Use: Analysis of systems, simple convolution, shorthand for e jw, stability. Motivation easier to write. Or X(z) = Z {x(n)}
|
|
- Sabina Ferguson
- 5 years ago
- Views:
Transcription
1 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 = e +jw, X(z) = x(n) e -jwn = F { x(n) } Can think of z as shorthand symbol for e +jw. Remember that z can equal r e +jw where r is any real number.
2 2 B. Convergence of z transform. Can converge even if Fourier transform doesn t. Convergence related to stability (absolute summability) Ex F{u(n)} = u(n) e -jwn is useless not absolutely summable, u(n) e -jwn = 1 = But if r >1 z transform u(n) (re +jw ) -n r -n = 1 ( 1- (1/r) )
3 3 1/r < 1 so converges For z = r > 1 Region of convergence (R.O.C.) drawn below. (1) Rational Functions in z common (usually closed form for transform of infinite length sequence)
4 4 Z( a n u(n) ) = a n u (n) z -n = 1 1- az -1 = z z - a if az -1 < 1 a < z Rational function poles and zeroes:
5 5 b(n) z -n B(z) H(z) = = = a(n) z -n A(z) K (1-z 1 z -1 ) (1-z 2 z -1 ) (1-z 3 z -1 ).. (1-z M z -1 ) (1-p 1 z -1 ) (1-p 2 z -1 ).. (1-p N z -1 ) = K Π (1-z i z -1 ) Π (1-p i z -1 ) H(z i ) = 0 for zeroes z i H(p i ) = for poles p i Theorem: If h(n) is real, the finite poles and zeroes of H(z) must be real or come in complex conjugate pairs. Partial Proof: (1) Note that: b( n ) = K n i= 1 z k ( i, n, j) where k(i,n,j) = index of the ith root in the jth term of b(n)
6 (2) h(n) real means b(n) and a(n) are real (3) Considering b(n), b(0) = K, so K is real (4) b(1) = -K( z 1 +z 2 + z z M ) = real so the imaginary parts of the zeroes sum to 0. One way this can happen is when complex roots come in conjugate pairs, and the other roots, if any, are real. (5) b(m) = ±K( z 1 z 2 z 3.. z M ) = real. One way this can happen is when complex roots come in conjugate pairs, and the other roots, if any, are real. 6
7 7 (2) Finite length sequences X(z) = x(n) z -n Will converge in some region if x(n) < Case1 n 1 = n 2 = 0 x(0)z -0 converges for entire z-plane. Case 2 n 2 > 0 x(n) z -n = x(n)/ z n In trouble if z = 0 so z 0 if n 2 > 0. Case3 n 1 < 0 have terms x(-1)z, x(-2)z 2 z if n 1 < 0.
8 8 (3) Right Sided Sequences X(z) = x(n)z -n + x(n)z -n x(n) = 0 for n < n 1 X(z) = x(n)z -n Ex x(n) = a n u(n) Converges for exterior of circle.
9 9 Large z makes z -1 small. If converges for z 1, x(n)z -n < Converges for z > R x _ except for z = if n 1 < 0. Right sided sequences have z-transforms which converge for exterior of a circle. Iff z-transform converges for exterior of a circle, it is right sided in time domain. (4) Left Sided Sequences x(n) = 0 for n > n 2
10 10 X(z) = x(n) z -n = x(n) z -n + x(n) z -n Converges if z < some value. Therefore converges for interior of circle except z=0 if n 2 > 0. x(n) z -n Ex x(n) = -(b n )u(-n-1) X(z) = - b n z -n = 1 - b -n z n z < b Draw region of convergence. Note no poles in the R.O.C.
11 11 (5) Two sided Sequences Extends from - to X(z) = x(n)z -n = x(n)z -n + x(n)z -n First converges for z large (exterior of circle z = R x _,) Second converges for z small (interior for circle R x + ) Common region of convergence is an annulus: R x _ < z < R x + and and z- transform exists.
