Question

Given an input signal (t)nd the output signal is y(t)-(e2 e)u(t), compute the transfer function H(s). Determine whether the system is stable and causal or not.

Answer 1

Input signal X(t) = e^-5 t converting in to Laplace x(s) = 1/s + 5 output signal y (t) = (e^-2 t - e^-3 t) u (t) converting into Laplace y (s) = 1/s + 2 - 1/s + 3 y (s) = s + 3 - s - 2/(s + 2) (s + 3) y (s) = 1/(s + 2) (s + 3) y (s) = x(s) H (s) H(s) = y(s)/x(s) = 1/(s + 2) (s + 2)/1(s + 5) h (s) = s + 5/(s + 2) (s + 3) y (s) = 1/s (s + 2) (s + 3) all poles lies in the left half of the s plare so given is stable y (t) (e^-2 t -e^-3 t) u (t) A system is causal if present output depends on present inputs and joint values of input y (0) = I - e = 1 - 1 = 0 Here o/p depending on present input only so the given system is causal