Linear Systems ECEN 3300 Spring 2019 M,W,F 2:00-2:50PM ECCR200. Signals

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1 Linear Systems ECEN 3300 Spring 2019 M,W,F 2:00-2:50PM ECCR200 Professor Kelvin Wagner phone: x office: ECEN 232 inside OCS ecen3300/ A.V. Oppenheim and A.S. Wilsky and S.H. Namib, Signals and Systems Goal Learn how to describe, understand, analyze, design LTI systems Continuous Time (CT) and Discrete Time (DT) Signals and Systems Transform domain analysis (Fourier first) Simulate, evaluate, plot, and analyze in MATLAB Balance between analysis and intuition Signals We begin with Signals and their mathematical representation Construction, manipulation, combination, and composition plots in MATLAB and sketches by hand Properties CT vs DT is different than analog vs digital Energy and Power finite duration vs infinite extent periodic vs nonperiodic parity real signals and complex transform domain deterministic vs random is left for a future course Kelvin Wagner, University of Colorado Linear Systems Kelvin Wagner, University of Colorado Linear Systems Systems Mathematical representations of operations on signals Properties and Categorization SISO vs MIMO Stability Invertibility Memory Linearity Time invariance Causality Integral and Differential equation representation of system LTI and convolutions Transform domains to describe signals and LTI system operation Fourier Series: Decomposition into a sum of harmonic sinusoids Fourier Transforms: Decomposition into continuous sinusoidal basis Discrete Time Fourier Transform: Decomposition into DT sinusoids Sampling Combs : The connection between CT and DT DFT and FFT: Algorithms for numerical evaluation of DFT Laplace Transform: Complex exponential basis decomposition Differential Equations, Block diagrams, pole/zero plots Z-transforms: Discrete time block diagrams Difference Equation Linear Feedback Systems Root locus analysis Kelvin Wagner, University of Colorado Linear Systems Kelvin Wagner, University of Colorado Linear Systems

2 ECEN Linear Systems Lectures: MWF 2:00-2:50 ECCR 200 Instructor: Kelvin Wagner Office: ECEE 232 (inside Optics Center) kelvin@colorado.edu phone: Office hours: Th 4:30-5:30 PM and by appointments Lectures: Monday, Wednesday and Friday 2:00-2:50pm in ECCR 245 Text: Alan V. Oppenheim, Alan S. Willsky, with S. Hamid Nawab, Signals & Systems, Second Edition, Prentice Hall, 1997, ISBN Prerequisites: APPM 2360, Intro to Linear Algebra and Differential Equations ECEN 2260, Circuits and Systems (min grade C-) Credit Hours:3 Topics Description: The concepts of signals and systems are abstractions that allow engineers and scientists to describe, analyze, synthesize and simulate a wide variety of naturally occuring and man-made processes within a common, implementation-independent framework. In traditional electrical engineering systems often originate from circuits consisting of lumped elements and/or active integrated circuits. In this case the signals are usually time-varying voltages and currents associated with the inputs and outputs of the circuit. Modern system implementations, on the other hand, increasingly rely on fast computer hardware to perform signal processing in the digital domain. In this case the input and output signals take on the form of discrete-time (DT) sequences that are often obtained by sampling continuous-time (CT) waveforms at regular time intervals. The digital systems or digital signal processors themselves generally consist of memory cells, adders, and multipliers. Of central importance are linear and time-invariant (LTI) systems, i.e., systems which satisfy the superposition principle and whose properties are independent of absolute time. Together with Fourier analysis, which models most physical signals or sequences of interest as linear combinations of spectral components, this leads to a "divide and conquer" approach for the analysis and synthesis of a large class of practically relevant processes. Examples include diverse topics including optics and imaging, RF/electromagnetics, antennas, acoustics, linear circuits, filters, and general signal and information processing that is used in communication systems, radar and lidar, optics and image processing systems, acoustics, quantum mechanics, linear feedback systems, robotics and controls. (1) Continuous time (CT) signals; (2) CT linear and time-invariant (LTI) systems: Linearity, time-invariance, memory, causality, Block diagrams; (3) Time domain analysis of CT LTI systems: Differential equations,unit impulse/step response, Convolution; (4) Frequency domain analysis of CT LTI systems: Laplace transform, pole/zero plots, Fourier transform, System function and frequency response; (5) Discrete time (DT) signals; (6) DT linear and time-invariant (LTI) systems:linearity, time-invariance, memory, causality, Block diagrams; (7) Time domain analysis of DT LTI systems: Difference equations, Unit impulse/step response, Convolution; (8) Frequency domain analysis of DT LTI systems: z-transform, pole/zero plots, DT Fourier transform, discrete Fourier series, System function and frequency response (9) Relationship between CT and DT signals and systems. Course Goals: Learn how to describe, understand, analyze, and design linear and time-invariant continuous-time (CT) and discrete-time (DT) systems for signal and information processing. Describe CT and DT signals and systems in the time and frequency domains. Understand the utility of the various transform domains including Fourier Series, Fourier Transfoirms, Discrete Time Fourier Transforms, Laplace transforms, and the Z-transform. Be able to simulate, evaluate, and analyze CT and DT processes in Matlab. Develop a balanced analytical and intuitive understanding of linear systems that allows you to analyze and solve a wide variety of engineering problems. Homework Quizes References Properties of Fourier Series, Fourier Transform, Laplace Transform 1 of 8 1/14/19, 3:09 PM 2 of 8 1/14/19, 3:09 PM

