(Solved): (a) Consider a stable LTIC system for which the input \( x(t) \) and output \ ...

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(a) Consider a stable LTIC 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. i) Determine the transfer function \( H(s) \) together with its ROC (region of convergence), and sketch the zero-pole plot of it (6 marks); ii) Determine \( h(t) \) if the system is stable ( 2 marks); iii) Sketch the magnitude response \( |H(\omega)| \) of the system using the geometric evaluation, based on the zero-pole plot obtained in i) ( 2 marks). 10 marks