Problem 4 - How do we decompose H(w)?

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Q: How is it possible to decompose \(S(\omega)\)?

A: There is a general observation to make. Given a transfer function \(H(s)=\frac{1}{s+a}\), its power spectral density will be \(S(\omega)=\frac{1}{-\omega^2+a^2}\). If we define \(\lambda:=-\omega^2\), then we see that we have \(S(\lambda)=\frac{1}{\lambda+a^2}\). Qualitatively, we can argue that poles of \(H(s)\) in \(-a\) are mapped to poles of \(S(\lambda)\) in \(-a^2\). Same for transfer functions having more than one pole and zeros.

In the exercise you should therefore substitute \(-\omega^2\) with \(\lambda\), find its poles and zeros, and then map back to a guess for \(H(s)\). Such guess will not be unique in general, but it is if one assumes certain properties regarding the phase!