Section 3 Discrete-Time Signals EO 2402 Summer /05/2013 EO2402.SuFY13/MPF Section 3 1
|
|
- Christian Reeves
- 5 years ago
- Views:
Transcription
1 Section 3 Discrete-Time Signals EO 2402 Summer /05/2013 EO2402.SuFY13/MPF Section 3 1
2 [p. 3] Discrete-Time Signal Description Sampling, sampling theorem Discrete sinusoidal signal Discrete exponential signal Unit impulse function Unit sequence function Signum function Unit ramp signal Periodic impulse function Even/Odd Products of Even and Odd Signals [p. 21] Symmetric Finite Summation [p. 22] Periodic Signal [p. 26] Energy & Power Signals 07/05/2013 EO2402.SuFY13/MPF Section 3 2
3 Discrete-Time Signal Description - Sampling - A discrete signal may be obtained by sampling a continuous signal - Sampling refers to acquiring values of a signal at discrete points in time xn [ ] xn ( ) xnt ( ), where T is the time s between samples s 07/05/2013 EO2402.SuFY13/MPF Section 3 3
4 Discrete Sinusoidal Signal xn [ ] Acos(2 f n ) Question: Is x(n) necessarily periodic? 0 07/05/2013 EO2402.SuFY13/MPF Section 3 4
5 Is x[n] a periodic signal? 07/05/2013 EO2402.SuFY13/MPF Section 3 5
6 Question: Is there a unique association between an analog signal and its discrete derivation? 07/05/2013 EO2402.SuFY13/MPF Section 3 6
7 07/05/2013 EO2402.SuFY13/MPF Section 3 7
8 Sampling Theorem Sampling Theorem: A unique association between an analog signal and its discrete derivation is guaranteed provided that the sampling rate f s used to sample the continuous-time signal x(t) is equal to at least twice the maximum frequency f max contained in the signal x(t) Nyquist rate: Sampling rate when selected as f s =2f max Definitions: When f s >2f max, signal is said to be oversampled When f s <2f max, signal is said to be undersampled (leads to aliasing) 07/05/2013 EO2402.SuFY13/MPF Section 3 8
9 Discrete Exponential Signal Real n x[ n] A Ae n Complex Real/imaginary plots 07/05/2013 EO2402.SuFY13/MPF Section 3 9
10 Discrete Exponential Signal, cont Real n x[ n] A Ae n Complex Magnitude/Phase plots 07/05/2013 EO2402.SuFY13/MPF Section 3 10
11 The Unit Impulse Function n 1, n 0 0, n 0 [n] is defined as the discrete-time unit impulse ( Kronecker delta function) [n] has a sampling property, n x x A n n n A n 0 0 [n] does NOT have a scaling property. n an, a 0. 07/05/2013 EO2402.SuFY13/MPF Section 3 11
12 Unit Sequence Function u n 1, n 0 0, n 0 07/05/2013 EO2402.SuFY13/MPF Section 3 12
13 The Signum Function 1, n 0 sgn n 0, n 0 1, n 0 07/05/2013 EO2402.SuFY13/MPF Section 3 13
14 Unit Ramp Function ramp n n, n 0 0, n 0 nu n n m um 1 07/05/2013 EO2402.SuFY13/MPF Section 3 14
15 Periodic Impulse Function N n m n mn 07/05/2013 EO2402.SuFY13/MPF Section 3 15
16 07/05/2013 EO2402.SuFY13/MPF Section 3 16
17 Even / Odd Signals - Definition: A Signal x[n] is odd if x[n]=-x[-n] A Signal x[n] is even if x[n]=x[-n] Property: Any signal x[n] can be decomposed into a sum of even and odd components x[ n ] xe [ n] x o [ n ] 0.5 x[ n] x[ n] 0.5 x[ n] x[ n] 07/05/2013 EO2402.SuFY13/MPF Section 3 17
18 Products of Even and Odd Functions Two Even Functions 07/05/2013 EO2402.SuFY13/MPF Section 3 18
19 Products of Even and Odd Function, cont An Even Function and an Odd Function 07/05/2013 EO2402.SuFY13/MPF Section 3 19
20 Products of Even and Odd Functions, cont Two Odd Functions 07/05/2013 EO2402.SuFY13/MPF Section 3 20
21 Symmetric Finite summation (Discrete equivalent of integration) Even function Odd function N N xn [ ] x[0] 2 xn [ ] xn [ ] 0 kn k1 N kn 07/05/2013 EO2402.SuFY13/MPF Section 3 21