12 12 Ex Find X(z) and its R.O.C. for x(n) = a n u(n) + b n u(-n) C. Inverse z -Transforms (1) Cauchy integral theorem leads to counterclockwise countour in region of convergence. x(n) = (1/2πj) X(z) z n-1 dz
13 13 We won t use this. Note; looks complicated but leads to partial fraction expansion often. (2) Power Series If we have X(z) = x(n)z -n and we have x(n), inverse transform by inspection. H(z) = 0.2z z -1 h(-1) = 0.2, h(0) = 1, h(1) = 0.3 (3) Partial Fraction Expansion for Rational Case H(z) = B(z) A(z)
14 14 = b(n)z -n a(n)z -n If we know poles p(n) of H(z), rewrite H(z) as K b(n)z -n Π (1- p(n)z -1 ) A 1 A 2 A N = + + (1- p(1)z -1 ) (1- p(2)z -1 ) (1- p(n)z -1 ) H(z) = A k (p(k)) n z -n h(n) = A k (p(k)) n u(n)
15 15 Ex Expansion of right sided sequence. x(n) = (1/2) n u(n) + (1/3) n u(n) 1 1 X(z) = /2 z /3 z /6 z -1 = /6 z /6 z -2 A B = /2 z /3 z -1 A = B =
16 16 (4) Long Division H(z) = B(z)/A(z) Rules (1) For a causal solution, put powers of z in descending order for both B(z) and A(z). (2) For an anti-causal solution, put powers of z in ascending order for both B(z) and A(z).
17 17 Comments (1) Solutions from the long division approach are identical to those from the partial fraction approach (2) The long division approach leads to a difference equation. Ex. H(z) = B(z)/A(z) = Y(z)/X(z). If x(n) = δ(n), X(z) = 1 and Y(z) = H(z). Normalize the coefficients so that a(0) = 1. Let A(z) = 1 + A (z). Then B(z) Y(z) = 1 + A (z) X(z) Y(z) = B(z)X(z) A (z)y(z). With x(n) = δ(n) and h(n) = y(n), h(n) = b(k) δ(n-k) - a(k)h(n-k)
18 D. Properties and Uses of Z-Transforms (1) Shift of a Sequence (most important basic use of z-transform). X(z) = x(0) + x(1)z -1 +x(2) z -2 z X(z) = x(0) z + x(1) +x(2) z -1 = Y(z) Z -1 (z X(z)) = y(n) = x(n+1) all n or Z -1 (z k X(z)) = x(n+k) all n, shift to left. Z -1 (z -k X(z)) = x(n-k) all n, shift to right. (2) Multiplication by Exponential Sequence y(n) = a n x(n) 18 Y(z) = a n x(n)z -n = x(n) (a -1 z) -n
19 19 = X(z) = X(a -1 z) If X(z 1 ) is special in original transform (poles, zero, etc) X(a -1 z 2 ) is special now. z 1 = a -1 z 2 or z 2 = a z 1 Expand or show z plane. (3) Convolution of Sequences (Most important Property) y(n) = h(k) x(n-k)
20 20 Z-transform both sides Y(z) = h(k) x(n-k) z -n z -n = z -k z -(n-k) Y(z) = h(k) z -k x(n-k) z -(n-k) Y(z) = H(z) X(z) ( Note: h(f(n)) g(z) -f(n) = X(g(z)) ) This implies that convolution is equivalent to multiplying polynomials in z -1 Ex (3z z -1 ) (2 + 3z -1 + z -2 ) h(n) = 3 δ(n+1) + δ(n) + δ(n-1) x(n) = 2 δ(n) + 3 δ(n-1) + δ(n-2)
21 21 Y(z) = 6z z z -2 + z -3 y(n) = 6δ(n+1) + 11δ(n) + 8δ(n-1) + 4δ(n-2) + δ(n-3) may be easier than straight convolution method. Ex h(n) = a n u(n), x(n) = b n u(n) H(z) = 1 1- az -1 X(z) = 1 1- bz -1
22 22 Y(z) = 1 (1-az -1 ) (1-bz -1 ) A B = + (1-az -1 ) (1-bz -1 ) 1 A = 1-bz -1 = 1 1-(b/a)
23 23 = a (a-b) 1 B = 1-az -1 = 1 1-(a/b) = b (b-a)
24 a b y(n) = [ ( ) a n + ( ) b n ]u(n) a - b b - a 24 Sometimes easier than time domain method. (4) System Function or Transfer Function. In Laplace Transforms, transfer function is Output(s)/intput(s) Also C(s) = M(s) R(s) Given M and R get C Also c(t) = m(t) r(t-t) dt Same in z transform and Fourier transform.
