ENGG1015 Homework 3 Question 1. ENGG1015: Homework 3


 Randolph Skinner
 1 years ago
 Views:
Transcription
1 ENGG0 Homewor Question ENGG0: Homewor Due: Dec, 0, :pm Instruction: Submit your answers electronically through Moodle (Lin to Homewor ) You may type your answers using any text editor of your choice, or you may submit a scanned copy of your hand written answer Acceptable file format: Text file (txt) OpenDocument file (odt) MS Word file (doc / docx) Portable Document File (pdf) Question Difference Equations Again In the previous homewor, we studied the difference equation y[n] = y[n ] + y[n ] + x[n] () with an impulse input and showed, without proof, that there is (liely) an analytical formula for y[n] In this exercise, you will derive this formula systematically Part(a) Derive the transfer function Y (z) of the system X(z) Computing the ztransform, Therefore, Y (z) = z Y (z) + z Y (z) + X(z) Y (z) X(z) = z z Part(b) Show that the poles are at ± Let the bigger one be r 0, and the smaller one be r The poles are the roots of the polynomial in the denominator In this case, applying the quadratic formula, the roots are ( ) ± ( ) 4()( ) () = ± Therefore, r 0 = + and r = EEE/ENGG0/0 () Page of 9
2 ENGG0 Homewor Question Part(c) Compute the partial fraction of the transfer function to turn it to a sum of two firstorder systems Let the partial fractions have coefficients A and B such that Thus, we need A r 0 z + B r z = A Ar z + B Br 0 z z z A + B = Ar + Br 0 = 0 Solving this simultaneous equation, we have Part(d) A = r 0 = + B = r = Show that, with x[n] = δ[n] (an impulse input), ( y[n] = + ) n+ ( ) n+ With an impulse input, y[n] = A(r 0 ) n + B(r ) n ( = + ) n+ ( ) n+ Part(e) The above difference equation, and the corresponding analytical expression, is nown as the Fibonacci sequence It occurs in certain biological settings in nature, such as branching in trees and the arrangement of leaves on a stem, as well as in mathematical constructions such as the Pascal s triangle The dominant pole, ϕ = +, is called the golden ratio There is a closely related Lucas number, defined for n 0 as if n = 0 L n = if n = L n + L n if n > Show that you can generate this sequence of numbers with a particular design of x[n] to the difference equation given in Eq () What is this x[n]? EEE/ENGG0/0 () Page of 9
3 ENGG0 Homewor Question Putting x[0] = in Eq () would give y[0] =, corresponding to L 0 For the next value, y[] = y[0] + x[], hence if we want y[] = L =, we need x[] = The other values of x[n] would be zero for n {0, }, so y[n] = y[n ] + y[n ] would correspond to L n = L n + L n Part(f) Maing use of the analytical formula for the Fibonacci sequence above, show that the analytical formula for the Lucas number is ( y[n] = + ) n+ ( ) n+ [( + ) n ( ) n ] ( + ) n ( ) n = + The input x[0] = would give a Fibonacci sequence where every number is multiplied by two The input x[] = would give another Fibonacci sequence where the sequence is delayed by one unit and is multiplied by negative one Combining the two, we have ( y[n] = + ) n+ ( ) n+ [( + ) n ( ) n ] ( + ) n ( ) n = + Question Complex Poles In the lecture, we considered the effect of a variable parameter in a twopole system We have only derived the case where the two poles are real; with the help of complex numbers, we can now consider other values that may lead to oscillations in the system By way of mathematical bacground, electrical and electronic engineers prefer to use j = as the imaginary number The reason is that the symbol i is normally reserved for electrical current Thus, a number of the form A = a + bj is a complex number (when a and b are real) The number A = a bj is called its conjugate It is also possible, and indeed sometimes more convenient, to represent the complex number A in polar form, written as A = re jθ, EEE/ENGG0/0 () Page of 9
