Question

A system consists of three systems as follows: System (1) has an impulse response of e~3'u(t); System (2) has a step response of te^u(t); System (3) has an output of 2te-2tu(t) to an input of 8(t) Find the output of the overall system xo (t) when the system has an input of Xi(t) = e-tu(t) h2(t) hi(t) ro h3(t)

Show the steps and find the solution clearly. Please identify the correct answers make it in the box

Answer 1

X_0(s)/X_i(s) = H_1 (S) H_2 (s) + H_1 (s) H_3 (s) h_1 (t) = e^-3t u(t) H_1 (s) = 1/s + 3 s_2 (t) = te^-2t u(t) h_2 (t) = d/dt s_2 (t) = 1 times e^-2t u(t) + t[d/dt (e^-2t u(t)] h_2(t) = e^-2t u(t) + t[(-2)e^-2t u(t) + e^-2t S(t)] = e^-2t u(t) + [-2t e^-2t u(t) + t e^-2t S(t)] f(t) S(t) = f(0) S(t0 \r\n h_2(t) = e^-2t u(t) - 2t e^-2t u(t) H_2 (s) = 1/s + 2 - 2/(s + 2)^2 = s + 2 - 2/(s + 2)^2 H_2(s) = s/(s + 2)^2 h_3(t) = 2t e^-2t u(t) H_3 (s) = 2/(s + 2)^2 x_i (t) = e^-t u(t) doubleheadarrow x_i (s) = 1/s + 1 X_0 (s)/X_i (s) = 1/s + 3 [s/(s + 2)^2 + 2/(s + 2)^2] = 1/s + 3 [s + 2/(s + 2)^2] X_0 (s)/X_i(s) = 1/s + 3 times 1/s + 2 X_0(s) = 1/(s + 1)