Isolate g(t) starting from the definition of power spectrum Starting from the following definition of power spectrum of a signal $g(t)$ in time:
$$S(f)=\int_{-\infty}^{+\infty}\rho(\tau)e^{-i2\pi f \tau} d\tau$$
where
$$\rho(\tau)=\int_{-\infty}^{+\infty}g(t)^{.}g(t+\tau)dt$$
Is it possible to isolate $g(t)$?
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.