6.003: Signals and Systems. Feedback, Poles, and Fundamental Modes
|
|
|
- Harold Hall
- 5 years ago
- Views:
Transcription
1 6.003: Sigals ad Systems Feedback, Poles, ad Fudametal Modes February 9, 2010
2 Last Time: Multiple Represetatios of DT Systems Verbal descriptios: preserve the ratioale. To reduce the umber of bits eeded to store a sequece of large umbers that are early equal, record the first umber, ad the record successive differeces. Differece equatios: mathematically compact. y[] =x[] x[ 1] Block diagrams: illustrate sigal flow paths. x[] + y[] 1 Delay Operator represetatios: aalyze systems as polyomials. Y =(1 R) X
3 Last Time: Feedback, Cyclic Sigal Paths, ad Modes Systems with sigals that deped o previous values of the same sigal are said to have feedback. Example: The accumulator system has feedback. X + Y Delay By cotrast, the differece machie does ot have feedback. X + Y 1 Delay
4 Last Time: Feedback, Cyclic Sigal Paths, ad Modes The effect of feedback ca be visualized by tracig each cycle through the cyclic sigal paths. X + Y p 0 Delay x[] =δ[] y[] Each cycle creates aother sample i the output.
5 Last Time: Feedback, Cyclic Sigal Paths, ad Modes The effect of feedback ca be visualized by tracig each cycle through the cyclic sigal paths. X + Y p 0 Delay x[] =δ[] y[] Each cycle creates aother sample i the output.
6 Last Time: Feedback, Cyclic Sigal Paths, ad Modes The effect of feedback ca be visualized by tracig each cycle through the cyclic sigal paths. X + Y p 0 Delay x[] =δ[] y[] Each cycle creates aother sample i the output.
7 Last Time: Feedback, Cyclic Sigal Paths, ad Modes The effect of feedback ca be visualized by tracig each cycle through the cyclic sigal paths. X + Y p 0 Delay x[] =δ[] y[] Each cycle creates aother sample i the output.
8 Last Time: Feedback, Cyclic Sigal Paths, ad Modes The effect of feedback ca be visualized by tracig each cycle through the cyclic sigal paths. X + Y p 0 Delay x[] =δ[] y[] Each cycle creates aother sample i the output.
9 Last Time: Feedback, Cyclic Sigal Paths, ad Modes The effect of feedback ca be visualized by tracig each cycle through the cyclic sigal paths. X + Y p 0 Delay x[] =δ[] y[] Each cycle creates aother sample i the output. The respose will persist eve though the iput is trasiet.
10 Geometric Growth: Poles These uit-sample resposes ca be characterized by a sigle umber the pole which is the base of the geometric sequece. X + Y p 0 Delay { y[] = p 0, if >=0; 0, otherwise. y[] y[] y[] p 0 =0.5 p 0 =1 p 0 =1.2
11 Check Yourself How may of the followig uit-sample resposes ca be represeted by a sigle pole?
12 Check Yourself How may of the followig uit-sample resposes ca be represeted by a sigle pole? 3
13 Geometric Growth The value of p 0 determies the rate of growth. y[] y[] y[] y[] z p 0 < 1: 1 < p 0 < 0: 0 < p 0 < 1: p 0 > 1: magitude diverges, alteratig sig magitude coverges, alteratig sig magitude coverges mootoically magitude diverges mootoically
14 Secod-Order Systems The uit-sample resposes of more complicated cyclic systems are more complicated. X + Y R 1.6 R 0.63 y[] Not geometric. This respose grows the decays.
15 Factorig Secod-Order Systems Factor the operator expressio to break the system ito two simpler systems (divide ad coquer). X + Y R 1.6 R 0.63 Y = X +1.6RY 0.63R 2 Y (1 1.6R +0.63R 2 ) Y = X (1 0.7R)(1 0.9R) Y = X
16 Factorig Secod-Order Systems The factored form correspods to a cascade of simpler systems. (1 0.7R)(1 0.9R) Y = X X + Y 2 + Y 0.7 R 0.9 R (1 0.7R) Y 2 = X (1 0.9R) Y = Y 2 X + Y 1 + Y 0.9 R 0.7 R (1 0.9R) Y 1 = X (1 0.7R) Y = Y 1 The order does t matter (if systems are iitially at rest).
