EECE 301 Signals & Systems Prof. Mark Fowler
|
|
- Franklin Crawford
- 6 years ago
- Views:
Transcription
1 EECE 30 Signals & Systems Prof. Mark Fowler Note Set #26 D-T Systems: Transfer Function and Frequency Response /
2 Finding the Transfer Function from Difference Eq. Recall: we found a DT system s freq. resp. H() by analying for input e jn or by taking DTFT of the Diff Eq. Here we can either analye the system for input n or take the ZT of the Diff Eq here we do the later: yn [ ] ayn [ ]... a yn [ N] bxn [ ] bxn [ ]... b xn [ M] N 0 M ZT ZT yn [ ] ayn [ ]... a yn [ N] ZT b x[ n] b x[ n ]... b x[ n M ] 0 N M Delay Prop. & Algebra Algebra Y( ) a... an N X( ) b0 b... bm M M b0 b... bm Y( ) X( ) N a... an So can just write H() by inspection of D.E. coefficients! From this H() we know how to compute the output: y = filter(b,a,x); 2/
3 Poles and Zeros of Transfer Function M b0 b... bm N a... an b b... b M M ( NM) 0 M N N a... an Define the polynomials A() and B() so that: ( NM) B ( ) A( ) Assume there are no common roots in the numerator B() and denominator A(). (If not, assume they ve been cancelled and redefine B() and A() accordingly) Poles of H(): The values on the complex -plane where H() Zeros of H(): The values on the complex -plane where H() = 0 The roots of the denominator polynomial A() determine N poles. The roots of the Numerator polynomial B() determine M eros. The term (N M) gives poles/eros at the origin according to: If N > M : N M Origin If N < M : M N Origin 3/
4 Example: Finding Poles and Zeros yn [ ] yn [ ] yn [ 2] 2 xn [ ] xn [ ] 2 4 p = ero at origin eros: 0, 2 poles: j 2 2 Using MATLAB: >> plane([2 ],[ -(/sqrt(2)) /4]) Coeff. Vectors MUST be rows Conjugate Pair Imaginary Part Real Part 4/
5 Impulse Response of System Sometimes looking at how a system responds to the impulse function (i.e., delta sequence) [n] can tell much about a system. Hitting a system with [n] is lot like ringing a bell to hear how it sounds [n] n LTI D-T system ICs = 0 Note: If system is causal, then h[n] = 0 for n < 0 h[n] n The symbol h[n] means the impulse response. Noting that the ZT of [n] = and using the properties of the transfer function and the Z transform: hn [ ] Z hn [ ] Z H( Z ) [ n] hn [ ] IDTFT H( ) From PFE and Poles/Zeros we see that a TF like this: will have an impulse response with terms like this: hn [ ] kpun [ ] k pun [ ] k p un [ ] n n n 2 2 N N ( NM) B ( ) A( ) Some simplifying assumptions made here! Now we almost always want this to decay (like a bell!): all poles p i < 5/
6 Stability of System Definition: A system is said to be stable if its output will never grow without bound when any bounded input signal is applied and that seems like a good thing!!! Without going into all the details a system with an impulse response that decays fast enough is said to be stable. From our exploration of the effect of poles on the impulse response we say that: j Im{} Re{} For a Stable System Poles must be inside unit circle Zeros can be anywhere j unit circle 6/
7 Relationship: Transfer Function and Freq. Resp. Recall: DTFT = ZT evaluated on Unit Circle if UC is inside ROC Fact: For causal systems UC is inside ROC if all poles are inside UC H ) H ( ) ( If all poles are inside the UC e j We saw how to use freq before to plot the Frequency Response this just shows how to plot the Frequency Response from the Transfer Function coefficients: b b b... b a a a a 2 M 0 2 M 2 N N num bb... b 0 den aa... a 0 omega -pi:?:pi M N H freq(num, denom, omega) plot(omega/pi, abs(h)) plot(omega/pi, angle(h)) must put any ero b i into the vector must put any ero a i into the vector Pick appropriate spacing 7/
8 Visualiing Relationship Between TF & FR = 0 j 0.3 j 0.8e 0.8e 0.3 = 0.8e j0.3 Pole-Zero Plot For This H() Im{} Re{} H() And we know that the Frequency Response is just the Transfer Function evaluated on the Unit Circle. H ( ) H ( ) e j 8/
