Question

1. Find the CTFT of the following signals 0 otherwise cos(40rt) sin(10Tt = e-10t (b) x(t) = ) ( c) x(t) u(t) + e10ta(-t + 1)

Answer 1

Given that CTFT of the signals (a) x (t) = {^5 -mod t mod (t) lessthanorequalto 5_0 otherwise d^2 x/dt^2 - delta (t + delta) - 2 delta (t) + delta (t - 5) apply Fourier transform (jw)^2 X (w) = e^j5w - 2 + e^-j5w (jw)^2 X (w) = 2 [cos 5w - 1] X (w) = -2/w^2 [cos 5w - 1] \r\n X (w) = 2/w^2 [1 - cos 5w] X (w) = 2/w^2 [2 sin^2 5w/2] implies 4/w^2 sin^2 5w/2 b) Given that x (t) = cos (40 pi t)sin (10 pi t)/t w k that IFT of rect (w/2a) implies sin at/pi t sin (10 pi t)/pi t rightarrow rect (w/20 pi) for any function y (t) cos w_0 t y (t) rightarrow Y (w - w_0) + Y (w + w_0)/2 pi cos 40 pi t sin 10 pi t/pi t rightarrow pi/2 [rect [10 - 40 pi/20 pi] + rect [w + 40 pi/20 pi]] \r\n Given that x (t)