4 ENGG0 Homewor Question where r = a + b is the magnitude, and θ = tan ( b a ) is the angle In this representation, we have made use of Euler s formula e jθ = cos θ + j sin θ A consequence of this formula is that we can express the cosine of an angle as cos θ = ejθ + e jθ For this question, consider a system of the form z + z, where in class we have derived that the poles are at ± 4 Part(a) When = 0, the two poles are at {0, } Setch in the diagram below the trajectories of the two poles as increases from 0 We have drawn one for you At what value of that the two poles meet? Imaginary The two poles meet when 4 = 0, or = 4 Real Part(b) When > 4, the term ( 4) inside the square root is negative We write, instead, that the poles are at ± 4 j Note that the two poles are now complex, and are conjugates of each other Draw the trajectories of the poles as increases from 4 What is the magnitude of the poles? As increases from 4, the real part of the poles stays at, while the imaginary part increases in magnitude Imaginary Real The magnitude of the poles is ( ) + ( ) (4 ) = EEE/ENGG0/0 () Page 4 of 9
5 ENGG0 Homewor Question Part(c) Now let us consider, specifically, that = Express the poles of the system in polar form Let r be the one with the positive imaginary value, and r its conjugate When =, the roots are at ± j The magnitude is ( ) + ( ) =, while the angle is ± tan / / = ± π 4 Therefore, Part(d) r = + j = e j π 4 r = j = e j π 4 Show that we can compute the partial fraction of the transfer function and arrive at We can combine the partial fractions so that r r z + r rz r r z + r rz = (r r z ) + (r r z ) ( rz )( r z ) We now ( rz )( r z ) = z + z because r and r are the roots of the quadratic equation In the numerator, ( r + r = + ) ( j + ) j =, and that Therefore, r + r = ej π + e j π = 0 r r z + r rz = z + z Part(e) Show that we can express the output y[n] to an impulse input x[n] = δ[n] as ( ) n cos(n ) π 4 The cosine function explains the oscillation, while ( ) n leads to a decay in the magnitude EEE/ENGG0/0 () Page of 9
6 ENGG0 Homewor Question With the partial fraction, we now that Part(f) y[n] = r(r ) n + r (r) n = ( ) n e j π 4 e j π 4 + ( e j π 4 e j π 4 ( ) n+ ( ) = e j(n ) π 4 + e j(n ) π 4 ( ) n+ ( = cos(n ) π ) 4 ( ) n = cos(n ) π 4 Now let = Show that the two roots are now the partial fraction becomes where r = e j π r = e j π, s r z + s rz s = e j π 6 s = e j π 6, and the output y[n] to an impulse input x[n] = δ[n] is y[n] = cos(n ) π 6 ) n Note that while there is oscillation due to the cosine function, but there is no decay Similar to the derivation above, r = + j = ej π r = j = e j π For the partial fraction, the numerator becomes s rs + s r s EEE/ENGG0/0 () Page 6 of 9
7 ENGG0 Homewor Question We note that e j π 6 + e j π 6 = cos π 6 = =, while rs + r s = e j( π + π 6 ) + e j( π + π 6 ) = e j π + e j π = 0 So the partial fraction given in the question is valid Finally, for an impulse input, Question y[n] = s(r ) n + s (r) n = e j π 6 e j π n + e j π 6 e j π n = e j(n ) π 6 + e j(n ) π 6 = cos(n ) π 6 Bloc Diagrams for Control A generic bloc diagram for feedbac control systems is shown below: reference + error + controller measurement input sensor system output In this problem, we will substitute different content for the various blocs and compute the transfer function The possible blocs include: (A) (B) delay () (C) + delay () (D) When these blocs are substituted for either controller or system, the flow is from left to right; when they are substituted for sensor, the blocs are flipped horizontally and the flow is from right to left Part(a) Let the controller be D, the system be A, and the sensor be B What is the overall transfer function and the value of the pole(s)? For this and subsequent parts, let controller = C(z), system = P (z), and sensor = G(z) The overall transfer function is System A is, B is z, C is C(z)P (z) + C(z)P (z)g(z), and D is z EEE/ENGG0/0 () Page 7 of 9
8 ENGG0 Homewor Question For this part, ()() + ()()(z ) = ( )z Hence, we have a firstorder system with one pole at Part(b) Let the controller be A, the system be C, and the sensor be D What is the overall transfer function and the value of the pole(s)? z = z + ( + ) z = Hence, we have a firstorder system with one pole at Part(c) z Let the controller be D, the system be C, and the sensor be B What is the overall transfer function and the value of the pole(s)? z + z z = ( )z Hence, we have a firstorder system with one pole at Part(d) Let the controller be B, the system be C, and the sensor be B What is the overall transfer function and the value of the pole(s)? z z z = + z z + z z Hence, we have a secondorder system with two poles at ± 4 EEE/ENGG0/0 () Page 8 of 9