9 Now plot just those values on the unit circle: Now Cut here and unwrap This shows after it has been cut and unwrapped and plotted on the axis: This shows the Frequency Response H() where is the angle around the unit circle this explains why H() is a periodic function of 9/
10 Effect of Poles & Zeros on Frequency Response of DT filters Im Note: Including a pole or ero at the origin Im doesn t change the magnitude but does change the phase Placing a ero at Im makes H() = 0 Im Placing more eros/poles Im gives sharper transitions. Figure from B.P. Lathi, Signal Processing and Linear Systems 0/
11 Cascade of Systems Suppose you have a cascade of two systems like this: x[n] H () H () y[n] H 2 () H 2 () [n] Y( ) H ( ) X( ) Y( ) H ( ) X( ) Z( ) H ( ) Y( ) Z( ) H ( ) Y( ) 2 2 Z( ) H ( ) H ( ) X( ) 2 Z( ) H ( ) H ( ) X( ) 2 Thus, the overall frequency response/transfer function is the product of those of each stage: H ( ) H ( ) H ( ) total 2 H ( ) H ( ) H ( ) total 2 Obviously, this generalies to a cascade of N systems: H ( ) H ( ) H ( ) H ( ) total 2 H ( ) H ( ) H ( ) H ( ) total 2 N N /
EECE 301 Signals & Systems Prof. Mark Fowler
EECE 3 Signals & Systems Prof. Mark Fowler Note Set #9 C-T Systems: Laplace Transform Transfer Function Reading Assignment: Section 6.5 of Kamen and Heck /7 Course Flow Diagram The arrows here show conceptual
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 3 Signals & Systems Prof. ark Fowler Note Set #28 D-T Systems: DT Filters Ideal & Practical /4 Ideal D-T Filters Just as in the CT case we can specify filters. We looked at the ideal filter for the
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 30 Signal & Syem Prof. ark Fowler oe Se #34 C-T Tranfer Funcion and Frequency Repone /4 Finding he Tranfer Funcion from Differenial Eq. Recall: we found a DT yem Tranfer Funcion Hz y aking he ZT of
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 informationSignals and Systems. Spring Room 324, Geology Palace, ,
Signals and Systems Spring 2013 Room 324, Geology Palace, 13756569051, zhukaiguang@jlu.edu.cn Chapter 10 The Z-Transform 1) Z-Transform 2) Properties of the ROC of the z-transform 3) Inverse z-transform
More informationUNIT-II Z-TRANSFORM. This expression is also called a one sided z-transform. This non causal sequence produces positive powers of z in X (z).
Page no: 1 UNIT-II Z-TRANSFORM The Z-Transform The direct -transform, properties of the -transform, rational -transforms, inversion of the transform, analysis of linear time-invariant systems in the -
More information7.17. Determine the z-transform and ROC for the following time signals: Sketch the ROC, poles, and zeros in the z-plane. X(z) = x[n]z n.
Solutions to Additional Problems 7.7. Determine the -transform and ROC for the following time signals: Sketch the ROC, poles, and eros in the -plane. (a) x[n] δ[n k], k > 0 X() x[n] n n k, 0 Im k multiple
More informationEE123 Digital Signal Processing. M. Lustig, EECS UC Berkeley
EE123 Digital Signal Processing Today Last time: DTFT - Ch 2 Today: Continue DTFT Z-Transform Ch. 3 Properties of the DTFT cont. Time-Freq Shifting/modulation: M. Lustig, EE123 UCB M. Lustig, EE123 UCB
More informationFrequency Response of Discrete-Time Systems
Frequency Response of Discrete-Time Systems EE 37 Signals and Systems David W. Graham 6 Relationship of Pole-Zero Plot to Frequency Response Zeros Roots of the numerator Pin the system to a value of ero
More informationEECE 301 Signals & Systems
EECE 30 Signals & Systems Prof. Mark Fowler Note Set #22 D-T Systems: DT Systems & Difference Equations / D-T System Models So far we ve seen how to use the FT to analyze circuits Use phasors and standard
More informationLecture 2 OKAN UNIVERSITY FACULTY OF ENGINEERING AND ARCHITECTURE
OKAN UNIVERSITY FACULTY OF ENGINEERING AND ARCHITECTURE EEE 43 DIGITAL SIGNAL PROCESSING (DSP) 2 DIFFERENCE EQUATIONS AND THE Z- TRANSFORM FALL 22 Yrd. Doç. Dr. Didem Kivanc Tureli didemk@ieee.org didem.kivanc@okan.edu.tr
