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