9 ENGG0 Homewor Question 4 Question 4 Binary Number Representation Part(a) What is the binary representation of the decimal number 7? 00 Part(b) Which of the following is the binary representation of the decimal number 4? (Hint: Spaces are inserted for ease of reading Each group contains exactly 6 binary digits) A B 0 C D 0 Part(c) Represent the number minus eighteen ( 8) in the following formats: (i) 6bit unsigned number (ii) 6bit s complement (iii) 6bit s complement (iv) 6bit signmagnitude, with a as the most significant bit representing a negative number Write your answer in the following table Indicate with N/A if the number cannot be using the specified format number 6bit unsigned 6bit s complement 8 6bit s complement N/A Part(d) B 6bit signmag For each pair of numbers A and B below, compare their values when they are treated as numbers in the four representations used in Part(c) Write down the larger of the two in the table below, ie if A > B, write A, if B > A, write B 6bit unsigned 6bit s complement 6bit s complement 6bit signmagnitude A : 000 B : 000 B A A A A : B : 0000 B B B A EEE/ENGG0/0 () Page 9 of 9
ENGG1015 Homework 1 Question 1. ENGG1015: Homework 1
ENGG1015 Homework 1 Question 1 ENGG1015: Homework 1 Due: Nov 5, 2012, 11:55pm Instruction: Submit your answers electronically through Moodle (Link to Homework 1). You may type your answers using any text
More informationENGG1015: Homework 1
ENGG1015 Homework 1 Question 1 ENGG1015: Homework 1 Due: Nov 5, 2012, 11:55pm Instruction: Submit your answers electronically through Moodle (Link to Homework 1). You may type your answers using any text
More informationSOLUTION. Homework 1. Part(a) Due: 15 Mar, 2018, 11:55pm
ENGG1203: Introduction to Electrical and Electronic Engineering Second Semester, 2017 18 Homework 1 Due: 15 Mar, 2018, 11:55pm Instruction: Submit your answers electronically through Moodle. In Moodle,
More informationHomework 1. Part(a) Due: 15 Mar, 2018, 11:55pm
ENGG1203: Introduction to Electrical and Electronic Engineering Second Semester, 2017 18 Homework 1 Due: 15 Mar, 2018, 11:55pm Instruction: Submit your answers electronically through Moodle. In Moodle,
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 202203 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 informationEE 3054: Signals, Systems, and Transforms Spring A causal discretetime LTI system is described by the equation. y(n) = 1 4.
EE : Signals, Systems, and Transforms Spring 7. A causal discretetime LTI system is described by the equation Test y(n) = X x(n k) k= No notes, closed book. Show your work. Simplify your answers.. A discretetime
More informationAlgebra II Standard Term 4 Review packet Test will be 60 Minutes 50 Questions
Algebra II Standard Term Review packet 2017 NAME Test will be 0 Minutes 0 Questions DIRECTIONS: Solve each problem, choose the correct answer, and then fill in the corresponding oval on your answer document.
More informationLecture 04: Discrete Frequency Domain Analysis (ztransform)
Lecture 04: Discrete Frequency Domain Analysis (ztransform) John Chiverton School of Information Technology Mae Fah Luang University 1st Semester 2009/ 2552 Outline Overview Lecture Contents Introduction
More informationENGG 1203 Tutorial_9  Review. Boolean Algebra. Simplifying Logic Circuits. Combinational Logic. 1. Combinational & Sequential Logic
ENGG 1203 Tutorial_9  Review Boolean Algebra 1. Combinational & Sequential Logic 2. Computer Systems 3. Electronic Circuits 4. Signals, Systems, and Control Remark : Multiple Choice Questions : ** Check
More informationAnalysis of Finite Wordlength Effects
Analysis of Finite Wordlength Effects Ideally, the system parameters along with the signal variables have infinite precision taing any value between and In practice, they can tae only discrete values within
More informationUse: Analysis of systems, simple convolution, shorthand for e jw, stability. Motivation easier to write. Or X(z) = Z {x(n)}
1 VI. Z Transform Ch 24 Use: Analysis of systems, simple convolution, shorthand for e jw, stability. A. Definition: X(z) = x(n) z z  transforms Motivation easier to write Or Note if X(z) = Z {x(n)} z
More informationZ  Transform. It offers the techniques for digital filter design and frequency analysis of digital signals.
