ELEG 305: Digital Signal Processing
|
|
- Carol Washington
- 5 years ago
- Views:
Transcription
1 ELEG 305: Digital Signal Processing Lecture 4: Inverse z Transforms & z Domain Analysis Kenneth E. Barner Department of Electrical and Computer Engineering University of Delaware Fall 008 K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 008 / 4 Outline Review of Previous Lecture Lecture Objectives 3 Inverse z Transform Distinct Poles Case Multiple Order Poles Case 4 Analysis of LTI Systems in the z Domain Response of Systems with Rational System Functions Causality and Stability Stability of Second Order Systems K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 008 / 4
2 Review of Previous Lecture Review of Previous Lecture The Rational z Transform conversion from difference equations, the system function (H(z)), poles and zeroes, the relation between pole location and time domain behavior The Inverse z Transform contour integration and power series expansion (function of z or z ) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 4 Lecture Objectives Lecture Objectives Objective Evaluate the inverse z transform using partial fraction expansion and table lookup; Analysis of LTI systems in the z domain (transient vs. steady state response and stability); Stability & response of second order systems Reading Chapter 3 ( ); Next lecture, the remainder of Chapter 3 (3.6) the one sided z transformation K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 4
3 Commonly Used Inverse z Transform Methods Direct evaluation of x(n) = X(z)z n dz πj by contour integration Power series expansion X(z) = n= x(n)z n 3 Partial fraction expansion and table lookup K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 4 Inverse z Transform Approach Express X(z) as a linear sum of components that have simple inverse transformations (that are known or can be looked up) If we can express X(z) as where it is known that Then, X(z) =α X (z)+α X (z)+ + α k X k (z) x k (n) =Z {X k (z)} x(n) =α x (n)+α x (n)+ + α k x k (n) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 4
4 Systems of interest: Rational functions that are proper Proper = numerator poly. order < denominator poly. order If function not proper, use division to write it as a polynomial and proper rational function Example Determine the inverse transformation of X(z) = + z + z 3 z + z X(z) is not a proper function Reverse the order of the polynomials Divide until remainder is proper z 3 z + z + z + K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 4 Inverse z Transform + 5z X(z) = + 3 z + z Note: The rational function is now proper. Next, factor the denominator 3 z + z = ( ) ( z z ) + 5z X(z) = + 3 z + z + 5z = + ( z )( z ) A = + ( z ) + A ( z ) }{{} partial fraction expansion K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 4
5 Approach: Solve for A and A by isolating the terms X(z) = + 5z + ( z )( z ) ( ) = A + ( z ) + A ( z ) ( ) A To isolate A : () multiply ( ) by the denominator of and () ( z ) evaluate at the associated root, z = ( X(z) ) z= z = ( ) z + A + A ( z ) ( z ) = 0 + A + 0 = A z= K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 4 Inverse z Transform Repeat the process for ( ), + 5z X(z) = + ( z )( z ) ( X(z) ) z= z = ( ) + z 5z + ( z ) z= + 0 = 0 + = 9 Combining the results for ( ) and ( ) A = 9 Next: Repeat the process for A, i.e., multiply ( ) and ( ) by ( z ) and evaluate at z =. K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 4
6 For ( ), ( X(z) z ) ( z= = z ) + A ( z ) + A z z= = A = A and ( ) ( X(z) z ) z= = ( z ) + 5z + z z= = = 8 Combining the results for ( ) and ( ) A = 8 Final Result: Combining results into final partial fraction expression: 9 X(z) = z z K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 008 / 4 Inverse z Transform Next, determine x(n) depends on X(z) and ROC 9 X(z) = z z Case : ROC: z > causal signal { } { } x(n) = Z {} 9Z + 8Z z z ( ) n = δ(n) 9 u(n)+8u(n) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 008 / 4
7 9 X(z) = z z Case : ROC: z < anti causal signal ( ) n x(n) =δ(n)+9 u( n ) 8u( n ) Case 3: ROC: < z < two sided signal 9 8 X(z) = + + }{{ z }} {{ z } causal anti-causal x(n) =δ(n) 9 ( ) n u(n) 8u( n ) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 4 Inverse z Transform General Partial Fraction Procedure Express X(z) in proper rational function form X(z) = + M i= b iz i + N i= a (M < N) iz i Factor the denominator X(z) = + M i= b iz i N ( i= pi z i) where the p i s are the roots of + N i= a iz i 3 Express as a sum of first order terms (assumes distinct poles) X(z) = N A i p i= i z i 4 Determine A i =( p i z i )X(z) z=pi (assumes distinct poles) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 4