22 Periodic Signal A discrete signal xn [ ] is periodic iff 0 xn [ ] xn [ mn], where N and m are integer N 0 is called the Period 0 07/05/2013 EO2402.SuFY13/MPF Section 3 22
23 Example: Find the period of the signal x[n]=cos(2n/36)+sin(10n/24) y[n]=cos(5n/13)+sin(8n/39) 07/05/2013 EO2402.SuFY13/MPF Section 3 23
24 07/05/2013 EO2402.SuFY13/MPF Section 3 24
25 07/05/2013 EO2402.SuFY13/MPF Section 3 25
26 Energy / Power Signals - Definition: The energy in a signal x[n] over the interval N=[N 1,N 2 ] is defined as E N N 2 k N 1 x[ k ] 2 - Definition: The energy in a signal x[n] is defined as: E x k x[ k ] - Definition: A signal with finite energy is called an energy signal 2 07/05/2013 EO2402.SuFY13/MPF Section 3 26
27 Signal Energy/Power, cont Find the signal energy of x n N N 1 n Note: r 1 r n0 1 r N 2 n 5 / 3, 0 n 8 0, otherwise, r 1, r 1 07/05/2013 EO2402.SuFY13/MPF Section 3 27
28 07/05/2013 EO2402.SuFY13/MPF Section 3 28
29 Energy / Power Signals, cont - Definition: The average power of a signal x[n] over the interval N=[N 1,N 2 ] is defined as P N k N N N 2 N 1 x[ k ] 2 - Definition: The average power of a signal x[n] is defined as: P x lim N 1 2 N k N 1 N x k - Definition: For a periodic signal, the average power of x[n] is defined as: 1 2 P x N k N x[ k ] - Definition: A signal with finite power is called a power signal 2 07/05/2013 EO2402.SuFY13/MPF Section 3 29
30 Signal Energy/Power, cont Find the average signal power of n x 2sgn[ n] 4 07/05/2013 EO2402.SuFY13/MPF Section 3 30
31 07/05/2013 EO2402.SuFY13/MPF Section 3 31
32 07/05/2013 EO2402.SuFY13/MPF Section 3 32
33 07/05/2013 EO2402.SuFY13/MPF Section 3 33
Sampling and Discrete Time. Discrete-Time Signal Description. Sinusoids. Sampling and Discrete Time. Sinusoids An Aperiodic Sinusoid.
Sampling and Discrete Time Discrete-Time Signal Description Sampling is the acquisition of the values of a continuous-time signal at discrete points in time. x t discrete-time signal. ( ) is a continuous-time
More informationSystem Processes input signal (excitation) and produces output signal (response)
Signal A funcion of ime Sysem Processes inpu signal (exciaion) and produces oupu signal (response) Exciaion Inpu Sysem Oupu Response 1. Types of signals 2. Going from analog o digial world 3. An example
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 informationChapter 1 Fundamental Concepts
Chapter 1 Fundamental Concepts 1 Signals A signal is a pattern of variation of a physical quantity, often as a function of time (but also space, distance, position, etc). These quantities are usually the
More informationDIGITAL 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 informationCommunication Signals (Haykin Sec. 2.4 and Ziemer Sec Sec. 2.4) KECE321 Communication Systems I
Communication Signals (Haykin Sec..4 and iemer Sec...4-Sec..4) KECE3 Communication Systems I Lecture #3, March, 0 Prof. Young-Chai Ko 년 3 월 일일요일 Review Signal classification Phasor signal and spectra Representation
More informationSEISMIC WAVE PROPAGATION. Lecture 2: Fourier Analysis
SEISMIC WAVE PROPAGATION Lecture 2: Fourier Analysis Fourier Series & Fourier Transforms Fourier Series Review of trigonometric identities Analysing the square wave Fourier Transform Transforms of some
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 informationECE 301 Fall 2011 Division 1. Homework 1 Solutions.
