A “triangle wave” can be defined by F(t) = 1 2|t|/ for / < t < +/, with F(t) def
ID: 3164148 • Letter: A
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A “triangle wave” can be defined by F(t) = 1 2|t|/ for / < t < +/, with F(t) defined at all other values of the time t by the property of having period 2/.
I need help with as much as possible, i dont know how to do this at all.
1. 120 pts A "triangle wave" can be defined by F(t) 1 2 lt/T for -T/w t k +T/w, with Ft) defined at all other values of the time t by the property of having period 2T/w (a) Find the Fourier series representation of this Ft). Express your answer BOTH in exponential form and in the form of sines and/or cosines. But get the exponential form first (it's easier and then get the s cosines version from that ines (b) Solve the driven damped oscillator equation 2Bi wor F(t) in the form of an infinite series. You will find the exponential form of your answer to part (a) most convenient, so your answer here will be a sum from n o to oo of terms proportional to e we t You can neglect the solutions to the homogeneous equation in which contains two arbitrary constants, because those "transient" effects go away like e-at if you wait long enough. (c) Convert your answer for part (b) to a sum of Sinin w t) and cos(wt) terms with coefficients that are obviously real (d) Explicitly write out the sin(n w t) and cos(n w t) terms in part (c) for n 1, n 2, and n 3 for the case w wo/3. Which term is most important if B is small? Math reminders: You can separate real and imaginary part using s a bi a bi a bi a2 be e' cos(a) i si (a)Explanation / Answer
A) From the fourier series formula, by substituting the limits and the given function we get the value of
Ck= - j cos k/
Substitute the value of Ck in the fourier series formula we get the spectrum equation as
X(t)= 1- j/ e^j42t/ + j/ e^j82t/ ….(exponential)
So the series exists only for all the values of k.
1-j/[ cos 42t/+ jsin 42t/] + j/[ cos 82t/+ jsin 82t/]+…..(Sines+Cosines)
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