25 25 For Linear Shift invariant system, System function or transfer function is z transform of impulse response. y(n) = h(k) x(n-k) Y(z) = H(z) X(z) If z = 1 or z = e jw, we get frequency response of system. Ex If H(z) = 2z z -1 H(e +jw ) = 2e jw e -jw = 1 + 4cos(w)
26 26 Ex : FIR Filters h(n) = 0 n < M or n > N H(z) = h(n)z -n is transfer function. Ex : IIR Filters a(k) y(n-k) = b(r) x(n-r) for all n z-transform eqn. A(z) Y(z) = B(z) X(z) Y(z) X(z) = B(z) A(z)
27 27 = b(r) z -r a(k) z -k = K Π (1 z r z -1 ) Π (1 p k z -1 ) Pole Zero pattern in z plane specifies the type of causality and stability for the sequence. If poles are inside unit circle, H(z) converges for z outside unit circle and have stable causal filter az -1 az -1 = 1 z = a
28 28 1 = a n z -n iff az -1 < 1 1- az -1 for z > a Z-Transform Examples Ex Find the transform of n x(n) / z ( x (n)z -n ) = - n x(n) z -(n+1) X'(z) = - z -1 n x(n) z -n = - z -1 n x(n) z -n
29 29 Therefore, Z { n x(n) } = -z X'(z) Ex Find the z transform of n a n u(n) 1 z z { a n u(n) } = = = X(z) (1-az -1 ) z - a X'(z) = [ ((z-a) z) / (z-a) 2 ] = -a (z-a) 2 -z X'(z) = a z (z-a) 2 Region of convergence? Same as for X(z)
30 30 Ex Find z-transform of sequence x(n) = n for 0 n N-1 N for n N Straightforward Way x(n) = n u(n) (n-n) u(n-n) r(n) r(n-n) z(n y(n) ) = -z d/dz Y(z) y(n) = u(n) Here, U(z) = z -n = 1 1- z -1
31 31 Y'(z) = - z -2 (1-z -1 ) 2 -zy'(z) = z -1 (1-z -1 ) 2 = R(z) X(z) = R(z) [ 1 z -N ] = z -1 z -N-1 (1-z -1 ) 2 Other way: x(n) = [u(n) u(n-n)] * u(n-1) X(z) = U(z) [ U(z) z -N U(z) ] z -1
32 32 = z -1 (1 z -N ) (1-z -1 ) 2 Ex h(n) = 2 n u(n) + (1/3) n u(n) (a) Find H(z) and its R.O.C. (b) Find a stable version of h(n) that has the same H(z), but with a different R.O.C. (c) Find a stable set of difference equations for the filter. (a) (b)
33 (c) 33
34 34 Ex Generalize the previous example K h(n) = c i a in u(n) i=1 (a) Find H(z) and its R.O.C. (b) If the 3 rd term, c 3 a 3n u(n) is unstable, find a stable version of it that leads to the same H(z), but with a different R.O.C. (c) How many versions of H(z) are there, which have the same form but with a different R.O.C.? (d) Out of these, how many are stable? (e) Which H(z) corresponds to H(e jw )? (f) Give a parallel form block diagram for this filter. (a)
35 35 (b) (c) (d) (e) (f)
36 36 Ex Z { cos(w 1 n) u(n) } Ex Given X(z) = e a z, find x(n)
37 Ex. An IIR digital filter has the impulse response (.5 n n h(n) = ) u( n) + (.2 ) u(n) (a) Find a stable H(z) in closed form, and its region of convergence. (b) Give a stable set of recursive difference equations for the filter. Use the parallel form. (c) Add the two H(z) terms of part (b ) together to get H(z) with one numerator and one denominator. Give the difference equation corresponding to this new direct form H(z). (d) In the pseudocode below, which is based upon H(z) in part (c), give correct expressions for A, B, C, and D, assuming that x(n) = y(n) =0 for negative n. 37
38 38 y(0) = A y(1) = B For n = C to N y(n) = D End 1.5z 1 H ( z ) = + (a) 1.5z 1.2 (b) z 1 1 (c) 1 z +.1z H ( z ) = 1.7 z +.1z (d)
39 E. Three Standard Implementations of H(z) Parallel Form H ( z ) = h + i z N /2 1 k k 1 2 k = 1 1+ fk z + gk z y ( n) = h x( n) + i x( n 1) k k k f y ( n 1) g y ( n 2) k k k k N /2 y( n) = y ( n) k = 1 k
40 40 2. Cascade Form H ( z ) = c + d z + e z N /2 1 2 k k k 1 2 k = 1 1+ fk z + gk z y ( n) = c x ( n) + d x ( n 1) + e x ( n 2) k k k k k k k f y ( n 1) g y ( n 2) k k k k x ( n) = x( n), 1 x ( n) = y ( n) k k 1 y( n) = y ( n) N /2
41 41 3. Direct Form Multiplying out the Cascade form or adding up the parallel form, we get a single transfer function: H ( z ) = a(0) = 1 M r= 0 N k = 0 M b( r) z a( k) z r k = Y ( z) X ( z) y( n) = b( r) x( n r) a( k) y( n k) r= 0 k = 1 N
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 informationDiscrete-Time Signals and Systems. The z-transform and Its Application. The Direct z-transform. Region of Convergence. Reference: Sections
Discrete-Time Signals and Systems The z-transform and Its Application Dr. Deepa Kundur University of Toronto Reference: Sections 3. - 3.4 of John G. Proakis and Dimitris G. Manolakis, Digital Signal Processing:
