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!