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Maybe your signal is finite, but the function you are transforming is infinite. So you can do your demonstration using the interval $[0,T]$ or $[-\infty,+\infty]$, the result should be the same, because if your signal is finite the FT threats it as an infinite signal with period T.


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Is it possible to isolate g(t)? No. At least not if you mean by "isolate", "can I calculate $g(t)$ from $S(f)$" You can recover $g(t)$ from it's Fourier Transform $G(f)$. This is related to the power spectrum through $$S(f) = |G(f)|^2$$ So you can get the magnitude of $G(f)$ but not the phase. Information is lost and you cannot recover $...


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If there was some code outlining this, I would be interested I don't have the expertise to describe pYin precisely, but I can point you to the original code for that paper https://code.soundsoftware.ac.uk/projects/pyin/repository There's also a more recent implementation which made it into the librosa library since this question was asked. The code and pull ...


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