Discrete-Time Signals and Systems. The z-transform and Its Application. The Direct z-transform. Region of Convergence. Reference: Sections

Size: px
Start display at page:

Download "Discrete-Time Signals and Systems. The z-transform and Its Application. The Direct z-transform. Region of Convergence. Reference: Sections"

Transcription

1 Discrete-Time Signals and Systems The z-transform and Its Application Dr. Deepa Kundur University of Toronto Reference: Sections of John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, 4th edition, Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 2 / 36 The Direct z-transform Region of Convergence Direct z-transform: Notation: X (z) x(n)z n n X (z) {x(n)} the region of convergence (ROC) of X (z) is the set of all values of z for which X (z) attains a finite value The z-transform is, therefore, uniquely characterized by:. expression for X (z) 2. ROC of X (z) x(n) X (z) Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 3 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 4 / 36

2 Power Series Convergence Power Series Convergence For a power series, For a power series, f (z) a n (z c) n a 0 + a (z c) + a 2 (z c) 2 + n0 f (z) n0 a n (z c) n a 0 + a (z c) + a 2 (z c) 2 + there exists a number 0 r such that the series convergences for z c < r, and diverges for z c > r may or may not converge for values on z c r. there exists a number 0 r such that the series convergences for z c >r, and diverges for z c <r may or may not converge for values on z c r. Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 5 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 6 / 36 Region of Convergence Region of Convergence: r > r 2 Consider X (z) n n x(n)z n x(n)z n + x( n )z n + n 0 }{{} ROC: z < r x(n)z n n0 x(n)z n n0 }{{} ROC: z > r 2 x(0) }{{} ROC: all z Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 7 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 8 / 36

3 Region of Convergence: r < r 2 ROC Families: Finite Duration Signals Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 9 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 0 / 36 ROC Families: Infinite Duration Signals z-transform Properties Property Time Domain z-domain ROC Notation: x(n) X (z) ROC: r 2 < z < r x (n) X (z) ROC x 2 (n) X (z) ROC 2 Linearity: a x (n) + a 2 x 2 (n) a X (z) + a 2 X 2 (z) At least ROC ROC 2 Time shifting: x(n k) z k X (z) ROC, except z 0 (if k > 0) and z (if k < 0) z-scaling: a n x(n) X (a z) a r 2 < z < a r Time reversal x( n) X (z ) r < z < r 2 Conjugation: x (n) X (z ) ROC dx (z) z-differentiation: n x(n) z r d < z < r Convolution: x (n) x 2 (n) X (z)x 2 (z) At least ROC ROC 2 among others... Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 2 / 36

4 Convolution Property x(n) x (n) x 2 (n) X (z) X (z) X 2 (z) Convolution using the z-transform Basic Steps:. Compute z-transform of each of the signals to convolve (time domain z-domain): X (z) {x (n)} X 2 (z) {x 2 (n)} 2. Multiply the two z-transforms (in z-domain): X (z) X (z)x 2 (z) 3. Find the inverse z-transformof the product (z-domain time domain): x(n) {X (z)} Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 3 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 4 / 36 Common Transform Pairs Common Transform Pairs Signal, x(n) z-transform, X (z) ROC δ(n) All z 2 u(n) z z > 3 a n u(n) az 4 na n u(n) az ( az ) 2 5 a n u( n ) az 6 na n az u( n ) 7 cos(ω 0 n)u(n) 8 sin(ω 0 n)u(n) 9 a n cos(ω 0 n)u(n) 0 a n sin(ω 0 n)u(n) ( az ) 2 z cos ω 0 z < a z < a 2z cos ω 0 +z 2 z > z sin ω 0 2z cos ω 0 +z 2 z > az cos ω 0 2az cos ω 0 +a 2 z 2 az sin ω 0 2az cos ω 0 +a 2 z 2 Signal, x(n) z-transform, X (z) ROC δ(n) All z 2 u(n) z z > 3 a n u(n) az 4 na n u(n) az ( az ) 2 5 a n u( n ) az 6 na n az u( n ) 7 cos(ω 0 n)u(n) 8 sin(ω 0 n)u(n) 9 a n cos(ω 0 n)u(n) 0 a n sin(ω 0 n)u(n) ( az ) 2 z cos ω 0 z < a z < a 2z cos ω 0 +z 2 z > z sin ω 0 2z cos ω 0 +z 2 z > az cos ω 0 2az cos ω 0 +a 2 z 2 az sin ω 0 2az cos ω 0 +a 2 z 2 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 5 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 6 / 36

