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 $$g(t)$$?

1 Answer

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.

• Ok. Thanks a lot for your answer. The calculation of the power spectrum by using the square of the absolute value of the spectrum, shouldn't be just an estimation? Shouldn't be the expected value of the square of the absolute value of the spectrum? Commented Apr 12, 2021 at 19:23
• yes, but the square of the absolute value "deletes" all phase information, exactly as Hilmar said: There's infinitely many signals with the same power spectrum. Commented Apr 12, 2021 at 20:04
• when you say "phase information", you mean information about the history of the signal? Commented Apr 12, 2021 at 20:12
• $G(f)$ is a complex number. It has a magnitude and a phase. You need both to reconstruct g(t) Commented Apr 12, 2021 at 20:23