Question

Question 1. Consider the LT system described by the differential equation d2y(t) Determine the system's impulse response h(t) by using the impulse matching method. No other method is accepted

Answer 1

Given d^2 y(t)/dt^2 + 2 y(t) = x(t) Apply Laplace transform LT[d^2 y(t)/dt^2] + 2 LT[y(t)] = LT [x(t)] S^2 y(s) + 2 [Y(s)] = X(s) Y(s) [s^2 + 2] = X(s0 Y(s) = X(s)/s^2 + 2 Y(S)/X(s) = 1/S^2 + 2 h(S) = Y(S)/X(S) = 1/S^2 + 2 h(S) = 1S^2 + 2 Apply inverse Laplace transform Formula LT^-1 [w/s^2 + w^2] = sin wt (u, t) LT^-1 [s/s^2 + w^2] = cos wt u(t) LT^-1 [h (s)] = LT^-1 [1/s^2 + 2] h(t) = 1/squareroot 2 [LT^-1 [squareroot 2/s^2 + (squareroot 2)^2] h(t) = 1/squareroot 2 LT^-1 (squareroot 2/s^2 + (squareroot 2)^2) h(t) = 1/squareroot 2 sin (squareroot 2 t) u(t) impulse response h(t) = 1/squareroot 2 sin (squareroot 2t) u (t)