Z  Transform The ztransform 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 informationComplex Numbers. James K. Peterson. September 19, Department of Biological Sciences and Department of Mathematical Sciences Clemson University
Complex Numbers James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 19, 2013 Outline 1 Complex Numbers 2 Complex Number Calculations
More informationComplex Numbers. Outline. James K. Peterson. September 19, Complex Numbers. Complex Number Calculations. Complex Functions
Complex Numbers James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University September 19, 2013 Outline Complex Numbers Complex Number Calculations Complex
More informationLaboratory handout 1 Mathematical preliminaries
laboratory handouts, me 340 2 Laboratory handout 1 Mathematical preliminaries In this handout, an expression on the left of the symbol := is defined in terms of the expression on the right. In contrast,
More informationSignals and Systems. Problem Set: The ztransform and DT Fourier Transform
Signals and Systems Problem Set: The ztransform and DT Fourier Transform Updated: October 9, 7 Problem Set Problem  Transfer functions in MATLAB A discretetime, causal LTI system is described by the
More informationLogic and Discrete Mathematics. Section 6.7 Recurrence Relations and Their Solution
Logic and Discrete Mathematics Section 6.7 Recurrence Relations and Their Solution Slides version: January 2015 Definition A recurrence relation for a sequence a 0, a 1, a 2,... is a formula giving a n
More informationECE503: Digital Signal Processing Lecture 5
ECE53: Digital Signal Processing Lecture 5 D. Richard Brown III WPI 3February22 WPI D. Richard Brown III 3February22 / 32 Lecture 5 Topics. Magnitude and phase characterization of transfer functions
More informationEEL3135: Homework #4
EEL335: Homework #4 Problem : For each of the systems below, determine whether or not the system is () linear, () timeinvariant, and (3) causal: (a) (b) (c) xn [ ] cos( 04πn) (d) xn [ ] xn [ ] xn [ 5]
More informationNumbers and Arithmetic
Numbers and Arithmetic See: P&H Chapter 2.4 2.6, 3.2, C.5 C.6 Hakim Weatherspoon CS 3410, Spring 2013 Computer Science Cornell University Big Picture: Building a Processor memory inst register file alu
More information6.003: Signals and Systems
6.003: Signals and Systems DiscreteTime Systems September 13, 2011 1 Homework Doing the homework is essential to understanding the content. Weekly Homework Assigments tutor (examtype) problems: answers
More informationHomework #1 Solution
February 7, 4 Department of Electrical and Computer Engineering University of Wisconsin Madison ECE 734 VLSI Array Structures for Digital Signal Processing Homework # Solution Due: February 6, 4 in class.
More informationThe ztransform Part 2
http://faculty.kfupm.edu.sa/ee/muqaibel/ The ztransform Part 2 Dr. Ali Hussein Muqaibel The material to be covered in this lecture is as follows: Properties of the ztransform Linearity Initial and final
More informationAs we know, the three basic trigonometric functions are as follows: Figure 1
Trigonometry Basic Functions As we know, the three basic trigonometric functions are as follows: sin θ = cos θ = opposite hypotenuse adjacent hypotenuse tan θ = opposite adjacent Where θ represents an
More informationLecture 19 IIR Filters
Lecture 19 IIR Filters Fundamentals of Digital Signal Processing Spring, 2012 WeiTa Chu 2012/5/10 1 General IIR Difference Equation IIR system: infiniteimpulse response system The most general class
More informationCHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION. Professor Dae Ryook Yang
CHBE320 LECTURE V LAPLACE TRANSFORM AND TRANSFER FUNCTION Professor Dae Ryook Yang Spring 2018 Dept. of Chemical and Biological Engineering 51 Road Map of the Lecture V Laplace Transform and Transfer
More informationDamped Oscillators (revisited)
Damped Oscillators (revisited) We saw that damped oscillators can be modeled using a recursive filter with two coefficients and no feedforward components: Y(k) =  a(1)*y(k1)  a(2)*y(k2) We derived
More informationECE538 Final Exam Fall 2017 Digital Signal Processing I 14 December Cover Sheet
ECE58 Final Exam Fall 7 Digital Signal Processing I December 7 Cover Sheet Test Duration: hours. Open Book but Closed Notes. Three doublesided 8.5 x crib sheets allowed This test contains five problems.