3 Unilateral and Bilateral Laplace Transform Tables Fourier Transform Table and Properties Matlab Intro Tentative Schedule Monday Wednesday Friday 01/14/19 01/16/19 01/18/19 1 Complex Numbes Signals 01/21/19 01/23/19 01/25/19 Entrance Quiz and Exponentials- SignalTransformations 2 MLK Holiday System Properties Causality & Linearity /28/19 01/30/19 02/01/19 Delta Functions and LTI CT convolutions LTI System Properties 02/04/19 02/06/19 02/08/19 Convolution Examples, d/dt and integration LTI DT convolutions & Derivatives of Convolution Block Diagrans and Difference Equations Constant Coefficient Linear Differential Equations 02/11/19 02/13/19 02/15/19 Diff EQ Impulse Response, exponential eigenfunctions CT Block Diagrams, Difference Eqns 02/18/19 02/20/19 02/22/19 LTI DT convolutions Differential Equation CC root pairs and example solutions. Zero State and Zero Input Difference Eqn Impulse Response 6 Midterm 1 FS vugraphs FS 7 FS FS/FT FT1 vugraphs 03/04/19 03/06/19 03/08/19 8 FT FT2 vugraphs FT 03/11/19 03/13/19 03/15/19 9 FT FT3 vugraphs FT 03/18/19 03/20/19 03/22/19 10 DTFT DTFT properties Midterm 2 03/25/19 03/27/19 03/29/19 Spring Break Spring Break Spring Break 04/01/19 04/03/19 04/05/19 11 DTFT DFT/FFT Sampling 04/08/19 04/10/19 04/12/19 12 Nyquist Thm Sampling /15/19 04/17/19 04/19/19 ROC and Inverse Laplace Transform ILT and PFX 04/22/19 04/24/19 04/26/19 Laplace Transform Properties Causality and Stability 04/29/19 05/01/19 05/03/19 Bilateral Laplace Transforms PFX CC root pairs LTI System ID and Butterworth 02/25/19 02/27/19 03/01/19 15 Z-transforms Linear Feedback Systems Classes Over, Reading Day 3 of 8 1/14/19, 3:09 PM 4 of 8 1/14/19, 3:09 PM