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 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 informationChapter Intended Learning Outcomes: (i) Understanding the relationship between transform and the Fourier transform for discrete-time signals
z Transform Chapter Intended Learning Outcomes: (i) Understanding the relationship between transform and the Fourier transform for discrete-time signals (ii) Understanding the characteristics and properties
More information(i) Represent discrete-time signals using transform. (ii) Understand the relationship between transform and discrete-time Fourier transform
z Transform Chapter Intended Learning Outcomes: (i) Represent discrete-time signals using transform (ii) Understand the relationship between transform and discrete-time Fourier transform (iii) Understand
More informationEE102B Signal Processing and Linear Systems II. Solutions to Problem Set Nine Spring Quarter
EE02B Signal Processing and Linear Systems II Solutions to Problem Set Nine 202-203 Spring Quarter Problem 9. (25 points) (a) 0.5( + 4z + 6z 2 + 4z 3 + z 4 ) + 0.2z 0.4z 2 + 0.8z 3 x[n] 0.5 y[n] -0.2 Z
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 informationSIGNALS AND SYSTEMS. Unit IV. Analysis of DT signals
SIGNALS AND SYSTEMS Unit IV Analysis of DT signals Contents: 4.1 Discrete Time Fourier Transform 4.2 Discrete Fourier Transform 4.3 Z Transform 4.4 Properties of Z Transform 4.5 Relationship between Z
More informationECE503: Digital Signal Processing Lecture 5
ECE53: Digital Signal Processing Lecture 5 D. Richard Brown III WPI 3-February-22 WPI D. Richard Brown III 3-February-22 / 32 Lecture 5 Topics. Magnitude and phase characterization of transfer functions
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 301 Signals & Systems Prof. Mark Fowler C-T Systems: Bode Plots Note Set #36 1/14 What are Bode Plots? Bode Plot = Freq. Resp. plotted with H() in db on a log frequency axis. Its easy to use computers
More informationThe Z-Transform. Fall 2012, EE123 Digital Signal Processing. Eigen Functions of LTI System. Eigen Functions of LTI System
The Z-Transform Fall 202, EE2 Digital Signal Processing Lecture 4 September 4, 202 Used for: Analysis of LTI systems Solving di erence equations Determining system stability Finding frequency response
More informationEEL3135: Homework #4
EEL335: Homework #4 Problem : For each of the systems below, determine whether or not the system is () linear, () time-invariant, and (3) causal: (a) (b) (c) xn [ ] cos( 04πn) (d) xn [ ] xn [ ] xn [ 5]
More informationZ Transform (Part - II)
Z Transform (Part - II). The Z Transform of the following real exponential sequence x(nt) = a n, nt 0 = 0, nt < 0, a > 0 (a) ; z > (c) for all z z (b) ; z (d) ; z < a > a az az Soln. The given sequence
More informationDigital Signal Processing Lecture 5
Remote Sensing Laboratory Dept. of Information Engineering and Computer Science University of Trento Via Sommarive, 14, I-38123 Povo, Trento, Italy Digital Signal Processing Lecture 5 Begüm Demir E-mail:
More informationz-transform Chapter 6
z-transform Chapter 6 Dr. Iyad djafar Outline 2 Definition Relation Between z-transform and DTFT Region of Convergence Common z-transform Pairs The Rational z-transform The Inverse z-transform z-transform
More informationDiscrete Time Systems
Discrete Time Systems Valentina Hubeika, Jan Černocký DCGM FIT BUT Brno, {ihubeika,cernocky}@fit.vutbr.cz 1 LTI systems In this course, we work only with linear and time-invariant systems. We talked about
More informationECE503: Digital Signal Processing Lecture 4
ECE503: Digital Signal Processing Lecture 4 D. Richard Brown III WPI 06-February-2012 WPI D. Richard Brown III 06-February-2012 1 / 29 Lecture 4 Topics 1. Motivation for the z-transform. 2. Definition
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 informationZ - Transform. It offers the techniques for digital filter design and frequency analysis of digital signals.