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 information3 What You Should Know About Complex Numbers
3 What You Should Know About Complex Numbers Life is complex it has a real part, and an imaginary part Andrew Koenig. Complex numbers are an extension of the more familiar world of real numbers that make
More informationMATH 1220 Midterm 1 Thurs., Sept. 20, 2007
MATH 220 Midterm Thurs., Sept. 20, 2007 Write your name and ID number at the top of this page. Show all your work. You may refer to one doublesided sheet of notes during the eam and nothing else. Calculators
More informationCITY UNIVERSITY LONDON. MSc in Information Engineering DIGITAL SIGNAL PROCESSING EPM746
No: CITY UNIVERSITY LONDON MSc in Information Engineering DIGITAL SIGNAL PROCESSING EPM746 Date: 19 May 2004 Time: 09:0011:00 Attempt Three out of FIVE questions, at least One question from PART B PART
More informationExercise Set 6.2: DoubleAngle and HalfAngle Formulas
Exercise Set : DoubleAngle and HalfAngle Formulas Answer the following π 1 (a Evaluate sin π (b Evaluate π π (c Is sin = (d Graph f ( x = sin ( x and g ( x = sin ( x on the same set of axes (e Is sin
More informationCMPT 318: Lecture 5 Complex Exponentials, Spectrum Representation
CMPT 318: Lecture 5 Complex Exponentials, Spectrum Representation Tamara Smyth, tamaras@cs.sfu.ca School of Computing Science, Simon Fraser University January 23, 2006 1 Exponentials The exponential is
More informationIB Mathematics HL Year 2 Unit 11: Completion of Algebra (Core Topic 1)
IB Mathematics HL Year Unit : Completion of Algebra (Core Topic ) Homewor for Unit Ex C:, 3, 4, 7; Ex D: 5, 8, 4; Ex E.: 4, 5, 9, 0, Ex E.3: (a), (b), 3, 7. Now consider these: Lesson 73 Sequences and
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 informationNumbers and Arithmetic
Numbers and Arithmetic See: P&H Chapter 2.4 2.6, 3.2, C.5 C.6 Hakim Weatherspoon CS 3410, Spring 2013 Computer Science Cornell University Big Picture: Building a Processor memory inst register file alu
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 informationLecture 11 FIR Filters
Lecture 11 FIR Filters Fundamentals of Digital Signal Processing Spring, 2012 WeiTa Chu 2012/4/12 1 The Unit Impulse Sequence Any sequence can be represented in this way. The equation is true if k ranges
More informationExample 9 Algebraic Evaluation for Example 1
A Basic Principle Consider the it f(x) x a If you have a formula for the function f and direct substitution gives the indeterminate form 0, you may be able to evaluate the it algebraically. 0 Principle
More informationNeed for transformation?
ZTRANSFORM In today s class Ztransform Unilateral Ztransform Bilateral Ztransform Region of Convergence Inverse Ztransform Power Series method Partial Fraction method Solution of difference equations
More informationIntroduction to Digital Signal Processing
Introduction to Digital Signal Processing 1.1 What is DSP? DSP is a technique of performing the mathematical operations on the signals in digital domain. As real time signals are analog in nature we need
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 informationMathematics 324 Riemann Zeta Function August 5, 2005
Mathematics 324 Riemann Zeta Function August 5, 25 In this note we give an introduction to the Riemann zeta function, which connects the ideas of real analysis with the arithmetic of the integers. Define
More informationExamples: Solving nth Order Equations
Atoms L. Euler s Theorem The Atom List First Order. Solve 2y + 5y = 0. Examples: Solving nth Order Equations Second Order. Solve y + 2y + y = 0, y + 3y + 2y = 0 and y + 2y + 5y = 0. Third Order. Solve
More informationAlgebra 2 Khan Academy Video Correlations By SpringBoard Activity
SB Activity Activity 1 Creating Equations 11 Learning Targets: Create an equation in one variable from a realworld context. Solve an equation in one variable. 12 Learning Targets: Create equations in
More information10/12/2016. An FSM with No Inputs Moves from State to State. ECE 120: Introduction to Computing. Eventually, the States Form a Loop
University of Illinois at UrbanaChampaign Dept. of Electrical and Computer Engineering An FSM with No Inputs Moves from State to State What happens if an FSM has no inputs? ECE 120: Introduction to Computing
More informationAlgebra 2 Khan Academy Video Correlations By SpringBoard Activity
SB Activity Activity 1 Creating Equations 11 Learning Targets: Create an equation in one variable from a realworld context. Solve an equation in one variable. 12 Learning Targets: Create equations in
More informationEE538 Final Exam Fall :20 pm 5:20 pm PHYS 223 Dec. 17, Cover Sheet
EE538 Final Exam Fall 005 3:0 pm 5:0 pm PHYS 3 Dec. 17, 005 Cover Sheet Test Duration: 10 minutes. Open Book but Closed Notes. Calculators ARE allowed!! This test contains five problems. Each of the five
More informationMath 312 Lecture Notes Linear Twodimensional Systems of Differential Equations
