Z Transform (Part - II)

Size: px
Start display at page:

Download "Z Transform (Part - II)"

Transcription

1 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 can be written as x(n)a n (n) where a > 0 Z Transform of given sequence is X(z) = Option (b) z z a ; z > a = az, z > a [GATE 990: 2 Marks] 2. A linear discrete-time system has the characteristics equation, z z = 0. The system (a) is stable (b) is marginally stable (c) is unstable (d) stability cannot be assessed from the given information [GATE 992: Marks] Soln. Characteristic equation is given z 3 0.8z = 0 z(z 2 0.8) = 0 z(z 0.9)(z + 0.9) = 0 So, poles are z = 0, 0.9 and 0.9 Note that all three poles are inside the unit circle, so the system is stable Option (a)

2 3. The z transform of a signal is given by z ( z 4 ) ( z. Its final value is ) 2 C(z) = 4 (a) /4 (b) zero Soln. Final value theorem for Z Transform is lim x[n] = lim ( N z z ) X(z) = lim. ( z ) z ( z 4 ) z 4 ( z ) 2 = lim z 4 z ( z 4 ) ( z ) = lim.. (z4 ) z 4 z 4 z (z ) z = lim z 4 (z 2 )(z 2 +) z 4 (z ) = lim z 4 z2 (z+)(z )(z2 +) (z ) (c).0 (d) infinity [GATE 999: 2 Marks] = lim z 4 z2 (z + )(z 2 + ) = = Option (a) 4. If the impulse response of a discrete-time system is h[n] = 5 n u[ n ]. Then the system function H(z) is equal to (a) z and the system is stable z 5 and the system is stable (b) z z 5 (c) z z 5 (d) z z 5 and the system is unstable and the system is unstable [GATE 2002: 2 Marks]

3 Soln. Impulse response h(n) = 5 n u[ n ] Above response is having left handed sequence whose z transform has standard form a n u(n)( n ) z ( a z ) ; ROC z < a Thus 5 n u(n)( n ) ( 5 z ) ; ROC z < 5 or, 5 n u(n)( n ) z z 5 ; ROC z < 5 since ROC is z < 5 so it includes unit circle, system is stable Option (b) 5. A causal LTI system is described by the difference equation 2y[n] = αy[n 2] 2x[n] + βx[n ] The system is stable only if (a) α = 2, β < 2 (b) α > 2, β > 2 (c) α < 2, any value of β (d) β < 2, any value of α [GATE 2004: 2 Marks] Soln. Casual LTI system is described by the difference equation 2 y[n] = α y[n 2] 2 x[n] + β x[n ] Taking z transform 2 Y[z] = α Y(z)z 2 2 X(z) + β X(z)z Y(z) = (β z 2) X(z) (2 α z 2 )

4 Or, H(z) = z(β 2 z) (z 2 α 2) It has poles at ± α 2 and zero at 0 and β 2 For stable system poles must lie inside the unit circle of z plane. But 0 can lie any where in plane. Thus β can be of any value Option (c) 6. The z transform X[z] of a sequence x[n] is given by X[z] = 0.5 2z. It is given that the Region of convergence of X[z] includes the unit circle. The value of x[0] is (a) 0.5 (b) 0 Soln. Given X(z) of a sequence x[n] X(z) = z Above transform is for left handed sequence with ROC z < 2 Corresponding sequence is x(n) = (0.5)2 n u( n ) So, x(0) = 0 (c) 0.25 (d) 0.5 [GATE 2007: 2 Marks] If, the value of x(0) is determined by initial value theorem then it will not be correct, since the sequence x(n) is defined for n < 0 Option (b) 7. A system with transfer function H(z) has impulse response h(n) defined as h(2) =, h(3) = - and h(k) = 0 otherwise. Consider the following statements. S : H(z) is low-pass filter