Z - Transform The z-transform is a very important tool in describing and analyzing digital systems. It offers the techniques for digital filter design and frequency analysis of digital signals. Definition
More informationDigital Filter Structures. Basic IIR Digital Filter Structures. of an LTI digital filter is given by the convolution sum or, by the linear constant
Digital Filter Chapter 8 Digital Filter Block Diagram Representation Equivalent Basic FIR Digital Filter Basic IIR Digital Filter. Block Diagram Representation In the time domain, the input-output relations
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 informationA system that is both linear and time-invariant is called linear time-invariant (LTI).
The Cooper Union Department of Electrical Engineering ECE111 Signal Processing & Systems Analysis Lecture Notes: Time, Frequency & Transform Domains February 28, 2012 Signals & Systems Signals are mapped
More informationZ-Transform. The Z-transform is the Discrete-Time counterpart of the Laplace Transform. Laplace : G(s) = g(t)e st dt. Z : G(z) =
Z-Transform The Z-transform is the Discrete-Time counterpart of the Laplace Transform. Laplace : G(s) = Z : G(z) = It is Used in Digital Signal Processing n= g(t)e st dt g[n]z n Used to Define Frequency
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 informationDigital Signal Processing Lab 4: Transfer Functions in the Z-domain
Digital Signal Processing Lab 4: Transfer Functions in the Z-domain A very important category of LTI systems is described by difference equations of the following type N a y[ nk] b x[ nk] M k k0 k0 k From
More informationDigital Signal Processing:
Digital Signal Processing: Mathematical and algorithmic manipulation of discretized and quantized or naturally digital signals in order to extract the most relevant and pertinent information that is carried
More informationVery useful for designing and analyzing signal processing systems
z-transform z-transform The z-transform generalizes the Discrete-Time Fourier Transform (DTFT) for analyzing infinite-length signals and systems Very useful for designing and analyzing signal processing
More informationChapter 13 Z Transform
Chapter 13 Z Transform 1. -transform 2. Inverse -transform 3. Properties of -transform 4. Solution to Difference Equation 5. Calculating output using -transform 6. DTFT and -transform 7. Stability Analysis
More informationDigital Signal Processing, Homework 1, Spring 2013, Prof. C.D. Chung
Digital Signal Processing, Homework, Spring 203, Prof. C.D. Chung. (0.5%) Page 99, Problem 2.2 (a) The impulse response h [n] of an LTI system is known to be zero, except in the interval N 0 n N. The input
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 informationDigital Signal Processing
COMP ENG 4TL4: Digital Signal Processing Notes for Lecture #21 Friday, October 24, 2003 Types of causal FIR (generalized) linear-phase filters: Type I: Symmetric impulse response: with order M an even
More informationLecture 7 - IIR Filters
Lecture 7 - IIR Filters James Barnes (James.Barnes@colostate.edu) Spring 204 Colorado State University Dept of Electrical and Computer Engineering ECE423 / 2 Outline. IIR Filter Representations Difference
More informationUse: Analysis of systems, simple convolution, shorthand for e jw, stability. Motivation easier to write. Or X(z) = Z {x(n)}
1 VI. Z Transform Ch 24 Use: Analysis of systems, simple convolution, shorthand for e jw, stability. A. Definition: X(z) = x(n) z z - transforms Motivation easier to write Or Note if X(z) = Z {x(n)} z
More informationLECTURE NOTES DIGITAL SIGNAL PROCESSING III B.TECH II SEMESTER (JNTUK R 13)
LECTURE NOTES ON DIGITAL SIGNAL PROCESSING III B.TECH II SEMESTER (JNTUK R 13) FACULTY : B.V.S.RENUKA DEVI (Asst.Prof) / Dr. K. SRINIVASA RAO (Assoc. Prof) DEPARTMENT OF ELECTRONICS AND COMMUNICATIONS
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 informationE : Lecture 1 Introduction
E85.2607: Lecture 1 Introduction 1 Administrivia 2 DSP review 3 Fun with Matlab E85.2607: Lecture 1 Introduction 2010-01-21 1 / 24 Course overview Advanced Digital Signal Theory Design, analysis, and implementation
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 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 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 informationShift Property of z-transform. Lecture 16. More z-transform (Lathi 5.2, ) More Properties of z-transform. Convolution property of z-transform