5 Why Rational? Poles and eros X (z) is a rational function iff it can be represented as the ratio of two polynomials in z (or z): X (z) b 0 + b z + b 2 z b M z M a 0 + a z + a 2 z a N z N zeros of X (z): values of z for which X (z) 0 poles of X (z): values of z for which X (z) For LTI systems that are represented by LCCDEs, the z-transform of the unit sample response h(n), denoted H(z) {h(n)}, is rational Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 7 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 8 / 36 Poles and eros of the Rational z-transform Poles and eros of the Rational z-transform Let a 0, b 0 0: X (z) B(z) A(z) b 0 + b z + b 2 z b M z M a 0 + a z + a 2 z a N z ( ) N b0 z M z M + (b /b 0 )z M + + b M /b 0 a 0 z N z N + (a /a 0 )z N + + a N /a 0 b 0 z M+N (z z )(z ) (z z M ) a 0 (z p )(z p 2 ) (z p N ) Gz N M M k (z z k) N k (z p k) X (z) Gz N M M k (z z k) N k (z p k) where G b 0 a 0 Note: finite does not include zero or. X (z) has M finite zeros at z z,,..., z M X (z) has N finite poles at z p, p 2,..., p N For N M 0 if N M > 0, there are N M zero at origin, z 0 if N M < 0, there are N M poles at origin, z 0 Total number of zeros Total number of poles Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 9 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 20 / 36

6 Poles and eros of the Rational z-transform Example: X (z) 2z + 6z 3 + 6z + 5 (z ( 2 (z 0) + j ))(z ( j )) (z ( + j 3))(z ( j 3))(z ( )) Pole-ero Plot Example: poles: z 4 ± j 3 4, 2, zeros: z 0, 2 ± j 2 unit circle poles: z 4 ± j 3 4, 2 zeros: z 0, 2 ± j 2 POLE ERO Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 2 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 22 / 36 Pole-ero Plot Graphical interpretation of characteristics of X (z) on the complex plane ROC cannot include poles; assuming causality... Pole-ero Plot For real time-domain signals, the coefficients of X (z) are necessarily real complex poles and zeros must occur in conjugate pairs note: real poles and zeros do not have to be paired up X (z) z2 2z + 6z 3 + 6z + 5 ROC ROC Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 23 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 24 / 36

7 Pole-ero Plot Insights For causal systems, the ROC will be the outer region of the smallest (origin-centered) circle encompassing all the poles. For stable systems, the ROC will include the unit circle. ROC Causal? Yes. Stable? Yes. For stability of a causal system, the poles will lie inside the unit circle. The System Function Therefore, h(n) time-domain impulse response y(n) x(n) h(n) H(z) H(z) Y (z) X (z) z-domain system function Y (z) X (z) H(z) Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 25 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 26 / 36 The System Function of LCCDEs The System Function of LCCDEs y(n) a k y(n k) + b k x(n k) k {y(n)} { {y(n)} Y (z) k0 a k y(n k) + k a k {y(n k)} + k a k z k Y (z) + k k0 b k x(n k)} k0 b k {x(n k)} k0 b k z k X (z) Y (z) + Y (z) [ a k z k Y (z) k + ] a k z k k H(z) Y (z) X (z) b k z k X (z) k0 X (z) b k z k k0 M [ k0 b kz k + ] N k a kz k LCCDE Rational System Function Many signals of practical interest have a rational z-transform. Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 27 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 28 / 36

8 Inversion of the z-transform Expansion into Power Series Three popular methods: Contour integration: x(n) X (z)z n dz 2πj C Expansion into a power series in z or z : X (z) x(k)z k k and obtaining x(k) for all k by inspection Partial-fraction expansion and table lookup Example: By inspection: X (z) log( + az ), ( ) n+ a n z n ( ) n+ a n n n n x(n) n { ( ) n+ a n n n 0 n 0 z n Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 29 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 30 / 36. Find the distinct poles of X (z): p, p 2,..., p K and their corresponding multiplicities m, m 2,..., m K. 2. The partial-fraction expansion is of the form: X (z) K k ( Ak + A ) 2k z p k (z p k ) + + A mk 2 (z p k ) m k where p k is an m k th order pole (i.e., has multiplicity m k ). 3. Use an appropriate approach to compute {A ik } Example: Find x(n) given poles of X (z) at p 2 and a double pole at p 2 p 3 ; specifically, X (z) X (z) z ( + 2z )( z ) 2 (z + 2)(z ) 2 (z + 2)(z ) 2 A z A 2 z + A 3 (z ) 2 Note: we need a strictly proper rational function. DO NOT FORGET TO MULTIPLY BY z IN THE END. Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 3 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 32 / 36

9 (z + 2) A (z + 2) + A 2(z + 2) (z + 2)(z ) 2 z + 2 z A (z ) 2 + A 2(z + 2) z A A 3(z + 2) (z ) 2 + A 3(z + 2) (z ) 2 z 2 (z ) 2 A (z ) 2 (z + 2)(z ) 2 z + 2 (z + 2) A (z ) 2 z + 2 A A 2(z ) 2 z + A 3(z ) 2 (z ) 2 + A 2 (z ) + A 3 z Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 33 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 34 / 36 (z ) 2 A (z ) 2 (z + 2)(z ) 2 z + 2 d dz (z + 2) [ ] (z + 2) + A 2(z ) 2 z + A 3(z ) 2 (z ) 2 A (z ) 2 + A 2 (z ) + A 3 z + 2 d [ ] A (z ) 2 + A 2 (z ) + A 3 dz z + 2 z A Therefore, assuming causality, and using the following pairs: X (z) 4 9 a n u(n) na n u(n) + 2z az az ( az ) 2 z + 3 x(n) 4 9 ( 2)n u(n) u(n) + 3 nu(n) [ ( 2) n n ] u(n) 3 z ( z ) 2 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 35 / 36 Dr. Deepa Kundur (University of Toronto) The z-transform and Its Application 36 / 36