ECE 3 Fall 2 Division. Homework Solutions. Reading: Course information handout on the course website; textbook sections.,.,.2,.3,.4; online review notes on complex numbers. Problem. For each discrete-time
More informationContents. Signals as functions (1D, 2D)
Fourier Transform The idea A signal can be interpreted as en electromagnetic wave. This consists of lights of different color, or frequency, that can be split apart usign an optic prism. Each component
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 informationContents. Signals as functions (1D, 2D)
Fourier Transform The idea A signal can be interpreted as en electromagnetic wave. This consists of lights of different color, or frequency, that can be split apart usign an optic prism. Each component
More informationContents. Signals as functions (1D, 2D)
Fourier Transform The idea A signal can be interpreted as en electromagnetic wave. This consists of lights of different color, or frequency, that can be split apart usign an optic prism. Each component
More informationContinuous Fourier transform of a Gaussian Function
Continuous Fourier transform of a Gaussian Function Gaussian function: e t2 /(2σ 2 ) The CFT of a Gaussian function is also a Gaussian function (i.e., time domain is Gaussian, then the frequency domain
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 informationCombinations and Probabilities
Combinations and Probabilities Tutor: Zhang Qi Systems Engineering and Engineering Management qzhang@se.cuhk.edu.hk November 2014 Tutor: Zhang Qi (SEEM) Tutorial 7 November 2014 1 / 16 Combination Review
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 informationBridge between continuous time and discrete time signals
6 Sampling Bridge between continuous time and discrete time signals Sampling theorem complete representation of a continuous time signal by its samples Samplingandreconstruction implementcontinuous timesystems
More informationChapter 1 Fundamental Concepts
Chapter 1 Fundamental Concepts Signals A signal is a pattern of variation of a physical quantity as a function of time, space, distance, position, temperature, pressure, etc. These quantities are usually
More 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 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 information1 Signals and systems
978--52-5688-4 - Introduction to Orthogonal Transforms: With Applications in Data Processing and Analysis Signals and systems In the first two chapters we will consider some basic concepts and ideas as
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Processing Pro. Mark Fowler Note Set #10 Fourier Analysis or DT Signals eading Assignment: Sect. 4.2 & 4.4 o Proakis & Manolakis Much o Ch. 4 should be review so you are expected
More informationContinuous-time Signals. (AKA analog signals)
Continuous-time Signals (AKA analog signals) I. Analog* Signals review Goals: - Common test signals used in system analysis - Signal operations and properties: scaling, shifting, periodicity, energy 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 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 informationAPPENDIX A. The Fourier integral theorem
APPENDIX A The Fourier integral theorem In equation (1.7) of Section 1.3 we gave a description of a signal defined on an infinite range in the form of a double integral, with no explanation as to how that
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 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 informationFourier transform. Stefano Ferrari. Università degli Studi di Milano Methods for Image Processing. academic year
Fourier transform Stefano Ferrari Università degli Studi di Milano stefano.ferrari@unimi.it Methods for Image Processing academic year 27 28 Function transforms Sometimes, operating on a class of functions
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 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 informationChapter 8 The Discrete Fourier Transform
