Lecture 3. Digital Signal Processing. Chapter 3. z-transforms. Mikael Swartling Nedelko Grbic Bengt Mandersson. rev. 2016
|
|
- Augustine Clement Robertson
- 5 years ago
- Views:
Transcription
1 Lecture 3 Digital Sigal Processig Chapter 3 z-trasforms Mikael Swartlig Nedelko Grbic Begt Madersso rev. 06 Departmet of Electrical ad Iformatio Techology Lud Uiversity
2 z-trasforms We defie the z-trasform of a impulse respose h() as H(z) h()z () We assume a causal impulse respose h(). Causal meas that h() 0 for < 0. The sum is therefore limited to H(z) h()z 0 where z r e jω is a complex umber. Complex umbers are ofte writte as a magitude ad a phase. The trasform H(z) is therefore a complex valued fuctio of a complex valued variable. Example Some examples of z-trasforms directly from the defiitio: Fuctio z-trasform h() H(z) h(0) + h()z + h()z + δ() { } 0... δ( k) z k h( k) z k H(z) h () { } 3 H (z) 3 + z + z h () { } 0 3 H (z) 0 + 3z + z + z 3 z ( 3 + z + z ) Proof for the time delay. y() x( ) Y (z) y()z (3) () x( )z (4) z x( )z ( ) (5) z m x(m)z m (6) z X(z) (7)
3 Example A IIR-system ad its z-trasform. h() u() H(z) z (8) 0 (z ) + z (9) h() a u() H(z) z if z > (ROC) (0) a z () 0 ( ) a z 0 () (a z ) + z (3) a z if z > a (ROC) (4) ROC meas regio of covergece: for which z the sum coverges. For a causal sigal the ROC becomes a regio z R mi. This is the ormal case i this course. Example of z-trasform of o-causal sigal (page 54) Give: ( ) x() for all (5) Fid: The z-trasform X(z) of x(). 3
4 Solutio: X(z) 0 0 x() z (6) ( ( ) z (7) ) z + ( ) z (8) 0 ( ) ( ) z + z (9) 0 ( z ) + z + ( z ) + z (0) z + () z ( ) ( z )( z ) if z < ad z > (ROC) () We are forced to check the ROC for each case idividually. Covolutio becomes multiplicatio Covolutio of the sequeces i time-domai becomes a multiplicatio of the polyomials i z-domai. y() h() x() Y (z) H(z)X(z) (3) Proof: Y (z) y()z (4) h(k)x( k)z (5) k h(k)x( k)z ( k) z k (6) k h(k)z k x( k)z ( k) (7) k H(z)X(z) (8) 4
5 Cascade or Serial couplig of systems Cascadig two systems gives: y () x() h () h () y() The impulse respose of the whole system is the covolutio of the impulse resposes for the idividual systems. h tot () h () h () (9) The system equatio of the whole system is the product of the system equatios for the idividual systems. H tot (z) H (z)h (z) (30) Covolutio i the time-domai is multiplicatio if the z-domai. See earlier proof for covolutio of system ad sigal. Example Calculate iterests usig the z-trasform, cotiued from chapter. determie a formula for the balace o the accout. We had: We ca ow y().05 y( ) + x() (3) x() 00 u() (3) Apply the z-trasform. Y (z).05 z Y (z) + X(z) (33) X(z) 00 Solve for Y (z). Y (z) z (34) X(z) (35).05 z.05 z ( z (36) ) 0.05 z z Apply iverse-trasforms to obtai the time-sequece. The balace at years,, 3, 5, 0 ad 000 is y() 00 (.05 0)u() (38) (37) { } (39) 5
6 Solutio to secod order differece equatios First order differece equatios were solved i chapter. Usig the z-trasform we ca ow easily solve higher order differece equatios. We solve for 0 ad assume that both y() ad x() are zero for egative (system i rest). Give a secod order differece equatio: y().7y( ) + 0.8y( ) x( ) x( ) (40) We ca z-trasform each term i the equatio ad we get (assumig both x() ad y() are causal). Y (z).7z Y (z) + 0.8z Y (z) z X(z) z X(z) (4) Solve for Y (z). Y (z) z z.7z X(z) H(z)X(z) (4) + 0.8z Usig tables of formulas for z-trasforms we ca also easily determie y() ad h(). I most cases we are oly lookig for the system properties from H(z). Example: Fiboacci sequece (page 0) The Fiboacci sequece is a sequece where a value is the sum of the two previous values. x() { } (43) Ca we fid a closed form equatio for this sequece? y() y( ) + y( ) where y(0) ad y() (44) Solutio usig impulse respose. y() y( ) + y( ) + δ() (45) Apply the z-trasform. Y (z) z Y (z) + z Y (z) + (46) z z (47) Partial fractio expasio gives where Y (z) p p A p z + A p z (48) ( + 5 ) ( 5 ) A A 5 5 (49) (50) 6