Digital Signal Processing

Digital 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 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

Discrete-Time Signals and Systems. Frequency Domain Analysis of LTI Systems. The Frequency Response Function. The Frequency Response Function

Discrete-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 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

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

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

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

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

SEL4223 Digital Signal Processing. Inverse Z-Transform. Musa Mohd Mokji

SEL4223 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 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

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

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

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

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

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

Topic 4: The Z Transform

Topic 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 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

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

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

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

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

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

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

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

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

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

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

Signal Analysis, Systems, Transforms

Signal 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 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

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

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

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

(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

# 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

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

! 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

Z Transform (Part - II)

Z 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

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

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

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

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

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 & Systems Handout #4

Signals & 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 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

The 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 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 information

If every Bounded Input produces Bounded Output, system is externally stable equivalently, system is BIBO stable

If 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 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

2.161 Signal Processing: Continuous and Discrete Fall 2008

2.161 Signal Processing: Continuous and Discrete Fall 2008 MIT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Massachusetts

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

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

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

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

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

ELEN E4810: Digital Signal Processing Topic 4: The Z Transform. 1. The Z Transform. 2. Inverse Z Transform

ELEN 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 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

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

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

Review of Fundamentals of Digital Signal Processing

Review 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 information

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

How to manipulate Frequencies in Discrete-time Domain? Two Main Approaches

How 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

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

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

Generalizing the DTFT!

Generalizing 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 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

ECSE 512 Digital Signal Processing I Fall 2010 FINAL EXAMINATION

ECSE 512 Digital Signal Processing I Fall 2010 FINAL EXAMINATION FINAL EXAMINATION 9:00 am 12:00 pm, December 20, 2010 Duration: 180 minutes Examiner: Prof. M. Vu Assoc. Examiner: Prof. B. Champagne There are 6 questions for a total of 120 points. This is a closed book

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

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

1. 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) + 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 information

Z-Transform. 清大電機系林嘉文 Original PowerPoint slides prepared by S. K. Mitra 4-1-1

Z-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 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

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

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

2.161 Signal Processing: Continuous and Discrete Fall 2008

2.161 Signal Processing: Continuous and Discrete Fall 2008 IT OpenCourseWare http://ocw.mit.edu 2.161 Signal Processing: Continuous and Discrete all 2008 or information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. assachusetts

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

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

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

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

Digital 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 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 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 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

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

Digital Signal Processing. Outline. Hani Mehrpouyan, The Z-Transform. The Inverse Z-Transform

Digital Signal Processing. Outline. Hani Mehrpouyan, The Z-Transform. The Inverse Z-Transform Digital Signal Processing Hani Mehrpouyan, Department of Electrical and Computer Engineering, California State University, Bakersfield Lecture (Digital Signal Processing) Feb 9 th, 206 The following references

More information

Digital Signal Processing Lecture 3 - Discrete-Time Systems

Digital 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 information

Chapter 13 Z Transform

Chapter 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 information

Digital Signal Processing

Digital Signal Processing Digital Signal Processing Discrete-Time Signals and Systems (2) Moslem Amiri, Václav Přenosil Embedded Systems Laboratory Faculty of Informatics, Masaryk University Brno, Czech Republic amiri@mail.muni.cz

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

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

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

Analog vs. discrete signals

Analog vs. discrete signals Analog vs. discrete signals Continuous-time signals are also known as analog signals because their amplitude is analogous (i.e., proportional) to the physical quantity they represent. Discrete-time signals

More information

7.17. 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.

7.17. 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 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

Discrete-Time David Johns and Ken Martin University of Toronto

Discrete-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 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

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

ENT 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 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 information

Introduction to the z-transform

Introduction to the z-transform z-transforms and applications Introduction to the z-transform The z-transform is useful for the manipulation of discrete data sequences and has acquired a new significance in the formulation and analysis

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

Digital Signal Processing. Lecture Notes and Exam Questions DRAFT

Digital Signal Processing. Lecture Notes and Exam Questions DRAFT Digital Signal Processing Lecture Notes and Exam Questions Convolution Sum January 31, 2006 Convolution Sum of Two Finite Sequences Consider convolution of h(n) and g(n) (M>N); y(n) = h(n), n =0... M 1

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

III. Time Domain Analysis of systems

III. Time Domain Analysis of systems 1 III. Time Domain Analysis of systems Here, we adapt properties of continuous time systems to discrete time systems Section 2.2-2.5, pp 17-39 System Notation y(n) = T[ x(n) ] A. Types of Systems Memoryless

More information