Math 2 Lecture Notes Linear Twodimensional Systems of Differential Equations Warren Weckesser Department of Mathematics Colgate University February 2005 In these notes, we consider the linear system of
More informationELEG 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 information21.4. Engineering Applications of ztransforms. Introduction. Prerequisites. Learning Outcomes
Engineering Applications of ztransforms 21.4 Introduction In this Section we shall apply the basic theory of ztransforms to help us to obtain the response or output sequence for a discrete system. This
More informationQ Scheme Marks AOs Pearson Progression Step and Progress descriptor. and sin or x 6 16x 6 or x o.e
1a A 45 seen or implied in later working. B1 1.1b 5th Makes an attempt to use the sine rule, for example, writing sin10 sin 45 8x3 4x1 States or implies that sin10 3 and sin 45 A1 1. Solve problems involving
More informationMethods for Synthesizing Very High Q Parametrically Well Behaved Two Pole Filters
Methods for Synthesizing Very High Q Parametrically Well Behaved Two Pole Filters Max Mathews Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA) Department of Music, Stanford
More informationZTRANSFORMS. 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. ZTRANSFORMS 5 ztransforms 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 information2.004 Dynamics and Control II Spring 2008
MT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control Spring 2008 or information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Reading: ise: Chapter 8 Massachusetts
More informationSample Problems for all sections of CMSC250, Midterm 1 Fall 2014
Sample Problems for all sections of CMSC250, Midterm 1 Fall 2014 1. Translate each of the following English sentences into formal statements using the logical operators (,,,,, and ). You may also use mathematical
More information1. COMPLEX NUMBERS. z 1 + z 2 := (a 1 + a 2 ) + i(b 1 + b 2 ); Multiplication by;
1. COMPLEX NUMBERS Notations: N the set of the natural numbers, Z the set of the integers, R the set of real numbers, Q := the set of the rational numbers. Given a quadratic equation ax 2 + bx + c = 0,
More informationIntroduction to Root Locus. What is root locus?
Introduction to Root Locus What is root locus? A graphical representation of the closed loop poles as a system parameter (Gain K) is varied Method of analysis and design for stability and transient response
More informationThis leaflet describes how complex numbers are added, subtracted, multiplied and divided.
7. Introduction. Complex arithmetic This leaflet describes how complex numbers are added, subtracted, multiplied and divided. 1. Addition and subtraction of complex numbers. Given two complex numbers we
More information14:332:231 DIGITAL LOGIC DESIGN. Why Binary Number System?
:33:3 DIGITAL LOGIC DESIGN Ivan Marsic, Rutgers University Electrical & Computer Engineering Fall 3 Lecture #: Binary Number System Complement Number Representation X Y Why Binary Number System? Because
More informationENGIN 211, Engineering Math. Complex Numbers
ENGIN 211, Engineering Math Complex Numbers 1 Imaginary Number and the Symbol J Consider the solutions for this quadratic equation: x 2 + 1 = 0 x = ± 1 1 is called the imaginary number, and we use the
More informationProblems for M 11/2: A =
Math 30 Lesieutre Problem set # November 0 Problems for M /: 4 Let B be the basis given by b b Find the Bmatrix for the transformation T : R R given by x Ax where 3 4 A (This just means the matrix for
More informationDigital Signal Processing I Final Exam Fall 2008 ECE Dec Cover Sheet
Digital Signal Processing I Final Exam Fall 8 ECE538 7 Dec.. 8 Cover Sheet Test Duration: minutes. Open Book but Closed Notes. Calculators NOT allowed. This test contains FIVE problems. All work should
More informationDO NOT DO HOMEWORK UNTIL IT IS ASSIGNED. THE ASSIGNMENTS MAY CHANGE UNTIL ANNOUNCED.
EE 537 Homewors Friedland Text Updated: Wednesday November 8 Some homewor assignments refer to Friedland s text For full credit show all wor. Some problems require hand calculations. In those cases do
More informationDifference Equations
6.08, Spring Semester, 007 Lecture 5 Notes MASSACHVSETTS INSTITVTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.08 Introduction to EECS I Spring Semester, 007 Lecture 5 Notes
More informationA Review of Complex Numbers Ilya Pollak ECE 301 Signals and Systems Section 2, Fall 2010 Purdue University
A Review of Complex Numbers la Pollak ECE 30 Signals and Sstems Section, Fall 00 Purdue Universit A complex number is represented in the form z = x + j, where x and are real numbers satisfing the usual
More informationFourier Series and Fourier Transforms
Fourier Series and Fourier Transforms EECS2 (6.082), MIT Fall 2006 Lectures 2 and 3 Fourier Series From your differential equations course, 18.03, you know Fourier s expression representing a T periodic
More informationWith this expanded version of what we mean by a solution to an equation we can solve equations that previously had no solution.