7 After iverse z-trasform the closed form equatio for the sequece becomes y() A p + A p for 0 (5) This ca also be solved as a differece equatio with iitial values (later i the course, see page 0). Iverse z-trasform For most of the cases, use tables of formulas. Defiitio (page 8) The defiitio of the iverse z-trasform is y() πj Y (z)z dz (5) C where C is a closed path aroud the origi ad iside the ROC. Solved usig residual calculus. Polyomial divisio (page 83) Y (z).5z + 0.5z (53) + 3 z z z 3 + (54) y() { } (55) Use table of kow trasforms This is the method we will use the most. First order Y (z) 0.9z (56) y() 0.9 u() (57) Secod order with real poles by partial fractio expasio Y (z) 3 z + z z z (58) ( ) y() u() u() (59) 7
8 Secod order with complex poles by table of formulas si ( ) π 4 z Y (z) cos ( ) π 4 z + 4 z (60) ( ) ) π y() si( 4 u() (6) Split the system ito first ad secod order polyomials by partial fractio expasio. For secod order polyomials, check first ad foremost if the poles are real or complex valued. More o this i the ext lecture Solvig differece equatios with iitial values First order differece equatios were solved earlier ad we assumed that the iitial values were zero for < 0. We ca use the z-trasform eve if y() 0 for < 0. We solve for 0 ad assume x() 0 for < 0 but y() 0 (system ot i rest). We defie the oe sided z-trasform as Y + (z) y()z 0 eve if y() 0 for < 0. Usig this defiitio the z-trasform of time shift is differet. Assumig y 0 () y( ) the z + -trasform becomes Y 0 + (z) y 0 ()z 0 (6) (63) y( )z (64) 0 y( ) + y( )z ( ) z (65) z Y + (z) + y( ) (66) We ow have to apply the iitial values y() for < 0 to get the correct aswer. Usig the oe sided z-trasform we ca ow solve differece equatios with iitial values. See the example i the book for a solutio of the Fiboacci-sequece (page 07). Commo: x() X + (z) (67) (68) 8
9 k x( k) z k X+ (z) + x( )z (69) [ x( k) + x( k + )z + + x( )z k+ + z k X + (z) ] (70) Example First order differece equatio with iitial values. Give: y() ay( ) + x() with a give iitial value for y( ). Fid: y() for 0. Solutio: Y + (z) a [z Y + (z) + y( ) ] + X(z) (7) az X(z) + a y( ) (7) az Iverse trasform with the give values of X(z) ad y( ). Direct form II (page 65) We ca illustrate differece equatios graphically. A simple diagram is show below, called Direct form II or caoical form. More about this i chapter 9. Give: System draw o the form: w() x() + + y() b 0 z a w( ) b Fid: The coectio betwee x() ad y(). Solutio: Itroduce the helper variable w() ad the otatios X(z), W (z) ad Y (z). Determie the sigal w(). W (z) a z W (z) + X(z) (73) X(z) (74) a z 9
10 Determie the output sigal y(). Y (z) b z W (z) + b 0 W (z) (75) b 0 + b z X(z) (76) a z H(z)X(z) (77) Therefore is Y (z) H(z)X(z) where H(z) b 0 + b z a z (78) 0
COMM 602: Digital Signal Processing
COMM 60: Digital Sigal Processig Lecture 4 -Properties of LTIS Usig Z-Trasform -Iverse Z-Trasform Properties of LTIS Usig Z-Trasform Properties of LTIS Usig Z-Trasform -ve +ve Properties of LTIS Usig Z-Trasform
More information6.003 Homework #3 Solutions
6.00 Homework # Solutios Problems. Complex umbers a. Evaluate the real ad imagiary parts of j j. π/ Real part = Imagiary part = 0 e Euler s formula says that j = e jπ/, so jπ/ j π/ j j = e = e. Thus the
More informationZ - Transform. It offers the techniques for digital filter design and frequency analysis of digital signals.