17 Factorig Secod-Order Systems The uit-sample respose of the cascaded system ca be foud by multiplyig the polyomial represetatios of the subsystems. Y X = (1 0.7R)(1 0.9R) = (1 0.7R) (1 0.9R) }{{}}{{} {}}{{}}{ = (1+0.7R R R 3 + ) (1+0.9R R R 3 + ) Multiply, the collect terms of equal order: Y =1+( )R +( )R 2 X +( )R 3 +
18 Multiplyig Polyomial Graphical represetatio of polyomial multiplicatio. Y =(1+aR + a 2 R 2 + a 3 R 3 + ) (1 + br + b 2 R 2 + b 3 R 3 + ) X 1 1 a R b R X a 2 R 2 + b 2 R 2 + Y a 3 R 3 b 3 R Collect terms of equal order: Y =1+(a + b)r +(a 2 + ab + b 2 )R 2 +(a 3 + a 2 b + ab 2 + b 3 )R 3 + X
19 Multiplyig Polyomials Tabular represetatio of polyomial multiplicatio. (1 + ar + a 2 R 2 + a 3 R 3 + ) (1 + br + b 2 R 2 + b 3 R 3 + ) 1 br b 2 R 2 b 3 R br b 2 R 2 b 3 R 3 ar ar abr 2 ab 2 R 3 ab 3 R 4 a 2 R 2 a 2 R 2 a 2 br 3 a 2 b 2 R 4 a 2 b 3 R 5 a 3 R 3 a 3 R 3 a 3 br 4 a 3 b 2 R 5 a 3 b 3 R 6 Group same powers of R by followig reverse diagoals: Y =1+(a + b)r +(a 2 + ab + b 2 )R 2 +(a 3 + a 2 b + ab 2 + b 3 )R 3 + X y[]
20 Partial Fractios Use partial fractios to rewrite as a sum of simpler parts. X + Y R 1.6 R 0.63 Y X = 1 1.6R +0.63R 2 = (1 0.9R)(1 0.7R) = 1 0.9R 1 0.7R
21 Secod-Order Systems: Equivalet Forms The sum of simpler parts suggests a parallel implemetatio. Y X = 1 0.9R 1 0.7R X + Y Y 0.9 R + Y R If x[] =δ[] the y 1 [] =0.9 ad y 2 [] =0.7 for 0. Thus, y[] =4.5(0.9) 3.5(0.7) for 0.
22 Partial Fractios Graphical represetatio of the sum of geometric sequeces. y 1 [] =0.9 for y 2 [] =0.7 for y[] =4.5(0.9) 3.5(0.7) for
23 Partial Fractios Partial fractios provides a remarkable equivalece. X + Y R 1.6 R 0.63 X + Y Y 0.9 R + Y R follows from thikig about system as polyomial (factorig).
24 Poles The key to simplifyig a higher-order system is idetifyig its poles. Poles are the roots of the deomiator of the system fuctioal whe R 1. z Start with system fuctioal: Y = = = X 1 1.6R+0.63R 2 (1 p 0 R)(1 p 1 R) (1 0.7R) (1 0.9R) }{{}}{{} p 0 =0.7 p 1 =0.9 1 Substitute R ad fid roots of deomiator: z Y 1 z 2 z 2 = = = X z 2 1.6z+0.63 (z 0.7) (z 0.9) }{{}}{{} z z z 0 =0.7 z 1 =0.9 The poles are at 0.7 ad 0.9.