Chapter 8 The Discrete Fourier Transform Introduction Representation of periodic sequences: the discrete Fourier series Properties of the DFS The Fourier transform of periodic signals Sampling the Fourier
More informationAnalog Digital Sampling & Discrete Time Discrete Values & Noise Digital-to-Analog Conversion Analog-to-Digital Conversion
Analog Digital Sampling & Discrete Time Discrete Values & Noise Digital-to-Analog Conversion Analog-to-Digital Conversion 6.082 Fall 2006 Analog Digital, Slide Plan: Mixed Signal Architecture volts bits
More information3.2 Complex Sinusoids and Frequency Response of LTI Systems
3. Introduction. A signal can be represented as a weighted superposition of complex sinusoids. x(t) or x[n]. LTI system: LTI System Output = A weighted superposition of the system response to each complex
More informationECE 301: Signals and Systems Homework Assignment #7
ECE 301: Signals and Systems Homework Assignment #7 Due on December 11, 2015 Professor: Aly El Gamal TA: Xianglun Mao 1 Aly El Gamal ECE 301: Signals and Systems Homework Assignment #7 Problem 1 Note:
More informationUNIT 1. SIGNALS AND SYSTEM
Page no: 1 UNIT 1. SIGNALS AND SYSTEM INTRODUCTION A SIGNAL is defined as any physical quantity that changes with time, distance, speed, position, pressure, temperature or some other quantity. A SIGNAL
More informationFourier Series Representation of
Fourier Series Representation of Periodic Signals Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it Email: ruiwang@tongji.edu.cn Outline The response of LIT system
More informationChapter 2: Problem Solutions
Chapter 2: Problem Solutions Discrete Time Processing of Continuous Time Signals Sampling à Problem 2.1. Problem: Consider a sinusoidal signal and let us sample it at a frequency F s 2kHz. xt 3cos1000t
More informationEE301 Signals and Systems In-Class Exam Exam 3 Thursday, Apr. 20, Cover Sheet
NAME: NAME EE301 Signals and Systems In-Class Exam Exam 3 Thursday, Apr. 20, 2017 Cover Sheet Test Duration: 75 minutes. Coverage: Chaps. 5,7 Open Book but Closed Notes. One 8.5 in. x 11 in. crib sheet
More informationFourier Series Summary (From Salivahanan et al, 2002)
Fourier Series Suary (Fro Salivahanan et al, ) A periodic continuous signal f(t), - < t
More information# FIR. [ ] = b k. # [ ]x[ n " k] [ ] = h k. x[ n] = Ae j" e j# ˆ n Complex exponential input. [ ]Ae j" e j ˆ. ˆ )Ae j# e j ˆ. y n. y n.
[ ] = h k M [ ] = b k x[ n " k] FIR k= M [ ]x[ n " k] convolution k= x[ n] = Ae j" e j ˆ n Complex exponential input [ ] = h k M % k= [ ]Ae j" e j ˆ % M = ' h[ k]e " j ˆ & k= k = H (" ˆ )Ae j e j ˆ ( )
More informationECE-700 Review. Phil Schniter. January 5, x c (t)e jωt dt, x[n]z n, Denoting a transform pair by x[n] X(z), some useful properties are
ECE-7 Review Phil Schniter January 5, 7 ransforms Using x c (t) to denote a continuous-time signal at time t R, Laplace ransform: X c (s) x c (t)e st dt, s C Continuous-ime Fourier ransform (CF): ote that:
More informationDiscrete-Time Signals: Time-Domain Representation
Discrete-Time Signals: Time-Domain Representation 1 Signals represented as sequences of numbers, called samples Sample value of a typical signal or sequence denoted as x[n] with n being an integer in the
More informationDiscrete-Time Signals: Time-Domain Representation
Discrete-Time Signals: Time-Domain Representation 1 Signals represented as sequences of numbers, called samples Sample value of a typical signal or sequence denoted as x[n] with n being an integer in the
More informationDiscrete-time Signals and Systems in
Discrete-time Signals and Systems in the Frequency Domain Chapter 3, Sections 3.1-39 3.9 Chapter 4, Sections 4.8-4.9 Dr. Iyad Jafar Outline Introduction The Continuous-Time FourierTransform (CTFT) The
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 information!Sketch f(t) over one period. Show that the Fourier Series for f(t) is as given below. What is θ 1?