Z - Trasform The -trasform is a very importat tool i describig ad aalyig digital systems. It offers the techiques for digital filter desig ad frequecy aalysis of digital sigals. Defiitio of -trasform:
More informationThe z-transform. 7.1 Introduction. 7.2 The z-transform Derivation of the z-transform: x[n] = z n LTI system, h[n] z = re j
The -Trasform 7. Itroductio Geeralie the complex siusoidal represetatio offered by DTFT to a represetatio of complex expoetial sigals. Obtai more geeral characteristics for discrete-time LTI systems. 7.
More informationGeneralizing the DTFT. The z Transform. Complex Exponential Excitation. The Transfer Function. Systems Described by Difference Equations
Geeraliig the DTFT The Trasform M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl 1 The forward DTFT is defied by X e jω = x e jω i which = Ω is discrete-time radia frequecy, a real variable.
More informationDefinition of z-transform.
- Trasforms Frequecy domai represetatios of discretetime sigals ad LTI discrete-time systems are made possible with the use of DTFT. However ot all discrete-time sigals e.g. uit step sequece are guarateed
More informationMassachusetts Institute of Technology Department of Electrical Engineering and Computer Science. Fall Problem Set 11 Solutions.
Massachusetts Istitute of Techology Departmet of Electrical Egieerig ad Computer Sciece Issued: Thursday, December 8, 005 6.341: Discrete-Time Sigal Processig Fall 005 Problem Set 11 Solutios Problem 11.1
More information3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,
3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [
More informationChapter 7: The z-transform. Chih-Wei Liu
Chapter 7: The -Trasform Chih-Wei Liu Outlie Itroductio The -Trasform Properties of the Regio of Covergece Properties of the -Trasform Iversio of the -Trasform The Trasfer Fuctio Causality ad Stability
More informationDigital Signal Processing
Digital Sigal Processig Z-trasform dftwave -Trasform Backgroud-Defiitio - Fourier trasform j ω j ω e x e extracts the essece of x but is limited i the sese that it ca hadle stable systems oly. jω e coverges
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science. BACKGROUND EXAM September 30, 2004.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Departmet of Electrical Egieerig ad Computer Sciece 6.34 Discrete Time Sigal Processig Fall 24 BACKGROUND EXAM September 3, 24. Full Name: Note: This exam is closed
More informationChapter 7 z-transform
Chapter 7 -Trasform Itroductio Trasform Uilateral Trasform Properties Uilateral Trasform Iversio of Uilateral Trasform Determiig the Frequecy Respose from Poles ad Zeros Itroductio Role i Discrete-Time
More informationSolutions of Chapter 5 Part 1/2
Page 1 of 8 Solutios of Chapter 5 Part 1/2 Problem 5.1-1 Usig the defiitio, compute the -trasform of x[] ( 1) (u[] u[ 8]). Sketch the poles ad eros of X[] i the plae. Solutio: Accordig to the defiitio,
More informationQuestion1 Multiple choices (circle the most appropriate one):
Philadelphia Uiversity Studet Name: Faculty of Egieerig Studet Number: Dept. of Computer Egieerig Fial Exam, First Semester: 2014/2015 Course Title: Digital Sigal Aalysis ad Processig Date: 01/02/2015
More informationEE Midterm Test 1 - Solutions
EE35 - Midterm Test - Solutios Total Poits: 5+ 6 Bous Poits Time: hour. ( poits) Cosider the parallel itercoectio of the two causal systems, System ad System 2, show below. System x[] + y[] System 2 The
More informationM2.The Z-Transform and its Properties
M2.The Z-Trasform ad its Properties Readig Material: Page 94-126 of chapter 3 3/22/2011 I. Discrete-Time Sigals ad Systems 1 What did we talk about i MM1? MM1 - Discrete-Time Sigal ad System 3/22/2011
More informationChapter 4 : Laplace Transform
4. Itroductio Laplace trasform is a alterative to solve the differetial equatio by the complex frequecy domai ( s = σ + jω), istead of the usual time domai. The DE ca be easily trasformed ito a algebraic
More informationSolutions. Number of Problems: 4. None. Use only the prepared sheets for your solutions. Additional paper is available from the supervisors.