25 Check Yourself Cosider the system described by y[] = 1 4 y[ 1] y[ 2] + x[ 1] x[ 2] 2 How may of the followig are true? 1. The uit sample respose coverges to zero. 2. There are poles at z = 1 2 ad z = There is a pole at z = There are two poles. 5. Noe of the above
26 Check Yourself y[] = 4 y[ 1] + 8 y[ 2] + x[ 1] 2 x[ 2] ( R 8 1 R 2 )Y =(R 2 1 R 2 )X R R 2 2 z H(R) = Y z 2 = = z = X 1+ 1 R z 4 8R 2 1 z 2 z 2 + z z 1 = ( )( 2 ) z +1 2 z The uit sample respose coverges to zero. 2. There are poles at z = 2 ad z = There is a pole at z = 1 X 4. There are two poles Noe of the above X X
27 Check Yourself Cosider the system described by y[] = 1 4 y[ 1] y[ 2] + x[ 1] x[ 2] 2 How may of the followig are true? 2 1. The uit sample respose coverges to zero. 2. There are poles at z = 1 2 ad z = There is a pole at z = There are two poles. 5. Noe of the above
28 Populatio Growth
29 Populatio Growth
30 Populatio Growth
31 Populatio Growth
32 Populatio Growth
33 Populatio Growth
34 Populatio Growth
35 Populatio Growth
36 Populatio Growth
37 Populatio Growth
38 Check Yourself What are the pole(s) of the Fiboacci system? ad ad ad oe of the above
39 Check Yourself What are the pole(s) of the Fiboacci system? Differece equatio for Fiboacci system: y[] = x[]+ y[ 1] + y[ 2] System fuctioal: Y 1 H = = X 1 R R 2 Deomiator is secod order 2 poles.
40 Check Yourself Fid the poles by substitutig R 1/z i system fuctioal. Y 1 1 z 2 H = = = X 1 R R z 2 z 1 2 Poles are at 1 ± 5 1 z = = φ, 2 φ z 1 z where φ represets the golde ratio 1+ 5 φ = The two poles are at 1 z 0 = φ ad z 1 = φ
41 Check Yourself What are the pole(s) of the Fiboacci system? ad ad ad oe of the above
42 Example: Fiboacci s Buies Each pole correspods to a fudametal mode. 1 φ ad φ φ ( ) 1 φ Oe mode diverges, oe mode oscillates!
43 Example: Fiboacci s Buies The uit-sample respose of the Fiboacci system ca be writte as a weighted sum of fudametal modes. φ 5 1 φ 5 Y 1 H = = = + X 1 R R 2 1 φr 1+ 1 R φ φ h[] = φ 1 + ( φ) ; 0 5 φ 5 But we already kow that h[] is the Fiboacci sequece f: f :1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,... Therefore we ca calculate f[] without kowig f[ 1] or f[ 2]!
44 Complex Poles What if a pole has a o-zero imagiary part? Example: Y 1 = X 1 R+ R 2 1 z 2 = = 1 z z 2 z +1 z 2 Poles are z = 2 1 ± 2 3 j = e ±jπ/3. What are the implicatios of complex poles?
45 Complex Poles Partial fractios work eve whe the poles are complex. ( ) Y e jπ/3 e jπ/3 X = 1 e jπ/3 R 1 e jπ/3 R = j 3 1 e jπ/3 R 1 e jπ/3 R There are two fudametal modes (both geometric sequeces): e jπ/3 = cos(π/3) + j si(π/3) ad e jπ/3 = cos(π/3) j si(π/3)
46 Complex Poles Complex modes are easier to visualize i the complex plae. e j2π/3 e jπ/3 = cos(π/3) + j si(π/3) Im e j1π/3 e j3π/3 e j0π/3 Re e e j4π/3 j4π/3 e j5π/3 e jπ/3 = cos(π/3) j si(π/3) Im e j5π/3 e j3π/3 e j0π/3 Re e j2π/3 e j1π/3
47 Complex Poles The output of a real system has real values. y[] = x[]+ y[ 1] y[ 2] Y H = = X 1 h[] = 1 1 R+ R = 1 e ( jπ/3 R 1 e jπ/3 R ) 1 e jπ/3 e jπ/3 = j 3 1 e jπ/3 R 1 e jπ/3 R ( ) e j(+1)π/3 e j(+1)π/3 2 ( +1)π = si j h[] 1 1
48 Check Yourself Uit-sample respose of a system with poles at z = re ±jω. Which of the followig is/are true? 1. r<0.5 ad Ω < r<1 ad Ω r<0.5 ad Ω < r<1 ad Ω oe of the above
49 Check Yourself Uit-sample respose of a system with poles at z = re ±jω. Which of the followig is/are true? 2 1. r<0.5 ad Ω < r<1 ad Ω r<0.5 ad Ω < r<1 ad Ω oe of the above
50 Check Yourself X + R R R Y How may of the followig statemets are true? 1. This system has 3 fudametal modes. 2. All of the fudametal modes ca be writte as geometrics. 3. Uit-sample respose is y[] :0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, Uit-sample respose is y[] :1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, Oe of the fudametal modes of this system is the uit step.
