT LTI: IR, Conv DT LTI: IR, Conv FREQ CTFS, CTFT, Laplace (sampling, modulation) DTFS, DTFT, Z CT LTI: FR, Multipl. DT LTI: FR, Multipl. 2/10
3 Signals in Time Domain Properties of signals: periodicity even/odd symmetry signal energy/power Time transformations of signals: shift reversal scaling combinations of these 3/10 a
4 Signals in Time Domain CT basic signals Sinusoids and phasor addtiion Complex exponentials e jωt Real exponentials e at δ(t) and u(t) x(t) δ(t τ) = x(τ)δ(t τ) x(t) δ(t τ)dt = x(τ) x(t) δ(t t 0 ) = x(t t 0 ) 3/10 b
5 Signals in Time Domain DT basic signals Unit Impulse δ[n] Exponential signals, especially e jωn Sinusoids cos(ωn + θ) Unit step function u[n] 3/10 c
6 Signals in FREQ domain: Fourier Analysis FS FT 1 1/5 7f 0 Spectrum 5f 0 1/3 3f 0 1/7 f sin(2π7f 0t) 1 5 sin(2π5f 0t) 1 3 sin(2π3f 0t) ω f 0 sin(2π f 0 t) e jωt t t 4/10 a
7 Signals in FREQ domain: Fourier Analysis CT DT Periodic (FS) x(t) = CTFS a k = 1 T 0 DTFS k= <T 0 > a k e jkω 0t x(t)e jkω 0t dt x[n] = N 0 1 k=0 a k e jkω 0n a k = 1 N 0 <N 0 > x[n]e jkω 0n Non-Periodic (FT) x(t) = 1 2π x[n] = 1 2π CTFT X(ω)ejωt dω X(ω) = x(t)e jωt dt DTFT π π X(Ω)ejΩn dω X(Ω) = k= x[n]e jωn 4/10 b
8 Signals in FREQ domain: Fourier Analysis Properties of CT Fourier Transform x(t) X(ω) Time Frequency Duality X(t) 2πx( ω) Linear ax(t) + by(t) ax(ω) + by (ω) Conv. x(t) y(t) X(ω) Y (ω) 1 Modu. x(t) y(t) 2π X(ω) Y (ω) Int. x(t) u(t) = t x(τ)dτ 1 jω X(ω) + πx(0) δ(ω) Diff x(t) δ (t) = d dtx(t) jω X(ω) derivative ω tx(t) j d dω X(ω) time shift x(t τ) X(ω) e jωτ freq shift e jω0t x(t) X(ω ω 0 ) 1 scaling x(at) a X ( ) ω a time reversal x( t) X( ω) conjugation x (t) X ( ω) Parseval x(t) y 1 (t)dt X(ω) Y (ω)dω F 2π 4/10 c
9 Signals in FREQ domain: Fourier Analysis FT of periodic signals Define one period of x(t) as x T0 (t), which is equal to x(t) over one period (any period), and zero outside that period. Then x(t) = x T0 (t) i= δ(t it 0). As a result, the spectrum is an envelop shape sampled by deltas: X(jω) = X T0 (jω) k= 2π δ(ω k 2π ). T 0 T 0 This gives the envelop of the spectrum This gives the location of the Deltas 4/10 d
10 Signals in FREQ domain: Sampling Impulse Train Sampling time frequency 5/10 a
11 Signals in FREQ domain: Sampling If x(t) is bandlimited so that Nyquist-Shannon Sampling Theorem X(jω) = 0 for ω > ω M, then x(t) can be uniquely determined by its samples {x(nt )} if ω s 2π T > 2ω M. 5/10 b
12 Signals in FREQ domain: Sampling Aliasing when sampling below Nyquist rate no aliasing aliasing 0 ωs ω s ω 2 ω 5/10 c
13 Signals in FREQ domain: Sampling Aliasing when sampling below Nyquist rate spectrum of cosine 2 2 f spectrum of impulse train spectrum of sampled signal /10 d
14 Signals in FREQ domain: Sampling Anti-Aliasing Filter analog LPF, ω s /2 anti-aliasing filter sample and hold ADC digital without anti-aliasing filter ω with anti-aliasing filter ω 5/10 e
15 Signals in FREQ domain: Sampling Reconstruction Ideal reconstruction Zero-order hold Linear interpolation h 0 (t nt ) h 1 (t nt ) 5/10 f
16 Signals in FREQ domain: Modulation DSB AM Modulation 6/10 a
17 Signals in FREQ domain: Modulation DSB AM Demodulation Other modulation: AM with carrier and envelop detector; SSB; Frequency-division multiplexing; Pulse amplitude modulation; PM, FM. 6/10 b
18 Systems in TIME domain System properties: Memory Invertibility Causality Stability Time Invariance Linearity Properties of LTI Systems Commutative Property, Distributive Property, Associative Property, System Memory, Invertibility, Causality, Stability, Unit Step Response 7/10
19 Systems in TIME domain Impulse Response input DT LTI systems DT LTI System h[n] output Convolution y[n] = x[n] h[n] = k= x[k]h[n k] Properties of convolution: linear, commutative, associative Step response: response to u[n] 8/10 a