Shift Property of -Transform If Lecture 6 More -Transform (Lathi 5.2,5.4-5.5) then which is delay causal signal by sample period. If we delay x[n] first: Peter Cheung Department of Electrical & Electronic
More informationEE Homework 5 - Solutions
EE054 - Homework 5 - Solutions 1. We know the general result that the -transform of α n 1 u[n] is with 1 α 1 ROC α < < and the -transform of α n 1 u[ n 1] is 1 α 1 with ROC 0 < α. Using this result, the
More 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 informationZ-Transform. 清大電機系林嘉文 Original PowerPoint slides prepared by S. K. Mitra 4-1-1
Chapter 6 Z-Transform 清大電機系林嘉文 cwlin@ee.nthu.edu.tw 03-5731152 Original PowerPoint slides prepared by S. K. Mitra 4-1-1 z-transform The DTFT provides a frequency-domain representation of discrete-time
More informationSolutions: Homework Set # 5
Signal Processing for Communications EPFL Winter Semester 2007/2008 Prof. Suhas Diggavi Handout # 22, Tuesday, November, 2007 Solutions: Homework Set # 5 Problem (a) Since h [n] = 0, we have (b) We can
More informationEE 313 Linear Signals & Systems (Fall 2018) Solution Set for Homework #7 on Infinite Impulse Response (IIR) Filters CORRECTED
EE 33 Linear Signals & Systems (Fall 208) Solution Set for Homework #7 on Infinite Impulse Response (IIR) Filters CORRECTED By: Mr. Houshang Salimian and Prof. Brian L. Evans Prolog for the Solution Set.
More informationELEN E4810: Digital Signal Processing Topic 4: The Z Transform. 1. The Z Transform. 2. Inverse Z Transform
ELEN E480: Digital Signal Processing Topic 4: The Z Transform. The Z Transform 2. Inverse Z Transform . The Z Transform Powerful tool for analyzing & designing DT systems Generalization of the DTFT: G(z)
More informationChapter 2 Time-Domain Representations of LTI Systems
Chapter 2 Time-Domain Representations of LTI Systems 1 Introduction Impulse responses of LTI systems Linear constant-coefficients differential or difference equations of LTI systems Block diagram representations
More informationDiscrete-Time Signals and Systems. The z-transform and Its Application. The Direct z-transform. Region of Convergence. Reference: Sections
Discrete-Time Signals and Systems The z-transform and Its Application Dr. Deepa Kundur University of Toronto Reference: Sections 3. - 3.4 of John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing:
More informationESE 531: Digital Signal Processing
ESE 531: Digital Signal Processing Lec 6: January 31, 2017 Inverse z-transform Lecture Outline! z-transform " Tie up loose ends " Regions of convergence properties! Inverse z-transform " Inspection " Partial
More informationVI. Z Transform and DT System Analysis
Summer 2008 Signals & Systems S.F. Hsieh VI. Z Transform and DT System Analysis Introduction Why Z transform? a DT counterpart of the Laplace transform in CT. Generalization of DT Fourier transform: z
More informationEE123 Digital Signal Processing
EE123 Digital Signal Processing Discrete Time Fourier Transform M. Lustig, EECS UC Berkeley A couple of things Read Ch 2 2.0-2.9 It s OK to use 2nd edition Class webcast in bcourses.berkeley.edu or linked
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 informationDSP-I DSP-I DSP-I DSP-I
DSP-I DSP-I DSP-I DSP-I Digital Signal Processing I (8-79) Fall Semester, 005 OTES FOR 8-79 LECTURE 9: PROPERTIES AD EXAPLES OF Z-TRASFORS Distributed: September 7, 005 otes: This handout contains in outline
More informationModule 4 : Laplace and Z Transform Lecture 36 : Analysis of LTI Systems with Rational System Functions
Module 4 : Laplace and Z Transform Lecture 36 : Analysis of LTI Systems with Rational System Functions Objectives Scope of this Lecture: Previously we understood the meaning of causal systems, stable systems
More informationDiscrete-time signals and systems
Discrete-time signals and systems 1 DISCRETE-TIME DYNAMICAL SYSTEMS x(t) G y(t) Linear system: Output y(n) is a linear function of the inputs sequence: y(n) = k= h(k)x(n k) h(k): impulse response of the
More informationTopic 4: The Z Transform
ELEN E480: Digital Signal Processing Topic 4: The Z Transform. The Z Transform 2. Inverse Z Transform . The Z Transform Powerful tool for analyzing & designing DT systems Generalization of the DTFT: G(z)
More informationLecture 11 FIR Filters
Lecture 11 FIR Filters Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/4/12 1 The Unit Impulse Sequence Any sequence can be represented in this way. The equation is true if k ranges
More informationEE 3054: Signals, Systems, and Transforms Spring A causal discrete-time LTI system is described by the equation. y(n) = 1 4.
