Problem 4 - How do we decompose H(w)?
Q: How is it possible to decompose ?
A: There is a general observation to make. Given a transfer function , its power spectral density will be . If we define , then we see that we have . Qualitatively, we can argue that poles of in are mapped to poles of in . Same for transfer functions having more than one pole and zeros.
In the exercise you should therefore substitute with , find its poles and zeros, and then map back to a guess for . Such guess will not be unique in general, but it is if one assumes certain properties regarding the phase!