Question

TC All answers should be in radians not degrees. 1) (30 pts total, Ch11.1 and 11.6) For the function f(x) = 1 for <x< and 0 for the rest of the period: a) Draw a sketch of the function. Is it even or odd? b) (10 pts) Find the Fourier series for f(x) which has a period of 2n for the terms up to sin5x and cos5x c) (10 pts) Find the error of your approximation 2) (30 pts total, Ch11.2 and 11.6) For the function, f(x) = x2 for – 1<x< 1 and P = 2 a) Draw a sketch of the function. Is it even or odd? b) (10 pts) Find the Fourier series for f(x) which has a period of 2L for the terms up to sin5x and cos5x 3) (40 pts, Ch 11.2) For the above Fourier series for both 1 and 2 find the E*, error if both the series have N =5. In the process, also find F(x) for both the functions and then calculate the error with respect to f(x).

Answer 1

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Given f(x)=1 for T/CAL / and o for rest of the period Sketch of the function a fox) -27 -3T TI 2 2 3T 2T ST 3T TO 2 2- SA Given signal is an even signal . Since food f(-x) 6 Total period of function = 2n fourier series is given by OLO food got E ancosnut te brisinnut ne) h ao Siar 21 an 1/2 Cosn wrdt T -722 1 sinnut 7/2 nu 12 2 T Th sin innt Ta T -7/

+ in "T T in an 2 n nw falat sin zani/ Sin 2Th / an asin nw 2.sin(nwT12) 76. bn 2 S" sinn.wt dr -T/ - cosnut 7/2 -Cosh with+cosnwil bno Total fowlen series asinnwnh cosnut COS2wt ws wal asin 1/2 2 Sinn cost t (v): n 27 T upto . cosswt forl= + ? Bin with loswt 2sin win +25in w 3 COS 3w+ + 2Sinwzī; Cosunt + 2 sin w 5/2 cossut ur W: 21 27 coszt 27 ť zsin 37/2 2 cosstt 2 Sin 27 cosut ut 30m + asin 55/ Los 5+.

+ 2 50 Cos 51 1. FCM): į tag cost tisk (0331); RMS value of given square wave signal RMS -0.707 RMS value of fourier series 1/ - 2 ao ti į art by be ¿[60.6366)*(0.21)* +6-12735] 0.25 2 t L RMS value = 0.3582 RM50.5984 0.707-0.5984 Error in approximation 0.707 - 15-417. erre) Given that fondant for CALI and P=2 a sketch the functioni. f(t) N

Here, if(-x)= 't»? a if(-x) = f(x) Hence fi in a even function Since food in a even function, the fourier series is f(x) = 1/9ot & an cos(na) do > n=1 bn=0 1 an I fondi 2L -2 L - 2 22 (dx 0 3 ao cina i l'incos com 7 2 / 2 (-1)" 24 (The ian ED" 4 The fourier series ū given by fam) ? + TÊ GO". Cos{ha.xyz] 3 NE n2