Question

Calculate the Fourier Series coefficients of x(t) = cos(2*pi*1*t) + 2*sin(2*pi*4*t). Based on your results which set of FS coefficients corresponding to the positive side of the spectra is correct.

  

a0=0, a1=1/2, a2=1/j, a3 = 0

   

a1=1, a2=2, a3 = 0

   

a1=1/2j, a2=1/2, a3 = 0

   

a1=1/2, a2=2/2j, a3 = 0

Answer 1

x t = cos Rit) + 2 sin (2 + 4 t) hl=27 W222H4 Wo= HerL WJ, W2] =HER [217, & IT] =25 To = 27 = 311 = 1 How XA) written ao X4) = loo wot +2 sin (4wot) // woon using Eular Adentity written as X4 = e ottoshoot +2 ja wot - ja wot X – swot ฟง & -jwot -jawot - e Colt

now exponential Fourier series defined ap- x 4) = { ac ev bokz K - O -j bott = ... +9_4² X janot shot +9- 2e +91e -j3100 fa-ze 4 taora, edroot 92 12 noot genot j4 wot taq e - - - - - compare ② and ③ we get, +9-4 = - 94= a = 72 ani = 1 / 2 107 and thes fore other toefficient are noro Positive coefficiente, as= 94- 9020, 9220, 43=0 other ax=0 * from given to option is correct. for sinusoidal tourier Penies- XA= not & tan copth wot) +by sinnwot] no X = 1 cop tot + 2 sin watt) compose both we get a sad 64 29 9020 & other wetticients que zero.