20 Systems in TIME domain DT LTI systems Simple convolutions x[n] δ[n] = x[n] x[n] δ[n n 0 ] = x[n n 0 ] x[n] u[n] = n k= x[k] Graphical approach of doing convolution Fix one signal, say x[k] Flip the other signal h[k] and shift right by n to obtain h[n k] (as a function of k) Multiply x[k] and h[n k] Sum over k Repeat for another n Can also do convolution using Fourier transform 8/10 b
21 Systems in TIME domain Causal LTI Systems Described by Difference Equations Example N k=0 DT LTI systems a k y[n k] = F [0] = 0, F [1] = 1 Important assumption for LTI and causality: initially at rest The complete solution can be written as the sum of two parts: homogeneous solution (natural response) particular solution M k=0 b k x[n k] F [n] F [n 1] F [n 2] = 0 8/10 c
22 Systems in TIME domain Impulse Response CT LTI Systems h(t) input CT LTI System output Convolution LTI y(t) = x(t) h(t) y(t) = x(t) h(t) = Step Response: response to u(t) x(τ)h(t τ)dτ 9/10 a
23 Systems in TIME domain CT LTI Systems Properties of convolution: linear, commutative, associative Simple Convolutions x(t) δ(t) = x(t) x(t) δ(t t 0 ) = x(t t 0 ) x(t) u(t) = t x(τ)dτ Graphical Approach Flip h(τ) to obtain h( τ) Shift h( τ) to right by t to obtain h(t τ) Multiple x(τ) and h(t τ) Integrate on τ to obtain x(τ)h(t τ)dτ Increase t and repeat 9/10 b
24 Systems in TIME domain Causal LTI Systems Described by Differential Equations Example: dy(t) dt N k=0 CT LTI Systems a k d k y(t) dt k = M k=0 b k d k x(t) dt k + 2y(t) = x(t), where x(t) = Ke 3t u(t) Important assumption for LTI and causality: initially at rest The complete solution can be written as the sum of two parts: homogeneous solution (natural response) particular solution 9/10 c
25 System in FREQ domain Frequency Response e jωn input DT LTI System H(e jω )e jωn output e jωt input CT LTI System H(ω)e jωt output Input-Output Relationship X(ω) H(ω) Y (ω) = X(ω)H(ω) 10/10 a
26 System in FREQ domain Relationship between IR and FR h(t) F H(jω) h[n] F H(e jω ) 10/10 b
27 System in FREQ domain Magnitude response and phase response Y (jω) = H(jω)X(jω) Y (jω) = H(jω) X(jω) Y (jω) = H(jω) + X(jω) Group delay τ(ω) = d dω { H(jω)} Bode plot for CT systems Magnitude in db, versus freq in log scale Phase in linear scale H(jω), versus freq in log scale 10/10 c
28 System in FREQ domain CT Ideal LPF DT FR IR inversely proportional overshoot SR ringing 10/10 d
29 System in FREQ domain Systems Characterized by Differential Equations N k=0 a k dk y(t) dt k = M k=0 b k dk x(t) dt k F { N k=0 a k dk y(t) dt k } = F { M } k=0 b k dk x(t) dt k N k=0 a k(jω) k Y (jω) = M k=0 b k(jω) k X(jω) H(jω) = Y (jω) X(jω) = M k=0 b k(jω) k N k=0 a k(jω) k Impulse response can be obtained by inverse FT 10/10 e
30 System in FREQ domain DSP of CT Signals If the sampling theorem condition is satisfied, namely, ω M < ω s 2 then the overall system is LTI, and Y c (jω) = H c (jω)x c (jω) H c (jω) = { H d (e jωt ), ω < ω s /2 0, ω > ω s /2 10/10 f
Homework 4. May An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt
Homework 4 May 2017 1. An LTI system has an input, x(t) and output y(t) related through the equation y(t) = t e (t t ) x(t 2)dt Determine the impulse response of the system. Rewriting as y(t) = t e (t
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 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 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 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 informationChapter 5 Frequency Domain Analysis of Systems
Chapter 5 Frequency Domain Analysis of Systems CT, LTI Systems Consider the following CT LTI system: xt () ht () yt () Assumption: the impulse response h(t) is absolutely integrable, i.e., ht ( ) dt< (this
More 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 informationProperties 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 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 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 informationESE 531: Digital Signal Processing
ESE 531: Digital Signal Processing Lec 8: February 12th, 2019 Sampling and Reconstruction Lecture Outline! Review " Ideal sampling " Frequency response of sampled signal " Reconstruction " Anti-aliasing