Quiz November 4th, 23 Sigals & Systems (5-575-) P. Reist & Prof. R. D Adrea Solutios Exam Duratio: 4 miutes Number of Problems: 4 Permitted aids: Noe. Use oly the prepared sheets for your solutios. Additioal
More informationECE-S352 Introduction to Digital Signal Processing Lecture 3A Direct Solution of Difference Equations
ECE-S352 Itroductio to Digital Sigal Processig Lecture 3A Direct Solutio of Differece Equatios Discrete Time Systems Described by Differece Equatios Uit impulse (sample) respose h() of a DT system allows
More informationFIR Filter Design: Part II
EEL335: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we cosider how we might go about desigig FIR filters with arbitrary frequecy resposes, through compositio of multiple sigle-peak
More informationfrom definition we note that for sequences which are zero for n < 0, X[z] involves only negative powers of z.
We ote that for the past four examples we have expressed the -trasform both as a ratio of polyomials i ad as a ratio of polyomials i -. The questio is how does oe kow which oe to use? [] X ] from defiitio
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 40 Digital Sigal Processig Prof. Mark Fowler Note Set #3 Covolutio & Impulse Respose Review Readig Assigmet: Sect. 2.3 of Proakis & Maolakis / Covolutio for LTI D-T systems We are tryig to fid y(t)
More informationThe Phi Power Series
The Phi Power Series I did this work i about 0 years while poderig the relatioship betwee the golde mea ad the Madelbrot set. I have fially decided to make it available from my blog at http://semresearch.wordpress.com/.
More informationChapter 3. z-transform
Chapter 3 -Trasform 3.0 Itroductio The -Trasform has the same role as that played by the Laplace Trasform i the cotiuous-time theorem. It is a liear operator that is useful for aalyig LTI systems such
More informationIntroduction to Signals and Systems, Part V: Lecture Summary
EEL33: Discrete-Time Sigals ad Systems Itroductio to Sigals ad Systems, Part V: Lecture Summary Itroductio to Sigals ad Systems, Part V: Lecture Summary So far we have oly looked at examples of o-recursive
More informationThe Z-Transform. Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, Prentice Hall Inc.
The Z-Trasform Cotet ad Figures are from Discrete-Time Sigal Processig, e by Oppeheim, Shafer, ad Buck, 999- Pretice Hall Ic. The -Trasform Couterpart of the Laplace trasform for discrete-time sigals Geeraliatio
More informationTime-Domain Representations of LTI Systems
2.1 Itroductio Objectives: 1. Impulse resposes of LTI systems 2. Liear costat-coefficiets differetial or differece equatios of LTI systems 3. Bloc diagram represetatios of LTI systems 4. State-variable
More informationSOLUTIONS TO EXAM 3. Solution: Note that this defines two convergent geometric series with respective radii r 1 = 2/5 < 1 and r 2 = 1/5 < 1.
SOLUTIONS TO EXAM 3 Problem Fid the sum of the followig series 2 + ( ) 5 5 2 5 3 25 2 2 This series diverges Solutio: Note that this defies two coverget geometric series with respective radii r 2/5 < ad
More information6.003: Signals and Systems. Feedback, Poles, and Fundamental Modes
6.003: Sigals ad Systems Feedback, Poles, ad Fudametal Modes February 9, 2010 Last Time: Multiple Represetatios of DT Systems Verbal descriptios: preserve the ratioale. To reduce the umber of bits eeded
More informationFrequency Response of FIR Filters
EEL335: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we itroduce the idea of the frequecy respose of LTI systems, ad focus specifically o the frequecy respose of FIR filters.. Steady-state
More informationFinite-length Discrete Transforms. Chapter 5, Sections
