Question

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LSM1 Problem (50 pts) Consider a causal continuous-time LTI system with input-output relationship dt+)t). (a) Find the transfer function H(s) of the system and specify its ROC. (b) Find the impulse response h(t) of the system. (12 pts) (12 pts) (c) Using the convolution property of the Laplace transform, find the output y(t) of the system in response to the input (t)ut) e2-u(t 1 (26 pts)

Answer 1

LTI system d/dt y(t) + y(t) = d/dt x(t) + 2x(t) Apply Laplace Transform sy(s) + y(s) = s x(s) + 2x(s) y(s) [s + 1] = x(s) [s + 2] y(s)/x(s) = s + 2/s + 1 (a) H(s) = s + 2/s + 1 ROC Re (s) > -1 (b) H(s) = s + 2/s + 1 H(s) = s + 1 + 1/(s + 1) H(s) = (s + 1)/(s + 1) + 1/(s + 1) H(s) = 1 + 1/1 + s h(t) = delta(t) + e^-t u(t) - impulse Response. \r\n (c) H(s) = y(s)/x(s) = s + 2/(s + 1) x(t) = u(t) + e^-z(t - 1) u(t - 1) By convolution theorem - y(t) = y(t) * x(t) y(t) = h(t) * x(t) y(s) = H(s) middot x(s) y(s) = [s + 2/(s + 1)] x(s) x(t) = u(t) + e^-2(t - 1) u(t - 1) x(s) = 1/s +