Question

(c) If the impulse response function of a linear time invariant (LTI) system is h0)-Se u(), compute the output of this system due to an input ) which is a 4 second pulse of height 3, as shown in Fig.1 below. x(t) t(sec) 0 Fig.1 Input signal 10 marks/

Answer 1

Given n(t) = 5e^-2t u(t) x(t) = 3u(t) - 3u(t - 4) Apply Laplace transform to h(t) & x(t) h(t) = 5e^-2t u(t) rightarrow 5/s + 2 = h(s) x(t) = 3u(t) - 3u(t - 4) rightarrow 3/s - 3e^-4s/s = x(s) y(s) = n(s)x(s) y(s) = 5/s + 2 times 3 - 3e^-4s/s = A/s + B/s + 2 = 0/s + -7.5(1 - e^8)/s + 2 s = 0 A = 0 s = -2 y(s) = -7.5(1 - e^8)/s + 2 B = 15(1 - e^+ 8)/-2 = -7.5(1 - e^8) Apply Inverse Laplace transform y(t) = -7.5 (1 - e^8)e^-2t u(t)