EE : Signals, Systems, and Transforms Spring 7. A causal discrete-time LTI system is described by the equation Test y(n) = X x(n k) k= No notes, closed book. Show your work. Simplify your answers.. A discrete-time
More informationEE482: Digital Signal Processing Applications
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu EE482: Digital Signal Processing Applications Spring 2014 TTh 14:30-15:45 CBC C222 Lecture 02 DSP Fundamentals 14/01/21 http://www.ee.unlv.edu/~b1morris/ee482/
More informationStability Condition in Terms of the Pole Locations
Stability Condition in Terms of the Pole Locations A causal LTI digital filter is BIBO stable if and only if its impulse response h[n] is absolutely summable, i.e., 1 = S h [ n] < n= We now develop a stability
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 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 informationDSP-I DSP-I DSP-I DSP-I
NOTES FOR 8-79 LECTURES 3 and 4 Introduction to Discrete-Time Fourier Transforms (DTFTs Distributed: September 8, 2005 Notes: This handout contains in brief outline form the lecture notes used for 8-79
More informationTransform Representation of Signals
C H A P T E R 3 Transform Representation of Signals and LTI Systems As you have seen in your prior studies of signals and systems, and as emphasized in the review in Chapter 2, transforms play a central
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 4 Digital Signal Prcessing Pr. ar Fwler DT Filters te Set #2 Reading Assignment: Sect. 5.4 Prais & anlais /29 Ideal LP Filter Put in the signal we want passed. Suppse that ( ) [, ] X π xn [ ] y[ n]
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 informationThe Z transform (2) 1
The Z transform (2) 1 Today Properties of the region of convergence (3.2) Read examples 3.7, 3.8 Announcements: ELEC 310 FINAL EXAM: April 14 2010, 14:00 pm ECS 123 Assignment 2 due tomorrow by 4:00 pm
More informationEECE 301 Signals & Systems Prof. Mark Fowler
EECE 31 Signals & Sysems Prof. Mark Fowler Noe Se #1 C-T Sysems: Convoluion Represenaion Reading Assignmen: Secion 2.6 of Kamen and Heck 1/11 Course Flow Diagram The arrows here show concepual flow beween
More informationLecture 19 IIR Filters
Lecture 19 IIR Filters Fundamentals of Digital Signal Processing Spring, 2012 Wei-Ta Chu 2012/5/10 1 General IIR Difference Equation IIR system: infinite-impulse response system The most general class
More information( ) John A. Quinn Lecture. ESE 531: Digital Signal Processing. Lecture Outline. Frequency Response of LTI System. Example: Zero on Real Axis
John A. Quinn Lecture ESE 531: Digital Signal Processing Lec 15: March 21, 2017 Review, Generalized Linear Phase Systems Penn ESE 531 Spring 2017 Khanna Lecture Outline!!! 2 Frequency Response of LTI System
More informationResponses of Digital Filters Chapter Intended Learning Outcomes:
Responses of Digital Filters Chapter Intended Learning Outcomes: (i) Understanding the relationships between impulse response, frequency response, difference equation and transfer function in characterizing
More informationECE503: Digital Signal Processing Lecture 6
ECE503: Digital Signal Processing Lecture 6 D. Richard Brown III WPI 20-February-2012 WPI D. Richard Brown III 20-February-2012 1 / 28 Lecture 6 Topics 1. Filter structures overview 2. FIR filter structures
More informationInterconnection of LTI Systems
EENG226 Signals and Systems Chapter 2 Time-Domain Representations of Linear Time-Invariant Systems Interconnection of LTI Systems Prof. Dr. Hasan AMCA Electrical and Electronic Engineering Department (ee.emu.edu.tr)
More informationNotes 22 largely plagiarized by %khc
Notes 22 largely plagiarized by %khc LTv. ZT Using the conformal map z e st, we can transfer our knowledge of the ROC of the bilateral laplace transform to the ROC of the bilateral z transform. Laplace
More informationLecture 04: Discrete Frequency Domain Analysis (z-transform)
Lecture 04: Discrete Frequency Domain Analysis (z-transform) John Chiverton School of Information Technology Mae Fah Luang University 1st Semester 2009/ 2552 Outline Overview Lecture Contents Introduction