Second Year Engineering Mathematics Laboratory Michaelmas Term 998 -M L G Oldfield 30 September, 999 Exercise : Fourier Series & Transforms Revision 4 Answer all parts of Section A and B which are marked
More informationFourier Series and Fourier Transforms
Fourier Series and Fourier Transforms EECS2 (6.082), MIT Fall 2006 Lectures 2 and 3 Fourier Series From your differential equations course, 18.03, you know Fourier s expression representing a T -periodic
More informationBME 50500: Image and Signal Processing in Biomedicine. Lecture 2: Discrete Fourier Transform CCNY
1 Lucas Parra, CCNY BME 50500: Image and Signal Processing in Biomedicine Lecture 2: Discrete Fourier Transform Lucas C. Parra Biomedical Engineering Department CCNY http://bme.ccny.cuny.edu/faculty/parra/teaching/signal-and-image/
More informationEE123 Digital Signal Processing
EE123 Digital Signal Processing Lecture 2 Discrete Time Systems Today Last time: Administration Overview Announcement: HW1 will be out today Lab 0 out webcast out Today: Ch. 2 - Discrete-Time Signals and
More informationTutorial Sheet #2 discrete vs. continuous functions, periodicity, sampling
2.39 utorial Sheet #2 discrete vs. continuous functions, periodicity, sampling We will encounter two classes of signals in this class, continuous-signals and discrete-signals. he distinct mathematical
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 informationConvergence of sequences and series
Convergence of sequences and series A sequence f is a map from N the positive integers to a set. We often write the map outputs as f n rather than f(n). Often we just list the outputs in order and leave
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 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 informationFundamentals of the DFT (fft) Algorithms
Fundamentals of the DFT (fft) Algorithms D. Sundararajan November 6, 9 Contents 1 The PM DIF DFT Algorithm 1.1 Half-wave symmetry of periodic waveforms.............. 1. The DFT definition and the half-wave
More informationThe Discrete-time Fourier Transform
The Discrete-time Fourier Transform Rui Wang, Assistant professor Dept. of Information and Communication Tongji University it Email: ruiwang@tongji.edu.cn Outline Representation of Aperiodic signals: The
More informationGeneralizing the DTFT!
The Transform Generaliing the DTFT! The forward DTFT is defined by X e jω ( ) = x n e jωn in which n= Ω is discrete-time radian frequency, a real variable. The quantity e jωn is then a complex sinusoid
More informationLet H(z) = P(z)/Q(z) be the system function of a rational form. Let us represent both P(z) and Q(z) as polynomials of z (not z -1 )
Review: Poles and Zeros of Fractional Form Let H() = P()/Q() be the system function of a rational form. Let us represent both P() and Q() as polynomials of (not - ) Then Poles: the roots of Q()=0 Zeros:
More informationChapter 6: Applications of Fourier Representation Houshou Chen
Chapter 6: Applications of Fourier Representation Houshou Chen Dept. of Electrical Engineering, National Chung Hsing University E-mail: houshou@ee.nchu.edu.tw H.S. Chen Chapter6: Applications of Fourier
More informationIt is common to think and write in time domain. creating the mathematical description of the. Continuous systems- using Laplace or s-
It is common to think and write in time domain quantities, but this is not the best thing to do in creating the mathematical description of the system we are dealing with. Continuous systems- using Laplace
More informationRadical Expressions and Graphs 8.1 Find roots of numbers. squaring square Objectives root cube roots fourth roots
8. Radical Expressions and Graphs Objectives Find roots of numbers. Find roots of numbers. The opposite (or inverse) of squaring a number is taking its square root. Find principal roots. Graph functions
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 informationLecture 7: z-transform Properties, Sampling and Nyquist Sampling Theorem
EE518 Digital Signal Proessing University of Washington Autumn 21 Dept. of Eletrial Engineering ure 7: z-ransform Properties, Sampling and Nyquist Sampling heorem Ot 22, 21 Prof: J. Bilmes
