Consider a continuous-time LTI system for which the input x(t) and output y(t) are

Chapter 9, Problem 9.31

(choose chapter or problem)

Consider a continuous-time LTI system for which the input x(t) and output y(t) are related by the differential equation

\(\frac{d^{2} y(t)}{d t^{2}}-\frac{d y(t)}{d t}-2 y(t)=x(t)\).

Let X(s) and Y(s) denote Laplace transforms of x(t) and y(t), respectively, and let H(s) denote the Laplace transform of h(t), the system impulse response.

(a) Determine H(s) as a ratio of two polynomials in s. Sketch the pole-zero pattern of H(s).

(b) Determine h(t) for each of the following cases:

1. The system is stable.

2. The system is causal.

3. The system is neither stable nor causal.