Fiite-legth Discrete Trasforms Chapter 5, Sectios 5.2-50 5.0 Dr. Iyad djafar Outlie The Discrete Fourier Trasform (DFT) Matrix Represetatio of DFT Fiite-legth Sequeces Circular Covolutio DFT Symmetry Properties
More informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More informationEECE 301 Signals & Systems
EECE 301 Sigals & Systems Prof. Mark Fowler Note Set #8 D-T Covolutio: The Tool for Fidig the Zero-State Respose Readig Assigmet: Sectio 2.1-2.2 of Kame ad Heck 1/14 Course Flow Diagram The arrows here
More informationWeb Appendix O - Derivations of the Properties of the z Transform
M. J. Roberts - 2/18/07 Web Appedix O - Derivatios of the Properties of the z Trasform O.1 Liearity Let z = x + y where ad are costats. The ( z)= ( x + y )z = x z + y z ad the liearity property is O.2
More informationThe z-transform can be used to obtain compact transform-domain representations of signals and systems. It
3 4 5 6 7 8 9 10 CHAPTER 3 11 THE Z-TRANSFORM 31 INTRODUCTION The z-trasform ca be used to obtai compact trasform-domai represetatios of sigals ad systems It provides ituitio particularly i LTI system
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More informationFind a formula for the exponential function whose graph is given , 1 2,16 1, 6
Math 4 Activity (Due by EOC Apr. ) Graph the followig epoetial fuctios by modifyig the graph of f. Fid the rage of each fuctio.. g. g. g 4. g. g 6. g Fid a formula for the epoetial fuctio whose graph is
More informationThe Z-Transform. (t-t 0 ) Figure 1: Simplified graph of an impulse function. For an impulse, it can be shown that (1)
The Z-Trasform Sampled Data The geeralied fuctio (t) (also kow as the impulse fuctio) is useful i the defiitio ad aalysis of sampled-data sigals. Figure below shows a simplified graph of a impulse. (t-t
More informationLinear time invariant systems
Liear time ivariat systems Alejadro Ribeiro Dept. of Electrical ad Systems Egieerig Uiversity of Pesylvaia aribeiro@seas.upe.edu http://www.seas.upe.edu/users/~aribeiro/ February 25, 2016 Sigal ad Iformatio
More information1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series.
.3 Covergece Theorems of Fourier Series I this sectio, we preset the covergece of Fourier series. A ifiite sum is, by defiitio, a limit of partial sums, that is, a cos( kx) b si( kx) lim a cos( kx) b si(
More informationx[0] x[1] x[2] Figure 2.1 Graphical representation of a discrete-time signal.
x[ ] x[ ] x[] x[] x[] x[] 9 8 7 6 5 4 3 3 4 5 6 7 8 9 Figure. Graphical represetatio of a discrete-time sigal. From Discrete-Time Sigal Processig, e by Oppeheim, Schafer, ad Buck 999- Pretice Hall, Ic.
More informationSignals & Systems Chapter3
Sigals & Systems Chapter3 1.2 Discrete-Time (D-T) Sigals Electroic systems do most of the processig of a sigal usig a computer. A computer ca t directly process a C-T sigal but istead eeds a stream of
More informationChapter 2 Systems and Signals
Chapter 2 Systems ad Sigals 1 Itroductio Discrete-Time Sigals: Sequeces Discrete-Time Systems Properties of Liear Time-Ivariat Systems Liear Costat-Coefficiet Differece Equatios Frequecy-Domai Represetatio
More informationAppendix: The Laplace Transform
Appedix: The Laplace Trasform The Laplace trasform is a powerful method that ca be used to solve differetial equatio, ad other mathematical problems. Its stregth lies i the fact that it allows the trasformatio
More informationSignal Processing in Mechatronics. Lecture 3, Convolution, Fourier Series and Fourier Transform
Sigal Processig i Mechatroics Summer semester, 1 Lecture 3, Covolutio, Fourier Series ad Fourier rasform Dr. Zhu K.P. AIS, UM 1 1. Covolutio Covolutio Descriptio of LI Systems he mai premise is that the
More informationThe z transform is the discrete-time counterpart of the Laplace transform. Other description: see page 553, textbook.