51 Check Yourself X + R R R Y How may of the followig statemets are true? 4 1. This system has 3 fudametal modes. 2. All of the fudametal modes ca be writte as geometrics. 3. Uit-sample respose is y[] :0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, Uit-sample respose is y[] :1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, Oe of the fudametal modes of this system is the uit step.
52 Summary Systems composed of adders, gais, ad delays ca be characterized by their poles. The poles of a system determie its fudametal modes. The uit-sample respose of a system ca be expressed as a weighted sum of fudametal modes. These properties follow from a polyomial iterpretatio of the system fuctioal.
53 MIT OpeCourseWare Sigals ad Systems Sprig 2010 For iformatio about citig these materials or our Terms of Use, visit:
6.003 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
Difference 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
The 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 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](/thumbs/82/85448534.jpg "The 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.
Lecture 3. Digital Signal Processing. Chapter 3. z-transforms. Mikael Swartling Nedelko Grbic Bengt Mandersson. rev. 2016
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 z-trasforms We defie the z-trasform
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
Signal 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
Name Date PRECALCULUS SUMMER PACKET
Name Date PRECALCULUS SUMMER PACKET This packet covers some of the cocepts that you eed to e familiar with i order to e successful i Precalculus. This summer packet is due o the first day of school! Make
Math 2412 Review 3(answers) kt
Math 4 Review 3(aswers) kt A t A e. If the half-life of radium is 690 years, ad you have 0 grams ow, how much will be preset i 50 years (rouded to three decimal places)?. The decay of radium is modeled
Chapter 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
Appendix F: Complex Numbers
Appedix F Complex Numbers F1 Appedix F: Complex Numbers Use the imagiary uit i to write complex umbers, ad to add, subtract, ad multiply complex umbers. Fid complex solutios of quadratic equatios. Write
ECE-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
Recursive Algorithms. Recurrences. Recursive Algorithms Analysis
Recursive Algorithms Recurreces Computer Sciece & Egieerig 35: Discrete Mathematics Christopher M Bourke cbourke@cseuledu A recursive algorithm is oe i which objects are defied i terms of other objects
Chapter 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
APPENDIX F Complex Numbers
APPENDIX F Complex Numbers Operatios with Complex Numbers Complex Solutios of Quadratic Equatios Polar Form of a Complex Number Powers ad Roots of Complex Numbers Operatios with Complex Numbers Some equatios
Math 312 Lecture Notes One Dimensional Maps
Math 312 Lecture Notes Oe Dimesioal Maps Warre Weckesser Departmet of Mathematics Colgate Uiversity 21-23 February 25 A Example We begi with the simplest model of populatio growth. Suppose, for example,
Discrete-Time Signals and Systems. Discrete-Time Signals and Systems. Signal Symmetry. Elementary Discrete-Time Signals.