M 74 An introduction to Complex Numbers. 1 Solving equations Throughout the calculus sequence we have limited our discussion to real valued solutions to equations. We know the equation x 1 = 0 has distinct
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 informationChapter 1.6. Perform Operations with Complex Numbers
Chapter 1.6 Perform Operations with Complex Numbers EXAMPLE WarmUp 1 Exercises Solve a quadratic equation Solve 2x 2 + 11 = 37. 2x 2 + 11 = 37 2x 2 = 48 Write original equation. Subtract 11 from each
More informationComplex Numbers. Basic algebra. Definitions. part of the complex number a+ib. ffl Addition: Notation: We i write for 1; that is we
Complex Numbers Definitions Notation We i write for 1; that is we define p to be p 1 so i 2 = 1. i Basic algebra Equality a + ib = c + id when a = c b = and d. Addition A complex number is any expression
More informationDiscreteTime Signals and Systems. The ztransform and Its Application. The Direct ztransform. Region of Convergence. Reference: Sections
DiscreteTime Signals and Systems The ztransform 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 informationNotice the minus sign on the adder: it indicates that the lower input is subtracted rather than added.
6.003 Homework Due at the beginning of recitation on Wednesday, February 17, 010. Problems 1. Black s Equation Consider the general form of a feedback problem: + F G Notice the minus sign on the adder:
More information3.2 Complex Sinusoids and Frequency Response of LTI Systems
3. Introduction. A signal can be represented as a weighted superposition of complex sinusoids. x(t) or x[n]. LTI system: LTI System Output = A weighted superposition of the system response to each complex
More informationSEVENTH EDITION and EXPANDED SEVENTH EDITION
SEVENTH EDITION and EXPANDED SEVENTH EDITION Slide 51 Chapter 5 Number Theory and the Real Number System 5.1 Number Theory Number Theory The study of numbers and their properties. The numbers we use to
More information15 th Annual HarvardMIT Mathematics Tournament Saturday 11 February 2012
1 th Annual HarvardMIT Mathematics Tournament Saturday 11 February 01 1. Let f be the function such that f(x) = { x if x 1 x if x > 1 What is the total length of the graph of f(f(...f(x)...)) from x =
More informationUNIT V FINITE WORD LENGTH EFFECTS IN DIGITAL FILTERS PART A 1. Define 1 s complement form? In 1,s complement form the positive number is represented as in the sign magnitude form. To obtain the negative
More informationProblem Set 2: Solution Due on Wed. 25th Sept. Fall 2013
EE 561: Digital Control Systems Problem Set 2: Solution Due on Wed 25th Sept Fall 2013 Problem 1 Check the following for (internal) stability [Hint: Analyze the characteristic equation] (a) u k = 05u k
More informationMATH 103 PreCalculus Mathematics Dr. McCloskey Fall 2008 Final Exam Sample Solutions
MATH 103 PreCalculus Mathematics Dr. McCloskey Fall 008 Final Exam Sample Solutions In these solutions, FD refers to the course textbook (PreCalculus (4th edition), by Faires and DeFranza, published by
More informationChap 2. DiscreteTime Signals and Systems
Digital Signal Processing Chap 2. DiscreteTime Signals and Systems ChangSu Kim DiscreteTime 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 informationEE292: Fundamentals of ECE
EE292: Fundamentals of ECE Fall 2012 TTh 10:0011:15 SEB 1242 Lecture 18 121025 http://www.ee.unlv.edu/~b1morris/ee292/ 2 Outline Review RMS Values Complex Numbers Phasors Complex Impedance Circuit Analysis
More informationDISCRETE RANDOM VARIABLES EXCEL LAB #3
DISCRETE RANDOM VARIABLES EXCEL LAB #3 ECON/BUSN 180: Quantitative Methods for Economics and Business Department of Economics and Business Lake Forest College Lake Forest, IL 60045 Copyright, 2011 Overview
More information1 1 0, g Exercise 1. Generator polynomials of a convolutional code, given in binary form, are g
Exercise Generator polynomials of a convolutional code, given in binary form, are g 0, g 2 0 ja g 3. a) Sketch the encoding circuit. b) Sketch the state diagram. c) Find the transfer function TD. d) What
More informationPrecalculus 12 Curriculum Outcomes Framework (110 hours)
Curriculum Outcomes Framework (110 hours) Trigonometry (T) (35 40 hours) General Curriculum Outcome: Students will be expected to develop trigonometric reasoning. T01 Students will be expected to T01.01
More informationCurve Fitting. Objectives
Curve Fitting Objectives Understanding the difference between regression and interpolation. Knowing how to fit curve of discrete with leastsquares regression. Knowing how to compute and understand the
More informationSTUDY GUIDE ANSWER KEY
STUDY GUIDE ANSWER KEY 1) (LT 4A) Graph and indicate the Vertical Asymptote, Horizontal Asymptote, Domain, intercepts, and y intercepts of this rational function. 3 2 + 4 Vertical Asymptote: Set the
More information7.17. Determine the ztransform and ROC for the following time signals: Sketch the ROC, poles, and zeros in the zplane. X(z) = x[n]z n.