The -Trasform 7. Itroductio The trasform is the discrete-time couterpart of the Laplace trasform. Other descriptio: see page 553, textbook. 7. The -trasform Derivatio of the -trasform: x[] re jω LTI system,
More informationThe z Transform. The Discrete LTI System Response to a Complex Exponential
The Trasform The trasform geeralies the Discrete-time Forier Trasform for the etire complex plae. For the complex variable is sed the otatio: jω x+ j y r e ; x, y Ω arg r x + y {} The Discrete LTI System
More informationComplex Numbers Solutions
Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i
More informationDifference Equation Construction (1) ENGG 1203 Tutorial. Difference Equation Construction (2) Grow, baby, grow (1)
ENGG 03 Tutorial Differece Equatio Costructio () Systems ad Cotrol April Learig Objectives Differece Equatios Z-trasform Poles Ack.: MIT OCW 6.0, 6.003 Newto s law of coolig states that: The chage i a
More informationPrecalculus MATH Sections 3.1, 3.2, 3.3. Exponential, Logistic and Logarithmic Functions
Precalculus MATH 2412 Sectios 3.1, 3.2, 3.3 Epoetial, Logistic ad Logarithmic Fuctios Epoetial fuctios are used i umerous applicatios coverig may fields of study. They are probably the most importat group
More informationGEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING
GEORGIA INSTITUTE OF TECHNOLOGY SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING ECE 06 Summer 07 Problem Set #5 Assiged: Jue 3, 07 Due Date: Jue 30, 07 Readig: Chapter 5 o FIR Filters. PROBLEM 5..* (The
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationSolution of EECS 315 Final Examination F09
Solutio of EECS 315 Fial Examiatio F9 1. Fid the umerical value of δ ( t + 4ramp( tdt. δ ( t + 4ramp( tdt. Fid the umerical sigal eergy of x E x = x[ ] = δ 3 = 11 = ( = ramp( ( 4 = ramp( 8 = 8 [ ] = (
More informationx x x Using a second Taylor polynomial with remainder, find the best constant C so that for x 0,
Math Activity 9( Due with Fial Eam) Usig first ad secod Taylor polyomials with remaider, show that for, 8 Usig a secod Taylor polyomial with remaider, fid the best costat C so that for, C 9 The th Derivative
More informationSolution of Linear Constant-Coefficient Difference Equations
ECE 38-9 Solutio of Liear Costat-Coefficiet Differece Equatios Z. Aliyazicioglu Electrical ad Computer Egieerig Departmet Cal Poly Pomoa Solutio of Liear Costat-Coefficiet Differece Equatios Example: Determie
More informationModule 18 Discrete Time Signals and Z-Transforms Objective: Introduction : Description: Discrete Time Signal representation
Module 8 Discrete Time Sigals ad Z-Trasforms Objective:To uderstad represetig discrete time sigals, apply z trasform for aalyzigdiscrete time sigals ad to uderstad the relatio to Fourier trasform Itroductio
More informationModule 2: z-transform and Discrete Systems
Module : -Trasform ad Discrete Systems Prof. Eliathamy Amikairajah ajah Dr. Tharmarajah Thiruvara School of Electrical Egieerig & Telecommuicatios The Uiversity of New South Wales Australia The -Trasform
More informationCh3 Discrete Time Fourier Transform
Ch3 Discrete Time Fourier Trasform 3. Show that the DTFT of [] is give by ( k). e k 3. Determie the DTFT of the two sided sigal y [ ],. 3.3 Determie the DTFT of the causal sequece x[ ] A cos( 0 ) [ ],
More informationCHAPTER 10 INFINITE SEQUENCES AND SERIES
CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece
More informationAnalog and Digital Signals. Introduction to Digital Signal Processing. Discrete-time Sinusoids. Analog and Digital Signals
Itroductio to Digital Sigal Processig Chapter : Itroductio Aalog ad Digital Sigals aalog = cotiuous-time cotiuous amplitude digital = discrete-time discrete amplitude cotiuous amplitude discrete amplitude
More informationf(w) w z =R z a 0 a n a nz n Liouville s theorem, we see that Q is constant, which implies that P is constant, which is a contradiction.
Theorem 3.6.4. [Liouville s Theorem] Every bouded etire fuctio is costat. Proof. Let f be a etire fuctio. Suppose that there is M R such that M for ay z C. The for ay z C ad R > 0 f (z) f(w) 2πi (w z)
More informationIn number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.
Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers
More informationUniversity of California at Berkeley College of Engineering Department of Electrical Engineering and Computer Sciences
A Uiversity of Califoria at Berkeley College of Egieerig Departmet of Electrical Egieerig ad Computer Scieces U N I V E R S T H E I T Y O F LE T TH E R E B E LI G H T C A L I F O R N 8 6 8 I A EECS : Sigals
More informationMathematical Description of Discrete-Time Signals. 9/10/16 M. J. Roberts - All Rights Reserved 1
Mathematical Descriptio of Discrete-Time Sigals 9/10/16 M. J. Roberts - All Rights Reserved 1 Samplig ad Discrete Time Samplig is the acquisitio of the values of a cotiuous-time sigal at discrete poits
More informationFIR Filter Design: Part I
EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I. Itroductio FIR Filter Desig: Part I I this set o otes, we cotiue our exploratio o the requecy respose o FIR ilters. First, we cosider some
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam 4 will cover.-., 0. ad 0.. Note that eve though. was tested i exam, questios from that sectios may also be o this exam. For practice problems o., refer to the last review. This
More information3 Gauss map and continued fractions
ICTP, Trieste, July 08 Gauss map ad cotiued fractios I this lecture we will itroduce the Gauss map, which is very importat for its coectio with cotiued fractios i umber theory. The Gauss map G : [0, ]
More informationDiscrete-Time Systems, LTI Systems, and Discrete-Time Convolution
EEL5: Discrete-Time Sigals ad Systems. Itroductio I this set of otes, we begi our mathematical treatmet of discrete-time s. As show i Figure, a discrete-time operates or trasforms some iput sequece x [
More informationDigital signal processing: Lecture 5. z-transformation - I. Produced by Qiangfu Zhao (Since 1995), All rights reserved
Digital sigal processig: Lecture 5 -trasformatio - I Produced by Qiagfu Zhao Sice 995, All rights reserved DSP-Lec5/ Review of last lecture Fourier trasform & iverse Fourier trasform: Time domai & Frequecy
More informationPAPER : IIT-JAM 2010
MATHEMATICS-MA (CODE A) Q.-Q.5: Oly oe optio is correct for each questio. Each questio carries (+6) marks for correct aswer ad ( ) marks for icorrect aswer.. Which of the followig coditios does NOT esure
More informationP.3 Polynomials and Special products
Precalc Fall 2016 Sectios P.3, 1.2, 1.3, P.4, 1.4, P.2 (radicals/ratioal expoets), 1.5, 1.6, 1.7, 1.8, 1.1, 2.1, 2.2 I Polyomial defiitio (p. 28) a x + a x +... + a x + a x 1 1 0 1 1 0 a x + a x +... +
More informatione to approximate (using 4
Review: Taylor Polyomials ad Power Series Fid the iterval of covergece for the series Fid a series for f ( ) d ad fid its iterval of covergece Let f( ) Let f arcta a) Fid the rd degree Maclauri polyomial
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationELEG 4603/5173L Digital Signal Processing Ch. 1 Discrete-Time Signals and Systems
Departmet of Electrical Egieerig Uiversity of Arasas ELEG 4603/5173L Digital Sigal Processig Ch. 1 Discrete-Time Sigals ad Systems Dr. Jigxia Wu wuj@uar.edu OUTLINE 2 Classificatios of discrete-time sigals
More informationWe are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n
Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at
More informationf(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim
Math 3, Sectio 2. (25 poits) Why we defie f(x) dx as we do. (a) Show that the improper itegral diverges. Hece the improper itegral x 2 + x 2 + b also diverges. Solutio: We compute x 2 + = lim b x 2 + =
More informationSeptember 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
More information10.6 ALTERNATING SERIES
0.6 Alteratig Series Cotemporary Calculus 0.6 ALTERNATING SERIES I the last two sectios we cosidered tests for the covergece of series whose terms were all positive. I this sectio we examie series whose
More informationCHAPTER 1 SEQUENCES AND INFINITE SERIES
CHAPTER SEQUENCES AND INFINITE SERIES SEQUENCES AND INFINITE SERIES (0 meetigs) Sequeces ad limit of a sequece Mootoic ad bouded sequece Ifiite series of costat terms Ifiite series of positive terms Alteratig
More informationMath 210A Homework 1
Math 0A Homework Edward Burkard Exercise. a) State the defiitio of a aalytic fuctio. b) What are the relatioships betwee aalytic fuctios ad the Cauchy-Riema equatios? Solutio. a) A fuctio f : G C is called
More informationENGI Series Page 6-01
ENGI 3425 6 Series Page 6-01 6. Series Cotets: 6.01 Sequeces; geeral term, limits, covergece 6.02 Series; summatio otatio, covergece, divergece test 6.03 Stadard Series; telescopig series, geometric series,
More informationCEMTool Tutorial. The z-transform
CEMTool Tutorial The -Trasform Overview This tutorial is part of the CEMWARE series. Each tutorial i this series will teach you a specific topic of commo applicatios by explaiig theoretical cocepts ad
More informationPractice Problems: Taylor and Maclaurin Series
Practice Problems: Taylor ad Maclauri Series Aswers. a) Start by takig derivatives util a patter develops that lets you to write a geeral formula for the -th derivative. Do t simplify as you go, because
More information6.003 Homework #12 Solutions
6.003 Homework # Solutios Problems. Which are rue? For each of the D sigals x [] through x 4 [] (below), determie whether the coditios listed i the followig table are satisfied, ad aswer for true or F
More informationWritten exam Digital Signal Processing for BMT (8E070). Tuesday November 1, 2011, 09:00 12:00.