8 Multiple Order Poles Case Note: For poles with multiplicity m >, m terms are needed Required Terms: For a pole ( pz ) m, required partial fraction expansion terms are A pz + A ( pz ) + + A m ( pz ) m where A i = { d m i [ (m i)!( p) m i dz m i ( pz) m X(z )] } z=p K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 4 Inverse z Transform Example Determine the inverse z transform of Expand into partial fraction form X(z) = B = X(z) = ( z )( z ) B + A z z + A ( z ) ( ) z X(z) = z= A = ( z ) X(z) = z= = ( ( ) ) = ( ) = K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 4
9 For A, we must use the derivative method: { d m i [ A i = (m i)!( p) m i dz m i ( pz) m X(z )] } z=p Note that X(z )=, p =, and m i = =. Thus, ( z)( z) d [ A =!( ) ( z) X(z )] = d dz z= dz ( z) z= = ( z) = ( ) = z= Combining the three partial fraction components, = X(z) = + z z + ( z ) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 4 Inverse z Transform Alternate Method for nd Order Poles B and A are easy to determine thus we know X(z) = Problem: Need to find A z + A z + ( z ) Observation: Only one unknown need one equation to solve Solution: Evaluate X(z) at value pole & solve for A, e.g. choose z = K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 4
10 Inverse Transformation of Partial Fraction Expansion Partial fraction expansion yields: X(z) = A p z + + A N p N z Inverse transform each component (and sum results) { } { Z A k Ak p p k z = k nu(n), ROC: z > p k A k pk nu( n ), ROC: z < p k Note: IfX(z) contains complex conjugate poles (ROC: z > p ) A pz + A p z [Ap n + A (p ) n ]u(n) = A r n cos(βn + α)u(n) where α and β are the phase components of A and p, i.e., A = A e jα and p = re jβ K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 4 Analysis of LTI Systems in the z Domain System Response: Suppose, X(z) = N(z) and Q(z) Response of Systems with Rational System Functions H(z) = B(z) A(z) Y (z) = H(z)X(z) = B(z)N(z) N A(z)Q(z) = A k L p k= k z + Q k q k= k z where p k are the poles of H(z) and q k are the poles of X(z) N L = y(n) = A k (p k ) n u(n) + Q k (q k ) n u(n) k= k= }{{}}{{} natural response forced response Observations: If poles of H(z) are inside the unit circle, then the natural response 0asn : transient If poles of X(z) are on the unit circle, the forced response is periodic: steady state K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 4
11 Analysis of LTI Systems in the z Domain Causality and Stability Causality and Stability Recall n h(n) < = BIBO stable Approach: Assume BIBO stability & evaluate H(z) on the unit circle H(z) = h(n)z n n h(n)z n n h(n) z n [ z n = ] n = n h(n) < [by BIBO assumption] H(z) BIBO stable ROC contains the unit circle (converse also true) Result: a LTI system is BIBO stable iff the ROC includes the unit circle Result: a causal LTI system is BIBO stable iff all poles are inside the unit circle K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 008 / 4 Analysis of LTI Systems in the z Domain Stability of Second Order Systems Consider the stability of a second order system y(n) = a y(n ) a y(n )+b 0 x(n) H(z) = Y (z) X(z) = b 0 + a z + a z p, p = a ± a 4a 4 Note: Three stable cases exist: Case : Real distinct (a > 4a ) Case : Real and equal (a = 4a ) Case 3: Complex Conjugates (a < 4a ) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall 008 / 4
12 Analysis of LTI Systems in the z Domain Stability of Second Order Systems Case : Time domain signal difference of exponentials h(n) = b ( ) 0 p n+ p p p n+ u(n) Case : Time domain signal ramp, exponential product h(n) =b 0 (n + )p n u(n) Case 3: Time domain signal oscillating exponential h(n) = b 0r n sin((n + )ω 0 )u(n) sin ω 0 K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 4 Lecture Summary Lecture Summary The Inverse z Transform partial fraction expansion; consideration of distinct and multiple order poles Analysis of LTI systems in the z domain response of systems with rational system functions (transient vs. steady state response), causality and stability (BIBO stable pole location restrictions), and stability & response of second order systems Next Lecture one sided z transformation (systems with non zero initial conditions) K. E. Barner (Univ. of Delaware) ELEG 305: Digital Signal Processing Fall / 4
Discrete-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 information6.003: Signals and Systems
6.003: Signals and Systems Z Transform September 22, 2011 1 2 Concept Map: Discrete-Time Systems Multiple representations of DT systems. Delay R Block Diagram System Functional X + + Y Y Delay Delay X
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 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 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 informationDigital Signal Processing
Digital Signal Processing The -Transform and Its Application to the Analysis of LTI Systems Moslem Amiri, Václav Přenosil Embedded Systems Laboratory Faculty of Informatics, Masaryk University Brno, Cech
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 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 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 informationNeed for transformation?