Discrete-ime Sigals ad Systems Discrete-ime Sigals ad Systems Dr. Deepa Kudur Uiversity of oroto Referece: Sectios. -.5 of Joh G. Proakis ad Dimitris G. Maolakis, Digital Sigal Processig: Priciples, Algorithms,
Course : Algebraic Combinatorics
Course 8.32: Algebraic Combiatorics Lecture Notes # Addedum by Gregg Musier February 4th - 6th, 2009 Recurrece Relatios ad Geeratig Fuctios Give a ifiite sequece of umbers, a geeratig fuctio is a compact
Introduction 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
Part I: Covers Sequence through Series Comparison Tests
Part I: Covers Sequece through Series Compariso Tests. Give a example of each of the followig: (a) A geometric sequece: (b) A alteratig sequece: (c) A sequece that is bouded, but ot coverget: (d) A sequece
CHAPTER 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
FIR Filters. Lecture #7 Chapter 5. BME 310 Biomedical Computing - J.Schesser
FIR Filters Lecture #7 Chapter 5 8 What Is this Course All About? To Gai a Appreciatio of the Various Types of Sigals ad Systems To Aalyze The Various Types of Systems To Lear the Skills ad Tools eeded
CALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
Notice that this test does not say anything about divergence of an alternating series.
MATH 572H Sprig 20 Worksheet 7 Topics: absolute ad coditioal covergece; power series. Defiitio. A series b is called absolutely coverget if the series b is coverget. If the series b coverges, while b diverges,
ECE 308 Discrete-Time Signals and Systems
ECE 38-5 ECE 38 Discrete-Time Sigals ad Systems Z. Aliyazicioglu Electrical ad Computer Egieerig Departmet Cal Poly Pomoa ECE 38-5 1 Additio, Multiplicatio, ad Scalig of Sequeces Amplitude Scalig: (A Costat
Z - 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:
Chapter 2 Feedback Control Theory Continued
Chapter Feedback Cotrol Theor Cotiued. Itroductio I the previous chapter, the respose characteristic of simple first ad secod order trasfer fuctios were studied. It was show that first order trasfer fuctio,
Generalizing 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.
5 Sequences and Series
Bria E. Veitch 5 Sequeces ad Series 5. Sequeces A sequece is a list of umbers i a defiite order. a is the first term a 2 is the secod term a is the -th term The sequece {a, a 2, a 3,..., a,..., } is a
CSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
SNAP Centre Workshop. Basic Algebraic Manipulation
SNAP Cetre Workshop Basic Algebraic Maipulatio 8 Simplifyig Algebraic Expressios Whe a expressio is writte i the most compact maer possible, it is cosidered to be simplified. Not Simplified: x(x + 4x)
Carleton College, Winter 2017 Math 121, Practice Final Prof. Jones. Note: the exam will have a section of true-false questions, like the one below.
Carleto College, Witer 207 Math 2, Practice Fial Prof. Joes Note: the exam will have a sectio of true-false questios, like the oe below.. True or False. Briefly explai your aswer. A icorrectly justified
Z ß cos x + si x R du We start with the substitutio u = si(x), so du = cos(x). The itegral becomes but +u we should chage the limits to go with the ew
Problem ( poits) Evaluate the itegrals Z p x 9 x We ca draw a right triagle labeled this way x p x 9 From this we ca read off x = sec, so = sec ta, ad p x 9 = R ta. Puttig those pieces ito the itegralrwe
MASSACHUSETTS 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
Sect 5.3 Proportions
Sect 5. Proportios Objective a: Defiitio of a Proportio. A proportio is a statemet that two ratios or two rates are equal. A proportio takes a form that is similar to a aalogy i Eglish class. For example,
Series: Infinite Sums
Series: Ifiite Sums Series are a way to mae sese of certai types of ifiitely log sums. We will eed to be able to do this if we are to attai our goal of approximatig trascedetal fuctios by usig ifiite degree
Solutions 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,
Math 113 Exam 4 Practice
Math Exam 4 Practice Exam 4 will cover.-.. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for
Definition 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
Infinite 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
FLC Ch 8 & 9. Evaluate. Check work. a) b) c) d) e) f) g) h) i) j) k) l) m) n) o) 3. p) q) r) s) t) 3.
