Question

4) a) Find and sketch the frequency spectrum of the half-wave rectified cosine wave form in Figure (a): 8 -6 -4 -2 2 4 6 8 10 (ms) b) What are the Fourier series coefficients of the above signal

Answer 1

fol1 at time laste) Soluhim: Giren periodie noore! Here fett is defined Over en periode fiyaf o :-4844-2 KT - -2<<<2 2 < t < 4 Here Time Peliod Y = am sec fondamental frequeny wosoft = II kad se Now, al ficts = filt) : fictt in eneen in nature for a signal to be even in nature The Fourier coefficients is giren as - 'ao may or may not be zero lan' eusts and I bn = o always. Calculcasion of Trigonometrie fourier coefficients - 96= + (fritidt for ficte to be even % • 400 g de cercet = [!ámpt +soxt] 403 (80 mq sino) - 16 x x Sinala ao=64 7X8

Calentains of land : an ser "fratemat at -712 the The Fourier coefficients for filt to be even in nature can be giren as- be from a comet a an = no casert et cosnoot at ama t cosnoot at t odt am= #locaçta comet at + 'ocx] an = p / coup & above, te at om z 2 wept wing 4 at onometue I den witryn its, usin 2 USA cosB = Cost A+B) + Cos(A-B) Cos (A+B) = COSA LOS B -Sin a sing . an = ( cosif(i tn't + corp (1 It) at for nas 1 2 3 4 [ {{cas p.at + los oj] at 9="[[cm. t + [at] 40 eurot + +17] 30,= 4 [3150g 2 -sin o) +(2-0)

0 - 9 = 4( 3 ( Sint-sino) + 2) As Sina = 0 2. = 14 x2 = 16x2 1682 for n>l : an 4 It + los q in-art fat an = 16 Siną (i+n)t + Sin (n-1) + For Itn) (n-1) Jo 2n = 9 (a sing count * ay * $ (cm) =]], au = 14 lite nal) e a man, she canare] oma spy they are mana sin (n-1)] oma . Sin intr t TA(n) I s (n-1) ana a Sin (n+1)I + 8 sin (n-1) I Tin-1) n (OH) 2 For never sin (n 4)] - Shang wą tecosmet sin a cosmety Sin (n-1) W2 = singur cases - Comte sine - Cosmo

Jer sin f(n+ 2) = -sin &(n-1) = cos mile =oh neeven to n= Odd nay and nog both are i sin g(n+1) = sin(n-1 = 0 enen. & to an = enen - an=8 sin(n+1) I to sin(n-1)7 1 (H) 1. A (na) om a entrega entre Sangras si - 9 -2.] covers * -, * (+9) 8 where na even 8 Tam=(-2) (-1) TOO-1) Trigonomessed coefficients - ao = 8/7 au =0 = in = odd (-2).(-1)/2 ł (n²-1) neeven bn =o (since filt is even in nature ) .

Now, Trigonometne representation of fiets in Terons of founier Coefficient fitt a aut an cermat tog sinament +) = 20+ a, casw,t + Ŝ an cash w.t n=1 9=2,4,6

n=enen f = 8 + 4085 2504 + set in tous ces also not where 't is in millisec. 6. = Rig komd see I xrooo rod Ise W. - 250 R rad /sec :

news, spereching the Amplitude spe Bam a fette 4o + Ar Coswort + S An Cos (not tou) na buen on = -tañil bra , Ao = 0o = 8/7=2.546 An= santan a laul 4-Wa* tim = 19,1 = 4 = 16 अ = 1.697 Ag = lal = 1 -16 y Gol 1 114-1) - 0.3395 Ag zo clou - | 8X-2 x (-1,92 Tal16-1) As =0 A6 - 100 - 31341 - 1959) = 0.1455 and so on...

sketching the Amplitude spectrum- I Am aj 2.546 1697 0-3395 0.1456 0 W. Wo 3W, VW, 5W, Glo .. no VOGO, Ib obiann Phase epocrmm - 0, ztañ'l ) = -17 (3) = 0 Page = -tamil hd ) = - tai liisa) =0° Pazos Pour = -tawie o ) = -tai ( 9 ) = -(180-tawi(- : -)--180° 1.850 = 0 r / Po = -taüll erine) = -tail Choparca and som--- cin of wo awa 3 wo wo Swo bee n wo -780° Phase spectrum Answer: