# 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!