5 Soln. S 2 : H(z) is an FIR filter Which of the following is correct? (a) Only S 2 is true (b) Both S is and S 2 are false (c) Both S is and S 2 are true, and S 2 is a reason for S (d) Both S is and S 2 are true, and S 2 is not a reason for S [GATE 2009: 2 Marks] Given h(2) = h(3) = h(k) = 0 otherwise The diagram of the response is as shown 3 0 k 2 - Note, that it has the finite magnitude values. So it is having finite impulse response. So, FIR But it is not low pass filter So, S is false Option (a) 8. The transfer function of a discrete time LTI system is given by 2 3 H(z) = 4 z 3 4 z + 8 z 2 Consider the following statements: S : The system is stable and casual tor ROC z > 2

6 S 2 : The system is stable but not causal for ROC z < 4 S 3 : The system is neither stable not causal For ROC 4 < z < 2 Soln. Which one of the following statements is valid? (a) Both S and S 2 are true (b) Both S 2 and S 3 true (i) (ii) We know: (c) Both S and S 3 are true (d) S, S 2 and S 3 are all true [GATE 200: 2 Marks] Causal System : A discrete time LTI system is causal if and only of the ROC of its system function is exterior of a circle, including infinity Stable System : A discrete time LTI system is stable if and only if the ROC of its system function includes the unit circle i.e. z = 2 3 H(z) = 4 z 3 4 z + 8 z 2 = ( 4 z ) + ( 2 z ) ( 4 z ) ( 2 z ) H(z) = ( + 4 z ) ( 2 z ) ROC z > 2 h(n) = ( n 4 ) u(n) + ( n 2 ) u(n) The system is causal, since ROC is exterior of the circle. ROC of H(z) includes unit circle so it is stable So, S is true For ROC : z < 4

7 ROC does not include the unit circle. So system is not stable. So, S 2 is not true For ROC : 4 < z < 2 ROC does not include unit circle. So system is not stable. Also ROC is not exterior of z =. So it not 2 causal So, S 3 is true Both S and S 3 are true 9. Two systems H (z) and H 2 (z) are connected in cascade as shown below. The overall output y(n) is the same as the input x(n) with a one unit delay. The transfer function of the second system H 2 (z) is x(n) H (z) = ( 0.4z ) ( 0.6z ) H 2 (z) y(n) (a) ( 0.6 z ) z ( 0.4 z ) (c) z ( 0.4 z ) ( 0.6 z ) (b) z ( 0.6 z ) ( 0.4 z ) (d) ( 0.4 z ) z ( 0.6 z ) Soln. Given y[n] = x[n ] Taking Z Transfer on both sides Y(z) = z X(z) or Y(z) X(z) = z For cascaded system H(z) = H (z). H 2 (z) [GATE 20: 2 Marks]

8 z = ( 0.4 z ) ( 0.6 z ). H z(z) Or, H 2 (z) = z ( 0.6 z ) ( 0.4 z ) Option (b) 0. Let x[n] = ( 9 )n u(n) ( 3 )n u( n ). The Region of Convergence (ROC) of the Z Transform of x[n] (a) is z > 9 (b) is z < 3 (c) is 3 > z > 9 (d) does not exist [GATE 204: 2 Marks] Soln. Given x[n] = ( 9 )n u(n) ( 3 )n u( n ) Let, x (n) = ( 9 )n u(n) and x 2 (n) = ( 3 )n u( n ) x (n) = ( 9 )n u(n) It is right sided sequence. X (z) = ( 9 ) z ROC : z 9 or z > 9 x 2 (n) = ( 3 )n u( n 3) It is left sided sequence X 2 (z) = ( 3 ) z ROC : z 3

9 When both these transform are added then ROC must be between Option (c) 3 > z > 9. The input-output relationship of a causal stable LTI system is given as y[n] = α y[n ] + β x[n] If the impulse response h[n] of this system satisfies the condition h[n] = 2, n=0 (a) α = β 2 (b) α = + β 2 the relationship between α and β is Soln. Given input output relation for causal, stable, LTI system. y[n] = α y[n ] + β x[n] Taking Z Transfer Also, Y(z) = α z Y(z) + β X(z) Y(z)[ α z ] = β X(z) Y(z) X(z) = H(z) = h(n) = β. α n u[n] β h(n) = 2 n=0 β α z i. e. β α n u[n] = 2 n=0 (c) α = 2β (d) α = 2β [GATE 204: 2 Marks] = 2 or β = 2 2α or α = β α 2 Option (a)