Math 100 Elemetary Algebra Sec 8.1: Radical Expressios List perfect squares ad evaluate their square root. Kow these perfect squares for test. Def The positive (pricipal) square root of x, writte x, is
, 4 is the second term U 2
Balliteer Istitute 995-00 wwwleavigcertsolutioscom Leavig Cert Higher Maths Sequeces ad Series A sequece is a array of elemets seperated by commas E,,7,0,, The elemets are called the terms of the sequece
Olli Simula T / Chapter 1 3. Olli Simula T / Chapter 1 5
Sigals ad Systems Sigals ad Systems Sigals are variables that carry iformatio Systemstake sigals as iputs ad produce sigals as outputs The course deals with the passage of sigals through systems T-6.4
Signal Processing. Lecture 02: Discrete Time Signals and Systems. Ahmet Taha Koru, Ph. D. Yildiz Technical University.
Sigal Processig Lecture 02: Discrete Time Sigals ad Systems Ahmet Taha Koru, Ph. D. Yildiz Techical Uiversity 2017-2018 Fall ATK (YTU) Sigal Processig 2017-2018 Fall 1 / 51 Discrete Time Sigals Discrete
GEORGIA 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
2.004 Dynamics and Control II Spring 2008
MIT OpeCourseWare http://ocw.mit.edu 2.004 Dyamics ad Cotrol II Sprig 2008 For iformatio about citig these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Massachusetts Istitute of Techology
Chapter 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
MAT 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,
Chapter 7: Numerical Series
Chapter 7: Numerical Series Chapter 7 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
TEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
Solutions. 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
3. 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 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,](/thumbs/82/85448051.jpg "3. 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 [
EE260: Digital Design, Spring n Binary Addition. n Complement forms. n Subtraction. n Multiplication. n Inputs: A 0, B 0. n Boolean equations:
EE260: Digital Desig, Sprig 2018 EE 260: Itroductio to Digital Desig Arithmetic Biary Additio Complemet forms Subtractio Multiplicatio Overview Yao Zheg Departmet of Electrical Egieerig Uiversity of Hawaiʻi
Appendix: 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
62. 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
Exponential Moving Average Pieter P
Expoetial Movig Average Pieter P Differece equatio The Differece equatio of a expoetial movig average lter is very simple: y[] x[] + (1 )y[ 1] I this equatio, y[] is the curret output, y[ 1] is the previous
The 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
Sequences 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
Question1 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
Once we have a sequence of numbers, the next thing to do is to sum them up. Given a sequence (a n ) n=1
. Ifiite Series Oce we have a sequece of umbers, the ext thig to do is to sum them up. Give a sequece a be a sequece: ca we give a sesible meaig to the followig expressio? a = a a a a While summig ifiitely
We 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
Signals and Systems. Problem Set: From Continuous-Time to Discrete-Time
Sigals ad Systems Problem Set: From Cotiuous-Time to Discrete-Time Updated: October 5, 2017 Problem Set Problem 1 - Liearity ad Time-Ivariace Cosider the followig systems ad determie whether liearity ad
Math 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
E. Incorrect! Plug n = 1, 2, 3, & 4 into the general term formula. n =, then the first four terms are found by
Calculus II - Problem Solvig Drill 8: Sequeces, Series, ad Covergece Questio No. of 0. Fid the first four terms of the sequece whose geeral term is give by a ( ) : Questio #0 (A) (B) (C) (D) (E) 8,,, 4
Chapter 10: Power Series
Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because
The 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/.
MAS160: Signals, Systems & Information for Media Technology. Problem Set 5. DUE: November 3, (a) Plot of u[n] (b) Plot of x[n]=(0.
![MAS160: Signals, Systems & Information for Media Technology. Problem Set 5. DUE: November 3, (a) Plot of u[n] (b) Plot of x[n]=(0.](/thumbs/94/122054675.jpg "MAS160: 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
P.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 +... +
Alternating Series. 1 n 0 2 n n THEOREM 9.14 Alternating Series Test Let a n > 0. The alternating series. 1 n a n.
