Question

Please answer the signals and systems question clearly :)

Assume the following LTI system and the input signal whose spectrum depicted below 2. x(t) Y(o) H(a) X(a) 2π 2π 3TT 3π Find x(t), Y(ω), and y(t) for each of the following filters.

Answer 1

y(w) = H(w) X(w) rightarrow y(t) = h(t) times x(t) X(w) = 2 pi/2 delta (w) + 2 pi/pi delta (w - w_0) + 2 pi/pi delta (w + w_0) - 2 pi/3 pi delta (w - 3 w_0) - 2 pi/3 pi delta (w + 3w_0) + 2 pi/5 pi delta (w - 5 w_0) + 2 pi/5 pi delta (w + 5w_0) we know that i doubleheadarrow^FT 2 pi delta (w) e^j w_0 t doubleheadarrow^FT 2 pi delta (w - w_0) apply inverse FT x(t) = 1/2 (1) + 1/pi [e^- j w_0 t + e^j w_0 t] - 1/3 pi [e^j3w_0 t + e^- j3w_0 t] + 1/5 pi [e^i 5 w_0 t + e^- j 5 w_0 t] d Y(w) = H(w) X(w) so Y(w) having frequency components as w_0, 3w_0 & 5w_0 y(w) = 2 pi/pi [delta (w - w_0) + delta (w + w_0)] - 2 pi/3 pi [delta (w - 3 w_0 + delta (w + 3w_0)] + 2 pi/5 pi [delta (w - 5w_0) + delta (w + 5w_0)] apply inverse Fourier transform y(t) = 1/pi[e^- jw_0 t + e^jw_0 t]