Question

Consider the function f[x] =2-x for -2<x<2 , periodically repeated.

Part A

a) Find the complex exponential series for f[x] . Retain terms up to e^(±i3πx)

Part B

What is the highest power e^(±inπx) that has to be retained if the maximum absolute error in the Fourier series approximation in the interval -1.8<x<1.8 is less than 0.15 ? Report the absolute value of n.

Answer 1

in X - Given f(x) = 8-2 for - 22x22 By the complex Forerier Series, we have few = Ě en eenthe -> (1) where cn = 1 a 5 fox) e entre de 1. Here L=2 - S (2-x) e a dx using integration by parts, we get 2-X -EN212 e atal é - in - 2 mil + En] T- - tl leite cot) + ent] E mpz (cskó tisomniekot tisonnt) shortcom

we have sinnt=0 - atat (0) + (camā + i siamo) = ((-1*+0) n=-of 27 cm = sin(-1** , M70 Sub. Ich nalue in (1), we get f() = Š 80 (-,**) e nto For m=0, Co = fins din = $ $ (2-x) dx = † (2x- = [4+4- C7-42] = = 2

| f x) = -----... - ] " 유기 . + + 2 .. 3M 1 올 2 L tat si le a 2 + 너 e ㅎ ] -. ene) ( - 2t ), - - - -
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