The \(\mathcal{Z}\)-transform of a discrete-time filter \(h(k)\) at a \(1 \mathrm{~Hz}\) sample rate is

\[H(z)=\frac{1+(1 / 2) z^{-1}}{\left[1-(1 / 2) z^{-1}ight]\left[1+(1 / 2) z^{-1}ight]}\]

(a) Let \(u(k)\) and \(y(k)\) be the discrete input and output of this filter. Find a difference equation relating \(u(k)\) and \(y(k)\).

(b) Find the natural frequency and the damping coefficient of the filter's poles.

(c) Is the filter stable?