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 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 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 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 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 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 informationChapter 5 Frequency Domain Analysis of Systems
Chapter 5 Frequency Domain Analysis of Systems CT, LTI Systems Consider the following CT LTI system: xt () ht () yt () Assumption: the impulse response h(t) is absolutely integrable, i.e., ht ( ) dt< (this
More 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 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 informationESE 531: Digital Signal Processing
ESE 531: Digital Signal Processing Lec 8: February 7th, 2017 Sampling and Reconstruction Lecture Outline! Review " Ideal sampling " Frequency response of sampled signal " Reconstruction " Anti-aliasing
More informationx[n] = x a (nt ) x a (t)e jωt dt while the discrete time signal x[n] has the discrete-time Fourier transform x[n]e jωn
Sampling Let x a (t) be a continuous time signal. The signal is sampled by taking the signal value at intervals of time T to get The signal x(t) has a Fourier transform x[n] = x a (nt ) X a (Ω) = x a (t)e
More informationLecture 13: Discrete Time Fourier Transform (DTFT)
Lecture 13: Discrete Time Fourier Transform (DTFT) ECE 401: Signal and Image Analysis University of Illinois 3/9/2017 1 Sampled Systems Review 2 DTFT and Convolution 3 Inverse DTFT 4 Ideal Lowpass Filter
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 informationDiscussion Section #2, 31 Jan 2014
Discussion Section #2, 31 Jan 2014 Lillian Ratliff 1 Unit Impulse The unit impulse (Dirac delta) has the following properties: { 0, t 0 δ(t) =, t = 0 ε ε δ(t) = 1 Remark 1. Important!: An ordinary function
More informationReview of Frequency Domain Fourier Series: Continuous periodic frequency components
Today we will review: Review of Frequency Domain Fourier series why we use it trig form & exponential form how to get coefficients for each form Eigenfunctions what they are how they relate to LTI systems
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 informationOverview of Sampling Topics
Overview of Sampling Topics (Shannon) sampling theorem Impulse-train sampling Interpolation (continuous-time signal reconstruction) Aliasing Relationship of CTFT to DTFT DT processing of CT signals DT
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 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 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 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 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 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 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 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 informationECE 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 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 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 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 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 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 informationELEN E4810: Digital Signal Processing Topic 11: Continuous Signals. 1. Sampling and Reconstruction 2. Quantization
ELEN E4810: Digital Signal Processing Topic 11: Continuous Signals 1. Sampling and Reconstruction 2. Quantization 1 1. Sampling & Reconstruction DSP must interact with an analog world: A to D D to A x(t)
More informationComplex symmetry Signals and Systems Fall 2015
18-90 Signals and Systems Fall 015 Complex symmetry 1. Complex symmetry This section deals with the complex symmetry property. As an example I will use the DTFT for a aperiodic discrete-time signal. The
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 informationJ. McNames Portland State University ECE 223 Sampling Ver