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 informationSOLUTIONS to ECE 2026 Summer 2017 Problem Set #2
SOLUTIONS to ECE 06 Summer 07 Problem Set # PROBLEM..* Put each of the following signals into the standard form x( t ) = Acos( t + ). (Standard form means that A 0, 0, and < Use the phasor addition theorem
More informationSignals and Systems. Lecture 14 DR TANIA STATHAKI READER (ASSOCIATE PROFESSOR) IN SIGNAL PROCESSING IMPERIAL COLLEGE LONDON
Signals and Systems Lecture 14 DR TAIA STATHAKI READER (ASSOCIATE PROFESSOR) I SIGAL PROCESSIG IMPERIAL COLLEGE LODO Introduction. Time sampling theorem resume. We wish to perform spectral analysis using
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 informationZ-Transform. x (n) Sampler
Chapter Two A- Discrete Time Signals: The discrete time signal x(n) is obtained by taking samples of the analog signal xa (t) every Ts seconds as shown in Figure below. Analog signal Discrete time signal
More informationChapter 2 z-transform. X(z) = Z[x(t)] = Z[x(kT)] = Z[x(k)] x(kt)z k = x(kt)z k = x(k)z k. X(z)z k 1 dz 2πj c = 1
One-sided -transform Two-sided -transform Chapter 2 -TRANSFORM X() Z[x(t)] Z[x(kT)] Z[x(k)] x(kt) k x(k) k X() k0 k x(kt) k k0 k x(k) k Note that X() x(0) + x(t ) + x(2t ) 2 + + x(kt) k + Inverse -transform
More informationInterchange of Filtering and Downsampling/Upsampling
Interchange of Filtering and Downsampling/Upsampling Downsampling and upsampling are linear systems, but not LTI systems. They cannot be implemented by difference equations, and so we cannot apply z-transform
More information! Circular Convolution. " Linear convolution with circular convolution. ! Discrete Fourier Transform. " Linear convolution through circular
Previously ESE 531: Digital Signal Processing Lec 22: April 18, 2017 Fast Fourier Transform (con t)! Circular Convolution " Linear convolution with circular convolution! Discrete Fourier Transform " Linear
More informationSIGNALS AND SYSTEMS I. RAVI KUMAR
Signals and Systems SIGNALS AND SYSTEMS I. RAVI KUMAR Head Department of Electronics and Communication Engineering Sree Visvesvaraya Institute of Technology and Science Mahabubnagar, Andhra Pradesh New
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 informationHow to manipulate Frequencies in Discrete-time Domain? Two Main Approaches
How to manipulate Frequencies in Discrete-time Domain? Two Main Approaches Difference Equations (an LTI system) x[n]: input, y[n]: output That is, building a system that maes use of the current and previous
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 informationFirst and Last Name: 2. Correct The Mistake Determine whether these equations are false, and if so write the correct answer.
. Correct The Mistake Determine whether these equations are false, and if so write the correct answer. ( x ( x (a ln + ln = ln(x (b e x e y = e xy (c (d d dx cos(4x = sin(4x 0 dx xe x = (a This is an incorrect
More information2. Typical Discrete-Time Systems All-Pass Systems (5.5) 2.2. Minimum-Phase Systems (5.6) 2.3. Generalized Linear-Phase Systems (5.
. Typical Discrete-Time Systems.1. All-Pass Systems (5.5).. Minimum-Phase Systems (5.6).3. Generalized Linear-Phase Systems (5.7) .1. All-Pass Systems An all-pass system is defined as a system which has
More informationFourier Transform for Continuous Functions
Fourier Transform for Continuous Functions Central goal: representing a signal by a set of orthogonal bases that are corresponding to frequencies or spectrum. Fourier series allows to find the spectrum
More informationInformation and Communications Security: Encryption and Information Hiding
Short Course on Information and Communications Security: Encryption and Information Hiding Tuesday, 10 March Friday, 13 March, 2015 Lecture 5: Signal Analysis Contents The complex exponential The complex
More informationSyllabus for IMGS-616 Fourier Methods in Imaging (RIT #11857) Week 1: 8/26, 8/28 Week 2: 9/2, 9/4
IMGS 616-20141 p.1 Syllabus for IMGS-616 Fourier Methods in Imaging (RIT #11857) 3 July 2014 (TENTATIVE and subject to change) Note that I expect to be in Europe twice during the term: in Paris the week
