Question

Problem #2 Consider a continuous-time LTI system given by: dy[ + 2y(t) = x(t). Using the Fourier transform, find the output y(t) to each of the following input signals: (a) x(t) = e-'u(t), and (b) x(t) = u(t).

Answer 1

a () لا < dt (JW) Y(w). fot 4 ) dy(t) F.T dylt) E-T n ۷ (ریال) dth (لا) اد -at e ult) t) * T .کم 1 atjw + é ult) f.T 1 له +۱ Given (ا الى + + R9 (t) =) apply fourier transform It iw ها (د) 2 + (دا)لا( مان) فائی = نماز به ) (لا نواز دینا = (0) لا زن سن بالا - IFT IFT (س) 4 9 (4)

(l+jw ) - > absolutely integrable IfT t -2t e ult) - Cult) (2+jw.) t y (t)= (-2dt)uct) 6 Criven alt)ult) Reason - Ult) is not 기 basically fourier transform of ult) cannot be determined but can it can be done with aid of laplace transform - 1-416y= Put s-jw to fairies transform directly tenis . get ult) (jw) d yet) dt'tay(t) = x(t) apply fourier transform -7 (sw) you) +a yetu (2+jwdy(w) = jw

=> Y (W)= (لازم) ( بال ). Y(w) ya بیان -1/2 (2+jw) Y(u) = kan tus (74jw) Ift Y(w) 97) IfT -at --- لای ۵ 1 2 stw a lot 341t) - euct) => 914)= { [l-e*)uct) Dutt) y it). -at re