Z-TRANSFORM In today s class Z-transform Unilateral Z-transform Bilateral Z-transform Region of Convergence Inverse Z-transform Power Series method Partial Fraction method Solution of difference equations
More informationy[n] = = h[k]x[n k] h[k]z n k k= 0 h[k]z k ) = H(z)z n h[k]z h (7.1)
7. The Z-transform 7. Definition of the Z-transform We saw earlier that complex exponential of the from {e jwn } is an eigen function of for a LTI System. We can generalize this for signals of the form
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 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 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 informationDIGITAL SIGNAL PROCESSING. Chapter 3 z-transform
DIGITAL SIGNAL PROCESSING Chapter 3 z-transform by Dr. Norizam Sulaiman Faculty of Electrical & Electronics Engineering norizam@ump.edu.my OER Digital Signal Processing by Dr. Norizam Sulaiman work is
More informationELEG 305: Digital Signal Processing
ELEG 305: Digital Signal Processing Lecture 19: Lattice Filters Kenneth E. Barner Department of Electrical and Computer Engineering University of Delaware Fall 2008 K. E. Barner (Univ. of Delaware) ELEG
More informationELEG 305: Digital Signal Processing
ELEG 305: Digital Signal Processing Lecture 1: Course Overview; Discrete-Time Signals & Systems Kenneth E. Barner Department of Electrical and Computer Engineering University of Delaware Fall 2008 K. E.
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationThe Z transform (2) Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 28 1
The Z transform (2) Alexandra Branzan Albu ELEC 310-Spring 2009-Lecture 28 1 Outline Properties of the region of convergence (10.2) The inverse Z-transform (10.3) Definition Computational techniques Alexandra
More information8. z-domain Analysis of Discrete-Time Signals and Systems
8. z-domain Analysis of Discrete-Time Signals and Systems 8.. Definition of z-transform (0.0-0.3) 8.2. Properties of z-transform (0.5) 8.3. System Function (0.7) 8.4. Classification of a Linear Time-Invariant
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 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 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 informationX (z) = n= 1. Ã! X (z) x [n] X (z) = Z fx [n]g x [n] = Z 1 fx (z)g. r n x [n] ª e jnω
3 The z-transform ² Two advantages with the z-transform:. The z-transform is a generalization of the Fourier transform for discrete-time signals; which encompasses a broader class of sequences. The z-transform
More informationELEG 305: Digital Signal Processing
ELEG 305: Digital Signal Processing Lecture 18: Applications of FFT Algorithms & Linear Filtering DFT Computation; Implementation of Discrete Time Systems Kenneth E. Barner Department of Electrical and
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 informationSEL4223 Digital Signal Processing. Inverse Z-Transform. Musa Mohd Mokji
SEL4223 Digital Signal Processing Inverse Z-Transform Musa Mohd Mokji Inverse Z-Transform Transform from z-domain to time-domain x n = 1 2πj c X z z n 1 dz Note that the mathematical operation for the
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 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 informationDiscrete Time Systems
1 Discrete Time Systems {x[0], x[1], x[2], } H {y[0], y[1], y[2], } Example: y[n] = 2x[n] + 3x[n-1] + 4x[n-2] 2 FIR and IIR Systems FIR: Finite Impulse Response -- non-recursive y[n] = 2x[n] + 3x[n-1]
More informationLecture 18: Stability
Lecture 18: Stability ECE 401: Signal and Image Analysis University of Illinois 4/18/2017 1 Stability 2 Impulse Response 3 Z Transform Outline 1 Stability 2 Impulse Response 3 Z Transform BIBO Stability
More informationECE-S Introduction to Digital Signal Processing Lecture 4 Part A The Z-Transform and LTI Systems
ECE-S352-70 Introduction to Digital Signal Processing Lecture 4 Part A The Z-Transform and LTI Systems Transform techniques are an important tool in the analysis of signals and linear time invariant (LTI)
More informationDiscrete-Time Fourier Transform (DTFT)
Discrete-Time Fourier Transform (DTFT) 1 Preliminaries Definition: The Discrete-Time Fourier Transform (DTFT) of a signal x[n] is defined to be X(e jω ) x[n]e jωn. (1) In other words, the DTFT of x[n]
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 informationZ-TRANSFORMS. Solution: Using the definition (5.1.2), we find: for case (b). y(n)= h(n) x(n) Y(z)= H(z)X(z) (convolution) (5.1.
