Question

14. An LTI system has the following transfer function, determine the output system response y(t) due to the input x(t) e u(t) H(s) s+2 S+7s+12 Answer: y(t)

Answer 1

y(s)/x(s) = H(s) y(s) = H(s) x(s) given x(t) = e^-2t u(t) take Laplace transform both side x(s) = 1/s + 2 given, H(s) = s + 2/s^2 + 7s + 12 y(s) = H(s) x(s) = (s + 2)/s^2 + 7s + 12 times 1/(s + 2) y(s) = 1/ s^2 + 7s + 12 y(s) = 1/s^2 + 4s + 3s + 12 y(s) = 1/(s + 4) (s + 3) \r\n y(s) = 1/(s + 4) (S + 3) = A/(s + 4) + B/(s + 3) A = (s + 4) y(s) Vertical-bar _s = -4 = 1/s + 3 Vertical-bar _s = -4 = 1/-4 + 3 Therefore A = -1 B = (s + 3) y(s) Vertical-bar _s = -3 = 1/-3 + 4 = 1 Therefore B = 1 y(s) = 1/s + 3 - 1/s + 4 y(t) = e^-3t - e^-4t