Isolate g(t) starting from the definition of power spectrum
Is it possible to isolate g(t)?

No. At least not if you mean by &quot;isolate&quot;, &quot;can I calculate $g(t)$ from $S(f)$&quot;
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 $g(t)$ uniquely. There are many different time domain signals that have the same power spectrum.