Overview of Sampling Topics (Shannon) sampling theorem Impulse-train sampling Interpolation (continuous-time signal reconstruction) Aliasing Relationship of CTFT to DTFT DT processing of CT signals DT
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 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 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 informationLecture 3 January 23
EE 123: Digital Signal Processing Spring 2007 Lecture 3 January 23 Lecturer: Prof. Anant Sahai Scribe: Dominic Antonelli 3.1 Outline These notes cover the following topics: Eigenvectors and Eigenvalues
More informationMathematical Foundations of Signal Processing
Mathematical Foundations of Signal Processing Module 4: Continuous-Time Systems and Signals Benjamín Béjar Haro Mihailo Kolundžija Reza Parhizkar Adam Scholefield October 24, 2016 Continuous Time Signals
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 informationContinuous-Time Fourier Transform
Signals and Systems Continuous-Time Fourier Transform Chang-Su Kim continuous time discrete time periodic (series) CTFS DTFS aperiodic (transform) CTFT DTFT Lowpass Filtering Blurring or Smoothing Original
More informationMEDE2500 Tutorial Nov-7
(updated 2016-Nov-4,7:40pm) MEDE2500 (2016-2017) Tutorial 3 MEDE2500 Tutorial 3 2016-Nov-7 Content 1. The Dirac Delta Function, singularity functions, even and odd functions 2. The sampling process and
More informationECE 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 information2.1 Basic Concepts Basic operations on signals Classication of signals
Haberle³me Sistemlerine Giri³ (ELE 361) 9 Eylül 2017 TOBB Ekonomi ve Teknoloji Üniversitesi, Güz 2017-18 Dr. A. Melda Yüksel Turgut & Tolga Girici Lecture Notes Chapter 2 Signals and Linear Systems 2.1
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 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 information4.1 Introduction. 2πδ ω (4.2) Applications of Fourier Representations to Mixed Signal Classes = (4.1)
4.1 Introduction Two cases of mixed signals to be studied in this chapter: 1. Periodic and nonperiodic signals 2. Continuous- and discrete-time signals Other descriptions: Refer to pp. 341-342, textbook.
More informationEECE 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 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 informationLecture Schedule: Week Date Lecture Title
http://elec34.org Sampling and CONVOLUTION 24 School of Information Technology and Electrical Engineering at The University of Queensland Lecture Schedule: Week Date Lecture Title 2-Mar Introduction 3-Mar
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 information2A1H Time-Frequency Analysis II
2AH Time-Frequency Analysis II Bugs/queries to david.murray@eng.ox.ac.uk HT 209 For any corrections see the course page DW Murray at www.robots.ox.ac.uk/ dwm/courses/2tf. (a) A signal g(t) with period
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 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: Continuous Fourier Transform
Review: Continuous Fourier Transform Review: convolution x t h t = x τ h(t τ)dτ Convolution in time domain Derivation Convolution Property Interchange the order of integrals Let Convolution Property By
More informationUniversity Question Paper Solution
Unit 1: Introduction University Question Paper Solution 1. Determine whether the following systems are: i) Memoryless, ii) Stable iii) Causal iv) Linear and v) Time-invariant. i) y(n)= nx(n) ii) y(t)=
More 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 informationENT 315 Medical Signal Processing CHAPTER 2 DISCRETE FOURIER TRANSFORM. Dr. Lim Chee Chin
ENT 315 Medical Signal Processing CHAPTER 2 DISCRETE FOURIER TRANSFORM Dr. Lim Chee Chin Outline Introduction Discrete Fourier Series Properties of Discrete Fourier Series Time domain aliasing due to frequency
More informationMaterial presented here is from the course 6.003, Signals & Systems offered by MIT faculty member, Prof. Alan Willsky, Copyright c 2003.