Techische Uiversiteit Eidhove Fac. Biomedical Egieerig Writte exam Digital Sigal Processig for BMT (8E070). Tuesday November, 0, 09:00 :00. (oe page) ( problems) Problem. s Cosider a aalog filter with
More informationMathematics 116 HWK 21 Solutions 8.2 p580
Mathematics 6 HWK Solutios 8. p580 A abbreviatio: iff is a abbreviatio for if ad oly if. Geometric Series: Several of these problems use what we worked out i class cocerig the geometric series, which I
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationXT - MATHS Grade 12. Date: 2010/06/29. Subject: Series and Sequences 1: Arithmetic Total Marks: 84 = 2 = 2 1. FALSE 10.
ubject: eries ad equeces 1: Arithmetic otal Mars: 8 X - MAH Grade 1 Date: 010/0/ 1. FALE 10 Explaatio: his series is arithmetic as d 1 ad d 15 1 he sum of a arithmetic series is give by [ a ( ] a represets
More informationMAS160: Signals, Systems & Information for Media Technology. Problem Set 5. DUE: November 3, (a) Plot of u[n] (b) Plot of x[n]=(0.
MAS6: Sigals, Systems & Iformatio for Media Techology Problem Set 5 DUE: November 3, 3 Istructors: V. Michael Bove, Jr. ad Rosalid Picard T.A. Jim McBride Problem : Uit-step ad ruig average (DSP First
More informationChapter 6 Overview: Sequences and Numerical Series. For the purposes of AP, this topic is broken into four basic subtopics:
Chapter 6 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals (which is what most studets
More informationPolynomial Functions and Their Graphs
Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively
More informationMAT 271 Project: Partial Fractions for certain rational functions
MAT 7 Project: Partial Fractios for certai ratioal fuctios Prerequisite kowledge: partial fractios from MAT 7, a very good commad of factorig ad complex umbers from Precalculus. To complete this project,
More information6.003 Homework #12 Solutions
6.003 Homework # Solutios Problems. Which are rue? For each of the D sigals x [] through x 4 [] below), determie whether the coditios listed i the followig table are satisfied, ad aswer for true or F for
More informationPRELIM PROBLEM SOLUTIONS
PRELIM PROBLEM SOLUTIONS THE GRAD STUDENTS + KEN Cotets. Complex Aalysis Practice Problems 2. 2. Real Aalysis Practice Problems 2. 4 3. Algebra Practice Problems 2. 8. Complex Aalysis Practice Problems
More informationFall 2011, EE123 Digital Signal Processing
Lecture 5 Miki Lustig, UCB September 14, 211 Miki Lustig, UCB Motivatios for Discrete Fourier Trasform Sampled represetatio i time ad frequecy umerical Fourier aalysis requires a Fourier represetatio that
More informationSection 11.8: Power Series
Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i
More information1 Generating functions for balls in boxes
Math 566 Fall 05 Some otes o geeratig fuctios Give a sequece a 0, a, a,..., a,..., a geeratig fuctio some way of represetig the sequece as a fuctio. There are may ways to do this, with the most commo ways
More informationQuiz. Use either the RATIO or ROOT TEST to determine whether the series is convergent or not.
Quiz. Use either the RATIO or ROOT TEST to determie whether the series is coverget or ot. e .6 POWER SERIES Defiitio. A power series i about is a series of the form c 0 c a c a... c a... a 0 c a where
More information