Question

1. Find the Fourier transform of 0 <t<2 (a) f(t) = 1+ -2<t<0 otherwise a > 0 (b) f(t) = Se-at eat t> 0 t < 0 () f(1) = { cost t> 0 t < 0 0 Answer: 1 - cos 20 (a) (b) 2a al + m2 (c) 1 + jo (1+0)2 + 1

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Answer 1

113 las - the o Given f(t) = 0Lt? 2. 2 1+t -2_7_o otherwise fourier transform of flt) is. Flw) - -jof f(t) e dt 2 -jwf dt * F(W) t att عمرانی (+1) e I ) -2 0 2 2 0 -juf » FEW) -inf at + -jwf dtt dt -jwf 이는 -2 -2 O 0 0 → FlW) = -jwf + 2 -jw -2 Le seringhe one fotoren e Tata [e sedintat - Star Se Cervia - 70 2 tle le

F(W) = - 2 หา" jaw 05 t t Rs 0 -5 (02)8 ) 0 8 8 0.5 - 0 Jan +++ .jok - Jam -jw -jw (jwa 4 |-jw -jw -jw te -jw * F(4) - 4 - 1 0 tjan jaw -jw (jwa -jw (jop -jaw - Jง ? - jw -jw (w)2 (jwa * F(9) - -) Jง - jw (w ) jo ( 9 ) jo + e + ejaw 4 et e - + 0 15 - -" 02 9 0 o.se { + 3 jo -jaw 0 : teva } 0:52 -jaw (jwa jw (jwa

0.5 FW) = 0.5 euan 2 0.5 e 0.5 w w2 + 2 wa f(w) 1 Triawieraj 1 w2 W2 a - - FlW) = Cosaw wa wa Flw) c ܥܙ cos (26) wa - at t>,0 Gêven, flt) - at e to

fourier transform of f(t) is, -jwf FLW) f(t) e dt . ►FW) = [cate dotat = Forum -at -jut dt HW) = s la-jult la-jw)t dt t se (a +jwt dt e 0 .:-(atjw) Flw) l e 2 (a-jwt e a-jw + -(atjw) ģ -00 0 6 >> FW) = e @ -e a-jw + (atju) → FW) = 1-0 a-jw - ( 0-1 atjo

1 + ► F(W) = 1 a-jw atjw >> Hw): a+fm ta-jw (a-jw) (atjo) → FlW) - qa a24 w2 e 9 cost ; t ), o o tro The Tourier transform of flt) is. F(W) -jwf f(t) e dt s F(W) -jwf cost.e dt - e et cost = (1+justat → FO) = coste 0

we Know, at Seat cos - e Here a= x So FlW) = at cosbt at a cosbt -b sin bt a²+62 -(1+jw), 681. -(17jw)t -(1+jw) cost - 1 sinf (1+jw)t11 -(1+jw) cost – sind (l+jw)2+1 e . * Fla) [e-(14jm)t e (itjw) coso 7748). ** Elves as custom biod] -e [- FlW) - (1+jw)2 + 1 0 + 1tjw -0 (1 + jw 2+1 1+jw 1 +(1+jW ) FlW) =