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

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!

--Elisa