Solutions to Additional Problems 7.7. Determine the transform and ROC for the following time signals: Sketch the ROC, poles, and eros in the plane. (a) x[n] δ[n k], k > 0 X() x[n] n n k, 0 Im k multiple
More information( ) + ( ) ( ) ( ) Exercise Set 6.1: Sum and Difference Formulas. β =, π π. π π. β =, evaluate tan β. Simplify each of the following expressions.
Simplify each of the following expressions ( x cosx + cosx ( + x ( 60 θ + ( 60 + θ 6 cos( 60 θ + cos( 60 + θ 7 cosx + cosx+ 8 x+ + x 6 6 9 ( θ 80 + ( θ + 80 0 cos( 90 + θ + cos( 90 θ 7 Given that tan (
More informationEE538 Digital Signal Processing I Session 13 Exam 1 Live: Wed., Sept. 18, Cover Sheet
EE538 Digital Signal Processing I Session 3 Exam Live: Wed., Sept. 8, 00 Cover Sheet Test Duration: 50 minutes. Coverage: Sessions 0. Open Book but Closed Notes. Calculators not allowed. This test contains
More information( ) John A. Quinn Lecture. ESE 531: Digital Signal Processing. Lecture Outline. Frequency Response of LTI System. Example: Zero on Real Axis
John A. Quinn Lecture ESE 531: Digital Signal Processing Lec 15: March 21, 2017 Review, Generalized Linear Phase Systems Penn ESE 531 Spring 2017 Khanna Lecture Outline!!! 2 Frequency Response of LTI System
More informationJUST THE MATHS UNIT NUMBER 6.1. COMPLEX NUMBERS 1 (Definitions and algebra) A.J.Hobson
JUST THE MATHS UNIT NUMBER 6.1 COMPLEX NUMBERS 1 (Definitions and algebra) by A.J.Hobson 6.1.1 The definition of a complex number 6.1.2 The algebra of complex numbers 6.1.3 Exercises 6.1.4 Answers to exercises
More information3 COMPLEX NUMBERS. 3.0 Introduction. Objectives
3 COMPLEX NUMBERS Objectives After studying this chapter you should understand how quadratic equations lead to complex numbers and how to plot complex numbers on an Argand diagram; be able to relate graphs
More informationState and Finite State Machines
State and Finite State Machines See P&H Appendix C.7. C.8, C.10, C.11 Hakim Weatherspoon CS 3410, Spring 2013 Computer Science Cornell University Big Picture: Building a Processor memory inst register
More information1 Sequences and Summation
1 Sequences and Summation A sequence is a function whose domain is either all the integers between two given integers or all the integers greater than or equal to a given integer. For example, a m, a m+1,...,
More informationAP Calculus Summer Packet
AP Calculus Summer Packet Going into AP Calculus, there are certain skills that have been taught to you over the previous tears that we assume you have. If you do not have these skills, you will find that
More informationADVANCED ALGEBRA (and Honors)
ADVANCED ALGEBRA (and Honors) Welcome to Advanced Algebra! Advanced Algebra will be challenging but rewarding!! This full year course requires that everyone work hard and study for the entirety of the
More informationDiscreteTime Fourier Transform (DTFT)
DiscreteTime Fourier Transform (DTFT) 1 Preliminaries Definition: The DiscreteTime Fourier Transform (DTFT) of a signal x[n] is defined to be X(e jω ) x[n]e jωn. (1) In other words, the DTFT of x[n]
More informationECE4270 Fundamentals of DSP Lecture 20. FixedPoint Arithmetic in FIR and IIR Filters (part I) Overview of Lecture. Overflow. FIR Digital Filter
ECE4270 Fundamentals of DSP Lecture 20 FixedPoint Arithmetic in FIR and IIR Filters (part I) School of ECE Center for Signal and Information Processing Georgia Institute of Technology Overview of Lecture
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 information