EE-295 Image Processing, Spring 2008 Lecture 1 Material presented here is from the course 6.003, Signals & Systems offered by MIT faculty member, Prof. Alan Willsky, Copyright c 2003. This material is
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 informationECGR4124 Digital Signal Processing Midterm Spring 2010
ECGR4124 Digital Signal Processing Midterm Spring 2010 Name: LAST 4 DIGITS of Student Number: Do NOT begin until told to do so Make sure that you have all pages before starting Open book, 1 sheet front/back
More informationCosc 3451 Signals and Systems. What is a system? Systems Terminology and Properties of Systems
Cosc 3451 Signals and Systems Systems Terminology and Properties of Systems What is a system? an entity that manipulates one or more signals to yield new signals (often to accomplish a function) can be
More 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 informationEE 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 informationLike 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 information1.1 SPECIAL FUNCTIONS USED IN SIGNAL PROCESSING. δ(t) = for t = 0, = 0 for t 0. δ(t)dt = 1. (1.1)
SIGNAL THEORY AND ANALYSIS A signal, in general, refers to an electrical waveform whose amplitude varies with time. Signals can be fully described in either the time or frequency domain. This chapter discusses
More informationFourier Representations of Signals & LTI Systems
3. Introduction. A signal can be represented as a weighted superposition of complex sinusoids. x(t) or x[n] 2. LTI system: LTI System Output = A weighted superposition of the system response to each complex
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 informationLecture 8: Signal Reconstruction, DT vs CT Processing. 8.1 Reconstruction of a Band-limited Signal from its Samples
EE518 Digital Signal Processing University of Washington Autumn 2001 Dept. of Electrical Engineering Lecture 8: Signal Reconstruction, D vs C Processing Oct 24, 2001 Prof: J. Bilmes
More informationSignal and systems. Linear Systems. Luigi Palopoli. Signal and systems p. 1/5
Signal and systems p. 1/5 Signal and systems Linear Systems Luigi Palopoli palopoli@dit.unitn.it Wrap-Up Signal and systems p. 2/5 Signal and systems p. 3/5 Fourier Series We have see that is a signal
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 information信號與系統 Signals and Systems
Spring 2015 信號與系統 Signals and Systems Chapter SS-2 Linear Time-Invariant Systems Feng-Li Lian NTU-EE Feb15 Jun15 Figures and images used in these lecture notes are adopted from Signals & Systems by Alan
More 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 informationFrequency-Domain C/S of LTI Systems
Frequency-Domain C/S of LTI Systems x(n) LTI y(n) LTI: Linear Time-Invariant system h(n), the impulse response of an LTI systems describes the time domain c/s. H(ω), the frequency response describes the
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 informationGEORGIA INSTITUTE OF TECHNOLOGY SCHOOL of ELECTRICAL and COMPUTER ENGINEERING
GEORGIA INSIUE OF ECHNOLOGY SCHOOL of ELECRICAL and COMPUER ENGINEERING ECE 6250 Spring 207 Problem Set # his assignment is due at the beginning of class on Wednesday, January 25 Assigned: 6-Jan-7 Due
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 informationSIGNAL PROCESSING. B14 Option 4 lectures. Stephen Roberts
SIGNAL PROCESSING B14 Option 4 lectures Stephen Roberts Recommended texts Lynn. An introduction to the analysis and processing of signals. Macmillan. Oppenhein & Shafer. Digital signal processing. Prentice
More informationEC6303 SIGNALS AND SYSTEMS
EC 6303-SIGNALS & SYSTEMS UNIT I CLASSIFICATION OF SIGNALS AND SYSTEMS 1. Define Signal. Signal is a physical quantity that varies with respect to time, space or a n y other independent variable.(or) It
More information2A1H Time-Frequency Analysis II Bugs/queries to HT 2011 For hints and answers visit dwm/courses/2tf
Time-Frequency Analysis II (HT 20) 2AH 2AH Time-Frequency Analysis II Bugs/queries to david.murray@eng.ox.ac.uk HT 20 For hints and answers visit www.robots.ox.ac.uk/ dwm/courses/2tf David Murray. A periodic
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 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 information6.003: Signals and Systems. Sampling and Quantization
6.003: Signals and Systems Sampling and Quantization December 1, 2009 Last Time: Sampling and Reconstruction Uniform sampling (sampling interval T ): x[n] = x(nt ) t n Impulse reconstruction: x p (t) =
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 informationFourier Analysis Overview (0B)
CTFS: Continuous Time Fourier Series CTFT: Continuous Time Fourier Transform DTFS: Fourier Series DTFT: Fourier Transform DFT: Discrete Fourier Transform Copyright (c) 2009-2016 Young W. Lim. Permission
More information2 M. Hasegawa-Johnson. DRAFT COPY.
Lecture Notes in Speech Production Speech Coding and Speech Recognition Mark Hasegawa-Johnson University of Illinois at Urbana-Champaign February 7 2000 2 M. Hasegawa-Johnson. DRAFT COPY. Chapter Basics
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 information4 The Continuous Time Fourier Transform
96 4 The Continuous Time ourier Transform ourier (or frequency domain) analysis turns out to be a tool of even greater usefulness Extension of ourier series representation to aperiodic signals oundation
More information