Answer Happy

Problem 3 [6 points] (Laplace transform) Consider a linear time-invariant (LTI) and causal system described by the following differential equation: ý" (t) +16(t) = z (t)+2x(t) where r(t) is the input of the system and y(t) is the output (recall that y" denotes the second-order derivative, and y' is the first-order derivative). Let h(t) be the impulse response of the system, and let H(s) be its Laplace transform. i) Compute the Laplace transform H(s), and specify its region of convergence (ROC). ii) Is the system BIBO stable? Justify your response. iii) Suppose that the system is at rest, and all the required initial conditions are 0. Compute the output of the system y(t) when the input is r(t) = u(t). Note: For the question iii), you are allowed to use the properties of the Laplace transform provided in the Table 9.1 of the book, and the transformation pairs in Table 9.2 of the book.