4 Readings out of Signals & Systems, Oppenheim & Wilsky 1. Week 1: P71-73, Sections Complex numbers and Signals 2. Week 2: Sections Systems 3. Week 3: Chapter 2, LTI Systems Exams Entrance quiz The entrance quiz will be given in class on Friday January 18, Closed book and no calculator. Laplace table provided. This will be counted as part of the 10-15% quiz score towards your grade. The quiz will consist of two problems chosen from the following topics: Operations on complex numbers, Circuits represented with differential equations, transfer functions. Laplace table provided Previous entrance exam solutions. Previous entrance exam and solutions part1, part2. Another previous Entrance exam and solutions Midterms Midterm 1 The first midterm will be on or about February 18. The midterm will count for 15% of your grade. Practice Exams that are representative of the type of midterm, but with different problems 1. Midterm from previous year -- solutions 2. Practice Midterm1 -- brief solns Midterm 2 The second midterm will be schedukled before Spring break. The midterm will count for 15% of your grade. 1. Practice Midterm 2 -- soln 2. Updated Practice Midterm > Covering Ch 3-5, Fourier Series, Fourier Transforms and DTFT Study Topics for Midterm 2 1. Fourier Series Representation of Periodic Signals 1. Definition and computation of FS 2. Properties and Transforms 3. CT and DT periodic signals and FS 4. FS and LTI systems 2. Fourier Transforms of CT signals 1. Definition and computation of FT 2. Properties and Transforms 3. Convolution 4. Linear Differential Equations and LTI systems 3. Discrete Time Fourier Transform 1. Definition and computation of DTFT 2. Properties and Transforms 3. Linear Difference Equations and LTI systems 4. Discrete Fourier transforms and the Fast Fourier Transform (FFT) Final The final exam will take place on the first day of Finals: Saturday May 4, 4:30PM-7:00 PM. The final will include all the topics covered in class during the semester for which homework has been assigned, as well as applications of the techniques learned to topics such as feedback, filters, and Z-transform analysis of DT systems, and will count towards 30% of your grade. 1. Practice Final Grading Computer Usage: All homework and course notes will be posted on the class website at Some homework will require the use of Matlab. Course Requirements: 1. Attend class. 2. Homework (25%): Weekly, usually due on Fridays at the beginning of class. Only one or two problem, selected at random, 5 of 8 1/14/19, 3:09 PM 6 of 8 1/14/19, 3:09 PM

5 will be graded. 3. Quizzes (10-15%): Approximately weekly to bi-weekly, on material covered in class and in the book. 4. Exam 1 (~15%): Approximately Mon. Feb 18. Closed book, closed notes. Transform tables provided when needed. 5. Exam 2 (~15%): Approximately Fri. Mar 22. Closed book, closed notes. Transform tables provided when needed. 6. Final exam (~30%): Saturday. May 4, 4:30PM - 7:00 pm. Closed book, closed notes. Transform tables will be provided. Homework, Labs, and Grades Nominal due dates Homework is due on Fridays in class. Homework will be graded using statistical sampling in which only 1 or 2 problems will be graded to determine your score on that assignment, but since you won't know which problems will be graded you should work on all the problems. Guidelines for getting full credit on Homework Assignments must be neat, organized and legible. In plain English: If we cannot read your assignment, you will not get credit for it. Typed assignments are welcome. At the start of each problem, write out a brief description of the problem including given information and what is to be found. Show your work enough to fully demonstrate your understating and your arrival at your answer. Write on only one side of the paper. Pages must stapled be in order (i.e. following the order in which the problems were assigned). Unless otherwise stated, plots must be computer-generated (in MATLAB, Excel, etc.). All plots must have a title and each axis should be labeled. Do not forget units. and values, critical peak amplitudes, widths (eg FWHM), unity gain crossings and corner frequencies, and other critical values. Put a box around all final answers. If coding is used in a problem, you must turn in a paper copy of your computer code along with your assignment. Attach the code immediately after each problem that required code. Computer code should be clearly commented to demonstrate your intent. You should go over your homework as soon as it has been graded and returned to you. Once an assignment has been returned to you, you only have TWO WEEKS from the return date to question the grading. Questions regarding the material are always welcome. External links: Linear Systems Tutorials Fourier Series Simulation website 3blue1brown Fourier Visualized Matlab Tutorials Official Matlab site a tutorial with applications to signals and systems Complex Numbers Tutorial Websites: top link from JHU site on review of complex numbers and phasors (and also overviews the Fourier Series) 3blue1brown e to the i pi = -1 January KW Sketches will refer to carefully hand drawn plots, include axis labels 7 of 8 1/14/19, 3:09 PM 8 of 8 1/14/19, 3:09 PM

