Consider what a pole-zero diagram can tell about the behavior of signals represented by their poles.

(a) For example, consider a single pole, say \(V(s)=V_{\mathrm{A}} /(s\) \(+\alpha\) ). Let \(\alpha\) vary from a large negative number towards zero and then from zero to a large positive number. Explain how the signal \(v(t)=V_{\mathrm{A}} e^{-\alpha t}\) varies as the pole moves along the \(\sigma\) axis.

(b) Now consider a pair of imaginary poles along the \(j \omega\) axis, say \(V(s)=V_{\mathrm{A}} /\left(s^{2}+\beta^{2}ight)\). Let \(\beta\) vary from zero to a large number. Explain how the signal \(v(t)=\left[V_{\mathrm{A}} \sin (\beta tight.\) )\(] / \beta\) varies as the poles move away from the origin.

(c) Finally, consider a pair of complex poles, say \(V(s)=V\)\({ }_{\mathrm{A}} \beta /\left[\left(s^{2}+\alpha^{2}ight)+\beta^{2}ight]\). Explain how the signal \(v(t)=V_{\mathrm{A}} e\) \({ }^{-\omega t} \sin (\beta t)\) varies as \(\alpha\) moves from a large negative number towards zero and then from zero to a large positive number. Lastly, explain how the signal varies as \(\beta\) increases.