More information! z-transform. " Tie up loose ends. " Regions of convergence properties. ! Inverse z-transform. " Inspection. " Partial fraction
Lecture Outline ESE 53: Digital Signal Processing Lec 6: January 3, 207 Inverse z-transform! z-transform " Tie up loose ends " gions of convergence properties! Inverse z-transform " Inspection " Partial
More informationDetermine the Z transform (including the region of convergence) for each of the following signals:
6.003 Homework 4 Please do the following problems by Wednesday, March 3, 00. your answers: they will NOT be graded. Solutions will be posted. Problems. Z transforms You need not submit Determine the Z
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 informationThe z-transform Part 2
http://faculty.kfupm.edu.sa/ee/muqaibel/ The z-transform Part 2 Dr. Ali Hussein Muqaibel The material to be covered in this lecture is as follows: Properties of the z-transform Linearity Initial and final
More informationIntroduction to DSP Time Domain Representation of Signals and Systems
Introduction to DSP Time Domain Representation of Signals and Systems Dr. Waleed Al-Hanafy waleed alhanafy@yahoo.com Faculty of Electronic Engineering, Menoufia Univ., Egypt Digital Signal Processing (ECE407)
More informationDIGITAL SIGNAL PROCESSING LABORATORY
L AB 5 : DISCRETE T IME SYSTEM IN TIM E DOMAIN NAME: DATE OF EXPERIMENT: DATE REPORT SUBMITTED: 1 1 THEORY Mathematically, a discrete-time system is described as an operator T[.] that takes a sequence
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 informationStability and Frequency Response of Linear Systems
ECE 350 Linear Systems I MATLAB Tutorial #4 Stability and Frequency Response of Linear Systems This tutorial describes the MATLAB commands that can be used to determine the stability and/or frequency response
More informationUniversiti Malaysia Perlis EKT430: DIGITAL SIGNAL PROCESSING LAB ASSIGNMENT 3: DISCRETE TIME SYSTEM IN TIME DOMAIN
Universiti Malaysia Perlis EKT430: DIGITAL SIGNAL PROCESSING LAB ASSIGNMENT 3: DISCRETE TIME SYSTEM IN TIME DOMAIN Pusat Pengajian Kejuruteraan Komputer Dan Perhubungan Universiti Malaysia Perlis Discrete-Time
More informationUniversity of Illinois at Urbana-Champaign ECE 310: Digital Signal Processing
University of Illinois at Urbana-Champaign ECE 0: Digital Signal Processing Chandra Radhakrishnan PROBLEM SET : SOLUTIONS Peter Kairouz Problem. Hz z 7 z +/9, causal ROC z > contains the unit circle BIBO
More informationLecture 3 Matlab Simulink Minimum Phase, Maximum Phase and Linear Phase Systems
Lecture 3 Matlab Simulink Minimum Phase, Maximum Phase and Linear Phase Systems Lester Liu October 31, 2012 Minimum Phase, Maximum Phase and Linear Phase LTI Systems In this section, we will explore the
More informationThe Impulse Signal. A signal I(k), which is defined as zero everywhere except at a single point, where its value is equal to 1
The Impulse Signal A signal I(k), which is defined as zero everywhere except at a single point, where its value is equal to 1 I = [0 0 1 0 0 0]; stem(i) Filter Impulse through Feedforward Filter 0 0.25.5.25
More informationMassachusetts Institute of Technology Department of Electrical Engineering and Computer Science. Fall Solutions for Problem Set 2
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science Issued: Tuesday, September 5. 6.: Discrete-Time Signal Processing Fall 5 Solutions for Problem Set Problem.
More informationChapter 3 Convolution Representation
Chapter 3 Convolution Representation DT Unit-Impulse Response Consider the DT SISO system: xn [ ] System yn [ ] xn [ ] = δ[ n] If the input signal is and the system has no energy at n = 0, the output yn
More informationCourse and Wavelets and Filter Banks. Filter Banks (contd.): perfect reconstruction; halfband filters and possible factorizations.
Course 18.327 and 1.130 Wavelets and Filter Banks Filter Banks (contd.): perfect reconstruction; halfband filters and possible factorizations. Product Filter Example: Product filter of degree 6 P 0 (z)
More information