6 Complex Numbers Complex Number Example Descarte 1637 Leibnitz 1702 j = 1 j 2 = 1 j 3 = j j 4 = Gauss described complex numbers as points/vectors in 2D plane z = x + jy x = R{z} = r cos φ y = I{z} = r sin φ x = 0 pure imaginary number y = 0 real number Complex numbers as vectors in the plane modulus Phase z = re iφ = x + iy Complex Plane r = x 2 + ( y 2 φ = tan 1 y ) + π? = tan 1 (y, x) = arg(z) x b y r φ (a,b) a x z = 3 + 2i r = = φ = tan 1 ( 2 3 ) =.588rad = 33.7 z = 3.61e i.588 = Use Cartesian form for sum/difference Addition of complex vectors: 2D shift Use polar form for product and ratio Multiplication of complex vectors: radial scaling and rotation Polar form for roots and powers Complex Plane π 2 y r=3.61 φ= x Kelvin Wagner, University of Colorado Linear Systems Kelvin Wagner, University of Colorado Linear Systems Polar to Cartesian: Use Euler s Formula Complex Conjugate R{z} = r cos φ Taylor series z = re iφ = r(cos φ + i sin φ) I{z} = r sin φ e x = 1 + x 1! + x2 2! + x3 cos x = 1 x2 2! sin x = x x3 3! 3! + = n=0 + x4 4! +... x n n! + x5 5! +... e ix = 1 + (ix) + (ix)2 + (ix)3 + (ix)4 + = 1 + ix 1! 2! 3! 4! 1! x2 2! ix3 ( ) ( ) 3! = 1 x2 2! + x4 4! +... x + i 1! x3 3! + x5 5! +... = cos x + i sin x Kelvin Wagner, University of Colorado Linear Systems x4 4! +. z = z = x jy = re jφ Complex conjugate of sum is sum of CC (z 1 + z 2 ) = [(x 1 + jy 1 ) + (x 2 + jy 2 )] = [x 1 + x 2 + j(y 1 + y 2 )] = x 1 + x 2 j(y 1 + y 2 ) = (x 1 jy 1 ) + (x 2 jy 2 ) = z 1 + z 2 Complex conjugate of Product is product of CC (z 1 z 2 ) = [r 1 e jφ 1r 2 e jφ 2] = [r 1 r 2 e j(φ 1+φ 2 ) ] = r 1 r 2 e j(φ 1+φ 2 ) = r 1 e jφ 1 r 2 e jφ 2 = z 1 z 2 Complex conjugate of ratio is ratio of CC ( ) ( ) z1 z = r 1 e jφ 1 2 r 2 e jφ = r ( ) 1 2 r e j(φ 1 φ 2 ) = r1 2 r e j(φ 1 φ 2 ) = r 1 e jφ 1 2 r 2 e jφ = z 1 2 z2 Kelvin Wagner, University of Colorado Linear Systems

2 + y 2 = r 2 1 z = 1 z z z = z z = 1 2 re = 1 jφ r e jφ = x 1x 2 + y 1 y 2 x 2 2 + y2 2 +j x 2y 1 x 1 y 2 x 2 2 + = r 1e jφ 1 y2 2 r 2 e =r 1 e j(φ 1 φ 2 ) jφ 2 r 2 z 2 = (x + jy)(x + jy) = x 2 y 2