84 5. Z-TRANSFORMS 5 z-transforms Solution: Using the definition (5..2), we find: for case (a), and H(z) h 0 + h z + h 2 z 2 + h 3 z 3 2 + 3z + 5z 2 + 2z 3 H(z) h 0 + h z + h 2 z 2 + h 3 z 3 + h 4 z 4
More informationECE 421 Introduction to Signal Processing
ECE 421 Introduction to Signal Processing Dror Baron Assistant Professor Dept. of Electrical and Computer Engr. North Carolina State University, NC, USA The z-transform [Reading material: Sections 3.1-3.3]
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 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 informationSignal Analysis, Systems, Transforms
Michael J. Corinthios Signal Analysis, Systems, Transforms Engineering Book (English) August 29, 2007 Springer Contents Discrete-Time Signals and Systems......................... Introduction.............................................2
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 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 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 informationELEG 305: Digital Signal Processing
ELEG 305: Digital Signal Processing Lecture : Design of Digital IIR Filters (Part I) Kenneth E. Barner Department of Electrical and Computer Engineering University of Delaware Fall 008 K. E. Barner (Univ.
More 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 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 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 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 information1. Z-transform: Initial value theorem for causal signal. = u(0) + u(1)z 1 + u(2)z 2 +
1. Z-transform: Initial value theorem for causal signal u(0) lim U(z) if the limit exists z U(z) u(k)z k u(k)z k k lim U(z) u(0) z k0 u(0) + u(1)z 1 + u(2)z 2 + CL 692 Digital Control, IIT Bombay 1 c Kannan
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 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 informationEE 521: Instrumentation and Measurements
Aly El-Osery Electrical Engineering Department, New Mexico Tech Socorro, New Mexico, USA November 1, 2009 1 / 27 1 The z-transform 2 Linear Time-Invariant System 3 Filter Design IIR Filters FIR Filters
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 informationDiscrete-Time Signals and Systems. Frequency Domain Analysis of LTI Systems. The Frequency Response Function. The Frequency Response Function
Discrete-Time Signals and s Frequency Domain Analysis of LTI s Dr. Deepa Kundur University of Toronto Reference: Sections 5., 5.2-5.5 of John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing:
More informationLecture 7 Discrete Systems
Lecture 7 Discrete Systems EE 52: Instrumentation and Measurements Lecture Notes Update on November, 29 Aly El-Osery, Electrical Engineering Dept., New Mexico Tech 7. Contents The z-transform 2 Linear
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 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 informationDetailed Solutions to Exercises
Detailed Solutions to Exercises Digital Signal Processing Mikael Swartling Nedelko Grbic rev. 205 Department of Electrical and Information Technology Lund University Detailed solution to problem E3.4 A
More informationReview of Fundamentals of Digital Signal Processing
Solution Manual for Theory and Applications of Digital Speech Processing by Lawrence Rabiner and Ronald Schafer Click here to Purchase full Solution Manual at http://solutionmanuals.info Link download
More 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 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 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 informationModeling and Analysis of Systems Lecture #8 - Transfer Function. Guillaume Drion Academic year
Modeling and Analysis of Systems Lecture #8 - Transfer Function Guillaume Drion Academic year 2015-2016 1 Input-output representation of LTI systems Can we mathematically describe a LTI system using the
More informationDigital Signal Processing, Homework 2, Spring 2013, Prof. C.D. Chung. n; 0 n N 1, x [n] = N; N n. ) (n N) u [n N], z N 1. x [n] = u [ n 1] + Y (z) =
Digital Signal Processing, Homework, Spring 0, Prof CD Chung (05%) Page 67, Problem Determine the z-transform of the sequence n; 0 n N, x [n] N; N n x [n] n; 0 n N, N; N n nx [n], z d dz X (z) ) nu [n],
More informationPROBLEM SET 3. Note: This problem set is a little shorter than usual because we have not covered inverse z-transforms yet.