More informationLinear Convolution Using FFT
Linear Convolution Using FFT Another useful property is that we can perform circular convolution and see how many points remain the same as those of linear convolution. When P < L and an L-point circular
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 information6.003: Signal Processing
6.003: Signal Processing Discrete Fourier Transform Discrete Fourier Transform (DFT) Relations to Discrete-Time Fourier Transform (DTFT) Relations to Discrete-Time Fourier Series (DTFS) October 16, 2018
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 informationCMPT 889: Lecture 3 Fundamentals of Digital Audio, Discrete-Time Signals
CMPT 889: Lecture 3 Fundamentals of Digital Audio, Discrete-Time Signals Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University October 6, 2005 1 Sound Sound waves are longitudinal
More informationFourier Transform 2D
Image Processing - Lesson 8 Fourier Transform 2D Discrete Fourier Transform - 2D Continues Fourier Transform - 2D Fourier Properties Convolution Theorem Eamples = + + + The 2D Discrete Fourier Transform
More informationsinc function T=1 sec T=2 sec angle(f(w)) angle(f(w))
T=1 sec sinc function 3 angle(f(w)) T=2 sec angle(f(w)) 1 A quick script to plot mag & phase in MATLAB w=0:0.2:50; Real exponential func b=5; Fourier transform (filter) F=1.0./(b+j*w); subplot(211), plot(w,
More information16.362: Signals and Systems: 1.0
16.362: Signals and Systems: 1.0 Prof. K. Chandra ECE, UMASS Lowell September 1, 2016 1 Background The pre-requisites for this course are Calculus II and Differential Equations. A basic understanding of
More informationReview of Fundamentals of Digital Signal Processing
Chapter 2 Review of Fundamentals of Digital Signal Processing 2.1 (a) This system is not linear (the constant term makes it non linear) but is shift-invariant (b) This system is linear but not shift-invariant
More informationRepresenting a Signal
The Fourier Series Representing a Signal The convolution method for finding the response of a system to an excitation takes advantage of the linearity and timeinvariance of the system and represents the
More informationEE123 Digital Signal Processing
EE123 Digital Signal Processing Lecture 2B D. T. Fourier Transform M. Lustig, EECS UC Berkeley Something Fun gotenna http://www.gotenna.com/# Text messaging radio Bluetooth phone interface MURS VHF radio
More informationSignals, Instruments, and Systems W5. Introduction to Signal Processing Sampling, Reconstruction, and Filters
Signals, Instruments, and Systems W5 Introduction to Signal Processing Sampling, Reconstruction, and Filters Acknowledgments Recapitulation of Key Concepts from the Last Lecture Dirac delta function (
More informationA523 Signal Modeling, Statistical Inference and Data Mining in Astrophysics Spring 2011
A523 Signal Modeling, Statistical Inference and Data Mining in Astrophysics Spring 2011 Lecture 6 PDFs for Lecture 1-5 are on the web page Problem set 2 is on the web page Article on web page A Guided
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 informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Processing Pro. Mark Fowler Note Set #14 Practical A-to-D Converters and D-to-A Converters Reading Assignment: Sect. 6.3 o Proakis & Manolakis 1/19 The irst step was to see that
More informationCSCE 222 Discrete Structures for Computing. Dr. Hyunyoung Lee
CSCE 222 Discrete Structures for Computing Sequences and Summations Dr. Hyunyoung Lee Based on slides by Andreas Klappenecker 1 Sequences 2 Sequences A sequence is a function from a subset of the set of
More informationMATH3283W LECTURE NOTES: WEEK 6 = 5 13, = 2 5, 1 13
MATH383W LECTURE NOTES: WEEK 6 //00 Recursive sequences (cont.) Examples: () a =, a n+ = 3 a n. The first few terms are,,, 5 = 5, 3 5 = 5 3, Since 5
More informationMathematical Description of Discrete-Time Signals. 9/10/16 M. J. Roberts - All Rights Reserved 1
Mathematical Descriptio of Discrete-Time Sigals 9/10/16 M. J. Roberts - All Rights Reserved 1 Samplig ad Discrete Time Samplig is the acquisitio of the values of a cotiuous-time sigal at discrete poits
More information