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

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Q: How is it possible to decompose ${\displaystyle S(\omega )}$?
A: There is a general observation to make. Given a transfer function ${\displaystyle H(s)={\frac {1}{s+a}}}$, its power spectral density will be ${\displaystyle S(\omega )={\frac {1}{-\omega ^{2}+a^{2}}}}$. If we define ${\displaystyle \lambda :=-\omega ^{2}}$, then we see that we have ${\displaystyle S(\lambda )={\frac {1}{\lambda +a^{2}}}}$. Qualitatively, we can argue that poles of ${\displaystyle H(s)}$ in ${\displaystyle -a}$ are mapped to poles of ${\displaystyle S(\lambda )}$ in ${\displaystyle -a^{2}}$. Same for transfer functions having more than one pole and zeros.
In the exercise you should therefore substitute ${\displaystyle -\omega ^{2}}$ with ${\displaystyle \lambda }$, find its poles and zeros, and then map back to a guess for ${\displaystyle H(s)}$. Such guess will not be unique in general, but it is if one assumes certain properties regarding the phase!