10 . The Z Transform of the sequence X[n] is given by X(z) = ( 2z ) 2, with the region of convergence z > 2. Then, X[2] is [GATE 204: 2 Marks] Soln. Given X(z) = Or, ( 2z ) 2 = z > 2 X(z) = ( 2z ) 2 = + 4 z + 2 z 2 + Expanding by binomial expansion Taking inverse z transform We get, x[n] = {, 4, 2, } So, x[2] = 2

EEL3135: Homework #4

EEL3135: 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 information

UNIT-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).

UNIT-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 information

Module 4 : Laplace and Z Transform Problem Set 4

Module 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 information

Need for transformation?

Need 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 information

LTI Systems (Continuous & Discrete) - Basics

LTI 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 information

Signals and Systems. Spring Room 324, Geology Palace, ,

Signals 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 information

GATE EE Topic wise Questions SIGNALS & SYSTEMS

GATE 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 information

8. z-domain Analysis of Discrete-Time Signals and Systems

8. 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 information

Solutions: Homework Set # 5

Solutions: 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

Discrete-time signals and systems

Discrete-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 information

Z-Transform. x (n) Sampler

Z-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 information

ESE 531: Digital Signal Processing

ESE 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 information

The Z transform (2) 1

The 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 information

z-transform Chapter 6

z-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 information

University of Illinois at Urbana-Champaign ECE 310: Digital Signal Processing

University 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 information

y[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)

y[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 information

Use: Analysis of systems, simple convolution, shorthand for e jw, stability. Motivation easier to write. Or X(z) = Z {x(n)}

Use: 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 information

Like bilateral Laplace transforms, ROC must be used to determine a unique inverse z-transform.

Like 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 information

Determine the Z transform (including the region of convergence) for each of the following signals:

Determine 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 information

ESE 531: Digital Signal Processing

ESE 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

EE 521: Instrumentation and Measurements

EE 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 information

QUESTION BANK SIGNALS AND SYSTEMS (4 th SEM ECE)

QUESTION 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 information

2. Typical Discrete-Time Systems All-Pass Systems (5.5) 2.2. Minimum-Phase Systems (5.6) 2.3. Generalized Linear-Phase Systems (5.

2. 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 information

Lecture 7 Discrete Systems

Lecture 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 information

EC Signals and Systems

EC 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 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.

# 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 information

Digital Signal Processing Lecture 4

Digital Signal Processing Lecture 4 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 4 Begüm Demir E-mail:

More information

LECTURE NOTES DIGITAL SIGNAL PROCESSING III B.TECH II SEMESTER (JNTUK R 13)

LECTURE 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

Stability Condition in Terms of the Pole Locations

Stability 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 information

! z-transform. " Tie up loose ends. " Regions of convergence properties. ! Inverse z-transform. " Inspection. " Partial fraction

! 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 information

Lecture 3 Matlab Simulink Minimum Phase, Maximum Phase and Linear Phase Systems

Lecture 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 information

EE Homework 5 - Solutions

EE 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 information

Let 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 )

Let 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 information

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. 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 information

ELEG 305: Digital Signal Processing

ELEG 305: Digital Signal Processing 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

More information

Signals and Systems Lecture 8: Z Transform

Signals 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 information

Z - Transform. It offers the techniques for digital filter design and frequency analysis of digital signals.

Z - 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 information

Very useful for designing and analyzing signal processing systems

Very 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 information

Lecture 19 IIR Filters

Lecture 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

Lecture 11 FIR Filters

Lecture 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 information

Digital Signal Processing. Midterm 1 Solution

Digital 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 information

Signals and Systems. Problem Set: The z-transform and DT Fourier Transform

Signals 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 information

DIGITAL SIGNAL PROCESSING. Chapter 3 z-transform

DIGITAL 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 information

EE 3054: Signals, Systems, and Transforms Spring A causal discrete-time LTI system is described by the equation. y(n) = 1 4.

