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Problem 1. Consider a periodic square wave defined by -T/2

Problem 1. Consider a periodic square wave defined by -T/2

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Problem 1. Consider a periodic square wave defined by -T/2≤t<T/2 by y(t)= -1,-T/2 < t < 0 1,0 < t < 1/2 0, t = T/2, 0 Show that the Fourier Series expansion of y(t) is given by sin 30t sin 5wt y(t) = [1₁ sin cot+ 3 5 where = 2л/T. = 27. The function squarewave.m, calculates and plots this Fourier series expansion for T Input arguments are used to specify the number of uniformly spaced points at which the series is calculated (nsamp) and the number of terms to sum in the series (nterm). (a) Set nsamp=1000 and progressively increase the nterm up to 100. How well does the Fourier series represent a square wave? Where is the fit worst and how does the character of the misfit change with the number of terms? Can you measure the maximum misfit (either using the zoom tool or the ginput function)? You are looking at an effect called the Gibbs phenomenon. (b) Set nsamp= 100 and nterm = 100. What does the misfit look like? (c) Set nsamp= 10,000 and keep nterm 100. Again what does the misfit look like? The difference can be ascribed to an effect known as aliasing.
Transcribed Image Text:Problem 1. Consider a periodic square wave defined by -T/2≤t<T/2 by y(t)= -1,-T/2 < t < 0 1,0 < t < 1/2 0, t = T/2, 0 Show that the Fourier Series expansion of y(t) is given by sin 30t sin 5wt y(t) = [1₁ sin cot+ 3 5 where = 2л/T. = 27. The function squarewave.m, calculates and plots this Fourier series expansion for T Input arguments are used to specify the number of uniformly spaced points at which the series is calculated (nsamp) and the number of terms to sum in the series (nterm). (a) Set nsamp=1000 and progressively increase the nterm up to 100. How well does the Fourier series represent a square wave? Where is the fit worst and how does the character of the misfit change with the number of terms? Can you measure the maximum misfit (either using the zoom tool or the ginput function)? You are looking at an effect called the Gibbs phenomenon. (b) Set nsamp= 100 and nterm = 100. What does the misfit look like? (c) Set nsamp= 10,000 and keep nterm 100. Again what does the misfit look like? The difference can be ascribed to an effect known as aliasing.
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