# 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 $$g(t)$$?

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 $$g(t)$$ uniquely. There are many different time domain signals that have the same power spectrum.
• $G(f)$ is a complex number. It has a magnitude and a phase. You need both to reconstruct g(t) – Hilmar Apr 12 at 20:23