0_0905.qxd //0 :7 PM Page SECTION 9.5 Alteratig Series Sectio 9.5 Alteratig Series Use the Alteratig Series Test to determie whether a ifiite series coverges. Use the Alteratig Series Remaider to approximate
Order doesn t matter. There exists a number (zero) whose sum with any number is the number.
P. Real Numbers ad Their Properties Natural Numbers 1,,3. Whole Numbers 0, 1,,... Itegers..., -1, 0, 1,... Real Numbers Ratioal umbers (p/q) Where p & q are itegers, q 0 Irratioal umbers o-termiatig ad
Created by T. Madas SERIES. Created by T. Madas
SERIES SUMMATIONS BY STANDARD RESULTS Questio (**) Use stadard results o summatios to fid the value of 48 ( r )( 3r ). 36 FP-B, 66638 Questio (**+) Fid, i fully simplified factorized form, a expressio
Apply change-of-basis formula to rewrite x as a linear combination of eigenvectors v j.
Eigevalue-Eigevector Istructor: Nam Su Wag eigemcd Ay vector i real Euclidea space of dimesio ca be uiquely epressed as a liear combiatio of liearly idepedet vectors (ie, basis) g j, j,,, α g α g α g α
EE / EEE SAMPLE STUDY MATERIAL. GATE, IES & PSUs Signal System. Electrical Engineering. Postal Correspondence Course
Sigal-EE Postal Correspodece Course 1 SAMPLE STUDY MATERIAL Electrical Egieerig EE / EEE Postal Correspodece Course GATE, IES & PSUs Sigal System Sigal-EE Postal Correspodece Course CONTENTS 1. SIGNAL
10.2 Infinite Series Contemporary Calculus 1
10. Ifiite Series Cotemporary Calculus 1 10. INFINITE SERIES Our goal i this sectio is to add together the umbers i a sequece. Sice it would take a very log time to add together the ifiite umber of umbers,
Solutions to quizzes Math Spring 2007
to quizzes Math 4- Sprig 7 Name: Sectio:. Quiz a) x + x dx b) l x dx a) x + dx x x / + x / dx (/3)x 3/ + x / + c. b) Set u l x, dv dx. The du /x ad v x. By Itegratio by Parts, x(/x)dx x l x x + c. l x
Lecture 3: Divide and Conquer: Fast Fourier Transform
Lecture 3: Divide ad Coquer: Fast Fourier Trasform Polyomial Operatios vs. Represetatios Divide ad Coquer Algorithm Collapsig Samples / Roots of Uity FFT, IFFT, ad Polyomial Multiplicatio Polyomial operatios
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,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
SCORE. Exam 2. MA 114 Exam 2 Fall 2016
Exam 2 Name: Sectio ad/or TA: Do ot remove this aswer page you will retur the whole exam. You will be allowed two hours to complete this test. No books or otes may be used. You may use a graphig calculator
ELEG 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
Chapter 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
Computing the output response of LTI Systems.
Computig the output respose of LTI Systems. By breaig or decomposig ad represetig the iput sigal to the LTI system ito terms of a liear combiatio of a set of basic sigals. Usig the superpositio property
Infinite Series. Definition. An infinite series is an expression of the form. Where the numbers u k are called the terms of the series.
Ifiite Series Defiitio. A ifiite series is a expressio of the form uk = u + u + u + + u + () 2 3 k Where the umbers u k are called the terms of the series. Such a expressio is meat to be the result of
Chapter 6: Numerical Series
Chapter 6: Numerical Series 327 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
Crash course part 2. Frequency compensation
Crash course part Frequecy compesatio Ageda Frequecy depedace Feedback amplifiers Frequecy depedace of the Trasistor Frequecy Compesatio Phatom Zero Examples Crash course part poles ad zeros I geeral a
LESSON 2: SIMPLIFYING RADICALS
High School: Workig with Epressios LESSON : SIMPLIFYING RADICALS N.RN.. C N.RN.. B 5 5 C t t t t t E a b a a b N.RN.. 4 6 N.RN. 4. N.RN. 5. N.RN. 6. 7 8 N.RN. 7. A 7 N.RN. 8. 6 80 448 4 5 6 48 00 6 6 6
= 4 and 4 is the principal cube root of 64.