More 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 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 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 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 (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 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 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 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 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 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 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 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 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 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 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 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 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 informationIII. 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 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 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 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 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 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 informationDigital Signal Processing Lecture 10 - Discrete Fourier Transform
Digital Signal Processing - Discrete Fourier Transform Electrical Engineering and Computer Science University of Tennessee, Knoxville November 12, 2015 Overview 1 2 3 4 Review - 1 Introduction Discrete-time
More 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 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. 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationHow to manipulate Frequencies in Discrete-time Domain? Two Main Approaches
How to manipulate Frequencies in Discrete-time Domain? Two Main Approaches Difference Equations (an LTI system) x[n]: input, y[n]: output That is, building a system that maes use of the current and previous
More 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 informationSolutions: Homework Set # 5
Signal Processing for Communications EPFL Winter Semester 2007/2008 Prof. Suhas Diggavi Handout # 22, Tuesday, November, 2007 Solutions: Homework Set # 5 Problem (a) Since h [n] = 0, we have (b) We can
More informationEE 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# 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 informationLECTURE NOTES DIGITAL SIGNAL PROCESSING III B.TECH II SEMESTER (JNTUK R 13)
LECTURE NOTES ON DIGITAL SIGNAL PROCESSING III B.TECH II SEMESTER (JNTUK R 13) FACULTY : B.V.S.RENUKA DEVI (Asst.Prof) / Dr. K. SRINIVASA RAO (Assoc. Prof) DEPARTMENT OF ELECTRONICS AND COMMUNICATIONS
More 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 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 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 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 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 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 informationDiscrete-time first-order systems
Discrete-time first-order systems 1 Start with the continuous-time system ẏ(t) =ay(t)+bu(t), y(0) Zero-order hold input u(t) =u(nt ), nt apple t
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 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 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 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 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 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 informationEC Signals and Systems
UNIT I CLASSIFICATION OF SIGNALS AND SYSTEMS Continuous time signals (CT signals), discrete time signals (DT signals) Step, Ramp, Pulse, Impulse, Exponential 1. Define Unit Impulse Signal [M/J 1], [M/J
More 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 informationDigital Signal Processing Lecture 3 - Discrete-Time Systems
Digital Signal Processing - Discrete-Time Systems Electrical Engineering and Computer Science University of Tennessee, Knoxville August 25, 2015 Overview 1 2 3 4 5 6 7 8 Introduction Three components of
More informationLike bilateral Laplace transforms, ROC must be used to determine a unique inverse z-transform.
