Question

The Fourier series of a periodic signal s(2) of period T can be expressed as k s(x) = cxexp ( 21 - where the coefficients Ck are given by 7/2 CR 1 T -T/2 | $(z) exp (-27 k -27=cdc T (i) Consider s(2) of period T = 6 and amplitude A= 2: 8(z) = 2 * |< T 2 Compute the Fourier coefficients ok. (ii) Use the identities exp(Trik) + exp(-rik) cos(Tk) = 2 sin(Tk) exp(Trik) – exp(-rik) 2i to write the Fourier series expansion of s(x). (iii) Explain if the Fourier series can be written with a finite number of ele- ments.

Answer 1

s(x) = & cx exp (27 i K x) Ka-a i) T=6, A=2 scal = 2A bl</2 T Sala 4 6 s(x) = 2x 3 (x = I l scx) exp(-27 ika) do 3 3 (k= į Į 2 x exp(-etike) dx 3 k = 1 9 + g exploring T2K² 2 (34) explement (9 4+* +9) Eis - [**] (k= T2 k² (k = (ink+1) elka (-i't+) elkim 2K² + ск - іттK TTK [eikt + elke reikt eike T²k² 00000

-2 i sintok) Ck = 2(TK Cos (TIK) T2K2 os/2iTtk Costik) - 21sin (KTT) S(a) = exp (T (kx) fam T1212 K=0 Since sin KTT = o for K & Integer ۱۱۱) Ck = 2 ' TTK Costik) ş since CONTK = (-11) TT 2k² Ck = zi (1) TK S (a) E ai (-1) K=-0 TK exp (Tika) = It is most simplified form Fourier series cannot be written with finite number of elemento