EE 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 information

Chapter Intended Learning Outcomes: (i) Understanding the relationship between transform and the Fourier transform for discrete-time signals

Chapter 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

Lecture 2 OKAN UNIVERSITY FACULTY OF ENGINEERING AND ARCHITECTURE

Lecture 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 information

Lecture 04: Discrete Frequency Domain Analysis (z-transform)

Lecture 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

EE 225D LECTURE ON DIGITAL FILTERS. University of California Berkeley

EE 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 information

The Z-Transform. Fall 2012, EE123 Digital Signal Processing. Eigen Functions of LTI System. Eigen Functions of LTI System

The 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 information

DIGITAL 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 information

Lecture 18: Stability

Lecture 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 information

(i) Represent discrete-time signals using transform. (ii) Understand the relationship between transform and discrete-time Fourier transform

(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 information

The z-transform Part 2

The 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 information

DSP-I DSP-I DSP-I DSP-I

DSP-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 information

Problem Set 2: Solution Due on Wed. 25th Sept. Fall 2013

Problem Set 2: Solution Due on Wed. 25th Sept. Fall 2013 EE 561: Digital Control Systems Problem Set 2: Solution Due on Wed 25th Sept Fall 2013 Problem 1 Check the following for (internal) stability [Hint: Analyze the characteristic equation] (a) u k = 05u k

More information

Question Paper Code : AEC11T02

Question 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 information

6.003: Signals and Systems

6.003: 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 information

Review of Discrete-Time System

Review 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 information

VI. Z Transform and DT System Analysis

VI. 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 information

EE123 Digital Signal Processing. M. Lustig, EECS UC Berkeley

EE123 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 information

Digital Signal Processing Lecture 10 - Discrete Fourier Transform

Digital 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 information

ECE503: Digital Signal Processing Lecture 4

ECE503: 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 information

Discrete Time Systems

Discrete 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 information

ECE503: Digital Signal Processing Lecture 5

ECE503: 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 information

Analog LTI system Digital LTI system

Analog LTI system Digital LTI system Sampling Decimation Seismometer Amplifier AAA filter DAA filter Analog LTI system Digital LTI system Filtering (Digital Systems) input output filter xn [ ] X ~ [ k] Convolution of Sequences hn [ ] yn [

More information

The z-transform and Discrete-Time LTI Systems

The 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 information

Discrete-Time Fourier Transform (DTFT)

Discrete-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 information

SIGNALS AND SYSTEMS. Unit IV. Analysis of DT signals

SIGNALS 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 information

Transform analysis of LTI systems Oppenheim and Schafer, Second edition pp For LTI systems we can write

Transform analysis of LTI systems Oppenheim and Schafer, Second edition pp For LTI systems we can write Transform analysis of LTI systems Oppenheim and Schafer, Second edition pp. 4 9. For LTI systems we can write yœn D xœn hœn D X kd xœkhœn Alternatively, this relationship can be expressed in the z-transform

More information

Module 4. Related web links and videos. 1. FT and ZT

Module 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 information

ECE 350 Signals and Systems Spring 2011 Final Exam - Solutions. Three 8 ½ x 11 sheets of notes, and a calculator are allowed during the exam.

ECE 350 Signals and Systems Spring 2011 Final Exam - Solutions. Three 8 ½ x 11 sheets of notes, and a calculator are allowed during the exam. ECE 35 Spring - Final Exam 9 May ECE 35 Signals and Systems Spring Final Exam - Solutions Three 8 ½ x sheets of notes, and a calculator are allowed during the exam Write all answers neatly and show your

More information

Your solutions for time-domain waveforms should all be expressed as real-valued functions.

Your solutions for time-domain waveforms should all be expressed as real-valued functions. ECE-486 Test 2, Feb 23, 2017 2 Hours; Closed book; Allowed calculator models: (a) Casio fx-115 models (b) HP33s and HP 35s (c) TI-30X and TI-36X models. Calculators not included in this list are not permitted.