7 Complex Algebra Complex Powers and Roots Addition Multiplication Division z 1 = x 1 + jy 1 = r 1 e jφ 1 z 2 = x 2 + jy 2 = r 2 e jφ 2 z 1 + z 2 = (x 1 + x 2 ) + j(y 1 + y 2 ) z 1 z 2 = (x 1 x 2 y 1 y 2 ) + j(x 1 y 2 + x 2 y 1 ) = r 1 r 2 e j(φ 1+φ 2 ) z 1 = z 1 z2 z 2 z 2 z2 = z 1z2 z 2 2=(x 1 + jy 1 )(x 2 jy 2 ) x y2 2 z + z = 2x R{z} = 1 2 (z + z ) z z = j2y I{z} = 1 2 (z z ) zz = z 2 = x 2 + y 2 = r 2 1 z = 1 z z z = z z = 1 2 re = 1 jφ r e jφ = x 1x 2 + y 1 y 2 x y2 2 +j x 2y 1 x 1 y 2 x = r 1e jφ 1 y2 2 r 2 e =r 1 e j(φ 1 φ 2 ) jφ 2 r 2 z 2 = (x + jy)(x + jy) = x 2 y 2 + j2xy z 3 = (x + jy) 3 = (x 2 y 2 + j2xy)(x + jy) = x 3 3xy 2 + j(3x 2 y y 3 ) z 4 = (x+jy) 4 = (x 3 3xy 2 +j(3x 2 y y 3 ))(x+jy) = (x 4 6x 2 y 2 +y 4 )+j(4x 3 y 4y 3 x) Way easier to use polar notation z n = r n e jnφ There are n roots to the 1 nth power [ z 1/n = r 1/n e j φ+k2π n ] k = 0, 1,..., n 1 So there are 2 square roots separated by π there are 3 cube roots separated by 2π/3 there are 4 fourth roots separated by π/2 there are n n th roots separated by 2π/n Roots of j j 3φ2φ φ r3 r r 2 j 1/2 j 1/3 j j 1/4 1/5 Kelvin Wagner, University of Colorado Linear Systems Kelvin Wagner, University of Colorado Linear Systems Other useful Complex Identities Modulus z z = z 2 = x 2 + y 2 z 1 z 2 = z 1 z 2 = x y2 1 x y2 2 Complex Conjugate of Sums, Products, and Ratios (z 1 + z 2 ) = z 1 + z 2 Phasors r(t) = u(t) = v 2 (t) + w 2 (t) u(t) = v(t) + jw(t) = r(t)e jφ(t) [ ] φ(t) = u(t) = tan 1 w(t) v(t) Harmonic Phasors are Linear Phase Factors, φ(t) = ω o t + φ o Ṽ =r o e jω ot+φ o =r o cos(ω o t+φ o )+jr o sin(ω o t+φ o ) t=0 r o e jφ o =r o cos φ o +jr o sin φ o ( z1 z 2 (z 1 z 2 ) = z1 ) z 2 ( ) = z 1 1 = 1 Real and Imaginary parts of Sums and Products z 2 R{z 1 + z 2 } = R{z1 } + R{z 2 } I{z 1 + z 2 } = I{z1 } + I{z 2 } But not for products R{z 1 z 2 } R{z1 } R{z 2 } = x 1 x 2 I{z 1 z 2 } I{z1 } I{z 2 } = y 1 y 2 z 2 z 2 Kelvin Wagner, University of Colorado Linear Systems Kelvin Wagner, University of Colorado Linear Systems

Spring 2014 ECEN Signals and Systems

Spring 2014 ECEN Signals and Systems Spring 2014 ECEN 314-300 Signals and Systems Instructor: Jim Ji E-mail: jimji@tamu.edu Office Hours: Monday: 12-1:00 PM, Room 309E WEB WeChat ID: jimxiuquanji TA: Tao Yang, tao.yang.tamu@gmail.com TA Office