PROBLEM SET 3 Issued: /3/9 Due: 2/6/9 Reading: During the past week we concluded our discussion DTFT properties and began our discussion of z-transforms, covering basic calculation of the z-transform and
More informationDigital Control & Digital Filters. Lectures 1 & 2
Digital Controls & Digital Filters Lectures 1 & 2, Professor Department of Electrical and Computer Engineering Colorado State University Spring 2017 Digital versus Analog Control Systems Block diagrams
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 informationUNIVERSITY OF OSLO. Please make sure that your copy of the problem set is complete before you attempt to answer anything.
UNIVERSITY OF OSLO Faculty of mathematics and natural sciences Examination in INF3470/4470 Digital signal processing Day of examination: December 9th, 011 Examination hours: 14.30 18.30 This problem set
More 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 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 informationCh. 7: Z-transform Reading
c J. Fessler, June 9, 3, 6:3 (student version) 7. Ch. 7: Z-transform Definition Properties linearity / superposition time shift convolution: y[n] =h[n] x[n] Y (z) =H(z) X(z) Inverse z-transform by coefficient
More information信號與系統 Signals and Systems
Spring 2013 Flowchart Introduction (Chap 1) LTI & Convolution (Chap 2) NTUEE-SS10-Z-2 信號與系統 Signals and Systems Chapter SS-10 The z-transform FS (Chap 3) Periodic Bounded/Convergent CT DT FT Aperiodic
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 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 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 informationGATE EE Topic wise Questions SIGNALS & SYSTEMS
www.gatehelp.com GATE EE Topic wise Questions YEAR 010 ONE MARK Question. 1 For the system /( s + 1), the approximate time taken for a step response to reach 98% of the final value is (A) 1 s (B) s (C)
More informationEE 225D LECTURE ON DIGITAL FILTERS. University of California Berkeley
University of California Berkeley College of Engineering Department of Electrical Engineering and Computer Sciences Professors : N.Morgan / B.Gold EE225D Digital Filters Spring,1999 Lecture 7 N.MORGAN
More informationIf every Bounded Input produces Bounded Output, system is externally stable equivalently, system is BIBO stable
1. External (BIBO) Stability of LTI Systems If every Bounded Input produces Bounded Output, system is externally stable equivalently, system is BIBO stable g(n) < BIBO Stability Don t care about what unbounded
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 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 informationDigital Signal Processing Lecture 9 - Design of Digital Filters - FIR
Digital Signal Processing - Design of Digital Filters - FIR Electrical Engineering and Computer Science University of Tennessee, Knoxville November 3, 2015 Overview 1 2 3 4 Roadmap Introduction Discrete-time
More informationTransform Analysis of Linear Time-Invariant Systems
Transform Analysis of Linear Time-Invariant Systems Discrete-Time Signal Processing Chia-Ping Chen Department of Computer Science and Engineering National Sun Yat-Sen University Kaohsiung, Taiwan ROC Transform
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 informationDiscrete-Time David Johns and Ken Martin University of Toronto
Discrete-Time David Johns and Ken Martin University of Toronto (johns@eecg.toronto.edu) (martin@eecg.toronto.edu) University of Toronto 1 of 40 Overview of Some Signal Spectra x c () t st () x s () t xn
More informationModule 4. Related web links and videos. 1. FT and ZT
Module 4 Laplace transforms, ROC, rational systems, Z transform, properties of LT and ZT, rational functions, system properties from ROC, inverse transforms Related web links and videos Sl no Web link
More informationThe z-transform and Discrete-Time LTI Systems
Chapter 4 The z-transform and Discrete-Time LTI Systems 4.1 INTRODUCTION In Chap. 3 we introduced the Laplace transform. In this chapter we present the z-transform, which is the discrete-time counterpart
More informationECE 438 Exam 2 Solutions, 11/08/2006.
NAME: ECE 438 Exam Solutions, /08/006. This is a closed-book exam, but you are allowed one standard (8.5-by-) sheet of notes. No calculators are allowed. Total number of points: 50. This exam counts for
More information