More information

EE482: Digital Signal Processing Applications

EE482: 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 information

Massachusetts 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. 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 information

DHANALAKSHMI COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING EC2314- DIGITAL SIGNAL PROCESSING UNIT I INTRODUCTION PART A

DHANALAKSHMI COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING EC2314- DIGITAL SIGNAL PROCESSING UNIT I INTRODUCTION PART A DHANALAKSHMI COLLEGE OF ENGINEERING DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING EC2314- DIGITAL SIGNAL PROCESSING UNIT I INTRODUCTION PART A Classification of systems : Continuous and Discrete

More information

ECE-S Introduction to Digital Signal Processing Lecture 4 Part A The Z-Transform and LTI Systems

ECE-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 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. 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 information

Solutions. Number of Problems: 10

Solutions. Number of Problems: 10 Final Exam February 4th, 01 Signals & Systems (151-0575-01) Prof. R. D Andrea Solutions Exam Duration: 150 minutes Number of Problems: 10 Permitted aids: One double-sided A4 sheet. Questions can be answered

More information

Solutions. Number of Problems: 10

Solutions. Number of Problems: 10 Final Exam February 2nd, 2013 Signals & Systems (151-0575-01) Prof. R. D Andrea Solutions Exam Duration: 150 minutes Number of Problems: 10 Permitted aids: One double-sided A4 sheet. Questions can be answered

More information

EE-210. Signals and Systems Homework 7 Solutions

EE-210. Signals and Systems Homework 7 Solutions EE-20. Signals and Systems Homework 7 Solutions Spring 200 Exercise Due Date th May. Problems Q Let H be the causal system described by the difference equation w[n] = 7 w[n ] 2 2 w[n 2] + x[n ] x[n 2]

More information

z-transforms Definition of the z-transform Chapter

z-transforms Definition of the z-transform Chapter z-transforms Chapter 7 In the study of discrete-time signal and systems, we have thus far considered the time-domain and the frequency domain. The z- domain gives us a third representation. All three domains

More information

Ch. 7: Z-transform Reading

Ch. 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

Module 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 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 information

Review of Fundamentals of Digital Signal Processing

Review 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

Examples. 2-input, 1-output discrete-time systems: 1-input, 1-output discrete-time systems:

Examples. 2-input, 1-output discrete-time systems: 1-input, 1-output discrete-time systems: Discrete-Time s - I Time-Domain Representation CHAPTER 4 These lecture slides are based on "Digital Signal Processing: A Computer-Based Approach, 4th ed." textbook by S.K. Mitra and its instructor materials.

More information

Digital Filters Ying Sun

Digital Filters Ying Sun Digital Filters Ying Sun Digital filters Finite impulse response (FIR filter: h[n] has a finite numbers of terms. Infinite impulse response (IIR filter: h[n] has infinite numbers of terms. Causal filter:

More information

/ (2π) X(e jω ) dω. 4. An 8 point sequence is given by x(n) = {2,2,2,2,1,1,1,1}. Compute 8 point DFT of x(n) by

/ (2π) X(e jω ) dω. 4. An 8 point sequence is given by x(n) = {2,2,2,2,1,1,1,1}. Compute 8 point DFT of x(n) by Code No: RR320402 Set No. 1 III B.Tech II Semester Regular Examinations, Apr/May 2006 DIGITAL SIGNAL PROCESSING ( Common to Electronics & Communication Engineering, Electronics & Instrumentation Engineering,

More information

EE102B Signal Processing and Linear Systems II. Solutions to Problem Set Nine Spring Quarter

EE102B 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 information

Digital Signal Processing Lecture 9 - Design of Digital Filters - FIR

Digital 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 information

Z-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.

Z-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 information

Discrete Time Systems

Discrete 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 information

Digital Control & Digital Filters. Lectures 1 & 2

Digital 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 information

Z-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) = 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 information

UNIVERSITY OF OSLO. Please make sure that your copy of the problem set is complete before you attempt to answer anything.

UNIVERSITY 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 information

X (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

X (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 information

Transform Analysis of Linear Time-Invariant Systems

Transform 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 information

Digital Signal Processing

Digital 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 information

Digital Signal Processing:

Digital 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 information

Chapter 7: The z-transform

Chapter 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 information

Responses of Digital Filters Chapter Intended Learning Outcomes:

Responses 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 information