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How to Use This DocumentPeriod and FrequencyQuizzing the ReadingExercisesSamplingQuizzing the ReadingExercisesSampling Analog SignalsQuizzing the ReadingExercisesAliasing Analog SignalsQuizzing the ReadingAliasing on the TI-83Quizzing the ReadingExercisesHow Things Can Go WrongQuizzing the ReadingExercisesObtaining the Correct ImageThe Moral of This TaleSolutions to Exercises0 0.005 0.01 0.015 0.02−4−2024Home PageTitle PageContentsJJ IIJ IPage 1 of 34Go BackFull ScreenCloseQuitAliasing of SinusoidsDavid ArnoldApril 30, 2000AbstractTo display a continuous signal (e.g., x(t) = A cos(ωt + φ)) on a computerscreen, one samples the continuous signal at discrete points. These discretepoints are plotted, then connected with line segments to produce a discreteapproximation of the continuous signal. However, it is possible that completelydifferent continuous signals can be mapped to the same discrete approximation,an artifact known as aliasing. When plotting sinusoids on a computer and/orcalculator, one needs to be aware of the effects of aliasing, which we will explorein this article.0 0.005 0.01 0.015 0.02−4−2024Home PageTitle PageContentsJJ IIJ IPage 2 of 34Go BackFull ScreenCloseQuitContents1 How to Use This Document 32 Period and Frequency 5Quizzing the Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 Sampling 9Quizzing the Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Sampling Analog Signals 11Quizzing the Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 Aliasing Analog Signals 16Quizzing the Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 Aliasing on the TI -83 18Quizzing the Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 How Things Can Go Wrong 25Quizzing the Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 Obtaining the Correct Image 329 The Moral of This Tale 34Solutions to Exercises 350 0.005 0.01 0.015 0.02−4−2024Home PageTitle PageContentsJJ IIJ IPage 3 of 34Go BackFull ScreenCloseQuit1. How to Use This DocumentThis document was prepared with LATEX and the exerquiz and pdfscreen packagescreated by Donald Story of Akron Unive rsity and C.V. Radhakrishnan of River-ValleyTechnologies. Dr. Story’s home page can be found athttp://www.math.uakron.edu/˜dpstory/.His exerquiz package is available athttp://www.math.uakron.edu/˜dpstory/webeq.html,where one can also find documentation describing the use of the exerquiz package.River-Valley Technologies is located athttp://www.river-valley.comand Dr. Radhakrishnan’s latest version of pdfscreen is available athttp://www.river-valley.com/download/.There is a navigation bar at the bottom of the page to help the reader navigatethrough the document efficiently. Clicking the first button takes you to the tableof contents, while the next two buttons take you to the beginning and end of thedocument, respectively. After that, there are two buttons that link to the previousand next page, respectively. The back button takes you to the last visited page, andthere are two buttons that are used to visit the previous and next documents, shouldthese exist. In this document they do not.This document is fully hyperlinked, with all hyperlinks marked a distinctive redcolor. For example, the links in the table of contents are active and clicking a linksuch as Figure 1 will take you to the referenced figure. In all cases, clicking the backbutton on the navigation bar will return you to your original spot where y ou cancontinue reading.At the end of each section, there is a short quiz to test the reader’s comprehen-sion of the material covered in the section. Click your mouse on ‘BeginQuiz’, thenselect the best answer to each question by clicking the corresponding checkbox with0 0.005 0.01 0.015 0.02−4−2024Home PageTitle PageContentsJJ IIJ IPage 4 of 34Go BackFull ScreenCloseQuityour mouse. When you are finished, click ‘EndQuiz’ to get your score, then click the‘Correct’ button t o have your test scored. A ✔ indicates that the student gave thecorrect response; a ✘, indicates an incorrect response and, in this case, the correctanswer is marked ●.You want to make sure that you understand the quiz questions at the end ofeach section before proceeding to read the next section. In addition, once the quizquestions are completely understood, there are some exercises that are to be com-pleted for homework. If the exercise label and number are colored green, then thiscombination is an active link that will send you to a solution of that particular exer-cise. Exercises with active p arts have solutions that are reached by clicking (a), (b),(c), etc. If the exercise links are colored blue, they are inactive, the purpose beingto hide solutions temporarily from the reader. Please return at a later date to see ifthese links have been activated.This document was typeset with Y&YTEX using the Lucida fonts from Y&Y.Enjoy.0 0.005 0.01 0.015 0.02−4−2024Home PageTitle PageContentsJJ IIJ IPage 5 of 34Go BackFull ScreenCloseQuit2. Period and FrequencyIn your first trigonmetry class, you studied sinusoids having equationsx(t) = A cos(ωt + φ), (1)where A determines the amplitude of the sinusoid, ω is the angular frequency, andφ determines the phase shift. An easy calculation shows that the period of thesinusoid (the time to complete one cycle) is 2π /ω.x(t + 2π/ω) = A cos(ω(t + 2π/ω) + φ)= A cos(ωt + 2π + φ)= A cos(ωt + φ)= x(t)However, engineers prefer to work with the frequency of a signal rather than theangular frequency. That is, engineers favor the formx(t) = A cos(2πf t + φ), (2)where f is the …


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CR MATH 25 - Aliasing of Sinusoids

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