Inversion of the z-transform Focus on rational z-transform of z 1. Apply partial fraction expansion. Like bilateral Laplace transforms, ROC must be used to determine a unique inverse z-transform. Let X(z)
More information2.161 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 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 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 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 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 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 informationYour 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 informationDigital Signal Processing Lecture 8 - Filter Design - IIR
Digital Signal Processing - Filter Design - IIR Electrical Engineering and Computer Science University of Tennessee, Knoxville October 20, 2015 Overview 1 2 3 4 5 6 Roadmap Discrete-time signals and systems
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 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 informationz-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 informationChap 2. Discrete-Time Signals and Systems
Digital Signal Processing Chap 2. Discrete-Time Signals and Systems Chang-Su Kim Discrete-Time Signals CT Signal DT Signal Representation 0 4 1 1 1 2 3 Functional representation 1, n 1,3 x[ n] 4, n 2 0,
More informationResponses of Digital Filters Chapter Intended Learning Outcomes:
Responses of Digital Filters Chapter Intended Learning Outcomes: (i) Understanding the relationships between impulse response, frequency response, difference equation and transfer function in characterizing
More informationLecture 8 - IIR Filters (II)
Lecture 8 - IIR Filters (II) James Barnes (James.Barnes@colostate.edu) Spring 2009 Colorado State University Dept of Electrical and Computer Engineering ECE423 1 / 27 Lecture 8 Outline Introduction Digital
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 informationAnalog 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 informationVU Signal and Image Processing
052600 VU Signal and Image Processing Torsten Möller + Hrvoje Bogunović + Raphael Sahann torsten.moeller@univie.ac.at hrvoje.bogunovic@meduniwien.ac.at raphael.sahann@univie.ac.at vda.cs.univie.ac.at/teaching/sip/18s/
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 informationINFINITE-IMPULSE RESPONSE DIGITAL FILTERS Classical analog filters and their conversion to digital filters 4. THE BUTTERWORTH ANALOG FILTER
INFINITE-IMPULSE RESPONSE DIGITAL FILTERS Classical analog filters and their conversion to digital filters. INTRODUCTION 2. IIR FILTER DESIGN 3. ANALOG FILTERS 4. THE BUTTERWORTH ANALOG FILTER 5. THE CHEBYSHEV-I
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 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
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 informationRecursive, Infinite Impulse Response (IIR) Digital Filters:
Recursive, Infinite Impulse Response (IIR) Digital Filters: Filters defined by Laplace Domain transfer functions (analog devices) can be easily converted to Z domain transfer functions (digital, sampled
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 informationEE482: Digital Signal Processing Applications
Professor Brendan Morris, SEB 3216, brendan.morris@unlv.edu EE482: Digital Signal Processing Applications Spring 2014 TTh 14:30-15:45 CBC C222 Lecture 05 IIR Design 14/03/04 http://www.ee.unlv.edu/~b1morris/ee482/
More informationVoiced Speech. Unvoiced Speech
Digital Speech Processing Lecture 2 Homomorphic Speech Processing General Discrete-Time Model of Speech Production p [ n] = p[ n] h [ n] Voiced Speech L h [ n] = A g[ n] v[ n] r[ n] V V V p [ n ] = u [
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 information! Introduction. ! Discrete Time Signals & Systems. ! Z-Transform. ! Inverse Z-Transform. ! Sampling of Continuous Time Signals
ESE 531: Digital Signal Processing Lec 25: April 24, 2018 Review Course Content! Introduction! Discrete Time Signals & Systems! Discrete Time Fourier Transform! Z-Transform! Inverse Z-Transform! Sampling
More informationAdvanced Training Course on FPGA Design and VHDL for Hardware Simulation and Synthesis
065-3 Advanced Training Course on FPGA Design and VHDL for Hardware Simulation and Synthesis 6 October - 0 November, 009 Digital Signal Processing The z-transform Massimiliano Nolich DEEI Facolta' di Ingegneria
More informationDigital Signal Processing, Homework 1, Spring 2013, Prof. C.D. Chung
Digital Signal Processing, Homework, Spring 203, Prof. C.D. Chung. (0.5%) Page 99, Problem 2.2 (a) The impulse response h [n] of an LTI system is known to be zero, except in the interval N 0 n N. The input
More informationDigital Filter Structures. Basic IIR Digital Filter Structures. of an LTI digital filter is given by the convolution sum or, by the linear constant
Digital Filter Chapter 8 Digital Filter Block Diagram Representation Equivalent Basic FIR Digital Filter Basic IIR Digital Filter. Block Diagram Representation In the time domain, the input-output relations
More informationLecture 7 - IIR Filters
Lecture 7 - IIR Filters James Barnes (James.Barnes@colostate.edu) Spring 204 Colorado State University Dept of Electrical and Computer Engineering ECE423 / 2 Outline. IIR Filter Representations Difference
More 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 information3.1. Determine the z-transform, including the region of convergence, for each of the following sequences: N, N::: n.
Chap. 3 Problems 27 versa. Specifically, we showed that the defining power series of the z-transform may converge when the Fourier transform does not. We explored in detail the dependence of the shape
More informationLecture 8 - IIR Filters (II)
Lecture 8 - IIR Filters (II) James Barnes (James.Barnes@colostate.edu) Spring 24 Colorado State University Dept of Electrical and Computer Engineering ECE423 1 / 29 Lecture 8 Outline Introduction Digital
More information