Chapter Real Numbers ad Radicals Day 1: Roots ad Radicals A2TH SWBAT evaluate radicals of ay idex Do Now: Simplify the followig: a) 2 2 b) (-2) 2 c) -2 2 d) 8 2 e) (-8) 2 f) 5 g)(-5) Based o parts a ad
3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials
Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered
6.003 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
Sail into Summer with Math!
Sail ito Summer with Math! For Studets Eterig Hoors Geometry This summer math booklet was developed to provide studets i kidergarte through the eighth grade a opportuity to review grade level math objectives
Topic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or
Topic : Sequeces ad Series A sequece is a ordered list of umbers, e.g.,,, 8, 6, or,,,.... A series is a sum of the terms of a sequece, e.g. + + + 8 + 6 + or... Sigma Notatio b The otatio f ( k) is shorthad
Series Review. a i converges if lim. i=1. a i. lim S n = lim i=1. 2 k(k + 2) converges. k=1. k=1
Defiitio: We say that the series S = Series Review i= a i is the sum of the first terms. i= a i coverges if lim S exists ad is fiite, where The above is the defiitio of covergece for series. order to see
Exam. Notes: A single A4 sheet of paper (double sided; hand-written or computer typed)
Exam February 8th, 8 Sigals & Systems (5-575-) Prof. R. D Adrea Exam Exam Duratio: 5 Mi Number of Problems: 5 Number of Poits: 5 Permitted aids: Importat: Notes: A sigle A sheet of paper (double sided;
Dynamic Response of Linear Systems
Dyamic Respose of Liear Systems Liear System Respose Superpositio Priciple Resposes to Specific Iputs Dyamic Respose of st Order Systems Characteristic Equatio - Free Respose Stable st Order System Respose
MATH 2300 review problems for Exam 2
MATH 2300 review problems for Exam 2. A metal plate of costat desity ρ (i gm/cm 2 ) has a shape bouded by the curve y = x, the x-axis, ad the lie x =. (a) Fid the mass of the plate. Iclude uits. Mass =
4x 2. (n+1) x 3 n+1. = lim. 4x 2 n+1 n3 n. n 4x 2 = lim = 3
Exam Problems (x. Give the series (, fid the values of x for which this power series coverges. Also =0 state clearly what the radius of covergece is. We start by settig up the Ratio Test: x ( x x ( x x
is also known as the general term of the sequence
Lesso : Sequeces ad Series Outlie Objectives: I ca determie whether a sequece has a patter. I ca determie whether a sequece ca be geeralized to fid a formula for the geeral term i the sequece. I ca determie
CSI 2101 Discrete Structures Winter Homework Assignment #4 (100 points, weight 5%) Due: Thursday, April 5, at 1:00pm (in lecture)
CSI 101 Discrete Structures Witer 01 Prof. Lucia Moura Uiversity of Ottawa Homework Assigmet #4 (100 poits, weight %) Due: Thursday, April, at 1:00pm (i lecture) Program verificatio, Recurrece Relatios
REVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains.
The mai ideas are: Idetities REVISION SHEET FP (MEI) ALGEBRA Before the exam you should kow: If a expressio is a idetity the it is true for all values of the variable it cotais The relatioships betwee
REVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains.
the Further Mathematics etwork wwwfmetworkorguk V 07 The mai ideas are: Idetities REVISION SHEET FP (MEI) ALGEBRA Before the exam you should kow: If a expressio is a idetity the it is true for all values
Properties and Tests of Zeros of Polynomial Functions
Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by
10.1 Sequences. n term. We will deal a. a n or a n n. ( 1) n ( 1) n 1 2 ( 1) a =, 0 0,,,,, ln n. n an 2. n term.
0. Sequeces A sequece is a list of umbers writte i a defiite order: a, a,, a, a is called the first term, a is the secod term, ad i geeral eclusively with ifiite sequeces ad so each term Notatio: the sequece