Question

2. Convolve the following: ふ(t) a) 초지 x(t) he) (七) c) e 上 xlt) 仕)

Please don't use the Laplace Transform Method. Thank you!

Answer 1

Given that x(t) delta (t), h(t) = e^-2t u(t) we know that x(t) times delta (t) = x(t) x(t) times h(t) = delta (t) times e^-2t u(t) = > e^-2t u(t) y(t) = x(t) times h(t) step: - 1 limits 0 + lessthanorequalto t lessthanorequalto 1 + 2 0 lessthanorequalto t lessthanorequalto 3 change of variable from t to T folding of h(t) shifting h(t) rightarrow h(t-t) casse: - is t lessthanorequalto 0, y(t) = x(t) times h(t) = 0 case: - if 0 < t < 1 y(t) = x(t) times h(t) = Vertical-bar ^t_0 x(t) h(t-1) dt = > Vertical-bar ^t_0 (t)(1) dt y(t) = t^2/2 Vertical-bar ^t_0 = > t^2/2, 0 < t < 1 case: - if 1 < t < 2 y(t) = x(t) times h(t) = Vertical-bar ^1_0 x(t) h(t-T) dt \r\n = > Vertical-bar ^1_0 T dt = > t^2/2 = > 1/2 case: - if rightarrow 2 < t < 3 y(t) = x(t) times h(t) = Vertical-bar ^1_t-2 T (1) dT = > T^2/2 Vertical-bar ^1_t-2